TSLAB V3.0 User Guidance
TSLAB V3.0 User Guidance
TSLAB V3.0 User Guidance
\
|
+
=
o
The deflection is also limited by the acceptable elongation of the reinforcement
as follows.
( )
30 2 . 19
2
1 2
l
h
l T T
w +
s
o
where:
(T
2
T
1
) is the temperature difference between the top and bottom surface
of the slab
L is the longer dimension of the floor design zone
l is the shorter dimension of the floor design zone
f
sk
is the characteristic strength of the mesh reinforcement
E is the modulus of elasticity of the steel
h is the overall depth of the composite slab
o is the coefficient of thermal expansion of concrete.
All of the available test evidence shows that this value of deflection will be
exceeded before load bearing failure of the slab occurs. This implies that the
resistance predicted using the design method will be conservative compared to
its actual performance.
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18
The load-bearing capacity due to the residual bending resistance of the
unprotected composite beams is then added to the enhanced slab resistance to
give the total load-bearing capacity of the complete system.
Integrity and insulation performance of the composite slab
TSLAB V3.0 does not check the insulation or integrity performance of the
floor slab. The designer must therefore ensure that the slab thickness chosen is
sufficient to provide the necessary insulation performance in accordance with
the recommendations given in BS EN 1994-1-2
[ 18]
.
To ensure that the composite slab maintains its integrity during the fire and that
membrane action can develop, care must be taken to ensure that the reinforcing
mesh is properly lapped. This is especially important in the region of
unprotected beams and around columns. Further information on required lap
lengths and placement of the reinforcing mesh is given in SCI publication P300
Composite slabs and beams using steel decking: Best practice for design and
construction (Revised edition)
[ 24]
.
Full depth crack Compression failure of concrete
Edge of slab moves towards centre
of slab and 'relieves' the strains in
the reinforcement in the short span
Yield-line pattern
Reinforcement in
longer span fractures
(a) Tensile failure of the reinforcement
Edge of slab moves towards centre
of slab and 'relieves' the strains in
the reinforcement in the short span
Yield-line pattern
Concrete crushing due
to in-plane stresses
(b) Compressive failure of the concrete
Figure 1.2 Failure modes in the floor design zone
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Part 2: Engineering Update
19
1.4.2 Fire design of beams on the perimeter of the floor design
zone.
The beams along the perimeter of the floor design zone, labelled A to D in
Figure 1.3, should achieve the fire resistance required for the floor plate, in
order to provide the required vertical support to the perimeter of the floor
design zone. This usually results in these beams being fire protected.
SIDE A
SIDE C
S
I
D
E
D
S
I
D
E
B
L
1
L
2
Unprotected
internal
beams
Protected
perimeter
beams
Figure 1.3 Layout of a typical floor design zone
TSLAB V3.0 calculates the design effects of actions on these perimeter beams
and the normal temperature moment of resistance of the beam, in order to
calculate the degree of utilisation for each perimeter beam, which is calculated
using the guidance given in BS EN 1993-1-2
[ 17]
, Clause 4.2.4, as shown below.
d,0 fi,
d fi,
0
R
E
=
where:
E
fi,d
is the design effect of actions on the beam in fire
R
fi,d,0
is the design resistance of the beam at time t = 0.
Having calculated the degree of utilisation, the software can compute the
critical temperature of the bottom flange of the perimeter beams. This critical
temperature is reported in the TSLAB V3.0 output for use when specifying the
fire protection required by each of the perimeter beams on the floor design
zone. Full details of the calculation method are given in Section 2.
For perimeter beams with floor design zones on both sides, the lower value of
critical temperature given by the design of the adjacent floor design zones
should be used to design the fire protection for that perimeter beam. The
method of design for a perimeter beam that is shared by two floor design zones
is illustrated in the worked example, see Section 3.
When specifying fire protection for the perimeter beams, the fire protection
supplier must be given the section factor for the member to be protected and
the period of fire resistance required and the critical temperature of the
member. Most reputable fire protection manufacturers will have a
multi-temperature assessment for their product, assessed in accordance with
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20
EN 13381-4
[ 22]
for non-reactive materials or EN 13381-8
[ 23]
for reactive
materials (intumescents). Design tables for fire protection which relate section
factor to protection thickness are based on a single value of assessment
temperature. This assessment temperature should be less than or equal to the
critical temperature of the member.
1.5 Reinforcement details
The yield strength and ductility of the reinforcing steel material should be
specified in accordance with the requirements of BS EN 10080
[ 19]
. The
characteristic yield strength of reinforcement to BS EN 10080 will be between
400 MPa and 600 MPa, depending on the national market. In order that the
reinforcement has sufficient ductility to allow the development of tensile
membrane action, Class B or Class C should be specified.
In the UK, national standards for the specification of reinforcement still exist
as non-contradictory complimentary information (NCCI)
[ 20, 21]
because a
common range of steel grades has not been agreed for BS EN 10080.
In composite slabs, the primary function of the mesh reinforcement is to
control the cracking of the concrete. Therefore, the mesh reinforcement tends
to be located as close as possible to the surface of the concrete while
maintaining the minimum depth of concrete cover required to provide adequate
durability, in accordance with BS EN 1992-1-1
[ 14]
. In fire conditions, the
position of the mesh will affect the mesh temperature and the lever arm when
calculating the bending resistance. Typically, adequate fire performance is
achieved with the mesh placed between 15 mm and 45 mm below the top
surface of the concrete.
Typically, sheets of mesh reinforcement are 4.8 m by 2.4 m and therefore must
be lapped to achieve continuity of the reinforcement. Sufficient lap lengths
must therefore be specified and adequate site control must be put in place to
ensure that such details are implemented on site. Guidance on fulfilling the
recommendations of BS EN 1992-1-1, Clause 8.7.5
[19]
are given in P300
[ 24]
.
The minimum lap length for mesh reinforcement should be 250 mm. Ideally,
mesh should be specified with flying ends, as shown in Figure 1.4, to
eliminate build up of bars at laps. It will often be economic to order ready fit
fabric, to reduce wastage.
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Part 2: Engineering Update
21
Flying
ends
Figure 1.4 Mesh with flying ends
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22
2 DESIGN OF FIRE RESISTING
PERIMETER BEAMS
The design method described in Appendix B: assumes that an envelope pattern
of yield lines will form in the slab at the ultimate limit state. In order for this to
occur, the beams on the perimeter of the floor design zone must have sufficient
bending resistance to prevent a beam and slab mechanism occurring at a lower
load level.
The perimeter beams which bound each floor design zone must be designed to
achieve the period of fire resistance required by the floor slab. This will ensure
that the pattern of yield lines and the associated enhancement due to tensile
membrane action which are assumed to occur in the design methodology
actually occur in practice.
For a typical floor design zone, as shown in Figure 2.1, two yield line patterns
have been considered which include the formation of a plastic hinge in the
perimeter beams. The yield lines may occur across the centre of the slab, either
parallel to the unprotected beams in the Span 1 direction with plastic hinges
forming in the perimeter beams on Sides A and C or perpendicular to the
unprotected beams in the Span 2 direction with plastic hinges forming in the
perimeter beams on Side B and D and in the unprotected beams.
SIDE A
SIDE C
S
I
D
E
D
S
I
D
E
B
L
1
L
2
Unprotected
internal
beams
Protected
perimeter
beams
Figure 2.1 Typical floor design zone
Using this pattern of yield lines shown in Figure 2.2 and Figure 2.3 and
equating the internal and external work for the mechanism, the moment
resistance of the perimeter beams required to achieve a load bearing capacity
equal to that for the floor slab may be determined. The derivation of
appropriate design equations is given below.
Having calculated the required bending resistance of these beams to ensure that
they provide sufficient support to allow development of the tensile membrane
enhancement of the slab load bearing resistance, a critical temperature for the
beams can be calculated and appropriate levels of fire protection can be applied
to ensure that this critical temperature is not exceeded during the required fire
resistance period.
