Precast Concrete Structures: Kim S. Elliott
Precast Concrete Structures: Kim S. Elliott
Precast Concrete Structures: Kim S. Elliott
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Kim S. Elliott
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Kim S. Elliott
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Chapter 3
This chapter introduces the basic principles and some of the design and analysis procedures,
involved in the design of precast concrete skeletal structures, essentially a beam–column
framework possibly braced using walls and/or cores, as well as briefly discusses precast
portal frames and wall frames. Eurocodes EC0, EC1 and EC2 used to determine the com-
binations and arrangement of gravity and horizontal loads acting on floors, beams and
structures are introduced. The design of reinforced and prestressed concrete elements, con-
nections and structures will follow in later chapters.
Preliminary structural design, which many people refer to as the feasibility stage, is more
often a recognition of the type of structural frame that is best suited to the form and func-
tion of a building than the structural design itself. The creation of a large ‘open plan’ accom-
modation giving the widest possible scope for room utilisation clearly calls for a column
and slab structure, as shown in Figure 3.1, where internal partitions could be erected to suit
any client’s needs. The type of structure used in this case is often referred to as ‘skeletal’ –
resembling a skeleton of rather small but very strong components of columns, beams, floors,
staircases, and sometimes structural (as opposed to partition) walls. Of course, a skeletal
structure could be designed in cast in situ concrete and structural steelwork, but here we
will consider only the precast concrete version.
The basis for the design of precast skeletal structures has been introduced in Figures 1.11
and 1.13. The major elements (the precast components) in the structure are shown in
Figure 3.2. Note that the major connections between beams and floors are designed and con-
structed as ‘pinned joints’, and therefore the horizontal elements (slabs, staircases, beams)
are all simply supported. They need not always be pinned (in seismic zones, the connections
are made rigid and very ductile) but in terms of simplicity of design and construction it is
still the preferred choice. Vertical elements (walls, columns) may be designed as continuous,
but because the beam and slab connections are pinned there is no global frame action and
no requirement for a frame stiffness analysis, apart from the distribution of some column
moments arising from eccentric beam reactions. The stiff bracing elements such as walls are
designed either as a storey height element, bracing each storey in turn, or as a continuous
element bracing all floors as tall cantilevers.
In office and retail development, distances between columns and beams are usually in the
range of 6–12 m (Figures 1.7 and 3.3) depending on the floor loading, method of stability
and intended use. In multi-storey car parks, where the imposed loading (vehicle gross weight
<30 kN according to the NA to BS EN 1991-1-1, Table NA.6) is 2.5 kN/m 2 it is around 16 m
for floor spans × 7.2 m for beams, giving three parking bays between columns (Figure 1.6).
The exterior of the frame – the building’s weatherproof envelope – could also be a skel-
etal structure, in which case the spaces between the columns would be clad in brickwork,
65
66 Precast Concrete Structures
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Figure 3.1 Precast skeletal structure showing large unobstructed spaces for the benefit of construction
workers and the client.
precast concrete panels, sheeting, etc. Alternatively, the envelope might be constructed in
solid precast bearing walls, which dispenses with the need for beams, and is referred to as
a ‘wall frame’ (Figure 1.14).
Examples of residential buildings where a precast wall frame would be the obvious
choice are shown in Figures 3.4 through 3.7 – the walls are all load-bearing and they
support one-way spanning floor slabs. There is less architectural freedom compared
to the skeletal frame, for example walls should (preferably) be arranged on a rectan-
gular grid and of fixed modular distance, usually 300 mm, which is quite important
Precast frame analysis 67
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Figure 3.4 Wall frames are best suited for apartments, hotels, schools, shopping units, as in this example at
Rhodes, near Sydney, Australia.
68 Precast Concrete Structures
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economically. A wall frame may be more economical and may often be faster to build,
especially if the external walls are furnished with thermal insulation and a decorative
finish at the factory. Figures 1.14 and 1.22 are good examples of this. Distances between
walls may be around 6 m for hotels, schools, offices and domestic housing, and 10–15
m in commercial developments. Given this description, wall frames appear to be very
simple in concept, but in fact are quite complicated to analyse because the walls have
Precast frame analysis 69
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Figure 3.7 Precast wall and slab frame at Strijkijzer, Den Haag, the Netherlands.
very large in-plane rigidity whilst the connections between walls and floors are more
flexible. Differential movement between wall panels and between walls and floors has
resulted in major serviceability problems over a 25+ year life, often leading to a break-
down in the weatherproof envelope and the eventual condemnation of buildings, which
are structurally adequate.
The third category of precast building is the ‘portal frame’ used for industrial buildings
and warehouses where clear spans of some 25–40 m I-section or T-section prestressed raf-
ters are necessary; Figures 3.8 and 3.9. Although portal frames are nearly always used for
single-storey buildings, they may actually be used to form the roof structure to a skeletal
frame, and as this book is concerned with multi-storey structures it gives us a reason to men-
tion them. The portal frame looks simple enough and in fact is quite rudimentary in design,
providing that the flexural rotations at the end of the main rafters, which we can assume
will always cause cracking damage to the bearing ledge, are catered for by inserting a flex-
ible pad (e.g. neoprene) at the bearing. As mentioned before, pinned connections between
the rafter and column are the preferred choice – they are easy to design and construct. But
the columns must be designed as moment-resisting cantilevers, which might cause a problem
in some structures as explained later in Section 3.6.2. A moment-resisting connection is
equally possible allowing some moment continuity into the column at the eaves. However,
unless the columns are particularly tall, say more than about 8 m, it is not worth the extra
effort.
Precast portal frames with flat (or shallow inclination) roof structures comprising pre-
stressed or reinforced beams of 6–8 m span supporting long-span precast folded plate roof
elements, spanning around 20 m. This is a popular option for industrial buildings, and in
the case of Figure 3.10 used in laboratory buildings. The overhang beam is an option for
sun or rain shading.
Table 3.1 reviews the various types of precast structures with respect to their possible
applications.
