Compressive Membrane Action in Bridge Deck Slabs

COMPRESSIVE
MEMBRANE
ACTION
::ln
BRIDGE DECK
SLABS
by
Paul Austin JACKSON

BSc CEng MICE MIStructE
Submitted to the
Council of National Academic Awards
in partial fulfilment of
the requirements of the degree of:
DOCTOR OF PHILOSOPHY
Sponsoring Establishment:
Polytechnic South West
Department of Civil Engineering
Collaborating Establishment:
British Cement Association
April 1989
COMPRESSIVE MEMBRANE
ACTION
:l.n
BRIDGE DECK SLABS
by
Paul Austin JACKSON
ABSTRACT
An elastic analysis of restrained slab strips shows that membrane action
enhances serviceability behaviour.
However, the enhancement is not as
great as for strength and serviceability is critical when membrane action
is considered in design.
A relatively simple form of non-linear firiite element analysis is developed

which is able to model bridge deck behaviour allowing for membrane action.
This reduces some of the disadvantages of non-linear analysis which have
prevented its use in practice. It uses line elements but, -because-of-novelfeatures of the elements and because it considers all six degrees of
freedom at each node, it is still able to model in-plane forces reasonably
realistically. It gives acceptable predictions for behaviour.
The tension stiffening functions used in non-linear analysis, which are
important to the prediction of restraint, are considered. Explanations are
proposed for several aspects of the behaviour and a new function is
developed.
This gives better results than previous expressions,
particularly for deflections on unloading and reloading.
Tests under full HB load have been performed on two half scale bridges.
These, and the analysis, show that conventional design methods for deck
slab reinforcement are very conservative.
They also show that the
restraint required to develop membrane action is not dependent on
diaphragms; it comes from under-stressed material surrounding the critical
areas. Thus, over much of a bridge's span, there is transverse tension in
the slab and membrane action does not significantly enhance the resistance
to global moments.
It is
Both bridge models failed by a wheel punching through the slab.
shown that these were primarily brittle bending compression failures which
were strongly influenced by global behaviour. This is confirmed both by
the analysis and by the higher wheel load at failure in single wheel tests.
Recommendations are made for using the results in design and assessment.
ACKNOWLEDGEMENTS
The research reported in this thesis was undertaken at the British Cement
Association, formerly the Cement and Concrete Association, whilst I was an
employee of that organisation.
former
directors
of
the
I wish to thank the Council, directors and
Association
for
instigating
the
project,
for
continuing to support it during a difficult period for the Association and

for allowing me to submit this thesis.
I should also like to thank Gifford
and Partners for practical help during the latter phase of the project,
particularly for the use of several hours of main-frame computer time.
Thanks are also due to my two' supervisors, Professor Robert Cope and
Doctor Andrew Beeby, for assistance with different aspects of the study.
In particular, I wish to thank Professor Cope for many helpful discussions
regarding the analysis.
Too many of the staff and former staff of the Association contributed to
the experimental phase of the project to name them all.
like to thank them all.
However, I should
Particular thanks are due to Colin Cook for the
instrumentation and to Daran Morahan and Ron Jewel, whose enthusiasm and
expertise enabled me to complete the testing in what became a very tight
timetable.
Thanks are due to my wife, Sue, for typing some of the text and for proof
reading the document.
Last, but by no means least, I should like to thank
our two sons Andrew and Simon, as well as Sue, for putting up with my
absence or preoccupation whilst working on the thesis.
CONTENTS
vii
LIST OF FIGURES
LIST OF TABLES
xi
1 INTRODUCTION
2 CURRENT DESIGN PRACTICE
2.1 Introduction
2.2 Outline Design
2.2.1 Choice of Form
2.2 .2 To Stress or not to Stress?
2.3 Detailed Design and Codes of Practice
2.3.1 The Importance of Codes of Practice
2.3.2 Sources of Code Clauses
2.3.3 Limit State and Working Stress Codes
2.3.4 BS 5400: Part 4: 1984

2.4 Analysis for Design - British Practice
11
15
2.4.1 Reasons for Linear Analysis
15
2.4.2 Section Properties
15
2.4.3 Global and Local Functions
16
2.5 Analysis for Design - North American Practice
17
2.6 Conclusions and Implications for this Study
18
3 PREVIOUS RESEARCH
20
3.1 Introduction
20
3.2 Reinforced and Plane Slabs
20
3.2.1 Bending Strength
20
3.2.2 Flexural Shear Strength
31
3.2.3 Punching Shear Strength
31
3.2.4 Ductility
44
3.2.5 Serviceability
47
3.2.6 Restraint
49
3.2.7 Global Behaviour
51
3.2.8 Empirical Design Rules
54
- i -
3.3 Prestressed Slabs
57
3.4 Conclusions
60
62
4 ELASTIC ANALYSIS
4. 1 Introduction
62
4.2 Assumptions
62
4.3 Stress
63
.4.4 Comparison with Other Analyses
'63
4.5 Crack Widths
64
4.6 Deflection
65
4c. 7 Effect of Restraint Flexibility
65
4.8 Conclusions
66
5 NON-LINEAR FINITE
El.EMENT ANALYSIS
67
5.1 Introduction
67
5.2 General Approach
67
5.3 Element Type
68
. 5.3.1 Slabs
68
5.3.2 Beams
69
69
5.4 Material Properties
69
5.4.1 Steel
70
-5.4.2 Concrete
5.5 Application to Membrane Action
72
5.6 Use in Design
73
5. 7 Conclusions
74
6 TENSION STIFFENING.
75
6.1 Introduction
75
6.2 Theory
76
6. 2. 1 Mechanisms
76
6.2 .2 Steel Stress
78
6.2 .3 Mesh Dependence
78
-
1i -
6.2.4 Cyclic Loading
79
6.2.5 Unloading
79
6.3 Analysis of Previous Tests
80
6.3.1 Direct Tension Tests
80
6.3.2 Flexural Tests
83
84
6.4 Tests
6.4.1 Design of Specimens
84
6.4.2 Loading Rig
85
6.4.3 Materials
86
6.4.4 Loading
88
6.4.5 Processing of Results
89
6.4.6 Results and Analysis
90
99
6.5 Conclusions
7 A SIMPLER NON-LINEAR ANALYSIS
100
7.1 Introduction
100
7.2 General Approach
101
7.3 Displacement Function
102
7.4 Element Initial Stiffness Calculation
106
7.5 In-Plane Forces
108
7.6 Large Displacements
111
7.7 Material Models
112
7.7.1 Steel
112
7.7.2 Concrete in Compression
113
7.7.3 Concrete in Tension
115
7.8 Stress Integration
119
7.9 Solution Scheme
120
7.9.1 Control
121
7.9.2 Initial Stiffness Method
121
7.9.3 Accelerators
122
7.9.4 Stiffness Recalculation
125
7.9.5 Convergence Criteria
127
7.10 Calibration
128
7.10.1 Duddeck's Slabs
129
- iii -
7.10.2 Taylor and Hayes' Slabs
131
7 .10.3 Batchelor and Tissington's Specimens
135
7.10.4 Kirkpatrick's Model
136
7.11 Conclusions
139
8 MODEL BRIDGE TESfS
140
8.1 Introduction
140
8.2 Design of Models
140
8.2.1 Scheme
140
8.2.2 Beams
143
8.2.3 Diaphragms
145
8.2.4 Slab Reinforcement
145
8.2.5 Bearings
148
148
8.3 Materials
8.3.1 Concrete
148
8.3.2 Reinforcement
154
8.3.3 Prestressing
155
8.4 Construction
155
8.5 Loading
157
8.5.1 Loads Applied
157
8.5.2 Loading Rig
160
8.6 Instrumentation
161
8.7 Tests on First Deck
163
8.7 .1 Global Service Load Tests
163
8.7.2 Global Failure Test
169
8.7.3 Local Failure Tests
177
181
8.8 Tests on Second Deck

8.8.1 Global Service Load Tests
181
190
196
8.9 Tests on Single Beam
198
8.10 Discussion and Conclusions
202
8.10.1 Service Load Tests
202
8.10.2 Failure Tests
202
- iv -
9 ANALYSIS OF MODEL BRIDGE TEsTs
205
9.1 Introduction
205
9.2 Conventional Analysis
205
9.2.1 Analysis for Design of Deck Slabs
205
9.2.2 Analysis for Design of Beams
207
9.3 Non-Linear Analysis
213
9.3.1 Single Beam
213
9.3.2 First Deck
214
9.3.3 Second Deck
229
9.4 Conclusions
233
10 USE OF MEMBRANE ACTION IN DESIGN AND ASSESSMENT
235
10.1 Introduction
235
10.2 Use in Design
235
10.2.1 M Beam Type Decks
235
10.2.2 Other Beam and Slab Decks
239
10.2.3 Other Types of .Deck
239
10.3 Use in Assessment
240
11 CONCLUSIONS AND RECOMMENDATIONS
241
11.1 Conclusions
241
11.2 Recommendations
243
11.2.1 Recommendations for Design and Assessment

I
1.2.2 Recommendations For Further Research
243
243
245
REFERENCES
APPENDICES
A. Restrained Slab Strip to Elastic Theory
AI
AI Stresses
AI
A2 Deflection
A3
A3 Effect of Restraint Flexibility on Stress
A6
B. Transverse Shear Deformation of Line Elements
AB
All
C. Large Displacements
Cl Example Showing Effect of Vertical Component of
AB
Axial Force
C2 Effect of Slope on Axial Extension
Al2
A13
D. Notation
- vi -
LIST
OF
FIGURES
3.1
Geometry of restrained slab strip
3.2
Load-displacement relationship of restrained
21
slab strip
22
3.3
Comparison of flow and deformation theory
25
3.4
Elastic-plastic theory (1/h = 10>
27
3.5
Elastic-plastic theory <1/h
= 30)
28
3.6
Kinnunen and Nylander's model
33
3.7
Effect of concrete strength
38
3.8
Seal's Model Two
39
3.9
Kirkpatrick's model
40
3.10
Effect of span
42
3.11
Compressive membrane action to resist global

moments
52
3.12
Transverse stresses in a wide compression flange
53
4.1
Restrained slab strip
63
4.2
Effect of restraint flexibility
66
6.1
Tension stiffening functions
75
6.2
Stresses in concrete as cracks develop
77
6.3
Effect of tension stiffening on analysis'in

direct tension
82
6.4
Analysis of Williams' Specimen
82
6.5
Analysis of Clark's Beam 4
84
6.6
Detail of test specimens
85
6.7
Half size specimen under test
86
6.8
Results of first full size test
92
6.9
Results of first half size test
92
6.10
Results of third half size test
93
6.11
Ideal tension stiffening function
95
6.12
Tension stiffening function adopted
96
7.1
Displacement function
104
7.2
Effect of change to displacement function
106
7.3
Transverse displacements of a line element
108
7.4
Effect of in-plane shear
110
- vii -
7.5
Steel properties
112
7.6
Properties used for concrete in compression
114
7.7
Effect of concrete tensile strength
117
7.8
Initial stiffness method
122
7.9
Modified initial stiffness method
123
7.10
Analysis of Duddeck's slab
129
7.11
Analysis of Duddeck's slab 3
130
7.12
Analysis of Taylor and Hayes' slab 2S4
132
7.13
Analysis of Taylor and Hayes' slab 2R4
134
7.14
Batchelor and Tissington's specimen
135
7.15
Analysis of Batchelor and Tissington's specimen
136
7.16
Analysis of Kirkpatrick's panel C2
137
8.1
Details of first deck
141
8.2
First deck under test
141
8.3
Details of second deck
142
8.4
Second deck under test
143
8.5
Comparison of full size T2 and half size M4 beams
144
8.6
Detail of reinforcement in slab of first deck
146
8.7
Reinforcement in corner of first deck
147
8.8
Stress-strain relationship for reinforcement
155
8.9
First deck under construction
156
8.10
Spreader beam assembly
160
8.11
Load positions for first deck
164
8.12
Transverse soffit strain under wheel 10

<Service load tests)
165
8.13
Deflection under wheel 9 <Service load tests)
166
8.14
Beam deflections
170
8.15
Deflection under wheel 9
170
8.16
Shear cracks in Beam B
172
8.17
Soffit cracks under wheel 4
173
8.18
First deck under 400kN per jack
173
8.19
View across deck as failure approached
174
8.20
Failure; Wheel 4 punched through deck
175
8.21
Failure cone viewed from below
176
8.22
Single wheel test rig
178
8.23
Result of single wheel test
178
8.24
Crack pattern under single wheel
179
- viii -
8.25
180
8.26
Load positions for second deck
181
8.27
Transverse strains adjacent to wheel 14

<Service load tests)
183
8.28
Deflection under wheel 14 <Service load tests)
184
8.29
Top cracks on completion of service tests
189
8.30
191
8.31
Beam deflections
192
8.32
Top cracks immediately prior to failure
193
8.33
Second deck after failure
195
8.34
Single wheel tests A and C
197
8.35
Local tests B and D
197
8.36
Single beam under test
200
8.37
Load-deflection response of single beam
200
8.38
Shear cracks in single beam
201
8.39
Beam after failure
201
9.1
Beam deflections of first deck (conventional

analysis)
208
9.2
Beam strains in first deck
209
9.3
Beam deflections of second deck <conventional

analysis)
211
9.4
Slab strains over beams in second deck
212
9.5
Beam soffit strains in second deck
213
9.6
Analysis of single beam test
214
9. 7
Beam deflections of first deck <from non-linear

analysis using a coarse element mesh)
9.8
Deflect ion under wheel 9 <from non-linear an lysis

using a fine element mesh)
9.9
215
219
Beam deflect ions of first deck (from non-linear

analysis using a fine element mesh)
220
9.10
Predicted force across centre-line of first deck
221
9.11
Restraint forces predicted under single wheel load
225
9.12
Elastic analysis of stresses around a hole
226
9.13
Beam deflections of second deck <from non-linear
9.14
analysis)
230
Predicted force across deck slab of second deck
232
- ix -
A!
Restrained slab strip under line load
Al
Bl
Plan of line element
AB
B2
Unrestrained deformation
A9
Cl
Three element strut
All
C2
Inclined element
A12
X -
LIST
OF
TABLES
87
6.1
Typical mixes
7.1
Stiffness matrix of an off-set beam element
107
8.1
Mixes for in situ concrete
149
8.2
Test results for slab concrete from first deck
150
8.3
Test results for slab concrete from second deck
151
8.4
Test results for other in situ concrete from

second deck
152
8.5
Mix for precast concrete
152
8.6
Test results for precast concrete from first deck
153
8.7
Test results for precast concrete from second deck
154
- xi -
CHAPTER
INTRODUCTION
If in-plane restraint prevents material in the tension region of a beam or
slab from expanding as load is applied, a compressive force is developed.
This
force
can
lead
to
greater
predicted by normal flexural theory.

enhancement
is
relatively
small
strengths
and
stiffnesses
than
are
In a simple steel beam, however, the

and
arises
only
when
the
in-plane
restraint is applied below, that is on the tension side of, mid-depth.
In
concrete, and also in masonry, the low tensile strength and consequent
cracking mean that the effect can arise even when the restraint is applied
at mid-depth.
The enhancement can also be very much greater since the
compressive force enables even unreinforced slabs to support large loads.

This effect, which is known as compressive membrane action, arching action
or dome effect, has been known since the earliest days of reinforced
concrete.
It was described by Westergaard and Slater <1 > in 1921 and as
early as 1909 Turner <2> wrote of his flat slabs "such a slab will act at
first somewhat like a flat dome and slab combined".
Turner built many
flat slabs with reinforcement designed by empirical means.
At the time
there was good reason to use empirical design methods; the theory of flat
plates was not well developed.
However, Turner's contemporaries used more
conservative design methods and Sozen and Siess<3> report that, in 1910,
the weight of_ steel required in the interior panel of a flat slab varied by
a factor of four according to the design method used.
As they put it
"design methods could not be correct if the variation in results was 4-00%".
When an analysis based on simple statics was published in 1914-<4>, it
suggested that Turner's slabs were grossly under-designed; yet they had
behaved well both in service and in load tests.
Lord(5) had even measured
strains in a load test which appeared to support Turner and defy the laws
of statics.
Compressive membrane action was an important reason for these
discrepancies although there were others, including tension stiffening <3>.

Despite the satisfactory behaviour of Turner's slabs, design methods which
can be justified by statics are now preferred and purely empirical methods
have tended to fall out of favour whenever more rational methods have
become available.
Thus even flat slabs are now designed using flexural
theory, although Sozen and Siess <3> report that the charige was gradual
whilst Beeby (6) has shown that it is sUll not complete.
-
1 -
Apart from indirect
(and very limited> use in the Soviet dfsign code <7>,
compressive membrane action seems to have been largely forgotten for many
years.
Thus, in Braestrup's words <8>, "it therefore came as a surprise
when Ockleston <9> tested a real structure in South Africa and recorded
collapse loads that were three or four times the capacities predicted by
yield-line theory".
In fact Guyon<lO> had found similar results slightly
earlier, when he tested a multi-bay continuous slab, but this seems to

have been considered a characteristic of prestressed concrete.
Ockleston's results stimulated research into compressive membrane action
which has continued ever since.
Despite this research, which will be
reviewed in Chapter 3, the effect is still not normally used in design.

Recently, however, new design rules for bridge deck slab reinforcement,
which do allow for the effect, have been developed and incorporated into
the Ontario Highway Bridge Design Code <11>.
These rules lead to major
savings
methods;
compared
with
conventional
design
reduction in main steel plus a saving in design time.
typicslly
70%
More recently still,
similar rules have been adopted in other parts of the World, including
Northern Ireland.
The rules used in Northern Ireland <12 > were proposed by
Kirkpatrick et al<13) for use in the whole of Britain but have not yet
been accepted on the mainland.
One objection to these rules is simply that they are empirical.
Existing
theory shows, as will be seen in Chapter 3, that slabs designed to the

rules will have ample strengths under local wheel loads, provided there is
adequate restraint.
It even suggests that there would still be ample
strength with no reinforcement at all.
This, however, is the limit of the
extent to which the rules are proven theoretically.
There is also an
apparently serious omission from the experimental work on which they are
based.
An extensive series of tests on laboratory specimens, model bridges
and real bridges was undertaken yet none of the tests produced anything
approaching the full design global load on a bridge.
Thus the integrity of
the deck slabs under combined global and local effects is unproven.
Also,
they msy not give the load distribution which is assumed in the design of
the
beams;
particularly
as
global
analysis
based
on
uncracked
slab
properties is recommended <11,1 ~> for use with the rules.

Concrete slab design has come full circle; bridge deck slsb design is now
in the position which flst slab design occupied in
-
2 -
191~.
On the one hand
there is an empirical design method which seems to work and which is very
economical yet which could be considered unproven: on the other there is
the conventional method which is supported by flexural theory but which
seems to be very uneconomical.
Just as in 1910 design methods for flat

des~n
slabs could not be correct when they differed by 400%, so
rules for
bridge deck slab reinforcement cannot be correct now when they differ by
300%.
There is clearly a need for further research.
In recent years the assessment of existing structures has assumed equal

importance to the
des~n
of new construction.
Current design standards
are used in these assessments, but they often suggest that structures
which have given many years of satisfactory service are unsafe.
In many
such cases, compressive membrane action offers the possibility of more

realistic
assessment
reconstruction
which
work.
could
Previous
avoid
expensive
research,
having
strengthening
concentrated
and
on
new
construction, does not enable this potential to be fully used.

Another
problem
which
has
reinforcement corrosion.
become more
important
in recent
years
is
Resistance to this can be greatly improved by
increasing cover or by using epoxy coated, or other special reinforcement.

Both these approaches would become more economical if membrane action
were considered in
des~n.
It has even been suggested that satisfactory
deck slabs could be built without any reinforcement at all, which would
certainly avoid the problem of reinforcement corrosion.
"Localised"
reinforcement
corrosion
is
believed
to
be
particularly
dangerous <15) but an interesting implication of membrane action, which has

not previously been considered, is that this may have no
s~nificant
effect
on the behaviour of slabs.

In the present study the behaviour of bridge decks is
order .to develop and justify a rational
des~n
invest~ated
in
and assessment method which
can be adopted in British practice but which takes as much advantage as

possible of compressive membrane action.
The approaches used in the study
include tests on large scale model bridges and a simple elastic analysis.
However, because model tests alone can produce only empirical results,
whilst the behaviour considered is too complex to analyse in full by hand,
non-linear computer analyses are also used.
3 -
CHAPTER
CURRENT
DESIGN
2
PRACTICE
2.1 INTRODUCTION
In order to direct this study towards those areas which are important in
design, and to ensure that the knowledge gained will be usable in practice,
it is necessary to begin the study with a good understanding of current
bridge deck design practice.
That is, of the way bridges are assumed to
behave for design purposes, of the way they are designed, and of the
criteria and codes of practice they are designed to.
provide such an understanding.
This chapter aims to
There are also more fundamental reasons
for respecting past practice which will be discussed.

Design practice, unlike the real behaviour of bridges, differs significantly
between countries.
It is not practical, or necessary, to review practice
throughout the world.
This study is aimed at improving British practice,
so this chapter will concentrate on British practice.
Much of the most
relevant previous research has, however, been undertaken in North America

against a background of North American design practice.
There are several
important differences between British and North American design practice

which have greatly influenced the research and render its application in
Britain more difficult than might be expected.
In order to appreciate
these problems it is necessary to review the relevant aspects of North

American design practice.
Only conventional design methods, which ignore
membrane action, will be considered here.
The newer empirical design
approach, which allows for membrane action, will be considered in the next
chapter, along with the research from which it was developed.
2.2 OU11.INE DESIGN
2.2.1 Choice of Form
Before the detailed design of a bridge can be started the form of the
bridge has to be decided; for example solid slab, voided slab, beam and
slab, box girder or arch.
experience.
In making this decision, engineers are guided by
For particular ranges of span and sets of circumstances,
certain forms of structure have been found to be most economical.
Over
the years these favoured forms change, usually because of changes in

construction
technology
rather
than
because
of
advances
Construction considerations are always very important <16>.
- 4 -
in
analysis.
The desired
erection method nearly always decides the form of the bridge, rather than
the reverse.
For example, one does not choose to use precast elements in
a bridge because it is a beam and slab bridge, one chooses a beam and slab
bridge because it is convenient to precast.
The few cases when advances in analysis have changed the form of bridges
have arisen when those advances have enabled the analysis of structures
which are physically simpler but analytically more complicated.
An example
of this is the virtual extinction of intermediate diaphragms in beam and

slab bridges since load distribution analysis has been in widespread use.
These diaphragms served not so much to distribute load between beams as
to enable this distribution to be analysed.
With modern analytical methods
they are eliminated, sometimes at the price of doubling the transverse

steel in the deck slab.
In terms of material cost this change may be
uneconomic, but the difficulty of forming diaphragms in the span is such

that eliminating them leads to significant overall savings.
Thus if re-
introducing these diaphragms solved a problem in using membrane action

<and
Chapter
shows
that
this
is
the
case)
it
would still not
be
economical.
The dominance of construction considerations in the choice of the form of
bridges
means
behaviour
bridges.
of
that
study
such
as
this,
which
completed structures,
is
unlikely
considers
to
alter
the
only
form
the
of
It is thus essentially concerned with detailed design rather than
with scheme design.

2.2.2 To Stress or not to Stress?
Another decision which has to be taken in the early stages of a concrete
bridge design is whether or not to prestress and if so whether to pre- or
post-tension.
Again
this
construction.
decision
is
often
dictated
by
practical
considerations
of
It is not possible to build a glued segmental bridge without
post-tensioning and it would be difficult to build any long-span bridge

<except an arch> of ordinary reinforced concrete.
On the other hand, a
small slab is obviously more conveniently reinforced and small precast

beams are more easily pre-tensioned on a long line bed.
Only over narrow
ranges of structures (such as large voided slabs) is the decision marginal,
- 5 -
and
therefore
sensitive
to
small
changes
in
the
relative
costs
or
quantities of steel required.

In Britain, and most of the rest of the world, it has been found that,
because of the extra operations involved, transverse stressing of bridge
deck slabs is rarely economical.
A large number of tendons have to be
fixed, threaded, stressed and grouted, usually with very difficult access.
Much of the cost of these operations is fixed so that, even if the
required force were greatly reduced, transverse stressing would still be
unattractive.
Because of this, the present study assumes that deck slabs
will not be transversely stressed.
Accordingly, the rest of this chapter
concentrates on the design of ordinary reinforced concrete.

however,
In reality,
<unlike in most codes of practice> prestressed and reinforced
concrete are not fundamentally different.

effectively
prestressed
behaviour of the deck.
in
the
Also bridge deck slabs are often
longitudinal
direction
by
the
global
Thus research on stressed slabs can be relevant
and some of it will be considered in Chapter 3.

2.3 Df:I'AILED DESIGN AND CODES OF PRACTICE
2.3.1 The Importance of Codes of Practice
Most major bridge owners, including all of those in Britain, require new
construction to be designed to specified codes of practice.
codes are also frequently specified for use in assessment.
The same
Because of
this, codes have an importance which they owe as much to their contractual
position as to their engineering merit.
This alone justifies the extensive
reference which is made to them throughout this chapter.
It also means
that a new design method, such as one which allows for membrane action,
will be much more easily put into practice if it can be used within
existing codes.
as
this
Despite this, it is arguable that a research thesis such
should
be
concerned
only
with
fundamental
requirements
of
structural behaviour, and not with the sometimes arbitrary provision of

codes of practice.
distinct
from
Codes do, however, have a considerable engineering, as
contractual,
significance
which
arises
from
their
two
different, but overlapping, types of source.

2.3.2 Sources of Code Clauses
The first of these sources is the philosophy, theory and test data on
which codes are based.
The second is the cumulative experience which they

-
6 -
represent.
The latter means that a code can be considered as a set of
arbitrary rules which have been found to produce satisfactory structures

in the past.
Paradoxically, this applies to new codes, as well as to long-
established ones, because they are adjusted to make significant changes to

past designs only where they are known to be at fault.
The two sources of code clauses each have their faults.
Our understanding
of structural behaviour and our stock of test results are too incomplete
to enable them to be used as the sole basis of a code of practice.
On
the
other hand experience, as a source of code clauses, allows no innovation

and shows only where provisions are inadequate, not where they are overconservative or even
unnecessary.
It
has
also
been
pointed out
by
Beeby <17) that experience is an unreliable guide to design practice when,

as is usual with bridges, the design life is long compared with the timescale of change in loading, materials and design methods.
Code clauses owe
their origins
to a complex mixture of
theory,
test
results, experience and the engineering judgement of the code writers.

Theories are fitted to test results and to experience.
New theories and
test results are used to design structures which become part of the stock
of experience.
Experience is reviewed in the light of new theories, whilst
structures which were designed using discarded theories remain in the

stock of experience.
Finally, when experience shows that a subject needs a
code clause but not what the clause should be, and when there is no clearcut theory or evidence to go on, the code committee makes an arbitrary
decision.
By now it is often difficult to tell what specific source, or
even what type of source, any particular code clause is based on.
This
may not matter to the ordinary user of the code, but it is important when
the code comes to be reviewed in the light of new discoveries.
Even when the source of a code clause can be identified it may be a
matter
of
opinion
whether
the
clause
requirement or an arbitrary rule.
is
logical
and
fundamental
A classic example of this is the no-
tension rule in prestressed concrete, which can easily be traced back to

Freyssinet <18).
This rule illustrates how the source of a code clause
<that is, whether it is a fundamental requirement or an arbitrary rule

which has been found to work) affects, or should affect, the way it is
reviewed in the light of new discoveries.
detail.
-
7 -
This will be considered in more
In the 1970's Emerson<19> observed that bridges were subjected to large
temperature differentials with non-linear distributions.

many
bridges designed
tensile stresses.
to
the
no-tension
rule
This implied that
experienced
s~ificant
If the .no-tension rule was a logical and fundamental
requirement this was an alarming discovery indicating that the prestress

in those bridges needed to be increased.
On the other hand, 1f the no-
tension rule was simply an arbitrary design criterion which has been found
to produce satisfactory structures in the past, the discovery that some of
those satisfactory structures do experience tension is no cause for alarm.
If anything, it implies that the remainder of the structures
des~ed
to
the rule, which do not experience tension, have more prestress than they
need.
It
is now widely accepted that
the no-tension rule is largely
arbitrary eg. see Low<20>l but at the time it was treated as though 1t
was a rational and necessary requirement.
The result was that from the
introduction of non-linear temperature distributions into bridge design

practice in 1973<21>, up to the implementation(22> of BS 5400(23) and the
use of a degree of so-called "partial prestressing" in 1983, many bridges
were provided with unjustifiably large amounts of prestress.
Research, by providing new theory and test results, can invalidate code
provisions which are based on theory and test results.
Where new research
provides sufficient understanding of the relevant aspects of behaviour 1t

can also supersede code provisions which are based on practical experience.
des~
criteria for bridges are difficult to
define, let alone check by analysis.
A bridge has to survive a long life
Often, however, the critical
in an adverse environment and to remain serviceable after experiencing a

complex history of loads: environmental, functional and accidental.
When
we cannot fully analyse these things we rely upon experience to fill in

the gaps.
This
inability
consequent
to
fully
analyse
all
dependence upon experience,
conservative.
If code provisions,
aspects
tends
of
behaviour,
and
the
to make bridge engineers
and hence design methods, are based
purely on experience how can we know if it is safe to reduce the steel

area in bridge deck slabs?
One might think that until we can fully
understand all aspects of the behaviour of structures, we have to .keep

using as much steel as we always have.
In truth, however, theory can be
used to extrapolate experience and to use experience of one type of

structure in the design of another.
- 8 -
If we can prove with new theory that the steel in deck slabs, which is
designed for a stress of 345N/mm 2

say, 80N/mm 2
,.
not
know,
actually experiences a stress of only,
it does not prove that it would be safe to design the slab
with the new theory.

do
Until we can understand all aspects of behaviour we
from
theory
alone,
that
it
behaviour of the slab would be satisfactory.

term
or
cyclic
stress
of
over
80N/mm 2
would
be safe or
that
the
Is it possible that a longwould
cause
problems?
Our
experience of slabs does not answer this question because all the slabs we
can observe were designed by the very conservative method which we are
trying to supersede.
Simply supported beams are statically determinate,
however, so we know they experience the stress they are designed for.
Thus we do not need to fully understand all the implications of allowing a
higher stress to know if it is safe; we know it works in beams.
Thus,
even if new theory cannot prove that a slab design will be satisfactory, it
can show that the maximum stresses the slab will experience are less than
.those experienced by beams whose behaviour we know to be satisfactory.
Thus it enables the reinforcement in slabs to be reduced, refining the
safety margins towards, but not below, those already found satisfactory in
beams.
2.3.3 Limit State and Working Stress Codes
The great
majority of bridges built
Department of Transport standards.

Part
2:
1978 <24)
as
in Great Britain are designed
to
The loading standard used is BS 5400:
implemented
<and
significantly
modified)
by
BD
14/82<25) whilst the design standard for concrete bridges is BS 5400: Part
4: 1984 <26) which is implemented by BD 24/84 <27 >.
These are limit state
codes but the Department has only recently changed from using its own
standards <21,28,29> which were based on the working stress approach.
It is
helpful to review what this change in concept means.

The basic idea of a limit state code is that the various ways in which a
structure could exhibit unacceptable behaviour are considered in turn.
structure which is on the limit of acceptable behaviour is said to have

reached a certain "limit state".
Thus a structure which has the maximum
acceptable deflection could be said to have reached the limit state of

deflection.
Checking a design involves checking each limit state in turn.
Partial safety factors and the concept of probabilistic design have been
introduced at the same time as limit state philosophy but they are not
central to the concept or definition of a limit state code.
-
9 -
A working stress code specifies allowable stresses.
Checking a design
involves using elastic theory to calculate the stresses which exist in the
structure under working loads.
allowable stress.
It
These stresses are then compared with the
is the code writer's responsibility
to set
the
allowable stress at a level such that compliance with the limit ensures
satisfactory behaviour of the structure.
At first sight the two approaches seem quite fundamentally different.
It
might also be thought, as some engineers have argued<30>, that the limit
state approach involves the designer in a great deal more calculation than
the working stress approach.
cut.
In practice the difference is far less clear-
This is largely because it has never been possible to develop
reasonable
stress
limits
which
structure in every respect.
ensure
satisfactory
behaviour
of
The result is that so called "working stressA
codes require separate checks on what are really limit states; such as
deflection and crack widths.
Conversely, it has been possible to write
many limit state codes in such a way that compliance with one limit state
<and perhaps some nominal rules as well> ensures compliance with other
limit states.
In CP110 <31) - now BS 8110 <32) - this has been taken to the
point where it is normally only necessary to check one limit state, the
ultimate limit state.
In principle a limit state code needs only to specify the design criteria
for each limit state.
It could leave the designer
method used to check compliance.
free to choose the
In practice limit state codes do give
methods for checking compliance, although these are often optional.
The
important point is that, in principle at least, the design criteria are

fundamental characteristics of structural behaviour <such as strength or
deflection> and are independent of the method used to check compliance.
This differs from the situation in a working stress code where the design
criterion is that the stress, as calculated using elastic theory, should
comply with the limits.
There the design criterion <stress> and the method
for checking compliance <elastic theory) are not independent.
The result
is that the adoption of limit state codes should make the introduction of
new methods of analysis and design into practice much easier than it was
under working stress codes.
It should be simply a case of using the new
method to check compliance with the existing fundamental design criteria.

In practice it is not this straightforward, because the design criteria in
limit state codes are not always truly fundamental or independent of the
- 10 -
methods used to check compliance.

for the
This will be considered in more detail
particular case of BS 5400: Part 4: 1984<26>.
2.3.4 BS 5400: Part 4: 1984
BS 5400 is a limit state code and, as far as reinforced concrete is

concerned,
the
major
limit
states
which
the
des~er
is
required
to
consider are the ultimate limit state and the serviceability limit states
of crack widths and stress limits.
durability,
deflection
and
There are other considerations, such as
reinforcement
normally critical in conventional design.
fatigue,
but
these are
not
The important limit states will
be considered in turn.
a. Ultimate Strength
The need for a check on the ultimate limit state <formerly, and arguably
more correctly, known as the limit state of collapse> is obvious.
The
consequences of failure at this limit state are clearly very serious so the
acceptable probability of failure is very low.
For this reason the partial
safety factors used in BS 5400 for both loads and materials are larger for
this limit state than for the serviceability limit state.
In principle, the design criterion for the ultimate limit state is simply
that the structure should not collapse under the specified loads.
a
This is
fundamental design criterion so, having specified loads and material
strengths, the code is able to give some freedom as to how it is checked.

The usual approach is to analyse the structure using methods which will be
discussed in 2.4 and then to check sections separately for bending and
shear.
The bending strength check is done by assuming that plane sections
remain plane and using the code specified stress-strain relationship for
concrete and reinforcement.
There is an additional proviso that
the
reinforcement should yield at failure which was introduced to ensure a

ductile failure mode.
proved difficult
As the clause is of questionable value, and has
to comply with in some sections, the code allows the
alternative of providing
15~
extra ultimate strength.
b. Crack Widths
The need for the two main serviceability limit states, crack width and
stress, is less obvious and requires some explanation.
It is desirable to
limit crack widths for aesthetic reasons but the restriction in BS 5400 is
unnecessarily severe for this purpose.
This has arisen because it has
- 11 -
...
been
assumed
that
there
reinforcement corrosion.
is
relationship
between crack widths
and
Beeby <33> and others have said that neither the
available test evidence nor the accepted theory of reinforcement corrosion

support such a relationship.
It seems likely, therefore, that the BS 5400
crack width restriction .is unnecessarily severe, although at present it has

to be complied with in design.
In principle a crack width is a fundamental design criterion which is
independent of the method used to calculate it.
available
crack
width
prediction
formulae
In practice, however, the
give
such
widely
different
results [see Beeby <34)] that the criterion and the method for checking
compliance are interdependent.
For this reason BS 5400 explicitly states
that its criterion is that the crack width as calculated using the code
method should not
exceed the specified values.
The particular formula
specified in BS 5400 is based on that given in CP 110.
The background to
this is given by Beeby <34>.

The code only requires crack widths to be checked for functional, not
environmental, loads.
considered,
not
the
It also only requires 25 units of HB load to be
full
design
value
of
up
to
45
units.
Tension
stiffening is not used if more than half of the bending moment in the
section is due to live load.
This is to allow for the effect of repeated
loading and for the possibility that a section could have been pre-loaded
to a higher load than that for which cracking is checked.
This differs
from CP 110 and makes the crack width prediction formula conservative.
Despite this, and unlike under BE1/73 (28>, it is rarely critical in the
design of the main steel for bridge deck slabs.
c. Stress Limits
The provision of stress limits in a limit state code is something of an
anomaly.
It is contrary to the basic concept of a limit state code.
If
the deflections, strengths and crack widths are satisfactory it is hard to

see how a structure can exhibit unacceptable behaviour due to stress.
The
stress
the
limits
author<35).
in BS 5400 have
This showed
that
been
the
their
linear-elastic structural behaviour.
subject
purpose
is
of a study by
to ensure reasonably
This is not a fundamental design
criterion either but it is desirable for two reasons.
Firstly, the methods
given in the code for checking the other serviceability criteria, such as
crack width and deflection, assume linear elastic behaviour.
- 12 -
Thus the
check is needed simply as a

checks.
elastic
Secondly,
range,
chec~.
on an assumption made in the other
in a structure which. went significantly out of the
transient
loads
would
cause
permanent
deformations.
This would mean that a structure could be influenced by the cumulative

effect of all the loads which it had experienced throughout its life.
would be extremely difficult, if not impossible, to assess.
This
It is much
simpler to assume that a structure recovers from transient loads and limit
stress so that this is approximately true.
It is only because of this
restriction that BS 5400 is able to ignore some load cases when checking
crack widths.
Because of cracking, the real behaviour of reinforced concrete structures
is not linear-elastic.
To ensure even approximately linear behaviour it
would be necessary to limit the tensile stress to the cracking stress of

the concrete, which is not considered practical.
This means that a precise
analysis of a reinforced concrete structure still requires an assessment

of
the
cumulative
throughout
needed.
that
effect
its life.
of
all
the
As this is not
loadings
which
it
experiences
possible some other approach is
The only rigorously safe approach which is practical is to assume
the cumulative damage is total, and hence to ignore the tensile
strength
of
concrete
completely.
This
is done
in
some calculations,
notably in assessing the ultimate strength of sections in flexure.
It is
also done in BS 5400 when assessing the crack widths which occur in
sections loaded predominantly by live load.
The approach is not, however,
followed rigorously and many of the calculation formulae provided in codes

of practice do depend on the tensile strength of concrete<6>.
The stress limits in BS 5400 are not fundamental design criteria as they
are not
independent of the method used to check compliance.
This is
particularly true of the concrete compressive stress limitation of 0.5fcu

Concrete is significantly non-linear at this stress but the code writers
considered it acceptable to allow some redistribution.
This means that the
actual maximum stress in a section with a calculated maximum of 0.5f cu

would be less than 0.5fcu
the code criterion.
Despite this it still only just complies with
In axial compression, where there is no scope for re-
distribution, the code specifies the much lower limit of 0.38f cu
This
inter-relationship between the code's criterion <the stress limit) and the
method
of
checking
compliance
<elastic
theory>
means
that,
if
an
alternative analytical method is to be used in design, the design criteria

- 13 -
_,
~-
_..
have to be reconsidered, a fact which has not always been appreciated by

non-linear analysts.
For example, see Edwards <36 ).
In routine design to BS 5400 it is not usual to check the stresses in

reinforced concrete.
analysis at
The code allows the check to be avoided provided the
the ultimate limit state is elastic without redistribution.
The writer<35) has shown that, for normal sections in flexure, this rule
results in designs which are similar to those which would be obtained by
checking the stress.
However, sections designed to this rule which are
either heavily reinforced or subjected to axial loads can have calculated

concrete stresses which are significantly above the 0.5f cu limit.
Despite
this, such sections behave satisfactorily.

d. Critical Limit State
It
can be seen
normally
critical
concrete.
This
from
the preceding sections that
in
the
conventional
has
led
some
design
researchers<37)
ultimate strength is
approach
to
the
for
conclusion
research on bridges should concentrate on ultimate strength.

however,
ultimate strength
is
critical
in design only
reinforced
that
In reality,
because of
the
conservative approach <elastic structural analysis) which is used to check

it.
This approach is, in effect, deliberately chosen in order to ensure
that ultimate strength is critical and hence to avoid the need to check
other considerations, such as the stress limits.
In the case of deck slabs, which are subjected to concentrated wheel loads,
elastic theory predicts high moment peaks.
distribute.
In reality these peaks re-
Because of this, a yield-line analysis of a typical deck slab
designed by conventional methods shows that the ultimate strength is twice

what is required.
If, however, the designer opted to use this analysis for
design he would have to check the service stress.
The writer(35> has
shown that, because of this, the maximum saving in steel area which can be
obtained from the use of yield-line analysis is only about 11%.
Thus, if
analysis taking account of compressive membrane action is to result in

significant economies, it must indicate improved serviceability behaviour
as well as strength.
- 14 -
2.4 ANALY515 FOR DESIGN - BRITISH PRACTICE

2.4.1 Reasons for Linear Analysis
Linear elastic analysis is nearly always used in design.

because of the code of practice.
methods at
This is partly
BS 5400 does allow the use of inelastic
the ultimate limit state but, as we have already seen for
reinforced concrete design, there is little to be gained from this.

analysis is also convenient for another reason;
Linear
the principle of super-
imposition applies and, with the great number of load cases which have to
be considered, this is a major advantage.
It also makes linear analysis
much easier to computerise than other methods.

Linear elastic analysis is so widely used in bridge design that designers
tend to forget that the real behaviour of reinforced concrete <particularly
lightly reinforced concrete) can be highly non-linear even at service loads.
However, linear elastic analysis does lead to safe lower-bound solutions
which
is
more
important
in
design
than
realism.
Also,
if
<as
was
suggested in 2.3.2) a major justification for the design criteria in codes

is the experience that they have led to satisfactory structures, the mere
fact that linear analysis was used in the design of those structures is
sufficient justification for using it.
2.4.2 Section Properties
Having opted to use linear elastic analysis to analyse a highly non-linear

material, such as reinforced concrete, it is necessary to make some gross
assumptions
to obtain the section properties.
designer considerable freedom.
Here BS 5400 gives the
It allows the use of the gross concrete
section, the gross concrete section plus reinforcement transformed on the

basis of the modular ratio, or the reinforcement <again transformed) plus
the concrete but ignoring concrete which is subjected to tension.
reinforced
concrete
frame
structure
it
makes
little
difference
In a
which
section properties are used because the relative stiffness of the members
is little changed.
Bridge decks, in contrast, are often prestressed <and
hence uncracked) longitudinally but lightly reinforced transversely.

transverse stiffness may differ by as much as a
methods
whilst
significantly
section
their
affects
properties
longitudinal
the
will
results
lead
to
stiffness
but,
a
theory <38 ).
- 15 -
factor of 8 between
is
unchanged.
fortunately,
safe
design
Their
any
This
assumption
according
to
of
plastic
Gross concrete sections are almost invariably used because this enables
the final structural analysis to be performed before it has been decided
how much reinforcement to provide.
It is not always appreciated that the
alternative
cracked transformed section
behaviour.
Moment is proportional to curvature, but only if any axial
force also varies proportionally.

sections
in
sagging
and
is not
strictly linear in its
Furthermore the cracked transformed
hogging
are
different
in
that,
even if the
reinforcement is symmetrical, the moment reversal moves the neutral axis.

It is this tendency of cracked sections to change their section properties
as moment is applied which causes compressive membrane action.

to
the possibility
This leads
that an elastic analysis using cracked transformed
sections would enable compressive membrane action to be used in design

within the existing code.
This would avoid the need to solve the complex
problem of assessing the cumulative effect of load history on non-linear

structures.
Such an analysis, which is only linear under proportional
loading, will be considered in later chapters.

2.4.3 Global and Local Functions
It is difficult to analyse a whole bridge in sufficient detail to design
the deck slab reinforcement.
It is convenient, therefore, to divide the
behaviour into "global" and Nlocal" functions.
The local function of the
deck slab is to support wheel loads spanning between the beams.

be analysed by a variety of elastic methods.
This can
These are all based on
isotropic plate theory which, as we shall see in later chapters, does not
model slab behaviour well.
The most popular methods are those due to
Westergaard(39) and Pucher<40>.

to
be
fully
fixed-ended
or
In this analysis, the slab may be assumed

simply
supported.
Alternatively,
an
intermediate case is sometimes used.

The global functions of the slab are to distribute load between the beams
and to act as the top flange of the beams.
A variety of elastic methods
have been used for global analysis including methods based on orthotropic
plate theory, such as the Morice Little method (41>, and several computer
methods.
The modern trend is to use computerised grillage analysis almost
exclusively <42 >.
As a bridge deck is not a true grillage, this requires
some approximations, particularly to represent the torsional behaviour, and

advice on these has been published by West <43>.
One fault of grillage
analysis for which it is difficult to correct is that it assumes that the

main beams are connected together only by transverse beams which are in
-
16 -
the same plane as the main beams.
The beams in real beam and slab decks,
in contrast, are connected together by their top flanges and the in-plane
shear stiffness of these tends to even out the stress between the beams.
Ignoring this is conservative and, although the effect on the slab stress
is significant [see Hambly <U.)J, the effect on the beam soffit stresses is
quite small.
As the latter are critical in design the effect of the error
on design is not important.

The
calculated
global
and
local
normally simply added together.
transverse
moments
in
the
slab
are
This is not strictly correct as the end
moments assumed in the local analysis should, theoretically, be applied to

the global analysis.
In the longitudinal direction the global behaviour imposes an axial force
on the slab.
reinforcement.
It is common practice to ignore this in designing the slab
The code explicitly allows this at the ultimate limit state,
apparently because it assumes sufficient redistribution capacity.
Where
the force is always compressive <that is in a simply supported deck) it

can
be
shown
that
it
is
conservative
to
ignore
it,
even
at
the
serviceability limit state.
2.5 ANALYSIS FOR DESIGN - NORTif AMERICAN PRACTICE
Bridge design throughout North America is strongly influenced, although not

always
controlled,
Transportation
<AASHTO>.
by
the
American
Association
Officials Standard Specification
of
for
State
Highway
and
Highway Bridges<45)
This differs from the British Standard in philosophy, detailed
design methods, loading specification and analytical method adopted.

these differences the last is the most significant to this study.
Of
It is
also the least well known so it alone will be considered in detail.

The AASHTO standard does allow global analysis
similar manner to that normally used in Britain.
to be performed in a
However, it is usual to
distribute the wheel loads between beams using a table of distribution

coefficients provided in the Standard and then to use simple beam theory.
This gives a less favourable distribution than the British approach.
If
the beams are not closely spaced it gives a static distribution, which is
certainly conservative.
- 17 -
A table of values is also provided for the local analysis.
This is based
on Westergaard<39), which is one of the methods often used in Britain, so

the results are similar.
The most significant difference from British
practice is that the main steel in the deck slab is designed only for the
local
moment.
The global
transverse
moments
are
not
calculated
or
specifically designed for at all.

Global transverse moments obviously do occur in American bridges so it is
interesting to assess their significance.
Where a static load distribution
is used in designing the main beams, global transverse moments are not
needed to maintain equilibrium.
Thus, according to plastic theory, the
American approach leads to designs with adequate ultimate strength.
This
does not necessarily ensure satisfactory service load behaviour, but the
writer is not aware of any cases of failures in American decks which can
be
attributed
to
global
moments.
This
can
be
explained
conservatism of the method used for local analysis.

designed only for
by
the
The reinforcement
local effects is adequate to resist global moments
because the global moments are smaller than the calculated local moments.
This would not apply to many British "M" beam deck designs
Reference 46 ).
<eg. see
The small close-spaced beams lead to higher global, but
lower local, moments than the larger wider-spaced beams which are used in
North
America.
The
British
HB
load
also gives
much
transverse moments than does the American design loading.
higher
global
Thus it seems
likely that the American design approach would not work for many British
bridges.
The author also understands that problems have been experienced
with some bridge deck slabs in the Middle East, apparently due to a
combination of designing British-style decks to AASHTO rules and very poor
control of vehicle and axle weights.
2.6 cotl.USIONS AND IMPLICATIONS FOR THIS snJDY
The
basic
form
of
bridges
is
largely
dictated
by
considerations, so it is unlikely to be changed by this study.
construction
Accordingly,
the remainder of the study will concentrate on the detailed design of the
forms of bridges in current use.
The conventional methods of analysis and design which have been reviewed
in this chapter assume structural behaviour which is often very different
from the real behaviour of reinforced concrete.
Nevertheless they have
produced structures which have behaved in a satisfactory fashion.

- 18 -
They
should be respected for the wealth of experience which they represent.

This does not mean that complicated structures, such as bridge deck slabs,
which appear to have been over-designed in the past, always have to be
over-designed in the future.
It is possible to refine analytical methods
within existing codes so that effects like compressive membrane action are
allowed for in design.
This amounts to reducing the safety margins in
such structures towards, but never below, the standards already accepted
<and found satisfactory> in simple statically determinate structures.
Even if a more radical approach,
based on first
adopted, this review has important lessons.

strength
is
not
sufficient
condition
principles,
is to be
It shows clearly that ultimate

for
satisfactory
structure.
Serviceability criteria and the effect of the complex load history of a

bridge have to be considered.
It shows too, that a deck slab design needs
to consider global, as well as local, effects.
Finally it shows that many
of the design criteria given in codes of practice are only strictly valid
in conjunction with the methods specified for checking them.
If other
methods are to be used the criteria have to be re-considered.
This
applies particularly to the serviceability criteria, such as crack width and

stress limits, which are less fundamental than ultimate strength.
- 19 -
CHAPTER
PREVIOUS
RESEARCH
3.1 INTRODUCTION
The stock of evidence showing that conventional flexural theory underestimates the strength of restrained slabs is vast.
real
structures <5,9,4-7,48,49>,
model
structures <10,13,50,51,52)
specimens (53,54-,55,56 > under
laboratory
both
been
attributed
particularly
membrane
since
action
theoretical
and
to
1955
has
compressive
when
been
Much of this extra strength
membrane
Ockleston<9>
the
experimental.
all the literature in detail.
action.
As
published his
subject
of
Research
has
countries over a long period of time.
and
concentrated <10,13,51,52,55 >
and uniformly distributed (5,9,50,54,56) loads.

has
It includes tests on
extensive
been
test
result,
results,
research,
undertaken
in
both
many
It is thus not practical to review
This chapter aims only to establish the
present state of knowledge of the subject as it affects, or could affect,

the design of bridge deck slabs.
Much of the experimental work which is most directly relevant to this
study, including most of the Canadian work mentioned in Chapter 1, has
been conducted in the last fifteen to twenty years.
Non-linear finite
element analysis, capable of allowing for compressive membrane action, has

been developed over much the same period.
Despite this, there is almost
no reference to the non-linear analytical work in the experimental studies

so it is convenient to consider finite element studies entirely separately
in Chapter 5.
3.2 REINFORCED AND PLANE Sl.ABS
3.2.1 Bending Strength
As
it
was
the
realisation
that
flexural
theory
under-estimates
the
strength of restrained slabs which promoted the interest in compressive

membrane
action,
bending strength.
it
was
natural
that
research
should
concentrate
on
Many researchers have extended flexural theories to
allow for in-plane forces.
Most have used Johansen's Yield-Line Theory<57>
as their starting point but a variety of approaches have been used.
It is
convenient to illustrate each in turn by considering the simplest possible

case;
symmetrical restrained
slab strip with equal top and bottom
reinforcement.
- 20 -
t'l
"0
I_
.I
Fig ure 3 .1:
Geometry o f r estrained slab strip
a. Rigid Plastic Deformation Theory

The simplest approach is to assume that the slab material is fully plastic
at the yield-lines but rigid elsewhere.
This gives the geometry shown in
Figure 3. 1 and it will be seen that :

=
by symmetry
say
Also, since both the restraint force, F, and the reinforcement area, A., are
the same at all sections:
dc2
de
AB2 + BC2
AC 2
<1/2 )2 + w2
Now;
Hence;
[1/2 + 2 (h/2 -dc).2w/.D2
Neglecting second order terms, this leads to;

d.:
h/2 - w/4.
Equation 3.1
Using a rectangular concrete stress block, considering a unit width of slab

and ignoring the tensile strength of concrete, the force in the concrete is
dc f c:; '
where fc ' is the "plastic" concrete stress; approximately 0.6fc: ....

The tensile force in the rein forcement is
- 21 -
A.fv
where A. is the steel area per unit width and fy is the yield stress.
Ignoring any reinforcement in the compression blocks,
this gives restraint
force, F
The total lever arm for the concrete in the compression blocks, taking the
forces to act horizontally, is
h - 2 <de: /2) - w
h/2 - 3w/4
<substituting for de:

from Equation 3.1)
and the total lever arm for the tensile steel forces
2 (d - h/2) + w
2d - h + w
So the total moment, that is the sum of the support and mid-span moments,
f.:' (h/2-w/4)(h/2-3w/4) + A. f Y <2d-h+w)
By using the virtual work approach, the load-displacement relationship of

the slab strip under any symmetrical load case can now be obtained.
The
result for some typical strips subjected to a single central load is shown
in Figure 3 .2.
Pll h2 30
20
10
p=0.5%
0.0
0.2
0. 4
0.6
0.8
1.0
1. 2
1.4
w/ h
Figure 3 .2:
Load-displacement relationship of restrained slab strip
<Rigid-Plastic Theory, f c ' = 20N/mm2
fy = 460N/mm2
d/h = 0 .8)
The unreinforced slab's maximum load, which occurs at zero displacement, is

as great as that of a simply supported slab with some 2% 1e inforcement.
It is also equivalent to an unrestlained, but still fixed-ended, slab with

about 1~ reinforcement.
The strength of the slab with

- 22 -
0.5~
reinforcement
is greatly enhanced by the restraint, but is only slightly greater than

that of the unreinforced slab.
The
load
on
unreinforced
the
slab
slabs
reduces
will not
greater than 0.67h.
as
support
tensile
any
displacement
load at
increases.
all at
The
displacements
The reinforced slabs reach a minimum load but then
the load starts to pick up again.

by
the
membrane
action;
This is because the slabs start to work
the
load
is
supported
component of the tension in the reinforcement.
by
the
vertical
Eventually the load carried
in this way can exceed the initial "ultimateu load.