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Part 2: Engineering Update
23
2.1 Floor design zone with edge beams on sides B
and D
2.1.1 Yield line parallel to unprotected internal beams
This case considers the required bending resistance of the perimeter beams on
Sides B and D of a floor design zone when these beams are also at the edge of
the slab. A single yield line is assumed to form across the centre of the floor
design zone in the Span 1 direction, as shown in Figure 2.2. In keeping with the
assumptions of the design method, the perimeter of the floor design zone is
assumed to be simply supported.
o
o
o
o
M
M
M
HOT
HOT
Edge
of slab
Edge
of slab
b,1
Yield line
Axis of rotation
Axis of rotation
Figure 2.2 Yield line parallel to the unprotected beams with edge beams on
Sides B and D
Considering a unit displacement along the yield line, the rotation of the yield
line can be calculated as follows:
Yield line rotation =
2
1
2
2
L
=
2
4
L
The internal work done due to the rotation of the yield line is given by:
Internal Work = ( )
2
1 , b eff 1,
4
2
L
M ML + =
2
1 , b
2
eff 1,
8 4
L
M
L
L M
+
where:
L
1,eff
is the effective length of the yield line discounting the effective width
of slab assumed to act with the perimeter beams where these are
designed as composite members.
M is the resistance moment of the slab per unit length of yield line.
For a uniform load on the slab, p, the external work due to the displacement is
given by:
External Work =
2 1
2
1
L L p
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Equating internal and external work gives:
2
b,1
2
eff 1,
2 1
16 8
L
M
L
ML
L L p + =
If the load on the slab is the load bearing capacity determined in accordance
with Appendix B:, the required minimum values of bending resistance for the
perimeter beams on Side B and D is given by:
16
8
eff 1,
2
2 1
b,1
ML L pL
M
=
where:
p is the uniformly distributed load to be supported by the floor design
zone in fire conditions.
2.1.2 Yield line perpendicular to unprotected beams
This case considers the required moment resistance of the perimeter beams on
Sides A and C of the floor design zone when these are at the edge of the slab.
A single yield line is assumed to form across the centre of the floor design zone
in the Span 2 direction, as shown in Figure 2.3. In keeping with the
assumptions of the design method, the perimeter of the floor design zone is
assumed to be simply supported.
o
o
o
o M
M
M
M
HOT
HOT
Axis of rotation
zero vertical
displacement
Axis of
rotation
Displacement
along yield line
equal to unity
b,2
b,2
Figure 2.3 Yield line perpendicular to the unprotected beams
Considering a unit displacement along the yield line, the rotation of the yield
line can be calculated as follows:
Yield line rotation =
2
1
2
1
L
=
1
4
L
The internal work done due to the rotation of the yield line is given by:
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Part 2: Engineering Update
25
Internal Work = ( )
1
HOT 2 , b eff 2,
4
2
L
nM M ML + +
=
1
HOT
1
2 , b
1
eff 2,
4
8
L
4
L
nM
L
M L M
+ +
where:
L
2,eff
is the effective length of the yield line discounting the effective width
of slab assumed to act with the perimeter beams where these are
designed as composite members and the composite unprotected
internal beams.
M is the resistance moment of the slab per unit length of yield line
The external work due to the slab displacement is given by:
External Work =
2 1
2
1
L L p
Equating internal and external work gives:
1
HOT
1
b,2
1
eff 2,
2 1
8
16 8
L
nM
L
M
L
ML
L L p + + =
If the load on the slab is the load bearing capacity determined in accordance
with Appendix B:, the required minimum values of bending resistance for the
perimeter beams on Side A and C is given by:
16
8 8
HOT eff 2, 2
2
1
b,2
nM ML L pL
M
=
where:
p is the uniformly distributed load to be supported by the floor design
zone in fire conditions.
2.2 Floor design zone with an edge beam on one
side
2.2.1 Yield line parallel to unprotected beams
This case considers the required moment resistance of the perimeter beams on
Sides B and D of the floor design zone when the beam on side B is an internal
perimeter beam. As the software only deals with an isolated floor plate, the
calculation of resistance for an internal perimeter beam must assume that the floor
design zone is adjacent to an identical area of slab on any side where an internal
beam has been specified. A single yield line is assumed to form across the centre
of the floor design zone in the Span 1 direction, as shown in Figure 2.4.
Considering a unit displacement along the yield line, the rotation of the yield
line can be calculated as follows:
Yield line rotation =
2
1
2
2
L
=
2
4
L
The internal work done due to the rotation of the yield line is given by:
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Internal Work = ( )
2
1 , b eff 1,
4
3 2
L
M ML + =
2
1 , b
2
eff 1,
12 8
L
M
L
L M
+
The external work due to the slab displacement is given by:
External Work =
2 1
2
2
1
L L p
Equating internal and external work gives:
2
b,1
2
eff 1,
2 1
12 8
L
M
L
ML
L L p + =
o
o
o
o
M M b,1
Axis of rotation
L
1
L
2
1
L
L
2
b,1
M
b,1
Yield line
with unit
displacement
along its length
Floor design zone
Figure 2.4 Yield line parallel to the unprotected beams, with edge beam on Side D
If the load on the slab is the load bearing capacity determined in accordance
with Appendix B:, the required minimum values of resistance moment for the
perimeter beams on Side B and D is given by:
12
8
eff 1,
2
2 1
b,1
ML L pL
M
=
where:
L
1,eff
is the effective length of the yield line discounting the effective width
of slab assumed to act with the perimeter beams where these are
designed as composite members
M is the resistance moment of the slab per unit length of yield line
p is the uniformly distributed load to be supported by the floor design
zone in fire conditions.
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Part 2: Engineering Update
27
2.2.2 Yield line perpendicular to unprotected beams
This case considers the formation of a single yield line across the centre of the
floor design zone in the Span 2 direction, as shown in Figure 2.5, when the
beam on side C is an internal beam.
o
o
o
o
M
Floor design zone
b,2
M
M
HOT
HOT
M
b,2
M
M
HOT
HOT
M
b,2
Figure 2.5 Yield line perpendicular to the unprotected beams edge condition on
Side A
Considering a unit displacement along the yield line, the rotation of the yield
line can be calculated as follows:
Yield line rotation =
2
1
2
1
L
=
1
4
L
The internal work done due to the rotation of the yield line is given by:
Internal Work =( )
1
HOT 2 , b eff 2,
4
2 3 2
L
nM M ML + +
=
1
HOT
1
2 , b
1
eff 2,
8
12
L
8
L
nM
L
M L M
+ +
The external work due to the slab displacement is given by:
External Work =
2 1
2
2
1
L L p
Equating internal and external work gives:
1
HOT
1
b,2
1
eff 2,
2 1
8
12 8
L
nM
L
M
L
ML
L L p + + =
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28
If the load on the slab is the load bearing capacity determined in accordance
with Appendix B:, the required minimum values of resistance moment for the
perimeter beams on Side A and C is given by:
12
8 8
HOT eff 2, 2
2
1
b,2
nM ML L pL
M
=
where:
L
2,eff
is the effective length of the yield line discounting the effective width
of slab assumed to act with the perimeter beams where these are
designed as composite members and the composite unprotected
internal beams.
M is the resistance moment of the slab per unit length of yield line
p is the uniformly distributed load to be supported by the floor design
zone in fire conditions.
2.3 Floor zone without edge beams
For zones where none of the perimeter beams are edge beams, it is
conservative to use the values determined by the expressions in Section 2.2.
2.4 Design of edge beams
It is common practice for beams at the edge of floor slabs to be designed as non
composite. This is because the costs of meeting the requirements for transverse
shear reinforcement are more than the costs of installing a slightly heavier non
composite beam. However, for fire design, it is important that the floor slab is
adequately anchored to the edge beams, as these beams will be at the edge of
floor design zones. For this purpose, if edge beams are designed as non
composite, they must have shear connectors at not more than 300 mm centres
and U-bars should be provided to tie the edge beam to the composite slab.