70 Precast Concrete Structures
Gable columns
Figure 3.9 Precast portal frame. (Courtesy David Fernandez-Ordoñez, Escuela Técnica Superior de
Ingeniería Civil, Madrid, Spain.)
Precast frame analysis 71
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Figure 3.10 Portal frame with folded plate roof units, the University of Sao Carlos, Brazil.
One of the most frequently asked questions is … how is a precast concrete structure ana-
lysed compared to a monolithic cast in situ one? The first response is to say that a precast
concrete structure is not a cast in situ structure cut up into little pieces making it possible
to transport and erect. It was mentioned in Chapter 1 that the passage of forces through
the prefabricated and assembled components in a precast structure is quite different to a
continuous (monolithic) structure. This is certainly true near to connections. It is therefore
72 Precast Concrete Structures
possible to begin a global analysis by first considering the behaviour of a continuous frame
and identifying the positions where suitable connections in a precast frame may be made.
A two-dimensional in-the-plane simplification is appropriate in the first instance. This is
defined in Figure 3.11 where there are no structural frame components, only simply sup-
ported floor units, connecting the 2-D in-plane frames together.
Figure 3.12 shows the approximate bending moments and deflected shape in a three-
storey continuous beam and column frame subject to vertical (gravity) patch loads and
horizontal (wind) pressure. Call this frame F1. The beam–column connections have equal
strength and stiffness as the members. The stability of F1 is achieved through the combined
action of the beams, columns and beam–column connections in bending, shear and axial.
This is called an ‘unbraced’ frame. There are points of zero moment (‘contraflexure’) in F1,
which depend on the relative intensity of the two load cases. If gravity loads are dominant,
beam contraflexure is near to the beam–column connection, typically 0.1 times the span of
the beam as shown in Figure 3.13; but if the horizontal load is dominant (more rare), con-
traflexure is at mid-span, with the final location for combined loading at about 0.15 × span.
In the column, contraflexure is always at mid-storey height, and this is a good place to make
a pinned (notionally = small moment capacity) connection between two precast columns.
Now, if the strength and stiffness of the connection at the end of the beam are reduced to
zero, whilst the column and the foundation are untouched, the resulting moments and deflec-
tions in this frame, called F2, are as shown in Figure 3.14. The columns alone achieve the
stability of F2 – the beams transfer no moments, only axial forces and shear. The foundations
must be moment-resisting (rigid). This is the principle of a pinned jointed unbraced skeletal
frame. In taller structures, > three storeys or about 10 m, the large sizes of the columns
Precast frame analysis 73
(a)
Sway deflection
all equal
F/2
Point of column
(b) contraflexure
Figure 3.12 Deformation and bending moment distribution in a continuous structure subjected to (a) gravity
loads and (b) horizontal sway load.
become impractical and uneconomic leading to bracing. The bracing may be used in the full
height, called a ‘fully braced’ frame, or up to or from a certain level, called a ‘partially braced’
frame. The differences are explained in Figure 3.15. The bracing could be located in the upper
storeys, providing the columns in the unbraced part below the first floor are sufficiently stable
to carry horizontal forces and any second-order moments resulting from slenderness.
Pinned connections may be formed at other locations. Referring back to frame F1, if the
flexural stiffness of the members at the lower end of a column is greater than that at the
upper end, the point of contraflexure will be near to the lower (stiffer) end of the column.
If the strength and stiffness of the lower end of the column are reduced to zero, whilst the
beam and beam–column connections are untouched, the resulting moments and deflections
in this frame, called F3, are as shown in Figure 3.16a. The stability of F3 is achieved by the
portal frame action of inverted U frames – clearly not a practical solution for factory cast
74 Precast Concrete Structures
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Figure 3.13 Beam half-joints at 0.1× span close to points of contraflexure in a continuous beam.
large spans so that this method is used for repetitious site casting. Therefore, a practical solu-
tion is to prefabricate a series of L-frames as shown in Figure 3.16b for long-span beams
and small-storey height columns in a parking structure. Foundations to F3 may be pinned,
although most contractors prefer to use a fixed base for safety and immediate stability.
The so-called H-frame is a variation on F3. Referring back to frame F1, if pinned con-
nections are made at the points of column contraflexure, structural behaviour is similar
Figure 3.14 Deformation and bending moment distribution in a pinned jointed structure subjected to
(a) gravity loads. (Continued )
Precast frame analysis 75
Sway deflections
all equal
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F/2
F
Zero beam
deformation
Figure 3.14 (Continued) Deformation and bending moment distribution in a pinned jointed structure subjected
to (b) horizontal sway load.
Unbraced
Braced
Braced
Unbraced
to a continuous frame as explained in Figure 3.17. Connections between frames are made
at mid-storey height positions. Although in theory the connection is classed as pinned, in
reality there will be some need for moment transfer, however small. Therefore, H-frame
connections are designed with finite moment capacity, this also gives safety and stability
to the H-frames, which by their nature tend to be massive. The foundation to half-storey
height ground floor columns must be rigid. The connection at the upper end of the column
may be pinned if it is located at a point of contraflexure. If not the connection must possess
flexural strength as shown in Figure 3.17, where the H-frame has been used in a number of
multi-storey grandstands.
76 Precast Concrete Structures
F/2
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Beam to upper
column pinned
Beam to lower
column monolithic
Column contraflexure
Foundations rigid
(preferable for erection
purposes) or pinned
(a)
Monolithic beam-
lower column
(b)
Figure 3.16 Structural systems for (a) portal U-frames and (b) portal L-frames.
Pinned connection
at beam end
Pinned connections at
column mid-height
Monolithic
(a) beam–column
Figure 3.17 H -frames (a) structural system, (b) deformation and bending moments.
techniques to be used to obtain these values. Figure 3.18 gives one such substructure, called
a ‘subframe’ – refer to (Bhatt et al. 2014) for further details. If the frame is fairly regular,
that is spans and loads are within 15% of each other, substructuring gives 90%–95% agree-
ment with full frame analysis.