Real slabs are not rigid between their yield-lines so they do not reach
their maximum compressive membrane load at zero displacement.
Thus the
real peak load is lower than shown in Figure 3.2, and occurs at
s~ificant
displacement.
However, apart from this, Figure 3.2 gives a good indication
of the behaviour of slabs, subject to certain conditions which will be

discussed in 3.2.2 to 3.2.4..
Researchers, such as Brotchie and Holley <56>,
have performed tests under displacement control and traced the descending
and ascending part of the curve after the ultimate compressive membrane
load is exceeded.
The ability of reinforced concrete slabs to support
tensile
membrane
exceptional
action
accident
may
loads.
occasionally
However,
useful
be
because
of
displacements required, it is of no practical use in the

decks.
load by
s~ificant
for
the
resisting
very
des~
large
of bridge
Slabs with realistic span to depth ratios become unserviceable long
before they enter the tensile membrane range.
In most practical bridge
deck slabs a deflection of 0.05h would be excessive.

Although the basic approach of rigid plastic deformation theory is simple,
the algebra becomes complicated when the yield-line patterns of two-way
spanning slabs are considered.
few cases.
One
Solutions have been published for only a
of the first to be solved was the axi-symmetrical case of
a fully restrained circular slab with isotropic reinforcement.

published by Wood (58>,
This was
who went on to use it to give an approximate
solution for square slabs.
He then compared the predictions of this
theory with the available test data.
Because of the elastic deformation,
the theory over-estimated the strengths.
Wood suggested that this could
be allowed for by multiplying the predicted loads by a reduction factor.

He found that the measured factors varied from 0.4. to 0.8; the smaller
- 23 -
factors occurring in the more lightly reinforced specimens.
This was
because heavily reinforced slabs are less sensitive to restraint.
If the
factor is calculated from the increase in load compared with that given by
yield-line theory, rather than from the total load, the range of observed
values is much smaller and there is no consistent trend with steel area.
Brotchie and Holley <56> used an alternative approach for correcting the
unsafe predictions of rigid-plastic theory.
Instead of multiplying the
load predicted for zero displacement by a reduction factor, they used the
load predicted for the displacement at which rigid-plastic theory gave the
same load as an elastic analysis.
Since both theories give upper-bound
solutions for the load at a given displacement on a structure composed of

elastic-plastic materials, it appears that this should over-estimate slab
strength.
This explatns why "theoretical maximum loads are slightly higher
than the test results for the thinnest slabs".
However, the theory tended
to be conservative for the thickest slabs, which had a span to depth ratio
of only 5.
This was because elastic flexibility has little effect on the
strength of such slabs whilst

greater than in shallow slabs.
the effect
of
triaxial enhancement
is
They attempted to allow for this but their
correction was conservative.

b. Rigid-Plastic Flow Theory
Plastic deformation
theory assumes
that
concrete develops
its plastic
compressive stress whenever it is subjected to compressive strain.
In
reality, not only does the strain have to be significant, it has to be

increasing; the stress reduces rapidly if the strain decreases.
Equation
3.1 predicts that the neutral axis moves closer to the compression face as
the deflection increases.
This implies that some concrete, near to the
neutral axis, experiences a reducing strain and so will not develop its
full
compressive stress.
The resulting error in the analysis can be
avoided by using "flow theory" which assumes that the full stress is
developed whenever the strain is increasing.
The derivation of Equation
3.1 is then replaced by its first differential with respect to displacement

or, more correctly, time.
do:
Braestrup(8) has shown that this leads to:
h/2 - w/2
and the load displacement relationship for the simple strip can then be
calculated in the same way as before.
- 24. -
Braestrup (8) noted that no clear distinction is made in the literature

between
flow
theory and deformation theory.
He suggested that, as a
result, much of the past research, which is based on deformation theory, is

in error.
In Figure 3.3, the results of flow and deformation theory are compared for
the
unreinforced strip which
was
considered
in Figure 3.2 .
At
large
displacements there is a very significant difference but in the practical

range of bridge deck slab deflections the difference is not significant .
Also, deformation theory is conservative because the extra concrete force
it predicts is on the wrong side of the undeflected centre-line and so
develops a couple which acts against the resistance moment .

p l/h 2 20
15
10
Deformation
0.8
0.6
0.4
0.2
0.0
w/h
Figur e 3 .3:
Comparison of flow and deforma tion theory
<p = 0,
f ,::'
= 20
N/mm2
Even in slabs which deflect 0 .5h or more before reaching peak load, the
difference between deformation and flow theory is not as great as Figure
3.3 suggests because the extra deflection is due to elastic deformation
and so does not have the same effect on the neutral axis position.
only
when
becomes
post- ultimate
important .
Since
behaviour
is
post- ultimate
considered
behaviour
that
is
the
of
no
It is
difference
practical
importance to the applications considered in this thesis, it is reasonable

to consider deformation and flow theory as interchangeable.
Mor ley <59) developed rigid-plastic flow theory so that, in principle, it is
general and can be applied to any case.
The algebra becomes complicated,
however, and he gave only a limited number of solutions.

for polygonal slabs.
One of these was
He compared this solution with some test results,
assuming that the true maximum load was that predicted for a deflection
of h/2 .
The choice of this deflection was based on work by Park(60) which
- 25 -
will be considered
later.
For the slabs which Morley considered the
predictions were reasonably good.

c. Elastic-Plastic Theory
Johansen's
Yield-Line
strengths
This
is
of
Theory<57)
unrestrained
possible
displacement.
because
Thus
deflections
slabs
without
it
gives
despite
predicts
elastic
very
predictions
ignoring
loads
deformations
affecting
good
strength.
elastic
which
can
When
are
for
the
deformations.
independent
significantly
membrane
of
increase
forces
are
considered, in contrast, there is a relationship between load and deflection

even in plastic theory.
Thus elastic deformations affect strength and it
is useful to consider them in an analysis.

The analysis is particularly sensitive to elastic shortening of the slab
because, as will be seen from Figure 3.1, small movements have a large
effect
on
the
behaviour,
particularly
at
small
deflections.
Ideally,
however, both in-plane and flexural deformations would be considered.
The
full equations for this have been formulated by Massonnet <61> and have
been applied to rectangular concrete slabs by Moy and May field <62 ).
The
mathematical complexity of the equations is such that hand solutions are

not practical so Moy solved the equations numerically by computer using a
non-linear
finite
reasonably
well,
programs.
difference
it
has
approach.
proved
Although
difficult
this
approach
to develop general
works
computer
Because of this the approach has been largely superseded by the
finite element method, which will be considered in Chapter 5.
It would be
particularly difficult to develop a finite difference program which could

be used by a non-specialist in a wide enough range of circumstances to
make it commercially viable.
far
Because of this the finite element method is
more suitable for direct
use in design and the finite difference
method will not be considered further in this study.

When elastic deformation is included in a hand analysis it is necessary to
make some gross approximations to simplify the mathematics.
The approach
adopted by Park <54), which has been followed by many other studies, was to
ignore the flexural deformation and to assume the axial strain to be
constant along the length of the strip.
Since flexibility in the in-plane
restraint
on
has exactly the same effect
the behaviour as
the axial
flexibility of the strip, it is both useful and convenient to include it in

the analysis.
- 26 -
Considering the same simple strip as before, and using deformation theory,
this leads to;
d.~
h/2 - w/4- - EF-/8w - t 1/4-w
where is the axial strain at mid-depth <which is taken to be constant) t

is the movement of each support, and the other notation is as before.
and,
Now
if
is
calculated
from
d._ f c: I
A.. f y
the
gross
concrete
properties
and
the
restraints are taken to be elastic such that;
Kt
d~
h/2
into
the
this leads to;
Substituting
this
result
+ f ,::' (P/8Erhw - 1/4-Kw)

expression
for
the moment,
which
is
otherwise the same as in the rigid plastic theory, the moment and hence
the load can be obtained.
In Figures 3.4 and 3.5 the result of this calculation is shown for slabs
with 0.5% steel and with span to depth ratios of 10 and 30 respectively.
In order to give an indication of the restraint stiffness required this is
expressed as a multiple of the axial stiffness of the slab strip.
Rigid- Plastic Theory
Pl/h 2 25
20
-I
-- Elastic-Plastic Theory
Unrestrained
K=E, h / 1
15
I
10
Kf= 0. lE.- h / 1
-1 I
I
---
0
0.0
0. 1
0. 2
0. 3
0 . 4-
0. 5
w/h
Figure 3. 4:
Elastic-plastic theory
(1/h = 10, p = 0.25%, f c: ' = 20N/mm.:: , f y = 460N/mm:.<: , d/h = 0.8)
- 27 -
At zero deflection there is no restraint force and the load is as given by

normal yield-line theory.
The load increases with deflection but starts to
reduce again before reaching the value predicted by rigid-plastic theory.

Less well restrained slabs support less load and reach peak load at higher
deflections.
The slab with the larger span to depth ratio is much more
sensitive to flexibility of the restraints but restraints which are far

less
stiff
than
the slab still have
very significant
effect
on its
strength.
Park <54) considered a more genera l case and used a different stress block
and notation but, apart from this, his theory is the same.
gave graphs
equivalent
to Figures
3.4 and
3.5
which
However, he
are significantly
different; for example, the load at zero deflection is not equal to that
given by yield-line theory.
This is due t o a small algebraic error in the
exa mple used to plot his graphs.

----------Rigid-Plasti c Theory
p l/h2 25
-
-Elastic-Plastic Theory
20
- Unrestrained
K=oo - - /
15
---
/K=E.=h / 1
_
/ / / ..- K= 0 . 3E.=h / l
10
I /........- -- -
K = 0 . 1E" h / 1
0 .0
0. 1
0. 2
Figure 3. 5:
0. 3
0. 5
0. 4
w/ h
Elastic- plastic theory
<I/ h = 30, p = 0. 25%, f c ' = 20N/mm 2
f y = 460N/mm-.' , d/h = 0. 8 )
Roberts<5 3) tested a series of restrained strips.
He compared the results
with theory similar to the above, which he attributed to Wood.
Because of
elastic flexibility in bending, the load at low deflections was less than
given
by
Roberts
the
theory.
attributed
this
The peak load exceeded

to
the
effect
of
the predicted value and
transverse
concrete at the supports which enhanced its strength.

proved
that
it
was possible
restraint
to
the
Supplementary tests
to develop stresses in excess of 0.67f '='-' "
After the peak load was passed the load reduced much more rapidly with
- 28 -
This was due to the
increasing deflection than the theory predicted.
difference between the real behaviour of concrete and the ideal plastic
behaviour assumed in the analysis.
Christiansen (63> developed a theory for restrained beams which is similar
except that he added the elastic bending deflection into the analysis.
calculated this using the uncracked section.
He
In addition to applying the
theory to beams he also applied it to slabs, including two-way spanning

slabs.
In
these
enhancement,
the
varies
deflection,
across
the
and
slab
hence
width.
the
effect
of
Christiansen
membrane
avoided
this
complication by "considering only arching action across the shorter span at

the centre of the longer span."
As expected
(and intended> this gave
conservative answers.
Park(54>
used
his
spanning slabs.
strip
theory
to
estimate
the
strength
of
two-way
He did this by assuming a central deflection and using
the strip theory to obtain the moment to use in the virtual work equations
obtained
deflection
from
of
normal
h/2,
yield-line
which
was
theory.
based on
He
a
chose
study of
to
use
test
central
results.
He
acknowledged that this deflection was conservative for slabs with span to
depth ratios below about 20.
He also acknowledged that it is a greater
deflection than his graphs, based on his strip theory, suggest.
In fact,
because of the error mentioned earlier, his graphs show peak loads which
are slightly lower than they should be and which occur at significantly
higher deflections than they should.
Thus the h/2 used by Park does not
agree with the strip theory but Park suggested that this was justified by
the elastic bending which the analysis ignores.
He showed that the theory
gave good predictions for
the strengths of slabs subjected to uniform
loads.
of
However,
because
the
use
of
deflection
of
h/2,
it
is
conservative for slabs with short span to depth ratios.

The
algebraic
complexity
of
this
elastic-plastic
theory
of
two-way
spanning slabs makes it difficult to use and gives it a false impression

of accuracy.
In fact, it is based on gross assumptions.
It is quite
different from the use of elastic-plastic material properties in non-linear

computer analysis.
It assumes that the whole depth of the slab is plastic
at the critical sections.
Elsewhere it is taken to be elastic for axial
behaviour but rigid in flexure.
The assumption that the axial strain of
the slab strips is constant at mid-depth can easily be shown to be wrong,
- 29 -
for example the neutral axis depth at the yield-line is always less than
h/2 which implies an axial shortening at this section.
Thus the major
justification for the equations developed by Park is not

which they are based
give
reasonable
the theory on
much as the fact that they have been shown to
50
results.
This
is significant
as
implies
it
that
the
approach is essentially empirical and thus may not be valid outside the
range of cases for which it has been tested.
Me Dowell et al<64> developed a different form of elastic-plastic analysis.
Although intended for use with masonry walls, it is equally applicable to
unreinforced
concrete.
It
used
the geometry shown
in Figure 3.1
and
assumed that the strain varied linearly in the span direction, from zero at
the crack to a maximum in the compressed region.
This was acknowledged
to be an arbitrary assumption, and it is easy to prove that it is not

Since the
correct, but it is just as reasonable as Park's assumptions.
total reduction in the slab length at any depth can be calculated from the
geometry shown in Figure 3.1, this enables the strain to be calculated at
any position.
Me Dowell used the strains at the yield-line positions to
calculate the stresses, and hence the bending moments, using an elasticplastic stress distribution.
He assumed that, once the plastic stress had
been reached, a subsequent reduction in strain would reduce the stress to

zero.
This made his approach equivalent to flow theory.
Rankin <65) has successfully applied the approach to unreinforced concrete

slabs.
He also adapted it to reinforced slabs by adding the effect of the
reinforcement.
Skates,
Rankin
and
Long<66)
used
similar
approach
although their method for combining the components of the moment capacity
due
to
arching
acknowledged
and
that
reinforcement
his
was
flexural and
slightly
arching
different.
analyses
Rank in
assumed
different
strain fields and the same is true of the approach used by Skates et al.
The main consequence of this is that the assumption that the reinforcement
yields
could
analysis.
be
inconsistent
with
the strains assumed
in
the arching
Although not stated in the other literature, this is a fault
which is shared with all the analyses considered in this section.

suggested
that
the
resulting
unsafe
predictions
could
be
Rankin
avoided
by
limiting the calculated moment capacity to the "balanced" capacity proposed

by_ Whitney <67) which is approximately 0.27f .,~bd2
This restriction appears
to be
taking d/h as 0.8,
maximum
conservative.
possible
Rankin
arching
pointed
moment
out
that,
capacity
- 30 -
of
an
unreinforced
the
slab
approximates
to
this
However,
capacity.
even
if
reinforcement does increase the moment capacity.

decks, d/h is significantly less than 0.8.
it
does
not
yield,
Also, in some bridge
Another conservative aspect of
Rankin's analysis is that although the reduction in the concrete lever arm
due to deflection is included, the increase in the steel lever arm is not.
_Thus the analysis would be conservative for shallow heavily reinforced
slabs . . Despite these faults, Rankin obtained good results and his approach
will be considered further in 3.2.4-.
3.2.2
F~exural
Shear Strength
The theories considered in 3.2.1

shear
failure:
With
few
assume that
exceptions,
flexural failure precedes
this assumption
literature without any particular justification.
is
made
in
the
It is therefore necessary
to investigate the validity of the assumption and again it is convenient

to consider the simple slab strip shown in Figure 3.1.
If the span to depth ratio is less than about 20, rigid plastic flexural
theory implies a shear force which exceeds the ultimate . shear strength
given
by
BS
54-00.
However,
this
ignores
the
fact
that
an
axial
compressive force enhances the shear strength of a concrete section.
simple correction for this, such as that given in the column clauses of
BS 5400, suggests that shear failures are only possible if the span to
depth ratio is less than about 6.
Since the code rules are conservative,
and shear strength is further enhanced if the shear span to depth ratio is
less than around 2.5
<which is equivalent to a flexural span to depth
ratio of 5 >, this means that shear failures in the type of strip shown in
Figure 3.1 are unlikely.
This argument can be extended to show that shear failures are unlikely in
practical restrained slabs subjected to uniform loads. and explains why no
such failures have been reported.
3.2.3 Punching Shear Strength
Even allowing
for
the limitation on the
load
imposed by the bending
strength of a slab, the shear stress in the-vicinity of a concentrated load

is much higher than under a uniform load.
Because of this, slabs subjected
to concentrated loads are likely to fail by punching and test results

confirm this <10,13,51,52,55 >.
Despite this, restrained slabs are stronger
. than unrestraiiled. slabs and, typically, five times stronger than suggested
- 31 -
by
conventional design rules
which assume
flexural
failures.
Several
attempts have been made to analyse the effect.

Aoki and Seki<68l have modified Moe's <69l equation for punching strength
to allow for membrane forces.
However, the correlation with their test
results was not particularly good and they obtained a better relationship
using a purely empirical formula.
Although this formula worked reasonably
well for their tests the author has found that it gives unsafe predictions
for many other restrained slabs and it will not be considered further.
The realisation, following research by Young (70), that bridge deck slabs
fail by punching at high loads prompted the Department of Highways and
Transportation
in Ontario
to sponsor a
major
research
programme
into
punching.
After largely experimental studies by Tong and Batchelor(51l and
Batchelor
and
Tissington<71l,
Hewitt
and
Batchelor<72l
endeavoured
to
develop a theoretical model by modifying an existing theory for punching

in unrestrained slabs.
They found that the best available theory for punching in unrestrained
slabs was that due to Kinnunen and Nylander<73l.
Kinnunen observed that
the punching failure modes of slabs were approximately axi-symmetrical,

even for rectangular specimens, so he used an axi-symmetrical analysis.
In
this model, which is illustrated in Figure 3.6, outer portions of the slab
bounded by a shear crack and two radial cracks are assumed to rotate as
rigid bodies.
The load is taken by the compressed conical shell above the
shear crack which is assumed to be shaped such that the concrete stress
is constant.
The system is taken to deform linearly with load until a
limiting strain
is reached and
the system
fails.
The stress
in the
compressed shell at failure is calculated allowing for the enhancement due

to the triaxial stress state.
Finally an empirical correction factor of 1.1
is applied to allow for dowel effect in the radial bars which the .analysis
ignores.
Hewitt and Batchelor applied the theory to 137 test results and. obtained
good results.
They said that they were better than Moe<69l obtained using
a purely empirical relationship.
However, since Kinnunen and Nylander used
empirical
strain,
factors
for
limiting
triaxial
enhancement
and
dowel
effect whilst Hew it t and Batchelor increased the factor for dowel effect
from 1.1 to 1.2 to improve correlation, the resulting "theoretical punching
load" is largely empirical.
In effect, the model was used only to give a

- 32 -
qualitative explanation of the behaviour and to give the form of the

equations; the actual values are empirical.
By adding the restraining force and moment acting at the outer edge of
the segment into the equilibrium equations, Hewit t was able to correct the
theory for known restraint forces.
Comparison with the results of tests
in which the known restraint was provided by unbonded tendons suggested

that the approach gave good predictions.
a)
SECTION SHOWING BOUNDARY FORCES
b)
FORCES ON SECTOR ELEMENT
Figure 3.6: Kinnunen and Nylander's model (73)
(as modified by Hewitt and Batchelor<72)J

By adding the restraining force and moment acting at the outer edge of
the segment into the equilibrium equations, Hewitt was able to correct the
theory for known restraint forces.
Comparison with the results of tests
in which the known restraint was provided by unbonded tendons suggested

that the approach gave good predictions.
For most practical slabs, the restraining force and moment are unknown.
Hewitt therefore proposed that the actual boundary forces and moments in
real slabs should
be expressed as a
maximum boundary forces and moments.
"restraint
factor",
R,
times
the
These maximum values were obtained
by using "The idealised geometry of displacement as used by Brotchie and

- 33 -
Holley (56>".
This gives a neutral axis depth at the support of h/2 - w/4
as in Equation 3.1.
However, this assumes that the neutral axis depths at
the support and mid-span are equal which conflicts with Kinnunen and
Nylander's assumptions.
There is also no reason why the actual restraint
force and moment should both be reduced by the same percentage relative
to their respective maximum values.
Thus Hew it t 's approach is, in effect,
largely empirical and he appeared to acknowledge this, saying "It is not

implied that the actual boundary restraint and distribution of stress are
known at the instant of failure".
Hewitt obtained the restraint factor values, R, for real slabs by backcalculation from observed failure loads.
Although he said "It is a fact
that
supported
R varies
from
zero
for
simply
slab
to
unity
idealised restraint" the highest value he observed was only 0.77.

appear to be two reasons for this.
with
There
Firstly Hewitt's analysis with full
restraint invariably gives a depth to root of crack which is greater than

h/2.
This is only geometrically compatible with his assumption that the
neutral axis depth at the support is h/2 if the supports are jacked closer
together.
Secondly, Hewitt assumed that the top steel at the supports
reaches yield which, except with large deflections, is incompatible with

Thus "full restraint" in his theory
the assumed neutral axis position.
appears to represent the ideal restraint forces, that is the forces which
r
\i"
lead to the highest
failure load, and not
<as some of his statements
imply> the forces which arise with ideal . <rigid) restraint; R = 1 could
only be obtained by prestressing.
Another oddity of the model is that it
assumes that a volume of concrete, bounded on one side by a shear crack,

rotates as a rigid body until a shear compression failure occurs; yet all
the descriptions of failures show that the shear crack does not appear
until the failure load is reached.
Clearly,
allhough
claimed
essentially empirical.
to
be
theoretical model,
the
approach
is
Hewitt claimed that it gave acceptable predictions
for the strengths of realistic bridge deck slabs and it has been used to
develop charts for assessing the strength of existing bridges <11 >.
In
order to ensure that these are safe, and to avoid the need for separate
charts
for
use
with
steel and
concrete
beams,
they are
based
on a
restraint factor of 0.5 even though tests on concrete bridges suggest that
values as high as 0. 7 give more accurate predictions.
- 34 -
Kirkpatrick, Rankin and Long<13) have developed an alternative analysis of

punching in restrained slabs.
Like Hewitt's approach, this was developed
by modifying a theory for punching in unrestrained slabs.
The theory used
was Long's<74-) "two-phase approach" which gave the strength of a slab

which fails in shear before the steel yields as;
4-(Ctd)d
0, 42(fcyl) 0
(10Qp) 0
26
(O. 75 + 4-cil>
where c is the side of the square loaded area,
f cv 1
is the cylinder
compressive strength and the other notation is as used previously.

Kirkpatrick et al took the denominator
<which is a correction for the
effect of the ratio ell> as constant at 1, arguing that

variation was small.
the effect of
This is reasonably true for the type of specimens
originally considered by Long, but the value of the denominator for some
of Kirkpatrick's slabs was as high as 1.6 so the stated reason for ignoring
this factor is unsatisfactory.
For fully restrained slabs they argued, by reference to test results, that
the effect of reinforcement was small and they took the term (100p)0 26 to
represent the influence of flexural strength on shear strength.
The value
of p which they used was the equivalent steel area p.; the area of steel
which would be required to give an unrestrained slab the same moment
capacity according to normal flexural theory which the fully restrained
slab had according to restrained strip theory.
The particular theory which
they used was that due to Me Dowell et al<64>, although it appears that
any of the methods described in 3.2.1
could be used.
Because of the
fourth root term, the choice of approach has little effect.

Kirkpatrick
empirical.
punching
appears
to
have
accepted
that
his
approach
was
largely
However Rankin<65) has developed a similar approach, to analyse

at
columns
theoretical basis.
in
flat
slabs,
and
he
attempted
to
give
it
He assumed that failure occurred when the compression
zone failed in shear.
Because compressive stress tends to enhance the
shear strength of concrete, he said that the critical position was at the
flexural neutral axis.
He calculated the shear strength of the compression
zone assuming an elastic stress distribution and a critical section at d/2

from
the
transmitted
face
of
the
across
the
loaded
shear
area.
crack
Then,
arguing
by . aggregate
- 35 -
that
shear
interlock and
was
dowel
forces,
he said that
the
total shear capacity was 2 to 5 times the
capacity of the compression zone.

There are many faults with this as a theoretical analysis.
Firstly, the
shear failure criterion at the flexural neutral axis was based on maximum
principal tensile stress.
This is not really a failure criterion for the
slab at all; it merely suggests that the shear force reduces the neutral
axis
depth,
fact
which
is
well
known
from
research
on
beams <75 ),
Secondly, if <as Rankin said and as the observed behaviour suggests> slabs
fail as soon as the shear crack appears, dowel forces cannot contribute
significantly
behaviour.
that
to
the
ultimate
strength;
only
to
the
post-ultimate
Thirdly, the geometry of the failure mode appears to suggest
there is no shear displacement across the shear crack; it merely
opens up.
Thus the aggregate interlock force must be small as Ghana <75 >
has found for beams.
However, in beams the load continues to increase
after a shear crack appears and Ghana found that the dowel effect was
very significant.
Using his approach, it is possible to quantify the force
for a punching failure.
Because <as Rankin noted> the inclination of the
shear crack means that the failure surface is very long at the position of
the reinforcement, the dowel force in slabs with conventional quantities of
reinforcement is large.
The assumption that this force is realised before
failure occurs is hard to reconcile with Kirkpatrick's observation

assumption> that reinforcement has little effect on strength.
<and
Although
Rankin was a co-author of Kirkpatrick's paper(13>, they appear to have

differed on this point.
Rankin(65) took the dowel force to represent 25%
of the shear strength of a reinforced unrestrained slab.
He therefore
assumed that the shear strength of a restrained unreinforced slab with the
same depth of concrete in compression at the critical section would be 25%
lower.
Kirkpatrick, like Skates <66) in a more recent paper, used the full
shear stress even in unreinforced slabs.
Despite this, differences in their
methods for estimating neutral axis depth make Kirkpatrick's formula
more
conservative than Rankin's for typical bridge deck slabs.

Kirkpatrick said that his formulae gave good predictions for test results
and it is informative to compare his approach with Hewitt's.
Both are
essentially empirical so they can only be compared by comparing their

predictions.
However, since they were calibrated using sets of data which
are not only very similar but which overlap, the .absolute value of their
predictions give little idea of the relative merits of the appro'aches.
- 36 -
As
might be expected, both give reasonably good predictions for typical slab
test results.
A better indication of their relative merits is given by the
predicted relationship between failure load and the important variables

which affect it.
These will now be considered in turn.
a. Loaded Area
Since Hewitt considered a critical section at the face of the loaded patch,
whilst Kirkpatrick considered a critical section at d/2 from the face, it
might be thought
that Hewitt's predictions would be more sensitive to
patch size than Kirkpatrick's.

corrections for
However, Kinnunen and Nylander's empirical
limiting strain and for triaxial enhancement more than
compensate for this.

Taylor and Hayes'<55> results enable the effect of patch size to be clearly
identified.
They suggest that Kirkpatrick's approach is remarkably good in
this respect.
However, including Long's original correction for ell makes
1t significantly worse, suggesting the factor was removed to improve the
results.
Hewit t 's analysis exaggerates the effect of patch size but it is
only with Taylor's smallest patch size
<2h/3) that
the error is really
significant and this is outside the range of c/h ratios which normally
occur in bridge deck slabs.
b. Concrete Strength
Because Hewitt and Batchelor's theory assumes a shear compression failure,
whilst Kirkpatrick et al's implies a shear tension failure,
significantly in their predictions for
Long's
two
phase
approach
gave
they differ
the effect of concrete strength.

square
root
relationship
<and
he
suggested that a coefficient of 0.4 was slightly better) but Kirkpatrick's

method of calculating p. increased this up to fc.., 0
to depth ratios.
theory itself.
75
for very short span
However, it is not clear if this is justified by the
It is generally accepted that such shear failure loads are
proportional to something between fc.., 0

the tensile strength of concrete
(as in BS 8110 and BS 5400) and
<approximately proportional to f c ... 0
Also, although Long's original paper implied that the term p0
25
5 ).
was purely
empirical, Rank in <65) suggested that 1t was used because, for the relevant
reinforcement ratios, the neutral axis depth is approximately proportional
to p0
25
If so, it would be more logical to use the neutrdl axis depth
given by the arching theory <as Rankin did), rather than going indirectly
to an approximate value via a hypothetical equivalent reinforcement area.
- 37 -
This would reduce the predicted sensitivity to concrete strength t o f c:u 0
for s hort s pan to depth ratios and less for longer span to depth ratios.
The shear compression failure mode consi dered by Hewit t and Bachelor might
be
expected
strength.
t his
to give
However,
failure
loads
which
are
proportional
the empirical expression for
sensit ivity.
Despite
this,
the
approach
to concrete
limiting strain reduces

predicts
s ignificantly
greater sensitivity to concrete strength than Kirkpatrick's as will be seen

from Figure 3 .7.
has
nol
Unfortunately, in the publis hed studies concrete strength
been varieJ sufficiently widely or
which is more realistic.
1000
Failure
Load <kN )
900
systematically
to determine
Hewitt
& Batchelor, R = 0.
Hewitt
& Bat chelor, R
800
700
= 0.5
Kirkpatrick et al
600
500
400
300
25
4.0
35
30
Figure 3 . 7:
4.5
50
f .. .... <N!Jnlli2)
Effect of concrete strength
<1 = 1.5m, h = 160mm, c = 320mm, p = O>
c. Reinforcement Area
The most obvious difference between Kirkpatrick's approach and Hewitt and
Batchelor's approach is that the former ignore s reinforcement whilst the
latter considers it .
aspect
of
However, although Batchelor <76) has criticised this
Kirkpatrick's
approach,
saying
that
reinforcement
is
an
"important consideration", his own theory predicts only a small effect for
well
rest rained
For
slabs.
typical
M beam
slab,
1% reinforcement
increases the predic ted strength by around 15%.

A
c urious
feature
reinforcement
is
of
Kirkpatrick's
ignored,
the
approach
prediction
- 38 -
is
is
that ,
affected
by
although
the
the
assumed
effective depth.
Since this is clearly illogical, the author has used a
hypothetical d of 0.75h for all calculations with the approach.

Analysis of the results of tests on bridge deck slabs appears to give
conflicting
evidence
for
the
significance
of
reinforcement
area.
Kirkpatrick <13) obtained virtually identical failure loads with 0.25%, 0.5%,
1.25% and
However both Beal<77) and Ba tchelor et
1.68% reinforcement.
al's <78> results suggest that Hewitt 's approach under-estimates the effect
of reinforcement.
Beal obtained average failure loads, for his model 2,
which is illustrated in Figure 3.8, of 11, 26.6 and 31.1 kN with 0, 0.23 and
0.35% reinforcement respectively.
He also obtained an average failure load
of 26.7kN with 0.35% bottom steel and no top steel.

the
apparent
contradiction
between
Beal
and
The possibility that
Kirkpatrick's
results
was
because small steel areas have a significant effect, whilst increases above
some critical area have no effect, can readily be eliminated by reference
to other tests such as Taylor and Hayes' (55 ).
They obtained a barely
s ignificant difference with 0, 0.9 and 1.8% steel.

Al e
114
82
C2
CJ
C5
D8
DlO
E8
mo. 35%
[1 0. 23% Ub
0 Unreinforced
0.35% t&b
Figure 3.8:
Beal's Model Two(77>
As Beal and Kirkpatrick's results appear to be so contradictory it is worth

considering them further.
bridges,
which
are
Accordingly the author has analysed the two
illustrated
in
Figures
3.8
and
3.9,
using
both
Kirkpatrick's and Hewitt 's approaches.

A major difficulty
in
interpreting
the results
is
that
Kirkpatrick varied the steel area within the same deck.

- 39 -
both Beal and

There are two
objections to this approach.
Firstly the restraint may be different in
different areas of the structure.
This is confirmed by Bea l's results, a s
tests conducted near the centre of t he bridge gave consi stently highe r
results t han t hose conducted near t he e dge.
36~
higher load than E2.
contributes
to
the
For example, 04 f a iled at a
The second objection is that
restraint ,
so
reinforcement
in
reinforcement
adjacent
contribute to the restraint available to the area under test.
bays
may
Again Beal's
tests show evidence of this as test 08, in the unreinforced a rea, was
stopped when the cracking extended into the adjacent reinforced bay, by
2~
which time the load was already
times that at which 010 later failed.
- --
~----..1..-- ----L------~.-!
~----T---- - -r------r----~
A3
C3
03
83
- --~- --- --- ~-----------"l

C----~-------~-----,----1 A5
I
C5
I
05
85 I
- - --j---- ----c ____

- ------1
--- -- --~-----j
------- 1
1 A2
C2
I
02
I
8'2 :
I ----+ - --- - ~--- -- -J - - - - - 1
~----T-----~- -- - -- T ----A4
I
C4
I
04
I
84 I
r -- ---- -r-----I- - - - I A,
I
C1
~--
t--- --
l.
-'"1---- ---- -__- ,_-- - - ..J

---- '
~ --
01
81
-l- - - - - -- .J---- -- -t------ -l
r - - - - ., - -- -- ---=t - - - - - -
A 168"1.
-_1----- -__]_
c- 0 49"z; o0 -25"1. B1 191~
0
Plan
3COO
r-
u6660 0666uSOJu666u
I
9X)
.,.
Section AA
Figure 3. 9:
Kirkpa trick's model ( 13)
Kirkpatrick's a rrangement of bays might be considered unfortunate as the

two lightly reinforced bays were near mid-span, where Beal's tests gave
higher failure loads, and they were surrounded by more heovily reinforced
areas.
Also
the bays were rather narrow relative to
However, by comparing the results for t he

separately
comparing
eliminated but
the
1.68Z and
0.25~
and
1.19% panels,
there is still no trend.

- 40 -
0.49~
the slab s pan.

panels and t hen
both effects can
be
Thus 1t appears that varying
reinforcement genuinely had no significant effect on Kirkpatrick's results.

They also show no sign of the difference between the centre bay (bay 2>
and the edge
(bays 1 and 3) whereas this effect is very significant in
Seal's results.
These differences between Kirkpatrick's and Seal's results could be because
Kirkpatrick's
stiff
concrete
beams,
diaphragms
and
parapet
upstands
provided adequate restraint whilst Seal's deck, with its flexible steelwork
and no upstands, was more dependent on the slab and its reinforcement for
restraint.
However,
restraint
factors
back-calculated
approach are little different for the two decks.
using
Hewitt's
Those for Kirkpatrick's
deck are in the range 0.5 to 0. 7 whilst those for Seal's reinforced panels
are in the range 0.45 to 0. 75.
Thus the greater effect of reinforcement
on Seal's results cannot be explained by lack of restraint, although it

seems likely that the steel girders in Seal's deck did provide less good
restraint.
The high restraint factors observed near the centre of his deck
appear to be the result of global effects which gave the centre portion of
the slab a significant biaxial compression.
Seal
noted
that
recommended
by
Hew it t 's
the
theory,
Ontario
with
code,
gave
restraint
factor
conservative
of
0.5
results.
as
The
predictions approximated closely to his results for the outer portions of

the deck.
but
the
strengths
He does not appear to have analysed the unreinforced sections

author
by
has
factor
found
of
that
up
Hewitt's
to
just
theory
over
2.
over-estimates
This
is
their
better
than
Kirkpatrick's predictions which are unsafe by a factor of up to nearly 3.

Kirkpatrick's predictions for the reinforced areas are slightly higher than
Hewitt's and are thus closer to the averageobserved values.
Clearly the effect of reinforcement on the strength of Seal's slabs was
greater than Hew it t 's theory predicts and much greater than was observed
by Kirkpatrick.
The reasons for this will be considered later.
Seal said that Hew it t 's theory ignores "compression steel" so the top steel
has no effect
on predicted strength.
compression steel but
he did consider
It
is
true that
top steel at
Hewitt
ignored
the support.
The
reason Seal's had little effect on the predicted strength was that it was
very close to mid-depth.
According to Hew it t 's model, mid-depth steel
should have no effect on the strength of a fully restrained slab.
However,
the fact that his theory still predicts no effect in partially restrained
- 41 -
slabs is merely a consequence of the assumption that the actual restraint

forces and moments are proportional to their respective maximum values.
d. Span
The two theories differ significantly in their prediction for the effect of
span to depth ratio.
beam slab.
This is illustrated in Figure 3.10 for a typical M
If Long's original expression for the effect of cl 1 is included
Kirkpatrick's analysis suggests that strength increases with span, which

seems improbable.
This suggests that Kirkpatrick removed it to improve
correlation rather than because the effect is small.
As the expression
was purely empirical this was a perfectly reasonable thing to do.

------ Hewitt
= 0. 7
Failure
1000
Load <kN )
900
& Batchelor
--Kirkpatrick et al
800
700
600
500
1. 0
Figure 3.10:
the
term,
smaller than Hewitt 's.
3. 0
Span (m)
Effect of span
<h = 160mm, f c '"' = 40N/mm 2

Even without
2.5
2.0
1.5
Kirkpatrick's
c = 320mm, p = 0)
predicted effect
of span
is much
However there is a fundamental difference between
the approaches in that Hewitt's includes a check on the moment equilibrium

of the system whilst Kirkpatrick's does not.
two-phase approach <74),
Within the logic of Long's
on which Kirkpatrick's analysis
is based,
it
is
clear that a separate check on bending strength should be made and this
would be more likely to be critical with longer spans.
Kirkpatrick did not
detail this check because he considered it would not be critical in normal

deck slabs.
However, Rankin <65) did detail such a check and this will be
considered in 3.2.4.
Batchelor also implied that his predictions were not
failures.
He
said
that
these
occur
- 42 -
with
low
valid for bending
reinforcement
and
poor
restraint but, like Kirkpatrick, he assumed that they would not occur in
realistic bridge deck slabs.
It is difficult to clearly identify the effect of span from test results.
As with reinforcement area, tests on model bridges appear contradictory.

Kirkpatrick obtained virtually identical results for his two span lengths
<1/h of 9.4 and 12.5 or 7.2 and 10.2 if only the clear span between the
stiff
beams
is
considered)
but
Batchelor<78) obtained
higher load with an 1/h of 13.7 than with 20.7.

varied
the
span within
single model,
an
average
43%
However, since Kirkpatrick
whilst
Batchelor
varied
it
by
testing three and four-beam models of the same width, it seems likely that
Kirkpatrick's longer spans were better restrained than his shorter spans
whilst Batchelor had the reverse situation.
bays,
Kirkpatrick's
analysis
longer spans by some 30%.
Even ignoring the unreinforced
over-estimates
the
strength
of
Batchelor's
It gives better predictions for the shorter
spans although it is still slightly (18%) optimistic for the unreinforced

bays.
Despite the differences in the restraint, and consequent difficulties of
interpretation, a trend can be detected from the analysis of Kirkpatrick,
Batchelor and Beal's results: increasing the span reduces the strength of
unreinforced
slabs
by
more
than
either
increases the effect of reinforcement.
theory
suggests
but
it
also
This could easily be explained if
the failures were flexural rather than shear failures.
In 3.2.1 we saw
that the greater deflections associated with longer spans reduce the area
of concrete in compression and reduce the lever arm at which it acts,
whilst increasing the lever arm at which the steel acts.
This effect is
allowed for by Hewitt's analysis but it is greatly under-estimated because

the
deflection
is
under-estimated.
Hewitt's
analysis
under-estimated
Kirkpatrick's small deflection at failure by a factor of 2 and Beal's, which

were of the order of h/2, by a factor of up to 10.
Kirkpatrick's analysis
does allow for the reduced concrete contribution with longer spans but,
because of the fourth root term, it. under-estimates the effect.
Neither
Hewitt's nor Kirkpatrick's analyses are capable of allowing for another

effect of deflection; it increases steel strain and hence, if the steel has
not reached yield, steel force.
It appears that both theories would become unsafe if they were applied to
slabs,
particularly
unreinforced
slabs,
- 43 -
with
very
large
span
to
depth
ratios.
However, neither are recommended by their originators for use
above a span to depth ratio of 18.

reasonably safe.
Within this restriction they are
Although Seal's slab was well within this llh ratio the
results which fell below the predictions were for panels which had neither
the nominal steel nor the edge stiffening recommended by both Kirkpatrick
and Batchelor.
It
should also be noted
that
both theories correctly
predict that the failure loads of such slabs are so
h~h
that their precise
values have no practical significance; the safety factors suggested by the

tests were in the range 5 to 30.
e. Multiple Loads
Another effect of increasing the slab span is to increase the effect of
the other
wheels of
the HB
load.
Neither
analysis enables this effect to be assessed.
Kirkpatrick's
nor Hewitt's
Kirkpatrick's choice of a
critical shear perimeter at d/2 from the loaded area implies that wheels
spaced by more than 2c
other.
d centre to centre should have no effect on each
However, the empirical nature of the approach makes this dubious
and Kirkpatrick's own tests confirm this: for the longer spans, two wheels
spaced by over twice this distance failed at only 40'4 more total load than
single wheels.
Hewitt 's analysis implies that wheels spaced by less than 1
could affect each other.
This is confirmed by Kirkpatrick's tests.
For his
shorter spans the HB wheel spacingcorresponded to 1.2 land there was no

effect.
For the longer spans the same spacing corresponded to 0.91 and
the effect was very
s~nificant.
However, since the presence of the second
wheel violates the assumption that the system is axi-symmetrical, Hewitt's

approach does not enable the effect to be quantified.
3.2.4 Ductility
Most of the membrane flexural theories considered in 3.2.1 are based on
plastic
theories,
such
as
Johansen's
yield-line
theory.
An
important
assumption of these theories is that the behaviour is ductile.

Reinforcement
although
is ductile,
in reality
there
whilst
is a
concrete is relatively brittle.

continuous
transition
Thus,
from ductile
to
brittle behaviour, the assumption of ductility is normally considered valid

provided that the tension reinforcement yields before the concrete crushes.
This means that sections are considered ductile provided the ratio dc/d
under ultimate moment is less than some critical value.
varies
sl~htly
according to the material properties.

- 44 -
The critical ratio
In the absence of axial forces, the neutral axis depth ratio is a function
of the reinforcement
percentage.
The requirement
for
ductility
reduces to a critical reinforcement ratio which is around 1.2% <79).
thus
This
includes most bridge slabs and nearly all building slabs.

In contrast. to the situation in unrestrained slabs, the theory considered
in 3.2.1 suggests that the neutral axis depth in a restrained slab is a
function of the in-plane restraint.
It
is also fixed relative to the
overall, rather than the effective, depth.
Simple calculations show that
realistic
bridge
requirement.
deck
slabs
almost
never
comply
with
the
ductility
In many cases, calculations suggest that the steel stresses
should still be quite low when the concrete crushes.
This is confirmed
by researchers who have found that such slabs fail in a brittle fashion
before the reinforcement, often even in the critical areas, has reached
yield.
Thus it appears that few of the theories considered in 3.2.1 are
valid in bridge deck slabs.

Building slabs tend to have larger span to depth ratios, relatively poor
restraint and higher effective depth to overall depth ratios.
Thus they
are more ductile than bridge deck slabs and hence their behaviour is
better predicted by plastic theories.
Despite this, calculations suggest
that the behaviour of some of the slabs which have been tested should be
brittle.
This was often supported by the behaviour at failure.
Yield-
line based theories did, however, agree reasonably well with failure loads.
To some extent this was mere coincidence; the theory under-estimates the
strength of strips so there is some margin for inability to re-distribute
the moments.
loaded
by
However, it is significant that all the test specimens were

uniform
loads.
Under
such
loads
the
yield-line
moment
distribution does not differ greatly from the elastic moment distribution
so plastic theories do not make great demande on rotation capacity.
Under
concentrated loads, in contrast, elastic theory predicts local moment peaks

so plastic theory depends on very high rotation capacity.
Because of this,
the theories considered in 3.2.2 tend to over-estimate strengths under

concentrated loads.
This is presumably why uniform loads were chosen to
test most of the theories, although this was not acknowledged.

Amongst
the few studies to acknowledge that, because of this lack of
ductility, yield-line based analyses may not be valid even in uniformly

loaded slabs, are those due to Skates, Rankin and Long<66> and also Niblock
- 45 -
and Long (80).

problem.
They developed a semi-empirical approach to overcome the
Jn this, the moment capacity of the critical section is calculated
as in their analyses considered in 3.2.1.
The relationship between the
failure load and the moment at the critical section is then calculated
using both elastic and plastic theory.
Only if the moment capacity is
zero, is the plastic relationship considered to be directly applicable.
If
the moment capacity is equal to that of a plastically balanced section the

elastic relationship is used.
these
two
extremes,
interpolation
an
according
balanced moment
For all realistic cases, which are between
intermediate
to
the
capacity.
solution
ratio
of
As might
the
is
obtained
moment
by
capacity
linear
to
the
be expected, since this approach
implies that yield-line theory is only valid in unrestrained slabs with

negligible steel areas, the result tends to be slightly conservative.
Skates, Rank in and Long (66) have applied this analysis to slabs subjected
to concentrated loads whilst Rankin(65) used it for flat slabs subjected to
uniform loads.
Because of the great difference between the elastic and
yield-line moment distributions for such cases, they are a severe test of
a simple linear interpolation.
The use of a strip-based method to obtain
the moment capacity is also questionable as there is no reason why the

distribution of membrane forces
throughout
the span.
across a section should be the same
Also, as Rankin acknowledged,
the slab analysis
implies a different support moment from the strip analysis.

perhaps surprising that
It is thus
they obtained a mean ratio of test result to
prediction of 1.16 and a standard deviation of only
10~.
However, to
achieve this, they used Kirkpatrick's approach as an upper limit imposed by

"shear mode failures".
Whilst many brittle bending failures are reported in the literature for
uniformly loaded slabs, few such failures are reported under concentrated
loads.
As there are theoretical reasons for thinking that slabs are more
likely
to
fail
in
bending
before
reaching
their
yield-line
moment
distribution under concentrated than under uniform loads, this may appear
surprising.
It is instructive to consider what such a failure would look
like.
Elastic theory predicts high moment peaks under the concentrated load.
Thus the highest concrete stress occurs in this region,
but here the
crushing stress is enhanced by the triaxial stress state so the first

- 46 -
- - - - - - - - - - - - - - - - - - -
-----
crushing is likely to occur around the edges of the loaded area.
It may
then extend along the potential yield-lines, 1n which case the failure will
However, the area where the concrete first
be described as "flexural".
reaches its crushing stress is subjected to a high shear stress.
Thus, as
it approaches its crushing stress, 1t is liable to fail suddenly under the

combined effect of shear and compression, in which case the load will
punch through the deck.
Thus there is no clear distinction between a
punching shear failure and a brittle bending compression failure so an

alternative interpretation of the failures considered in 3.2.3 is that they
are essentially flexural failures with shear playing a comparatively minor
role.
It has already been noted that some aspects of the test results can
be explained by flexural theory.
It is also clear from the descriptions of
failure that the characteristic conical shear cracks do not appear until
failure.
Thus this interpretation is worth further investigation and it
will be considered in later chapters.

If the failures considered in 3.2.3 were primarily brittle bending failures,
it
provides
another
explanation
for
the
reinforcement on Kirkpatrick's failure loads.
small
effect
of
varying
Unlike the other researchers,
he used the same secondary steel throughout; he varied only the main
steel.
Increasing this did significantly reduce the deflection at failure,
apparently
sections.
due
to
the
reduced
ductility
of
more
heavily
reinforced
With constant secondary steel this implies that the moments in
the secondary direction at failure must have been greater in the more
lightly reinforced panels.
This in turn implies that the distribution of
the primary moments must have been more favourable in the more lightly
reinforced panels and this tended to compensate for the reduced strength.
3.2.5 Serviceabillty
Because of
compressive membrane action,
crack widths,
slabs.
restrained slabs have smaller
deflections and steel stresses than similar unrestrained
Holowka <81 ), Cairns <82) and others have measured steel strains of
the order of a tenth of those predicted by conventional flexural theory

whilst Kirkpatrick <4-9) observed a similar effect on crack widths.
There is
also wide agreement that compressive membrane action delays the formation
of
the
first
crack,
presumably
because
concrete's stress-strain
departs from linearity before cracks become visible.
curve
However, the effect
of restraint on acceptable service load is not as great as on strength.

Because of
this,
nearly all
the
researchers
- 4-7 -
who have considered
the
implications of using membrane action in design have acknowledged that

serviceability
criteria
would
become
critical.
Despite
this,
the
theoretical studies have concentrated almost exclusively on the prediction

of strength.
The design rules which will be considered in 3.2.8 do depend
on the enhancement of serviceability due to membrane action but, in this

respect, they have no quantitative theoretical basis at all.
Hewitt's approach, which was considered in 3.2.4, is one of the few to have
been applied to behaviour at service loads, specifically to the prediction
of deflection.
However it is very unsatisfactory for this purpose.
It is
an axi-symmetrical model whilst, although the failure modes of deck slabs

are approximately axi-symmetrical, the behaviour at service loads is not.
Also the model considers a compressed volume of concrete which is almost
entirely arbitrary except in its area at the critical section.
In view of
these and other faults, some of which were considered in 3.2.4, it appears
that any resemblance between the deflections predicted by this approach
and
those
However,
which
occur
because
in
the
practice
analysis
is
little
more
linear
assumes
than
coincidental.
load-displacement
relationship, whilst the observed behaviour is often highly non-linear, it

does not
under-estimate deflections under service loads as much, or as
consistently, as at failure.
Although
compressive
membrane
action
tends
to
improve
the
ultimate
strength of restrained slabs more than their service load behaviour, there
are situations in which it may be useful at service loads but not at
failure.
Yield-line theory assumes that the full plastic bending moment is
developed
across
wide
width
of
slab.
This
means
that
helpful
compressive force across this critical section can only be developed by

restraint
which is external to the slab, or at
material well away from the loaded area.

appropriate
to service
load
However, elastic theory is more
behaviour and
bending moment under concentrated loads.
least which comes from
this predicts high peaks of

Thus a beneficial compressive
force across these critical areas could be developed by adjacent areas of

less heavily stressed slab.
----~---stresses
could
be
reduced
This means that maximum crack widths and

by
compressive
unrestrained slabs, such as slab bridges.
membrane
action
even
in
This possibility does not appear
to have been considered before, presumably because of the concentration on

strength and the historical development of compressive membrane theory
from yield-line theory.
- 48 -
3.2.6 Restraint
Compressive membrane enhancement depends on the availability of adequate

restraint strength and stiffness.
Thus the prediction of this restraint is
important.
One reason why membrane action has been so little used in
design
the
is
feeling
that
the
restraint
available
to
slabs
is
unpredictable and perhaps unreliable.

Park is one of the few researchers to give restraint the at tent ion it
deserves.
His work considered building-type slab and beam systems and he
tested many nine-panel specimens <83).
When only the centre panel was
loaded,
before
peak
load
was
achieved
provided the restraint) cracked.

the
concrete
restraint.
in
just
the
outer
panels
<which
This shows that the tensile strength of
the surrounding structure contributed greatly to the
Park assumed that this tensile strength should be considered
unreliable for design purposes, as is usual.
Thus, when Hopkins and
Park (50) designed a nine-panel floor system allowing for membrane action,
they provided extra reinforcement in the beams to resist the restraint
forces.
They showed that this steel was heavier than that which they had
saved by considering membrane action in the design of the slab so they

suggested that design using membrane action was uneconomic.
This arises
because building slabs are designed for all bays fully loaded so the same
load case is critical for all bays.
Bridge decks, in contrast, are designed
for moving loads and hence a different load case is critical for each part
of the slab.
This means that the critical area is always surrounded by
areas for which a different load case is critical.
Thus there is always
under-stressed steel available to provide the restraint and no extra steel

is needed.
This means that the scope for economy from using membrane
action in design is much greater in bridges, and other structures which

are designed for moving loads, than it is in buildings.
This is why recent
research into membrane action, including this study, has concentrated on

bridges.
Park analysed his specimens using his strip approach, which was described
in 3.2.1c.
resist
the
He consistently recommended that steel should be provided to

full
restraint -force- but-he-was-less--consistenc -inhls- - --
assessment of the contribution of concrete to restraint stiffness.
In
reference 83 he used only the steel in assessing axial stiffness, but

ignored
lateral
bowing of
the outer slab panels.
Theoretically this
approach should be conservative where there are wide lightly reinforced

- 4-9 -
outer
panels
panels.
but
unsafe
where
there
are
narrower
This is confirmed by the test results.
heavily
reinforced
In reference 50 Hopkins
and Park used gross concrete properties for assessing restraint stiffness
but compensated for this by arbitrarily increasing the axial flexibility of
the loaded panel by a factor of 4.
This shows that the prediction of
restraint flexibility is highly approximate so it is fortunate that, as was

seen
in
3.2.1c,
the
strength of
slabs
is
not
sensitive
to the exact
stiffness of the restraints.