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Part 2: Engineering Update
29
3 WORKED EXAMPLE
In order to illustrate the application of the output from TSLAB V3.0, this
Section contains a worked example, based on a realistic composite floor plate.
The building considered is a four-storey steel-framed office building. As the
height from ground level to the uppermost floor of the building does not
exceed 18 m, the building requires 60 minutes fire resistance in accordance
with the requirements of Approved Document B to the UK Building
Regulations.
The floor plate for each storey consists of a composite floor slab constructed
using Kingspan MD60 trapezoidal metal decking, normal weight concrete and
a single layer of mesh reinforcement. The slab spans between 9 m long
secondary beams designed to act compositely with the floor slab. These
secondary beams are in turn supported on composite primary beams of 9 m and
12 m spans. The beams on the edge of the building are designed as
non-composite beam in accordance with BS EN 1993-1-1
[ 15]
.
The construction of the floor plate is shown in Figure 3.1 to Figure 3.4.
Figure 3.1 shows the general arrangement of steelwork at floor level across the
full width of the building and two bays along its length. It is assumed that this
general arrangement is repeated in adjoining bays along the length of the
building. The columns are 305 UKC 158, designed as non-composite columns
in accordance with BS EN 1993-1-1.
The floor loading considered was as follows
Variable action due to occupancy: 4 kN/m
2
Variable action due to light weight partitions: 1 kN/m
2
Permanent action due to ceilings and services: 0.7 kN/m
2
Self weight of beam: 0.5 kN/m
2
For the edge beams, an additional cladding load of 2 kN/m was considered in
their design.
The beam sizes required to satisfy the normal stage verification for these values
of actions are shown in Figure 3.1. The internal beams are composite and the
degree of shear connection for each beam is shown in Table 3.1.
Figure 3.2 shows a cross section through the composite slab. The slab is
C25/30 normal weight concrete with overall thickness of 130 mm. The slab is
reinforced with A142 mesh reinforcement with a yield strength of 500 MPa;
this meets the requirements for normal temperature design but the mesh size
may need to be increased in size if the performance in fire conditions is
inadequate.
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2 3 1
A
D
C
B
457x191x74 UB 457x191x74 UB
533x210x109 UB 533x210x109 UB
533x210x109 UB 533x210x109 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
457x191x74 UB 457x191x74 UB
5
3
3
x
2
1
0
x
9
2
U
B
5
3
3
x
2
1
0
x
8
2
U
B
5
3
3
x
2
1
0
x
9
2
U
B
5
3
3
x
2
1
0
x
8
2
U
B
6
8
6
x
2
5
4
x
1
4
0
U
B
6
8
6
x
2
5
4
x
1
4
0
U
B
6
8
6
x
2
5
4
x
1
4
0
U
B
5
3
3
x
2
1
0
x
8
2
U
B
5
3
3
x
2
1
0
x
8
2
U
B
9000 9000
9
0
0
0
9
0
0
0
3
0
0
0
1
2
0
0
0
Figure 3.1 General arrangement of steelwork at floor level
Table 3.1 Beam details
Beam Section
(S355)
Location of beam
Construction
Type
Degree of
Shear
Connection
(%)
Number of
shear studs
per group and
spacing
457191x74 UKB Secondary internal
beam
Composite 54 1 @ 323 mm
533210109 UKB Secondary edge beam Non composite N/A
53321082 UKB Primary internal beam Composite 96 2 @ 323 mm
53321092 UKB Primary edge beam Non composite N/A
686254140 UKB Primary internal beam Composite 100 2 @ 323 mm
686254140 UKB Primary edge beam Non Composite N/A
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Part 2: Engineering Update
31
130
30
60
Mesh A142 MD60 decking Normal weight
concrete
Figure 3.2 Construction of floor slab
All joints between the main steelwork elements use flexible end plate details
and are designed as nominally pinned in accordance with EN1993-1-8
[ 16]
.
Figure 3.3(a) shows the joint used between the primary beams and the
columns. The beam to column joints for secondary beams are as shown in
Figure 3.3(b). Figure 3.4 shows the endplate connection between the secondary
beams and the primary beams.
60
30
130
A142 mesh MD 60 decking
50
40
5 x 70
140
430 x 200 x 10 thick
end plate
6mm
fillet
weld
40
(a) Primary beam to column joint
60
30
130
A142 mesh MD 60 decking
50
40
3 x 70
90
290 x 150 x 8 thick
end plate
6mm
fillet
weld
40
(b) Secondary beam to column joint
Figure 3.3 Beam to column joints
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30
130
40
40
3 x 70
A142 mesh
MD 60 decking
90
280 x 150
x 8 thick
50
6mm
fillet
weld
60
Figure 3.4 Secondary beam to primary beam connection
Figure 3.5 shows the floor plate divided into floor design zones. It is likely that
floor design zones A and B will give the most onerous design conditions. The
design of both of these zones will be considered.
9000 9000
3
0
0
0
9
0
0
0
1
2
0
0
0
9
0
0
0
A
D
C
B
2 3 1
A
B
D
E
F
C
533x210x109 UB
533x210x109 UB
533x210x109 UB
533x210x109 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
457x191x74 UB
5
3
3
x
2
1
0
x
9
2
U
B
5
3
3
x
2
1
0
x
9
2
U
B
6
8
6
x
2
5
4
x
1
4
0
U
B
5
3
3
x
2
1
0
x
8
2
U
B
5
3
3
x
2
1
0
x
8
2
U
B
6
8
6
x
2
5
4
x
1
4
0
U
B
6
8
6
x
2
5
4
x
1
4
0
U
B
5
3
3
x
2
1
0
x
8
2
U
B
5
3
3
x
2
1
0
x
8
2
U
B
457x191x74 UB
Perimeter of floor design zones
Figure 3.5 Floor design zones (A F)
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Part 2: Engineering Update
33
3.1 Design of composite slab in fire conditions
The following design checks carried out on the floor design zones are based on
the floor construction required for normal temperature design checks. If this
construction proves to be inadequate for fire conditions then the mesh size
and/or the floor depth will be increased to improve the performance in fire
conditions. As the design zone B seems more critical than design zone A due to
its larger span, design zone B is considered first.
3.1.1 Floor design zone B
Figure 3.6 and Figure 3.7 show the output from the TSLAB V3.0 for floor
design zone B, which is 9 m by 12 m with the mesh size of A142. Within this
floor design zone, there are 3 unprotected composite beams.
From the output, the load bearing capacity of the slab based on the lower
bound yield line mechanism (Slab yield), is seen to be 0.46 kN/m
2
. This
capacity is enhanced due to the effect of membrane action to give a slab
capacity of 2.86 kN/m
2
at 60 minutes. The enhancement factor was based on an
assumed displacement of 636 mm which will be a conservative estimate of the
true slab displacement.
The load bearing capacity of the composite beams within the floor design zone
is added to that of the slab to give the total load bearing capacity. The beam
capacity is based on the temperature of the unprotected beams at each time
step. At 60 minutes, the beam capacity for the unprotected internal beams is
2.12 kN/m
2
. Thus, the total load bearing capacity of the floor design zone is
2.86 + 2.12 = 4.98 kN/m
2
, which is less than the design load 6.33 kN/m
2
, hence
a further case must be analysed with the mesh size increased to A252.
Figure 3.6 Input data of floor design zone B using TSLAB V3.0
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Figure 3.7 Results for the resistance of floor design zone B using TSLAB V3.0
Figure 3.8 and Figure 3.9 show the output from the TSLAB V3.0 software for
floor design zone B, which is 9 m by 12 m with mesh size of A253. Within this
floor design zone, there are 3 unprotected composite beams.
From the output, the load bearing capacity of the slab, based on the lower bound
yield line mechanism is seen to be 0.78 kN/m
2
. This capacity is enhanced due to
the effect of membrane action to give a slab capacity of 5.02 kN/m
2
at 60
minutes. The enhancement factor was based on a slab deflection of 636 mm.