Substructuring is also carried out in precast frame analysis, except that, where pinned con-
nections are used, no moment distribution or redistribution is permitted. Figure 3.19 shows
78 Precast Concrete Structures
Use column EI
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I
½ EI
L L L
e
Upper column
h3
Beam
h2
h1
Ground column
L1 L2 L3 L4 L5
h3/2
L1/2 L2 L3/2
(a)
Figure 3.19 Substructuring methods for internal beam and columns in a pinned jointed frame (a) internal beam.
(Continued )
Precast frame analysis 79
1/2 column EI
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Max.
h3/2
Min.
1/2 column EI
Column to be
analysed = EI
h2/2
Max. Min.
h2
Min. Max.
Column to be analysed = EI
or 0.75 EI if foundation is
pinned
h1
h1/2
1/2 column EI
Figure 3.19 (Continued) Substructuring methods for internal beam and columns in a pinned jointed frame
(b) upper floor column and (c) ground floor column.
subframes for internal beam and upper and ground floor columns where all beam–column
connections are pinned. For rigid connections, refer to Figure 3.18. Horizontal wind loads
and sway forces due to imperfections are not considered in subframes because the bending
moments due to horizontal loads in an unbraced frame (there are no column moments due
to horizontal loads in a braced frame) are additive to those derived from subframes. Elastic
analysis is used to determine moments, forces and deflections, but a plastic (ultimate) section
analysis is used for the design of the components. Clearly, some inaccuracies must be accepted,
but according to ‘Designer’s Guide to EN 1992-1-1 and EN 1992-1-2’ (Narayanan and Beeby
2005), a design using the partial safety factors (PSFs) and methodologies in the Eurocodes
(design philosophy and materials) is “likely to lead to a structure with a reliability index
greater than the target value of 3.8 stated in the code for a 50-year reference period.”
This book will refer to the UK NAs to Eurocodes EC0 (NA to BS EN 1990 2002), EC1
(NA to BS EN 1991-1-1 2002), EC2 (NA to BS EN 1992-1-1 2004) and briefly to the
NA to Eurocode 3 where steelwork, inserts, welding, etc. is required (NA to BS EN 1993-
1-1 2005). Appendix 3A (at the end of this chapter) summarises the content of Eurocodes
EC2 Parts 1-1 and 1-2, together with the specific clauses related to precast and prestressed
concrete elements in the NA to BS EN 1992-1-1. Reference will also be made to the UK’s
Published Document PD 6687-1 (PD 6687-1 2010) that gives guidance on some specific
items that were not published in the concrete Eurocodes or were in need of additional or
noncontradictory additional information. The main items in the PD relating to the design of
precast concrete structures are listed in Appendix 3B.
These documents give the magnitude and combinations of loads, loading patterns, and
PSFs γf (in BS EN 1990) for gravity and horizontal loads in frames and beams. Four conditions
are considered, each with their own values of γf follows
However, each condition varies depending on the nature of the loads. These are called
‘actions’ in the Eurocodes, and those applicable to the super-structure are as follows:
Dead, live and wind loads are based on the 95% characteristic value for uniformly distrib-
uted load (UDL) known as gk, qk and wk [kN/m 2] and for line/beam loads and point loads
as Gk, Qk and Wk [kN/m or kN].
The self-weight of plain concrete made with normal-weight aggregates (approx.
2600 kg/m3) is taken as 24 kN/m3, according to BS EN 1991-1-1, Table A.1, unless it
is shown by the manufacturer that the characteristic self-weight of elements is different.
An additional 1 kN/m3 is made for reinforcement and prestressing tendons, although it is
unlikely that tendons will add this amount, for example 10 no. 9.3 mm strands in a 1200 ×
150 deep solid slab add only 0.22 kN/m3. The density of wet concrete is taken as 25 kN/m3.
The densities or self-weight of other building materials and stored materials in warehouses,
etc. are given in BS EN 1991-1-1, Tables A.2 through A.12. Note that the self-weight of
masonry units are given in BS EN 771 (BS EN 771 2011) and not in the masonry code (BS
EN 1996-1-1 2005).
The design values of actions for each of the limit states depend on the nature of the
load (i) to (iii), the use of the floor slabs (e.g. residential, parking, storage) and the num-
ber and location of the variable loads. Statistically, it is improbable that all imposed loads
will be acting at their characteristic value Qk1, Qk2 … Qki and at the same time, that is full
live loads will not act at all floor levels in a multi-storey building, or live, wind and snow
Precast frame analysis 81
loads will not act at the same time. Exceptions to this obviously apply and the designer
must be aware of the certain simultaneous combination, such as full live loads acting on
a staircase and landing at the same time, in which case the characteristic load will be
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a. for function and damage to structural and non-structural elements (e.g. partition
walls, etc.), the characteristic combination, for example stress, strength
b. for comfort to user, use of machinery, avoiding ponding of water, etc. the frequent
combination
c. for appearance of the structure, the quasi-permanent combination, for example defor-
mation, deflections
Combination ψ0 Qk
Imposed load Q
Frequent ψ1 Qk
Quasi-permanent ψ2 Qk
Time
These are according to Expressions 6.14b, 6.15b and 6.16b of EN 1990 as follows.
The design service moment M s, shear force Vs and end reaction Fs are based on the design
service load = characteristic load × set of load factors ψ as follows:
Gkj “+” P “+” Qk,1 “+” ψ0 Qk,1 with j ≥ 1 and i > 1 (3.1)
Gkj “+” P “+” ψ1 Qk,1 “+” ψ2 Qk,1 with j ≥ 1 and i > 1 (3.2)
+ + + +
+ + + =
Figure 3.21 E xample of the characteristic, frequent and quasi-permanent combinations of service stresses
in a fictitious prestressed concrete section.
Precast frame analysis 83
where ψ1 and ψ2 are the frequent and quasi-permanent load factors in Table 3.2, for example
for domestic usage, ψ1 = 0.5 and ψ2 = 0.3.