Apart from Park's study, very little work has been done on the prediction
of restraint.
The approach adopted in the Canadian study was to measure
the restraint available; not by direct measurement of restraint stiffness

or strength, but by observing the behaviour of the slab under a load and
back-calculating the "restraint factor" needed in their theory to predict
the observed behaviour.
A disadvantage of this approach is that it is only possible to measure the
restraint available at the time of the test.
prediction means
that
it
is not
The lack of an analytical
possible to predict
restraint which might occur in the future.
any reduction in
In view of the importance,
according to Park's work, of the tensile strength of concrete in providing

the restraint this is significant; cracking due to loads previously applied
in other positions, or to shrinkage,
Canadians
were
aware
of
this
so
could reduce
they
the restraint.
conducted
tests
where
loads <84) or pre-loading to failure <78) had occurred in adjacent
The
cyclic
bays.
This seems to have had little effect.

Hew it t obtained restraint factors for laboratory specimens and models by
back-calculating from the observed failure loads.
However, in the field
tests on full size bridges <8D, it was not practical to test to failure so
the restraint factors were estimated from the deflections at lower loads.
In view of the doubts expressed in 3.2.5 about the validity of Hewitt's
method
for
predicting
deflections,
this
approach
is
less
satisfactory.
However, the results were similar to those obtained from models although
the variation was much greater.
Kirkpatrick assumed rigid restraint which is obviously an unconservative
assumption.
However, since his approach is essentially empirical and was
calibrated with
tests on real structures which had less than perfect
restraint, this is unimportant from a practical viewpoint except that, as

- 50 -
with the Canadian work, there is no way of allowing for possible future
reductions in restraint.
3.2.7 Global Behaviour
Compressive membrane action in bridge decks is normally considered as a
mechanism for resisting local wheel loads spanning between webs.
However,
as was noted in 2.4.3, bridge decks are also subjected to global flange
forces
and
moments.
Since slab
behaviour
is not
linear-elastic
<and
compressive membrane action depends on this non-linearity> the principle of

superimposition
does
not
apply.
Similarly,
because
the
behaviour
of
restrained slabs under concentrated loads is not ductile, it is not safe to

assume
that
global
stressed areas.
forces
will
re-distribute away
from
locally over-
Thus separate studies of global and local effects cannot
prove that behaviour will be satisfactory under combined effects so the

interaction of the effects has to be considered.
A global flange force which is compressive has the effect of prestressing
the slab.
Thus,
unless the stress is so high that concrete crushing
becomes a problem <which is unusual), it improves the behaviour and can

safely be ignored; as it has been by all the previous research.
Tensile flange forces might be expected to have a detrimental effect on
slab behaviour.
simulated
the
Because of this the Ontario study included tests<78> which

support
region
of
continuous
bridge.
The
resultant
tensile flange force had remarkably little effect on the behaviour which
was still entirely satisfactory.
It can also be shown that tensile flange
forces are unlikely to be serious for another reason: the critical design
load case for global flange tension does not impose any local wheel loads
in the critical area.
Thus, when local wheel loads are imposed, any loss
of longitudinal compressive membrane action due to flange forces is more

than
compensated
for
by
reinforcement
provided
to
resist
the
non-
coexistent worst global moment.

Global transverse moments present a more difficult problem.
It was noted
in 2.4 that, in some types of deck, these moments can be even greater than
the local moments predicted by elastic theory.
these
large
moments
in
combination
with
local
It
is conceivable that
effects
could
premature failures in the very lightly reinforced slabs proposed.
cause
Previous
experimental studies, although comprehensive in other respects, have not

- 51 -
investigated this possibility.

bridges
moments.
with
Many of the tests were on steel composite
cross-frames which greatly reduce
the global
transverse
Most of the tests were performed using single wheels and some,
because of propping off the laboratory floor or reacting against the

adjacent beams, did not even model the global transverse moments which a
single wheel could cause.
Those studies which have considered whole
vehicles used loads which were much less severe than HB.
It would be possible to virtually eliminate global transverse moments by
providing intermediate diaphragms or cross-frames.
However, for reasons
discussed in 2.2.1, this solution is unlikely to be economic in concrete

bridges, except in the rare cases when beam and slab bridges are built
entirely in-situ on falsework.
It is more practical in bridges with steel
girders where cross frames are, in any case, often required to provide
restraint to the compressive flange in construction.
Both KirkpatrickC13)
and the Ontario code<11) require such frames to be provided between steel
girders although they say that this is primarily to provide restraint.
Figure 3.11:
HB Whul
load
Compressive membrane action to resist global moments (J3)
Although Kirkpatrick, like all the other researchers, failed to model full
global effects in his tests, his background in British practice meant that
he was more aware of the problem.
He suggested that compressive membrane
action improves the ability of slabs to resist global, as well as local,

transverse moments as shown in Figure 3.11.
This may be true but there
are no tests to prove it and there are several reasons for believing that
the effect is less pronounced.
One of these is clear from Figure 3.11;
resisting global moments requires the slab to effectively span at least

twice as far as resisting local effects.
This doubles the span to depth
ratio which reduces the effectiveness of compressive membrane action and,

as was shown in 3.2.1c, makes it more sensitive to restraint flexibility.
Another reason is that, unlike local moments, global moments act over a
- 52 -
substantial portion of the span length.

restraining
structure
to
structure
This means that the ratio of
requiring
restraint
is
far
less
favourable.
There is also another effect which is likely to reduce the contribution of
compressive membrane action to resisting global transverse moments.
It
was noted in 2.4.3 that the connection between the top flanges of adjacent
beams tends to even out the compressive stresses.
most heavily stressed beam
<the one which most
This means that the

requires support
from
global transverse moments) effectively has a top flange which is wider

than the beam spacing.
moment
equilibrium
required
as
transverse
In order to keep a wide compression flange in
about
the
vertical
shown
in
Figure
3 . 12.
tension
at
mid-span
and
axis,
These
transverse
put
compression
the
at
stresses
are
flange
into
support:
the
top
the
opposite of what is required to develop compressive membrane action.
Figure 3.12:
Although
action
Transverse stresses in a wide compression flange
it seems likely that the contribution of compressive membrane
to resisting global transverse moments
necessarily mean that

unserviceable.
is small,
the slabs themselves will become unsafe or even
Once a slab cracks, its stiffness reduces and the global
transverse moment starts to redistribute away.

deterioration
this does not
in
However, this leads to a
the distribution properties of
increase in the moment in the critical beam.
the deck and hence an
This could be a
problem as
both Kirkpatrick and the Ontario Code use analysis based on

section
properties
Paradoxically,
the
to
obtain
Ontario
Code
the
design
introduced
- 53 -
moments
this
in
analysis
uncracked
the
<which
beams.
is
departure from conventional North American practice> at the same time as

introducing the empirical slab design method.
Theoretically,
design
the problem of worst
value when
conventional
the slab cracks
methods.
However,
beam moment
increasing above the
also arises in decks designed
the
cracked
and
uncracked
by
elastic
stiffnesses differ by a factor of around three compared with around ten in.
very lightly reinforced slabs designed allowing for membrane action.
the effect is much smaller.
Thus
Also, conventional design methods provide a
safe solution according to plastic theory.
Thus, if a beam did start to
fail, redistribution would bring the transverse moments back into play.
There is no guarantee that slabs designed to the Ontario rules, or even
the Northern Irish rules, will be able to act in this way.
3.2.8 Empirical Design Rules
Both the Ontario and the Northern Irish study noted that the available
"theory" for restrained slabs predicted only their strength, which is not a
critical design criterion.
design method.
Thus there was no theoretical basis for a
However, they considered that there was no need for one
either: the observed load-carrying capacity of deck slabs was so high that
simple,
and
probably
very
conservative,
empirical
design
rules
would
suffice.
Batchelor et al<78) noted that tests suggested that unreinforced slabs
would have adequate strength so they initially recommended 0.21 isotropic
reinforcement
AASHT0<45>.
in each
face;
the
minimum reinforcement
recommended
by
This was later amended to 0.3S for reasons which are unclear.
Curiously, the percentage is based on the effective depth: there is no

logical reason why less steel should be required if it is further from the
face.
However, this is a fault which is shared with the minimum steel
rules in many other codes, including BS 5400 and CP 110 but not BS 8110.
The
Ontario
Code
requires
extra
steel
to
be
provided
in
some
circumstances.
The reinforcement is doubled in the end regions of highly
skewed decks.
Also, but only in decks with box girders, reinforcement
designed by normal means to resist global transverse moments is added to

the nominal steel.
This rule is rather odd since these moments can be
just as great, and just as important, in other types of deck.
- 5.4 -
Also, for
reasons noted in 3.2. 7, it still does not prove that behaviour will be
satisfactory under combined effects.
Kirkpatrick gave specific recommendations for only one slab thickness;
160mm.
This was to provide T12s at 150mm which is approximately 0.6%.
The reason for specifying more steel than the Ontario Code was that
Kirkpatrick realised that the reinforcement required to resist calculated
global transverse moments alone could exceed
covered by his rules.
0.5~
in some slabs which were
It is not clear why he specified the same steel in
the longitudinal direction and his rules appear to be unduly conservative

in this respect: the author has designed a deck slab to conventional rules
which had less longitudinal steel.
Both sets of rules require reinforcement for any deck cantilevers to be
designed by conventional methods, which means they are likely to require
substantially more steel than the rest of the deck.
In his own design
Kirkpatrick<13,49> avoided the resulting awkward detailing by not having
any cantilevers at all.
This was an economic solution for his particular
case because the cantilever formwork and reinforcement would have been
expensive compared with the cost of an extra beam;
This would not apply
to longer span bridges and the need to provide extra reinforcement for the
cantilevers is a significant limitation on the advantage of using the
rules.
The major disadvantage of empirical design rules is that there must be
restrictions on their range of applicability.
These will now be considered.
a. Span and Depth

Both Kirkpatrick and the Ontario Code specify a limiting span to depth
ratio of 15 for the use of their empirical rules.
evidence
that
the
theories
on
which
they
Although there is some

are
based
<particularly
Kirkpatrick's) become unsafe by this span to depth ratio, the observed and
predicted strengths of slabs are so high that the limit is conservative.
However,
it
seems
to have been considered that
this was unimportant
because the limit covered normal practice, at least for beam and slab
decks.
This is not entirely logical; the reason shallower slabs are not
used is that
they are uneconomical, or even impossible, to design to
conventional rules because the reinforcement required increases rapidly

with span to depth ratio.
This does not apply in slabs designed to the

- 55 -
empirical rules,
thickness.
indeed
Thus,
the
required
reinforcement
reduces
with
slab
if one was designing a bridge to these rules from
scratch <that is, without the restrictions imposed by using an existing

range of standard beams>,
it appears that
the optimum solution would
always have wide beam spacings and the maximum allowable slab span to
depth ratio .
.The Ontario Code specifies a minimum slab thickness of 225mm but the
Commentary makes it clear that this is not for structural reasons but
because shallower slabs are not advised for durability reasons; in Ontario,
as in many states in the USA, bare concrete decks are the norm. The
restriction on minimum depth, which is not applied in the assessment of
existing decks, has the unintended advantage of limiting the problem of
global transverse moments since the author's analysis shows that theseare
most significant in shallow slabs on close-spaced beams.
The Ontario Code also specifies a
maximum slab span of 3.7m.
This
requires a slab depth of only 247mm, compared with the absolute minimum
of 225mm,
so the range of slab depths which are likely to be designed to
the rules is very narrow.
There is no advantage in using more than the
minimum slab thickness.

A restriction on span is probably justified because longer spans introduce
effects
which
have
not
yet
been
researched;
significant
deadweight
stresses and a much greater interaction between the effect of several

wheels.
However, even with the Ontario Code's allowance for haunches, 3.7m
is a modest slab span by the standards of modern long-span concrete box

girder bridges.
the
Thus the Ontario rules will not be used for these, indeed
limiting span
to depth
ratio
makes designing
them
to the rules
uneconomic anyway as the extra weight would more than cancel out the
saving in reinforcement.
There is scope for economy in the design of this
type of deck from using membrane action, particularly if this could justify
even longer slab spans or shallower slabs than at
present, but
this
requires further research.

b. Restraint
The two sets of rules are very similar in their requirements to ensure
adequate restraint.
are used.
Both require intermediate cross-frames if steel beams
Both require diaphragms at the supports if concrete beams are

- 56 -
used.
Kirkpatrick also suggests the use of concrete support diaphragms
even with steel beams.
Both require parapet upstands or edge cantilevers
to ensure adequate lateral restraint.

The test results suggest that these requirements are sufficient but give
little indication as to whether they are necessary.
Seal's
deck,
which
did
not
comply
with
the
The outer bays of
requirements
stiffening, did show lesser (but still adequate> strength.
for
edge
All the other
decks tested complied with the requirements, as do most of the decks

currently designed.
Some concrete decks have, however, been built without
diaphragms and this has been advocated by Grans ton <85) because of the
costs of forming diaphragms.
3.3 PRESTRESSED SLABS
One
of
the
earliest,
comprehensive
and
and
in
some
influential,
respects
studies
of
still
the
one
effect
of
of
the
most
compressive
membrane action on the behaviour of bridge deck type structures was

conducted by Guyon <10).
His study is worth reviewing even though he
considered only prestressed slabs whilst, for reasons given in 2.2.2, the
remainder
of
this
thesis
assumes
that
bridge
deck
slabs
will
be
Guyon's slab was cast integral with longitudinal and transverse beams.
It
constructed of ordinary reinforced concrete.
was stressed transversely by concentric wires giving a stress of 1.5N/mm"',

whilst
tendons
2.~N/mm 2
located
in
the
beams
gave
longitudinal
stress
of
These stresses are very low, much lower than the longitudinal
stress applied by global effects to many slabs which are not normally
considered as being "prestressed".
A jack was used to apply a single. central concentrated load to each bay in
turn.
It reacted, via steel girders, against the beams adjacent to the
loaded bay of the slab.

Several
conventional
Pucher's<~O>,
Thus only local moments were applied.

elastic
methods,
including
were used to analyse the slab.
Westergaard's <39 > and
The results were reasonably
consistent both with each other and with the initial behaviour of the slab.
The strain gauge readings started to show some
approximately
the . load
for
which
~he
signs-.~f
calculated stress equalled
-measured flexural tensile strength of the concrete.

- 57 -
non-linearity at
the
However, despite the
use of "a powerful microscope", no cracks were visible until the load was
increased by a further 30 to 40%.
Once formed, the cracks extended very slowly in both width and length;
much more slowly than conventional elastic flexural theory would suggest.
Guyon attributed this to a combination of moment re-distribution away from
the cracked region and redistribution of the prestress force towards the
cracked strips, that is compressive membrane action.
confirmed this explanation.
Strain gauge readings
Initially only the central part of the slab
was subjected to a compressive force and tension in the remainder helped

to restrain it.
As the load increased, the area in compression extended
until the whole of the loaded bay was in compression.

Guyon considered that the behaviour was acceptable from a serviceability
viewpoint up to a load of over 211! times that at which the calculated
stress equalled the measured tensile stress, or 10 times the load given by
Freyssinet's no-tension rule.
Removal of the load at this stage caused the
cracks to close up, but this is the one aspect of the behaviour of such a
lightly stressed slab which could be significantly different from that of a
reinforced slab.
With further increases in load the existing cracks grew wider and new
radial cracks developed.
The load was then carried by "a system of
concrete struts", that is pure compressive membrane action.
A brittle
punching failure occurred at a load of some 25 times the 'no tension" load
or twice the load given by Johansen's yield-line theory.
In add it ion to this qualitative description, Guyon developed some simple
analyses.
He
acknowledged
that
these
were
based
on
"debatable
assumptions" and in many respects they have been superseded by more

rigorous analyses such as those given in 3.2 and Chapter 5.
are still useful as
descriptions of
behaviour.
His
However, they
analysis
of
the
behaviour of strips of slab at relatively low loads is largely confirmed by

the form of analysis considered in Chapter 7.
Although it is difficult to
use in any quantitative way, it is significant because none of the more

recent theoretical studies explore the behaviour at low loads.
Guyon extended Johansen's theory to allow for compressive membrane force.
Instead of calculating the membrane force required to maintain lateral
displacement compatibility, like most of the analyses considered in 3.2.1,
- 58 -
he estimated the maximum available restraint force.
His analysis over-
estimated the failure load but he attributed this to the fact that he
ignored the effect of the vertical displacements on the lever arm at which
the restraint force acts.
Back-calculation confirms this explanation.
He
acknowledged that his analysis would not be valid for a slab with a very
large area of surrounding restraining concrete and he attempted, largely
unsuccessfully, to analyse such a case.
Guyon also gave an axi-symmetrical analysis of the punching failure based
on the assumption of rigid lateral restraint.
This assumed that the radial
struts were elastic and uncracked, except at the outer edge and at the
edge of the loaded area, which is analogous to the elastic-plastic analyses
considered in 3.2.1 c.
It also assumed that the force in the struts was
constant over their length.
This implies that there are no circumferential
forces but this was neither mentioned nor justified.
Guyon assumed that,
at failure,
the whole depth of the slab adjacent to the load was in
compression.
This seems unlikely as there is no mention of cracks closing
up in the description of behaviour.
Another fault in the analysis is that
the calculated concrete stress on the critical section at failure is some

130N/mm2 and a very large portion of the slab is stressed up to more than
the elastic limit.
This shows that there were circumferential forces and
this axi-symmetrical analysis appears to be the least satisfactory aspect

of the study.
As a result of the study Guyon developed a simple design method which he
acknowledged to be ''much too conservative".
This was to analyse the slab
using elastic theory but taking Poisson's ratio as zero and taking the slab
to be simply supported.
The resultant mid-span moment is then shared
between the support and mid-span sections and resisted by bending of the
prestressed sections.
These are analysed ignoring the tensile strength of
concrete and the lateral redistribution of prestress, but allowing cracking

to extend to the level of the centre-line of the cable.
If the cable is at
mid-depth of the slab, this gives twice the allowable moment given by the
no-tension rule.
In the case of a simply supported slab it also gives
twice
load.
the design
In fixed-ended slabs the difference is much
greater because Guyon's method allows designers to take as much moment as

they like at the support, whereas conventional elastic theory only allows a
reduction in the mid-span moment of some 15".
uniform
fixed-ended
slab,
Guyon's
method
- 59 -
The result is that, for a
requires
only
36"
of
the
prestress
that
the no-tension
rule requires.
In
haunched slabs
the
difference can be greater.

Although less radical than the Ontario approach, this design method is
more useful in longer span slabs as it requires no limit on span to depth
ratio.
The reduction in stressing force is so great that, despite the
reservations in 2.2.2, it has had a significant effect on the relative

economy of transverse stressing and ordinary reinforced concrete;
This
has meant that many deck slabs mainly <but not exclusively> in France have
been designed using Guyon's rules.
These now represent a very significant
number of bridge-years of satisfactory experience.

Slabs designed to Guyon's rules are very lightly stressed.
crack
long
Thus
their
before the concrete's

behaviour
is
not
They would
compressive stress became excessive.
fundamentally
different
from
that
of
reinforced slabs so the experience of their satisfactory behaviour is

significant to this study.
However, there is one respect in which stressed
and reinforced slabs differ in their behaviour.
Once cracked, a rein'forced
slab's stiffness is greatly reduced for all subsequent applications of

tensile stress, however small, but a
significantly affected when
prestressed slab's stiffness is only
the applied
tension exceeds the prestress.
Thus slabs designed to Guyon's rules may have better restraint than those
designed to, for example, the Ontario rules.
3.4 CONCLUSIONS
Previous research shows that bridge deck slabs are far stronger than
conventional design methods imply.
Slabs designed to the Ontario rules,
for example, have very much less steel yet they have behaved well both in
service and in load tests.
Two "theories" have been proposed which claim to "predict" the ultimate
punching shear strength of bridge deck slabs subjected to wheel loads.
These
theories
are
essentially
sometimes significantly in error.
empirical
and
their
predictions
are
There is also some indication that the
assumption that the observed failures were "shear" rather than "flexunil"
failures could be incorrect.
However, the observed strengths of bridge
deck slabs are so high that these faults have no practical significimce;
typically it is a question of whether the factor of safety is 5 or 7.
In
practical terms, the only questions over the strength of slabs which are
- 60 -
restrained and which are subjected to single wheel loads relate to span to
depth ratios above those for which these theories clatm to be valid.
Although it
is clear that
the restraint available in bridge decks is
adequate to develop compressive membrane action, there is no quantitative

explanation for this.
Similarly, there is no quantitative theory to explain
the observed satisfactory service load behaviour of decks designed to the

empirical rules discussed.
Since service load behaviour is critical in
design, this means that there is no theoretical basis for a design method.
Another
aspect
reinforced
of
slabs
the
which
behaviour
of
has
been
not
bridge
proven
decks
with
very
theoretically
is
lightly
their
performance under combined global and local effects.
This appears to be
been
far
more
serious
omission
since
experimentally either.
- 61 -
it
has
not
investigated
CHAPTER
ELASTIC
ANALYSIS
4.1 INTRODOCTION
Chapters
and
showed
that
serviceability
criteria,
strength, are critical in the design of bridge deck slabs.
not
ultimate
Elastic theory
is more appropriate to the analysis of serviceability than plastic theory

but the elastic theory of restrained slabs has not been developed.
The complexity of the behaviour of realistic slabs, particularly under
concentrated loads, is such that it is not practical to obtain rigorous
analytical solutions, either elastic or elastic-plastic.
considered
in 3.2.1
empirical factors.
Thus the solutions
all contained gross approximations,
assumptions or
It is, however, possible to determine elastic solutions
for simple cases by making reasonable assumptions.
These cases are not
realistic but they do indicate the sensitivity of the behaviour to the

relevant
variables.
Also, by comparison with conventional analyses of
similar cases, they give some indication of the significance of membrane

action in practical cases.
In addition they can be used for checking
computer programs which can then analyse more realistic cases.

Since these simple analyses cannot be used directly in design, there is
little point in considering a wide range of cases.
Thus, in this chapter,
only one simple case will be considered; the unreinforced symmetrical slab
strip which was considered in 3.2.1, subjected to a single central point
load.
4.2 ASSUJIPTIONS
The analysis is based on conventional elastic engineer's beam theory.

Plane sections are assumed to remain plane and compressive stress is taken
to be proportional to strain whilst concrete is taken to have no tensile
strength.
Unlike the analyses considered in 3.2.1, the deflection is taken
to be small relative to the slab thickness but
the validity of this
assumption will be checked.

Although these assumptions are just as arbitrary as those used in the
analyses considered in 3.2.1, this analysis is more rigorous in the sense
that the assumed moment, stress and strain fields are made consistent
- 62 -
throughout the structure.
The analyses considered in 3.2 . 1 either used
different material properties at the critical sections from elsewhere, as

in Park's approach, or, like Me Dowell, only checked that the assumed strain
field
and material properties
were consistent with the forces at
the
critical sections.
4.3 STRESS
Since the assumed slab system has no tensile strength, 1t can only resist
vertical forces by virtue of the vertical component of the restraint force.
It is thus convenient to consider the system in terms of the line of
thrust
of the
restraint
force.
This must
be straight except at
the
supports and the point of application of the vertical load.

Because of the assumptions, the slab cracks if the line of thrust goes
outside the middle third of the section.
Where the slab is cracked, the
line of thrust must act at the edge of the middle third of the effective,
uncracked, sect ion.
This leads to the geometry shown in Figure 4.1.
P/2
Cracked concrete
Figure 4.1:
P/2
Restrained slab strip
<elastic theory>
In Appendix Al it is shown, by consideration of displacement compatibility
assuming rigid restraint, that the depth of concrete in compression at the
supports and at mid-span is 0.222h and the maximum stress in the concrete
is 2.64-P1/h2
Thus, with a concrete stress fc:, the load P
4.4 COMPARISON Wim OTIR AHALYSES
The rigid- plastic analysis considered in 3.2.1a gave the strength of the
equivalent strip as
f c 1 h2 / 1
- 63 -
Using the BS 5400
des~n
rectangular stress block this gives;
Using the.BS 5400 elastic stress limit (0.5fcu> the elastic solution gives;
p
at the serviceability limit state.
The ratio of design ultimate to
des~n
service load is a function of the load factors and in BS 5400(23> it is
Y3 x y,L <ultimate limit state>

Y<3 x Y<L <serviceability limit state>
Considering the case of HB load and load combination 1 <which is usually

critical in deck slabs) this is;
1.1
1.3
1.0
1.1
This means that a section on the limit of the allowable elastic service
stress would have a design ultimate load
which is only 62% of its strength, confirming that the serviceability check
is critical even without allowing for redistribution.
Using the simple BS 5400
des~n
method, the reinforcement required in each
face to resist this load would be approximately 0.6%.

s~nificant
This is a very
amount of reinforcement, confirming that membrane action is
worth considering.
4.5 CRACK WID11IS
Unlike most other crack width prediction formulae, the BS 5400 formula can
be applied to unreinforced concrete.
With the maximum allowable service
load derived in 4.4 the calculated crack width for our case
"'
0.00027h
For a 160mm deep slab this is 0.43mm.
If it is assumed that the maximum
allowable crack width had to be complied with on the surface <which is not
strictly required as there is no reinforcement> the limiting value would be
0.25mm.
However, this does not have to be complied with under the full HB
load; only under 25 units of HB.
The result is that crack widths would

- 64 -
not be a limitation in a deck designed for '5 units of HB but they would
be in a deck
des~ed
for a lower load.
,,6 DEFLECTION
In Appendix A2 it is shown that the analysis given in ,,3 and Appendix A1

leads to a central deflection
0.173 PP/Eh3
for the load corresponding to a stress of 0.5feu this is

"'
4.5 X 10- 5 P/h
Now the membrane force acts at a total lever arm

=
(1 -
"'
0.852h
0.222
2/3)h
For small deflection theory to be valid the displacement has to be small

compared with this.
101. with ,3,6;
The error is 1" with an 1/h of 13.8, 5" with 30.9 and
In practical terms, this means that small displacement
theory is valid for the serviceability analysis of local effects.
However,
if membrane action were used for resisting global transverse moments the
effect of displacements could be significant.

4. 7 I014X:I' OF RESTRADI1' FLEXIBD...ITY
In Appendix A3 the effect of in-plane restraint flexibility is added into
the analysis.
The result is shown in Figure ,,2 by plotting the load for a
stress of 20N/mm2 against the restraint stiffness expressed as a multiple

of the axial stiffness of the uncracked slab.
The elastic analysis is very
much more sensitive to restraint stiffness than the analysis considered in

3.2.1.
Thus restraint is an even more important factor than Chapter 3
suggested.
- 65 -
Load
120
CkN / m)
100
80
60
40
20
0 --~----~----r---~----,-----r----,
0. 125
0. 25
Figure 4.2 :
Cl
0. 5
Effect of restraint flexibility
= 1.5m,
= 160mm,
= lm>
4..8 CONCLUSIONS
The simple analysis considered in this chapter shows;

1. Compressive membrane action is significant in the elastic range.
2. It is less significant than in plastic analysis, hence serviceability
criteria
are
likely
to
be
more
critical
in
design
allowing
for
compressive membrane action than in conventional design.

3. Small displacement theory is valid for considering local behaviour at
the serviceability limit state.
4. The behaviour under service loads is more sensitive than ultimate
strength to restraint flexibility.
Despite the extreme simplicity of the case considered, the various test
results considered in Chapter 3 suggest that all these conclusions are
likely to remain valid for more realistic cases.
- 66 -
CHAPTER
NON-LINEAR
FINITE
ELEMENT
ANALYSIS
5.1 INTRODUCTION
The major difficulty with analysis allowing for membrane action, at least
as far as flexure is concerned, is not conceptual; it is the complexity of
the mathematics.
This fact,
which is clear from 3.2.1
and Chapter '
suggests that the subject should be amenable to solution by numerical

analysis and the finite element method is the most convenient way of doing
this.
The analysis of membrane action has to be non-linear; even the simple
analysis considered in Chapter 4 is only linear under proportional loading.
Non-linear finite element analysis, NLFEA, is only practical with powerful
computers so it is a comparatively recent method which was not applied to
concrete until the 1960's <86>.
Despite this the literature is extensive
and, although only a tiny fraction is aimed at the analysis of membrane

action, much of it is relevant.
work in detail.
It is thus not possible to review all the
This chapter aims only to introduce the principles and
problems of the method.
A particular, relatively simple, form will be
considered in more detail in Chapter 7 whilst readers requiring a more

comprehensive coverage of the state of the art
should consult recent
specialist works such as reference 87.

5.2 GENERAL APPROACH
The analytical method adopted should be capable of resolving all
problems identified in Chapter 3.
the
Two of the most important of these, the
prediction of restraint and the analysis of global effects, require the

analysis to consider the whole bridge.
Even with very powerful computers,
this puts a severe restriction on the form of analysis which can be used.
In
this
study,
therefore,
only
the
"smeared
layered approach" will be considered.

modelled;
the
cracks
infinitesimal cracks.
are
smeared
crack,
distributed
steel,
In this, individual cracks are not
out
into
an
infinite
number
of
Similarly, individual bars are not represented; the
steel is distributed evenly across the element width.
The significance of
layering is that it enables the material state to be varied over the

element depth whilst still using a two-dimensional element.
The stresses
are calculated independently for each layer as a function of the strains

- 67 -
which are calculated from the displacements at the "reference plane"; the
level at which the elements are tmplicitly located.
The element forces are
then calculated by integrating the stresses over the element volume.
Thus
the forces are calculated directly from the displacements but the correct
displacements can only be obtained from the loads by an iterative solution
scheme.
Although linear-elastic analyses of slabs, particularly for bridge design,
are often performed using alternative structural idealisations, such as
grillage analogy, non-linear analysts have assumed it necessary to use
plate finite elements.
This is because of their desire to produce rigorous
and accurate analyses.

5.3 ELEMENT TYPE
5.3.1 Slabs
Early finite element analyses of slab systems used classical thin plate
theory which assumes that
normal.
This
appr:oach
lines normal to the reference plane remain

is
being
"gradually
supersededn<87)
by
the
Mindlin <88) form which assumes that lines normal to the reference plane
remain
straight
but not
necessarily
normal.
This
enables
shear
deformations to be included in the analysis so the theory is sometimes

described as "thick plate theory".
in reinforced concrete and
straight
prevents
the
However, shear causes diagonal cracks
the assumption
realistic
that
vertical
modelling of shear
lines
failures.
remain
Indeed,
according to Chana<75), shear failures are sensitive to dowel behaviour at

the crack which tmplies that they cannot be realistically modelled by any
form of smeared crack, distributed steel analysis.
Despite this, Mindlin's
theory does give more realistic predictions than classical theory for the
shear forces at a free edge, as has been illustrated by Cope <89).
1t
appears
that
the main reason
for
adopting
it
However,
is one of analytic
convenience; it requires a lower order of displacement continuity across

the element boundaries <87 ).
The
nodal
forces
in
the
elements,
due
calculated using the virtual work approach.
to
nodal
displacements,
are
To do this it is necessary to.
assume a displacement field for the whole element from the known nodal
displacements.
A wide variety of elements can be developed, according to
the number of nodes, the displacement field assumed and the method of
- 68 -
integrating
the
stresses
and
strains
over
the
element
volume.
discussion of their relative merits is outside the scope of this study.

5.3.2 Beams
The beams in beam and slab decks can be modelled using either simple beam
elements or an assemblage of plate elements.
The latter is far more
expensive but 1t enables inclined web cracking and the transverse bending
stiffness
of
the
beam
to
be
modelled.
Edwards(36>
found
the
two
approaches gave very similar results in a bridge with rectangular beams.

However, the transverse bending stiffness of the flange of, for example, an
M beam is much greater so there may be more advantage in using plate
elements for these.
Buckle and Jackson<90> have developed a form of beam
element which can model transverse bending.
However, because 1t assumes
that plane sections remain plane, 1t cannot model the warping stresses
which contribute to the resistance to torsion.
The beam elements are rigidly attached to the plate elements at the nodes.
Since the mesh size is decided by the requirement to model the local slab
behaviour, 1t is smaller than is required to model the beam behaviour.
Thus the analysis is not sensitive to the type of beam element used.
Buckle and Jackson<90) used a displacement function which will be shown in
Chapter 7 to have serious faults, whilst Edwards <36> used a displacement
function
which
was
not
consistent
with
that
used
for
the
slab.
Calculations suggest that neither of these faults had a significant effect

on the results.
5.4 MATERIAL PROPER'IIES
5.4.1 Steel
The reinforcing and prestressing steel is assumed to be fully bonded to

the concrete and to exhibit uniaxial behaviour; that is, it is stressed only
by strain in the direction of the bars.
Any stress-strain relationship can be defined numerically and incorporated
into a program but it is more usual to use elastic-plastic properties,
sometimes with linear strain hardening.
prestressing,
departs significantly
from
Modern reinforcement, and all

this
advantages in using more realistic properties.
- 69 -
assumption so there are
When unloading is considered, it is usual to use a straight line parallel

to the inif:ial portion of the stress-strain curve which gives a permanent
set equal to the previous departure from linearity.
5.4.2 Concrete
a. Uncracked
The
enhancement
compression,
and
of
concrete's
compressive
strength
due
to
the reduction due to orthogonal tension,
biaxial
is normally
considered but the effect of vertical stress cannot be modelled in the

form of analysis considered here.
Despite this, the enhancement can be
significant; up to approximately 20%.

A variety of stress-strain relationships have been used.
Abdel Rahmen<87>
used a simple elastic-plastic relationship with a straight cut-off at a

limiting
strain whilst
Edwards<36>
used
Popovics'
formula(91)
for
the
uniaxial case in beams and Nilson's<92> approach for the biaxial case.
As with steel, unloading is usually modelled with a straight line.
unloading part of the properties are sometimes
under
monotonically
increasing
loads
The
specified even in analyses
in order
to avoid
deformation theory which was mentioned in 3.2.1 b.
the
fault
of
this is done, the
maximum strains have to be stored for all the sampling stations.

The variability of
analysis.
concrete
is a
major difficulty
in a
deterministic
This variability is particularly significant to failures, such as
punching failures, which are affected by local rather than average concrete
strength.
The effect
stress-strain
curves
is
large compared with
and
this,
relationships are used many
combined
the difference between
with
thousands of times
the
fact
that the
in the course of an
analysis, encourages the use of simple relationships.
It also means that
the predictions are unlikely to be precise.

b. Cracked
Although
smeared
crack
infinitesimal spacings,
centres.
analysis
implies
infinitesimal
cracks
real structures have discrete cracks at
at
finite
The concrete between the cracks is able to resist tension and
this stress contributes to the stiffness of the structure.
This effect,
which is known as "tension stiffening, is very significant, particularly in

lightly reinforced elements and at
low loads.
- 70 -
It
is modelled by an
empirical stress-strain curve which has a descending branch after the

concrete has cracked.
Cracks first form in the direction of the maximum principal tensile strain.
If this direction subsequently changes, a shear stress is developed across
the crack.
The shear stiffness is reduced by the crack and can be
modelled by another empirical factor called a "shear retention factor".

However, even with this reduced shear stiffness, the analysis can imply a
tensile stress
in other directions
which exceeds
the cracking stress.
Cope et al<93l used an alternative approach in which the "crack" direction

rotates to follow the principal strain direction.
This appears to give

It
may
formed
but
better results in cases where. the rotation is significant (94).

appear
illogical
that
cracks
can
rotate
after
they
have
presumably the explanation is that the crack direction in a smeared crack

analysis represents only the average or active crack direction so it can
rotate as new cracks form.
The few analysts who have considered unloading in cracked concrete have
used widely different assumptions (95,96,97> reflecting the lack of data in
this area.
Whatever tension stiffening function is used, the predicted behaviour is
very sensitive to the assumed cracking stress.

even
wider
random
variation
than
In addition to having an
compressive
according to strain rate, strain gradient, curing

number of load repetitions and many other factors.
strength,
this
varies
load duration,
r~gime,
This makes accurate
deterministic predictions of behaviour impossible.

Unlike
the
non-linearity
due
to
reinforcement
yielding
and
concrete
crushing, that due to concrete cracking is significant under service loads.

Thus it is the only non-linearity which is important to the design of
structures for which serviceability criteria are critical.
Also, even in
strength analysis, it is not realistic to consider only the cracking due to

a single monotonically increasing load case since cracking could have been
caused by many different service load cases.
In particular, cracking due
to wheel loads previously applied in other positions could reduce the

restraint
available
considered.
particularly
Thus
for
to develop
membrane
action
under
the
case
the lack of an agreed tension stiffening
unloading,
is a
serious obstacle to
being
function,
the use of the
analysis in design so the subject will be considered further in Chapter 6.

- 71 -
5.5 APPLICATION TO MEMBRAHE ACTION

Several analysts have applied NLFEA to slab systems.
Most have considered
only monotonically increasing loads, which restricts the application of

their analyses, but they do give a useful insight into behaviour.
Several
analysts have claimed good predictions for the behaviour of restrained

slabs but some of these might be considered slightly surprising.
For
example, Jackson <98> obtained good predictions for Roberts' tests <53) but
his
analysis
used
small
displacement
theory
and
simple
calculations
suggest that including the effect of the observed <and predicted> large
displacements would have reduced the
predicted strength by some
20~.
Despite these doubts, non-linear analysis has proved better able to predict
the behaviour of complicated slab systems than other methods.
At service
load levels, the predictions for complicated structures actually appear to

be better than those for simple "fully restrained" laboratory specimens.
The reason for this appears to be a fault in the tests rather than the
analysis;
it
is
difficult
to develop
full
restraint
and
service load
behaviour is very sensitive to restraint as was demonstrated in 4.7.

It is not practical or necessary to consider all these analyses in detail
but it is useful to consider a particularly relevant example; that of Cope

and Edwards<99).
They analysed several of the tests which were considered
in Chapter 3, including those of Kirkpatrick.
In view of the inherent
variability of results which are sensitive to local concrete behaviour,

they
considered
their
predictions
to
be
good.
However,
Kirkpatrick
produced enough results to enable the variability to be estimated and this

appears to be remarkably small and certainly smaller than the discrepancy
between the test results and the non-linear analysis.
Despite this, the
analysis is reasonably good with the worst error in the failure prediction
being some 30% with 15% being more typical .. This may not sound that good
compared with Hewitt's or Kirkpatrick's "analyses" but they are largely
empirical
whilst
the
non-linear
analysis
obtained
the
restraint
and
strength only from the geometry and material properties of the specimen
with no empirical correct ions.
The brittle nature of the "punching shear failures was ale~ correctly
predicted even though the analysis is incapable of modelling shear.
This
appears to confirm the suggestion in 3.2.4 that such failures are primarily __
brittle bending compression failures although the analysis did tend to
over-estimate strength slightly, implying that the high shear stress in the
- 72 -
critical region reduced its compressive strength.
In practical terms, the
good predictions for stresses and deflections at lower loads are more
significant as they suggest that the approach is valid for the critical
serviceability analysis.
Cope and
Edwards'
study suggests that
NLFEA is able
to successfully
predict restraint and analyse the local behaviour of bridge deck slabs
allowing for membrane action.
of
global
considered
and
local
this.
In theory it can also model the interaction
moments
He
but
analysed a
only
Edwards(36>
hypothetical
appears
to
have
bridge with rectangular
reinforced concrete beams and with deck slab reinforcement designed to the
empirical rules considered in 3.2.8.
reinforcement
would
His analysis suggested that this
be over-stressed under combined global and
moments, confirming the doubts expressed in 3.2.7.
local
However, because of the
lack of test data, there is no proof that the analysis was realistic in
this respect.
The form of deck he considered was also unrepresentative of
modern practice since ordinary reinforced beams are rarely used and the
moment redistribution behaviour would be very different with prestressed
beams.
5.6 USE IN DFSIGN
Although NLFE has proved capable of predicting the behaviour of reinforced
concrete slab structures,
it has rarely
(if ever> been used in their
design; either directly or for validating simpler design methods.

and Kotsovos<lOO> have said
'~he
Bedard
main reason for this appears to be a lack
of agreement concerning the numerical description of material behaviour".

This reason is supported by 5.4.2b but, in the case of the slabs considered
here, it is not a sufficient reason; the analysis would still produce more
economical designs
than conventional methods if the most conservative
conceivable material properties were
used.
Thus there must
be more
fundamental reasons and these will now be considered.

a. Cost and Complexity
A non-linear analysis of a given structure with a given element mesh is at
least an order of magnitude more expensive in computer time than the
equivalent linear analysis.
Also, because the principle of SUPerimposition
does not apply, every load combination has to be analysed separately.

Similarly, global and local effects cannot be superimposed so the whole
structure has to be analysed with a fine enough mesh, at least in the
- 73 -
critical areas, to model local behaviour.

computer
time
is
several
orders
The result is that the cost in
of
magnitude
analytical methods considered in Chapter 2.
higher
than
for
the
More seriously, the analysis
is also much more expensive in engineer's time.
This discourages its use,
particularly under design fee competition.

A related disadvantage, which is perhaps more serious, is the conceptual
difficulty; NLFEA is difficult for the ordinary designer to fully understand
or control.
This
makes
it
potentially dangerous
as
(at
least at
the
present state of the art> NLFEA is neither fully automatic nor foolproof.
b. Load History Dependence
For
reasons
which
were
discussed
in
2.3.4
and
5.4,
the
behaviour
of
concrete structures, and hence the realistic analysis of such structures,

is
load
history
dependent.
Since
it
is
impossible
to
predict
and
impractical to analyse the load history of a bridge over its entire design
life, this could be a serious problem.
c. Incompatibility with Codes
Existing
codes
methods
in
of
practice
mind.
If
were
the
written
critical
with
design
conventional
criteria
were
analytical
clear-cut
fundamental requirements, such as ultimate strength, this would not be a

major problem.
However, Chapter 2 showed that the fundamental critical
design criterion for bridges is the very ill-defined one that they should
remain "serviceable" for their design life.
It is very unclear what this
means in non-linear analysis terms, except that it appears to confirm that

a whole life analysis is required.
5.7 CONCLUSIONS
Non-linear
finite
element
analysis
is a
powerful analytical
tool which
sheds some light on the fundamental behaviour of slab systems and which
can give reasonably good predictions for their behaviour.
The reported
analyses support the suggestion in 3.2 .4 that "punching shear" failures may
be primarily
flexural.
One also appears
global behaviour expressed in 3.2.7.
to confirm the doubts about
There are, however, major difficulties
in using the analysis in design.
- 74 -
CHAPTER
TENSION
STIFFENING
6.1 INTRODUCTION
In Chapter 5 it was noted that
the lack of an agreed expression for
tension stiffening is a serious obstacle to the use of NLFEA.
Tension
stiffening is particularly significant in lightly reinforced elements and at

low loads; that is, at service load, rather than at failure.
Since bridge
deck slabs designed using membrane action are very lightly reinforced, and
since
serviceability
criteria
are
critical
in
stiffening is particularly important to these.

structures,
tension stiffening
may still be
their
design,
tension
Also, unlike in most other

important at
higher
loads
because of its contribution to the restraint.

Cope et al <37)
Stress
1.0
Cr ac ki ng Stress
Gilbert and Warner
( 101 )
0. 5
15
10
Stra in/Crac king St rai n

Figure 6.1 :
Tension stiffening functions
The tension stiffening functions used in non-linear analyses are purely

empirical.
Many such relationships have been used,
illustrated
in
fundamental
Figure 6.1.
analytical
method,
stiffening
function
appears
analytical
methods
depend
properties.
However
The dependence of
to
NLFEA,
be a
ultimately
major
on
apparently rigorous
totally
in not
empirical
weakness.
empirically
tension stiffening differs from,
tensile strength of reinforcement

property.
on
an
two of which are
being a
tension
Admittedly,
derived
material
for example,
the
fundamental material
It is a property of the composite material, reinforced concrete,
or even of the structure, not of the concrete or reinforcement.

therefore
all
decided
to
investigate
this
subject
at
slightly
It was
more
fundamental level than is justified by the relatively simple analytical
- 75 -
method subsequently adopted in Chapter 7.
The tests reported in this
Chapter were also used to calibrate the analysis used in Chapters 7 and 9
as well as to investigate the effect of scale in the half scale models
considered in Chapter 8.
6.2 TIIEORY
6.2.1 Mechanisms
The tension stiffening functions used in non-linear analysis represent
stress which is transmitted to the concrete between cracks by two, or
perhaps
three,
mechanisms.
The
reinforcement and concrete.
first
of
these
is
the
bond between
This enables some of the force, which is
carried across the cracks by the reinforcement, to transfer to the concrete

between the cracks.
The second mechanism, which only applies to sections
in flexure, is the shear connection between the compression zone and the
teeth of concrete between the cracks.
The third mechanism which affects

Mart he <102 >
tension stiffening is the ductility of concrete in tension.
has shown that even when the strain exceeds that at which- the peak stress
is developed and
cracks
significant tension.
have
started
to
form,
concrete
can transmit
However the effect is often ignored, which may be
justified as the stress is only significant over a narrow range of smeared

strains.
Consideration
of
these
mechanisms
might
suggest
that
particular
empirical expression for tension stiffening stress would only be valid in a

narrow range of circumstances.
The bond contribution, for example, might
be expected to be sensitive to the bond characteristics, size, quantity and

orientation
<relative to the cracks) of the reinforcement.
In practice,
however, many non-linear analysts have obtained satisfactory results using

the same function in a wide range of circumstances.
this apparent
An explanation for
paradox can be obtained by considering the way cracks
develop in a region of constant moment or constant direct stress.

Prior to the formation of the first crack, the bulk of the load is taken by
the concrete <Figure 6.2a).
As the stress approaches the effective tensile
strength of the concrete, f.:t a crack forms at the weakest point.
Here
most of the stress is transferred to the steel but beyond a distance, S0

the stress is unaffected <Figure 6.2b).
cause another crack to form.
A further increase in load will
This cannot occur within S0
crack because the stress is too low.
of the first
Finally, when all the cracks have
- 76 -.
formed <Figure 6.2c>, no two adjacent cracks will be more than 250
apart
because otherwise there would be a section between them subjected to a

stress in excess of
spacing becomes 1.350
Beeby (34) has shown that the average crack

.
Stres~
Cracki ng
------------------------..-- Stres~
----------------------a ) Before First Crac k
n\V
So
----------
b> After First Crack

--cracking Stress
c> After Last Crack
Flgure 6.2:
This description
Stresses 1.n concrete as cracks develop
implies that
anything which improves the transfer of
stress to the concrete on either side of the cracks reduces the final
crack spacing.
It thus reduces the wavelength of the stress distribution
shown in Figure 6.2c, but has no effect on either the amplitude or the
average value, which is the stress used in smeared crack analysis.
The above description can be used to obtain an estimate for the tension
stiffening when all
necessary
cracks.
the cracks have first
to assume
Vetter <103),
shape
for
formed.
To do this
the stress distribution
in an analysis
it is
between
intended for a different
the
purpose,
assumed that the concrete stress increased linearly either side of the
crack.
He also assumed that the rate of increase of stress either side of
the crack was unaffected by the formation of further cracks.

deduced that the average stress was approximately 0.5fc t
From this he
However, this
was based on the incorrect assumption that the average crack spacing was
2.050
Using Beeby's crack spacing of 1.350
becomes 0.3350
the average concrete stress
It has often been assumed that a further increase in strain reduces the
tension stiffening stress but it is not clear why it should.

where
bond
inconsistent
is
the
with
dominant
the
usual
mechanism,
design
the
assumption
assumption
- 77 -
that
For example,
appears
to
be
bond strength
is
independent of strain up to at least the yield strain of the reinforcement.

It
may be argued,
therefore,
that
the smeared stress
should remain constant after cracking.

code
of
practice
formulae<104>
and
in the concrete
This assumption is used in some

is
supported
by
some
researchers
including Hartl <105>.

6.2.2 Steel Stress
An unfortunate consequence of smearing the concrete strain is that the
steel strain is also smeared.
occurs
at
the
cracks,
is
This means that the peak steel strain, which

not
modelled
reinforcement yields is over-estimated.
so
the
load
at
which
the
This has not previously been a
serious problem because the tension stiffening functions used have meant
that the effect became insignificant well before the reinforc'ement became
non-linear.
as
If, however, a constant tension stiffening stress were used,
suggested
in
6.2.1,
the
problem
would
become
more
serious.
Cervenka <106 > avoided this by calculating the steel strain independently,
ignoring tension stiffening.
This approach introduces the reverse error;
that is, it is assumed that all the steel is subjected to the strain which
only really occurs at the crack position and thus the non-linearity is
over-estimated.
It appears that it would be more correct to use some form
of averaging process between the strains <or stresses> calculated with, and
without, allowing for tension stiffening.
However, this would be even more
inconvenient than Cervenka's approach.

Because of these problems, an analysis using one of the tension stiffening
functions shown in Figure 6.1 could give better results than an analysis
using a constant tension stiffening stress, even if the constant stress is
more representative of the real behaviour of the concrete.
This, and the
tendency of researchers to concentrate on behaviour at high loads, could

explain the preference for the type of tension stiffening function shown
in Figure 6.1.
6.2.3 Mesh Dependence
The formation of a crack affects the stress over a distance which is
related to the final crack spacing, but in a finite element analysis it
affects the stress over a distance which is related to the element size.
Because of this it has been suggested that the tension stiffening function
should be varied with the element size so that the energy released by a
crack is independent of the mesh.
However, this was not done in the
- 78 -
analysis considered here and it was found that the results were entirely
independent
of mesh size.
This was because only regions of ,:onstant
moment or constant direct
tension were considered so all the elements
cracked simultaneously, unless a variable tensile strength was used.
Thus,
when a fine element mesh was used, the resulting under-estimate of the
energy released at the position of the real cracks was compensated for by
the over-estimate of the energy released elsewhere.
Consideration of this
behaviour shows that this would not occur in a region of varying moment.
Ideally, therefore,
the tension stiffening stress should be varied both
with element size and with the stress state in the adjacent elements.
This would be very difficult to do in a general solution procedure so it
is fortunate that experience shows that, unless the mesh size is small
compared with the crack spacing, a constant function can be used.
of
the
variability
additional
accuracy
significance.
of
tensile
obtained
strength
from
and
finer
tension
mesh
In view
stiffening,
would
have
no
the
real
The problem does, however, prevent the use of smeared crack
analysis in the study of behaviour which is very local compared with crack
spacing.
6.2.4 Cyclic Loading
The contribution of tension stiffening tends
loading.
zero.
to reduce under repeated
Indeed the crack width clauses in BS 54-00 assume it reduces to
Cope and Rao<107) modelled the reduction by reducing the length of
the tail of their tension stiffening function, leaving the value of the
tensile
strength
unchanged.
This
approach
cannot
be
constant tension stiffening stress suggested in 6.2.1.
used
It
with
the
implies that
cyclic loads reduce the tension stiffening stress but do not cause any new
cracks.
However,
it
is known that concrete is susceptible to fatigue
failures in tension, indeed the conventional design methods for concrete

pavement (108> are based on quite well established fatigue relationships.
Thus an alternative way of modelling the effect of cyclic loads on tension
stiffening would be to reduce the tensile strength used in the tension
stiffening expression but to leave the form of the expression unchanged.
6.2.5 Unloading
The
bulk
of
research
into
both
tension
stiffening
concentrated on monotonically increasing loading.
and
NLFEA
has
This means that
the
tension stiffening functions assume that the tensile strain currently being
- 79 -
In a
experienced is the greatest the concrete has ever experienced.
complex non-linear structure, this assumption may not be valid even when
the structure itself is experiencing a monotonically increasing load.
importantly,
for reasons discussed in 5.4.2b,
it
is not
More
reasonable to
consider only monotonically increasing loads in the type of structure

considered in this study.
Once concrete has cracked, it never re-acquires its tensile strength.
Thus
the tensile properties of cracked concrete are not reversible; a separate

unloading curve is needed.
It seems reasonable that once the crack has
fully closed the compressive stiffness of the concrete will be largely

unaffected by the crack.
It remains only to decide the stress required to
close a crack and the amount of strain, if any, which becomes permanent.
Although, in reality, the unloading curve may have a complex shape the
other errors in the analysis and variability in the behaviour mean that
the use of such a
function
relationship will be used.
cannot be justified.
A simple bi-linear
Unfortunately, at present the values to be used
in this relationship can only be obtained empirically.

A variety of expressions have been used.
Bazant <97), have used a straight
cracks
close
completely
at
Some researchers, such as
line to the origin implying that the
zero
stress.
At
the
other
extreme,
Crisfield<95) used a straight line parallel to the initial, linear, part of

the stress-strain curve, implying that the cracks do not close at all.
This
seems
Cope<96>
extremely
used
the
corresponding
permanent.
data.
to
unlikely,
more
that
particularly
reasonable
at
which
if
assumption
the
concrete
the
cracks
that
only
first
are
the
cracks
wide.
strain
becomes
The wide range of these expressions indicates the lack of
However many structures, because of relatively heavy reinforcement,
or only monotonic loading, are insensitive to the assumptions.

6.3 AHALYSIS OF PREVIOUS TESTS
6.3.1 Direct Tension Tests
Some analysts have derived their tension stiffening functions by obtaining

the
best
structures.
fit
to
This
the
load-displacement
approach
is
not
very
response
of
satisfactory
quite
complex
because
tension
stiffening is only one of many factors which affect the response.
Thus
tension stiffening functions are liable to become "fiddle factors" which .

compensate
for
a wide variety of errors in the analysis.