The load bearing capacity of the composite beams is added to that of the slab to
give the total load bearing capacity. The beam capacity is based on the
temperature of the unprotected beams at each time step. At 60 minutes, the
beam resistance of the three unprotected beams is 2.12 kN/m
2
. Thus, the total
load bearing resistance of the floor design zone is 5.02 + 2.12 = 7.14 kN/m
2
,
which is greater than the design load; hence the floor slab is adequate
Figure 3.8 Input data of floor design zone B using TSLAB V3.0
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Part 2: Engineering Update
35
Figure 3.9 Results for the resistance of floor design zone B using TSLAB V3.0
TSLAB also provides a critical temperature for each of the perimeter beams, as
shown in Figure 3.10. The fire protection applied to these beams should be
sufficient to ensure that the temperature of the beams in a fire does not exceed
this critical temperature for the required period of fire resistance. The degree of
utilisation quoted for each beam is the ratio between the effect of actions on the
beam in the fire condition divided by the moment resistance of the beam
calculated in fire conditions at time zero (room temperature), as explained in
Section 1.4.2.
Figure 3.10 Requirements for the resistance of the perimeter beams of floor design
zone B, given by TSLAB V3.0
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3.1.2 Floor design zone A
Figure 3.11 and Figure 3.12 show the output from TSLAB for floor design
zone A, which is 9 m by 9 m. Within this floor design zone there are 2
unprotected composite beams.
From the output, the load bearing capacity of the slab based on the lower
bound yield line mechanism is seen to be 1.01 kN/m
2
. This capacity is
enhanced due to the effect of membrane action to give a slab capacity of
5.50 kN/m
2
at 60 minutes. The enhancement factor at 60 minutes was based on
a slab deflection of 573 mm.
The load bearing capacity of the composite beams is added to the slab capacity
to give the total load bearing capacity. The beam capacity is based on the
temperature of the unprotected beams at each time step. At 60 minutes, the
bending resistance of the two unprotected beams is 2.12 kN/m
2
. Thus, the total
load bearing resistance of the floor design zone is 5.50 + 2.12 = 7.62 kN/m
2
Figure 3.11 Input of floor design zone A using TSLAB V3.0
Figure 3.12 Results for the resistance of floor design zone A using TSLAB V3.0
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Part 2: Engineering Update
37
TSLAB V3.0 also provides a critical temperature for each of the perimeter
beams, as shown in Figure 3.13. The fire protection applied to these beams
should be sufficient to ensure that the temperature of the beams in a fire does
not exceed this critical temperature for the required period of fire resistance.
The degree of utilisation quoted for each beam is the ratio between the effect of
actions on the beam in the fire condition divided by the moment resistance of
the beam calculated in fire conditions at time zero (room temperature), as
explained in Section 1.4.2.
Figure 3.13 Requirements for the resistance of the perimeter beams of floor design
zone A, given by TSLAB V3.0
3.2 Reinforcement details
Since the output confirms that the load bearing capacity of zones A and B are
both adequate, the A252 mesh provided is adequate for fire design. This mesh
has an area of 252 mm
2
/m in both directions and has 7 mm wires spaced at
150 mm centres in both directions.
The mesh in this example has a yield strength of 500 N/mm
2
. For fire design
the Class of reinforcement should be specified as Class B or C in accordance
with BS EN 10080, to ensure that the mesh has adequate ductility.
At joints between sheets the mesh must be adequately lapped in order to ensure
that its full tensile resistance can be developed in the event of a fire in the
building. For the 7 mm diameter bars of the A252 mesh the minimum lap
length required would be 250 mm. In order to avoid the build up of bars at
lapped joints, sheets of mesh with flying ends should be specified as shown in
Figure 1.4.
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Additional reinforcement in the form of U-shaped bars should be provided at
the edge beams to ensure adequate tying between these beams and the
composite slab.
3.3 Fire design of perimeter beams
3.3.1 Internal perimeter beams
The internal perimeter beams to each zone are part of more than one floor
design zone. For example, consider the beam on Gridline B between gridlines 1
and 2: it can be seen from Figure 3.5 that this is the perimeter beam on Side C
of floor design zone A and is the beam on Side A of floor design zone B. The
fire protection applied to this member must therefore be based on the lower
value of critical temperature given by the results from these two floor design
zones. In this case, floor design zone B is the more critical.
For this member, the following information (taken from the requirements listed
in Figure 3.13) should be given to the fire protection manufacturer in order to
determine the required thickness of fire protection.
Fire resistance period 60 minutes
Section size 457 191 74 UB
Critical temperature 641C
For this size of beam the section factor, determined in accordance with
BS EN 1993-1-2, is:
Section Factor 115 m
-1
for box protection heated on 3 sides
155 m
-1
for profiled protection heated on 3 sides
3.3.2 Edge beams
In this example the edge beams were designed to be non-composite. However,
for the fire design case these beams should be adequately tied into the
composite slab. This is achieved by providing U-bars and shear studs on the
beam. Studs should be provided at 300 mm centres where the decking is
parallel to the beam and in every trough of the decking profile where the
decking spans perpendicular to the beam.
The fire protection required for the edge beams should be specified in the same
way as for internal perimeter beams.
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Part 2: Engineering Update
39
3.4 Fire protection of columns
Fire protection should also be specified for all of the columns in this example.
The following information should be provided when specifying the fire
protection.
Fire resistance period 60 minutes
Section size 305 305 158 UC
Section Factor 65 m
-1
for box protection heated on 4 sides
90 m
-1
for profiled protection heated on 4 sides
Critical temperature: 500C or 80C less than the critical temperature
calculated on the basis of the BS EN 1993-1-2 design
rules, whichever is the lower.
The applied fire protection should extend over the full height of the column, up
to the underside of the composite floor slab.
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4 REFERENCES
1. NEWMAN, G.M., ROBINSON, J.T. and BAILEY, C.G.
Fire safe sesign: A new approach to multi-storey steel framed buildings
(P288, 2
nd
edition)
The Steel Construction Institute, 2006
2. JOHANSEN, K.W.
The Ultimate strength of Reinforced Concrete Slabs
International Association for Bridge and Structural Engineering, Final
Report, Third Confress, Liege, September 1948
3. OCKLESTON, A.J.
Load tests on a 3-storey reinforced concrete building in Johannesburg.
Struct Eng 1955;33(10):304-22
4. BAILEY, C.G. and MOORE, D.B.
The structural behaviour of steel frames with composite floor slabs
subjected to fire: Part 1: Theory
The Structural Engineer, June 2000
5. BAILEY, C.G. and MOORE, D.B.
The structural behaviour of steel frames with composite floor slabs
subjected to fire: Part 2: Design
The Structural Engineer, June 2000
6. WOOD, R. H.
Plastic and elastic design of slabs and plates, with particular reference to
reinforced concrete floor slabs
Thames and Husdon, London. 1961.
7. HAYES, B.
Allowing for membrane action in the plastic analysis of rectangular
reinforced concrete slabs
Magazine of concrete research Vol. 20 No. 81 Dec 1968. pp 205-212.
8. BAILEY, C. G. and TOH, W.S.
Behaviour of concrete floor slabs at ambient and elevated temperature
Fire Safety Journal, 42, pp425-436, 2007
9. BAILEY, C.G.
Efficient arrangement of Reinforcement for membrane behaviour of
composite slabs in fire conditions
Journal of Constructional Steel Research, 59, 2003, pp931-949.