Continuing the example mentioned earlier, the final frequent stress fb = +12.0 – 5.0 – 0.5 ×
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Continuing the example mentioned earlier would not be meaningful as the ‘quasi-permanent’
combination is used for calculating deflections, but for completeness fb = +3.4 N/mm2 (com-
pression). It is clear from these three examples that the conditions of stress are less onerous with
each successive combination, and this is a reflection of the diminishing effect of viscoelastic
deformations according to the use of buildings and the effect of specific creep. Note that in
Table 3.2 for storage the ψ factors are between 0.8 and 1.0, indicating a higher specific creep.
6.10b ξ γG,j Gk,j “+” γP P “+” γQ,1 Qk,1 “+” Σ γQ,i ψ0,i Qk,I (3.5)
To satisfy the ULS design, the three load combinations must be used to determine the maxi-
mum end reactions, and bending moments and shear forces at all points along the span.
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where
h is the depth of slab (including topping)
Lb2 and Lb1 are the bearing lengths at either end
For cantilevers, Lb2 is also the width of the support. For continuous elements after comple-
tion of the continuity (i.e. stage 2 loading), leff = distance between beam centres. If a bearing
medium (pad, plate) is provided, leff is to the centre of the pad, and this is also the case for
beams supported on steel inserts, cleats and plates, etc.
Example 3.1
Calculate the maximum ultimate end reaction FEd in a simply supported beam of effective
span 6.0 m subjected to Gk = 30 kN/m, Qk1 = 20 kN/m, and Qk2 = 40 kN point load at
mid-span. ψ0 = 0.7.
Solution
Exp. 6.10a wEd = 1.35 × 30 +(0.7 × 1.5) × 20 = 61.5 kN/m
PEd = 1.05 × 40 = 42 kN.
Then F Ed = 61.5 × 6.0/2 + 42/2 = 205.5 kN.
Exp. 6.10b = 1.25 × 30 + 1.5 × 20 = 67.5 kN/m + 42 kN (UDL dominant)
Then F Ed = 67.5 x 6.0/2 + 42/2 = 223.5 kN. Maximum.
Exp. 6.10b = 1.25 × 30 + 1.05 × 20 = 58.5 kN/m + 1.5 × 40 = 60 kN (point load
dominant)
Then F Ed = 58.5 × 6.0/2 + 60/2 = 205.5 kN.
6.10 γG,j Gk,j “+” γQ,1 Qk,1 “+” Σ γQ,i ψ0,i Qk,I (3.7)
γQ = 0 (favourable)
At installation refer to BS EN 1990, Table A2.4(A) as follows:
γG,j = 1.05 (unfavourable) and 0.95 (favourable)
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Example 3.2
Calculate the minimum end reaction FEd in a simply supported beam of effective span 6.0
m with a 3.0 m span overhanging cantilever at one end subjected to Gk = 30 kN/m and
Qk = 20 kN/m.
Solution
6.11 Gk,j “+” P “+” Ad “+” ψ1,1 Qk,1 “+” Σ γQ,i ψ2,i Qk,I (3.8)
1. All spans loaded with the maximum ultimate load wEd,max = γG Gk + γQ Qk (for BS EN
1990, Exp. 6.10a or b).
2. Alternate (‘pattern’) spans loaded with wEd,max on one span and the minimum wEd,min = 1.0 Gk
on the adjacent span.
For frame analysis with sway, horizontal loads Wk are combined with gravity Gk and Qk
load combinations 6.10a and 6.10b for three situations:
Table 3.3 Partial load factors and safety factors for gravity and horizontal loads
Permanent load Imposed load
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The fundamental PSF for wind load (notation used here γW) is as NA to BS EN 1990, Table
NA.A1.2 (B) – Design values of actions (STR) (Set B) γW = 1.5, and is modified by ψ0 = 0.5
(see Table 3.2) in the same way as for gravity loads. The values for γ and ψ are summarised
in Table 3.3.
Example 3.3
Calculate the maximum ultimate bending moment M Ed at the lower end of the columns
of height h = 4.0 m in Figure 3.22. The beam–column connections are pinned, and the
foundation is rigid. The distance from the edge of the column to the centre of the beam
end reaction is 100 mm. Characteristic beam loading is Gk = 40 kN/m and Qk = 30 kN/m,
and the wind pressure equates to a horizontal load Wk = 12 kN. The carry-over moment
at the lower end of the column is equal to 50% of the upper end moment due to beam
eccentricity. Let ψ0 (gravity load) = 0.7.
Solution
Eccentricity of beam reaction R from the centre of column e = 300/2 + 100 = 250 mm.
Pinned jointed
L = 8.0 m
Moment at the lower end of each column due to wind load, M Ed = W Ed h/2 (because
there are two columns)
Ultimate load combinations and moments are summarised in the following table.
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Then M Ed,max = 65.5 kNm using Exp. 6.10b for all loads (it is interesting to note that
according to BS 8110 values for all loads, wult = 1.2 × 70 = 84 kN/m, Mu (gravity) = 42 kNm,
Wu = 1.2 × 12 = 14.4 kN, M u (wind) = 28.8 kNm. Total M u = 70.8 kNm).
Figure 3.23 Inclined staircase imposing horizontal forces to the structure at Bella Sky Hotel, Denmark.
(Courtesy Ramboll, Denmark.)
88 Precast Concrete Structures
exactly imperfections but demonstrate the point of transferring inclined gravity loads into
horizontal forces.
BS EN 1992-1-1 defines imperfections as ‘possible deviations’ in geometry and ‘positions’
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of loads in Section 5.2 and as quantified by code Exp. 5.1 through 5.4 as an ultimate hori-
zontal force Hi = Nθi due the inclination θi according to the height l and number m the
elements contributing to the imperfection, recognising the improbability that imperfection
will be the same in all elements. The value attributed to θi is related to ‘Class 1 execution
deviations’ according to BS EN 13670 (BS EN 13670 2009) and is taken into account at the
ULS and accidental design situations, but not at serviceability. Deviations in cross section
dimensions are taken into account in material safety factors. Imperfections and deviations
should not be included in structural analysis as their effect is additional to first-order bending
moments and shear forces, but not when checking deflections.