- 80 -
A better
approach,
which has
been
used
by Cope et
al <37 ),
is
to
test simple
statically determinate beams with long constant-moment regions.

these specimens, however,
displacement
it
relationships
Even with
is possible to obtain very similar load-
with
different
tension
stiffening
functions.
Only tests which subject the whole specimen to the same smeared strain,
that is direct
tension tests, give unambiguous results.
Unfortunately,
because there are theoretical reasons for believing that tension stiffening
could be different in flexure and direct tension, direct tension tests
cannot be used as the sole basis for deriving tension stiffening functions.
However they do give some useful information.
Williams <109) has tested a series of fifteen large slabs in direct tension.
The response was approximately linear until the first crack appeared.
This
occurred at a stress of 0.5 to 0. 7 times the tensile strength of the

concrete as measured by the split cylinder test.
This difference between
the effective tensile strength and the split cylinder strength is partly
due to the random variation of the tensile strength of concrete; cylinders
are constrained to fail on a pre-defined plane whilst a slab is free to
crack at its weakest section.
Statistical analysis suggested, however, that
this alone could not explain the difference.
The remainder was presumably
due to restrained differential strains which had a greater effect on the

slabs than on the cylinders.
A restrained strain equal to only- some 5% of
the total likely shrinkage is sufficient to explain the difference so it

could be due to differential shrinkage across the section.
After the first
crack appeared, the extension increased rapidly with a
relatively slow increase in load.
However, tension stiffening remained
significant even at a load such that the steel behaviour was non-linear.
All the tension stiffening functions previously used by non-linear analysts
under-estimate this effect; indeed most <including both of those shown in
Figure 6.1> have no effect at all on the results of an analysis performed
under load control, as can be seen from Figure 6.3.
The specimens were re-analysed using a constant tension stiffening stress
as suggested in 6.2.1.
This analysis under-estimated the load in the slabs
at extensions just above that required to cause the first crack.
This
could suggest that the stress between existing cracks was higher at low
strains but it seems more likely that it was because of the variation of
the tensile strength of the concrete; that is, because not all the cracks
- 81-
developed at the same load.

split
into
several
equal
To investigate this, the computer model was

elements
distribution of tensile strengths.
and
these
were
given
normal
The number of elements used was ten,
which was approximately the final number of major cracks.
This approach
implicitly assumes that the strengths of the ten elements are independent
variables, whi ch is not strictly correct, but it does give a good indication
of the effect of concrete variability.
Force
Only
----Analysis <displacement control)
---Analysis <load control)
Strain
Figure 6.3: Effect of tension stiffening on
analysis in direct tension
Force
400
Steel Area
Limit of Linearity
<Nimm2
-------
300
-~----
--------------)(
Test
Analysis
-Steel Only
0. 2
0.4
0.6
0.8
1.2
1.0
1.4
Strain x 108
Figure 6. 4:
Analysis of Williams' Specimen 1
<1% steel>
Analysis with a coefficient of variation equal to that obtained for the
split cylinder tests (10%) gave results such as those shown in Figure 6.4.
To
obtain
this
excellent
relationship,
however,
the
average
tensile
strength used in the analyses had to be adjusted for each specimen.
In
the more normal situation, where this cannot be done, the best estimate
for
the
effective
average
tensile strength of
approximately 0.8 times the split cylinder strength.

- 82 -
the
concrete would
be
The actual range used
was 0.7 to 0.9.

effective
The effect of small errors in the value used for the
tensile
strength
of
concrete
is
large
compared
with
the
difference between the analysis and test results shown in Figure 6.4 so
there
is
no
practical advantage
in
making
further
refinement
to
the
tension stiffening function.

10~
Although a coefficient of variation of
gave good predictions for the
load displacement response, a higher variation and a skewed distribution of

strengths were needed to make the analysis model the actual development
of the cracks.
tension
Analyses which did this exaggerated the rate of decay of
stiffening with
increasing strain.
In
the
tests,
developed with no discernible effect on tension stiffening.
new cracks
This seems to
suggest that the stress in the concrete between cracks, even cracks which
are within 250
of each other, increases with strain.
This gives further
confirmation of the suggestion in 6.2.1 that the tension stiffening effect

does not reduce with strain.
It is the development of new cracks which
causes the apparent decay.

Hartl<105) has also concluded that the tension stiffening stress in direct
tension remains constant once the concrete has cracked.
suggested a tension stiffening stress of 0.4 fct
He said his tests
However, because re did
not consider concrete variability in his analysis, this conclusion is closer

to the Author's than it may at first appear.
In effect, Hartl concluded
that the average tension stiffening stress is 40% of the initial cracking
stress.
The Author has concluded that
stress is approximately 30% of the
the average tension stiffening
average cracking stress.
For the
analysis considered here this comes to over 35% of the initial cracking
stress.
In view of the other variables in the analysis this is remarkably
close.
6.3.2 Flexural Tests
Clark and Spiers <11 0) have tested a series of beams and slabs with long
constant-moment
regions.
As
it
was
these tests which were used
to
develop Cope's tension stiffening function<37>, it is not surprising that

his function gives a good fit to the results.
r~analyse
However, it was decided to
some of the specimens using the constant tension stiffening
function suggested in .6.3.1.
The only concession made to the difference
between direct tension and bending was to increase the effective tensile
strength from 0.8 to 1.0 times the cylinder strength.
- 83 -
Predictions using
the two tension stiffening functions are compared with the results of one
of Clark's tests in Figure 6 .5 .
Cope's function appears to give a better fit to the results but it is not
possible to tell conclusively from Clerk's r esults which function is more
realistic.
0
Mome nt 40
(kNm)
30
20
<!
Test
---Cope' s Function
10
-
Constant Function
0
0
Curvature x 10 6
Figure 6.5:
<mm - )
Analysis of Clark's Beam 4-
6.4. TESTS
6.4.1 Design of Specimens
Because the tests would serve to calibrate the analysis and to investigate
the effect of scale for the model bridge tests which will be considered in
Chapter 8 , it was desirable to make
possible
to a
strip of
the test specimens as similar as
the proposed slab.
perform identical tests at full and half size.
It
was also desirable to
In order to facilitate the
cyclic loading of the specimens they were designed so that the full size
specimens would fit into a Mayes testing machine.
This left very little
choice in the design of the specimen which is illustrated in Figure 6.6.

The specimens,
like the deck slabs considered in this thesis, were very
lightly reinforced which made tension stiffening more significant to their

behaviour.
They were provided with 0.49% reinforcement.
This compares
with Clark's most lightly reinforced specimen which had 0.4-4-%.

is
conventional,
this
percentage
is
calculated
from
A. l bd.
However, as
A better
indication of the significance of tension stiffening is given by the ratio

of uncracked to cracked stiffness.
This is a function of bh 3 Ec /A.E.d2
According to this relationship, the specimens were some 40% more lightly
reinforced than any of Clark's.
Some slabs used in practice are, however,

- 84 -
more
lightly
reinforced still,
particularly in
the secondary direction.
Consideration was given to testing more lightly reinforced specimens.
It
appeared, however, that they would not crack until normal service loads
were exceeded so little use could be made of the results.
A single specimen was
tested
typical of the reinforcement
with
used
higher
reinforcement
in current
area, more
practice, to see if the
expression derived from the lightly reinforced specimen was applicable to

these.
800
Load
~c
"'
0
_.
TA
. I
_.._
"'
0
Support
35Cover
1500
f700
390
A-A
.I
-r_j
7T12-0'2-250EF
Figure 6.6:
2T12-0 1 -200EF
Detail of test specimens
<dimensions are for full size specimen)
6.4.2 Loading Rig
All the specimens were tested in the Mayes machine simply supported under
two point loading as illustrated in Figure
under test in Figure 6.7.
6.6.
A specimen is illustrated
Consideration was given to applying known in- 85 -
plane
restraint
applicable to
results
to
the specimens
membrane action.
would
then
be
to
make
However,
extremely
the
results
more
analysis suggested
sensitive
to
the
directly
that
the
of
the
stiffness
restraint and it was not possible to control this well enough to obtain
useful results.
All the tests were performed under load control.
information
could
have
displacement
control.
been
obtained
from
Theoretically, more
tests
performed
under
However, the test rig was not stiff enough to
achieve true displacement control.
Figure 6. 7:
Half size specimen under test
6.'-.3 Materials
a. Reinforcement
The reinforcement used was GKN ''Tor Bar" obtained from normal commercial
sources in the required sizes; 6mm and 12mm for the main tests and 16mm
for the more heavily reinforced specimen.
A stress-strain curve was obtained for samples of the bar using the same
Mayes machine which was used for the tests.
b. Concrete
The
mix
used
for
the
full
scale
specimens
was
intended
to
be
representative of normal practice for bridge deck slabs and to give a

strength close to the nominal design strength.
be mutually exclusive;
These objectives proved to
mix which complied with the minimum cement
content normally specified and which had a reasonable workability always

- 86 -
The exact mix
produced significantly more than the nominal strength.
proportions were varied between specimens as an attempt was made to get

closer to the desired results, but a typical mix is detailed in Table 6. 1.
Quantity (per nominal m3
Material
Half Size
Full Size
10-20mm Thame s Valley Gravel
780kg
5-lOmm Thames Valley Gravel
390kg
995kg
Sand <Thames Valley, zone 2 grading)
610kg
726kg
Ordinary Portland Cement
325kg
365kg
~1901
Water
Table 6.1:
~2101
Typical mixes
The mix used for the half scale tests was intended to be as close as
practical to a half scale model of the mix used for the full scale tests.
A lOmm maximum size aggregate was used and the proportion of fines was
increased
to get
close
to
the scaled grading curve.
greater surface area of aggregate in

content was needed.
the
finer
mix,
Because of the
a
higher cement
The water cement ratio was also increased so that
the strength, particularly the tensile strength, would be no higher than

for the full scale specimens.
A typical mix is detailed in Table 6.1 .
As
with the full size mix, modifications were made over the course of the
test sequence.
However, in order to ensure that this did not affect the
relationship between the full and half size mixes, the same modifications
were made to both mixes.
The small size of the specimens meant that it would have been practical to
mix
the
concrete
in
the
laboratory
using
dried
aggregate.
However,
because it was intended to use the tests to develop a mix design for the
model bridge tests for which this would not be practical, it was decided
to use the same 0.25 cubic metre pan mixer and batching plant which would
be used for the model bridge tests.
Cube tests and split cylinder tests for all the mixes were performed using
150mm cubes and 150mm diameter cylinders.
Some 150mm diameter cylinders
were also tested for Young's modulus in the Mayes machine.
- 87 -
All the cubes
and cylinders were cured with the test specimens, under plastic for seven
days and then in the laboratory.
The split cylinder tests suggested that the tensile strength of the full
and half size mixes were similar.
size mixes at test age
The mean tensile strength of the full
<approximately 28 days> was 3.30N/mm"' compared
with 3.14 for the half sized mixes.
Theoretically, it is more correct to
compare results for the full size mix tested with 150mm diameter cylinders
with results for the half size mix tested with 75mm cylinders.
However,
since no 75mm cylinder moulds were available, this was not done.
Instead,
some IOOmm cylinders from the full size mix and some 50mm cylinders from
the half size mix were tested.
3.94
and
3.49N/mm
The mean results from these tests were
respectively.
Thus,
changing
from
150
to
100mm
specimens for the full size mix gave a 19% higher strength whilst changing
from
150 to 50mm with the half size mix gave only an
Interpolating
between
the
results
for
the
150
and
11% increase.
50mm
cylinders
suggested that the strength of the half size mix measured using 75mm
cylinders would be 3.36N/mm"'; 2% higher than the measured strength of the
full size
The real significance of
mix.
these results is that
they
indicate that both the scale effect and the difference between the two
mixes were small compared with the random variation in the results.
The compressive strength of the full and half size mixes were also similar
to each other but the latter did tend to be slightly lower.
Typical
figures (actually those for the first pair of specimens and for the mix
detailed in Table 6.1) were 54.7N/mm"' for the full size mix and 47.0 for
the half size, both measured with 150mm cubes.
increased the latter to 48.4N/mm
Using half size cubes
6.4..4 Loading
The first pair of specimens, one full size and one half size, were loaded
to a load corresponding to the maximum service moment which BS 5400 would
allow.
the
Next, they were subjected to many cycles of a lower load <55% of
first
BS 5400,
load)
that
corresponding
is 25
to
the
units of HB in a
maximum
HA
equivalent
bridge designed
for
load
in
45 units.
However, the number of cycles <over 100,000> and the intensity of the load
were deliberately excessive.
It was hoped to use the test
to justify
using a much smaller number of cycles in the model bridge tests.
- 88 -
After
the cyclic loads had been completed the specimens were loaded to full
service load, unloaded, then loaded to failure.
The second pair of specimens were treated in the same way except that
they were first loaded to only the reduced, 55%, load.
The third full size specimen was treated in the same way as the second,
except that it was tested upside down to see if this altered the results.
When the third half size specimen was tested, an unloading expression had
been developed which gave reasonable results.
It was realised, however,
that these results were not affected by the stress which was assumed to
exist
in cracked concrete which was subsequently compressed.
It was
desirable, therefore, to test a specimen under reversed moment but the

apparatus did not enable this to be done.
The solution adopted was to
load the speciuien in the same way as the first but to turn it over after
10,000 cycles and start the test again.
It was considered that only the
half. size specimens could reliably be tested in this way because the dead
weight stress involved in turning over the full size specimen would be too
great.
The more heavily reinforced specimen was treated in the same way as the
first
specimen,
the
loads - being
increased
to
allow
for
the
extra
reinforcement.
6.4-.5 Processing of Results
Three columns of "demec" points were 'fixed to one side of the constant
moment regions.
The strain was averaged over the three demec readings in
a row and then a linear regression over the rows was performed to give an
average curvature and extension.
The curvature was also estimated from
deflection readings taken from rows of dial gauges.
These curvatures
differed, typically by 10% but sometimes by as much as 30%.
The curvature
estimated from a row of dials along the edge of the slab on the side to
which
the demec studs were attached
<that
is the three dial gauges
nearest the camera in Figure 6. 7 >, was only marginally closer than that
estimated from a row at the longitudinal centre-line or on the far side of
the- slab.
It was therefore conCluded that
the discrepancy was due to
variation in curvature over the length of the constant moment region,

rather - than_ over the width of the slab.
Since the analysis assumes a
constant curvature and the regression- gives a true average curvature,

89 -
whereas the curvature estimated from the deflection readings is weighted

towards the curvature at the mid-span of the slab, it was decided to use
the demec readings in preference to the dials.
The discrepancy does,
however, give an indication of the relatively low accuracy which can be

expected in the analysis of tension stiffening.
The regression analysis calculated the deviation of the strain readings
from the straight line.
The root mean square deviations varied between
specimens from less than 1% of the maximum strain to over 20%.

former
figure
The
indicated that, on average over the gauge length, plane
sections had remained plane
<as the theory assumes> whilst
the latter
indicated that they had not.
The difference is due to the random nature
of the cracking and the fact that the constant moment region was only
long enough to accommodate some three main cracks.
The best fit was
obtained in specimens for which both ends of the gauge length happened to
be mid-way between cracks, giving the theoretically desirable exact integer
number of cracks.
The worst fit occurred in a specimen in which a sloping
crack crossed the end of the gauge length.
This problem could be reduced
by using a longer constant moment region.
However, because of the effect
of variability of concrete tensile strength, this would give a misleading

impression of the shape of the tension stiffening function.
The main reinforcement in most of the specimens was also provided with
electrical resistance strain gauges.
This provided some useful information
but
that
the short
gauge length meant
the results were not directly
applicable to smeared crack analysis and the gauges were not fitted to the
final specimen.
6.4.6 Results and Analysis
a. First Loading
All the tests were analysed using a non-linear program.
specimens
were
essentially
beams,
in
that
they
were
Because the
subjected
to
constant moment over their width and they were not wide enough to be
forced
to
bend
cylindrically
(rather
than
anti-elastically>
convenient to use beam elements for the analysis.
it
was
The program used will
be described in Chapter 7.
The results of three of the tests are shown in Figures 6.8 to 6.10 along
with the results of analyses using the tension stiffening function which
~
90 -
was
eventually
adopted and
which
is
illustrated
in Figure 6.12.
To
facilitate direct comparison between the figures, the results of the half
size tests are expressed as equivalent results at full size.
The
experimental
predictions
results
obtained
were
using
initially
variety
compared
of
tension
with
the
analytical
stiffening
functions.
Because of the low steel area, and because the predictions for both the
curvature and for the axial extension at mid-depth were considered, this
gave a better indication of the shape of the tension stiffening function
than previous tests.
However, it was still not possible to .obtain totally
unambiguous results.
Both Cope's function and that proposed in 6.2.1 gave
reasonably good predictions.
A close study of the results suggested,
however, that immediately after cracking the true ,tension stiffening stress
was higher than suggested by either function.
up
to and above yield) it
explanation
for
this
appeared
behaviour,
and
At high strains <apparently
to be around 0.1
its
apparent
to 0.2f et
difference
from
An
the
behaviour in direct tension, is proposed.

When the peak concrete stress is reached in direct tension, a crack forms
and the load reduces.
A significant increase in extension is needed to get
back to the load which caused the crack and no new cracks can form until
this has happened.
If the specimen was held at an extension just above
that at which the peak concrete stress was developed, there would be a
significant tensile stress in the concrete even at the sections where the
cracks were forming.
a
test
unless
it
However, this stress has no effect on the results of

is
performed
under
true displacement
control,
which
requires a very stiff testing rig.

In a section in flexure, in contrast, there is always a region near the top
of a crack where there is a significant tensile stress due to the ductility
of concrete.
This gives the observed higher tension stiffening stress at
lower strains.
of
cracks
and,
It also increases the stress in the concrete on either side
as
we
saw
in
6.2.1,
this
reduces
the
crack
spacing.
Paradoxically this means that when the strain subsequently increases, and
the stress at the cracks reduces to zero, the tension stiffening stress is
lower than it would have been without the ductility of concrete in tension.
This explains why, at high strains, the tension stiffening stress is .lower
in flexure than in direct tension.
- 91 -
Moment 10
<kNm )
8
-Analysis
0
2
><
Test (first cycle)
Test (after cycling)
10
12
14
Curvature x 10
Fi!fure 6.8:
16
18
<mm-
Results of first f ull size test
Momen t 10
<kllm)
8
0
0
6
lr
--Ana~ysi s
xTest <first c ycle)
Test <after cycling)
><
0
0
10
12
14
Curvature x 10 6
Figure 6.9:
Theoretically,
this
effect
16
18
<mm-
1 )
Results of first half size test
should
be
more
pronounced
where
the
crack
s pacing is controlled by the depth of concrete in tension, rather than by

the reinforcement .
According to Beeby's theory <34), this means it will be
more pronounced where the reinforcement is widely spaced.
Thus it appears
that the tension stiffening stress at high strains should reduce as the
bar spacing increases.
This has been observed by Clark and Cranston (111 ),
further confirming the theory.

- 92 -
The
ductility
of
c onc rete
in
tension
also
explains
why
the
drop
in
stiffness visible in Figures 6 .8 to 6 . 10 when the concrete first cracks is

far less abrupt than analysis using any normal tension stiffening function
suggests.
This might have been partly explained by the variabi lity of
concrete which means that not all the cracks formed at once.
However,
analysis using an approach similar to that cons idered in 6.3. 1 s howed tha t
this explanation was not suffi cient.
Moment 8
<kNm>
6
4
-20
- 15
10
-1 0
-'
15
<mm-
Curvature x 10 6
-2
Analysi s
(first cyc le >
)(
Tes t
Test <after c yc ling >
..',
Test (after inve rted
-6
..
-8
c yc ling )
Fig ure 6.10:
Resul ts of third half size test
The f irst vis ible crack did not appear in the half s cale specimens until
the strain was
significantly greater than in the full scale specimens;
approximately 300 microstrain compared with 200.
However, there was not a
corresponding differen ce in the effective tensile strength.

and
half s ize
s pecimens
exhibited
significant
Both the full
non-linearity
before
the
cracks became visible but t his was more pronounced with the half scale
specimens.
It was considered that this might have arisen solely because
the cracks in the half scale specimens were half as wide and so did not
become vis ible
until their s cale size was greater.
To eliminate this
pos sibility, the later half scale specimens were inspected thoroughly for
- 93 -
cracks using a magnifying glass and a crack microscope, whilst the full
scale ones were inspected only with the naked eye.
This did not alter the
conclusion.
The results suggested that the half scale tests would give a reasonably
good indication of the load-displacement relationship, and hence of the
stresses, in a full scale specimen.
tension
However,
stiffening
the
most
function
could
obvious
fault
They also suggested that the same

be
used
of
at
the
full
analysis,
and
at
its
half
size.
tendency
to
exaggerate the abruptness of the loss of stiffness as the concrete cracks,

is greater with the half scale model.
The results also suggest that the
use of a half size model to predict the behaviour of a full size bridge is
liable to over-estimate the load at which cracking first appears.
The non-linearity observed before the cracks could be seen suggested that
both moment re-distribution and compressive membrane action could start to
act before cracks become visible.
Thus compressive membrane action should
delay the format ion of the first visible crack.
This has been observed by
both Guyon <10> and
it
Guyon's
specimen,
Kirkpatrick (49).
being
small
However,
scale
model,
now seems likely that

exaggerated
the
effect.
Similarly Kirkpatrick's slab, being only 160mm thick, would have shown a
more pronounced effect than a thicker slab.
Another implication of this non-linearity before cracking is that, in an
analysis which ignores the effect, reinforcement could significantly affect
the apparent
tensile strength.
This is confirmed by the fact
that
the
best fit to the results for the lightly reinforced specimens was obtained
using an effective tensile strength of approximately 0.8 times the split
cylinder strength whilst, for the more heavily reinforced specimen and for
Clark's tests, the full split cylinder strength gave better results.
These effects could be modelled by including some non-linearity before
cracking in the analysis.
It was found, however, that if sufficient non-
linearity was included to model these effects, the non-linearity in the

moment-curvature response prior to cracking was greatly exaggerated.
explanation for
this is that
The
the non-linearity was due to local micro-
cracking which had little effect on the strain averaged over a long gauge
length.
- 94 -
The best
interpretation of the results seemed
to be that
the tension
stiffening function should have the form shown in Figure 6.11.
This ie
essentially a constant tension stiffening function with an expression for

the tensile stress transmitted across the crack added on.
Theoretically,
the latter stress is a function of the width of the crack.
This implies
that, when it is expressed as a function of strain, it should be affected

by crack spacing.
However, the variability of the results and the narrow
range of crack spacing in the specimens made it impossible to detect this

trend.
f.~
Stress
Strain
Figure 6.11:
Because
the stress
Ideal tension stiffening function
reduces significantly on cracking,
and
because
the
subsequent stress is taken to be a function of the initial cracking stress,

the predicted response is very sensitive to the assumed tensile strength.
Small changes in the tensile strength have a much greater effect on the
results
than quite
large changes in
stiffening
function.
considered
in 6.2.2, encourages the continued use of tension stiffening
fun et ions
s tiffening
of
This,
the assumed shape of the tension
the type
stress
reinforcement
in
could
combined with
shown
in Figure 6.1.
concrete
delay
the
the
problem of steel stress
Even with
between
the
yielding
of
neutral
the
these,
axis
tension
and
reinforcement
the
but,
fortunately, the effect is not significant in realistic cases.

For the present study, therefore, it was decided to use the function shown
in Figure 6.12.
This is essentially a compromise between the constant
function s uggested in 6.2.1 and the type of function shown in Figure 6.1
and it was found to give marginally better results than either.
In most
cases, the results could be improved further by stopping the reduction in

stress at approximately 0.1
f e t rather than at zero but, because of the
risk of artificially delaying reinforcement yielding, this was not done.
- 95 -
Stress
FTENSHct/Er
<FTENS normall y = 20)
"'
Stra in
Unloading
/
/
Per man en t s e t = 0 .
Figure 6. 12:
5 *f c t/E~
Tens i on stiffening f un c ti on adopted
In any particular tes t, the results can be improved by minor adjustments

to the tension stiffening function but the behaviour is too variable to
justify this.
An indication of the lack of repeatability of the tests is
given by comparing the maximum curvature on first loading in Figures 6.9

and 6.10.
The curvature of two nominally identical specimens subjected to
identical
loads
differed
by
47%,
even
though
their
measured
material
properties and dimensions were almost identical.

The
tension
stiffening
expression
derived
for
the
lightly
reinforced
specimens appeared to work equally well in the one more heavily r e inforced
specimen.
However, because the behaviour of this was much less sensitive
to tension stiffening,
and
because only one specimen was
tested,
this
result wa s not conclusive.
b. Cyclic Loading
The effect of even 160,000 cycles to 55% of the peak load experienced was
considered to be too small, compared with the other variables and errors
in the analysis, to be worth including.
As expected, the effect of the
same load cycles on specimens which had not previously been subje cted to a
higher load was much greater.
The static load used in this test was not

- 96 -
sufficient
to
produce a
fully
developed
developed during the cyclic tests.

fatigue theory.
crack pattern and new cracks
This is as would be expected from
Analysis using a reduced tensile strength to allow for
fatigue appeared to give good results but tests to a wide range of load
cycle intensities would be needed to check this properly.
It is also not
entirely clear that the degradation was due to fatigue; it could have been
largely due to creep since the tests were conducted under sinusoidal load
variations which gave a mean moment some 55% of the maximum.
However,
since no long term static tests were performed, it was not possible to
separate the effects of fatigue and creep.
A single cycle to a high stress had a much greater effect than many cycles
to a lower stress.
In terms of bridge deck design and analysis this
suggests that it is reasonable to consider only the worst load cases, the
HB load cases, in the stress history analysis and to ignore cyclic loads
completely.
This is fortunate as it means the stress history of a given
point in the structure can, for practical purposes, be recorded by a single

number; the maximum historic strain.
c. Unloading and Re-loading
There was a difference between the unloading and the re-loading path, as
can be seen from Figures 6.8 to 6.10.
this
was
assumed
path,
small
compared
with
However, it was decided that since
either
the
effect
fct or the difference between the first
it
was reasonable to ignore
it.
of
small
changes
in
loading and unloading
Thus the ability to store the
relevant stress history as a single number was preserved.
It should be
noted, however, that this approach may not be valid in a dynamic analysis
because the difference in the paths, the hysteresis loop, represents energy
absorbed by the structure and contributes to the damping.
None
Cope's
of
the
for
unloading
example
expressions
<which
was
the
previously
best
of
used gave good

them)
results.
under-estimated
the
curvature which remained when the load was removed; typically by a factor
of three.
Since these functions were based on data which was either very
inadequate or derived from structures which, because of relatively heavy

reinforcement, were not sensitive to the expression used, it was decided to
ignore them completely.
After trying various relationships that shown in
Figure 6.12 was adopted, the slope of the unloading path being 3.5 times
the slope of the tension stiffening function, that is ci 2 equals 3.5a,
- 97 -
in
Figure 6.12.
This gave reasonably good results such as those shown in
Figures 6.8 and 6.9.

A significant
deformation
implies
when a
that,
remained after
load
is applied
the
load
which
is
was removed.
small relative
This
to the
maximum previously applied load, the strains and deflections relative to

the
initial
<unstressed) condition can
conventional
concrete.
elastic
analysis
which
be greater than predicted by a

ignores
the
tensile
strength
of
It was found that the deformations <both observed in the tests
and predicted by the non-linear analysis) became equal to those predicted

ignoring concrete in tension at a load which would correspond to 25 units
of HB if the section was fully stressed under 45 units of HB.
Since crack
widths in BS 5400 are checked under a load of 25 units of HB, this implies
that
BS
5400
is
calculations in
justified
in
ignoring
bridges designed
for
45
tension
stiffening
in
crack
units of HB load even though
significant tension stiffening was observed under full load after over a
hundred thousand cycles of normal service load had been applied.
However,
in structures designed for lower HB loads, the assumption is conservative.

Unfortunately, the one specimen which was inverted during the tests was
the only
assumed
one
to
0.5f ct/Ec.
which
become
was
sensitive to the amount
permanent
after
unloading,
of strain which was
which
was
taken
to
be
The results for this specimen are shown in Figure 6.10 in which
moments due to loads applied before the specimen was inverted are shown
as positive.
The biggest discrepancy in the unloading and re-loading part
of the plot
is that
the analysis
failed to predict the earlier initial
cracking load under negative moments, that
is in the inverted position.
This earlier cracking appears to have been the result of the cracks formed
by the previously applied positive moments acting as crack inducers since
the new cracks all joined the previous cracks.
this
earlier
cracking
in
the
one
specimen
There was no evidence of

which
was
tested
inverted
throughout.
d. Failure
On completion of the tests, all the specimens were loaded to failure and
they all failed
in
flexure.
The only unusual feature of the failure
behaviour was that the low steel area combined with the low d/h ratio
meant they did not reach peak load until the top steel yielded in tension.
This was predicted by the analysis.
- 98 -
The tension stiffening function used implied that unloading and re-loading
to the same load would have no effect on the deflections.
did have some effect, as can be seen from
Figure 6.8.
In practice it
However, when the
loading was further increased the tension stiffening appeared to recover

and the discrepancy was considered acceptable.
6.5 CONCLUSIONS
Because of its sensitivity to a highly variable quantity, the effective
tensile
strength
of
concrete,
tension
stiffening
cannot
be
predicted
accurately.
However, the functions developed in this chapter appear to be
significant
improvements over
those used
in
the
past,
particularly for
unloading.
The studies of similar full and half size strips indicated that a half
scale model will give a good indication of all aspects of behaviour except
the load to produce the first visible crack.
function can be used as at full size.
- 99 -
The same tension stiffening
CHAPTER
A
SIMPLER
NON-LINEAR
ANALYSIS
7.1 INTRODOCTION
Section 5.6 may give the impression that analysis for design is far more
difficult
than
specimens.
analysis
for
the
behaviour
of
laboratory
However, designers have a major advantage; they have no need
accurate predictions,
for
predicting
they
need
only
safe predictions.
Realistic
predictions are desirable because they lead to more economical designs but
errors which would be considered excessive to researchers are acceptable
to designers, provided they act in the safe direction.
Research on non-
linear analysis has concentrated on obtaining accurate predictions for the

load-displacement response of structures monotonically loaded to failure.
From the point of view of the design of the type of structure considered
here,
this
is unfortunate; neither deflection nor ultimate strength are
critical design criteria, loads do not increase monotonically and "accuracyu

is
neither
obtainable
Batchelor<78)
and
nor
necessary.
Kirkpatrick<l3>,
Indeed,
slabs
with
according
the
minimum
to
both
practical
reinforcement have over three times the required ultimate strength.
If
this is true, an analysis which under-estimates strength by a factor of

three is not merely adequate for predicting strength; it is as good as one
which is accurate to 0.001%.
The analytical methods considered in Chapter 5 contrast sharply with those
currently used in design and considered in Chapter 2.
sophisticated
and
expensive
potentially
but
able
to
The former are

give
realistic
predict ions based on realistic behaviour models even if, at the present
state of the art, they are not totally reliable.
The latter are cheap and
simple but based on unrealistic models of behaviour.
Their predictions are
not as realistic as those of NLFEA but they are more reliable; they are
always safe.
In the extreme case of restrained slabs, the two forms of
analysis may differ by factors of 5 or even 10 on strength.

therefore, some intermediate form of analysis
understand
and
considered
in
more
Chapter
compatible
5
but
Chapter 2) would be useful.
with
more
<safer, cheaper, easier to
codes
realistic
Clearly,
of
practice
than
those
than
those
considered
in
There is vast scope for making conservative
simplifying assumptions compared with the analysis considered in Chapter 5,

whilst
still
maintaining
greater
realism
- 100-
than
the
forms
of
analys~_s
This Chapter aims to develop such a form of
analysis which could be used in design and assessment.

The
Chapter
also
aims
to
develop
program
which
can
be
used
assessing simpler design methods, such as those considered in 3.2.8.
for
The
same program will be used as an analytical tool for investigating the

behaviour of bridge decks, including the models which will be considered. in
Chapter 8.
However, because of the fundamental difference between the
safe estimate analysis needed for design and the best estimate analysis
needed in research to facilitate direct comparisons with test results some

details, including the material models, will differ.
7.2 GENERAL APPROACH

The analysis is essentially a simplification of the approach considered in
Chapter 5; that is, it is a non-linear analysis using the smeared crack,
distributed steel approach.
However, in order to simplify it as much as
possible and to make it more similar to the grillage analyses with which
most bridge engineers are familiar, simple line elements are used to model
both the beams and the slab.
This greatly reduces the size of the program
enabling it to run on a desk top computer; it is perhaps the first time

this form of analysis has been performed on such a machine.
A disadvantage of this "grillage" type of analysis is that it treats the
stresses in the two directions as independent so it has to use uniaxial
material properties.
It thus cannot model the increase in stiffness due to
the Poisson's ratio effect in concrete subjected to biaxial compression, nor

can
it
model
reduction
in
the
enhancement
tensile
strength,
of
due
concrete's
to
biaxial
compressive
stress.
strength,
These
or
faults,
however, generally act in the safe direct ion and are considered acceptable
in design, indeed they are shared with all the analytical methods normally
used in
design.
Another fault is that this form of analysis can only check stresses in the
element direction so the maximum principal stress is not modelled if its
direction does not coincide with an element direction.
faults,
Unlike the other
this one is not acceptable because it could lead to significant
over-estimates of the load to cause cracking or failure.
To avoid this,
torsionless elements are used forcing the principal moment directions to

align with the elements.
This is also sometimes done in conventional

- 101-
linear grillage analyses because it enables the computed moments to be

used directly to design the reinforcement without the need to transform
them
to
the
reinforcement
direction.
Because
the
program
uses
this
principle, the element direction has to be the same as the reinforcement

direction.
This, and the desirability of using an approximately orthogonal
mesh so that the concrete stresses in the two directions are independent,
leads to what is probably the major practical limitation on the use of the
program; it is difficult to use it to model highly skewed bridges.
The problem of the principal moment direction does not arise in the downstand beams so these can be given torsional stiffnesses and elastic values
are
used
for
this.
However,
because
the program assumes
that
plane
sections remain plane and normal to the reference plane, warping stresses
and the effect of transverse bending in the flanges cannot be represented.
In the type of beams considered in this study, the predicted torques were
not excessive and the increase in stiffness due to the transverse bending
stiffness of the bottom flange exceeded any reduction due to cracking.
Thus the errors resulting from using elastic torsional properties were
conservative as well as small.
However,
this would not apply in all
structures and the program has been altered to enable a limiting value for
the torsional strength of beams to be specified <112 ).
This feature was
used in the analysis of the second of the models which will be considered
in Chapters 8 and 9 and the limiting moment was reached in the diaphragms
although not the main beams.
The particular program used was developed from one written by Edwards<36>,
although the modifications are so extensive that analyses have little more
than some basic principles in common.
7.3 DISPLACEMENT FUNCTION

In a beam element which is loaded only at the ends, the axial force is
constant whilst the bending moment varies linearly over the length.
The
displacement function used by Edwards matched thiE by using constant axial

strain and a linear v11riation in curvature over element length.
linear-elastic beam element,
force
In a
is proportional to axial strain and
moment is proportional to curvature so this shape function is ideal.
In a
non-linear element 1t is not quite as good because the stiffness varies

over the length which tends to result, for example, in the analysis underestimating the moment variation over element length.
- 102-
For use in non-
linear analysis, however, there is a more fundamental fault in the shape

function which does not appear to have been considered by other analysts,
such as Buckle and Jackson<90), who have used this type of element in nonlinear analysis.
The displacements are defined at the reference "plane" or <more correctly
for a line element) the reference "line" which, in this study, is at middepth of the slab.
assumed
to
be
This means that the strain at the reference plane is
constant
whilst
at
any
other
variation in axial strain along the element.

to
be
proportional
level
there
is a
linear
This variation is constrained
to the vertical distance
from
the
reference plane.
Since the level of the reference plane is largely arbitrary, this is not
satisfactory; even with a perfectly uniform and linear-elastic element, the
correct displacement field can only be reproduced if the reference plane is
at the neutral axis.
In a typical cracked slab element the actual neutral axis, the level at
which there is no axial strain, is well above <that is, on the compressive
side of) mid-depth.
In the real structure the variation in axial strain
along the element is proportional to the distance from the neutral axis
but, in the computer model, it is proportional to the distance from middepth.
Thus, if the variation in curvature over length is correct, the
variation
in
strain
along
the element
material below the reference plane.
is
under-estimated
for
all
the
As a result, unless the element mesh
is so fine, that the variation in curvature over length is insignificant,

the analysis can
fail
steel
can
and
hence
to predict
reinforcement yielding in the tension
over-estimate strength.
There
is
also a
region
between the actual neutral axis and the reference plane where the real
strain becomes more tensile in the direction of increasing curvature but
that in the computer model becomes more compressive.
the top of the cracks are in this region
This means that, if
<which is often the case> the
computer model will indicate that the extent of cracking will reduce over
element length in the direction of increasing curvature, which is clearly
incorrect.
One solution
to this problem would be to keep the same displacement
function but to put the reference plane at the neutral axis.
This is not
practical because the neutral axis moves as the concrete cracks.
The
effect can, however, be obtained by introducing a linear variation in axial

- 103 -:
strain, as well as in curvature, over element length.

strain
due
to
reproduced at
linear
variation
all depths
in
curvature
in the element,
can
The variation in
then
be
correctly
rather than at only one.
In
addition, a linear variation in neutral axis depth over element length can
be modelled.
Introducing
the
linear variation in axial strain gives
the displacement
function illustrated in Figure 7 . 1.
Since the stresses are calculated at
only
of
two
sections
in
the
length
the
elements,
it
simply
means
increasing the axial strain at one section and reducing it at the other by
the same amount.
However, because this linear variation does not alter
the element length, it cannot be defined from the displacements of the two
end nodes.
The modification effectively amounts to introducing a
third
node at mid-length with only one degree of freedom; the axial displacement
o"'
in Figure 7.1.
-8,
Node 2
Node 1
jo, 1 1
L-----------------~
1.
Due to
o,
10:2/ 1
~--------------~-
2 . Due to 0 2
4-0 c /]
~40c/ ]
3. Due to Oc
Axial Strain
<at reference level>
Axial Displacement
Figure 7.1:
Displacement fun ction
- 104-
Introdt1cing
this
complicated
the
node
into
analysis
the
and
globsl
stiffness .matrix
increased
the
computer
would
storage
have
space
required.
This was avoided by considering the internal equilibrium of the
elements.
The fault in the original displacement function meant that the
axial forces calculated for the two sampling sections were not necessarily
equal.
This leads to a criterion for the correct value of liei the value
which equalises the forces.

comparing the forces at
calculation.
However, this value can only be obtained by
the two sections which requires an iterative
Performing this iterative calculation for every element each
time the forces in the structure are calculated would have greatly slowed
down the analysis.
To avoid this, the number of iterations for lie is
limited to two, but a vector of lie for all the elements is stored and used
as the first estimate the next time the element forces are calculated;
that is in the next iteration of the whole structure.
The modification has
effectively increased the number of degrees of freedom in the analysis by

some
30%
without
proportional
increase
in
the
required
computer
capacity.
The modified version of the program was tested by analysing the simple
case considered in Chapter 4 and the results are shown in Figure 7.2.
the Figure the percentage error in predicting the restraint
In
force or
displacement, whichever is greatest, is shown for analyses using different

numbers of elements.
For comparison, the same case was also analysed
using the previous version of the program.
Because this beam has no
tensile strength, and hence the formation of a crack does not release any
energy, the problem of mesh dependence which was considered in 6.2.3 does
not arise.
Thus, as the mesh is refined, both programs converge on the
"exact" analytical solution which was derived in Chapter 4.
However, the
modified form of the program converges very much more quickly and 3
elements with this give better results than 6 with the original program.
In most of the structures considered in this study, the improvement is
more fundamental because, with the old program, the mesh size required to
reduce the discretisation errors to acceptable levels is too fine by the
criteria considered in 6.2.3.
Having adopted the principle of defining extra degrees of freedom by
considering internal equilibrium of elements,
extend it to develop higher order elements.
it would be possible to
For example, one could use a
quadratic variation in both axial displacement and curvature.

- 105-
This would
give two extra degrees of freedom which could be defined by calculating

both the axial force and the bending moment at a third section at midlength and checking that
sections.
they were consistent with those at the other
This would undoubtedly enable a beam to be modelled with a
coarser element mesh.
However, Figure 7.2 shows that with the existing
program a remarkably coarse mesh gives satisfactory results.

only 3
elements
in a
half span
in which
there
is
Even with
complete moment
reversal, the worst error is around 3% which is small compared with the
variability
of
behaviour observed
in Chapter
6.
Also,
to
model slab
behaviour with a grillage (even a linear grillage ) a finer mesh would be

r equired.
Thus
the extra complication of higher order elements is not
justified.
Vcrs t Err or
<% )
30
---Orig1nal Program
- - - - Modified Program
20
\
10
'-...
1
32
16
Number of Elements
<in ha l f model)
Figure 7.2:
Effect of change t o di splacement function
7.4. ELEMENT INITIAL STIFFNESS CALCULATION

The program cal culates the initial stiffness matrix elastically, as in a
conventional linear grillage, using the gross- concrete section properties.
Althoug h the displacements are defined from the reference plane at middepth of the slab, Edwards' program cal culated the initial stiffnesses of
the down-stand beams about their own neutral axes.
It then treated them
as though they were calculated about the reference plane.

lead
to
errors
in
the
final
results
because
the
This did not
non-linear
force
calculation correctly calculated the forces from the displacements allowing

for the eccentricities.
However, because the initial stiffness matrix did
not model composite action between the beam and slab, and thus did not
- 10 6 -
represent the true behaviour even in the elastic range, it did result in
very slow convergence.
With the simple solution scheme used by Edwards,
analyses of structures in which the beams were large compared with the
slab would not converge at all.
The solution to this was to use a rigid
body transformation and this was done by defining the element stiffness
about the reference plane by using the stiffness matrix for an off-set
beam which is g iven in Ta ble 7.1.
x,
w,
Reference Level
= - -- -
8,
X
= EA
Axial Stiffness
Flexural Stiffnes s = El
Length
M,
R,
F,
6EI I P
- EAXI l
=1
4EI 1 1
8,
+
EA X2 11
w,
R2
F..,
2EI/l
- 6EI I J2
EAXI l
- 12EI IJ2
EAP 11
12EI/ P.
x,
M2
EA /1
6EI I P'
EAX I l
- EA i l
4EI 11
e."
<sy mmetri cal )
+
EAX2 1 l
w2
- 6EI IF
12EI/ F
X :.'
-EAX I l
EA/1
Table 7.1:
Stiffness matrix of an off-set beam element
<For simplicity an element in a plane frame is illustrated)

- 107 -
7.5
~PLANE
FORCES
Although Edwards described his program as a "grillage" this is not strictly

The non-linear analysis leads to axial forces in the elements
correct.
In order to distribu te these forces
which a true grillage cannot model.
correctly, it is necessar y to consider horizontal displacements and the inplane shear in the slab.
Thus Edwards' program considered five degrees of
freedom per node instead of three as in a true grillage.

The
in-plane
transverse
shear
in
displacement
the
elements
of
the
was
nodes;
calculated
the
from
horizontal
the
relative
displacement
perpendicular to the element direction of the node at one end of the

element relative to the node at the other end.
This implied that all of
this transverse displacement was resisted by shear even though it may

have actually been largely due to rotation of the whole element about the
vertical axis with no shear deformation, that is as shown in Figure 7 .3b
rather than 7.3a.
It also meant that
the complimentary shear and the
resulting axial forces, such as the transverse forces illustrated in Figure

3.12, were not modelled.
This led to errors in the treatment of in-plane
forces which were serious, not so much because they were large <although
they could be), as because they tended to act in the unsafe direction.
In
the finite element programs considered in Chapter 5, this fault is avoided

because there are enough nodes in an element to define its horizontal
shear
deformation
from
the
horizontal
displacements
of
the
nodes.
However, with only two nodes per element, the shear deformation can only
be defined if the rotation of the nodes is also known.
Thus, it was
necessary to introduce this sixth degree of freedom, rotation about the

vertical axis, into the program.
Trans verse displacement
b. Due t o rotation
a . Due to shear deformation
c. Due to uniform bending

Figure 7.3:
d. Due to non- uniform bending
Transverse displacements of a line element
- 108-
If
the
slab
was
modelled
with
normal
line
elements,
as
used
in
conventional space frame analysis, the transverse shear force would have
caused
transverse
bending
in
the
individual
elements
as
shown
in
Figure 7 .3d; it would have introduced Vierendeel frame type displacements.

These displacements do not arise in the real slab because the "elements"
cannot bend independently; they act compositely.
Thus the displacements
due to this local transverse bending of the elements had to be suppressed

in
the
computer
model.
To
achieve
this,
it
is
assumed
that
if
the
elements are subjected to a transverse displacement without rotation of

the nodes <that is as shown in Figure 7.3a and d> the only deformation is
due to shear flexibility and the deformation is as shown in Figure 7 .3a.
The shear force is calculated from this shear deformation, as in Edwards'
program, but the moments required to keep the element in equilibrium about
the vertical axis <an equal and opposite moment at each end) are applied.
In Edwards' program, these moments were not applied to the structure.
In
effect they were resisted by a totally artificial restraint to rotation of

the nodes about the vertical axis.
In order to preserve the basic simplicity of the elements, the individual
elements are assumed not
to provide any resistance to uniform bending
about the vertical axis; they do not resist the form of deformation shown
in Figure 7.3c.
This means that
the stress state can be taken to be
constant across the element width and avoids the need to perform a stress
integration over width as well as over depth and length.
The relatively
small moments required to maintain equilibrium with the in-plane shear are
the only moments about the vertical axis within the elements.
The bending
stiffness of the structure about the vertical axis is, however, modelled by
the differential axial forces in the elements.
The approach is to split
the transverse deformation of the elements into two components; a uniform

bending about
the vertical axis as shown in Figure 7.3c, which is not
resisted,
and
shear
deformation
as
shown
in
Figure
7.3a
which
is
resisted by the transverse shear stiffness of the concrete in the slab.

The mathematics of the assumed deformation state are given in Appendix B.
In
practice,
nominal
bending
stiffness
was
added
because
otherwise
rotation about the vertical axis is completely unrestrained in some models.

This treatment of in-plane forces is inherently approximate.
It might also
be argued that including in-plane shear is inconsistent with the reasons

given in 7.2
for
using torsionless elements since it

- 109-
implies that
the
maximum principal tensile stress may not align with the element direction.
However, the program is intended for modelling structures whose behaviour
is primarily flexural so it is appropriate to use a lower order of analysis
for the in- plane forces .
Load 60
<kN >
- - - - - 6 D. 0. F.
20
--6 D.O.F <2X s he a r modu l us )
10
15
Defle ction
Figure 7. 4- :
Figure
7.4
(actually
modified
shows
the
the
program
original program.
with
two
( DID)
Eff ect of in-plane shear
results
structure which
20
of
the
will
analysis
of
be considered
different
shear
a
in
moduli
simple
structure
7.10 .3) using
and
also
using
the
the
The effect of doubling the shear modulus is small, which
implies that errors in the treatment of in-plane shear have little effect
and justifies the use of approximate analysis for in-plane shear.
Even the
apparently fundamental fault in Edwards ' program has only a small effect
on this particular structure, although the artificial restraint to rotation
about
the vertical axis
modulus.
is equivalent
to more than doubling the shear
However, it is possible that the effect could be greater in some
other structures s o it was considered prudent to use the modified program

for all subsequent analyses to ensure that the results would be safe.
For
the same reason, and unlike in Edwards' program, a reduced shear modulus
is used for cracked concrete.
Because the elements are fixed together at slab level, and because inplane forces in the slab are represented, the program is able to model
both shear-leg and the effect of the shear connect ion between the beams
which was discussed in 2.4.3 and 3.2.7.
checks
moment
equilibrium about
Unlike Edwards' program, because it
the vertical axis,
it
also models
resulting transverse stresses which were illustrated in Figure 3.12.

failure
of
the
original
program
to
model
effect further justifies the modification.

- 110-
this
potentially
the
The
significant
7.6 LARGE DISPLACEMENTS
When a slab deflects relative to the restraining beams, the le1er arm at
which the restraint force acts is reduced.
significant
compared with
the
Once the deflection becomes
thickness of the slab,
reduces the slab's load carrying capacity.
this significantly
Curiously, most of the NLFEA
studies mentioned in Chapter 5 did not consider this effect whereas all
the <otherwise far less sophisticated) analyses considered in 3.2 did.
this study,
was originally decided
it
to follow
In
the NLFEA studies and
ignore the effect and, because the analysis is conservative in other ways,
the predictions still tended to err on the safe side.
reasons,
it was eventually decided to modify the program to make some
allowance
always
However, for three
for
leads
large displacements.
Firstly,
ignoring the effect nearly
to
in
unsafe
errors
which
act
the
direction
so
it
is
undesirable in a design situation even if the errors are relatively small.

Secondly, some of the tests on model bridges which were considered in
3.2.3,
notably Seal's <77>,
reached
such
large deflections
before
failing
<around h/2) that an analysis of these which assumes the deflection to be

small relative to slab thickness is clearly invalid.
Thirdly, for reasons
discussed in 3.2.8a, it would be desirable to be able to use membrane

act ion in the design of slabs with longer span to depth ratios than the
empirical design rules allow.
However, financial and time restrictions on
this project prevented an experimental study of such slabs.

design
was
to
be
justified
purely
by
analysis,
it
was
Thus, if their
particularly
important to ensure that the analysis was safe and, since longer span to
depth ratios increase the significance of deflections, this meant allowing
for the effect of deflect ions in the analysis.
Because
of
the
use
of
line
elements,
include the deflection in the analysis.
it
was
comparatively
simple
to
It was done within the elements by
adding the vertical component of the axial force to the shear force.
As
is illustrated for a simple case in Appendix Cl, this has the effect of
modelling
the
moment
in
the
elements
<that
is,
about
the
deflected
reference level) due to the axial force acting at the undeflected reference
level: it models what in a column would be called the "buckling", "added". or
"P6" moment.
The vertical component of the axial force is calculated only
from the difference in the vertical displacements of the two nodes.
The
effect of curvature over the length of the element is not included but
this is only significant if an excessively coarse element mesh is used.
- 111 -
To
maintain consistency, the vertical component of the in-plane shear was also
added and this is calculated from the rotation about the longitudinal axis
averaged for the two ends of the element.
As an additional allowance for
finite displacements, the axial strain in the elements is also corrected

for the effect of the slope as detailed in Appendix C2.
That is, the axial
strain used to calculate the forces in the element allows for the increase
in length of the element due to its slope.
These effects are modelled only in the non-linear force calculation, not in
the stiffness matrix.
This must reduce the convergence rate but was
considered acceptable.
7. 7 MATERIAL MODELS
7.7. 1 Steel
A tri-linear stress-strain relationship is used for steel as indicated in

Figure 7 .5.
In analyses for research purposes, the factors are chosen to
give the best approximation to the actual stress-strain curve of the steel.
The pre-strain is included primarily to enable prestressing to be modelled
but,
in simple slabs,
shrinkage.
a negative pre-strain can be used to represent
When modelling prestress, the pre-strain has to be reduced to
allow for losses because the program does not consider long-term effects.
Stress
f..,1t
Evlt
0. 2%
Strain
<including Pre-Strain)
Figure 7.5:
Steel properties
The same properties are used in compression as in tension, except for a

limit on strain hardening in compression.
function only of the present strain.
The stress is taken to be a
It would be simple to adjust the
- 112-
program to allow for the permanent set in steel which has been stressed
beyond its elastic limit but this would require the maximum strains to be
stored for all the steel layers in all the elements.
structures analysed
under
the
final
had steel stressed above
failure
load
case
when
Since none of the
its elastic
the
strain
limit, except
was
increasing
monotonically, the facility to model permanent set was not implemented.

In analyses for serviceability design, in order to avoid the problem of
stress history dependence as much as possible, the steel is taken to be
linear-elastic.
To
justify
this
assumption,
service stress has
limited to the elastic limit and this becomes a design criterion.
to
be
Thus no
advantage can be taken of re-distribution due to reinforcement yielding

under service loads.
is
considered
However, this is not a disadvantage as such yielding
undesirable
anyway.
The approach
has
the advantage of
making the analysis more compatible with current codes of practice.

In analysis for design at the ultimate limit state, the tri-linear stressstrain
relationship
can
be
used
to
represent
properties or the code specified properties.
either
the
actual
steel
It is normally assumed that
only reinforcement yielding due to the load case being analysed needs to
be considered.
This is justified if one assumes that only one load case
above design service level is applied.
However, in a bridge deck slab, this
is not very logical since the design vehicle cannot get to the critical
position without
first
being applied in other positions which are only
marginally less severe.

strength analysis) has
Fortunately,
this problem
(like all aspects of
little practical significance since serviceability
criteria are critical.
7.7.2 Concrete in Compression

The
stress-strain
relationship
illustrated in Figure 7.6.
used
for
concrete
in
compression
is
Various curves have been proposed which are
more realistic, but when one allows for the variability of concrete the
improvements
are
not
significant
and
Abdul-Rahmen<87)
used
the
even
simpler elastic-plastic relationship.