10. BAILEY, C.G.
Membrane action of lightly reinforced concrete slabs at large displacements
Engineering Structures, 23, 2001, pp470-483
11. BS EN 1990:2002 Eurocode Basis of structural design
BSI
12. BS EN 1991-1-1:2003 Eurocode 1. Actions on structures. Part 1-1: General
actions. Densities, self-weight, imposed loads for buildings
BSI
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Part 2: Engineering Update
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13. BS EN 1992-1-2:2004 Eurocode 2. Design of concrete structures. Part 1.2:
General rules. Structural fire design,
BSI
14. BS EN 1992-1-1:2004 Eurocode 2. Design of concrete structures. Part 1.1:
General rules and rules for buildings
BSI
15. BS EN 1993-1-1:2005 Eurocode 3. Part 1.1: Design of steel structures.
General rules and rules for buildings
BSI
16. BS EN 1993-1-8:2005 Eurocode 3. Part 1.8: Design of steel structures.
Design of joints
BSI
17. BS EN 1993-1-2:2005 Eurocode 3. Design of steel structures. General
rules. Structural fire design
BSI
18. BS EN 1994-1-2:2005 Eurocode 4. Design of composite steel and concrete
structures. Structural fire design
BSI
19. BS EN 10080:2005 Steel for the reinforcement of concrete - Weldable
reinforcing steel General, BSI.
20. BS 4483:2005 Steel fabric for the reinforcement of concrete. Specification.
BSI
21. BS 4449:1:2005 Steel for the reinforcement of concrete. Weldable
reinforcing steel. Bar, coil and decoiled product. Specification
BSI
22. EN13381-4 Test methods for determining the contribution to the fire
resistance of structural members. Applied passive protection to steel
members
CEN, (To be published 2010)
23. EN13381-8 Test methods for determining the contribution to the fire
resistance of structural members. Applied reactive protection to steel
members
CEN, (To be published 2010)
24. COUCHMAN, G. H., HICKS, S. J. and RACKHAM, J. W.
Composite slabs and beams using steel decking: Best practice for design
and construction. Revised edition (P300)
The Steel Construction Institute, 2008
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APPENDIX A: Thermal analysis
TSLAB V3.0 uses a 2D finite difference heat transfer method to predict the
temperature distribution within the composite slab. This method has been used
for many years by SCI to predict the temperature distributions in steel and
steel-concrete composite cross sections and has been shown to be able to give
reasonably accurate predictions of the behaviour of sections in fire resistance
tests. The model to be analysed must be defined on a rectangular grid of cells.
The method can also analyse the sloping sides of trapezoidal or re-entrant
composite slabs by using configuration factors. The thermal properties of steel
and concrete used by TSLAB are based on the values given by
BS EN 1994-1-2
[ 18]
. The thermal actions are calculated on the basis of the net
heat flux,
net
h
to which the surface of the member is exposed. The net heat flux
is determined considering the heat transfer by convection and radiation.
r net, c net, net
h h h
+ = (1)
The net convective heat flux component is determined as follows:
( )
m g c c net,
u u o = h
(2)
where;
c
o is the coefficient of heat transfer by convection
g
u is the gas temperature
m
u is the surface temperature of the member.
When carrying out a thermal analysis for a member exposed to the standard
temperature-time curve the coefficient of heat transfer by convection on the
exposed face is taken as
c
= 25 W/m
2
K. For natural fire models, the
coefficient of heat transfer by convection is increased to
c
= 35 W/m
2
K. On
the unexposed side of the slab, the net heat flux is based on heat transfer by
convection, but the coefficient of heat transfer by convection is taken as
c
= 9 W/m
2
K, to allow for the effects of heat transfer by radiation, which are
not considered explicitly in the model. The net radiative heat flux is determined
from the following formula:
( ) ( ) | |
4
m
4
r f m r net,
273 273 + + = u u o c c h (3)
where:
is the configuration factor
m
c is the surface emissivity of the member
f
c is the emissivity of the fire
o is the Stephan Boltzmann constant (5.67 x 10
-8
W/m
2
K
4
)
r
u is the effective radiation temperature of the fire (C)
m
u is the surface temperature of the member (C).
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Part 2: Engineering Update
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The emissivity of the fire is taken as 0 . 1
f
= c , in accordance with the
recommended value in BS EN 1994-1-2. The emissivity of the member may be
determined from Section 0.
A.1 Configuration factors
For steel decking profiles, the following configuration factors are used to
modify the net heat flux incident on each surface. The locations in which the
following factors are applied are shown in Figure A.1 for trapezoidal decking
profiles and in Figure A.2 for re-entrant decking profiles.
Trapezoidal Profiles
The bottom flange of the trapezoidal profile is assumed to have a configuration
factor of 1.0. For the top flange the configuration factor,
TOP
, is calculated as
follows.
( )
2
tan 2
1
1
TOP
|
|
.
|
\
|
=
b p
h
Similarly, for the sloping web of the trapezoidal profile, the configuration
factor,
SIDE
, is calculated as follows,
y x
L
+
= 5 . 0
SIDE
Re-entrant Deck
The bottom flange of re-entrant steel profiles is assumed to have a
configuration factor of 1.0. The configuration factor for the surfaces of the re-
entrant dovetail is calculated as follows,
y x
L
+
= 3 . 0
INT
| = 1.0
|
|
h
b
Element
L
y
x
i
TOP
SIDE
1
i
Element
Figure A.1 Configuration factors for trapezoidal decks
|
|
INT
= 1.0
Figure A.2 Configuration factors for re-entrant decks
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A.2 Material properties for steel and concrete
The following material properties are used for steel and concrete. These values
are based on the recommendations of EN1994-1-2. Table A.1 shows the values
of surface emissivity, density and moisture content used for steel, normal
weight concrete and light weight concrete.
Table A.1 Material properties of steel and concrete for thermal analysis
Steel NWC LWC
Emissivity,
m
c
0.7 0.7 0.7
Density, 7850 2300 1850
% moisture by mass 0 4 4
The specific heat capacity of steel, C
a
, for all structural and reinforcing steel is
given by the following temperature dependant formulae:
3 2
a
00000222 . 0 00169 . 0 773 . 0 425 u u u + + = C
(J/kg K)
C 600 C 20 s s u
( ) 738
13002
666
a
=
u
C
(J/kg K)
C 735 C 600 s s u
( ) 731
17820
545
a
=
u
C
(J/kg K)
C 900 C 735 s s u
C
a
= 650 (J/kg K)
C 1200 C 900 s s u
The following temperature dependant values of specific heat capacity, C
c
, are
used for normal weight dry concrete with siliceous or calcareous aggregates.
C
c
= 900 (J/kg K) C 100 C 20 s s u
C
c
= 900 + ( 100) (J/kg K) C 200 C 100 s s u
C
c
= 1000 + ( 200)/2 (J/kg K) C 400 C 200 s s u
C
c
= 1100 (J/kg K) C C s s 1200 400
As recommended by EN1994-1-2 the following temperature independent value
of specific heat capacity is assumed for lightweight concrete.
C
c
= 840 (J/kg K) for all temperatures
The thermal conductivity of steel is defined using the following temperature
dependent relationship.
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Part 2: Engineering Update
45
For normal weight concrete the upper limit of thermal conductivity defined by
BS EN 1994-1-2 has been used. The thermal conductivity for normal weight
concrete is determined from the following temperature dependent relationship.
( ) ( )
2
c
100 0107 . 0 100 2451 . 0 2 u u + =
(W/mK)
The thermal conductivity of lightweight concrete is also temperature dependent
and is given by the following formula.
( ) 1600 1
c
u = but not less than 0.5 (W/mK)
A.3 Internal heat transfer by conduction
The thermal analysis computes the conducted heat transfer between a cell and
the four cells above, below and to the sides (Figure A.3). No other cells are
involved.
Figure A.3 Basis of conductive heat transfer
The heat transferred per unit time depends on the sizes of the cells, the
temperature of each cell and the thermal conductivity of each cell. Each pair of
cells is considered in turn and the net heat transferred into or out of a cell is
computed. The basic conduction model is illustrated in Figure A.4.