BS EN 1992-1-1 distinguishes between (a) whole braced or unbraced structures, known
as ‘global analysis’, (b) isolated columns in braced or unbraced structures, and (c) floor and
roof diaphragm action (see Chapter 8) as follows:
(a) For structures, the horizontal force Hi is applied to the bracing system, for example
cores or shear walls, at each floor level as the horizontal component of the total ulti-
mate gravity load at that floor level. Referring to Figure 3.24a (adapted from BS EN
1992-1-1, Fig. 5.1b).
where θi = θ0 αh αm (3.10)
N3
Reactions
ei,3 Hi,3
Hi,3
Na N2
ei,2 Hi,2
Hi,2
Total ei,1 N1 H
Nb i,1
height l
Hi,1
θi
l0 to first floor
Mi
Figure 3.24 E xamples of the effect of geometric imperfections. (a) Bracing system and (b) isolated column
in unbraced structure. (Adapted from BS EN 1992-1-1. 2004, Eurocode 2: Design of Concrete
Structures – Part 1-1: General rules and rules for buildings, BSI, London, February 2014,
Fig. 5.1a1 and b.)
Precast frame analysis 89
ei = θi l 0/2 (3.13)
where l0 is effective length (m) of the column at the floor level that ei is considered,
that is at the second floor level l0 is based on the height to the second floor, etc.
αh = 2/3 ≤ (2/√l) ≤ 1, where l = actual length of column at the level that ei is acting
(m) and m = αm =1
(c) For isolated columns (and minor axis of walls) in a braced structure αh is simplified
such that αh = m = αm = 1
ei = l 0/400 (3.14)
Example 3.4
Calculate the horizontal forces at each roof and floor level and the over-turning moment at
the foundation due to imperfection in the braced skeletal structure shown in Figure 3.25.
The number of columns in each line is six, and there are five rows of column. There are
two sets of shear walls in each of the external rows of columns. The total ultimate gravity
load per floor = 15,000 kN and at the roof = 7,000 kN.
Hi,roof
3.25 m 3.25 m
Hi,floor
Hi,floor
4.00 m
Columns unbraced in
Example 3.5 only
Solution
l = 10.5 m, then αh = 2/√10.5 = 0.617 use 2/3
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Example 3.5
Calculate the horizontal forces at each roof and floor level and the over-turning moment
at the foundation due to imperfection in one of the internal columns, if the same structure
shown in Figure 3.25 is unbraced. The effective length factor for the columns may (in this
example) be taken as 2.2. The total ultimate gravity load per column per floor = 900 kN
and at the roof = 500 kN.
Solution
At the roof level. l = 10.50 m, then αh = 2/√10.50 = 0.617 use 2/3. l 0 = 2.2 × 10.5 =
23.1 m
ei = (1/200) × (2/3) × 23.1/2 = 0.039 m
Then M i due to ei = 500 × 0.039 = 19.5 kNm which equates to Hi = 19.5/10.5 =
1.85 kN
t the second floor level. l = 7.25 m, then αh = 2/√7.25 = 0.742. l 0 = 2.2 × 7.25 =
A
15.95 m
ei = (1/200) × 0.742 × 15.95/2 = 0.030 m
Then M i due to ei = 900 × 0.030 = 27.0 kNm which equates to Hi = 27.0/7.25 =
3.72 kN
At the first floor level. l = 4.00 m, then αh = 2/√4.00 = 1. l 0 = 2.2 × 4.00 = 8.80 m
ei = (1/200) × 1 × 8.80/2 = 0.022 m
Then M i due to ei = 900 × 0.022 = 19.8 kNm which equates to Hi = 19.8/4.00 =
4.95 kN
Total Mi = 19.5 + 27.0 + 19.8 = 66.3 kNm per column.
(Note that M i for isolated columns is greater per column than if the total M i for the walls
was divided over the total number of columns).
End reactions in the adjacent beams are R1 = wEd,minL1/2 and R3 = wEd,minL3/2 (3.17)
assuming that R1 < R 3 and h1 > h3. Figure 3.26a shows the final moments.
e e
R1 R2
L1 L2 L3
(a)
e4 e5
Mcol, upper
R4 R5
Mcol, lower
(b)
Figure 3.26 Bending moments in a pinned jointed frame for (a) internal beams, (b) upper floor columns.
(Continued)
92 Precast Concrete Structures
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½Mcol, upper
Zero
EI2
h2
Mcol ,upper = (R4e4 - R5e5)
EI2 EI3
+
h2 h3
EI1
h1
Mcol ,upper = (R4e4 - R5e5) (3.19)
EI1 EI2
+
h1 h2
where R4 and R 5 are given in Equations 3.2 and 3.3. Figure 3.26b shows the final moments.
Note that patch loading produces single curvature in the columns.
is rigid (moment resisting), the moment at the upper end of the designed column is given by
Equation 3.5 with appropriate notation. The carry-over moment at the lower end is equal to
50% of the upper end moment. If the foundation is pinned, the upper end moment is given by
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EI 1
0.75
h1
Mcol , upper = ( R4 e4 - R55e5 ) (3.20)
EI1 EI2
0.75 +
h1 h2
and the lower end moment is zero. Figure 3.26c shows the final moments. Patch loads pro-
duce single curvature in the columns.
Example 3.6
Determine, using substructuring techniques, the bending moments in the beam X and
columns Y and Z identified in Figure 3.27. The beam–column connections are pinned,
and the foundation is rigid. The distance from the edge of the column to the centre of
the beam end reaction is 100 mm. Characteristic beam loading is Gk = 40 kN/m and
Qk = 30 kN/m.
Solution
wEd,max = max{1.35 × 40 + 1.05 × 30; 1.25 × 40 + 1.5 × 30} = max {85.5; 95.0} = 95.0
kN/m; wEd,min = 40 kN/m.