It
is
assumed
that
when
the
concrete
is unloaded,
parallel to the initial part of the loading diagram.
it
follows
line
Thus it takes on a
permanent deformation which is equal to the departure from linearity on

loading.
- 113-
Stress
0.7fc
0.5fcu
0+---_.------------~-------------r-----------
0.0025
0.0045
Strain
Figure 7. 6:
Properties used for concrete in compression
The strain at which the stress is taken to start to reduce is lower than
in many other analyses.
This was a reflection of the results obtained
f r om the cylinder tests and may have been due to the relatively fast speed
at which these tests were performed.
The rate of reduction of stress
after
intended
the
peak
has
been
passed
was
to
represent
the
true
behaviour of the concrete, which can only be observed with a very stiff
testing machine, rather than its apparent behaviour.
However, because of
the nature of most of the structures considered, analyses performed with

the more usual form of curve, with a longer plateau followed by a more
abrupt cut-off, gave very similar results.
The same basic approach to serviceability design is used as for steel; the
material is taken to be linear-elastic and a stress limit is imposed to
ensure that this is reasonably true.
However, although this limit is also
given in codes of practice, it is far

limit .
As
noted
in
2.3.4c
and
less satisfactory than the steel
demonstrated
in
reference
35 ,
even
structures designed to BS 5400 can be stressed well above the limits, yet
their behaviour is satisfactory.
This presents a problem.
If the stress
limit is imposed on structures designed using non-linear analysis it is

unduly conservative; if it is not imposed it will be more difficult to get
the approach accepted and it is also difficult to decide what the design
criteria should be.
A possible compromise is to impose a limit but t o
make it less conservative.
This can be done without departing from the
principle of linearity if the increase is justified by the biaxial stress

state in the critical area.
enhancement
due
to
There is a precedent for explicitly considering
multiaxial
stress
- 114-
states
in
BS
4975 <113).
An
alternative approach is to make a totally arbitrary increase in allowable

stress and then to justify it by comparison with test results.
For analysis for design ultimate strength the code specified stress block
could
be
used
but
tri-linear
conventional design methods,
the
approximation
was
employed.
In
characteristic material properties are
used in the analysis of the structure and the design strengths, with the
partial safety factors applied, are used only for
critical sections.
the analysis of the
In non-linear analyses, the analyses of the structure
and of the critical sections are not separated so this approach, although
recommended by BS 8110, is not appropriate.
safety
factors
were applied
to all
viewpoint, this is not justified.
In this study, therefore, the
the material.
From a
statistical
However, it is conservative (except for
some cases where restraint stresses are dominant> and, since serviceability
criteria are critical, this is acceptable.
A disadvantage of this approach
is that in most codes, including BS 5400, the design ultimate stress in

concrete is less than the limit of linearity used at serviceability.
Thus,
unlike in analyses for research, completely separate analyses have to be

performed for serviceability and for ultimate strength.
7.7.3 Concrete in Tension
The properties used for research analyses are illustrated in Figure 6.12
and were discussed in the last chapter.
In choosing properties for analyses for design the major problem is that
the
desirable
characteristics
of
the
properties,
that
they should
be
reasonably representative of real behaviour and that they should not be

strain history dependent, are mutually exclusive.
The simplest solution to
this problem is to abandon realism in favour of avoiding strain history

dependence and ignore the tensile strength of concrete completely.
As was
noted in 2.4.2, this approach has the major practical advantage of being
compatible with current codes.
disadvantage
is
that
it
is
It is also normally conservative, indeed a
liable
to
lead
to
an
unduly
prediction of the distribution of moments between the beams.
pessimistic
However, it
is not possible to prove that the approach is always conservative.

peculiarity of membrane action is that
the restraint force,
the effect
which leads to the enhanced behaviour, is a direct result of cracking.

Thus tensile strength, by reducing the extent of cracking, can reduce the
restraint force and hence the degree of enhancement.
- 115-
To investigate this,
a simple slab was analysed using concrete tensile strengths of zero and
3N/mm~.
The
latter
analysis
used
the
material
properties
Figure 6.12 and the results are shown in Figure 7.7.

used and the load quoted is that on the half model.
given
in
A half model was

For comparison, the
results of a conventional analysis ignoring the restraint as well as the

tensile strength are also shown.
The stresses in the slab which was analysed using a tensile strength of
3N/mm2
are calculated in two ways.
The lower lines give the stresses
directly from the computer program; that is the smeared stresses.

are
always
less
than
those
calculated
ignoring
the
tensile
although, once the concrete has cracked, the margin is small.
These
strength
The reason
for the discontinuous plot is that in the numerical analysis the cracking
advances, both in depth and along the slab, in discrete steps.
In the real
structure, the cracks can grow more smoothly in depth but the cracked zone
can only advance along the slab in discrete steps as individual cracks
form.
In order to make the analysis as realistic as possible, the element
length was matched to the estimated crack spacing giving five elements in
a half model.
It
was found that an analysis using a
finer mesh
(20
elements in place of 5> predicted very similar behaviour, the deflections

being within 2%.
However, the smeared steel stress was up to 50% higher ..
A study of the results revealed that there were two reasons for this.
The
first was that, on first cracking, the fine mesh predicted unrealistically
localised
cracking
and
hence
an
unrealistically
small
restraint
force.
However, because of the tension stiffening function used, the extension on

initial cracking was very limited even in the coarse model.
The effect of
mesh size was therefore far less pronounced than in an earlier analysis
performed with concrete tensile strength but without tension stiffening.
The second reason for the effect of mesh size is that the analysis gives
the
stress
section.
only
at
the
last
integration station,
not
at
the
critical
In the coarse mesh, the last integration station is 40mm from
the critical section and thus is subjected to a 4% lower moment.
This
might normally be expected to make only a 4% difference to the stress.

However, the concrete properties used in the analysis make the momentsteel stress relationship non-linear whilst the 4% difference in moment is
not accompanied by a difference in the enhancing axial force.
The peak
stress predicted by the fine mesh is very localised and the effect is far
- 116-
less pronounced in the analysis of real slabs because of the finite width
o f the applied loads.
Stress 300
<Nimm2
200
100
I
f~t
10
15
=3
20
25
<smeared stress)
30
Load <kN/m width)

a . Steel Stress
=0
Stress 20
<unrestrained)~c~
=3
<stress at crack)
=0
(N/ID1!f2)
15
10
=3
<smeared stress)
10
15
20
25
30
Load <kN/m width)

b. Concrete Stress
Figure 7. 7: Effect of concrete tensile strength

<Central line load, simply supported but with rigid in- plane restraint
1
= 2000,
= 160,
= 119,
T12-250 reinforcement)
It was noted in 6.2.2 that smeared crack analysis under-estimates the peak
stress in the reinforcement.
This is clearly undesirable if stress is to

- 117-
be used as a design criterion so it appears to be more appropriate to

calculate the peak stress at the crack at the critical position using only
the forces, not the stresses, given by the computer analysis.
This also
has the advanfage of eliminating one of the effects of mesh size.

stresses calculated in this way are given by the upper solid lines.
The
In
accordance with normal practice, the critical sect ion was analysed ignoring
both the tensile strength of the concrete and the effect of the top steel.
Separate
calculations
relatively small,
confirmed
provided that
that
the
effect
of
these
would
be
the post-peak part of the stress-strain
relationship used for the analysis of the structure was not included.
Until the analysis predicts cracking, the stress calculated in this way has
no real physical meaning and is not
plotted.
When the analysis first
predicts cracking, the extent of the cracking is very limited.
Thus the
restraint force is small and the calculated stress at the critical section
is similar to that given by the conventional analysis and substantially
greater
than
completely.
is
predicted
ignoring
the
tensile
strength
of
concrete
As the load increases, the extent of cracking <and hence the
restraint force) increases disproportionately.
Because of this, the steel
stress calculated for the critical section does not increase substantially
until concrete non-linearity comes into effect and the plot is discontinued
because the elastic sect ion analysis used is invalid.
The difference between the various calculation methods is much less for
the
concrete
stress
which,
using
critical for the restrained slabs.
BS
5400
serviceability
criteria,
is
The restrained analyses also converge
to give similar failure loads of around 70kN compared with 25kN for the
unrestrained
analysis.
Nevertheless,
the
difference
in
the
allowable
service loads implied using t"he stress at crack approach, 21kN, and the
smeared crack approach, 30kN, is disturbingly large and it appears prudent
to
use
the
former
approach.
It
should
be
noted,
however,
that
substantial part of the difference is due to the aforementioned effect of

the difference between the stress at the critical section and at the last
integration station.
This has two important
implications.
Firstly the
effect will be less pronounced under patch loads <as opposed to point or
line loads) so the difference between the two approaches will normally be
less than implied by this study.
Secondly, the peak concrete stress is too
localised for normal material models, based on the behaviour of specimens
- 118-
in a uniform state of stress, to be valid.
Because of this the use of the
stress at the critical section is conservative.

The analysis of these slabs appears to confirm that ignoring the tensile
strength
of
concrete
strength
of
concrete
may
not
does
be
have
conservative.
a
major
However,
beneficial
the
effect
tensile
on
real
structures which does not arise in the rigidly restrained slabs considered
here;
it improves the restraint.
Thus the tensile strength of concrete is
far less likely to have a detrimental effect on the stresses in realistic

bridge deck slabs.
7.8 SfRESS INTEGRATION
The
element
forces
element volume.
are obtained
by
integrating
the stresses over
the
The stress is taken to be constant over element width and
the integration over length is performed using two integration stations at

the Gauss points, 21% of element length from each end.
The forces at the
nodes are then obtained as a function of the forces at the integration

stations using the shape functions.
In analyses of this type, it is usual to perform the stress integration
over
depth
numerically
sometimes as
smooth
few as
curve
with
high
order
integration
five sampling stations.
between
the
stations.
As
the
function
This effectively
stress
and
fits
functions
used,
particularly for concrete in tension, are highly discontinuous it appears

that this could lead to significant errors and Ganaba and May<114) have
confirmed this.
In many of the sections considered in this study, with
their very light reinforcement, a five point integration scheme gave only
one station in uncracked concrete.
This, combined with the fact that the
tension stiffening function used was more discontinuous than that favoured
by
Ganaba
and
May,
suggested
that
the
integration
errors
would
be
particularly significant.
Two solutions to this problem were used.
For analyses which did not
consider stress history, an exact analytical integration was developed.
In
addition to eliminating integration errors, this was significantly faster

than
numerical
integration.
However,
neither
this
solution
nor
that
suggested by Ganaba and May <splitting the integration at the r.oot of the
crack),
could
be
used
for
stress history analyses.
It
was
therefore
decided to increase the number of integration stations from five to eight,

- 119-
to improve accuracy, and to change from high order Newtonian integration

to trapezoidal integration to reduce the effect
of the discontinuities.
The stresses in the down-stand beams were integrated separately using the
same integration scheme.
Comparison with the exact analytical version of the program showed that
these changes made the integration errors in the analyses of structures
insignificant compared with the other errors.
However, it appeared that
this was partly because the errors were essentially random and so tended
to cancel out;
the error in the forces calculated for a single element
could still be significant.
This tendency of the errors to cancel out
explains why, despite the large errors observed by Ganaba and May in the
forces calculated for individual elements, other analysts [such as Abdel
Rahmen<87)l have found their results to be insensitive to the number of
integration stations used.
In the analyses of the constant moment regions considered in Chapter 6
there was no scope for the integration errors to cancel out so they could
be
more
significant.
Because
of
this,
and
because
"accuracy"
was
considered more important for a fundamental study of tension stiffening, a

special version of
trapezoidal
rule
the program was developed which employed 32
integration.
This was
used
for
all
point
the analyses
in
Chapter 6, except those which did not consider stress history and so could
be performed with analytical integration.
eliminated integration errors completely.
For practical purposes, this

Indeed, since they were spaced
at only a quarter of the maximum aggregate size, the integration points

were unrealistically close.
However, because the maximum historic strains
are stored for all the integration stations, this version of the program
required more storage space as well as more computer time and it was not
used for the analysis of more complex structures.
7. 9 SOUJI'ION
~HEME
In non-linear analysis,
the
forces can be calculated directly from the
displacements but the displacements can only be obtained from an iterative

solution
scheme.
Incremental
iterative
schemes
are normally
used to
enable the behaviour of the structure under increasing loads to be studied

and also because the behaviour is sometimes "path dependent" so analyses
using very large increments could give incorrect solutions.
- 120-
A detailed study of solution schemes is beyond the scope of this thesis.

However, some problems were experienced which are peculiar either to the
type of structure considered or to the form of analysis used.
These will
now be considered along with a brief review of the scheme adopted.
7.9.1 Control
Solution schemes using displacement<97) or arc-length(ll5) control are now
favoured
by
analysts
but
this
has
arisen
primarily
because
of
the
It is difficult to
concentration on ultimate and post-ultimate behaviour.
achieve convergence with analyses using load control as failure approaches

and
impossible
through".
to
model
softening,
post-ultimate
behaviour
or
"snap-
However, with the type of structures considered in this study,
ultimate strength is a secondary consideration and neither "post-ultimate

behaviour"
because
nor
the
true
"displacement
failures
are
local and
energy is stored in the beams.

under
perfect
control"
displacement
suddenly and completely.
have .much
brittle whilst
physical
most
of
meaning
the strain
Thus, even if the bridges had been tested
control,
the
slabs
would
still
have
failed
Also, the temporary reduction in load which can
occur under monotonically increasing displacements as cracking occurs has

no
practical
control.
significance since
real
structures are
loaded
under
load
There is thus little practical advantage in departing from using
load control, at least in a pragmatic study such as this.
As structures
are designed for specified loads, and neither strength nor displacement are
critical
design
criteria
for
the
type
of
structures
considered
here,
analysis under load control is far more convenient for use in design.
7.9.2 Initial Stiffness Method

As
serviceability
criteria
are
critical
there
is
no
need
to
take
an
analysis for design up to failure, only to design ultimate load which is

just 30% above design service load in BS 5400.
Since this is normally
well below the actual collapse load, the demands on the solution scheme
are
comparatively
modest
so
relatively
simple
scheme
can
be
used.
Edwards used the simplest possible scheme; the initial stiffness method
with no accelerators.
degree of
freedom
In this approach, which is illustrated for a single
system
matrix is used throughout.
in
Figure
7.8,
the
initial elastic stiffness
The displacements are calculated from the
loads using the inverted initial stiffness matrix.
The forces are then
calculated
non-linear
from
these
displacements,
- 121 -
using
the
material
properties, and compared with the applied loads.
The difference
(which
represents the forces released by cra cking, crushing and yielding> is then
used to calculate a new set of displacements which are added to the first
set.
A new set of forces is then calculated for these displacements and
the whole process is repeated until the forces match the applied loads.
results
The
are
then
printed
out
and
the
next
increment
of
load
is
applied.
The approach has the advantage of being numerically stable and reliable as
well as simple.
However, if the actual tangent stiffness matrix of the
structure is substantially different from the initial stiffness matrix, the

convergence rate is very slow.
This normally occurs as failure approaches.
However, in the case of some of the slabs considered in this study, the
very low steel areas meant that cracking changed the stiffness so much
that
the
convergence
rate
service load was reached.
became
excessively slow
even
before design
In a typical analysis of a 25 node model, over a
hundred iterations per increment were needed.
This, and the desire to use
the analysis as a research tool <which meant that failure behaviour had t o
be considered ) me an t t hat the convergence rate had to be improved.
!..oad
:n.:r eroem:s )
---------- Analysis
-
- Str uc ture
0 ~----------------------------~--------------
Displacement
Figure 7.8:
Initial stiffness method
7.9.3 Accelerat ors

The first modification to improve convergence was to use a simple approach
suggested by Cope et al<37>.
In this, the displacements due to the last
load increment are used as the first estimate for the displacements due to
the
present
load increment
as
illustrated in Figure 7 .9.

- 122-
Because the
stiffness
of
the
structures
considered
tended
to
degrade
reasonably
progressively, this meant that the first estimate was much closer than it
would have been if calculated from the initial stiffness matrix.
Thus the
number of iterations required to achieve convergence was much reduced.

This approach is particularly effective in analyses using very low or zero
tensile
strength
for
concrete
because
the
displacements
due
to
each
increment are then equal until reinforcement yielding or concrete crushing

occurs.
Although
this
modification
greatly
reduced
the
number
of
iterations
required, it was still excessive so a number of acceleration schemes were

considered.
Some were found to be very effective on some structures but
they had erratic results and prevented analyses of other structures from
converging
altogether.
procedure instead.
stiffness
optimised.
matrix
Eventually, it was decided to use a "line search"
In this, the displacement vector calculated from the

is multiplied
by
scalar
factor
and
this
factor
is
In co-ordinate geometry terminology, the stiffness matrix is
used only to obtain the search direction in "n" dimensional space and the
line search attempts to find a scalar multiplier for this vector such that
the component of the error energy in that direction is zero.
Load
~ ncr<?ments>
- - - - - Ana l ysis
- Struc ture
0 ~------------------------------------Displac ement
Figure 7. 9:
Modi f ied initial stiffness method
Since the value of this scalar can only be obtained iteratively, which
involves
calculating
all
the
element
forces
calculation would require excessive computation.
for
each
iteration,
exact
However, by using a very
slack optimisation criterion, the number of iterations or "searches" can be

- 123-
reduced.
better
In this study, the criterion used was that attempts to obtain a

scalar
factor
were
made
only
if
the
sum
of
the
error
forces
multiplied by the ... r respective iterational displacements, that is the error

energy
in
the
search
direction,
was
in
excess
of
60%
of
similar
summation performed using the error forces from the previous iteration.
The
line
required.
search
procedure
greatly
reduced
the
number
of
iterations
However, because of the computer time used in the line searches,
the effect on the time to achieve convergence was less dramatic although
still very significant.
Perhaps more importantly, the procedure means that
when the structure has failed the analytical deflections become very large.
With a pure initial stiffness scheme, failure was sometimes indicated only
by
failure
of
distinguish
the
failure
analysis
of
the
to
converge
structure
which
from
made
it
difficult
numerical problems
with
to
the
program.
Line
searches
are
used
in
most
recent
NLFEA
programs,
sometimes
combination with other more sophisticated acceleration schemes.
in
However,
for the analysis of cracking, they do have a theoretical fault which does
not appear to have been fully resolved.
When a crack first occurs, the
true displacement is greater than predicted by the stiffness matrix so a

line search factor substantially greater than one is applied to all the
displacements.
This can cause cracking in elements which were previously
uncracked and in perfect equilibrium.
The cracking leads to error forces
which are eventually reduced by the iterative solution scheme.
However,
this could be done by increasing the deformations until the force is taken
up
by
the
uncracked
reinforcement,
state.
The
rather
than
fundamental
by
returning
problem
is
the element
that
there
can
to
its
be
two
different deformation states in a section which give the same forces; one
cracked and one uncracked.
The initial stiffness method always under-
estimates displacements and so always arrives at the uncracked equilibrium

state first.
However, once a line search is included in the analysis, it is
theoretically possible
for
the analysis to predict
cracking in concrete
which has never been stressed up to its tensile strength.
In practice it
was found that this did not occur in the analysis of highly redundant slab
systems; analyses with the line search converged on the same solution as
those without.
However, it did arise in the analysis of direct tension
tests using variable tensile strength.

was not used in the analyses for 6.3.1.
- 124-
For this reason, the line search
Line searches, and other accelerators, can give displacements
<and hence
strains) within an iter at ion which exceed the final equilibrium values.
If
these strains were used in the stress history analysis, false results could
be obtained.
For example, concrete could be taken to have cracked, and
thus to have lost most of its tensile strength, as a result of strains

which only occurred in iterations which had over-shot the true solution.
To avoid this, the maximum strains are updated only after convergence has
been achieved.
7.9.4 Stiffness Recalculation

With these improvements, the convergence rate was acceptable for small
problems and for analyses for design.
However, it was still too slow to
use the program to analyse large computer models up to failure.
It was
therefore decided to depart from using the initial stiffness method and a
numerical
recalculation
program.
Ideally, this should calculate the exact tangent stiffness for
the
current
represents
of
deformation
the
the
stiffness
state
structure's
so
response
that
to
matrix
the
was
added
stiffness
small
changes
of
into
the
matrix
truly
load.
Some
analyses <98) have been performed using a "Newton-Raphson" approach,
in
which the stiffness matrix is recalculated for every load increment or

even every
iteration.
This
approach gives
much reduced number of
iterations but the computer time required to recalculate and invert the
stiffness matrix more than uses up that saved by reducing the number of
iterations.
In the analyses of cracking, the true current stiffness matrix
can also contain negative diagonal terms which would lead to numerical
instability.
For
these
reasons,
in
the
present
study
the
tangent
stiffness
was
calculated only infrequently and approximately and the concrete was always
given a significant positive stiffness; usually not less than 3% of the
full elastic value.
It appears that most studies have attempted to obtain
a closer estimate and used a lower tangent stiffness for cracked concrete.
This
is
possible
in
an
analysis
under
monotonically
increasing
loads.
However, when unloading is considered, it leads to complications since the

material
models
used
give
different
tangent
whether the strain is increasing or decreasing.
stiffnesses
according
to
Thus the exact tangent
stiffness matrix can only be calculated if the direction of change, as well

as the value, of the strain is known for all the sampling stations.
- 125-
It
proved
much
simpler
to
use
only an
approximate
calculation giving
stiffness matrix which could be used for both loading and unloading.
Having
adopted
periodic
recalculation
of
the
stiffness
matrix,
necessary to adopt a criterion to decide when to do this.

approach
is
to recalculate at
the
beginning of an
it
is
The usual
increment
if some
"current stiffness parameter" is substantially different from that implied

by the stiffness matrix currently in use.
it
However, in the present study,
was found that this approach did not work very well.
cracking occurred
in
particular
increment
the
always recalculated for the next increment.
If extensive
stiffness
matrix
was
However, if little further
cracking occurred in that increment, the use of the displacements due to

the last increment as a first estimate for the displacements due to the
. current increment meant that the analysis would converge quickly whatever
stiffness matrix was used; provided the previous increment had converged.
Thus recalculating the stiffness merely wasted time.
If, however, the
previous increment had not converged, it would have been better to make it
converge by.' recalculating the stiffness matrix earlier.
Thus it was found
more satisfactory to recalculate during the increment.
It was decided to
do this at
iteration eight
if the convergence rate was slow and the
remaining errors were significant.

The_ choice of iteration eight was a compromise between early recalculation,
which could
mean
unnecessary
recalculation,
and
delaying
recalculation
until much computer time had been used up in iterations using the old
stiffness matrix.
However, late recalculation has the advantage that the
deformation state, and hence the calculated tangent stiffness, is closer to

that in the final equilibrium state so the final convergence tends to be
faster.
The
stiffness
matrix
recalculation
improved
the
although not by as much as the line search.
rate
of
convergence
However, the greater effect
of the line search may not indicate that it is a superior method; rather it
appeared to be due to the line search having been incorporated first.
recalculations
had
much
greater effect
analyses performed without the line search.

effect of
structures,
recalculating
being
the
generally
cracking in the beams.
stiffness
greatest
the
This implies that
convergence
rate of
It was also apparent that the
matrix
where
- 126-
on
The
the
varied
greatly
softening
was
between
~ue
to
the details of the pptimum
solution scheme, such as when to recalculate the stiffness matrix, are

different for different structures.
It was thus clear that the solution
scheme adopted was not the optimum for all the structures considered and
there was certainly scope for improvement.
However, the convergence rate
achieved was considered acceptable for the project.
7.9.5 Convergence Criteria

In an iterative solution scheme, it is necessary to adopt a criterion to
decide when the solution is sufficiently accurate to stop the iterations,
without
knowing
the exact
forces,
iterational displacements
energy> can be used.

local
convergence.
solution.
or
Criteria based on out-of-balance

the
product
of
the
two
<that
is
It is also possible to consider either overall or
Analysts
tend
to
favour
overall
energy
criteria,
primarily because finite element analysis is an energy based approximation

method and there is a useful norm with which to compare the error energy;
the work done by the loads on the structure.
However, in the present
study two difficulties were experienced with energy criteria.
Firstly, as
Cope and Cope<94> have noted, the in-plane forces tend to be the last to
converge and,
flexural
since the
stiffness,
significant
in-plane
the
in-plane stiffness is large compared with the

energy
error
associated
forces
can
converged according to energy criteria.
with
these
remain
in
is
small.
analyses
which
Thus
have
Although these forces might be
considered unimportant, since eliminating them usually has little effect on

the displacements, they can represent a significant force in the critical
elements.
Thus, if local stresses are to be used as design criteria, it is
important to limit the error forces.

The second problem encountered is a peculiarity of the type of structure
considered.
The failures were local and brittle.
The energy associated
with a failure, an individual wheel load multiplied by the displacement of

the slab relative to the beams, thus represented only a small fraction,
typically
1~,
of the total work done by the loads.
The combined effect of
these problems was that there could be significant local force errors in
an analysis when the error energy was less than
0.0001~
of the work done
by the loads.
Another disadvantage of both energy and displacement criteria is that they
depend on the iterational displacements which (unlike the out-of- balance
- 127-
forces) are a
function of the solution scheme used as well as of the
displacement.
desirable
Thus,
to
use
iterational
if
the
tighter
displacements
initial
stiffness
energy
convergence
systematically
method
is
used,
criterion
under-estimate
it
is
because
the
the
true
displacement errors.
The major difficulty with force criteria is defining a norm with which to
compare
the
out-of-balance
forces.
The
standard
of
comparison
for
moments and axial force has to be different otherwise the criteria become
dimension dependent.
The out-of-balance moments could be compared with
the maximum element moment.
However, in the type of structure considered
here, this would lead to either unduly slack criteria for the slabs or
unduly severe criteria for
the beams.
Comparing
in-plane forces
with
maximum element forces is even less satisfactory because the axial force
in a slab element is obtained from the difference between similar tensile
and compressive forces.

cracking
is
not
In the early stages of a slab analysis, when the
extensive,
the
in-plane
forces
are
very
small
so
criterion based on a percentage of these forces would be unduly severe.

Conversely, in an analysis of a beam and slab deck, the axial forces in the
down-stand beams are too large to use as a standard of comparison.
Consideration of these problems led to the decision to use both an overall
energy
and
local
force
convergence
criterion.
stopped only when both criteria were satisfied.

based
on
comparison
structure whilst
avoid
dimensional
with
the
total
work
The
iterations
The energy criterion was
done
by
the
loads
on
the force tolerances were specified by the user.

problems, separate
force
were
and
moment
crite~ia
the
To
were
specified.
Despite
using
very
tight
energy criterion,
typically
0.01~,
and the
slackest force criterion considered reasonable, the in-plane force criterion

was nearly always the last to be satisfied.
7.10 CALIBRATION
Although
the program
was not
intended
to be highly accurate,
it
was
considered desirable to check it by comparison with test results and other

analyses to ensure that the results were reasonable.
In addition to the
studies mentioned in 7.3, 7.7, and also Chapter 6, as well as a. check

against
linear
grillage
to
ensure that
- 128-
the
program
was 'at
least
numerically correct,
number
of structures which
had been
tested
by
others were analysed to investigate the behaviour.
7.10.1 Duddeck's Slabs
Duddeck<116) tested a series of three square corner supported slabs under

single
central
point
loads.
These
have
been analysed
by
both
Abdel
Rahmen<87) and Cope and Cope(94) using non-linear plate finite element
programs.
Thus they enabled the program to be compared both with test
results and with more sophisticated analyses.

Two of the slabs were analysed using a four by four node quarter model.
The results for the first slab, which had 0. 7% isotropic reinforcement, are
shown in Figure 7.10 and, for comparison, Abdel Rahmen's results are also
shown.
Duddeck gave little material data so Abdel Rahmen used Mueller's
estimate <117) for the tensile strength.
In order to make the analysis
directly compara ble the author used the same figure, but it is improbably
low for the quoted compressive strength which probably explains why Abdel
Rahmen's analysis under-estimates stiffness.
study
under-estimates
stiffness
still
At
more.
low loads, the present

This
might
be
expected
because of the torsionless elements, particularly as the principal moment

direction
in
the
critical
area
is
at
45
to
the
elements,
the worst
possible direction.
Both analyses give good predictions for the failure load.
The present
study is marginally conservative but this is not significant compared with

material variability.
--
:..oad <kN ) 60
50
40
30
- - - -Test
20
10
-----Abdel Rahmen's Analys is
- Author's Analysis
10
20
30
40
50
Deflection (mm)
Figure 7. 10:
- 129-
Duddeck's second and third slabs had the same total quantity of steel as
the first but in orthotropic arrangements.
The results for the third slab,
which had 1 0% steel in one direction and 04-% in the other, are shown in
Figure 7.11.
Abdel Rahmen's results for this are far less satisfactory;
they significantly over-estimate the strength.
In contrast, the present
study gives better results for this slab than for the first.
Both these
changes are due to the fact that, as failure approached, the direction of
maximum princ ipal moments rotated towards the direction of the heavier
reinforcement.
Since the use of torsionless elements in the present study
means that the principal moments in the analysis act in this direction
from
the
outset,
the
rotation
Conversely, Abdel Rahmen's
improves
the
realism
of
the
analysis.
analysis correctly predicted that the principal
moments in the uncracked slab, and hence the initial cracks, would be at
45 to the reinforcement.
was then fixed.
rotated,
However, it assumed that the crack direction
As failure approached, and the principal moment direction
the shear retention
factor
in the analysis gave a significant
shear stress across these cracks which implies a significant tension in the
reinforcement
approximately
direction.
In
perpendicular
to
yielded, they became very wide.
fact,
the
new
cracks
formed
secondary steel and,
which
as
were
this steel
Thus the real concrete was incapable of
resisting tension in this direction so the slab was weaker than Abdel
Rahmen's analysis suggested.
Cope and Cope<94) have shown that this fault
can be avoided, either by using a shear retention factor which becomes

very
low at high smeared tensile strains or by using a rotating axis
material model.
However, Abdel Rahmen failed to identify the cause of the
error and it illustrates the danger of using this form of analysis in the
design o f even simple s labs without some calibration against tests.
45
Load <kN > 40

30
- -...;Test
20
---Author's Analysis
10
--- -- Abdel Rahmen's Analysi s

0
10
20
30
40
Deflect ion (mm)

Figure 7.11 :
- 130-
Abdel Rahmen noted that both the tests and his analysis gave failure loads
for all three slabs which were higher than predicted by yield-line theory.
He attributed this to the contribution of the tensile strength of concrete.
However, even with the tensile strength set to zero, the author's analysis
gave
failure loads which were higher than yield-line predictions.
The
reason for this is that, as in the slab strips considered in Chapter 6, the
depth of concrete in compression was substantially less than the depth to
Thus the strength was enhanced by the tensile force in the
the top steel.

top steel.
Although, with the top steel removed, the non-linear analysis gave almost
identical failure
loads to yield-line theory, it did not give the same
moment distribution.
At peak load, it predicted a moment in the element
under the load which was substantially above the yield line value; the
extra strength coming from a net compressive force on the element.
force was resisted by tension
moments.
in outer elements which resisted lesser
Thus the analysis suggested that compressive membrane action
affected the behaviour of even these unrestrained slabs.

be
This
confirmed
by
other
test
results.
For
example,
This appears to
Regan
and
Rezai-
Jorab1<118J measured strains in the reinforcement of a one-way spanning

slab
subjected
transverse
to
single
reinforcement
transverse curvature.
concentrated
indicated
that
load.
there
The
was
strains
very
in
the
significant
Thus the longitudinal curvature must have varied
significantly
over
the
slab width;
reinforcement
did
not
vary
yet
the
significantly
strain
over
in
slab
the
longitudinal
width.
The
only
possible explanation for this appears to be that the neutral axis depth
varied across the slab width because of the compressive membrane force in
the centre of the slab and the tension at the edge.
7.10.2 Taylor and Hayes' Slabs
Taylor and Ha yes (55 J tested a series of square slabs under single central
concentrated
loads.
These
enable
the
program
to
be
assessed,
by
comparison with test results, for both restrained and unrestrained slabs.
a. Unrestrained Slabs
Taylor and Hayes tested a series of slabs with two different reinforcement
percentages
<0.9~
and 1.8%) under patch loads of three different sizes.
These were all analysed using both a four by four and a five by five node
quarter
model.
A typical
load
displacement
- 131 -
relationship
is
shown
in
Figure 7 . 12.
predicted
will be seen that
It
well.
The
two
the displacements at
analyses
gave
very
similar
low loads are
deflections
but,
because it modelled the stress concentration under the load, the finer
mesh
always gave
slightly
lower
failure
load;
the
difference being
greater with the smaller load patches and the heavier r einforcement.
Load <lrN J 100
80
60
;""
/
40
Test
/
/
----
20
Ana lysis
0
6
10
Deflect ion (mro )

Figure 7. 12:
Analysis of Taylor and Hayes' slab 254
The predicted failure loads were generally lower than the actual failure
loads,
which
discrepancy
is
was
Nevertheless,
desirable
up
to
in
30%,
an
analysis
which
f or
might
design.
be
However,
considered
the
excessive.
the analysis still gave failure loads which were typically
30% higher than are implied by the elastic analyses currently used in
bridge design.
use
of
the
Thus, even though arguably excessively conservative, the
analysis
in
bridge design
would still l e ad t o significant
economies compared with current practice.

Although
the
ratio of predicted to actual failure
cons i s tent,
with
s ystematic
fault
coefficient
of
variation
of
load was reasonably

approximately
7%,
could be observed in the res ults ; the analysis under-
estimated the effect of load patch size.
It gave errors in the unsafe
direction in only one case; the heavily reinforced slab with the very small
load pat ch.
This was a rather extreme example with a load patch only
50 mm acr oss , compared with the slab thickness of 75 mm.
This, combined
with the heavy reinforcement, resulted in very high stresses round and
under the loaded area at
failure.
The calculated shear stress on the
critical section at the fa ce of the load reached 6N/mm 2

s tress under the load was 31N/mm2
analysis
which
ignores
these
and the vertical
It is thus not surprising that an
stresses
over-estimates
the
strength.
However, an alternative explanation is that since even the finer mesh gave
- 132-
elements which were over twice as wide as the load patch, the analysis had
failed to model the stress concentration round the load.
To test this,
the slab was re-analysed using a nine by nine node quarter model.
gave a significantly lower failure load, below the actual value.
This
For other
slabs, with larger loaded areas, it gave only a very slight reduction in
failure load.
It
thus appears that in order to correctly model failure
load without a separate shear check there is an additional criterion for

the size of the elements in the critical area; they should not be much
bigger than the loaded area.
This criterion would be difficult to comply
with in the analysis of complicated structures.
However, further tests
showed that a rather coarser mesh can safely be used if the concentrated
load
is applied at a single node, rather than being distributed in an
attempt to model the patch size as it was in the analysis of Taylor and
Hayes' slabs.
In contrast to Duddeck's slabs,
the failure loads for Taylor and Hayes'
unrestrained slabs were lower than predicted by yield-line theory.
They
said that this was because the slabs failed in punching shear, rather than
flexure.
However,
the
analysis
suggested
that
essentially brittle bending compression failures.

grade concrete <typically 30N/mm2
higher
steel
percentage,
the
slabs
were
than Duddeck's.
However,
Petcu
Stanculescu's<79>
ductility
theory.
failures
were
Because of the lower
compared with 43N/mm 2
reinforced
and
the
effectively
as well as the
),
far
more
heavily
they were still only just outside

requirement
for
using
yield-line
The analysis predicted, apparently correctly, that the behaviour
would be less ductile

critical
section,
under
compressive force.
than Petcu and Stanculescu assumed because the

the
load
patch,
would
be
subjected
to
net
This effect does not appear to have been considered
previously, apparently because this type of failure has been attributed to

shear.
loss
The analysis slightly over-estimated the detrimental effect of this

of
ductility
on
strength.
Other
reasons
why
the
analysis
was
conservative for these slabs, and more so than for Duddeck's, include the
under-estimate of concrete crushing strength due to ignoring the multiaxial
stress
state
<which
has
greater
effect
in
more
heavily
reinforced slab), the torsionless elements' failure to model the diagonal

hogging moments in the corners <which do not arise in a corner supported
slab) and the use of a finer element mesh relative to load patch size in
analysing Taylor and Hayes' slabs.
Despite all these faults, the analysis

- 133-
was entirely satisfactory from a design point of view and its use would
lead to significant economies compared to current practice.
b. Restrained Slabs
In addit i on t o the unrestrained slabs, Taylor and Hayes tested restrained
s labs .
They tested a series to match the unrestrained set plus a set of
othe rwis e s imilar unre inforced slabs .
These were also analysed and t ypical
results are s hown i n Figure 7 .1 3.

:..oad ( kN ) 140
120
100
80
I/~
60
40
--
Test
- - - - Analys is <full restraint )

- - - --Analysis ( in plane r estraint onl y )
20
0
10
Deflection (mm)
Fig ure 7.13:
In
the tests,
the
Analysis of Taylor and Hayes ' s lab 2R4
restraint greatly
increased the
init ial analys i s predic ted a much smaller effec t .
failure
loads.
The
This appe ars to be due
t o differences between the real and assumed restraint conditions.
Taylor
and Hayes used a steel frame to provide the restraint and the slab was
ins erted j ust prior to the test, the gaps being packed out with mortar.
This
was
apparently
intended
to
give
full
in- plane
restraint
with
negligible rotational restraint so these restraint conditions were used in

the analysis.
In fact, there clearly was significant rotational restraint;
hence the greater than predicted increase in strength.
However, when full
rotational restraint was used in the analysis, it over- estimated strength.

It appears that the steel frame gave partial restraint to both in-plane
and rotational movement.
This could not be predicted satisfactorily by the
analys is.
- 134-
7.10.3 Batchelor Md Tissington's Speciaens
The analysis of Taylor and Hayes' slabs confirmed, as had been found in
designing the specimens for Chapter 6, that it is difficult to produce
known restraint conditions artificially.
not
It therefore appeared that it was
possible to check the analysis of compressive membrane action for
simple
laboratory
specimens
before
complicated bridge structures.
going
However,
on
to
use
it
to
analyse
Batchelor and Tissington have
tested a series of simple bridge models with only two beams each and
these provided a useful intermediate case.
They also had the advantage of
having been analysed by Cope and Edwards(99) so they enabled the analysis
to be compared with a plate type finite element program.
Ba t chelor and Tissington's largest specimen is illustrated in Figure 7.14
and the load-deflection response under a single central load is illustrated
in Figure 7.15.
It will be seen that Cope and Edwards' analysis gives good
results whilst the author's, a s expected and intended, is conservative.