1
1 1
2
w w
d
T T , ,
2 2
T
Figure A.4 Basic conduction model
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46
The temperature of each cell is defined at its centre (T
1
, T
2
). The temperature of
the interface between the cells is T. The heat transfer from cell 1 to the
interface is the same as the heat transfer from the interface to cell 2. The
thermal conductivities of each cell are
1
and
2
.
The heat transfer per unit time from the centre of cell 1 to the interface is:
( )
1
1
1
2
T T
w
D
h =
This is equal to the heat transfer per unit time from the interface to the centre of
cell 2:
( ) T T
w
D
h =
2
2
2
2
Thus, by eliminating the interface temperature, T:
( )
|
|
.
|
\
|
+
=
2
2
1
1
1 2
2 2 D
w
D
w
T T
h per unit time
This equation is used to compute the heat transfer between all cells. For each
cell, the value of:
D
w
2
is calculated. The value of thermal conductivity will often vary with
temperature and is calculated at preset intervals (normally 30 seconds) to speed
up computation.
A.4 Design temperatures for unprotected steel
beams
The calculated design temperatures of the unprotected steel beams are based on
the simple method given in BS EN 1994-1-2, Section 4.3.4.2.2. The increase in
steel temperature during a small time interval is calculated using the following
equation.
t h
V
A
c
k A
|
|
.
|
\
|
|
|
.
|
\
|
= A
net
i
i
a a
shadow t a,
1
u
where:
shadow
k is the correction factor for shadow effect
a
is the density of the steel
t A is the time interval
i i
V A is the section factor for part i of the cross section.
TSLAB V3.0 calculates the steel temperature for the bottom flange of the
section for time increments of 2.5 seconds. The correction factor for the
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Part 2: Engineering Update
47
shadow effect is taken as 1.0. The section factor for the bottom flange is
expressed as a function of flange thickness, e
1
, as follows
1
2000
e
V A
i i
=
The net heat flux is calculated as shown in Equation 1, with the convective and
radiative components calculated as shown by Equations 2 and 3 respectively.
When calculating the radiative heat flux using Equation 3 the configuration
factor should be taken as 1.0.
The main change to the temperature distribution used in TSLAB V3.0 is the
method of determining the mesh temperature. In TSLAB V2.4, the mesh
temperature was based on a weighted average value calculated from the results
of thermal analysis. This has been changed to the maximum value of
temperature of the mesh given by the thermal analysis.
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48
APPENDIX B: SIMPLE DESIGN METHOD
Since Johansens work on yield line analysis
[ 2]
researchers have observed the
beneficial effects of membrane forces in improving the load bearing capacity
of concrete slabs, compared to estimates of capacity based only on flexural
behaviour
[ 3]
.
A number of experimental and theoretical investigations have been carried out
to investigate the beneficial effects of in-plane forces at room temperature,
leading to a good theoretical understanding of the behaviour. Following the
experimental work carried out at Cardington, this theory has been extended to
fire design scenarios, as discussed below.
The experimental work at Cardington and evidence from other real fires in
building structures had served to illustrate that there are significant reserves of
strength in composite steel concrete buildings, which means that the
performance of the structure in fire exceeds the expectations created by
standard fire tests on individual structural elements. Cardington demonstrated
that it was possible to leave the composite steel beams that supported the
concrete floor slab unprotected; work commenced to investigate suitable
design models to allow structural engineers to justify the fire design of a floor
slab supported by unprotected steel beams.
Researchers at the Building Research Establishment (BRE), with funding from
the Steel Construction Institute, developed a simple design method for
composite steel concrete floor slabs following the experimental work at
Cardington
[ 4, 5]
. The BRE model has been validated against the Cardington
large scale fire test results and previous experimental work conducted at room
temperature. This method is presented and discussed in detail in Section B.1.
This simple design method differs from the simple design procedures provided
in design codes
[ 13, 14]
, as it considers the behaviour of an assembly of structural
members acting together, rather than individual elements. While it would also
be technically possible to use non-linear finite elements to determine the load
bearing capacity in fire, that is a more expensive solution requiring a
significant amount of expertise and prior knowledge. The method presented in
this document is more accessible to structural engineers with only a basic
appreciation of fire engineering.
B.1 Calculation of resistance of composite floors in
accordance with the simple design method
This Section describes the development of a simple design method that can be
used to calculate the resistance of rectangular composite floor plates. The
method has developed over a number of years. The initial development
[ 4, 5]
of
the method for use with isotropic reinforcement only considered one failure
mode, due to fracture of the mesh across the short span, as shown by
Figure B.1(a). Later developments
[ 8, 9]
included a more general derivation
allowing the use of orthotropic reinforcement, and the inclusion of
compression failure of the concrete at the slab corners (see Figure B.1(b)).
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Part 2: Engineering Update
49
B.1.1 Calculation of resistance
The load bearing capacity of a two-way spanning simply supported slab, with
no in-plane horizontal restraint at its edges, is greater than that calculated using
the normal yield line theory. The enhancement of the resistance is as a result of
tensile membrane action developing in the slab at large displacement and due
to the increase of the yield moment in the outer regions of the slab, where
compressive stresses occur across the yield lines (see Figure B.2).
The enhancement of the resistance determined as a lower bound solution for
yield line failure is based on the assumption that at ultimate conditions the
yield line pattern will be as shown in Figure B.1(a) and that failure will occur
due to fracture of the mesh across the short span at the centre of the slab. A
second mode of failure might, in some cases, occur due to crushing of the
concrete in the corners of the slab where high compressive in-plane forces
occur as shown by Figure B.1(b). This mode of failure is discussed in
Section B.2.
Full depth crack Compression failure of concrete
Edge of slab moves towards centre
of slab and 'relieves' the strains in
the reinforcement in the short span
Yield-line pattern
Reinforcement in
longer span fractures
(a) Tensile failure of mesh reinforcement
Edge of slab moves towards centre
of slab and 'relieves' the strains in
the reinforcement in the short span
Yield-line pattern
Concrete crushing due
to in-plane stresses
(b) Compressive failure of concrete
Figure B.1 Assumed failure modes for composite floor
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The first failure mode will occur when the compressive strength of the concrete
exceeds the ultimate strength of the mesh in tension, leading to fracture of the
mesh. The second failure mode will occur in cases were the ultimate strength
of the mesh exceeds the compressive strength of the concrete, resulting in
compression failure of the concrete at the corners of the slab.
Compression
Tension
Element 2
Element 1
L
nL
l
Figure B.2 Rectangular slab simply supported on four edges showing in-plane
forces across the yield lines due to tensile membrane action
Figure B.2 shows a rectangular slab simply supported on its perimeter and the
expected lower bound yield line pattern that would develop due to uniformly
distributed loading. The intersection of the yield lines is defined by the
parameter n calculated using the general yield line theory and given by:
( ) 1 1 3
2
1
+ = a
a
n
where:
a is the aspect ratio of the slab (L/l)
is the ratio of the yield moment capacity of the slab in orthogonal
directions (should always be less than or equal to 1.0)
The shorter span should be defined by the span with the lower moment
resistance resulting in coefficient of orthography () being always less than, or
equal to one. Therefore n would be limited to maximum of 0.5 resulting in a
valid yield line pattern.
The resistance of the mechanism which occurs due to the formation of these
yield lines is given by the following equation:
P =
( )
2
2 2
1 1
3
24
(
(
+
a' a' l
M
where:
a = a
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Part 2: Engineering Update
51
Hayes
[ 7]
noted that assuming rigid-plastic behaviour, only rigid body
translations and rotations are allowed. Further assumptions that the neutral axes
along the yield lines are straight lines and that the concrete stress-block is
rectangular, means that the variations in membrane forces along the yield lines
become linear, as shown in Figure B.3. These assumptions and the resulting
distribution of membrane forces were also adopted by Bailey
[ 4, 10]
.
nL
L
E
C
A
S
|
2
T
b K T
o
Element 1
F
T
1
k b K T
o C
D
S
T
b K T
2
o
C
L
Element 2 l
T
M
0
0
KT
M
0
0
Resistance in
long span =
Moment =
Resistance in
short span =
Moment =
Figure B.3 In-plane stress distribution for the elements 1 and 2
Derivation of an expression for parameter k
Considering the equilibrium of the in-plane forces T
1
, T
2
and C acting on
Element 1 allows the following relationships to be derived:
| | cos ) ( sin
2
T C S = and
2
sin ) ( cos
1
2
T
T C S = | |
Therefore,
) ( sin
2
2
1
T C
T
= | (1)
where:
| is the angle defining the yield line pattern.