Beam subframe
e = 450/2 + 100 = 325 mm
Equation 3.1. M1 = 95.0 × (8.000 – 2 × 0.325)2 /8 = 641.5 kNm
Column Y subframe
Beam end reactions R1 = 95.0 × 8.000/2 = 380.0 kN; R2 = 40 × 6.000/2 = 240.0 kN
e 1 = e 2 = 300/2 + 100 = 250 mm
h2 = 3.2 h3 = 3.2 h4 = 3.2
300
300
Y
Section Y
X
450
h1 = 5.0
Z
300
Section Z
L1 = 8.0 L2 = 6.0
50 mm below top
of foundation
but EI1/h1 = EI 2 /h 2
Equation 3.5. At upper and lower ends, Mcol = (380.0 – 240.0) × 0.250 × 0.5 =
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17.5 kNm
Column Z subframe
Beam end reactions as before. e 1 = e 2 = 450/2 + 100 = 325 mm
Given that E is constant
300 ´ 4503
I1 /h1 = = 451 ´ 103 mm2
12 ´ 5050
300 ´ 3003
I2 /h2 = = 211 ´ 103 mm2
12 ´ 3200
Equation 3.5. At upper end, Mcol,upper = (380 – 240) × 0.325 × 451/(451 + 211) = 31.0 kNm
At lower end, Mcol,lower = 50% × 31.0 = 15.5 kNm.
Connections form the vital part of precast concrete design and construction. They alone can
dictate the type of precast frame, the limitations of that frame, and the erection progress.
It is said that in a load-bearing wall frame the rigidity of the connections can be as little as
1/100 of the rigidity of the wall panels −200 N/mm 2 per mm length for concrete panels ver-
sus 2.7–15.0 N/mm 2 per mm length for joints (Straman 1990). Moreover, the deformity of
the bedding joint, that is the invisible interface where the panel is wet bedded onto a mortar,
between upper and lower wall panels can be 10 times greater than that of the panel.
The previous paragraph contained the words connections and joints to describe very simi-
lar things. Connections are sometimes called ‘joints’ – the terminology is loose and often
interposed. The definition adopted in this book is as follows:
Connection: is the total construction between two (or more) connected components: it
includes a part of the precast component itself and may comprise several joints.
Joint: is the part of a connection at individual boundaries between two elements (the ele-
ments can be precast components, in situ concrete, mortar bedding, mastic sealant, etc.)
For example in the beam–column assembly shown in Figure 3.28, a bearing joint is made
between the beam and column corbel, a shear joint is made between the dowel and the
angle, and a bolted joint is made between the angle and column. When the assembly is
completed by the use of in situ mortar/grout, the entire construction is called a connection.
This is because the overall behaviour of the assembly includes the behaviour of the precast
components plus all of the interface joints between them. Engineers prove the capacity of the
entire connection by assessing the behaviour of the individual joints.
Structurally, joints are required to transfer all types of forces – the most common of these
being not only compression and shear, but also tension, bending and occasionally torsion.
The combinations of forces at a connection can be resolved into components of compressive,
tensile and shear stress, and these can be assessed according to limit state design. Steel (or
other materials) inserts may be included if the concrete stresses are greater than permissible
Precast frame analysis 95
Tension
joint
Shear
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joint
V M
Bearing and
shear joint
values. The effects of localised stress concentrations near to inserts and geometric disconti-
nuities can be assessed and proven at individual joints. However, connection design is much
more important than that because of the sensitivity of connection behaviour to manufactur-
ing tolerances, erection methods and workmanship.
It is necessary to determine the force paths through connections in order to be able to
check the adequacy of the various joints within. Compared with cast in situ construction,
there are a number of forces which are unique to precast connections, namely frictional
forces due to relative movement causes by shrinkage, etc pretensioning stresses in the con-
crete and steel, handling and self-weight stresses. In the example shown in Figure 3.29, a
reinforced concrete column and corbel support a pretensioned concrete beam. The figure
shows that there are 10 different force vectors in this connection as follows:
The structural behaviour of the frame can be controlled by the appropriate design of connec-
tions. In achieving the various structural systems in Section 3.3.2, it may be necessary to design
and construct either/both rigid and/or pinned connections. Rigid monolithic connections can
only truly be made at the time of casting, although it is possible to site cast connections that
have been shown to behave as monolithic, for example cast in situ filling of prefabricated soffit
96 Precast Concrete Structures
K
K F
H
E
G
K+J
J
B
D Beam
A
C
Column corbel
beams before and after casting as shown in Figure 3.30a and b. The advantages lost to in situ
concreting work (cold climates in particular), the delayed maturity, the increase in structural
cross section, and the reliance on correct workmanship, etc. detract this solution in favour of
bolted or welded mechanical devices. Rigid connections may be made at the foundation where
there is less restriction on space as shown in Figure 3.31. In very simple terms, a bending
moment is generated by the provision of a force couple in rigid embedment, that is no slippage
when the force is generated. Pinned connections are designed by an absence of this couple,
although many connectors designed in this way inadvertently contain a force couple, giving
rise to spurious moments which often cause cracking in a region of flexural tension.
To gain an overview of the various types, Figure 3.32 and Table 3.4 show the locations,
classification and basic construction of connections in a precast structure.
In theory, no connection is fully rigid or pinned – they all behave in a semi-rigid manner,
especially after the onset of flexural cracking. Using a ‘beam-line’ analysis, Figure 3.33, we
can assess the structural classification of a connection. Although the beam-line approach was
developed for structural steelwork in c1936, research carried out since 1990 has shown that
the method is appropriate to precast connections (Elliott et al. 1998, Elliott et al. 2003a,b,
Ferriera et al. 2003, Elliott and Jolly 2013).
The moment–rotation (M-θ) diagram in Figure 3.33 is constructed by considering the two
extremes in the right hand part of Figure 3.33. The hogging moment of resistance of the beam
at the support is given by MRd > wL2/12, and the rotation of a pin ended beam subjected to a
UDL of w is θ = wL3/24EI. The gradient of the beam line is 2EI/L. The M-θ plot for plots 1
and 2 give the monolithic and pinned connections, respectively. In reality, the behaviour of a
connection in precast concrete will follow plots 3, 4 or 5, etc. If the M-θ plot for the connec-
tion fails to pass through the beam-line, that is plot 5, the connection is deemed not to possess
sufficient ductility and should be considered in design as ‘pinned’. Furthermore, its inherent
Precast frame analysis 97
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(a)
(b)
Figure 3.30 Cast in situ concrete topping over precast soffit beams forms a fully rigid connection.