0 )J"I. osotropoc
r eonforcement
IDI~Dilf~.,..
I
L060
~~~-J'\ As=920mml
fy : )10N/ mm 1
u
Fi gure 7. 14:
Ba t chelor and Tissington's specimen
The analysis also gave good
predictions
for
the cracking response. It
predicted cracking due to hogging moments along the edge of the slab,
where these moments are resisted only by torsion in the beams.
It also
predic ted, as observed in the tests, that just before peak load was reached
the main beams would crack right through under the restraint forces.
Unlike Taylor and Hayes' slabs Batchelor and Tissington's, with their large
span to depth ratio, reached deflections which were significant compared
with
their
thickness
before
failing .
Thus
the
displacements
had
significant effect on the lever arm at which the restraint force acted.
However, in order to make the analysis directly comparable with Cope and
Edwards', the correction for this, which was described in 7.6, was not used
in the analysis shown in Figure 7 .15.
The analysis was, however, repeated
- 135 -
with the correct ion.
This increased the deflection at a load of 20kN by
only 1% and at 30kN by 6%.

25%.
However, it r educed the failure load by some
This implies that both the author's and Cope and Edwards' analyses
under-estimated the basic static strength of the slab.
This may have been
due partly to under-estimating the material strengths since Batchelor and

Tissington gave little data for this; for example, they gave no indication
as to whether the reinforcement strain hardened.
Lo ad <kN )
80
60
4- 0
- - - - Test
Author's Analysis
20
- - - - Cope & Edwards ' Ana l ysis
10
15
20
Deflec t ion (nun )

Figure 7. 15:
Analysis of Bat chelor and Tissington 's Specimen
7 .10.4. Kirkpatrick's Model

Kirkpatrick's bridge model, which was considered in Chapter 3, enabled the
program's prediction of local bridge deck slab behaviour to be compared
both with test results and with Cope and Edwards' analysis.
A major problem with the analysis of this type of structure is the size of
the computer model required.
In order to model local effects safely the
element mesh has to be fine.
In order to obtain the correct restraint the
whole of the bridge should be represented, even in an analysis for local

effects.
which
To reduce the computer time required,
restricted
it
to
the
analysis
of
a half model was used
symmetrical
load
cases.
The
computer model was also banded so that only the half of the loaded slab
span was modelled with a fine enough mesh to represent local behaviour.
Despite this, a model with 288 nodes was required.
The banding, combined
with the variety of reinforcement areas and different slab spans used by
Kirkpatrick, meant that the model required 53 element types.
The result of the analysis of bay C2, which had 0.5% reinforcement, is
shown
in
Figure 7. 16.
The analysis gave

- 136-
conservative
predictions for
deflection and strength.
It was slightly more conservative than Cope and
Edwards', although this was partly due to including the effect of large
displacements.
The analysis of bay A2, which had 1. 7% reinforcement, also
gave conservative deflection predictions but over-estimated failure load by

some 10%.
Although a 10% error would normally be considered acceptable,
as it is small compared with the variability of concrete behaviour, in

combination
program
with
the 20% under-estimate for C2
over-estimated
the
effect
of
it
indicated that
increasing
the
the
reinforcement
However, a study of the results revealed that much of the
percentage.
increase was due to the contribution of the top steel, not only in tension
over the beams but also in compression under the load.
The latter is
unusual; in a thin bridge deck slab the cover required usually means that
the steel on the compression face is too near the neutral axis to make a
sig nificant contribution.
Load CkN ) 100
--
80
60
40
- -- - -:rest
Author's Analysis
20
- -- - Cope & Edwards' Analysis
Deflec tion <mm >

Figure 7. 16:
The
steel
in
Analysis of Kirkpatrick's panel C2
Kirkpatrick's
model
was
given
only
6mm
cover
which
is
equivalent to 18mm at full size; approximately half the cover required by

BS 54-00.
Kirkpatrick was careful to maintain and check the bottom cover
but attributed less significance to the top cover.
Thus the true top
cover is uncertain and the effect of the top steel is very sensitive to
its position.
The steel would need to drop only a few millimetres to
explain the discrepancy.
Another effect of having significant compression
steel is that, unlike the other analyses, the analysis of this bay was
- 137-
sensitive
to
the assumption
compressive strains.
made
for
the stress
continued
to
reinforcement.
increase
as
compressive
However, in this bay the

force
relatively low strain.

effect
of
analyses.
transferred
to
the
Thus the unsafe prediction could have been avoided by
using a concrete model which gave an abrupt
the
high
Other analyses failed soon after the. compressive
stress in the critical region started to reduce.

load
in concrete at
reduction in stress at a
This may explain why the tendency to exaggerate
reinforcement
was
less
pronounced
The greater ductility given
by the
in Cope and Edwards'

relationship shown
in
Figure 7.6 may be more representative of the behaviour of concrete loaded

uniaxially under displacement control.
However, the critical concrete in
these slabs was subjected to biaxial compression and also to shear which
reduced its ductility.
slab,
to
impose
compressive stress.
Thus it may be prudent, in the analysis of such a

limit
on
the
strain
at
which
concrete
can
carry
A limit of approximately 0.0045, which is still higher
than used in most analyses, would eliminate the unsafe predict ion for this
bay but have little effect on any of the other analyses.
that
However, given
this bay had more effective compression reinforcement and as much
tensile reinforcement as any practical bridge deck. slab, the analysis can
be considered safe for practical slabs despite over-estimating the effect
of steel.
It also appears, once again, that a flexural analys-is has proved
capable of predicting a "punching shear" failure.
This suggests that the
failures were primarily brittle bending compression failures, although the

shear force did precipitate the final collapse.
Although
the reasonably good predict ions for the failure
load of these
slabs are reassuring, they have little practical significance.
Kirkpatrick
acknowledged that design should be controlled by serviceability criteria.

Applying conventional 85 5400 stress criteria to the element forces from
the
analysis
of
approximately 22kN.
bay
C2
suggested
an
allowable
service
load
of
This compares with a design service 45 unit HB wheel
which, at this scale, is 13.75kN.
This is interesting as the reinforcement
in this bay was similar to Kirkpatrick's eventual recommendation.

analysis has given further support to Kirkpatrick's proposals.
Thus the
However, it
remains to consider the influence of global transverse moments.

From Kirkpatrick's observations of the behaviour of his model, and of his
full scale bridge,
it
would appear
that,
in the absence of any global
effects, the behaviour of this bay would certainly be satisfactory under a

- 138-.
service
load
significantly above
22kN.
analysis was unduly conservative.

it
should
be
noted
that
Thus
it
may appear
that
the
However, to put this into perspective,
conventional
analysis
of
this
bay,
using
Westergaard and BS 5400, gives an allowable service load of only 9kN and
implies a failure load of less than 14kN.
7.11 cor:L.USIONS
The
form
gives
satisfactory
predictions for the behaviour of realistic slab structures.
Provided an
element
of
mesh
concentrations
analysis
is
used
around
considered
which
the
is
applied
in
this
fine
chapter
enough
loads,
it
to
model
appears
local
to
stress
give
'~
safe
predictions for the failure loads even of slabs which fail in "punching
shear".
In some cases the analysis under-estimated strengths by up to
30~.
The
allowable service loads calculated from the analysis also appear to be

conservative.
However, despite this, the use of the program in design and
assessment would still lead to significant economies compared with current

practice.
- 139-
CHAPTER
MODEL
BRIDGE
TESTS
8.1 INTRODUCTION
The analytical methods considered in Chapters 5 and 7 are potentially very
useful, but they have not yet reached the point where they can justify
radical changes in design practice without some calibration against tests.
It was therefore necessary to perform some tests.
These were designed to
investigate the key areas identified in Chapter 3 as requiring further

research;
service
load
behaviour,
restraint
and
the
effect
of
global
moments.
8.2 DESIGN OF MODEl.S
8.2.1 Scheme
Although small scale models have proved successful for
predicting the
strength of slabs <51>, the cracking behaviour of concrete does not scale
well.
Thus,
in order
to obtain
reliable
predictions of
service
behaviour, it is desirable to use the biggest practical scale.

full size models would be used.
load
Ideally,
However, financial constraints on this
project, combined with the need to model a whole bridge and a whole HB
load, made this impractical.
It was therefore decided to use half scale
models of relatively small M beam type bridges.

these were
Analysis suggested that
the type of structures in which global transverse moments
would be most significant.

The first model, which is detailed in Figure 8.1 and illustrated in Figure
8.2, was designed to be a worst case for restraint so it had four beams
<the minimum practical number for a bridge of this type>, no parapet upstands and no diaphragms.
The last point is particularly significant since
3.2.8 noted that previous researchers have said that diaphragms are needed
to
provide
the restraint,
yet
no tests have
without diaphragms to confirm this.
been performed on decks
Also, analysis using the program
described in Chapter 7 suggested not only'that diaphragms were not needed

to provide the restraint but also that, because of the effect illustrated
in Figure 3.12 and discussed in 3.2.7, the slab near the ends of the bridge
would be subjected to transverse compression when the full HB load was
applied near mid-span.
- 140-
I
250 I
6000
250
ELEVATION
500
1000
10 00
1000
500
SECTION
Figure 8.1:
Details of first deck
Figure 8.2:
First deck under test
To give a worst case for local effects, the maximum practical beam spacing
was used with the standard slab thickness,
160mm at
full size.
The
spacing of the beams was li.mited by their shear strength under the design
~5
unit HB vehi.cle and the slab's span to depth ratio, although greater
than normal for this type of deck, was qui.te modest at 12.5.
However,
analysis suggested that larger, wider spaced beams would be a less severe
test because of the smaller global transverse moments.
As these moments
were a key area requiring investigation, it was considered better to use a

deck which was a severe case for these.
- 141 -
c
L
ELEVATION
50
, .... 225
1000
-(-- --.....
~-
..........
8000
400
1000
1000
SECTION
Figure 8 .3:
De tails of second d eck
1000
1!.
250
1 150
I
400
The second model, which is detailed in Figur e 8 .3 and illustrated in Figure

8 .4, was designed to be more typical of current practice so it had parapet
upstands, support diaphragms and an extra beam.
The same beam spacing
was used as for the first deck and the overall width approximated, at full
size,
to
that
shoulders.
of
two
lane
bridge
for
footways
nor
hard
It thus represented the narrowest bridge which is likely to be
designed for 45 units of HB load.

case
with neither
restraint,
analysis
Although a narrow bridge is the worst
suggested
that
would have been greater in a wider deck.
global
transverse
moments
However, it was considered that
the behaviour of a wider deck could safely be predicted with the aid of
the results of tests on the deck and the program described in Chapter 7 .
It was not possible to test a wider deck in the laboratory.
Figure 8.4:
Second deck under test
After the two models had been tested, a single beam with the appropriate
width of slab was tested on its own to help calibrate the analysis.
8.2.2 Beaas
Since the slab behaviour was the main concern of the project,
modelling of the beams was not required.
perfect
However, pre- tensioned beams
were used so that the global behaviour was reasonably similar to that of
the prototype bridge.
half scale models of
Standard inverted T beams were used as approximate

M beams.
These had the s ame advantage in the
research project which they have in practice; the multiple use of formwork
makes them much cheaper than specials.
- 143-
In Figure 8.5 the section of the inverted T beam is compared with a true
half scale M beam.
The thicker web of the inverted T beam was considered
an advantage as it was desirable to avoid shear failures.
However, the
lack of rebates for the slab formwork was a disadvantage, not only because
they were needed to support the formwork, but also because their absence
improved the support to the slab.
Thus non-standard rebates were provided
as shown in Figure 8.5.
25
~L
0
n
'
.. ''
103
"'
100
!9
0
Ll"l
40
53
0
0
0
Ll"l
,------
0
0
0
~
......,
C)
C)
Ll"l
N
C)
<X)
'\.
'
248
Figure 8.5:
When
243
T2
M4
<full size>
<half size)
Comparison of full size T2 and half size M4 beams
the design of the beams was fixed,
which was very early in the
project, it was considered desirable to avoid global failures so the beams

for the first deck were provided with approximately 25% more prestress
than the conventional BS 5400 based design method required.
The beams
for the second deck were provided with the same prestress which, because
of the improvement in distribution properties due to the diaphragm, meant
they had approximately 35% more steel than BS 5400 would have required.
Because of the interest in the interaction of global and local effects.
under service loads, it was desirable to provide a realistic beam size near
- 144-
the minimum which could be used, within the code, for this type of deck.
Thus
the beam size was not
increased
to match
over~provision
the
of
prestress so the beams were stressed at transfer to a higher stress than

would normally be allowed.
The
shear
reinforcement
was
designed
to
the
normal
BS
5400
rules.
Because Hughes(1!9) has found that these are conservative for this type of
beam,
the
shear
reinforcement
was
not
increased
to
match
the
over-
provision of prestress.
The beams
for
the second deck were provided with standard transverse
holes to accommodate reinforcement for the diaphragms.
In order to get
the diaphragms down to the correct scale size, the holes had to be nearer
to the end than is recommended by Green<120>, so extra links were provided
to control the expected cracking.
6.2.3 Diaphragms
The diaphragms
for
the second deck were designed to the conventional
BS 5400 rules.
However, a considerable variety of approaches are used for
calculating the torsional inertia used in the analysis to obtain the design
moments.
This significantly affected the design.
It was decided to follow
the recommendations of Clark and West <121> and use half the Saint Venant
value for the gross-concrete sect ion.
6.2.4 Slab Re:lnforcement
Because
global
investigation,
reinforcement
behaviour
it
was
and
restraint
considered
were
that
using
percentages was undesirable.
major
areas , requiring
bays
with
different
The more heavily reinforced
bays would have provided extra restraint and distribution which would have
given an optimistic impression of the behaviour of the lightly reinforced
bays.
This meant the choice of steel area was important.
The original idea was to provide the first deck with 6mm high tensUe
steel bars
secondary
at
steel
lOOmm
in
centres
both
<that
faces.
is T6-100) main
This
compares
steel and T6-125

with
Kirkpatrick's
recommendations <13) which are equivalent to T6-75 at this scale.
However,
later analysis suggested that even .T6-100 was slightly excessive and it
was decided to reduce the ma:ln steel to T6-125 as well.
deck
was
20%
less
heavily
reinforced
- 145-
than the- 'strips
This meant the.

considered
in
Chapter 7.
However, since doubling the reinforcement appeared to have had
little effect on the tension stiffening, this was unimportant.

that the secondary steel could also have been reduced.
reinforcement
125mm
was
was
the
considered
largest
undesirable
spacing
which
for
It appeared
However, smaller
practical reasons
complied
with
the
code
whilst
maximum
<300mm at full size) and which kept the reinforcement spacing in phase
with the beam spacing.
The reinforcement is detailed in Figure 8.6.
The reason for providing an
8mm longitudinal bar over each beam was to provide a proper anchorage for
the 8mm links projecting from the beams.
Since these links stopped short
of the edge of the top flange of the beam <as is usual because they have
to fit inside the formwork when the beams are cast) it was considered that
they would not greatly affect the slab's flexural behaviour and they were
ignored in its analysis.
i,.TB-01-IOOOT
l7 T6 -03- 115'
l7T60U08-11S'ff
fA
63T6 -03 -I 25'
-- -- -
1-1-
- -. k- 1-
1'-
- - - - - - - - - --
63T6-0I-/15' Er
2Th-(JL-IOOF f
- "" - j;- - -
- - - - - - - - -
--
m.
-
2TI0-05 -25EF
Quarter Plan
Ol C? Ol
le
og
07 Ol
oe
02 08 Ol
?" 1
08
Ol
02
07 01 09
I i
A-A
01
~!
01
01
r
\
01
01.
01.
; ; ; ~ ; ~ J.tm.
01.
01.
Ol
Figure 8.6:
Detail of reinforcement in slab of .first deck
- 146-
O!i 05
O!i 05
The real bridge would have to support wheel loads right up to the end of
the deck, where there were no diaphragms and where membrane enhancement
would be reduced.
It was therefore assumed that extra reinforcement would
be required.
It was also considered desirable to loop the secondary steel
round
bars
these
to
provide
the
correct
detailing
for
free
edge.
Because of the very thin slab the resulting detail, which is shown in
Figures 8 .6 and 8 .7, was slightly awkward.
Figure 8 . 7:
Reinforcement in corner of first deck
It was clear that the diaphragms in the second deck would make it stronger
than the first.
Since the first slab had behaved well, it was decided to
reduce the steel area in the second.
The opportunity was taken to look
into the possibility of using only one layer of steel each way.
This has
advantages for durability, since it greatly increases the cover, and it also
halves the steel fixing cost.
It had been rejected by Beal<122) but it
seemed probable that the thinner slab in the type of deck considered here
would make it more viable.
The main steel was increased to T8- 125 giving nearly 90% of the total
steel area in this direction used in the first deck.
However, since both
the first test and the analysis suggested that the secondary steel would
be very lightly stressed, this was not increased and just one layer of T6125 was used.
One effect of diaphragms is to apply a support moment to the most heavily
stressed beam.
This relatively small moment is
frequently
ignored in
design and the steel in the deck slab would normally be ample to resist
it.
However, the longitudinal reinforcement in this deck was so light that

- 147-
it was necessary to provide some designed reinforcement to resist this

moment.
It was also decided to use the second deck to conduct a small test, which
has
been
described
briefly
elsewhere<l23),
reinforcement corrosion on deck slabs.
on
the
effect
of
local
To simulate the effect of severe
local corrosion, eight adjacent main slab bars were cut right through at
mid-span of the slab.
slab was cast.
This was done using using bolt croppers before the
The position of the cuts was chosen so that the "damage"
would have the minimum effect on the behaviour under the load case used
for the initial failure test.
However, it would be possible to conduct a
service load test with one wheel of the HB vehicle immediately over the
cut bars.
It was also hoped to perform a
failure test using a single
wheel over
the area if it was in reasonably good condition after the
failure test.
8.2.5 Bearings
The beams were supported on normal commercial laminated bearings which
were PSC"370132"<124>; the smallest size of this type made.
These had a
greater movement capacity than a single span bridge of this type would
require.
As a result, they were less stiff than a true half scale model of
bearings for a single span bridge.
Their behaviour was close to that of
the bearings which would be required for a two-span bridge.

The stiffnesses of the bearings were checked in the Mayes machine, first
under concentric loading, which gave results very similar to the specified
stiffness, then under eccentric loading to measure the flexural stiffness.
8.3 MATERIALS
8.3.1 Concrete
The mixes used for the deck slabs and for the other in situ concrete were
similar to those used in the half scale beam strips considered in Chapter
6 and the nominal mixes are detailed in Table 8.1.
The mix for the first
deck used a realistic cement content but even with a high water content,
giving a very wet-looking mix with a slump of some lOOmm, it gave a 28
day cube strength of 44N/mm 2
model.
obtained from 150mm cubes stored with the
The cement content was reduced for the second mix givihg a .28 day
strength of 33N/mm2 with a lower water content and a more typical slump
of around 40mm.
The change in properties between the two mixes was much

- 148-
greater than the change in the nominal mix proportions would s uggest .
However, this was a consequence of the long delay between the tests <which
meant that both the cement and the aggregate came from different batches)
and
the
use of a
normal
commercial type
batching
plant
without
the
advantage of using dried aggregate as in smaller scale tests.
Quantity <per nominal
Material
First Deck
113 )
Second Deck
5-10mm Thames Valley Gravel
875kg
905kg
Sand
900kg
930kg
Ordinary Portland Cement
300kg
275kg
et1601
=1651
Water
Table 8.1:
Mixes for in situ concrete
Similar control specimens were used as for the beam strips considered in
Chapter 6 .
In addition, two sets of six 150mm cubes were tested, one
cured in a tank in accordance with BS 1881 <125> and the other cured with
the specimens.
The second deck used 12 batches and a pair of cubes, one
for each of these sets, was taken from every other batch.
results are shown in Tables 8.2 to 8.4-.
The test
The BS cured 150mm cubes gave
higher crushing stresses than the dry cured cubes of either 70 or 150mm
size showing that curing had a greater effect than size.
Because the beams were 500mm deep, compared with only 80mm for the slab,
and
because
the
precise
reproduction
of
their
behaviour
was
less
important, it was considered acceptable to use 20mm aggregate for these so

a normal commercial mix design was used.
This is detailed in Table 8.5,
and the results for the control specimens are given in Tables 8.6 and 8. 7.
Although the beams were cast in four separate pours, there was so little
difference between the test results for the different pours they have all
been considered together in the tables.
Although it was
nominally a
50N/mm2 mix, the actual strengths were much higher with a characteristic
strength of over 65N/mm2
This arose in part from the specified minimum
cement content and in part from the mix being designed to achieve transfer
strength, 40N/mm 2 , in the minimum time.
- 149-
Age
Test
(days>
28
Size
Number
(N/mn,2
<mm>
Veriation
Mean
Cube
150
'3
43.8
Cube
150
53.2
150
46.0
70
48.2
Indirect
1500
3.66
tension
1000
3.42
500
3.66
1500
27000
150
49.8
70
45.3
150
60.3
Indirect
1500
3.66
tension
1000
3 .20
500
3 .76
1500
3.92
1500
29000
(~)
(wet cured)
61
Cube
<start of
tests)
Elastic
modulus
90
Cube
<end of
5 .0
tests)
Cube
1.6
<wet cu.r ed)
Indirect
tension
<wet cured>
Elastic
modulus
Table 8.2:
Test results for slab concrete from first deck
- 150 -
Age
Test
<days)
28
Number
Cube
Variation
Me1jn
(N/m[l} 2
<mm)
(%)
150
33.4
70
33.0
150
35.1
4.5
Indirect
1500
2.52
14.8
tension
500
3.42
Elastic
1500
28500
150
36.2
70
33.5
Indirect
1500
3.01
tension
1000
3.20
Elastic
1500
24400
<start of
tes ts)
Size
5 .2
f---
Cube
<wet cured)
modulus
43
Cube
<end of
test>
modulus
Table 8.3:
Test results for slab concrete from second deck
- 151 -
Element
Test
Size
Number
(N/mm2
<mm)
Diaphragm
Mean
Cube
100
34.2
end of deck
Elastic
1500
23100
in figures>
modulus
Diaphragm
Cube
100
37.2
Cube
100
33.3
Elastic
1500
23500
<right-hand
<left-hand end)
Parapet
modulus
Table 8.4:
Test results for other in situ concrete from second deck
<all tested at end of test; approximately 40 days old)
Material
Quantity
<per nominal m3)
10- 20mm Crushed Limestone
819kg
5-lOmm Crushed limestone
352kg
Fines; Crushed Limestone
629kg
Rapid Hardening Portland Cement

P2 additive
400kg
Water
1. 121
::!1701
Table 8.5:
Mix for precast concrete
- 152-
Age
Test
Size
Number
<Nimm2
<mm>
(days>
Variation
Mean
)
(%)
2-5
<transfer)
Cube
100
43.6
6.0
28
Cube
<wet cured)
100
12
71.6
2.6
180+
Cube
150
70.1
Indirect
1500
4.08
1500
35500
Cube
150
71.5
Indirect
1500
3.92
1500
37900
<start of
tests)
tension
Elastic
modulus
210+
2.2
<end of
tests>
tension
Elastic
modulus
Table 8.6:
Test results for precast concrete from first deck
- 153 -
Age
Test
Size
<days)
Number
(N/mm2
<mm>
2- 4
Mean
Variation
)
<r.)
Cube
100
43.6
6.0
Cube
100
12
71.6
2.6
150
72.4
2. 4
1500
3.87
1500
39500
<transfer)
28
(wet cured)
320+
Cube
<end of
tests)
Indirect
tension
Elastic
modulus
Table 8. 7:
Test results for precast concrete from second deck
8.3.2 Reinforcement
The reinforcement for the deck slabs was GKN Tor- Bar in 6, 8 and lOmm
sizes.
The first of these sizes is no longer available commercially and
sufficient steel was in stock for the main steel of the first deck only.
For the remaining 6mm steel, all secondary steel, hard drawn wire was
used.
In order to give reasonably similar bond characteristics to the
normal steel, this was specially indented.

Stress-strain
curves
for
all
the steel were obtained using
the Mayes
testing machine and these are shown in Figure 8.8 in which each line
represents the average of three test results.
The hard drawn wire had a
significantly higher yield stress than the equivalent Tor-Bar.
Although it
appeared that this would have little effect on the behaviour of the slab
because the secondary steel was lowly stressed, the different properties
were modelled in the computer analyses.
- 154-
Stress
600
(JI/JDJ!i2)
500
400
- - - - 6mm Indented bArd drawn wire
300
<ultimate stress
200
- - - - 8mm Tor-Bar
<622)
6mm Tor-Bar
<594)
= 6511/~)
100
0
Figure 8 .8 :
0.6
0.4
0.2
Stress-strain relationship for reinforcement
8.3.3 Prestressing
The prestressing was provided by 12.7mm Bridon Dyform strand stressed up
to 70% of characteristic strength at transfer.
It was intended to take a
stress- strain curve for this using exactly the same procedure as for the
reinforcement.
However, two problems were experienced with this.
Firstly
the steel was too hard for the points on the clip-on 50mm gauge length
strain gauge.
It was
Plastic Padding as in
therefore necessary to attach demec points with

the concrete tests.
machine caused premature failures.
Secondly,
the jaws on the
This problem could have been solved by
using normal commercial wedge anchors but these would not fit in the jaws
of the testing machine.
properties
stress
at
for
1~
the
It was therefore decided to obtain the material
computer analysis
elongation
manufacturer's certificates.
and
the
from
the 0.2% proof stress,
ultimate
strength
given
on
the
the
The results obtained in the laboratory were
used only for the elastic modulus, E,..
In the event, this was the only
important property since the structures failed before the steel was fully
stressed.
8.4 CONSTROCTION
The
beams
were cast
commercial manner.
by Costain Concrete, South Wales,
in
the normal
Demec points were attached to the centre section of

- 155-
the beams and were read before and after stressing, as well as at the
start of the tests, to enable prestress losses to be estimated.
The beams
were transported to the laboratory, stored outside until required, and

placed on
the bearings
in
accordance with
the bearing manufacturer's
instructions <124).
The slabs were cast on plywood formwork supported off the beams.
Thus
the stresses due to the normal unpropped construction were reproduced but,
because of the lack of deadweight compensation, they were under-estimated
by a factor of two compared with a full size bridge.
In the case of the second deck, the diaphragms were poured first, then the
slab and finally the parapet up-stands.
In both cases, the deck slab was
cast in one pour and this required some 12 batches of concrete, slightly
more than would be used in a real deck where the batches would normally
be 6m3 truck mixer loads.
The concrete was placed by skip and Figure 8. 9
shows the first deck under construction.
Figure 8.9:
First deck under construction
It was considered very important
to give the slab an even and correct
thickness since analysis suggested that the local strength would be very
sensitive to this.
Two spare beams of each type were cast and those used
were selected for equal camber.
In the event the cambers of the beams,
although greater than normal due to the high prestress, were unusually
equal and this precaution was unnecessary.
The formwork was adjusted to
give as flat a soffit as possible and the concrete was finished using a
- 156-
screeding rail which spanned the full width of the deck.
After completion
of the tests, the thickness of the slab was checked by drilling a number
of holes and measuring through.
measured
on
the
second
deck
The mean thickness of the
was
79.65mm
and
although
17 depths
there
was
significant variation, from 76 to 84mm <which was a considerably greater

variation
than
was
observed
in
the
first
deck),
this
appeared
to
be
entirely random with the mean depths for four bays being 80.0, 80.5, 79.8
and 78.3mm.
It was therefore decided to base the analysis on the nominal
dimensions.
The
top of
the
concrete
was
covered
in
plastic
for
seven days
then
uncovered whilst the soffit formwork was struck after a minimum of four
days.
Real
bridge decks of
this
type are normally
constructed using
permanent formwork but access to the soffit was required to enable the
cracking to be observed and the surface strain gauges to be attached.
For
the same reason, neither water-proofing nor surfacing were provided.
This
made the tests conservative and Cairns<82> has found that surfacing alone
reduces the live-load steel stress by some 30%.
No
attempt
was
made
to
match
experienced in a real bridge.
the
curing
conditions
which
would
be
The lack of permanent formwork or water-
proofing, the small scale, the unusually wet concrete <particularly in the
first deck) and the dry laboratory air all had the effect of increasing the
shrinkage of the slab whilst the beams were rather older than usual when
they were placed.
Thus the shrinkage of the slab, and the differential
shrinkage between the slab and the beams, was significantly greater than
in a real bridge.
Because of this, if
<as has been suggested) shrinkage
has an adverse effect on the development of membrane action, the test

results would be conservative.
8.5 LOADING
8.5.1 Loads Applied
Since the slab behaviour was of prime concern, and since analysis indicated
that HA load would have a relieving effect on the slab whilst dead weight
would have an insignificant effect, only the HB load was applied with no
HA load or dead weight compensation.
These loads would have increased the
moments in the beams, thus the degree of over-strength in the beams was
slightly greater than that due to the over-provision of prestress.
- 157-
The loading sequence was designed to first apply the design service HB
load in a critical position, then to simulate the full load history due to
the service life of a real bridge before returning the load to its original
position.
The service load would then be re-applied, enabling the effect
of cracking and loss of restraint due to other load cases to be assessed.

The load on the HB rig would then be increased until failure occurred.
Whilst the design static service load which should be applied to the deck
was well established,
simulate
the
and defined
service
life
of
in BS 5400,
bridge
Part 10 <126) defines fatigue loads.
the loading required to
was
less
clear.
BS
5400
However, these are intended for use
with defined fatigue relationships for steelwork details whilst the primary
concern in this project was the cracking behaviour of the concrete.
This
is much more sensitive to small numbers of large load cycles, as has been
found in Chapter 6.
Thus if the fatigue loads had been used, they would
have been used well outside the range for which they were intended or
calibrated.
When relatively small numbers of cycles are considered, the
design fatigue loads can be locally more severe than the design ultimate
load.
small
This does not matter in normal fatigue assessment, since these

numbers
calculations.
of
cycles
have
little effect
on
the
cumulative
damage
However, it is clearly illogical to require a structure to
resist a thousand cycles of a load in excess of design ultimate.
Since
bridge deck slabs are likely to be most sensitive to the few loads of near
design service level which are applied in their life, it was decided to
base the cyclic loads on BS 5400: Part 2 loads.
Unlike the long span HA loading, the HB loading and the short span HA
loading, which are relevant to these decks, have no statistical base<127>.
It
was therefore necessary to make some gross assumptions in order to
decide how many cycles, and of what magnitude, to apply.

assumed that
It was initially
the design service loads should have the same chance of
occurrence as their
long span HA equivalents;
occurring once in 120 years< 12 7 ).

loading should be applied.
that
is a 5% chance of
This implied that only one cycle of this
However, it was decided to apply a more severe
sequence to ensure that the tests would be conservative.

Another difficulty with simulating the load history of a bridge was that
real bridges are subjected to rolling loads whilst the loading rig was only
able to apply pulsating loads at di?crete positions.
- 158-
In order to ensure
that this would be at least as severe as applying the intended load at all
positions along the length of the deck, the test load . was increased.
load of 1.2 times design service load was therefore applied to all the
positions.
The original intention was to apply two cycles of this load
followed by 10000 cycles of a reduced load, simulating 25 units of HB

<again with a 20% excess) then 100 cycles of design service HB.
The
significance of the 25 unit HB load is that it was used to represent HA in

the then current loading standards <23,24-).
Finally, a cycle of 1.2X design
service load . would be applied, enabling the effect of the cyclic loads to
be assessed by comparing the behaviour then with that under first loading.
In the event, the 10000 cycles had very little effect so, after the first
position, the number applied was reduced to 5000.
critical
parts
of
the
slab
were
subjected
to
However, because the
wheel
loads
under
two
different load positions, these were subjected to at least 10000 cycles of

wheel loads.
The maximum load applied in the service load tests was approximately equal
to the design ultimate load.
<which
This, combined with the nature of HB load
is particularly severe
for
this type of structure and probably
unrealistic), the lack of surfacing and the large number of load cycles
applied, meant that the load history to which the bridges were subjected
was
excessively
severe
and
made
the
tests
conservative,
as
intended.
However, Perdikaris and Beim's work<128>, which was published after these
tests were completed, suggests that rolling loads are more severe than
fixed pulsating loads.
They suggested that one passage of a rolling load
could have the same effect on the fatigue life of a slab as 34- to 1800
cycles of a fixed load.
As they considered the number of cycles to 60% of
static strength to cause failure, whilst the tests considered here are
investigating the effect of cycles of service load level, their conclusions
may not be applicable here.
be
related
to
the
crack
Also, the difference they observed appeared to

patterns;
pulsating
loads
gave
local
patterns whilst rolling loads gave extensive grid-iron patterns.

a consequence of the use of large single wheel loads.
radial
This was
Under the HB
service loads used in the author's tests, the cracking extended over a
greater length of the bridge but was purely longitudinal.
no
reason
to
anticipate
that
rolling
fundamentally different crack pattern.
loads
would
There is thus
have
led
to
However, even if <as Perdikaris and
Beim suggested for this type of reinforcement> one pass of a rolling load
was equivalent to 34 cycles of a static load, the use of 20% over-load
- 159-
meant that the load history used was still conservative.

of
1.2X design service
load had more effect
than
One application
100 applications of
design service load .
8.5.2 Loading Rig

The
16
wheels of the HB load were loaded by
four one-hundred tonne
hydraulic jacks acting through spreader beam systems which are illustrated
in Figure 8 . 10.
The four jacks were interconnected and connected to a
hydraulic pump system which enabled cyclic loads to be applied.
This
s ystem was rated at only 40% of the hydraulic pressure for which the
jacks were designed.
applied, equivalent
Although this enabled a load of 400kN per jack to be

to 2. 7 times the design ultimate load, calculations
suggested that a slightly higher load would be required to fail the deck.
Separate hand pumps were therefore provided for the final failure test.
Figure 8.10:
Spreader beam assembly
- 160-
The jacks reacted against
two large steel universal beams which were
supported by double channel stanchions bolted down to the strong floor of

the laboratory.
The loading frames can be seen in Figure 8.2.
Because the
anticipated loads were close to the calculated capacity of the floor, it

was necessary to position the bridge to minimise the moments in the floor
under the load case which would be used for the failure test.
It was also
desirable to spread the load on each leg of the frame evenly between four
floor bolts.
Since the standard spacing of the HB bogies did not match up
with the bolt centres, it was only possible to achieve this for the legs of
one of
the
bogies.
The
load
from
each of
the other
two
legs was
therefore spread unevenly amongst six bolts.

The HB bogies could easily be moved sideways to any required position by
moving the spreader beams and jacks.
However, to move them longitudinally
it was necessary to move the whole loading frame.
It could have been
moved to any position but this would have required a re-arrangement of

the anchorage system.
In practice it proved adequate to move the bogies
only by multiples of the bolt spacing.

8.6 INSTRUMENTATION
The loads were measured using four 800kN load cells located below the
jacks.
Separate figures were recorded for the four cells but no facilities
to adjust the relative loads were incorporated in the system.

A
50 mm travel linear voltage displacement transducer was provided under
the centre
provided
of each
over
beam.
each
In addition,
bearing
and
under
10mm travel
transducers
some
positions.
wheel
were
The
transducers under the wheels were supported off the top flanges of the
beams and thus measured only the slab displacement relative to the beams.
Vibrating wire strain gauges were used both on the surface and in the
concrete at selected positions.
These have the advantage of remaining
stable over long periods, which was important as it was intended to record
the
total
positions.
strains
due
to
the
application
of
several
different
load
However, their strain capacity was not sufficient to use them
to measure smeared strains in cracked concrete.
Thus "portal" gauges
developed by Cook<129) were used in positions where cracking was expected.

Because it was considered undesirable .to estimate curva.tures or extensions
in concrete sections
from
top and bottom gauges with different gauge

- 161-
lengths and other characteristics, portal gauges were also used in some
positions where cracking was not expected.
A disadvantage of surface strain gauges is that
because their thermal
inertia is much less than that of the specimen, they are very sensitive to
temperature
coefficient
changes;
of
unlike
demecs,
expansion since
the
portals
have
significant
they are made of aluminium.
However,
although the laboratory was not air conditioned, it proved possible to keep
the temperature constant to within some 2c for the tests for which the
strain data was used.
In order to avoid the problem of sunlight warming
the gauges directly, all the blinds on the South side of the laboratory
were closed for the duration of the tests.
Because portal gauges have not previously been used for long-term tests,
it was decided to monitor their long-term performance using readings off
demec points mounted as close as possible to each portal.
The original
idea was to use the portals only to record the change of strains during a
test and to add these on to long term changes recorded by the demecs.
In
practice, the changes of reading in the portals were close to those in the
demecs so this extra complication proved unnecessary.
The reinforcement under one wheel in the first test and two in the second
was also strain gauged, using electrical resistance gauges.
Unfortunately,
some of these gauges were damaged during the construction of the deck and
few of the results were usable.
Two gauge lengths were used for the portal gauges: 200mm for the beams,
which is the largest size made, and lOOmm for the slab.
The latter length
was a compromise between the requirement for a short gauge length, to

monitor local peaks in the bending moment distribution, and a long gauge
length
to
make
the
results
comparable
with
smeared
crack
analysis.
However, the latter objective was not achieved since the crack spacings
were greater than lOOmm.
as
indicating only
gauges
on
the
Thus it is more realistic to consider the gauges
the movement
reinforcement
of
individual cracks.
represented
only
the
Similarly,
strain
at
the
their
particular location and were not directly comparable with smeared. crack
analysis.
All the electronic instrumentation, a total of 74 channels,
wa~
connected
to a ''Compulog" data logging system which converted the results to digital

- 162-
strain, displacement and force readings before storing it on disc and tape
for later processing.
Some key strain and deflection readings, as well as
the load cell readings, were printed out whilst the tests were in progress.
8. 7 TESTS ON FIRST DECK
8.7.1 Global Service Load Tests

a. First Load Position
The loading
frame
was
first
positioned to apply the HB load in the
position indicated in Figure 8.11.
The design service load was then
applied in ten approximately equal increments.

examined for cracks after each increment.
critical areas
of
the slab with an
The structure was carefully
However, despite studying the
illuminated
magnifying glass,
cracks were seen until the full load had been applied.
no
Under the previous
increment the strain measured by the portal gauge immediately under the
wheel nearest the centre of the deck was 575 microstrain; some three times
the strain at which cracking normally first becomes visible.
This was
partly a consequence of the thin slab and high strain gradient.
However,
this also applied to the half scale specimens considered in Chapter 6

which cracked at lower strains.
Another explanation is that under the
concentrated load the scope for stress redistribution, both by moment

redistribution and by membrane action, was so great that the concrete in
the critical area was effectively being stressed under strain control, even
though the structure was loaded under load control.
Thus the cracks did
not become visible until the concrete stress had dropped significantly
below the normal cracking stress.
The crack widths were measured using a crack microscope.
Under full
service load the maximum width, which occurred under wheel 10 in Figure
8.11, was 0.05mm;
equivalent to O.lmm at full size.
This would certainly
be acceptable in practice and may appear to be very small considering that

conventional design methods implied that the slab should have failed by
this
stage.
However,
other
studies,
notably
Kirkpatrick et
al's <49>,
suggested that the slab should have been uncracked under this wheel load.
The fact that the outer bay of the slab <where global transverse moments
were
less
significant>
was
indeed
uncracked,
suggested
that
global
transverse moments were the reason for this difference from Kirkpatrick's
result.
- 163-
After
this
test,
the
deck
was
unloaded
and
the
cracks closed
completely that they were invisible even using the microscope.

Figures 8.12 and 8.13
indicate that
up
so
However,
the local strains and deflectionss
under the critical wheels did not fully recover.
The strain reading under
wheel 10 was marginally greater than under wheel 9 and the reason for
plotting the deflection under wheel 9 <rather than 10) in Figure 8.13 was
that the displacement transducer under wheel 10 failed .
.900
1056
J50
900
2250
1056
1056
1056
Beam A
BeamS
Beam(
Beam 0
I
0
HB wheel positions for first and last service load test

and f or failure test.
1(
Other HB wheel positions.
Single wheel test positions.
Figure 8. 11:
Load positions for first deck
- 164-
Load
150
<lri/Jaclr)
Cycle 2
125
/
Cycle 1
/
/
100
~After
/
75
Loading in
Other Positions
/
/
/
50
Loading
U1
- -
- Unloading
25
/
/
0
0
500
1000
1500
2000
2500
Strain x 106
Figure B. 12:
Transvers e sofflt s train under wheel 10

<Servi c e l o ad tests)
Load
150
(kJI/Jac k>
Cyc le 2
125
Af ter Loading in
Cycle 1
Other Positions
lOO
/
/
75
Loading
O'l
en
50
Unloading
25
0 ~~~~---,----------~--------~~--------~---------,
0.0
0. 2
0. 4
0. 6
0. 8
1.0
Deflection <mm relative to beams>
Figure 8 . 1 3 :
De fle c t i o n unde r
<Serv ic e l o ad tests )
wheel
The deck was then loaded to 1.2 times design service load.
As the load
increased above design service level, the cracks under wheels 9 and 10
grew longer and at a load of 140kN per jack they joined up.
maximum load
<150kN/jack> the crack width under both these wheels was
approximately 0.13mm whilst

wide.
been
Under the
mid-way between them the crack was 0.08mm
Since the maximum crack width under the design service load had
less
than 0.08mm,
this
<and
the similar relationship between the
strain readings> suggested that applying the increased load in this one
position was at
least equivalent, as far as this area of the deck was
concerned, to rolling the service load 0.9m along the deck.
If similar
relative widths occurred in subsequent tests <which they did) this meant
that applying 1.2 times service load in just the three positions along the
deck illustrated in Figure 8. 11 would be equivalent to rolling the service
load along its full length.
After the cyclic loads described in 8.5.1 had been applied, a load of 1.2
times design service load was again applied.
The change in the strains
and displacements, compared with the load application before the cyclic
tests had been performed, was so small that this application could not be
plotted on
Figure 8.13
without
making
it
illegible
and,
for
the same
reason, only the peak part of this load cycle is shown in Figure 8.12.
The
strain measured at the start of the cycle was marginally smaller than that
measured at the end of the second cycle to 1.2. times service load.
Thus
the 10000 cycles to "HA" service load plus 20% and 100 cycles to full HB
service load had had a small effect on the behaviour compared with just
two
cycles
to
1.2
times
HB
service
load;
the
structure
had
actually
recovered some of its strain whilst the cyclic loads were being applied.
Under full load, the cracks were not significantly wider than under the
first
load application and no cracks were visible on the top surface of
the slab; the only visible cracks were the four longitudinal soffit cracks,
one
under each
pair of wheels.
On
unloading,
the cracks were again
invisible even with the microscope.
b. Other Load Positions

On completion of the tests in the first
moved sideways by 500mm.
position, the loading rig was
The same load sequence was applied in this and
subsequent positions, except for the reduction from 10000 to 5000 cycles
of the HA equivalent load.
As can be seen fr.om Figure 8.11 some of the

- 167-
wheel positions in this load case were the same as in the first case. Thus
cracks were visible under these wheels at a much earlier stage than in the
previous test.
Cracks were also visible in the newly loaded bay one load
stage earlier than they had been in the first test.
However, apart from
this the behaviour was very similar.

On completion of these tests, the whole loading frame was moved along the
deck by 1056mm to apply the HB load in the third position illustrated in
Figure 8.11.
similar.
The same load sequence was applied and the behaviour was
Because the instrumentation had been positioned to suit the first
load position,
the behaviour could not be monitored so closely.
Crack
widths were measured, however, and they were marginally greater than under
the
first
load
position;
the maximum width
being 0.15mm against 0.13.
Since the first load position was a worse case for global effects, and
both were identical for local effects, this suggested that
the loss of
restraint and distribution due to the cracking caused by the previous load
cases was affecting the behaviour.
The
same
load
sequence
was
then
applied
in
the
remaining
positions
By the completion of these tests, all three
bays of the deck slab had cracked along almost the full length of the
bridge.
seen.
However, these three cracks were the only cracks which had been
They were visible with a
magnifying glass
when
the deck was
unloaded, with a maximum width of 0.05mm and a more typical width of

0.02mm.
c. Return to First Load Position

For the final service load test, the loading frame was returned to its
original position and the load was re-applied.
As will be seen
from
Figures 8.12 and 8.13, the deformations were greater than under the first
applications but still not excessive.
The maximum measured crack width
was 0.2mm which, as in all the tests, was slightly <25%) less than would
be assumed from the strain gauge reading, indicating that the concrete on
either
side
of
the
cracks
was
still
under
significant
tension.
The
maximum crack width was equivalent to 0.4-mm at full size, compared with an
allowable width of 0.25mm in BS 54-00: Part 4-.
However, that document only
requires crack widths to be checked .under a much lower load; 25 units of

HB compared with
subjected.
the
1.2
times 4-5
unit
load to which the model was
Under the load used . for crack width calculation, the measured
- 168-
crack width was 0, 12mm; the scale equivalent of 0.24mm compared with the
allow ab le width of 0.25mm.
Although it is unreasonable to expect this
level of precision in crack width predictions, and a model is likely to

under-estimate crack widths, the many conservative features of the tests
which have been mentioned earlier mean that it is reasonable to conclude
from this that the crack widths would be acceptable in a full size bridge.
Thus,
by
this
criterion,
the
service
load
behaviour
of
the
deck
was
satisfactory although it clearly did not have the enormous margin of overcapacity which previous research implied it should have.
will
be
considered
in
the
next
chapter,
and
also
Analysis, which
observation of
the
behaviour suggested that this difference was due to the global transverse
moments resulting from the use of full HB load in this study, compared
with only single wheels in other studies.
conclusively
that
it
was
not
due
to
However, it remained to prove

the
absence
of
the
diaphragms
recommended by others.
It
was noted in Chapter 2 that crack widths are an unsatisfactory, and
perhaps unnecessary, design criterion.
However, by any other fundamental
design
deformations>
criterion
satisfactory.
<such
as
permanent
behaviour
was
Similarly, the stresses estimated from the strain readings
were well within the BS 5400 criteria.
Thus the behaviour of this very
lightly reinforced deck slab was clearly satisfactory.

the
the
over-provision
of
prestress
and
the
However, because of
conservative
nature
of
conventional design rules for prestressed concrete, the lack of cracks in

the beams did not prove that the distribution properties of the deck were
either satisfactory or similar to those which had been assumed in the
design
of
investigated
the
by
beams.
detailed
This
aspect
comparison
of
the
with
behaviour
analyses
could
only
be
thus
will
be
and
After the service load tests had been completed, the design ultimate HB
load was re-applied and
the HB
load
was
then
increased
in
steps of
approximately 25kN per jack, that is 17t of design ultimate load.
The
displacements of the beams are shown in Figure 8.14 whilst that of the
slab under wheel 9 is shown in Figure 8.15.
The loading to failure was
not continuous and the points where the load was removed and re-applied
are indicated by breaks in the plots in the figures.
- 169-
Beam ~
(k~~~=ck)
:::
Beam~
Beam A
300
250
200
150
100
50
0 4-----,-----~----~------~----~----0
10
20
30
40
50
60
Deflection (mm)
Figure 8. 14:
Load
Beam deflect1ons
4 00
' kN/Jac k )
350
:wo
250
200
150
lOO
50
0
0
Deflection <rn1n )
Figure 8.15:
As the load increased the longitudinal cracks under the slab grew wider
but , at a
load of 245k.N per jack
cracks were visible.
(1.67
times design ultimate), no new
At this stage the largest strain recorded by the
portal gauges across the crack in the centre bay of the slab was 4600
microstrain, 80% higher than under design ultimate load,
indicating
(as
will be seen from Figure 8. 15> that the slab was beginning to depart from
- 170-
the near linear behaviour it had exhibited since the completion of the
service load tests.
this crack was
Even mid-way between the bogies, the strain across
1900 microstrain but, in contrast, the highest
tensile
strain recorded by the portals on the top of the slab over the edges of
the webs of the beams was 300 microstrain adjacent to wheel 10.
The
gauge over the web of Beam B on the outside, that is adjacent to wheel 2,
was reading 136 microstrain compression indicating that in this region the
sagging moment due to transverse global effects was greater than the
local moment.
Longitudinal portal gauges positioned under wheels 9 and 10
were showing very small strains but

microstrain tension,
that under wheel 12 showed 600
the difference presumably being due to the lower
global compression in this area.

A number of transverse strain gauges and demec points had been positioned
in the slab near the expected points of transverse contraflexure in an
attempt to estimate the membrane forces.
Due to the small and erratic
readings, the proximity of cracks and the transverse strains resulting from
the
Poisson's
ratio
extremely difficult
effect
of
to interpret.
the
global
However,
flange
forces,
these
were
there did appear to be a
transverse compression adjacent to the wheels.

After
this load stage,
the
load was removed and there was over .80%
recovery on all the significant readings.

increased further.
The load was then re-applied and
At 250kN per jack, a shear crack appeared in the right
hand end of Beam B <as shown in the figures) and a flexural crack was also
just visible in the soffit of the same beam under wheel 6.
and flexural cracks formed in the same regions at 275kN.
Further shear
At this stage, a
shear crack also appeared in the right hand end of Beam C and in the left
end of Beam B.
There was also a very fine horizontal crack running along
the outside of the web to Beam A adjacent to wheels 1 and 2, due to the
beam's action in restraining the hogging moment in the slab.
A second
longitudinal crack had appeared in the soffit of the slab but still no
cracks were visible on its top surface.
At 300kN per jack, twice design ultimate load, the first shear crack which
had appeared in Beam B extended right through the bottom flange.
What
looked like shear cracks also appeared in the left end of Beam C between
the support and wheel 9.
However, cracks on the opposite side of the web
sloped the opposite way, indicating that the cracks were largely due to
- 171-
torsion although there were only the flexible bearings and thin deck slab
available to resist, or apply, the torque.
Longitudinal cracks in the top of the slab also became visible at this
stage.
These then extended over most of the length of the deck on either
side of Beam B and on one side of Beam C.
A crack also appeared over the
inside edge of the web to Beam A, adjacent to wheels 1 and 2, but this
only extended a short distance either side of the bogie.
At a
load of 2.39 times design
through
the
flange
of
Beam
discontinuity of approximately
ultimate
(350k.N/jack),
became
1mm across
the shear crack
very
wide
with
vertical
it.
This was due
to bond
failure with the strands which could be seen to have drawn in by some
2mm.
than
Since the strands used were larger in size and smaller in number
in
true half scale
relative to a true model.
model,
this
may
have occurred prematurely
The shear cracks are illustrated in Figure 8 .16.
Figure 8.16:
Shear cracks in Beam B
By this stage, flexural cracks extended over 2.2m of the length of Beam B
and had also developed in Beam C.
There were now five longitudinal cracks
in the soffit of the outer bay of the slab under wheel 4.

out
beyond
These fanned
the wheel towards the beams and the end of the deck as
illustrated in Figure 8 . 17.
This <and less pronounced fanning at the far
end of the same bay beyond wheel 1) was the only sign in the slab of
transverse cracks or of the characteristic radial crack pattern observed by
other researchers in single wheel tests.
- 172 -
Figure 8.1 7:
Figure 8.18:
Soffit cracks under wheel 4
First deck under 400kN per jack
By this stage, Beam A had developed a substantial rotation which is visible

in Figure 8.18.
This was largely due to
the differential deflection
between
the still uncracked Beam A and the much more heavily loaded
Beam B.
The local transverse hogging moment due to the wheels may have
also contributed, but the crack pattern showed clearly that the slab was
subjected to a net transverse sagging moment right out to Beam A adjacent
to wheels 3 and 4.
The slab in this region was thus contributing to
restraining the rotation; the global transverse moment was dominating over
the local moment.
The rotation of the beam about its longitudinal axis,
combined with the resulting transverse movement, was well in excess of the
intended capacity of
the bearing.
At
- 173-
load of 400kN per jack the
resulting transverse force
became too much for the bearings under the
other beams and the whole deck suddenly moved sideways by some 20mm.
The resulting unintended eccentricity led to elastic torsional buckling in
the main girders of the four load spreading rigs.
The load was therefore
removed and the structure recovered remarkably welli

the
local deflections
and
90~
of
the
approximately
beam deflections.
80~
However,
of
the
maximum strain recorded by the gauge under wheel 10 reduced only from
11500 to 4.400 microstrain.
On re-loading, the deck settled down to its new position but the buckling
re-occurred so the load was removed again and the ball bearing under one
end
of
the
offending
girders
was
replaced
with
rocker
bearing.
Calculations showed that because of the low torsional stiffness of the

girder <which had been the cause of the problem) the resulting unevenness
of the load distribution between the four wheels would not be significant.
The modified rig was therefore used for all subsequent tests, including
the service tests on the second deck.
By this stage the limiting pressure of the electric pump had been reached
so
the
movement
loading
of
was
resumed
the deck had
using
moved
hand
pumps.
Because
the bearings off
the
sideways
their seatings,
the
pronounced step in the plots in Figure 8.14 may have little significance;
the displacement transducers over the bearings had come off their points
so this could not be checked.
However, there clearly was a deterioration
in the distribution properties of the deck as well as in the stiffness of

Beam B.
The large differential deflections across the deck are clearly
visible in Figure 8.19.
Figure 8.19:
View across deck as failure approached
- 174-
When the load level was only marginally higher than before, a number of
"bangs" were heard from around the right hand end of Beams A and B; one
so loud an observer assumed it to be due to rupture of the links either in
Beam B Cdue to shear) or Beam A (due to separation from the slab).
However,
when the concrete was
proved not to be the case.
later broken out
in this region,
this
It may have been due to further bond failure
in the end of Beam B as the draw- in was now approaching lOmm.

By this stage difficulty was being experienced in holding the deck up to
load, indicating that failure was imminent .
The appearance of the right
hand end of Beam B, combined with the loud bangs, suggested to some
observers that this would take the form of a shear failure in this beam.
However, a line of crushing concrete was just discernible between wheels 3
and 4-.
At a
load of approximately 4-14-kN per jack
ultimate> wheel 4- punched through the deck.
(2.83 times design
The resulting release in the
load on the beams caused the beam deflections to reduce; hence the wheel,
although
loaded
only
by a
jack and
thus
under
displacement
control,
punched right through the slab as shown in F'igure 8 .20, rupturing the
steel as it went.
,.
Figure 8.20:
Failure; Wheel 4 punched through deck
The hand pumped hydraulic systems used for the two bogies were separate.
The reduction in beam deflection caused by the failure therefore led to an
increase in the load on the other bogie.
Several minutes later, when a
reading was taken, the load on this was some 25% higher than that which
- 175 -
had caused failure under the first bogie.
Because of the high creep in
concrete structures approaching failure, this indicates that the failure

load would have been substantially higher.
After the structure had been unloaded, the concrete remaining under the
wheel which had punched through was removed.
This revealed the classic
conical form of a punching shear failure as can be seen in Figure 8 .21.

However, the suggestion in earlier chapters that such a failure can be
precipitated by concrete crushing had been reinforced by the fact that t he
line of crushing concrete
noticed
before
the
failure
(which
is
visible
occurred.
It
in Figure 8 .20) had

also
appeared
that
been
global
transverse moments had significantly reduced the local strength of the

slab.
The rotation of the edge beam <which was due to the differential
beam deflections at
mid-span) had clearly led to a transverse sagging
moment in the slab over the web near the wheel which failedi a region
where the local moment would have been sagging.
transverse moment
Near mid- span, where the
in the slab was causing rather than restraining the
rotation of the beam, there was a transverse hogging moment over the
beam; hence the higher local strength.
Figure 8.21:
Further confirmation of the significant reduction in strength due to the

global moments <and hence, by implication, of the flexural nature of the
- 176-
failures)
methods
was
given
considered
estimated
when
in
the
slab
3.2.3.
the _strength
by
strength
Kirkpatrick
nearly
80%
was
et
whilst
estimated
al's
using
approach(13)
Hewitt
and
the
over-
Batchelor's
approach <72>, using a restraint factor of 0.6, was only marginally better.
The
restraint
factor
back-calculated
from
the
failure
load
was
approximately 0.2, compared with the "conservative" figure of 0.5 used in

the Ontario Code<1ll for assessing existing decks.
Thus, although the test
results did not suggest that decks designed to the empirical rules would
be unsafe,
could be.
they did
imply that
the Ontario assessment recommendations
However, since all previous research into compressive membrane
action in bridge decks had suggested that support diaphragms are needed
<or at least desirable> to provide the restraint, a plausible alternative
explanation for the reduced strength was that the restraint in this deck
was inadequate.
It was decided to perform local tests to investigate this.

Two single wheel tests were performed using the test rig illustrated in
Figure 8.22.
The position of these, which is shown in Figure 8.11, was
chosen for convenience in testing and also to avoid areas of the slab
which had been significantly damaged in the previous tests.
However, the
slab around the wheel tests had been cracked by the previous tests whilst
the adjacent bay, which could be important to the restraint, had apparently
been loaded very close to failure.
Thus the test situation was extremely
unfavourable compared with the normal situation in a real bridge deck.
It
was considered that the behaviour at low loads was so greatly affected by
. this
that
it
had
no
real
significance.
Thus
serviceability
was
not
considered in such detail as in the global tests and the slab was loaded
monotonically to. failure.
The same instruments were used as . for the global tests but some were
repositioned
and
all
were
re-zeroed..
Thus
strain
and
displacement
readings were taken relative to the start of the test, rather than relative
to the initial <uncracked) state as in the global tests.
were experienced with
the logger during the first
Some difficulties
test whilst in the
second test the displacement transducer under the wheel stopped working.
Since the behaviour in the two tests was very similar, only the behaviour
of the second test will be described in detail but the load-deflection
response for the first test is illustrated in Figure 8.23.
- 177-
9
Fig ure 8.22:
Single wheel test rig
Load <k!D 2 00
150
100
50
0
0
8
Deflection
Figure 8.23:
10
( :mm)
Result o f single wheel test
A longitudinal crack was visible under the wheel before the test started
and by a l oad of 43kN <which corresponds to 1.15 times the design ultimate
wheel load ) this crack was 0.08mm wide.
This crack was thus significantly
narrower than under the same load per wheel in the global test, despite
the pre-cracking.
At the s ame stage, the crack on the top of the slab
over the web was 0 .05mm wide, whilst the equivalent crack did not appear
in the global test until nearly twice the load per wheel had been applied.
However, this difference was probably largely due to damage sustained in
- 178-
previous tests; the crack was visible before the load was applied.
transverse crack was just visible in the soffit under a load of 67kN.
In
the global tests, such a crack had not appeared until the wheel load was
some 30% higher.
This difference could not be explained by pre-cracking;
it was due to the global flange force prestressing the slab in the local
tests.
Figure 8.24:
Crack pattern under single wheel
At a load of 113kN, by which point failure had occurred in the global

tests, the maximum crack width was 0.3mm.
By 145kN the cracks in the
soffit had taken on the characteristic radial form which is illustrated in

Figure 8.24.
At a load of 202kN the crack in the top of the slab along
the web of beam 3 had joined the similar crack due to the previously
performed
single
wheel
test.
However
there
was
no
sign
of
this
interaction reducing the strength; the wheel finally failed under a load of
approximately
226kN
compared
with
204kN
in
the
previous
test.
comparison, the failure load in the global test was 103.5kN per wheel,
- 179-
For
The wheel punched a neat hole through the top of the deck, coming to rest
only
some
10mm
below
the
top
surface
of
the
slab.
However,
this
difference from the failure mode under full HB was not indicative of any
fundamental difference in the local behaviour.
different
post-failure
behaviour
caused
It was purely due to the
by
the
much
smaller
displacement and force which was released by the local failure .
global
Removal
of the loos e concrete from under the wheel revealed the classic conical
failure s urface illus trated in Figur e 8 .25 which is very similar to that
observed in the global tests.
The behaviour was also very similar to the
global tes ts in that although there was plenty of warning of failure, in

the sens e that the structure was clearly unserviceable when subjected to
only half its final f ailure load, the final collapse was very sudden.
with
Even
the advantage of the strain readings, and of having observed an
iden t i cal test only hours earlier, it was not easy to tell when failure was
imminent.
Figure 8.25:
The difference between the two results may not be significant as it is not
unusual to obtain
much greater strength differences between nominally
identic al concrete
spec imens.
However,
it
may
have
been due
to
the
inferior restraint available to the first test which was nearer the end of
the deck.
The failure loads were, however, close to the 185kN predicted by
Kirkpatrick et al's approach; the ratios being 1.22 and 1.10 compared with
an
average
of
1.19
for
Kirkpatrick's
own
tests<13 ).
Similarly,
the
res traint factor back-calculated using Hewitt and Batchelor's approach, at

approximately 0 .65, was close t o t heir findings.
- 180-
This confirms that both
approaches give good predictions for the failure load under single wheel s
and suggests that the restraint needed to develop this local strength is
not dependent on diaphragms.
Thus the lower local strength observed when
all 16 wheels of the HB l oad had been applied must have been the result
of global effects.
8.8 TESTS ON SECOND DECK

8 .8 . 1 Global Service Load Tests
a . First Load Position
The second deck was s ub jected to a very similar load history to the first.
However, only 5000 cycles of the reduced load were used and, because of
the greater width of the deck, t he load was applied in a maximum of four
different posit i ons across the width instead of two.
The loading positions
are illustrated in Figure 8 .26.
1-
r
I
)(
--
)I')(
r
)(
~-
I
,_
f--
-~
~
-
s
~
)(
- -=- ~
..
)0(
Jt
I"
I"'
111(
-~
J(
l' I
-- - --
-:it
~- - --*"- .
10
14
1(-
.If)(
..
-
,.
- --
-7
le@
--,
)(
- - -....... -
--
)(
4
.
-8~
0
12
-=-o
e
t-----i
-_;_-?t---..:..-
X::
BeamB
-.- Beam(
Beam 0
Beam E
HB whe e l pos i t i ons for firs t and last s ervi ce l oad test
and f or f ailur e t est.
Beam A
l
J
-=-~ -0- l
1(1(
~ .
11
-xr~-
'&.
)('
)(
""3
-- - -_-:Jt
- - r-(
lt:=
}(
13
.q
XI(
"
-~
Ot he r HB wheel positions.
Wheel posi t i ons fo r local tes t s.
Line o f cut s i n re inforcement .
Fig ure 8. 26:
Load p ositions for second deck
- 181 -
Like the first, this deck was initially subjected to design service load
then to two cycles of a 20% higher load.
A soffit crack appeared at
approximately the same stage as before and under full design service load
it
had
width
of
0.1 mm:
equivalent
stage
in
the
approximately
previous
test.
twice
the
However,
concrete are very variable, particularly at
width
as
at
the
the crack widths
in
loads just above that which
causes cracking, so this may have had little significance or may have been
due to the lower tensile strength of the concrete.
The difference in crack width was less pronounced at the higher load of .
the second loading cycle.
A more important difference from the behaviour
of the first deck was that a top crack appeared at the same time as the
soffit crack, whereas in the first it had required a load some 150% higher.
The measured local deflect ions, crack widths and transverse strains were
greatest adjacent to wheel 14.
These strains are shown in Figure 8.27,
whilst the deflection of wheel 14 relative to the beams is shown in Figure

8.28.
Initially the soffit strain exceeded the top strain by some 50~ but
this percentage reduced once the behaviour departed from linearity and the
top strain overtook the soffit strain when the cracks became visible.
remained greater
greatest
<even
throughout
in
the subsequent
absolute
terms>
when
tests,
the
It
the difference being
structure
was
unloaded.
However, these high strains were confined to the region over the inside of
the web to Beam D which was the location of the only top crack.
Figure
8.27 shows that the strain over the edge of the adjacent Beam C was very
much lower.
Indeed, nearly all the tensile strain in that region could. be
explained by the Poisson's ratio effect of the longitudinal compressive

strain due to the global flange force.
The difference
particularly
analysis
between
significant
would
treat
the strains at
when
the
either end of the slab span is
it
is
realised
that
slab
as
symmetrical
and
conventional
so
would
local
predict
identical strains at either end, whilst an apparently more sophisticated

local analysis <treating the slab as continuous over simple supports) would
predict a greater hogging moment over Beam C than over Beam D.
- 182-
150
Load
<kN/ Jack>
Cycle 3
Cycle 2
;I
;I
;I
125
100
I
/
Af ter Loading in
Ot her Positions
/
/
/I
75
/I
():)
.p.
I
50
~//
25
~/
'l
/
/
//
Loading
//
Unloading
Deflection <mm relative to beams>
Figure 8 . 28:
Derlection under wheel 14
<Ser v i ce l oa d tests )
The explanation .for the greater strain over Beam D was that there the
global transverse moment was hogging, adding to the local moment, whereas
over Beam C the global moment was sagging and therefore acting against
the local effect.
Further evidence to suggest that the top crack over
Beam D was largely due to global effects was given by the length of the
crack.
As soon as it appeared, it extended over most of the length of the
deck.
The
width
even
mid-way
between
the
bogies,
where
Pucher's
charts (40) indicated that the local moment should have been sagging, was
some two thirds of the maximum width adjacent to wheel 14.
In contrast,
as in the first deck, separate soffit cracks appeared initially under each
wheel.
The soffit cracks formed by the two wheels of a bogie, such as
wheels 13 and U, joined together as the load increased but the cracks
formed by the two bogies did not join until the load had been applied in
other positions along the length of the deck.
Thus it appeared that the
soffit cracks were primarily due to the local effect whilst the top crack
was largely due to the global effect.
This
also explains
soffit.
strain
Initially,
was
behaviour
smoothed
concrete.
as
greater
departed
out
as
the difference
predicted
than
from
force
the
in strain behaviour of the
by elastic
maximum
linearity
top
the
redistributed
theory,
strain.
stress
peak
top and
the maximum soffit

However,
in
the
once
the
soffit
was
to the surrounding under-stressed
There was less scope for redistribution of the top stresses
because global moments are relatively uniform over the length of the deck;
hence
the
rather
greater
increase
in
strain
on
cracking.
Uncracked
concrete surrounding the local cracks and trying to push them closed would
also lead to better recovery of the soffit strain on unloading.
However
the much inferior recovery of the top strains <the top strain after each
cycle of the first load position .was over double the soffit strain) was
undoubtedly exaggerated by the reinforcement detailing since the single
layer of main steel was located some 10mm below mid-depth.
The reason why the top cracks had not appeared in the first deck until a
much higher load was applied was that the lack of diaphragms meant that
the beams were free to rotate.
Thus local hogging moments were relieved
by rotation and differential displacements of the beams led to lesser

global transverse moments.
The diaphragms in the second deck contributed
to its superior distribution properties and the maximum beam displacement
- 185-
was some 25% lower than in the first
deck even though a static load
distribution would predict identical deflections.

As with the first deck, the application of 5000 cycles of a reduced load
had remarkably little effect on the behaviour.
b. Other Load Positions

After completion of the tests in the first position, the HB loading rig was
moved a metre sideways in the direction towards the top of Figure 8.26 to
apply load in the second position.
that in the first position.
The behaviour was generally similar to
The one new top crack appeared over the edge
of the centre beam when the design service load was applied, whilst the
new soffit crack appeared in the bay at the top of Figure 8.26 under the
full load; that is 1.2 times design service load.
The original intention had been to apply the HB load in a total of three
different positions across the width of the deck.
This loading sequence
was designed to induce all the soffit cracking which was likely to occur
in service.
However, it was now apparent that top cracks due to global
effects could be equally significant.
The intended load sequence would
have failed to induce top cracks over the inside of the web of Beam B.
Since such cracks could .be significant to the behaviour of the adjacent
bay of slab
<that between Beams B and C> and since that bay would be
loaded in the final test, it was considered that this was a fault of the
sequence.
An intermediate load position was therefore used.
The loading
rig was moved back 1.5m towards the bottom of Figure 8.26 to apply the
same load position as in the first test but opposite hand.
Apart from the
effect of the pre-cracking, which led to a softer initial response, the

behaviour was very similar and the maximum crack widths were similar.
The
maximum strains and deflections were also similar although few direct
comparisons could be made as few gauges were in equivalent positions.
The
global deflections were very similar; within 5%.

After completion of the tests
moved a
in this position,
the loading frame was
further metre sideways to the position nearest
shown in Figure 8.26.
the bot tom as
This position was particularly significant as it
included a wheel directly over the region of the cut reinforcement.

Under the maximum load which had previously been applied, there were no
cracks visible in the region of the cut reinforcement.
- 186-
This was slightly
surprising as the load case was identical to one which had already been
tested, apart from being opposite hand, and cracks had appeared in that
test at the equivalent position and load stage.
It was also unfortunate
as it had been intended to use the test to investigate the behaviour,

under
cyclic
reinforcement.
loads,
of
cracked
bridge
deck
slab
with
damaged
It was therefore decided to increase the load slightly to
crack the slab; the strain of 500 microstrain measured by the portal gauge
under the wheel indicated that
cracking was imminent.
However a
10%
increase in load failed to produce visible cracks despite giving a strain

of 610 microstrain.
the
reinforcement
Curiously, the strain under the adjacent wheel, where

was
intact,
was
986
microstrain which
indicated,
comparison with other cases, that cracks would have been visible.
by
These
were not noticed although the area was not inspected as thoroughly.
Clearly, cutting the main steel had not advanced the formation of cracks:
indeed it appeared to have delayed it although, in view of the variability
of concrete behaviour, this was probably not significant.
A further increase in load would have applied a significantly higher global
load
than
had
been
intended
and
may
significantly alter the behaviour under
have
caused
the later
enough
damage
load cases.
It
to
was
considered that the intended load sequence was over-severe and a further
increase would have made it too unrealistic.
It was therefore decided to
unload the structure and disconnect three of the four jacks.

a
higher
load
to
be
applied
on
four
of
the
This enabled
wheels without
causing
significant damage in regions where it was likely to affect the behaviour

under the later load cases.
At a load of 163kN, a marginally lower local load than that which had
previously failed to crack the region, a 0.05mm crack was visible in the
soffit under the cut reinforcement.
The load was then increased to 178kN,
equivalent to 1.22 times the design ultimate HB wheel load, which increased
the crack width to 0.1mm.
It also induced a crack, approximately 0.05mm
wide and 1.5m long, in the top of the slab over Beam D.
The bridge was then unloaded, the other three jacks re-connected and the
cyclic loads applied as in the previous positions.
Despite the initial
over-loading, the behaviour was entirely satisfactory and appeared similar

to that when the load case had been applied opp'osite-hand . over .intact
- 187-
re in f orcemen t.
There was thus no evidence that cutting the steel had
adversely affected the behaviour.

For the nelCt load position the loading frame was moved 1056mm along the
deck and back to its original transverse position.
and
subsequent
positions
was
very
similar
to
positions so it will not be described in detail.
The behaviour in this

that
in
the
previous
Because the load cases
were less severe on the beams, the top cracks were not greatly elCtended.
It was therefore decided that it was not necessary to apply the load in as
many transverse positions to simulate all the damage which could occur in
practice and only two were applied.
It appeared that applying 1.2 times
service load in just one longitudinal position was equivalent, as far as

top
cracks
were
concerned,
longitudinal positions.
to
applying
service
load
in
all
possible
However, it was still necessary to apply the load
in all the positions shown in Figure 8.26 in order to ensure that the
soffit cracking would be correctly simulated.
For the final loading position, the bogies were positioned to give wheel
loads 250mm off-centre to the slab span.
A new crack formed under some
wheels but not until the load elCceeded design service load.
appeared
that
loading
only at
mid-span of
the slabs,
as
It therefore
in all
the
previous tests, had given a reasonable representation of the extent of

cracking which would have occurred if the service load had been applied in
all positions.
c. Return to First Position
The loading frame was returned to its original position and the full load
re-applied.
As will be seen from Figure 8.28, the local deflection under
wheel 14 <the greatest local deflection recorded> was substantially greater

than under the first loading.
The maximum local deflection was also 42%
greater than the equivalent deflection at the same stage in the tests on
the first deck, whilst the deflection on unloading was over four times
greater.
It will also be seen, by comparing Figures 8.28. and 8_.13, that
loading in other positions had had a greater effect on this deck than it
had on the first.
One reason
for
the greater effect
on
this deck of loading in other
positions can be inferred from Figure 8.27:. it had opened .a crack over the
web of Beam C.
This crack tended to close as load was applied, indicating

- 188-
that the global sagging moment in this region under this load case was
greater
than
remained
the
local
s~nificant
hogging
moment.
thus
continued
and
However,
to
the
tensile
contribute
to
strain
the
local
deflection throughout the load cycle.

The maximum crack width recorded was for
the crack over the edge of
Beam D and was equivalent to approximately 0. 7mm at full size, whilst the
maximum soffit cra ck width was equivalent to approximately 0.4mm.
The
pat tern of top cracks after the completion of the service load tests is
Under the reduced load used for checking crack
widths in BS 5400, the soffit crack width was equivalen t to 0.2mm compared
with
the allowable width of 0.25mm.
The top crack was equivalent to
approximately 0.4mm but the top cover was over twice the "C""'"' " required
by BS 5400.
Thus, since BS 5400 allo ws the crack widths to be calculated
"on a hypothetical surface at a distance
Cn o rro
from the outermost bar", the
crack width was, at worst, very close to being acceptable.
Nevertheless
the condition of the slab was not as obviously satisfactory as that of the
first deck had been.
the
permanent
occurred in
The permanent deformations were greater and although
soffit
the
cracks were
first
deck at
equivalent, even on unloading .
no
wider
top cracks,
all, were up
which had
to 0.3mm wide
not
full scale
It was thus somewhat debatable whether the
condition of the slab could be considered acceptable.
f-
r
I
--
- --
.---
- --
Beam A
-----=-- --
- - -
"---
- --
---
,.-
-l J
----- - l
___, J
--=
- -:. 1--
"------,-
BeamS
Beam C
Beam 0
J BeamE
- - -.;_-_
,_
Figure 8.29:
Top cracks on completion of service tests
The fact that top cracks had occurred in t he second deck but not in the
first
was
undoubtedly
due
to
the greater
resulting from the presence of diaphragms:

- 189-
transverse
hogging moments
the load which was required to
induce top cracks in the first deck was so much greater <over two and a
half times as great) that none of the other differences between the decks
could have been more than minor contributory factors.
for
the
poorer recovery of
However, the reason
the second deck on unloading,
greater maximum crack widths, was less clear.
and for its
It could have been due to
the nature of global as opposed to local moments, but the difference in

the main reinforcement and in the tensile strength of the concrete may
also have been important.
essentially
parallel
to
However, since the significant cracks all ran

the
beams,
it
appeared
that
the
substantial
reduction in the secondary steel could not have been a major factor.
Unfortunately, it was not possible to test a third deck so the relative
significance
of
the
differences
investigated analytically.
between
the
decks
could
only
be
Analytical investigation was also essential to
assess the distribution properties and to see how they compared with those
which would normally have been assumed in design.
Nevertheless, it was
clear that the distribution properties were superior to those of the first
deck since, despite the lower grade concrete in the slab, the deflect ion of
the heaviest loaded beam had been consistently some 25% lower.

On completion of the service load tests, the load on the model was reapplied and then increased.
The strains adjacent to wheel 14 are shown in
Figure 8.30 whilst the beam deflections are shown in Figure 8.31.
Both are
plotted relative to the original zeros, which explains why they do not pass
through the origin, and the break in the plot indicates a point where the
load was removed before being re-applied.
The initial strain response shown in Figure 8.30 is approximately linear.
However, once the load exceeded 150kN per jack, the highest load which had
previously been applied,
significant departure
from
observed as the cracks extended in depth and width.
linearity can be
At this point, the
tensile strain over Beam C, which had previously been reducing slightly,
began to increase slightly.
. This would appear to indicate that the local
moment in this region was increasing as moment redistributed away from

the more heavily cracked regions.
The load-deflect ion response of Beams B and C is dist in et ly non-linear
from a load of approximately 150kN per jack.
- 190-
However, this was clearly
due
to
deterioration
softening
of
deflections
these
does
of
the
beams,
not
approximately 250kN.
dist r ibution
since
become
the
properties,
plot
noticeably
of
the
rather
sum
non-linear
of
until
t han
the
to
beam
load
of
As the load increased, the proportion which the outer
two beams carried clearly reduced and, above a load of 275kN per j ack, the
deflection of Beam E began to reduce in absolute, as well as in relative
terms.
to
the
This deterioration of the distribution properties was largely due

extensive
diaphragm,
cracking
wh i ch began
contributed .
Load
500
<kN /J ack )
450
400
in
the
to appear
slab
but
fr om a
torsional
cracks
in
the
load of 225kN per jack, also
Soffit at Kid- Span
over Beam C
------
350
300
250
Top over Beam D
2 00
150
100
50
0
10
12
16
14
Strain x 10 3
Figure 8 .30:
Although, despite the lower grade concrete in the slab, the maximum global
deflec tions
were s maller than at equivalent stages in the test s on the
first deck, a comparison of Figures 8 .31 and 8.14 shows that they began to
depart
from linearity at an earlier stage.
This was apparently largely
because the earlier formation of top cracks meant that the distribution
properties
began
to
deteriorate
at
an
earlier
stage.
However,
an
additional but closely- related reason was that the distribution properties
of
the
first
deck had been so poor.
Thus
the heaviest
loaded beams
carried s uch a high percentage of the load which a static distribution

would predict that even a total loss of distribution properties would have
had little effect on the deflec tions.
- 191 -
Beam
Beam C
Load
450
<kN/Jack)
400
350
300
250
200
150
Sum of all Beams
100
50
0
10
20
30
40
50
60
Deflection (mm)
Figure 8 .31:
Beam deflections
As the load increased, new cracks formed.
The distinct kink in the plot of
the soffit strain in Figure 8.30 at a load of around 260kN per jack is due
to the formation of another longitudinal crack close to, but outside, the
gauge length.
At a load of 300kN, there were four longitudinal cracks in
each of the loaded bays of the deck.
They extended over much of the
length of the deck; three of the initially separate cracks caused by the
two bogies in each bay having joined together.
These cracks fanned out in
a radial pattern at either end of the deck, between the bogies and the
diaphragm, but there were no transverse cracks in the soffit and only one
longitudinal crack in each bay deviated
from the longitudinal direction
between the bogies to meet the webs.

Whilst the pattern of soffit cracks was generally similar to that in the
first deck at equivalent load stages, the pattern of cracks in the top of
the slab, which is illustrated in Figure 8.32, was very different .
The
longitudinal crack over the web to Beam A formed at a load of 225kN per
jack whilst the diagonal cracks at either end of this appeared at between
250 and 400kN per jack.
Since there was no load at all applied directly
to this bay of the slab, it is quite clear that these cracks were the
result of global effects.
in other bays.
Global effects also dominated the crack pattern
Thus the cracking in the bay between Beams C and D, which

- 192 -
had very different deflections, was far more extensive than that between
Beams
and
whose
deflectionss
were
more
similar.
The
crack
perpendicular to Beams B and C at the right hand end of the Figure was
clearly due largely to global hogging in these beams as torsion in the
diaphragm <which was restrained by the lightly loaded Beams A, D and E>
attempted to resist the rotation of Beams B and C.
Similarly, th e extreme
asymmetry <about the deck's longitudinal axis) of the crack pattern in the
bay between Beams D and C indicated t hat 'here too, global effects were
dominant.
Beam A
- -0:
BeamS
- --
CE
_- 0:::::
Beam C
Beam 0
~~~~~~~~~~~~~
Figure 8.32:
BeamE
Top cracks immediately prior to failure
<note: cracks which were formed by other load positions and