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C
T
2
o
o
D
kbKT
bKT
(k/[l+k])
2 2
([nL] + l /4)
nL
2 2
([nL] + l /4) l/(l+k)
C
l/2
Figure B.4 In-plane stress distribution along yield line CD
Figure B.4 shows the geometry of the stress distribution along yield line CD.
Considering Figure B.3 and Figure B.4,
) 2 (
0 1
nL L bKT T =
4
) (
1
1
2
2
2 0
2
l
nL
k
bKT
T + |
.
|
\
|
+
=
4
) (
1 2
2
2 0
l
nL
k
k kbKT
C + |
.
|
\
|
+
=
4
) (
sin
2
2
l
nL
nL
+
= |
where:
b, k are parameters defining the magnitude of the membrane force,
0
KT is the resistance of the steel reinforcing mesh per unit width,
n is a parameter defining the yield line pattern
Substituting the above values into Equation (1) gives,
4
) (
1
1
2 4
) (
1 2
4
) (
2
) 2 (
2
2 0
2
2 0
2
2
0
l
nL
k
bKT l
nL
k
k kbKT
l
nL
nL nL L bKT
+
|
.
|
\
|
+
+
|
.
|
\
|
+
=
+
=
a n
n na
k (2)
Derivation of an expression for parameter b
Considering the fracture of the reinforcement across the short span of the slab,
an expression for the parameter b can be developed. The line EF shown in
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Part 2: Engineering Update
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Figure B.5 represents the location of the mesh fracture, which will result in a
full depth crack across the slab. An upper bound solution for the in-plane
moment of resistance along the line EF can be obtained by assuming that all
the reinforcement along the section is at ultimate stress (f
u
) and the centroid of
the compressive stress block is at location E in Figure B.5.
It is assumed that,
y u
f f 1 . 1 =
where:
y
f is the yield stress.
Taking moment about E in Figure B.5,
o
sin L/2
T
2
1 (L/2 - nL) / cos
cos L / 2
cos L/2 - (L/2 - nL)/cos
E
F
S
C
nL
L / 2
1.1T / 2 l
|
|
|
| |
|
|
T /2
Figure B.5 In-plane stress distribution along fracture line EF
( )
( )
8
1 . 1
2 2
1
2
cos
2
4 1 3
sin
2
4 1
1
3
1
tan
1
cos
2
cos
2
2
1
2
2
2
2
2
l T
nL
L T L
S
l
nL
k
k l L
C
l
nL
k
nL
L
L
T
o
=
(
|
.
|
\
|
+
(
(
+ |
.
|
\
|
+
+
(
(
(
(
+ |
.
|
\
|
+
|
|
|
|
.
|
\
|
|
.
|
\
|
|
|
| |
|
(3)
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56
Contribution of membrane forces to load bearing capacity.
a) Element 1
According to Figure B.6, the moment about the support due to membrane force
is given by:
Membrane Force
w
M
1m
Figure B.6 Calculating the moment caused by the membrane force
|
|
.
|
\
|
+
|
|
.
|
\
|
+
+
+ =
2
3
0
2
0 0 1
) 1 ( 3 ) 1 ( 3
2 3
) 2 (
k
k
nLw bKT
k
k
nLw bKT w nL L bKT M
m
where:
m
M
1
is the moment about the support due to membrane forces for
element 1.
The expression reduces to:
|
|
.
|
\
|
+
+
+ =
2
3
0 1
) 1 ( 3
) 2 3 (
) 2 1 (
k
nk k n
n Lbw KT M
m
.
The above formulation defines the contribution from the membrane forces to
the load bearing capacity that needs to be added to the contribution due to the
enhanced bending capacity in the areas where the slab is experiencing
compression forces. For simplicity, the contribution from the membrane forces
and enhanced bending action is related to the normal yield line load. This
allows an enhancement factor to be calculated for both the membrane force and
the enhanced bending moments. These enhancement factors can finally be
added to give the overall enhancement of the slab due to membrane action.
Dividing
m
M
1
by L M
o
, the resistance moment of the slab, when no axial
force is present, allows the effect of tensile membrane action to be expressed as
an enhancement of yield line resistance (Figure B.7).
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Part 2: Engineering Update
55
which can be rearranged to give,
( ) ( )
( )
( )
( )
K
l nL L
nL
L
b k
n
bl
l
nL
k
k nL
k
k b
l
nL
k
l
nL
nL
nL
L
n
l
k
b
8
1 . 1
2 4 2
1
16
4 1 3 2 1 2
4 1
1
3
1
4
2
8 1
1
2
2 2
2
2
2 2
2
2
2
2
2
= |
.
|
\
|
|
.
|
\
|
+
(
(
|
|
.
|
\
|
+
+
|
|
.
|
\
|
+
+
(
(
(
(
|
|
|
|
|
.
|
\
|
|
|
.
|
\
|
+ |
.
|
\
|
+
|
|
.
|
\
|
+
|
.
|
\
|
|
.
|
\
|
+
(4)
Equation (4) can be rewritten as,
K
l
Db Cb Bb Ab
8
1 . 1
2
= + +
Whence:
( ) D C B A K
l
b
+ +
=
8
1 . 1
2
(5)
where:
( )
( ) ( )
(
(
|
|
.
|
\
|
+
|
.
|
\
|
+
|
|
.
|
\
|
+
|
.
|
\
|
+
=
4 1
1
3
1
4
2
8 1
1
2
1
2
2
2
2
2
l
nL
k
l
nL
nL
nL
L
n
l
k
A ,
( )
(
|
|
.
|
\
|
+
+
|
|
.
|
\
|
+
=
4 ) 1 ( 3 2 1 2
1
2
2
2 2
l
nL
k
k nL
k
k
B
,
( ) 1
16
2
= k
n
l
C
,
|
.
|
\
|
|
.
|
\
|
=
2 4 2
nl L
nL
L
D
.
The parameters k and b, which define the in-plane forces, can be calculated
using equations (2) and (5) respectively.
Membrane forces
The load bearing capacity for Elements 1 and 2 of the slab can be determined
by considering the contribution of the membrane forces to the resistance and
the increase in bending resistance across the yield lines separately as shown
below. These effects are expressed in terms of an enhancement factor, to be
applied to the lower bound yield line resistance. Initially, the effects of the in-
plane shear S (Figure B.3) or any vertical shear on the yield line was ignored,
resulting in two unequal loads being calculated for Elements 1 and 2
respectively. An averaged value was then calculated, considering contribution
of the shear forces.
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56
Contribution of membrane forces to load bearing capacity.
a) Element 1
According to Figure B.6, the moment about the support due to membrane force
is given by:
Figure B.6 Calculating the moment caused by the membrane force
|
|
.
|
\
|
+
|
|
.
|
\
|
+
+
+ =
2
3
0
2
0 0 1
) 1 ( 3 ) 1 ( 3
2 3
) 2 (
k
k
nLw bKT
k
k
nLw bKT w nL L bKT M
m
where:
m
M
1
is the moment about the support due to membrane forces for
element 1.
The expression reduces to:
|
|
.
|
\
|
+
+
+ =
2
3
0 1
) 1 ( 3
) 2 3 (
) 2 1 (
k
nk k n
n Lbw KT M
m
.