(a) Prepared for casting and (b) as cast and awaiting structural topping to also cover the slabs.
stiffness (given by the gradient of the M-θ plot) is ignored. Conversely, if the M-θ behaviour
follows plot 3 (the gradient must lie in the shaded zone and the failure takes place outside the
shaded zone), the effect of the connection will not differ from a monolithic by more than 5%.
Structural stability and safety are necessary considerations at all times during the erection
of precast concrete frames. The structural components will not form a stabilising system
until the connections are completed – in some cases, this can involve several hours of matu-
rity of cast in situ concrete/grout joints, and several days if structural cast in situ toppings
98 Precast Concrete Structures
M
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50 ignored
F
1.5h approx
F
Split level
beams to
column face
Beam to deep Continuous beam
Mid height column corbel Beam half joint to column head
column splice
Floor level
Half joint
column
to column arm
splice
Beam to Beam to
Beam to shallow shallow
column face column carbel haunch
Beam to
column splice Continuous beam
to column splice
are used to transfer horizontal forces. A stabilising system must comprise two things as
shown in Figure 3.34:
2 1
MR = wL Rigid M
12
3
Pinned M=0
Failure of
connection θ
Semi
Be
rigid M
am
5
-li
θ
ne
2
θ
3
θR = wL
24EI
Loa
din
gx
Ch
gy ord
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n
adi x
Lo rd
y
o
Ch
y
rd
cho
Tie Bracing x
Strut x
Strut y
Bracing y
Reactions x
Reactions y
The horizontal system is considered in detail in Chapter 8 where reference is also made to
the many code regulations on this topic. When subjected to horizontal wind or lack-of-
plumb forces, the floor slab acts as a deep beam and is subjected to bending moments Mh
and shear forces Vh (h being the subscript used for horizontal diaphragms). The basic design
method is shown in Figure 3.35. The design is a three-stage approach:
1. The floor diaphragm is analysed as a long, deep beam which is supported by a number
of shear walls, shear cores, deep columns (wind posts), or other kinds of bracing such
as steel cross bracing. Figure 3.35a.
2. If there are only two supports (bracing), the analysis is statically determinate and MEd,h
and VEd,h may be calculated directly. If there are more than two supports, irrespec-
tive of where they are positioned, the analysis is statically indeterminate. The support
reactions must first be found by a technique which considers the relative stiffness and
position of each support, and the horizontal (e.g. wind load) pressure distribution.
The derivation is given in Section 8.1 after which MEd,h and VEd,h may be calculated.
3. The area of reinforcement required to resist M Ed,h and V Ed,h is determined as follows:
where
0.8 B is the assumed lever arm between the compression zone and the tie steel (the
assumption is known to be conservative)
f yk /γm is the design stress in the tie steel with γm = 1.15
High tensile rebar with fyk = 500 N/mm2 or standard helical strand with fpk = 1770 N/mm2 is
used (super strand or Dyform tend to be too stiff to handle) – the reasons are given in Section 8.4.
where μ is the coefficient of friction as given in BS EN 1992-1-1, clause 6.2.5(2). Hollow core
slabs are considered as being untreated and smooth, then μ = 0.6 with no special, that is ex-
factory, edge preparation (see Section 8.2). For ex-steel mould with smooth surfaces, μ = 0.5.
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4. The tie steel A shd must be placed everywhere moments occur. The tie steel A svh must
be placed only where the shear force is greater than a certain value. This is found by
checking that the interface shear stress vEdi = V Ed,h /B (D – 30 mm) does not exceed
vRdi ≤ 0.15 N/mm 2 for smooth and rough surfaces (as in the case of machine cast hollow
core units and as-cast precast planks) or vRdi ≤ 0.10 N/mm 2 for very smooth surfaces
cast against steel moulds, according to BS EN 1992-1-1, clause 10.9.3(12). (The reason
for the deduction of 30 mm is explained in Section 8.4.1.)
(b)
Figure 3.35 Diaphragm floor action. (a) Deep beam analogy. (b) Reinforced structural topping in double-
tee floors. (Continued)
102 Precast Concrete Structures
30 mm minimum
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500 mm
Figure 3.35 (Continued) Diaphragm floor action. (c) Perimeter reinforcement in hollow core floors.
Diaphragms may be reinforced in several ways. In Figure 3.35b, a reinforced cast in situ top-
ping transfers all horizontal forces to the vertical system – the precast floor plays no part but
for restraining the topping against buckling. In Figure 3.35c, there is no cast in situ topping.
Perimeter and internal tie steel resists the chord forces resulting from horizontal moments.
Coupling bars are inserted into the ends of the floor units, and together with the perimeter
steel provides the means for shear friction generated in the concrete-filled longitudinal joints
between the units.
Example 3.7
Determine the shear wall reactions and diaphragm reinforcement in the floor shown in
Figure 3.36a. The precast units are 150 mm deep hollow cored and have an ex-factory
edge finish. The characteristic wind pressure on the floor wk = 3 kN/m. Tie steel is high
tensile ribbed bar f yk = 500 N/mm 2 . Suggest some reinforcement details.
Precast frame analysis 103
Floor units
3 kN/m
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Shear walls
or bracing
5.0 m
24.0 m 6.0 m
(a)
4.2 kN/m
47.2 kN 78.8 kN
Mmax at V = 0
11.24 m
Figure 3.36 Detail to Example 3.7. (a) Plan view of floor diaphragm, (b) Wind loading diagram and shear
force diagram in the floor diaphragm.
Solution
NA to BS EN 1990, Table NA.A1.2 (B) – Design values of actions (STR) (Set B) γW = 1.5
Design ultimate wind load = 1.5 × 3.0 = 4.5 kN/m.
From Figure 3.36b, support reaction R1 = 50.62 kN; R 2 = 84.38 kN.