which were closing under this load case are not shown)
At a load of 300kN per jack, twice the design ultimate load, the first
crack was observed in a beam; a flexural crack in the soffit of the centre
beam under wheel 10 .
A shear crack appeared in the right hand end of the
same beam (as shown in the Figures ) at a load of 325kN.
A shear crack
appeared in Beam B at a load of 350kN and a flexural crack followed at

approximately 375kN.
By this stage, the capacity of the hydraulic system
had
The
been
reached.
bridge
was
therefore
unloaded
and
the
jacks
connected to a new electric pump which had not been available at the time
of the tests on the first deck.
The load was then re-applied.
By this stage, the cracking in Beams B and C had caused a significant

reduction
in
their
stiffness
as
can
be seen
from
Figure 8.31.
The
resulting increase in the differential deflections of the beams led to

further cracking in the top of the slab and the strain represent ed by the
- 193-
crack over the web to Beam D had reached the limit of the capacity of the
portal gauge.
Figure 8.32
By a
load of 400kN per jack, all the cracks shown in
had formed except
the
longitudinal one over Beam E.
The
widest of the cracks, as throughout the test, was that over Beam D.
By a
load of 425kN this crack was some 3mm wide and the crack over Beam A was
1.5mm wide.
By a load of 460kN per jack, 10% higher than the failure load of the first
deck, there were flexural cracks over much of the length of the centre
beam joining the shear cracking at the right hand end.
The flexural cracks
crossed the soffit of the beam at right angles to its longitudinal axis.
In contrast, those in the adjacent Beam D crossed at an angle of up to 45
degrees and tended to form first on the outside, that is away from. the
loaded bay.
Since the very wide longitudinal crack over one side only of
the web to Beam D suggested that it was subjected to a very substantial

torque, this was not surprising.
However, the crack pattern implied that
this torque was almost entirely resisted by transverse bending and shear
in the bottom flange and not by torsion as such; unlike in the first deck,
the shear cracks on opposite sides of the web sloped in the same direction
indicating relatively low torsional stresses in the web.
The asymmetrical
loading of the beam had, by this stage, caused a longitudinal crack in the
web on the outside of the beam.
Up to this stage, despite the very extensive cracking, there had been no
difficulty in loading the deck or in holding it up to load.
However, as
the load increased further this became increasingly difficult, indicating

that failure was imminent.
At a load of approximately 475kN per jack, a
line of crushing concrete could be clearly seen on the soffit of the slab
extending
along
the edge of Beam D for some one and a half metres
adjacent
to wheels
13 and
14.
reaching
the
of
moment
limit
its
This section of the slab was clearly

capacity
and
it
is
presumably
the
resulting redistribution of local moments which caused the increase in the

strain over Beam C which can be seen in Figure 8.30.
At a load of approximately 490kN per jack, 3.35 times design ultimate load
and some
18% higher than the failure load of the first
deck,
failure
occurred in the form of wheel 14 punching through the deck.. The. resulting
sudden
reduction
in
the
global
load
on
the
deck
reduced .the global
deflections and hence increased the load on the other three jacks.
- 194-
The
local behaviour was so brittle and the failure so sudden that this caused
wheels 5, 8 and 16 <one under each jack) to punch through as well, despite
the fact that the four jacks were inter-connectedi they punched through in
such quick succession there was not time for the hydraulic pressures to
equalise.
The deck after failure is illustrated in Figure 8.33.

I
Figure 8 .33:
Second deck after failure
After the failure, a line of crushed concrete could be seen on the top of
the slab between wheels 9 and 10 as well as between wheels 13 and 14-.
There was also more localised crushing adjacent to the other two wheels
which punched through.
However, despite a thorough inspection at the final
load stage before the failure,
this had not been observed until after
failure although there had been some sign of very local crushing by wheel
U.
It appeared that the soffit crushing which had been observed before
failure had been the root cause of the collapse yet, despite this, the
failure once again looked like a classic "punching shear" failure.
This
further confirmed that such failures could be caused by flexural effects.

The fact that the critical section of slab had so clearly reached the limit
of its moment capacity when the strains and crack widths at the equivalent
position at the other side of the local slab span were so modest, as can
be
seen
from
Figure
8.30,
again
confirmed
the
importance
of
global
transverse moments.
Another
interesting
feature
of
the results
was
that, despite a
lower
concrete strength and less reinforcement, this deck slab had failed at an
18Z higher load than the first.
through
the
slab
at
As in the first deck, wheels had punched
substantially
- 195 -
lower
load
than
predicted
by
Kirkpatrick et al's approach but the margin was much smaller; approximately
a
factor
of
1.22
against
l. 78.
It
appeared
that
this
was
due
to
differences in the global behaviour but again an alternative explanation

was the superior restraint to in-plane forces in the second deck with its
diaphragms.
It was decided to perform some local tests to investigate
this.
Despite the extensive damage caused by the global tests, it was considered
that the two outer bays of the slab were in sufficiently good condition
for
local failure
tests
to
be
useful.
perform a total of four local tests.
This gave
the opportunity to
It was decided to perform two tests
identical to those which had been performed on the first deck.
These
would enable the failure loads to be compared with that in the global
tests and in the tests on the first deck.
controls for
single
the other two
wheel
test
over
In addition, they would act as
tests which would be performed;
the
region
with
the
cut
investigate its effect, and secondly a two wheel test.
firstly a
reinforcement,
to
The latter was
considered important as it was not clear how much of the reduction in

local strength which had occurred in the global tests could be attributed
to interaction of the local effects of adjacent wheels.
Since the slab was cracked much more extensively in the bay between Beams
A and B than on the other side of the deck, it was decided to perform both
control tests in this bay.
This meant that any reduction in the failure
load per wheel of the other tests could be clearly identified as due to
the effect being investigated, rather than due to the effect of previous
damage.
The positions used for the tests are illustrated in .Figure 8.26.
The first test performed was the control single wheel test at position A
in Figure 8.26.
This was followed by the test over the cut reinforcement
at position C in the Figure and the load-deflection response of both these

tests is illustrated in Figure 8.34.
The behaviour in the two tests was
very much alike and also very similar, apart from the lower failure load,
to that in the equivalent tests on the first deck.
The fact that most of
the main steel had been cut right through beneath the wheel at B appeared
to have had very little effect, indeed the initial response was softer iri
the test with intact reinforcement although this was probably due to the
greater damage sustained by this region of slab in the global tests.
- 196-
As
in the tests on the f1rst deck, the failure loads were very similar to that
predicted by Kirkpatrick et al's approach.
184-kN
Load \. kN ) 200
150
100
.....:::
50
----------- A<intact
reinforcement )
C (c ut reinforc emen t >
.1.
0
0
10
Deflect ion <mm >

Figure 8.34:
Singl e wheel
t es t s A and C
176kN
200
Load
<kNt wh eel)
150
...-
100
...-
U6kN
B <single wheel)
- D <two Wheels)
50
10
Deflection (mm)
Figure 8 .35:
The
next
test
performed was
Local tests B and D
the
control test
expected, this behaved in a very similar fashion.
at
B and,
as might
be
For the final test, the
two wheel test, a new loading arrangement was required.
Since neither the
time nor the money was available to fabricate this, one of the four load
spreader rigs used for the global tests was employed, the main beam being
repositioned so that over 95% of the load was applied to two of the
wheels .
The load-deflection response for the two tests is illustrated in
Figure 8.35 and it will be seen that the presence of the second wheel had
a significant effect on both the deflection and the failure load although
the failure once again took the form of one wheel punching through the
- 197 -
deck.
The reduction in strength was approximately 17'1. <or 19'1. using the
average of the two control tests) which is slightly less than experienced
by Kirkpatrick(13) in bays with equivalent span and depth.
However, he
suggested
due
that
the
reduction
particular support conditions.

a
minor
contributory
he
observed
may
have
been
to
his
It now appears that this was probably only

If,
factor.
as
his
approach
appears
to
imply,
punching failures are caused directly by excessive shear stresses in the

region immediately round the wheel, it is hard to explain why the presence
of a second wheel should affect strength.
However if, as suggested here,
the failures are primarily flexural one would expect that any load which
increased the bending moment would reduce the strength.
Although
the
presence of a
second wheel
reduced
the
failure
load
per
wheel, it remained substantially (23%) higher than in the global tests.
It
thus appeared that the lower failure load per wheel in the global tests
was indeed partly due to global effects although the interaction of the
local effects of the two wheels was also significant.
As in the first deck, the single wheel tests were remarkably consistent
with
Kirkpatrick's,
the average ratio of
failure
load
being 1.20 compared with 1.19 in his own tests(13).

single wheel tests,
ratio was
that
1.183 with a
remarkably good
result,
to his
Considering all the
is those performed on both decks,

coefficient
of variation of 0.04.33.
even
lftrgely
for
predict ion
empirical
the average
This is a
formula developed
from tests on structures which were similar to those considered here.

small variation means that
observed
when
more
wheels
variation can be eliminated.
The
the possibility of the reduction in strength

were
applied
being
purely
due
to
random
However, paradoxically, even the fact that an
approach based on shear stresses gave such good predictions could be used
as an argument for saying that the failures are primarily flexural: shear
failure loads are inherently more variable than flexural failure loads and
it
would be extremely unusual to obtain such a small variation in the
shear strength of even apparently identical specimens.
8.9 TESTS ON SINGLE
BEAM
Although this is primarily a study of deck slab behaviour, it is clear from

the previous section that global behaviour is important to this.
behaviour of
important,
the
before
Thus the
beams has a significant effect and it was considered

analysing
the
complicated
- 198-
decks,
to
ensure
that
the
analysis was capable of modelling the relatively simple beam behaviour

correctly.
It was therefore decided to test a beam on its own.
The precast beam which was tested was similar to those used in the first
deck, and had been cast in the same batch.
It was provided with a one
metre wide in situ top flange cast in the same way as the models using a
similar mix which gave a cube strength at time of loading of 45. 7N/mm 2
The reinforcement provided in this flange was like that used in the second
deck.
It was considered that the flexural behaviour, which was of prime
concern,
would
be
very similar
for
the two
types of
beams so their
features were chosen for convenience.

Although it may appear obvious that the single beam should have a flange
width to match the beam spacing in the decks, this is less obvious when
the in-plane shear stiffness of the deck slab is considered:
in 3.2.7
that
the heaviest
loaded beam in a
it was noted
beam and slab deck can
effectively have a flange which is wider than the beam spacing.
This
means that concrete crushing failures may be less likely in bridge deck
tests than in single beam tests but no allowance was made for this effect.
The beam was positioned on bearings in the same way as in the deck tests.
The loads were applied with the same loading frame and jacks, although the
loading rig was modified to bring the wheels closer to the longitudinal
centre-line of the beam to avoid over-loading the slab.
To make the
results directly comparable, the longitudinal position of the loads was

kept the same as in the global failure tests.
Since neither the bearings
nor the loading rig provided significant restraint to rotation about the
longitudinal axis, a steel beam was placed across the top of the beam and
held down to the floor.
However, it proved acceptable to allow this system
to go slack in the test.

Two 100mm travel displacement transducers were provided to measure the
deflection.
These were mounted over the top of the beam to avoid damage
in the event of the sudden failure which was anticipated.
A number of
demec points were provided but no electronic strain gauges were used.
The
beam under test is illustrated in Figure 8.36.

Since the previous tests indicated that the beam behaviour was, for all
practical purposes, perfectly linear elastic until well above design service
load, there was little point in applying complicated load histories.
- 199-
The
beam was therefore loaded to 150kN

monotonically to failure.
load in Figure 8.37.
p~r
jack then unloaded, then loaded
The mid-span deflection is plotted against the
The response was linear elastic up to a load of
approximately 200kN per jack so the first load cycle cannot be seen in the
Figure.
Figure 8 .36:
Single beam under test
Load
500
<kN/Jack)
4.00
300
200
lOO
20
40
60
80
lOO
120
Deflection <mm >

Figure 8.37:
Load-deflection response of single beam
The behaviour prior to failure was remarkably similar in many respects to

that of Beam B of the first deck.
The crack patterns were very similar,
- 200-
including the shear cracks which are illustrated in Pigure 8.38.
Even the
loads at which the cracks appeared, expressed in kN per jack, were similar.
However,
the deflections were slightly higher at each load stage which
suggests that the load distribution in the first deck was slightly better
than that predicted by simple statics.
Figure 8.38:
Shear cracks in single beam
Despite the extensive shear cracks and the draw-in of the tendons which
had occurred by a deflection of 60mm <the global beam deflection at which
failure
occurred in the first
deck) the load increased by another 24%
before a sudden explosive failure occurred at a deflection of 110mm.
The
only warning of this was the increasing creep and the failure was so total
that
meant
the beam fell some 500mm even though the hydraulic system used
that
the load was
removed as soon as failure occurred.
After
failure, as illustrated in Pigure 8 .39, there was no concrete left at all in

the critical section.
Figure 8.39:
Beam after failure
It was clear that the failure had been caused by concrete crushing in the
top flangei what had been observed was the classic brittle bending failure
- 201-
of an over-reinforced <or in this case over-prestressed) concrete section.

However, it is easy to imagine that a two-dimensional version of such a
failure,
that
is in a slab rather than in a beam, would look like a
punching shear failure.

crushing concrete
Indeed, in a sense it is a shear failure because
fails
on
inclined planes.
Thus
there is no clear
distinction between the two types of failure.

8.10 DISCUSSION AND CONCLUSIONS
8.10.1 Service Load Tests
The condition of the first deck was satisfactory at the completion of the
tests.
Its behaviour during the tests was also apparently satisfactory
although it remains to check the distribution properties by comparison

with analysis.
The behaviour of the second deck was less satisfactory because relatively
large cracks opened in the top of the slab on loading and failed to close
on unloading.
Although the failure of these cracks to close may have been
due to the reinforcement detailing, the fact that they occurred only in the
second
deck
was
clearly
result
of
the
higher
transverse
moments
resulting from the presence of the diaphragms.

It appears, although it remains to check this by comparison with analysis,
that
compressive
membrane
action
did
not
greatly
contribute
resistance of either deck to global transverse moments.
to
the
However, as
predicted by previous researchers, compressive membrane action clearly did

contribute
suggestions,
diaphragms.
to
the
this
resistance
to
contribution
did
local
not
moments.
depend
Contrary
on
the
to
their
presence
of
The result was that the cracks in the top of the second deck,
the feature which led to its behaviour being considered less satisfactory,
were the direct result not only of an effect which has been largely
ignored by previous research <global transverse moments>, but also of a
feature which had
been positively recommended
<diaphragms>.
Although
these diaphragms improved the distribution properties, they appear to have

had a detrimental effect on the serviceability of the slab.
8.10.2 Failure Tests
The most significant aspect of the failure tests was not so much the
individual
failure
modes
or
loads
as
the
relationship
between
them.
Failure occurred in the first test when a single wheel punched through the
-202-
deck under a wheel load which represented only half the local strength of
the slab: not only as predicted by previous research but also as measured
by the subsequent single wheel tests.
Despite its diaphragms the second
deck, with its weaker concrete and smaller steel area, was weaker in the
single
wheel
tests
yet
it
was
able
to
resist
higher
global
load.
Clearly, the failures under full HB load were greatly influenced by the
global behaviour but this does not mean that they did not occur until beam
failure was imminent.
The maximum beam deflection in the global tests was
little more than half that at which failure occurred in the beam tested
alone.
That
beam had reached only 80% of
its failure
load when
its
deflection matched that at which the decks failed.

All this fits the hypothesis that the slab failures were primarily brittle
bending compression failures.
In the global tests, the global transverse
moments induced by the beam's differential displacements or <in the case

of the first deck) rotations used up some of the bending strength of the
slab.
though
Because, by virtue of the membrane forces,

locally
rein forced,
heavily
reinforced
even
though
their local behaviour was brittle.
the slabs behaved as

actually
very
lightly
Thus redistribution was
very limited and the safe theorem of plastic design did not apply.
slabs
failed
under
combined
global
and
local
The
transverse moments
even
though, at the failure load, the global transverse moments were not needed
to maintain equilibrium.
The single wheel tests confirmed the work of previous researchers.
they
suggested
that
the
enhancement
to
local
strength
Indeed
caused
by
compressive membrane action is remarkably tolerant of features which might

be expected to reduce restraint.
All the tests were performed 1n outer
slab bays which had already been extensively cracked for their full length
by previous tests; half were performed after other failure tests in the
same bay.
Two of
remaining tests
the tests were
1n
a deck without diaphragms;
the
(all those on the second deck) were performed close to
points where wheels had punched through the adjacent bay of the slab.
Despite all this, the behaviour had been entirely satisfactory.
The only
thing which significantly reduced the strength relative to that predicted

by previous research was the presence of an adjacent wheel under load.
Even with this, the failure load for the very lightly reinforced slab was
equivalent to four times design ultimate load.
-203-
The global tests confirmed that the contribution of compressive membrane

action to resisting global transverse moments is, at best, substantially
less than the contribution to local behaviour.
Because the local behaviour
was so enhanced by membrane action, the crack pattern prior to failure

tended
to
effects.
be
greatly
influenced,
and
in
places
dominated,
by
global
As in the service load tests, global- transverse moments <which
have been largely ignored by previous research) had a major influence on

behaviour.
However, despite the large reduction in local strength caused
by this, the failure loads were still very high; a minimum of 2.83 times
design ultimate load.
It might, therefore, be thought that the reduction
had no practical significance.
This may not be the case.
The failures occurred when the combined local and global moments became
too great for the slab.
local
moments
are
This has important implications because, whilst
direct
effect
of
the
load
on
the
slab,
transverse moments are only indirectly an effect of this.

direct effect of the differential deflections of the beams.
which
increases
these
differential
strength of the slab.
deflections
could
The implication is that
global
They are a
Thus anything
reduce
the
local
if the beams had been
weaker, or less stiff, the slab would have failed in the same way but at a
lower load.
Perhaps the most important conclusion from the tests is that, as predicted
in 3.2.7, global and local behaviour are not independent.
Most previous
research on bridge deck behaviour has implicitly <or sometimes explicitly)

assumed that they are.
The result is that most of the previous research
on membrane action in bridge deck slabs is only strictly applicable under

single
wheel
loads.
This
does not
necessarily
mean
that
recommendations resulting from that research are unsafe.

only
because
of
the
large
reserve
of
strength
of
the design
Indeed, even if
prestressed
beams
designed to current rules, it seems likely that bridges designed using the
empirical rules discussed in 3.2.8 will have more than adequate strength.
Nevertheless it does mean that caution is required.
bridge
assessed
to
the
Ontario
assessment
It appears that a
rules(ll)
as
having
just
adequate global and local strength could actually have a much lower safety
factor
than
intended.
As
for
service
investigation is required.
-204-
load
behaviour,
analytical
CHAPTER
ANALYSIS
OF
MODEL
9
BRIDGE
TESTS
9.1 INTRODUCTION
The tests described in the last chapter provide useful empirical evidence
for the contribution of compressive membrane action to the behaviour of
bridge
deck
slabs.
They
could
help
with
the
development
of,
and
justification for, empirical design rules such as those considered in 3.2.8.

However, in order to appreciate the significance of the behaviour, it is
necessary to compare it with analyses.
conventional
analyses,
to
quantify
the
Firstly it will be compared with

potential
savings
from
using
membrane action in the design of deck slab reinforcement and to see if

the distribution properties predicted by conventional methods for global
analysis, based on uncracked slab properties, were realised.
will be compared with the form of analysis considered in
Secondly it
Chapter 7, both
to see if that analysis would have provided a suitable design method for
the models and to obtain some understanding of the behaviour.
9.2 CONVENTIONAL ANALYSIS

9.2.1 Analysis for Design of Deck Slabs
The deck slabs of both the bridges were checked using BS 54.00 and the
analytical methods which would normally be used with it and which were
considered
in
2.4.
A linear grillage model
was
used
for
the global
analysis and Westergaard's formula(39) was used for the local analysis.
a. First Deck
The
allowable
load
on
the
approximately 14. units of HB.
first
deck
slab
using
this
approach
was
For the intended design load of 4.5 units of
HB the reinforcement required was Tt0-87.5, the odd spacing giving the
minimum steel area and being equivalent to 175mm at full size.
nearly four
times
the steel area actually provided.
This is
The failure
load
implied for the reinforcement provided (setting all Ym values to 1.0) was
14..3kN per wheel compared with the actual failure load of approximately
103.5kN per wheel in the global tests.
The implied failure load under a
single wheel was 21 kN compared with the actual failure load of over 200kN.
It is thus clear that the conventional analytical approach under-estimates
local strength, apparently by a factor of up to ten.
- 205-
It was nofed in
Chapter 2, however, that the use of linear analysis at the ultimate limit
state is merely a convenient way of avoidirig the requirement to check
stresses
under
service
loads.
It
may
therefore
be
considered
more
realistic to compare the failure loads with the predictions of yield-line

theory.
This approach, which is normally considered to give an upper-bound
solution, under-estimated the failure load in the single wheel tests by a

factor of approximately two.
The failure load of the slab in the global
tests was reasonably close to that predicted by yield-line theory although

the failure mechanism was so different that this can be little more than
coincidence.
Even
if
yield-line
reinforcement
analysis
used
in
design
to
BS
5-iOO,
the
provided would be little reduced since the stress limits
would become critical<35).

reinforcement
approach.
were
as
It
the basis of
is thus reasonable to use the Tl0-87.5

comparison with
the conventional design
Since it has been noted in previous chapters that the critical
criteria are serviceability criteria under full global load,
it
is most
realistic to compare the observed behaviour with the conventional design

approach on
considered
this basis.
As
just satisfactory,
the serviceability of the first
deck was
the best comparison with the conventional
design approach is to say that it over-estimated the steel required by a

factor of nearly four.
The steel area provided approximated very closely to that required by the
conventional approach to resist the global transverse moments alone.
b. Second Deck
The allowable load calculated for .the second deck was approximately 16
units of HB, which is slightly higher than for the first deck.
However,
the calculation was based on the strength of the slab in sagging.
It is
common, when designing this type of slab using Westergaard's approach, to

analyse only sagging and then to provide.the same steel in the top.
Since
the single layer of steel provided was lOmm below mid-depth, this approach
was not valid for this deck and analysis of hogging would have given a
lower allowable load.
It was not possible to design a single layer of steel to resist 45 units
of HB using normal design methods because the calculated bending moments

exceeded the concrete capacity.
Using two layers of steel, the requirement

-206-
was !ust marginally higher than for the first deck, due to slightly greater
global transverse moments.
However, this reinforcement was designed using
a nominal vnlue for concrete cube strength of 40N/mm 2
which is the normal
value used in deck slabs and was slightly conservative for the concrete
used in the first deck.
The concrete in the second deck was weaker and
using
strength
the
actual
cube
in
the
design
calculations
made
it
impossible to design the reinforcement for 45 units of HB.

The steel area provided in this deck also approximated very closely to
that required by the conventional British design approach to resist global
transverse
sagging moments.
However,
due
to
the reinforcement
being
located below mid-depth, the moment capacity in hogging was only some 50%
of the maximum transverse global moment given by the grillage.
As in the first deck, the failure loads in the single wheel tests were
approximately double the values predicted by yield-line theory.
9.2.2 Analysis for Design of Beams

Since
the beams were provided with more prestress
satisfactory
behaviour
in
the
service
tests
proves
than normal,
very
little.
their
The
distribution properties of the deck can only be investigated by comparing

predicted and measured displacements or strains.
a. First Deck
In Figure 9.!, the maximum mid-span beam deflection in the test on the
first deck is compared with the prediction of the linear grillage analysis
which used 9 nodes per beam.
load distribution is also shown.
For comparison, the prediction of a static

In order to eliminate the effect of error
in predicting the stiffness of the beams, as opposed to error in predicting

the distribution properties, the displacement is expressed as
the average displacement of the four beams.
whilst
the
behaviour
of
the
beams
is
are shown for the first
factor of
This approach is only valid
linear-elastic
throughout the range plotted in the Figure.
a_
but
this applied
The distribution properties
time that each load level was applied and the
breaks in the plot indicate points where, as described in Chapter 8, the

loading was not continuous.
Figure 9.1 appears to show that the grillage prediction using the grossconcrete properties for the slab is very good.
-207-
Initially, the distribution
was slightly better than the analysis predicts but this might be expected
because the analysis ignores the shear connection between the top flanges
of
the
beams.
As
the
load
increased,
the
distribution
deteriorated
slightly due to the reduction in transverse stiffness caused by concrete's

non-linearity and cracking in tension.
By 120kN per jack, the design
service load, the distribution was worse than the analysis suggests but
the deflection of the heaviest loaded beam was only 3.7% higher than the
prediction and, even under design ultimate load, the discrepancy was less
than 5%.
The global transverse moments predicted by the grillage implied
concrete stresses in excess of
5N/ mm ~
in the slab concrete over the beams
but this concrete was apparently uncracked.

Static Load Distribution
of Beam B 2.0
t:;
Mean
(~:;
= mid-span
~:;
beam 1.8
deflection)
Test
from start of
(t:;
this loading)
1.6
--- - - - -~---- - ----
Linear Grillage
-~--
1.4
Test (/:; from start of tests)

1.2
1.0
25
50
75
100
125
150
175
200
Load (ki/Jack )
Figure 9. 1.
Beam deflections of first deck
<conventional analysis )
The final part of the plot in the Figure, that for loads above 150kN/jack,
relates t o the load application at the end of the service load tests when
the load was returned to its original position.
By this stage, there was a
significant permanent deflection and, as will be seen from the Figure, it

makes a difference to the apparent distribution properties whether the
total deflections <that is, the deflections relative to the original zero)
or
the deflections only from
considered.
of this
load application are
The latter probably gives a better indication of the live- load
stresses so
continues
the start
to
it
appears
deteriorate
that,
as
might
as
the
load
be expected,
increases.
the distribution
Despite
this,
distribution remains very much better than the static load distribution.
- 208-
the
Deflections
are
frequently
used
as
properties of beam and slab decks.
an
indicator
of
the
distribution
However, it is the tensile stresses in
the soffits of the beams which are normally the critical criteria in design
so these are a better indicator.
In Figure 9.2, the predicted strains on
the top and the bot tom of each beam at mid-span are compared with the
measured strains for a particular load level.
Because the beam behaviour
was linear- elastic at this stage, it is reasonable to assume that stress

was proportional to strain.
As in Figure 9.1, the results are expressed as a factor of the average
value f or the four beams.
This makes the grillage predictions identical
for the top and soff it.

Strain at Beam
Mean Strain at Beams
Test <top of slab )
Test <soffit >

Linear Grillage
1. 5
1.0
+
0
0.5
+
0.0
Beam A
Figure 9.2:
Beam C
Beam B
Beam D
Beam strains 1n firs t deck
(end of service load tests: load
= 150kN/jack)
Because the distribution properties of the deck are poor, the analysis
predicts
beams.
very significant
differences
between
the stresses in adjacent
This is reasonable for the soffit stresses because there is no
direct connection between the bottom flanges of the beams.
However, the
analysis implies that the stress distribution across the slab is literally
as shown in the Figure; with large discontinuities between adjacent pieces
of concrete.
This is impossible and shear in the slab evens out the
longitudinal stresses in the slab as can be seen in the Figure.

this does not even out the soffit stresses.
- 209 -
However,
Indeed, because it reduces the
differential dieflections of the beams and hence reduces the contribution

of global transverse moments to distribution, it can cause a deterioration
of the distribution properties as expressed by soffit stresses.
The result
is that whilst Figure 9.1 suggests that the distribution at this load stage
is only 5% worse than the grillage prediction, Figure 9.2 shows that it is
16t worse.
This may suggest that the distribution properties of this deck, with its
very light reinforcement, were unsatisfactory and that the fears expressed
in 3.2.7 are confirmed.
However, the increase in the percentage load
carried by the heaviest loaded beam was only approximately
5~
between the
initial linear condition and the load stage considered in Figure 9.2 which
is the design ultimate load.
This implies that, even if the deck slab had
been so heavily reinforced
that
it remained effectively linear-elastic
after cracking, the grillage based on gross-concrete slab properties would

have
under-estimated
the maximum soffit
stress by approximately
10~.
Thus most of the discrepancy was due to a normally accepted error in

analysis for design, not the reduced steel area.
b. Second Deck
In Figure 9.3, the greatest mid-span beam deflection in the tests on the
second deck is compared with the prediction of a linear grillage.
The
prediction of the static distribution is not shown but it corresponds to a

value of 2.5 in the Figure.
As in Figure 9.1, the maximum beam deflection
is expressed as a factor of the average deflection of all the beams.

However, because of the parapet up-stands in this deck, the four beams
were not identical.
The approach is not, therefore, quite such a reliable
guide to distribution because errors in predicting the difference between

the stiffness of the edge and inner beams would show up in the Figure.
The longitudinal strains in the slab over the beams are illustrated in
Figure
9.~
Figure 9.5.
first
deck,
and
the
soffit
strains
in
the
beams
are
illustrated
in
The reason for using two figures, rather than one as for the
is that
because of the different edge beams the grillage
predictions in the two figures are different.
The reason for choosing a
different load stage to illustrate is related to difficulties experienced in

both tests with the strain gauges.
- 210-
A of Beam C 2 . 0
Mean
(A=
Test
from start of
(A
this loading>
mid-span beam 1.8
--
deflection)
1.6
Linear Grillage
1.4
Test
(A
from start of tests)
1. 2
1. 0
75
50
25
100
125
150
175
200
225
Load <kN/ Jack)

Figure 9.3:
Beam deflections of second deck
<conventional analysis)
Figure
9.3
appears
to
suggest
that
the
distribution
significantly better than the grillage prediction.
was
initially
Assuming that this
difference was due to the shear connection between the top flanges the
extent of the difference is surprising: theoretically it should be less
than
for
the
first
deck.
However,
a detailed study of
the results
suggested that part of the discrepancy may have been because the analysis
exaggerated the effect of the parapet up-stands and hence exaggerated the
stiffness of the edge beams.
This was despite the analysis using the
measured
beam
and
than
the
values
E"'
substantially
more
for
the
different
strength difference would suggest.
parapet
nominal
concretes
which
(or
the
even
were
actual>
The reason for the small effect of the
parapets was probably that they were cracked due to plastic settlement .
The first break in the plot in Figure 9.3 corresponds to the end of the
first
load cycle to service load.
At the end of this cycle the more
heavily loaded beams did not return to their original positions.
The outer
beams
permanent
had
small
negative
deflections
suggesting
that
the
deflection was mainly due to some of the transverse curvature becoming

permanent.
This implies that there were locked-in stresses in the beams.
Since the next part of the plot relates to a load application immediately
after the first, these stresses would not have been relieved greatly by
creep.
Thus the solid line, relating to the distribution calculated from
the total deflections,
probably gives the best

- 211 -
indication of the load
distribution in the next section of the plot.
In contrast, the permanent
de flections at the s t art of the load application represented by the fina l

part of the plot were relatively uniform across the deck, implying that
they were
largely due
dotted line gives a

stage.
to permanent beam curvature and
thus that
the
better indication of the load distribution at this
It will thus be seen that the distribution properties deteriorated
progressively throughout the test and that the deterioration was greater
than
for
the
first
deck.
The greater deterioration
might
have
been
expected both from the extent of apparently global cracking in the slab
and from the fact that the steel provided was inadequate to resist the
global transverse moments predicted by the analysis.
Stra in at Beam
2.0
Test
Linear Grillage
1.5
1.0
0.5
0.0
Beam A
Figure 9.4:
Be a m B
Beam C
Beam E
Beam D
Slab strains over beams in second deck
(f irst loading;
load = 120kN/jack)
Figure 9.4 shows that, as for the first deck and for the same reason, the
stresses
predicts.
in
the
slab
were
more
evenly
distributed
than
the grillage
Figure 9.5 shows that the maximum soffit stress was higher than
the analysis predicts.
The margin is so small it would not normally be
considered significant.
However, Figure 9.3 shows that there was further
deterioration
in
the
Figure 9.5 relates.

that
the
worst
distribution properties after
the stage to
which
By the completion of the service load tests it appears
soffit
stress
was
approximately
10%
higher
than
the
grillage prediction although still some 30% lower than the static load
distribution suggests.
- 212-
2. 0 .
Stra in at Beam
Test
Linear Grillage

-t-
1. 5
+
1.0
...
-
+
0.5
0.0
Beam A
Figure 9.5:
Beam B
Beam C
Beam D
Beam E
Beam soffit strains in second deck
(first loading: load = 120kN/jack)
9.3
NON-L~
ANALYSIS
9 .3.1 Single Beam
The s ingle
beam
test
served
to
check
that
the
analysis modelled the
behaviour of the beams correctly, and thus to ensure that any errors in
the predictions for the behaviour of the model bridge decks were not due
to failure to model the behaviour of the beams.
Because of this it is
convenient to consider these tests first.

The predicted and observed load-displacement relationships are shown in
Figure 9 .6 .
They are as close together as can reasonably be expected;
larger discrepancies are frequently observed between the behaviour of two

nominally identical concrete specimens even when they both come from the
same batch of concrete.
- 213-
Load
500
<kN/Jack)
400
~
300
200
- - - - Analysis
-Test
100
60
40
20
80
100
120
Deflect ion
Figure 9.6:
<mm>
Analysis of single beam tesc
9.3.2 First Deck
a. Global Tests; coarse mesh analysis

The deck was initially analysed using a 15 by 12 node model.
This gave
four
considered
transverse
reasonable.
elements
between
each
beam,
which
was
However, these elements were 727mm wide so it was assumed
the model would be too coarse to model the local behaviour correctly.
The
reason for using such a coarse mesh was that this was the largest model
which would fit in the 386 desk- top computer used.
The computer model was loaded monotonically to failure under the load case
used in the tests and the predicted central deflections of the beams are
shown along with the test results in Figure 9. 7.
Considering the many
approximations in the analysis, the complexity of the behaviour and the

usual variability of the behaviour of concrete structures, the predictions
are
reasonably
good.
Because
the
analysis
was
performed
under
load-
control using relatively large increments, it did not pin-point t he failure

load precisely.
However, the best estimate of the failure load whi ch could
be obtained from the analysis was 400kN per jack compared with the actual
value of 414.
The analysis also predicted correctly several features of
the behaviour which might otherwise have been considered surprising.
It
predicted that the first cracking would occur in the soffit of the slab
under wheel 10 (as shown in Figure 8.11) but that at later stages of the
loading the slab would be much more highly stressed in the outer bay, that
is under wheels
to 4.
It
also predicted correctly that, as
failure
approached, the slab adjacent to wheel 4 would be subjected to transverse

- 214-
sagging moments right out to Beam A.

at
failure
and
the
mode
of
Finally, it predicted both the load
failure
surprisingly well;
in
the
final
increment plotted in the Figure, concrete was crushing most extensively

under wheel 4. and it was
converge
in t his regi on t hat the anal ys is
failed to
gave l arge deflect i ons in the next increment.
an~
Beam D
400
Load
<kN/ Jack >
Beam A
Beam C
Beam B
350
300
250
2 00
150
Analysis
100
Test
50
0
10
20
30
40
50
60
Deflection <mm>
Figure 9. 7:
Beam deflections o f first deck
(from non-linear analysis using a coarse element mesh )

With such a
coarse element
mesh,
the correct
prediction of such an
apparently local failure might be considered surprising.

be three explanations for it.
There appear to
Firstly, although the final collapse took the
form of local punching due to the local shear stress, it was apparent that
the concrete was crushing along a line which extended from wheel 3 to
slightly
beyond wheel 4.
This crushing,
which apparently caused the
failure and which was predicted by the analysis, thus extended over the
width of two elements of the computer model enabling the resulting failure
to be predicted.
Secondly, as was found in Chapter 7, the analysis tends
to be conservative in its prediction for punching failure loads and this

cancelled out the failure to model the peak of the stress concentration
under the wheel; thus a finer mesh would have led to an under-estimate of
failure load.
model,
the
Thirdly, in an attempt to cancel out the fault of the coarse

loads
were
applied
as
points
distribution over the patch area.
- 215-
with
no
allowance
for
the
This
ability
of
the
analysis
to
predict
failure
load
and
mode
is
reassuring; in particular, the fact that it still gives low predictions even
when a coarse mesh is used means that
is safe for
it
use in design.
However, the model tests suggest that the limiting service load for the
bridge was equivalent
to around 120 to 150kN per jack.
In BS 5400, a
design service HB load of !50kN corresponds to a design ultimate HB load

of 204kN*.
should
be
requiring
Even allowing for the fact that the material safety factors
applied
an
debatable),
in
extra
it
the
analysis
15% strength
is
clear
that
for
for
the
ultimate
brittle
serviceability
failure
is
limit
state,
mode
critical.
and
(which
is
Thus
the
there was local soffit cracking under wheel 10 at
the
predictions for the lower load stages are more important.

In the analysis,
first increment, 50kN per jack, and there was limited top cracking in the
slab by the second increment, IOOkN per jack.
In the tests, such cracking
was not observed until loads of 110 and 300kN per jack respectively.
main
reason
for
this
very
large
discrepancy
appears
to
be
that
The
the
analysis assumed the concrete to be linear-elastic until cracking, whereas

in fact
there clearly was a significant departure from linearity before
cracking.
The fault was exaggerated by the use of a low tensile strength
for the concrete in the analysis, 0.67 times the split cylinder strength.
This was chosen because it gave the best results in Chapter 6 for lightly
reinforced specimens.
However, because of the scope for redistribution to
the steel, the use of the full split cylinder strength gave better results
in heavily reinforced specimens.
lightly
reinforced
there
was
Although the slab of this bridge was
great
scope
for
redistribution.
It
thus
appears that in this respect, as in their failure mode and load, lightly
reinforced
restrained
slabs
behave
like heavily
reinforced
unrestrained
slabs.
*Footnote
The ratio of design ultimate to design service load implied by this is
higher than that used in Chapter 8.
This arises from the Author's
interpretation of the factor y~ 3 in BS 5400.
y," is a partial safety
factor for errors in analysis which, in BS 5400: Part 4, is applied to the
loads. Since the form of analysis considered here is not elastic, a factor
of 1.15 is used at the ultimate limit state as specified by the code.
However, when considering the load to be applied to models, the author has
assumed that, since no analysis is involved, y; 3 can be 1.0. This might be
considered debatable since y, 3 also covers errors in dimensions. However
applying y,,. to the loads used in Chapter 8 would not alter any of the
conclusions.
- 216-
The
behaviour
linearity of
in
the
tests
concrete
in
was
clearly greatly
tension
tension after cracking.
and
Although
by
affected
its ability
to
by
the
non-
transmit
some
the analytical prediction of cracking
under wheel 10 by a load of 50kN per jack was greatly premature, the
measured strain in this region at this stage was 100 microstrain which,
with
E~
the measured
value,
implied a
stress of approximately 3N/mm"'
Assuming concrete to be linear elastic in tension until it cracks, this

would
certainly
imply
that
cracking
was at
least
imminent.
In
fact,
although non-linearity is visible in Figure 8.12 from approximately this

stage and becomes very pronounced by a load of 75kN per jack, cracking was
not visible until a load of 1l0kN.
Although the analysis exaggerated the amount of cracking, there was not a
corresponding exaggeration of the stresses in the slab.
Using the analysis
in the same way as a conventional linear analysis, that is calculating the

stresses from the element
forces given by the program using a cracked
elastic section analysis ignoring the tensile strength of the concrete, the
allowable service load from the BS 54.00 criteria was approximately 1l0kN;
the critical criterion being the steel stress.
Although the actual steel
stress in the model was unknown and probably substantially lower than the
34.5N/mm2 implied by this, it was concluded in Chapter 8 that the behaviour
was just acceptable for the load history applied which had been intended
to simulate the life of a bridge with a design service load of 120kN.
Thus the analysis was conservative although it still allowed nearly three
times the load on the deck that a conventional analysis would allow.
For reasons discussed
material
models
concrete's
to
ductility
in Chapter 6,
make
in
development of cracking.
it
the
analysis
tension
and
is not
possible to adjust
reproduce
hence
to
the
full
predict
the
effect
correctly
of
the
Indeed, it is debatable whether this is desirable
since the effect is probably size dependent and thus an analysis which did
this for the model would be incorrect for a full size bridge.
analysis appeared
development
of
to be as good as possible.
cracking
did
have
an
However,
undesirable
effect;
Thus the
the premature
it
made
the
analysis exaggerate the rate of decay of the distribution properties, a

trend which can be observed from Figure 9. 7.
Before
cracking,
substantially
the
better
predictions
than
those of
for
a
- 217-
the
strain
conventional
in
the
beams
linear grillage;
were
not
because of non-linear it ies but
because of modelling
shear connection of the top flanges of the beams.
the effect
of the
However, by a load of
150kN per jack, the error in the predicted soffit strain was as great as
that of the linear grillage although, unlike for the linear analysis, the
error was in
properties
analysis.
the safe direction.
was
due
to
the
The premature decay of distribution
premature
development
of
cracking
This was not entirely due to the material model used.
partly due to the failure of the analysis

finite width of the beam webs.
in
the
It was
to model the effect of the
The predicted hogging moment in the slab
at the position corresponding to the face of the web in the model was
little more than half
that over the centre-line of the beams.
Thus a
length of transverse element which was, in fact, uncracked and effectively

very deep was modelled as being shallow and cracked.
b. Global Tests; fine mesh analysis

The model bridge was re-analysed using a finer mesh to give a better
indication of the behaviour.
Because the previous analysis had shown that
the most highly stressed regions of the slab were not confined to small
areas
and
that
the
most
highly stressed region
moved as
the
loading
progressed, it was undesirable to restrict the fine mesh to local critical

areas,
as
Chapter 7.
had
been
done
in
the
analysis
of
Kirkpatrick's
tests
in
Because of this the computer model was too large to run on
the desk top computer and it was transferred to a Vax 111750 machine.
This
greatly
increased
the
space
available
but
the
machine
was
significantly slower than the 386 and this imposed a practical limit on
the size of model which could be analysed.
Six transverse elements were used between each beam and those adjacent to
the beam were made shorter so that their ends coincided with the face of
the web.
They were given a full width lOOmm deep web to represent the
presence of the top flange of the beam.
32 nodes were used along the
span of the bridge giving 258mm wide elements and a total of 672 nodes.
The full split cylinder value was used for the effective tensile strength
of the concrete and, unlike in the coarse mesh analysis, the finite size of
the load patches was represented.
The
load history of
the service
tests
was
simulated
removing the test load from the six different positions.

- 218-
by applying and
However, in order
to keep the computer time used within reasonable limits, only the first
and last loadings were analysed in detail.
The loads were applied to and
removed from the other positions in single increments.

The predicted displacement of wheel 9 relative to the beams is shown in
Figure 9.8.
On first loading, the analysis still under-estimated the load
to cause visible cracking although not by as large a margin.
It also
over- estimated the loss of stiffness caused by this cracking.
This is
again due to its failure to model the ductility of concrete in tension.

This
is
far
more
significant
factor
in
this
highly
indeterminate
structure than in the simple strips considered in Chapter 6.
In a simple
beam, despite the ductility of concrete in tension, as soon as cracks form

they extend well above the soffit.
despite
the
relatively
brittle
In the analysis of this bridge deck,

concrete
model
used,
the
scope
for
redistribution meant that it was common for cracks to form at only one of
the eight integration stations through the depth of the slab.
This meant
that the area of concrete which is strained out of the linear range but
still resisting tension is larger and further from the neutral axis.
It is
thus far more significant t o the behaviour.
250
Load
<kN/ Jac k )
200
---Analysis
150
-Test
100
50
loading in other positions
0
0
0.5
1.0
1.5
2.0
Deflection <mm relative to beams)
Fig ure 9.8:
(from non-linear analysis using a fine element mesh)

The over-estimate of the deflection in the final loading is also probably
due to the failure to model the effect of the ductility of concrete in
- 219-
tension.
However, considering the many unknowns in the analysis and the
complexity of the behaviour, the prediction is remarkably good.

Unfortunately, the analysis could not be taken up to failure.
300kN
per
jack,
when
the
effect
of
cracking
in
the
By a load of
beams
became
significant, the convergence rate became excessively slow and it was not
possible to obtain a sufficiently accurate solution using a reasonable
amount of computer time.
However, the analysis appeared to confirm that
the allowable service load on the deck was approximately 120kN per jack
and the design ultimate load corresponding to this is only 172 .5kN per
jack.
Thus the analysis had shown that the deck had at least 50% more
ultimate strength than could be used in design.

The beam deflections predicted by the analysis are shown in Figure 9.9.
Because of the use of a higher tensile strength for the slab concrete, and
because of the modelling of the finite width of the beams, this analysis
predicted better distribution properties than the coarse analysis.
predicted
beam
deflections are as close to
the
test
The
results as can
reasonably be expected.
Beam D Beam A Beam C
Load
<kN /J ack )
25 0
Beam B
200
150
Analysis
100
--Test
50
0
5
10
15
Deflection
Figure 9.9:
20
(mm)
Beam deflections of first deck
(from non- linear analysis using a fine element mesh)

Since the computer models had given good predictions for the behaviour of
the deck, it seemed reasonable to use them to obtain some insight into the
mechanism
by which
this behaviour was achieved.