The above formulation defines the contribution from the membrane forces to
the load bearing capacity that needs to be added to the contribution due to the
enhanced bending capacity in the areas where the slab is experiencing
compression forces. For simplicity, the contribution from the membrane forces
and enhanced bending action is related to the normal yield line load. This
allows an enhancement factor to be calculated for both the membrane force and
the enhanced bending moments. These enhancement factors can finally be
added to give the overall enhancement of the slab due to membrane action.
Dividing
m
M
1
by L M
o
, the resistance moment of the slab, when no axial
force is present, allows the effect of tensile membrane action to be expressed as
an enhancement of yield line resistance (Figure B.7).
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Part 2: Engineering Update
57
1m
Enhancement factor due
to membrane forces (e )
for a given displacement (w )
1
1
w
Load capacity based
on yield line theory
Load capacity based
on membrane forces
Displacement ( ) w
Load
Figure B.7 Enhancement factor due to membrane force
The value of
o
M is obtained by considering Figure B.8.
(g h)
1
1
1
1 0
z
1
C
d
h
h
KT
0
(g h)
0
z
C
d
h
h
0
2
2
2
2
2
T
Figure B.8 Calculation of the moment resistance
The bending moments M
0
and
o
M per unit width of slab in each orthogonal
direction are given by:
( )
|
|
.
|
\
| +
=
4
3
1 0
1 0 0
g
d KT M
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( )
|
|
.
|
\
| +
=
4
3
2 0
2 0 0
g
d T M
where:
( ) ( )
2 0 1 0
, g g are parameters which define the flexural stress block in the
two orthogonal directions (see Figure B.8)
1
d , d
2
are the effective depths of the reinforcement in each direction.
The enhancement factor,
m
e
1
, is given by:
( )
( )
( )
|
|
.
|
\
|
+
+
+
|
|
.
|
\
|
+
= =
2
3
1 1 0 0
1
1
) 1 ( 3
2 3
2 1
3
4
k
nk k n
n
d
w
g
b
L M
M
e
m
m
(6)
b) Element 2
The moment about the support due to the membrane forces is given by:
( )
|
|
.
|
\
|
+
+
=
2
3
0 2m
1 6
3 2
k
k k
lbw KT M
where:
2m
M is the moment about support due to membrane force for element 2.
The effect of tensile membrane action can be expressed as an enhancement of
yield line resistance by dividing the moment about the support due to
membrane action, M
2m
by the moment resistance in the longitudinal direction,
when no axial force is present, l M
0
, which results in,
( )
( )
|
|
.
|
\
|
+
+
|
|
.
|
\
|
+
= =
2
3
2 2 0 0
2m
2m
1 6
3 2
3
4
k
k k
d
w
g
bK
l M
M
e (7)
The effect of the membrane forces on the bending resistance along the yield
lines is evaluated by considering the yield criterion when axial load is also
present, as given by Wood
[ 6]
. In the case of the short span the bending moment
in the presence of an axial force is given by:
2
0
1
0
1
0
1
|
|
.
|
\
|
|
|
.
|
\
|
+ =
KT
N
KT
N
M
M
N
| o
(8a)
where:
( )
( )
1 0
1 0
1
3
2
g
g
+
= o
and
( )
( )
1 0
1 0
1
3
1
g
g
+
= |
Similarly for the long span,
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Part 2: Engineering Update
59
2
0
2
0
2
0
1
|
|
.
|
\
|
|
|
.
|
\
|
+ =
T
N
T
N
M
M
N
| o
(8b)
where:
( )
( )
2 0
2 0
2
3
2
g
g
+
= o
and
( )
( )
2 0
2 0
2
3
1
g
g
+
= |
Effect of membrane forces on bending resistance
a) Element 1
The effect of the membrane forces on the bending resistance is considered
separately for each yield line.
For the yield line BC, the membrane force is constant and equals bK
0
T and
therefore:
2
1 1
BC
0
1 b b
M
M
N
| o =
|
|
.
|
\
|
For the yield line AB (Figure B.9),
C
T
2
o
kbKT
o
bKT (l+k)
A
o
bKT
B
x
|
Figure B.9 Forces applied to element 1, yield line CD
The membrane force across the yield line, at a distance of x from B is given
by:
( )
( )
|
|
.
|
\
|
+
=
+ + =
1
1
1
0 x
0 0 x
nL
k x
bKT N
bKT K
nL
x
bKT N
Substitution into Equation (8a) gives, for yield lines AB and CD:
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( ) ( )
} }
(
(
|
.
|
\
|
+
|
.
|
\
|
+
+ =
nL nL
dx
nL
k x
b
nL
k x
b dx
M
M
0 0
2
2
1 1
0
1
1
1
1
1 2 2 | o
This results in:
( ) ( )
}
(
+ + =
nL
k k
b
k
b
nL dx
M
M
0
2
2
1 1
0
1
3
1
2
1 2 2
| o
The enhancement of bending resistance due to membrane forces on Element 1
is given by:
( ) ( ) ( )( )
2
1 1
2
2
1 1
0
1
1 2 1 1
3
1
2
1 2 b b n k k
b
k
b
n
L M
M
e
b
| o
| o
+
(
+ + = = (9)
b) Element 2
Referring to Figure B.10 for element 2, the force at a distance y from B can be
expressed as:
( )
0 0
1
2
bKT k
l
y
bKT N
y
+ + =
C
o
kbKT
o
bKT (l+k)
o
bKT
y
l
|
B
A
Figure B.10 Forces applied to element 2
By rearranging
( )
|
|
.
|
\
|
+
= 1
1 2
0
l
k y
bKT N
y
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Part 2: Engineering Update
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Substitution into Equation (8b) gives:
( ) ( )
} }
(
(
|
.
|
\
|
+
|
.
|
\
|
+
+ =
2 1
0
2 1
0
2
2
2 2
0
1
1 2
1
1 2
1 2 dy
l
k y
K b
l
k y
bK dy
M
M
| o
Resulting in,
(
(
+ + =
}
) 1 (
3
) 1 (
2
1 2
2
2
2 2
2 1
0
0
k k
b
k
b
l dx
M
M | o
which gives the enhancement factor due to the effect of the membrane forces
on the bending resistance according to the following formulation:
) 1 (
3
) 1 (
2
1
2
2
2 2
0
2
+ + = = k k
K b
k
bK a
l M
M
e
b
|
(10)
Equations (6), (7), (9) and (10) provide the contribution to the load bearing
capacity due to the membrane forces and the effect of the membrane forces on
the bending resistance of the slab.
Consequently, the combined enhancement factor is obtained for each element
as follows
b m
e e e
1 1 1
+ =
b m
e e e
2 2 2
+ =
As stated earlier, the values
1
e and
2
e calculated based on the equilibrium of
elements 1 and 2 will not be the same and Hayes
7]
suggests that these
differences can be explained by the effect of the vertical or in-plane shear and
that the overall enhancement is given by.
2
2 1
1
2 1 a
e e
e e
+
=
B.2 Compressive failure of concrete
The enhancement factor in Section B.1.1 was derived by considering tensile
failure of the mesh reinforcement. However, compressive failure of the
concrete in the proximity of the slab corners must also be considered as a
possible mode of failure, which in some cases may precede mesh fracture. This
was achieved by limiting the value of the parameter b, which represents the
magnitude of the in-plane stresses.
According to Figure B.3, the maximum in-plane compressive force at the
corners of the slab is given by
0
kbKT . The compressive force due to bending
should also be considered. By assuming that the maximum stress-block depth
is limited to 0.45d, and adopting an average effective depth to the
reinforcement in both orthogonal directions results in:
P390 TSLAB V3.0 User Guidance and Engineering Update
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62
|
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= |
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45 . 0 85 . 0
2
2 1
ck
0 0
0
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f
T KT
kbKT
where, f
ck
is the concrete cylinder strength.
Solving for the constant b gives:
|
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45 . 0 85 . 0
1
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2 1
ck
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K
T
d d
f
kKT
b
(11)
The constant b is then taken as the minimum value given by the Equations (5)
and (11).
P390 TSLAB V3.0 User Guidance and Engineering Update
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