- 220-
In Figure 9.10 the
transverse restraint force across the longitudinal centre-line of the deck

predicted by the computer model is shown for a particular load stage.
As
might be expected from previous research, there is a compressive force in

the region of the wheels.
However,
the compressive force
in the end
regions of the deck (which is apparently due to the effect illustrated in

Figure 3.12) is much greater.
of the deck is in tension.
Thus, with these forces to resist, the rest
The behaviour is different from that described
or implied by other researchers: with significant compressive force in the

end regions their assumption that diaphragms are needed to resist tension
cannot be correct.
The tension required to resist the compression in the
critical areas comes

critical areas.
from
material which
is
relatively
close
to
those
The analysis also suggests that t he restraint force is
more localised than previously supposed in the transverse direction.
The
plot in Figure 9.10 is based on the average of the forces in the elements
on
either
side
of
the
centre-line.
predicted for a section at
the
The
transverse
restraint
face of the web is markedly different,
barely going into tension at all adjacent to the wheel positions.

appears to confirm
force
the suggestion in 3.2.5
This
that membrane action under
service loads is not dependent on external restraint and could still be

significant in unrestrained slabs.
It is also clear from Figure 9.10 that the resistance to global transverse
moments is not enhanced by compressive membrane action since much of the

relevant area of the deck is in tension.
.Wheel Posmons
Resrraint 200
Force (kNAN
1
100
-100
Figure 9.10:
Predicted force across centre-line of first deck
(from fine analysis; load
= 250kN/jack)
The coarse mesh analysis suggested that the area of slab in compression
around the wheels extended in both directions as failur e approached.
- 221-
The
restraint
force also increased disproportionately as the load increased.
However, the general form of Figure 9.10 remained unchanged and the end
regions of the deck were subjected to an increasing compressive force.
As the predicted distribution of restraint forces was so different from
that
implied
or
described
by
previous researchers,
was considered
it
highly desirable to check it by attempting to measure the restraint forces

in the model.
concrete,
Because of the many variables in the behaviour of cracked

was
it
considered
best
to do
this
for
section where the
concrete would be uncracked.
Accordingly, a series of transverse demec
points
top
were
attached
to
the
and
bottom of
the slab at
matching
positions close to the assumed point of transverse contraflexure in the

slab.
Unfortunately,
results.
Firstly,
sufficient
to
be
there
were
although
the
highly
many
in
interpreting
the
indicated
in
Figure
are
forces
significant
reinforced cracked section,
difficulties
to
the
they represent
concrete section, typically 1N/mm 2
behaviour
of
9.10
a
lightly
low stress on the gross-
which makes them difficult to measur.e.
This was made worse by the significant longitudinal compression in the

deck due to global moments.
It was necessary to correct for the Poisson's
ratio effect of this and, although longitudinal demec points were provided
to enable the longitudinal strains to be measured, the corrections were
inevitably inaccurate if only because of the uncertainty in the Poisson's
ratio used.
Since the correction was often significantly greater than the
measured transverse strain, errors in the correction had a large effect on

the estimated transverse forces.
A second difficulty was caused by the effect of cracking.
Although there
were no visible cracks within the gauge lengths, some of the measured
tensile strains were in excess of 100 microstrain which would normally be
taken to imply that there would be some non-linearity in the behaviour.
More seriously, cracks outside the gauge length but close enough to affect
stresses within
considered
in
it
<that
6.2.1)
could
concrete transferring it
making
the
is cracks within S0
restraint
release
some of
to the steel.
force
estimated
of
the
the gauge
tensile
length as
stress
in
the
This would have the effect of

from
compressive than the actual restraint force.
the
strain
readings
It appears that
more
this must
have been a significant effect since integration of the restraint forces

estimated
from
all
the
readings
appeared
to
imply
that
significant net transverse compression across the bridge.

-222-
there
was
Since the beams
were
restrained
impossible.
only
by
the
flexible
elastomeric
bearings,
this
was
However, it is perhaps significant that if the bridge had been
provided with
diaphragms
one
might
reasonably have supposed
that
the
"compression" in the slab was resisted by tension in the diaphragms.

The one case where the demec readings did give a reasonable indication of
the transverse force w11s for the ends of the slab when the
bridg~
was
loaded in the first position.
Since this was a free edge, the longitudinal
stress was clearly zero so no Poisson's ratio correction was required.

Similarly, since the nearest visible crack was over a metre away it seemed
reasonable to suppose that the strain readings could not have been much
affected
by
cracks.
Another
advantage
was
that
it
was
possible
to
position demec points at mid-depth of the slab as well as on the top and
bottom surface.
Unfortunately, where this was done, the mid-depth gauge
gave a strain which was significantly different from the mean of the top
and bot tom gauges.
behaviour
plane.
which
This was presumably due to non-linearities in the
invalidated
the
assumption
that
plane sections
remain
The maximum measured tensile strain, over 150 microstrain, also
implied that concrete non-linearity was possible.
These difficulties meant
that, even for the. ends of ttie slab; the restraint

_estimated to within plus or minus some 50%.
force could only be
Nevertheless the results were
significant; all four_ demec sets showed a compressive strain which was
approximately as predicted by the analysis.
Given that previous research
implied that this region should be in -tension, this alone appeared to be

_sufficient to show that Figure 9.10 was closer to reality than were the
implications of previous research ..
The restraint
behaviour
but
enhancement
forces
not,
predicted by the
on
their
own,
analysis. are significant
sufficient
t_o
explain
the
to the
enormous
relative to the predictions of conventional design methods.
An equally significant mechanism is moment redistribution.

factor here is the orthotropic nature of the cracked slab.
An important
It has already
been noted that, with such light reinforcement, the cracked stiffness is
only some 10% of the uncracked stiffness.
rt is clear from the figures in
Chapter 6 that the tangent stiffness of the cracked section is lower still.
Under
global _load,
the deck slab was subjected
to a
very significant
longitudinal compressive stress which delayed the_ formation of transverse

cracks,
value.
hence the longitudinal stiffness remained at
its full uncracked
The result was that the distribution of the transverse moments,

-223-
both in the real slab and in the analysis, was very much more uniform than
implied by a conventional analysis.
This redistribution of the moments is
due to cracking hence, unlike redistribution due to reinforcement yielding,

it
starts to take effect before there is any material damage which is
unacceptable under service loads.
c. Local Test
The
distribution
of
restraint
forces
indicated
in
Figure
9.10
significantly different from that implied by previous research.

the
load
case
considered
was
also
significantly
different
is
However,
from
that
investigated by previous researchers in that 16 wheel loads were applied

It is possible that this was the reason for
instead of only one or two.

the difference.
To investigate this, it was decided to re-analyse the deck
for a single wheel load.
The wheel was positioned in approximately the
position of single wheel A in the tests but the analysis was not directly
comparable with the test.
In the analysis the load was applied to the
undamaged bridge whereas in the tests it was not applied until after the
bridge had been loaded to failure under full global load.
Because
of
the
stiffer
initial
uncracked
response
slab
than
the
was
analysis
observed
predicted
in
the
significantly
tests.
As
failure
approached, the crack pat tern in the test began to be dominated by the
single wheel and consequently the difference between the test and analysis
reduced.
The analysis converged under a load of 220kN but indicated that
failure due to local concrete crushing round the wheel would occur before
230kN.
This is remarkably good agreement with the failure load in the
test which was approximately 226kN.
However, although very fine for the
analysis of a whole bridge deck, the element mesh used was still slightly
too coarse for a local analysis as is indicated by the large difference
between the restraint forces in adjacent elements in Figure 9.11.
It is
likely that a finer mesh would have given a slightly lower failure load.
It is also possible that
higher
if
failure.
the failure load in the test would have been
the deck had not
been damaged by
the
previous
loading
to
However, other tests and analyses suggest that this effect would
have been very small.

In Figure 9.11 the restraint force predicted across the centre of the bay
of slab between Beams D and E is illustrated for two different load levels.
The
first
of
these,
60kN;
is
close
to
- 224-
the
wheel
load
considered
in
Figure 9. 10.
For this relatively low load, the forces are plotted only for
the region around the wheel.

Wheel Position
FEs traint 400

force (k NJrrJ
200
-200
Fig ure 9.11:
Restraint forces predicted under single wheel load
There is a compressive force in the end regions of the deck but this is
very much smaller relative to the wheel load than in Figure 9.10.
confirms
that
the
This
force is due to global effects which are much less
significant with only one wheel
loaded.
The compressive force in the
region of the wheel under 60kN is greater than in Figure 9.10.
This is
partly due to the lack of the global tension force near mid-span.
However,
a comparison of Figures 9.10 and 9.11 reveals another explanation.

compressive
restraint
superimposed on
the
force
tensile
under
force
each
wheel
in
Figure
due to an adjacent
wheel.
9.10
The
is
This is
another reason why behaviour in single wheel tests is an unreliable guide

t o behaviour under multiple wheel loads.
As
the
load
increases,
the
restraint
force
in
Figure
9.11
increases
disproport ionately, particularly for the elements either side of the wheel.
However, even as failure approaches, the area in compression around the
wheel is comparatively localised.
The restraint required to resist this
compression comes from the slab i mmediately on either side of the wheel
and the end regions continue to be subjected to compression.
the global tes ts, the analysis shows
provide the restraint .
Thus, as in
that diaphragms are not needed to
The source of restraint is substantially different
from that implied by previous research.
However, if the cr itical region is
considered to be applying a compressive force to the r est of the sla b, t he

rest of the slab is analogous to a slab with a hole in it across which a
compressive force has been applied.
Figure 9.11 is remarkably consi stent
- 225-
with the r e sult of an elastic analysis of such a case which is shown in

Figure 9.12.
St ress Across Centre -Line
Unit Applied St ress
Figure 9.12 :
Elastic analysis of stresses around a hole
[from finite element analysis by Mehkar-Asl <130 )]

Under 220kN, the analysis was very close to f ailure and the stress in the
concrete immediately below the wheel was j ust starting to reduce from its
peak va lue a s the concrete began to crush.
The maximum restraint force
predicted was only approximately half of that suggested by the rigidplastic strip theory considered in 3.2.1.
This was
entirely due to the
requirem en t for compatibility and to the lack of ductility of the concrete.

The fact
that
the slab failed in the analysis before reaching the full
plastic momen t capa city could not have been due to shear as the analysis
does not model this effect.
As with the global tests, an attempt was made to ascertain whether the
real restra int forces were as predicted by the analysis.
was
already
cracked
difficult to do this.
before
the
tests were st arted,
Because the slab

it
was
even more
All that could be established with any certainty was
that the general form of Figure 9.11 is reasonable.
d. Mod ified Decks

In the last chapter it was suggested that if the beams had been provided
with less pres tress, the deck could still have failed in the same way but
at a l ower l oad.
To investigate this, the bridge was re- analysed using
on ly 60% of the prestress area.
To reduce the computer time required, and
to enable the analysis to be taken up to f ailure, the coarse element mesh

was used.
The beams were not predicted to crack until the load was 175kN per jack
and up to this point the behaviour was not affect ed by the reduction in
- 226-
prestress.
load
Thus the analysis confirmed that the behaviour in the service
tests
would
have
been
identical
without
the
over-provision
of
prestress.
The analysis predicted the same form of failure as before but at a lower
load of approximately 320kN per jack.
failure
was greater,
indicating that
The maximum beam deflection at

the slab could withstand greater
differential beam deflections when subjected to smaller local loads.

The load at which the slab was predicted to fail was below that predicted
by yield-line theory.
This is entirely consistent with the explanations of
the behaviour given earlier in this thesis.
It also suggests that the
supposedly conservative approach to design allowing for membrane action of

using yield-line theory, which was proposed by Tong and Batchelor(51>, is
potentially unsafe.
Analysis using the coarse element mesh was also used to investigate the
effect of varying the quantity of reinforcement in the slab.
Two analyses
were performed, one using the actual quantity of secondary steel with
double the quantity of main steel and another in which both the main and
the secondary steel were reduced to half that which was actually provided.
In both cases, the prestress and also the additional transverse bars in the
end regions of the slab were as provided in the model.
The allowable
service load implied by these two analyses were approximately 190 and
60kN per jack respectively whilst the failure loads were approximately 440
and 375kN.
The service loads were obtained from normal BS 5400 criteria using the
worst stress at a crack calculated ignoring the concrete in tension; the
"stress at crack approach" described in 7.7.3.
The steel area in the
lightly reinforced slab was so low that this approach predicted high steel
stresses as soon as the concrete cracked, that is before the cracking was
extensive enough to develop much membrane action.
service
load
of 60kN predicted
probably too low.
in this
way
Because of this the
is very approximate and
However, this behaviour might be taken to imply that
the steel area was below the desirable absolute minimum.
At 0.18% it was
above the code nominal steel area but the low d/h ratio in a thin deck
slab means that the minimum steel area expressed as a percentage of the
net section should be higher than normal.
-227-
The analysis with increased steel area suggested that doubling the area of
main steel increased the service load by over 50%.
Although well below
the near linear relationship given by normal design methods, this is a

greater effect than implied by previous research.
This is because the
steel contributes to the resistance to global moments and also to the

restraint.
Under single wheel loads, as tested by previous researchers,
the global moments are insignificant and

available
to
provide
the
restraint.
there is relatively more slab

Thus
these
effects
are
less
deck
slab
pronounced.
A
final
analysis
was
performed
using
single
layer
of
reinforcement as provided in the second deck, although still with the extra
bars in the end regions of the slab.
higher allowable service load than
This analysis suggested a slightly

for
the steel actually provided,
in
contrast to the analysis of the second deck which will be described in the
next sect ion.
The
failure
load was, however,
reduced to approximately
375kN per jack.

All the analyses
predicted the same
punching through the deck.
failure mode with the same wheel
However,
the failures were clearly greatly
influenced by global transverse moments.

steel
area
was
to
improve
the
A major effect of increasing the
distribution
properties
of
the
decks,
particularly in the later stages of the analysis as failure approached.

Thus, although the most heavily reinforced slab had the smallest rot at ion
capacity,
and hence failed
when
the differential beam deflections were
relatively small, it failed at the highest load.
At failure, the predicted
deflection of the heaviest loaded beam <Beam B> was similar to that in the
test and analysis of the actual model but the deflections of all the other
beams were significantly greater.
e. Analysis with no Concrete Tensile Strength

The
coarse mesh analysis
was also used
to investigate the effect
reducing the tensile strength of concrete to zero.
of
This analysis gave a
failure load of approximately 375kN which is a reduction of less than 10%

compared with the original analysis.
by a similar percentage.
The implied service load was reduced
These relatively small reductions indicate that
the tensile strength of concrete is not as important to the restraint as

might have been supposed.
that
the
slab
was
This arises because the global moments ineant
cracked
over
much
-228-
of
its
length, reducing
the.
It implies that much of the restraint
contribution of concrete in tension.

actually
comes
from
the
under-stressed
reinforcement
away
from
the
critical areas and confirms that reinforcement is necessary in deck slabs.

It
was
noted
in Chapter 7
that
this
form
of analysis has
practical advantage of not being load history dependent.
the major
In this case, it
did provide conservative answers and would have been a reasonable design
approach.
However,
always be the case.
it was also noted in Chapter 7 that
this may not
Another disadvantage of this form of analysis is that
it gives over-conservative predictions for the distribution properties.

this case it
over~estimated
In
the worst beam moment under service loads by
10%.
9.3.3 Second Deck
Only one computer model was used for the second deck.
This used the same
width of transverse elements as the fine mesh analysis of the first deck,
258mm, but it used only four elements across a slab span.
total of 608 nodes.
This gave a
The use of six elements across a slab span, as in the
fine mesh analysis of the first deck, would have required 864 nodes and a
significantly greater band width,
amount
elements
of
computer
across
time.
slab
which would have needed an excessive
The major disadvantage of using only

span
is
that
it
prevented
the
model
four
from
representing the finite width of the beam webs.

As with the fine mesh analysis of the first deck, a complete load history
analysis was performed.
The predictions for local deflection were not as
good as
deck with
for
consistently
the
first
over-estimated,
the deflection
typically
by
50%.
under wheel 14
This
was
being
undoubtedly
largely due the failure to represent the finite width of the beam web.
The analysis implied a transverse moment at
the face of the beam web
which was only some 50% of that over the centre-line of the beam.
was more significant
than in the
first
This
deck because the critical slab
section was over the beam rather than at mid-span of the slab.
The
analysis
predicted
lower
steel
stresses
than
in
the
first
deck.
However, using conventional BS 5400 design criteria, the allowable service

load would still have been lower at just under 100kN per jack compared
with 120kN for the first deck.
The critical criterion was the concrete
stress in the soffit of the slab over Beam D adjacent to wheel 14.
- 229-
This
contrasts with the hypothetical analysis of the first deck with only one
steel layer which gave a higher service load.
This is due to the greater
hogging moments in the second deck and the fact that the single layer of
steel was below mid- depth.
Another factor was the higher concrete grade
in the first deck.

By
150kN
per
jack
the
concrete
in
the
critical
significantly beyond the limit of linear elastic

computer model.
region
was
stressed
behaviour used in the
This, and the more extensive cracking in this bridge,
meant that the analysis predicted a much greater difference between the
behaviour under the first and last applications of the service load than
for the first deck.
This reflected the real behaviour of the bridge and
confirmed the implication of both the analysis and the test results that
the 150kN per jack applied in the tests was above the desirable service
load for this structure.
Beam E
4-50
Load
<k N/Jack )
4-00
Beam B
/
,...........Beam
.,/"
.,/"
350
300
250
200
150
Analysi s
100 -
Test
50
0
0
10
20
30
40
50
60
Deflect ion (mm)

Figure 9. 13:
Beam deflections of second deck
<from non-linear analysis)

On completion of the analysis of the service load tests,
model
was
loaded
monotonically
to
deflections are shown in Figure 9.13.
failure
and
the
the computer
predicted
beam
As for the first deck, the analysis
correctly predicted the failure mode; it predicted concrete crushing on the

slab soffit over Beam D followed by excessive local deflection under wheel
- 230-
14.
However, the prediction of failure load was not quite as good, the
analysis under-estimating
this by over
10%.
Indeed it would be more
realistic to say that the analysis under-estimated failure load by nearly

20% since
plotted
in
it
did not
Figure
converge properly under the last
9.13
and
at
this
stage
it
load increment
also gave
an
excessive
deflection under wheel 14 of over 20mm relative to the beams.

This greater conservatism of the analysis compared with the coarse mesh
analysis of the first deck might have been attributed to the finer element
mesh or to the greater significance to this deck of the failure to model
the web width.
However, the fact that the analysis predicted significant
increases in the deflections of Beams B and D as well as C in the final

load increment plotted in Figure 9.13 suggests that it was the prediction
of the beam behaviour which was at fault.
load had
been due
to
If the low prediCted failure
the analysis under--estimating slab strength and
consequently under-estimating distribution properties, the analysis should

have under-estimated the deflection of Beam D.
estimated the deflection of that beam.
In fact it slightly over-
It appears that
the reason the
analysis predicted an earlier failure than actually occurred was that it

predicted that the concrete in the slab would start to crush due to the
global flange forces under a lower load than was the case.
In the tests,
there was no obvious sign of this crushing although it seems likely that
it
was
beginning
to
occur
when
the
bridge
failed.
The
happened at a lower deflection than in either the first
reason
this
bridge or the
single beam test was that the slab concrete was significantly weaker and
the analysis appears to have exaggerated the effect of this.
However,
although this global crushing was a major reason for the failure in the
analysis, the analysis still correctly predicted that
the final collapse
would look like a local failure; once again it showed that global and local
behaviours are not independent.
Since the predicted failure load, although 20% below the actual failure
load,
was
nearly
four
times
the
allowable
service
analysis its value had no practical significance.

predictions for
the behaviour at
load given by
the
The reasonably good
lower loads are more important.
The
analysis suggested that the slab as tested was inadequate for the intended
load and this confirms the findings from the tests suggesting that the
analysis would have provided a satisfactory design method.
- 231-
As
with
the
sufficiently
first
good
deck,
the
predict ions
analysis
of
considered
was
to
behaviour
to
suggest
have
that
it
given
was
reasonable to use it to obtain some insight into how that behaviour was
obtained.
In Figure 9 . 14. the predicted transverse force across the centre of the
slab span between Beams C and D is shown for a particular load stage in
the final loading.
In order to make the plot directly comparable with
Figure 9.10 <the equivalent plot for the first deck) the same load level is
used.
It will be seen that
the restraint
force in the region of the
wheels is much greater and much less localised than for the first deck.
The diaphragm at the right hand support, which is relatively close to a
wheel,
is
resisting
significant
tension
as
implied
by
previous
researchers but that at the opposite end of the deck is resisting very
little axial force.
The distribution of forces might be considered less
different from that implied by previous research than was that predicted
for the first deck.
However, the central portion of the deck between the
two bogies of the HB vehicle is still subjected to a significant transverse

tens i on, showing that compressive membrane action is not contributing to
the resistance to global transverse moments.
Wrea Pos;t;cns
Res tram 200

Force lkNiml
100
-100
Figure 9. 14:
Predicted force across deck slab of second deck
<load = 250kN/jack )
At first sight. the obvious reason for the greater compressive membrane
forces in this deck than in the first deck is that the diaphragms provided
better restraint
to these
forces.
However,
this
is not
a satisfactory
explanation; adding in-plane transverse stiffness to a region of the first

deck which which was subjected to significant compression could not have
this effect.
Also, Figure 9 .14 shows that only one of the diaphragms was
resisting
significant
tension.
There
are
two
other
explanations.
Firstly, the weaker concrete and less effective reinforcement in the second
- 232-
deck meant that it was more extensively cracked at this load stage and
consequently there was more membrane action.
Secondly, the diaphragms
reduced
in
the
difference
between
the
moments
adjacent
beams
and
consequently the global effect which led to tension at mid-span of the

first deck was less pronounced.
As with the first deck, an attempt was made to see if the membrane forces
predicted by the analysis were realised in practice.
However, the more
extensive cracking made this even more difficult.
All that could be
determined was that the form of Figure 9. a was reasonable.

Although
the
compressive
membrane
forces
shown
in
Figure
9.14
are
sufficient to cause a very significant enhancement in the behaviour they

are not, on their own, sufficient to explain all the difference between the
actual behaviour of the slab and that predicted by normal design methods.
The re-distribution of moments away from the critical region is equally
significant.
9.4 CONCLUSIONS
Conventional
analyses
of
the
models,
as
expected,
conservative predictions for the slab behaviour.

gross-concrete
slab
properties
are
used
the
give
extremely
Also as expected, if
predicted
distribution
properties are slightly better than were realised in practice.
However,
the discrepancies are relatively small and no greater than other faults of
conventional analysis which are normally considered acceptable.
The non-linear analyses gave reasonably good predictions for behaviour and
appear to give a reasonably good basis for design.
insight into the behaviour.
develop
compressive
They also give a good
They suggest that the restraint required to
membrane
action
comes
relatively close to the areas being restrained.
from
material
which
is
This explains why, as was
clear from the results of the tests on the first deck, membrane action is
not
\
dependent
on
the
presence of diaphragms.
It
also
confirms,
as
suggested in the last chapter, that membrane action does not contribute to
the resistance to global transverse moments.
The
analyses
also
confirm
that
the
failures observed were
primarily
brittle bending compression failures and that they were greatly influenced
by global behaviour.
They suggest that a large part of the difference

- 233-
between the real behavic.ur of bridge deck slabs and that predicted by
conventional elastic analysis
is actually due
rather than to pure membrane act ion.
to moment
redistribution
This redistribution, like membrane
action, is not dependent on reinforcement yielding; it starts to occur as

soon as the behaviour of the concrete becomes non-linear in tension which
is well before the slab becomes unserviceable in any way.
-234-
CHAPTER
USE OF
10
MEMBRANE
ACTION
IN
DESIGN AND
ASSESSMENT
10.1 OO'RODUCTION
Previous chapters have shown that membrane action, and the closely related
mechanism of moment redistribution, have a
the behaviour of bridge deck slabs.
s~nificant
beneficial effect on
They have also shown that the effect
is sufficiently reliable to justify its use in design and assessment.

chapter will consider the use of the effect in
des~n
This
and assessment.
Only the application to concrete bridges will be considered as steelconcrete composite bridges are considered to be outside the scope of this
thesis.
10.2 USE IN DESIGN
10.2.1 M Beall . Type Decks
Under present
des~n
slabs of otherwise
rules, the quantity of main reinforcement in the deck

identical bridges designed
for
identical loads
in
Northern Ireland and in the rest of Britain differ by a factor of over two.
This is clearly unsatisfactory and should be resolved.
It appears that non-linear analysis such as the form of analysis described
in Chapter 7, is needed to give a realistic prediction of the behaviour of
a deck slab under full HB load.
Although it is feasible to use this form
of analysis in design, it is probably not justified for such a routine,

simple and relatively standardised structure as the deck slab of an M beam
type bridge.
That standardisation enables simple prescriptive rules to be
developed.
Although Chapter 8 showed that deck slabs can fail at substantially lower
wheel loads than are predicted by the research on which the Northern Irish
rules are based, the rules are so conservative compared with that research
that they remain adequate.
Indeed they appear to be over-cautious.
The
first of the two decks tested in this study remained serviceable after the
deliberately excessively severe load history had been applied, despite
having only 60t of the steel area recommended by the Northern Irish rules.
- 235-
It also had more than adequate ultimate strength.
It might be argued,
considering the observed significance of global and local interaction, that

the tests were unrealistic because of the over-provision of prestress.
However, since the beams' behaviour remained linear elastic up to some 1.3
times design ultimate load, the behaviour under service loads would have
been virtually identical with substantially less prestress.
strength undoubtedly would have been lower.
The ultimate
However, the analysis in
9.3.2d suggested that even with less than the normal amount of prestress,
the bridge would have been over twice as strong as was required.
It thus
appears that Tl2-250 reinforcement is adequate compared with the Tl2-150

specified by the Northern Irish rules.
Nevertheless, and allowing for the
fact that analysis shows that global transverse moments could be greater
in a wider deck, it is prudent to continue to specify Tl2-150 main steel.
If this reinforcement is provided in M beam deck slabs there is no need to
do any analysis for the design of the slab.
Although this steel area can be justified from the test results alone,
there may be a preference for a design method which is based on some form
of analysis.
Such a method can be obtained by consideration of the tests
described in Chapter 8.
The first deck, whose. behaviour was considered
satisfactory, was provided with just enough transverse reinforcement to

resist the global transverse moments predicted by a conventional grillage
analysis.
The second deck, whose behaviour was less satisfactory, was
provided with substantially less steel.
A possible design approach is thus
to require that the reinforcement be designed for the global transverse

moments only; the opposite of the conventional North American approach.
This would give very light steel areas in some decks so a minimum nominal
area would also have to be specified.
Although one might put a case for
using the steel percentage specified by the Ontario Code, it is considered

prudent to specify a minimum of Tl2-250 which corresponds to the steel
area used in the first test deck and is the lightest steel area which has
been
demonstrated
requirement
to
be
satisfactory
by
tests.
In
practice,
the
to resist global moments means that the main steel would
normally be slightly heavier than this.

It is more difficult to justify the continued specification of the same
quantity of secondary steel.
It appears that the Ontario researchers had
two reasons for specifying isotropic reinforcement.
Firstly, their research
used an axi-symmetrical analysis and so they chose to use isotropic

-236-
reinforcement to get closer to the assumptions of the analysis.
Secondly,
and more significantly, they began by specifying main steel equal to the
minimum nominal steel required by their code so they could hardly have
specified less secondary steel.
When Kirkpatrick et al wisely specified a
larger area of main steel, to allow for global transverse moments, they
rather arbitrarily decided
Both
the
tests
reported
to continue to use isotropic reinforcement.

in
Chapter
and
the
analysis
reported
in
Chapter 9 suggest that the secondary steel in the deck slab of a simply
supported M beam deck is very lightly stressed and contributes little to
the behaviour.
It could be reduced to that provided in the first of the
models considered here, equivalent
to T12-250 at
approach is used for the design of the main steel.
full size, whichever

Even this is probably
over-conservative; there is no evidence from this study that any secondary

reinforcement is required.
The same basic approach to deck slab reinforcement design is valid in
regions of global longitudinal hogging.
This is
clear from previous
research and also because, as was discussed in 3.2.7, the critical load
cases for global longitudinal hogging do not impose any wheel loads in the
region of the slab which is in tension.
It is prudent, although probably
conservative,
longitudinal
to
additional to that
require
the
nominal
slab
steel
to
be
required for global moments and also to require a
proportion of the latter, say 30%, to be placed close to the bottom face of
the slab.
The reason for this restriction is that, although intended only
to resist
local effects, the nominal steel will be stressed by global
effects.
Thus the reserve strength available for local effects could be
very small if only this very small quantity of already highly stressed
steel was provided in the soffit.
The basic limitations imposed by Kirkpatrick et al on the use of the
empirical rules appear to be reasonable; one could debate the limiting span
given but since, with M beams, this is well above the limitation imposed by
web shear strength there is little to be gained by so doing.
The one
restrict ion which is worth reconsidering is the requirement for diaphragms.

The analyses and tests reported in this thesis show that diaphragms are
far less important to the development of compressive membrane action than
has previously been believed.
The empirical rules could be extended to
cover bridges with only nominal diaphragms, or with no diaphragms at all.

-237-
In the latter case the end section of the slab would not receive the full
benefit
of
restraint
and
would
require
extra
reinforcement.
is
It
suggested that a strip of slab extending 0.5m from the end of the deck
should be provided with enough reinforcement to enable it to support a
wheel acting as a single beam.
This is marginally more steel than was
provided in the first test deck considered in Chapter 8.

From a purely theoretical viewpoint, the use of these empirical design
rules in combination with global analysis based on gross-concrete section
properties cannot be justified.
However, analysis shows that the use of
cracked transformed transverse properties is conservative and in 9.2.2 it

was found that the errors resulting from the cracking are no greater than
other normally accepted faults of grillage analysis.
is to
use half
the
concrete properties.
transverse stiffnesses
A reasonable approach
calculated for
the gross-
The economic consequences of the slightly worse
distribution properties resulting from this compared with the conventional

approach are extremely small and significantly less than might be inferred
from the results of the tests considered in this thesis.
This is because
the tests considered HB alone, the worst case for the slab, whilst the
critical load case for the beams is HA plus HB.
Improving the distribution
properties reduces the effect of the HB load in the critical area but it
increases the effect of the associated HA.
Due to a continuing increase in
the HA load which is applied in combination with the HB load, the benefits
of good
distribution
standard
introduced
properties
in
Britain
have
since
reduced
the
with
1950s.
every new
loading
Nevertheless,
the
suggestion that reduced transverse properties should be used does imply

that the beams of bridges designed to the existing Northern Irish rules
could be subjected to slightly greater moments than those for which they
were designed.
In the author's view this is unimportant since the design
criteria currently used for this type of beam (class 1 and 2 criteria in
combination with an extremely severe service load) are unduly conservative.
However, a discussion of this subject is outside the scope of this thesis.
Where half the gross-concrete transverse properties are used in the global
analysis, it appears prudent to continue to require the transverse steel to
be capable of resisting the global transverse moments predicted by a
conventional analysis based on gross-concrete properties.
To avoid the
need to perform two separate analyses, these can be taken conservatively
-238-
to
be
double
the
moments
calculated
using
half
the
gross-concrete
properties.
10.2.2 other Beam and Slab Decks
The slabs of bridges built with U Beams or the proposed new Y Beams are
so similar to those of M beam decks that the same design rules can be
applied.
The only modification required being that, with U beams, the main
steel may have to be increased to enable it to act as part of the torsion

links of the beams.
Other types of beam and slab bridges normally have thicker deck slabs with
wider-spaced beams.
This means that the global transverse moments are
likely to be less significant but it is difficult to prove this.
It is
prudent, therefore, to recommend a check that the main steel in the deck
slab is always sufficient to resist the transverse moments given by the
global analysis.
The suggested minimum steel area to be specified is 0.3"
of the gross-section.
This corresponds to T12-250 <the minimum suggested
for M Beam slabs> for a thickness of 160mm so the rules are consistent.
These suggestions are more conservative than the Ontario rules but this is
justified due to the significance of global transverse moments noted in
this thesis and by the nature of the HB load which is exceptionally severe
for this effect.
The restriction on the use of these rules can be as for the Ontario rules
except for relaxing the requirement for diaphragms as with M Beam decks.
However, where these restrictions are not complied with it does not mean
that membrane action cannot be used in design; merely that the empirical
rules are not applicable.
could still be used.
Analysis such as that described in Chapter 7
Where the span to depth ratio is outside that
required to use the empirical rules the analysis should consider large
displacements.
10.2.3 other Types of Deck
It has been noted in earlier chapters that compressive membrane action is
potentially significant to other types of bridges, apart from beam on slab

structures.
These range from simple slab decks to major concrete box
girder structures.
The detailed consideration of these is considered
beyond the scope of this thesis and, in any case, they are probably not
- 239-
sufficiently standardised to enable prescriptive rules to be developed;

non-linear analysis would be required.
However,
a simple conservative
approach can be developed using normal analytical methods.
This approach
could also be used for beam and slab decks if desired.

The mechanism by which the behaviour of deck slabs is enhanced relative to
the predictions of elastic plate theory is essentially one of stresses
redistributing
Chapter
away
that
from
the
the
critical areas.
restraint
force
It
required
was demonstrated
to
develop
in
compressive
membrane force comes from material which is relatively close to these

critical areas; not
from the diaphragms.
This suggests a very simple
over-conservative way of allowing for the effect.
Design could be based
on a normal elastic slab analysis but ignoring, or rather smoothing out,

the peaks in the moment over a finite width.
If it
was only moment
redistribution which was being considered this width would be related to

the span and to the ductility of the sections.
however, the critical factor is the depth.
elastic analysis
could be
used
with
With arching action,
It is suggested, therefore, that
the
design
based
on
the
moment
averaged over a width equal to the lesser of 6d or half the slab span.
This is undoubtedly extremely conservative; it was demonstrated in 8.8.3
that removing the steel completely over a width of 12h had little effect
on behaviour.
10.3 .ASSESSMENTS
The approaches suggested in 10.2 are equally applicable to the assessment

of
existing
bridges.
However,
purely
empirical
approaches
are
less
suitable for assessment because it is not possible to adjust the structure

to
fit
the
limitations
imposed
for
the
rules.
It
will
therefore be
necessary to resort to non-linear analysis more frequently than in design.

The use of the assessment approach given in the Ontario Highway Bridge
Design Code< 11 >, which relies on the strength predictions of Hewitt and
Batchelor's approach,
is not normally advised.
This is because of its
failure to consider global transverse moments.
However, in assessing a
bridge which has intermediate diaphragms, it is reasonable to assume that

the global transverse moments in the deck slab are insignificant and so
the approach is more reliable.
Even then, if the spacing of the design
wheel loads is less than the slab span, some allowance should be made for
the effect of the second wheel.
- 240-
CHAPTER
CONCLUSIONS
11
RECOMMENDATIONS
AND
11.1 CONCLUSIONS
The first conclusion to be drawn from this study is that bridge deck slabs
are able to support loads by compressive membrane action and, as a result,
that they are able to support very much greater loads than is suggested
by conventional design methods which are based on flexural theory.
Judged
against the background of the research which was reviewed in Chapter 3,

this conclusion is unremarkable.
However, the conclusions to be drawn from
any study depend as much on the the background against which the study is
assessed
as
on
the
study
itself.
Judged
against
the
background
of
conventional design practice, which was reviewed in Chapter 2, the enormous

strengths
of
deck
slabs,
particularly
lightly
reinforced
deck
slabs,
compared with the predictions of conventional flexural theory remains the

most significant
conclusion.
is re-stated here to put some of the
It
other conclusions into perspective; it should be remembered, for example,

that when the deck slab of the first model considered in Chapter 8 failed
at little over half the load which might have been expected from some
previous research,
it
was
resisting some
five
times its ultimate load
according to normal design methods.

The remaining conclusions are:
1.
Compressive
moment
membrane
action
and
the
closely
allied
mechanism
of
redistribution start to enhance the behaviour of deck slabs
relative
to
the
predictions
of
linear
analysis
as
concrete's behaviour becomes non-linear in tension.

thin slabs,
soon
as
the
This, at least in
is well before there are visible cracks.
It
does not
depend on any material behaviour which is unacceptable under service

loads.
Because of this, membrane action significantly increases the
service load, as well as the ultimate load, which a slab can carry.
2.
Compressive membrane act ion is sufficiently reliable to justify its

consideration in design and assessment.
The model tests described in
Chapter 8 were an .exceptionally severe test yet

substantially
better
than
could
be
analysis.
- 241 -
anticipated
the behaviour was

by
purely
flexural
3.
The restraint required to develop compressive membrane action comes

from
the under-stressed reinforcement and concrete surrounding the
critical areas of the slab.
It is not dependent on the presence of
diaphragms.
4.
Compressive
membrane
action
could
even
enhance
behaviour of slabs with no external restraint.
the
service
However,
load
it cannot
increase the failure load of such slabs above that predicted by yieldline theory.
5.
Compressive membrane action does not greatly enhance the resistance

to global transverse moments.
6.
Because of 3 and 5 above, and contrary to the implications of some

earlier
research,
reinforcement
is
needed
in
bridge
deck
slabs.
However, because it is required to resist global transverse moments

and to provide restraint
<rather than to resist local moments), the
behaviour is not sensitive to the exact position of the reinforcement.

Thus the behaviour of bridge deck slabs is remarkably insensitive to
local reinforcement corrosion.
7.
The failure loads of bridge deck slabs subjected to single wheel loads
are reasonably well predicted by the approaches which were considered
in 3.2.3.
The cases where these approaches gave unsafe predictions
were restricted to impractically lightly reinforced slabs with large

span to depth ratios and relatively poor restraint.
The methods do
not, however, give good predictions of other aspects of behaviour; for

example, Hewitt's approach under-estimated the deflection at failure by
a factor of up to 10.
8.
Non-linear analyses of the forms considered in Chapters 5 and 7 are

also capable of predicting these failure loads and are better able to
predict other aspects of behaviour.
The form considered in Chapter 5
is theoretically more rigorous and realistic than that considered in

Chapter 7 but the latter has many practical advantages in a design
situation;
it
is simpler, more compatible with design standards and
also appears to be more consistently safe.

9.
The local failures observed in deck slabs are primarily brittle bending
compression failures.
They can be predicted by analyses which do not
consider shear, the load at which they occur can be reduced by the
- 242-
presence of other moments <such as global transverse moments) and, in

many cases, crushing concrete is visible before failure.
10. Bridges which are subjected to multiple wheel loads, such as HB, can
fail by wheels punching through their slabs at wheel loads which are
substantially
below
the
local
strength
of
their
slabs;
both
as
measured in single wheel tests and as predicted by the approaches

developed_by previous research.
11. The form of failure considered in 10 above can .occur even when the
beams have a reserve of strength and the global transverse moments
are thus not needed to maintain equilibrium.
This is contrary to the
safe theorem of plastic design but the behaviour is too brittle for
this to apply.
12. Non-linear analysis is capable of predicting the behaviour of bridge
decks reasonably well.
In particular, it appears to be the only form
of analysis which is capable of modeliing the interact ion of global

and local effects and of predicting the restraint.
11.2~RECO~ATIONS
11.2.1 Recommendations for Design and Assessment
Less conservative 'design methods for bridge deck slab reinforcement should
be
introduced
which
membrane action.
allow
for
the
beneficial
effects
of
compressive
Possible details of these methods were considered in
Chapter 10 and will not be discussed here.
11.2.1 Recommendations for Further Research
There are many aspects. of the behaviour of the type of slabs considered in
detail in this and previous studies which could be considered to require
further research.
However, such research is not needed to justify the use
of membrane action in design or assessment.
To recommend it would merely
serve to perpetuate the use of conventional design methods which have

been shown to be extremely unrealistic and conservative.
This study has suggested that membrane action could have a significant
beneficial effect on the behaviour of a wide range of bridge deck slabs in
addition to those for which it has so far been investigated in detail.
Any
future studies of membrane action in bridge deck slabs should consider

types. of slab which have not previously been researched.
- 243-
This includes
the thin long-span slabs typical of longer span concrete bridges.
However,
this study has shown that the restraint required to develop compressive
membrane
action
comes
from
critical areas of the slab.
under-stressed
material
of
simply
the
At service load levels, which are critical in
design, it is not dependent on any external restraint.

behaviour
surrounding
supported
and
even
It follows that the
cantilever
slabs
could
significantly enhanced by the effect and this should be investigated.
- 24.4-
be
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APPENf)ICES
A. RESTRAINED SLAB STRIP TO ELASTIC THEORY
Al Stresses
112
~----------------------------------~p
u
"0
X:z
Figure: Al:
Restrained slab strip under line load
<half section: elastic theory>

Consider the slab strip shown in Figure A.l.
For convenience the origin of
the x axis is located at the intersection of the extended line of thrust

of the restraint force and the projection of the soffit of the slab.
Since the support is fully fixed in rotation, and the section at mid-span
cannot rotate either because this would violate the symmetry of the
system;
=
x,
and
1/2
Now, from the geometry of the line of thrust:
Therefore
Now, from x
=0
to x
P/2
=x
the depth of concrete in compression, de,
=
=
since
hx/x2
3hx/x4
=
A 1
Now the stress on the compression face of the section, f.,.,,

=
2F/d.,
2F
3hx/x4
Substituting for F, this gives:

=
f.,.,
Now, the strain at mid-depth of the slab
Note: For a wide slab, the Young's modulus, E.,, of the concrete should
strictly be replaced by E.,/ <1-v 2
since the slab is forced to bend
cylindrically because the transverse strains, which occur in a narrow

slab due to the Poisson's ratio effect, are prevented.
Substituting for f.,c, d., and F, leads to:
Px 4 2
3E., h2 x
=
Px 4 2
3E.,h 2 x
=x
/3 to x
at mid-depth.
= 2x
h
2<3hx/x..,)
/3 the section is uncracked so its centroid is
Hence the stress there is:
=
so
[1x - 6xx... J
3E.,h 2
From x
p
p- ~;]
E.,
F/h
Now the slab's extension from the centre-line to the support;
x./2
J E.,
-2
-2Px 4
E.,h 2
dx
x,
[f"
f" ]
x...
! - x... dx + 1
6x 2
x,
x./3
x./3
-2Px 4
E h2
"
A
x... [lnx + x...

3
6x
x,
2
x/2
[~]
x./3
~2]
=
=
-2Px42 [ln x4 - ln3 + 3 - x4

~
x,
4 6x,
-2Px 42 [ln<x 4 /x 1 )
3E.,h 2
0. 3486 - <x4/6x,
If the slab is rigidly restrained this must be equal to zero.
and numerical solution of this equation leads to;

=
13.54
so, at the support and at mid-span;

de
= ...L!L
13.54
now;
=
=
=
=
0.222h
1/2
2 [h-(2/3)d.,]
.p
1/2
h[l- (2/3)
Pl/3. 41h
and the maximum concrete stress

=
2F/d.,
2 Pl
3. 41x0. 222h 2
2. 64P l/h 2
A2 Deflection
From x
=0
to x
= x~/3
=
and;
now, the curvature
f cc /Ec
a;;
A 3
0.2221
>l
Hence;
Px4 2
x4
Ec3h'"x 3hx
From x = x4 /3 to x = x4 /2,
curvature
where the section is uncracked,
where the eccentricity, e;
=
=
and
Thus the curvature
Px4
2h
12Px4
2Ech 3
6P
Ech 3
[i - :J
X]
[~4 -
From x = x, to x = x4/3 the slope

X
curvature dx
x,
since it is zero at
thus the slope
x,
!X
dx
Px4"'
9Ech 3 X2
x,
X
Px4"'
9Ech 3
[-~]
x,
slope
Px4"'
9Ech 3
[_!_~
X
X
1
<slope at x
= x4 /3)
+ J curvature dx
x./3
A '
the
.!. [ X,
3J +
X
6P <x... /2 - x) dx
Ech 3
x./3
[ .!..x,
- 2x2].
~] +
x...
x./3
=
=
From x = x, to x
=x
/3, the deflection
= I
=
slope dx
x,
"'J
Px ...
9Ech 3
1 - 1 dx
X,
x,
=
at x
=x
Px ... "' [xx,- lnx]

9E c h"'
x,
/3 this is
=
Px ... "' [x... - lnx ... - 1 + lnx,J

9Ech 3 3x,
3
From x
=x
/3 to x
=x
/2, the deflection relative to that at x
=X
= I
slope dx
x./3
fx . "'
X
Ech 3
X4
+ 3x 4 x - 3x2 dx
9x 1
x./3
Ech 3
at x
=x
/2 this is
A 5
xx ... "' [ 9x,
XX 4
+ 3x4 x2
-2-
x"']
x./3
/3
[X
Ech
1BX 1
x4:a -
2
=
3x4 "'
8--
x,."' + x.. "' + x.. "' - x .. "'

6 + ;;""]
27x,
3
Px,."'
Ec h"'
[ x,. - 5
54x,
108
so the total deflection, w

=
2Px,.'" [ x,. - 1 + 1 ln 3x, + x,. - 5 J

x,.
54x, 108
9
9
Ech 3 27x,
2Px,."'
Ech"'
13. 54
[~
18x,
17 + ..!_ ln 3x,J
108
9
x..
now, with full restraint,

(from Appendix A1)
substituting this into the expression for deflection gives;
now
0.8547 Px,."
Ech 3
1/2
x4[1 -
2~,
x..
so
1. 7046 x..
and
Pl"'
Ech"'
0. 1726 pp
Ech 3
0.8547
l. 7046"'
Note. This expression can also be obtained by an algebraically simpler

method using the virtual work approach.
A3 Effect of Restraint Flexibility on Stress
In Appendix A1 it was shown that the slab's extension from the centre-line
to the support
This is equal to the lateral movement of each support so, if the supports
develop a restraint force, F, of K times the movement whilst stili giving
full rotational restraint, this leads to;
F/K
-2Px,."'[ln<x 4 /x,
3Ech2
>-
0.3486- <x.. l6x,
and, substituting for F using Appendix A1, this leads to;

A 6
>]
Px 4
2Kh
-2Px.. "' [ln(x 4 /x,) - 0.3486 - (x 4 /6x, >]

3Ech 2
3Ech
-4Kx 4 [ln<x.. tx,>- 0.3486 - <x.. l6x, >J
-3E.,h
4x.. [ln <x.,lx,) - 0.3486
Therefore:
Therefore
now
Therefore,
1/2
substituting
<x 4 /6x, )J
x... [ 1 - 2x, /x4 J
for
X4
and
expressing
the restraint
stiffness
relative to the axial stiffness of the uncracked slab strip:

=
Kl
-3[1 - 2x. /x.J
E.,h
Numerical solution of this equation gives a value of x,/x 4 for any given
restraint
stiffness.
By
substituting
this
into
the
expressions
Appendix A1 the restraint forces and the stresses can be obtained.
A 7
in
APPENDIX B. TRAHSVERSE SHEAR DEFORMATION OF LINE ELEMENTS
Undeflected cent re-line
R...,.,
Figure Bl :
Plan of line element
Assume the element illustrated in Figure Bl has only a uniform curvature,

C, and shear deformation, S.
Then:
and:
Rw, + Cl
t., + CP/2 + Sl
Substituting for Cl gives
=
Rearranging gives:
Therefore:
Sl
This deformation is used to calculate the shear force, F, using the elastic
shear stiffness of the slab.
In the program described here, this stiffness
is based on the gross area of uncracked concrete plus one third of the
area of cracked concrete.
To maintain equilibrium, the moments
=
=
are applied to the nodes.
Fl/2
This automatically results in the complimentary
shears being applied to the orthogonal

A 8
element~
The element provides no resistance to the uniform curvature, C.
Thus the
structure provides no resistance to the form of deformation shown in

Figure B2 and this deformation would not affect the results.
could lead to numerical instability.
However, this
To avoid this a nominal resistance to
uniform curvature is added.
Figure 82: Unrestrained deformation

The relevant terms of the stiffness matrix are then as follows;
M...,
R... ,
F,
ASG/2
-ASG/2
ASG/1
ASG/ 2
<symmetrical>
-ASG/1
-ASG/2
ASG/1
Where:
AS
is the the effective transverse shear area of the slab taken as the
width of the slab in the element multiplied by [de: + <h - de )/3J;
that is the uncracked slab area plus one third of the cracked area.
is the shear modulus of the concrete.

A 9
El"
is
the
nominal
transverse
bending
stiffness
element which is set to a very low value,

elastic transverse bending stiffness.
F
is the transverse force
and the other notation is as used previously.
A 10
of
the
slab
in
the
less than 1% of the
APPENDIX C. I..ARGJ: DISPI..ACEMENTS
Cl Example Showing Effect of Vertical Component of Axial Force
~4 P
a. deflected shape
b. bending moment about deflected centre-line
PMl
PMl
c. vertical component of axial force applied to nodes
Figure Cl:
Assume
the
initially
Three element strut
horizontal
strut
subjected only to the axial force P.
illustrated
in
Figure
Cla
is
Clearly, the true bending moment
about the deflected centre-line of the strut is as shown in Figure Cl b.

The axial force in the strut is equal to P and, in an analysis using small
displacement
theory,
this
is
taken
as
acting
along
the
line
of
the
elements.
The vertical component of the axial force in the outer elements is Pt:./ 1,
where 1 is the length of each element.
If this force, which is ignored in
an analysis using s mall displacement theory, is applied to the nodes of a

computer
model
<which
otherwise
uses
A 11
small
displacement
theory)
the
resulting vertical forces are as shown in Figure C1c.
These give the
bending moments which are illustrated in Figure Clb and which are the true
bending moments in the real strut.
They also give shear forces in the
outer
the resultant
two elements
of P6/ 1.
Thus
line of thrust
acts
horizontally because, as in the real strut, the vertical component of the

thrust in the line of the elements is equal and opposite to the shear in
the elements.
Thus adding the vertical component of the axial force has
reproduced the true forces in the elements.
C2 Effect of Slope on Axial Extension
Figure C2: Inclined element
Consider the element shown in Figure C2.
For convenience the left hand
node is assummed to be undeflected.

According to small displacement theory, the axial extension of the element
is equal to the x displacement of the right hand node.
an additional extension due to the slope.
Taking the horizontal length of
the element as 1 and the inclined length as L. we obtain:
=
Therfore
1,.
Neglecting second order terms gives:

1,.
Hence the total axial extenxsion of the element
A 12
However, there is
APPENDIX D. NOTATION
Because of the many references to BS 5400, the notation used has been
made consistent with that document wherever possible.
The main symbols
used are as follows:

reinforcement area
b
width of section
diameter of circular contact area of load

depth to tension steel
depth of concrete in compression
Young's modulus of concrete
Young's modulus of steel
stress
f
fc
f., t
concrete stress on rectangular stress block (0.6f c >

tensile strength of concrete <normally effective value)
cube strength of concrete
cylinder strength of concrete
yield stress of reinforcement
force <normally restraint force)
overall depth of section
restraint stiffness
span <also used as element length>
bending moment
load
restraint factor (used in reference 72)

distance over which a crack affects the stress
vertical deflection
)(
x direction (always horizontal, normally along element>
y direction (horizontal and perpendicular to x)
Y<L
partial safety factor for loads

partial safety factor for errors in analysis
partial safety factor for materials
displacement
strain
reinforcement area (as percentage of concrete area, bd)
A 13

Compressive Membrane Action in Bridge Deck Slabs

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What is the thesis about?

What is the thesis about?

What type of analysis is developed in the thesis?

What type of analysis is developed in the thesis?

COMPRESSIVE

Paul Austin JACKSON

BRIDGE DECK SLABS

Paul Austin JACKSON

A relatively simple form of non-linear firiite element analysis is developed

I wish to thank the Council, directors and

continuing to support it during a difficult period for the Association and

I should also like to thank Gifford

Particular thanks are due to Colin Cook for the

Last, but by no means least, I should like to thank

2 CURRENT DESIGN PRACTICE

2.2 Outline Design

2.2.1 Choice of Form

2.2 .2 To Stress or not to Stress?

2.3 Detailed Design and Codes of Practice

2.3.1 The Importance of Codes of Practice

2.3.2 Sources of Code Clauses

2.3.3 Limit State and Working Stress Codes

2.3.4 BS 5400: Part 4: 1984

2.4.1 Reasons for Linear Analysis

2.4.2 Section Properties

2.4.3 Global and Local Functions

2.5 Analysis for Design - North American Practice

2.6 Conclusions and Implications for this Study

3.2 Reinforced and Plane Slabs

3.2.1 Bending Strength

3.2.2 Flexural Shear Strength

3.2.3 Punching Shear Strength

3.2.7 Global Behaviour

3.2.8 Empirical Design Rules

3.3 Prestressed Slabs

.4.4 Comparison with Other Analyses

4.5 Crack Widths

4c. 7 Effect of Restraint Flexibility

5.2 General Approach

5.3 Element Type

5.4 Material Properties

5.6 Use in Design

6.2 .2 Steel Stress

6.2 .3 Mesh Dependence

6.2.4 Cyclic Loading

6.3 Analysis of Previous Tests

6.3.1 Direct Tension Tests

6.3.2 Flexural Tests

6.4.2 Loading Rig

6.4.5 Processing of Results

6.4.6 Results and Analysis

7.2 General Approach

7.3 Displacement Function

7.4 Element Initial Stiffness Calculation

7.5 In-Plane Forces

7.6 Large Displacements

7.7 Material Models

7.7.2 Concrete in Compression

7.7.3 Concrete in Tension

7.8 Stress Integration