Shear span from LHS (distance to zero shear and hence point of maximum
moment) = 50.62/4.5 = 11.25 m.
MEd,h,max = 50.62 × 11.25/2 = 284.74 kNm; VEd,h,max = 57.38 kN at RHS of 24 m span.
A shd = 284.74 × 106/0.8 × 5000 × (500/1.15) = 164 mm 2 . Use 2 no. H12 bars (226).
Interface shear stress vEdi = 57.38 × 103/5000 × (150 – 30) = 0.096 N/mm 2 < 0.15
N/mm 2 allowed. No shear reinforcement needed.
Vertical stabilising systems are dictated by the necessary actions of the structural system,
that is skeletal, wall or portal frame. Column effective lengths depend on the type and direc-
tion of the bracing. However, there is a broad classification as the structure is
1. Unbraced frame, Figure 3.37, where horizontal force resistance is provided either by
moment resisting frame action, cantilever action of columns, or cantilever action of
wind posts (deep columns)
2. Braced frame, Figure 3.38, where horizontal force resistance is provided either by
cantilever action of walls or cores, in-plane panel action of shear walls or cores, infill
walls, cross bracing, etc. or
3. Partially braced frame, Figure 3.39, which is some combination of (1) and (2)
104 Precast Concrete Structures
Column lo = 2.083 l
lo ≈ 2.2 l
Figure 3.37 Alternative sway mechanisms and resulting column effective length factors.
Figure 3.38 Alternative full-height bracing mechanisms and resulting column effective length factors.
The type of stabilising system may be different in other directions. The floor plan arrange-
ment and the availability of shear walls/cores will dictate the solution. The simplest case is a
long narrow rectangular plan where, as shown in Figure 3.40a, shear walls brace the frame
in the y direction only, the x direction being unbraced. In other layouts, shown for example
in Figure 3.40b, it is nearly always possible to find bracing positions. Precast skeletal frames
Precast frame analysis 105
Column lo = 2.083 l
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Figure 3.39 Alternative partial height bracing mechanisms and resulting column effective length factors.
Floor units
y Shear walls
Column and
or other bracing
(a) x beam frame
Stairs
Service core
lifts etc.
Stairs
(b)
Figure 3.40 Positions of shear walls and cores in alternative floor plan layouts. (a) Positions of shear walls,
(b) positions of shear cores or walls around stairs and lift shafts. (Continued)
106 Precast Concrete Structures
Braced Unbraced
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Shear core
Construction and
movement gap
(c)
Figure 3.40 (Continued) Positions of shear walls and cores in alternative floor plan layouts. (c) bracing meth-
ods and positions in partially braced irregular or non-symmetrical buildings.
of three or more storeys in height are mostly braced or partially braced. This is to avoid hav-
ing to use deep columns to cater for sway deflections, which give rise to large second-order
bending moments. Section 6.2.6 refers in more detail.
It is not wise to use different stabilising systems acting in the same direction in different
parts of a structure. The relative stiffness of the braced part is likely to be much greater than
in the unbraced part, giving rise to torsional effects due to the large eccentricity between the
centre of external pressure and the centroid of the stabilising system, as explained in Figure
3.40c. The different stabilising systems should be structurally isolated – Figure 3.40d.
In calculating the position of the centroid of a stabilising system, the stiffness of each com-
ponent of thickness t and length L is given by Ecm,long I, where Ecm,long = long-term Young’s
modulus (usually taken as 15 kN/mm 2) and I = tL 3/12. First moments of stiffness are used
to calculate the centroid as explained in Example 3.8.
Example 3.8
Propose stabilising systems for the five-storey skeletal frame shown in Figure 3.41a.
The beam–column connections are all pinned, and the columns should be the minimum
possible cross section to cater for gravity loads. Wind loading may be assumed to be uni-
form over the entire façade. Use only shear walls for bracing.
Hint: the grid dimensions around the stairwell may be taken as 4 m × 3 m, and at the lift
shaft 3 m × 3 m.
Solution
A braced frame is required up to the fourth floor, after which a one-storey unbraced
frame may be used. It would not otherwise be possible to satisfy the requirement of
minimum column sizes for gravity loads. To avoid torsional effects (see Figure 3.40c),
the centroid of the stabilising system should be as close as possible to the centre of exter-
nal pressure, that is at x ≈ 24 m and y ≈ 16 m. It is necessary to first consider the two
orthogonal directions.
Stability in y-direction
The centroid of the stability walls x′ ≈ 24 m.
Precast frame analysis 107
Stairs
Columns
Opaque obstruction
Lift shaft
Stairs
4.0
16.0
x΄ = 26.3
3.0
12.0
y΄ = 16.0
Shear centre
4.0
(b) 48.0
Figure 3.41 Detail to Example 3.8 (dimensions in metres). (a) Plan view and cross-section of framed struc-
ture, (b) Plan view showing positions of shear walls.
Select walls as shown in Figure 3.41b. On the assumption that the material and con-
struction of all walls is the same, Young’s modulus and thickness of wall are common to
all walls and need not be used in the calculation.
(43 ´ 0) + (43 ´ 3) + (33 ´ 33) + (33 ´ 36) + (43 ´ 45) + (43 ´ 48)
Centroid of stiffness x¢= = 25.0 m ,
(4 ´ 43) + (2 ´ 33)
which is sufficiently close to the required point to eliminate significant torsional effects.
108 Precast Concrete Structures
Stability in x-direction
The centroid of the stability walls y′ ≈ 32/2 = 16 m.
(33 ´ 0) + (33 ´ 16) + (33 ´ 32)
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To assist the transition between the British code BS 8110:1997 and the Eurocodes EC0, EC1
and EC2, the following precast reinforced and prestressed concrete elements are designed:
wu = 1.4 × 44.32 + 1.6 × 30 = 110.1 kN/m {Exp. 6.10a; 6.10b} wEd = max{1.35 × 44.5 + 0
0.7 × 1.5 × 30; 1.25 × 44.5 + 1.5 × 30} = 100.7 kNm
Effective span {3.4.1.2} {5.3.2.2, Fig. 5.4a}
lo = min{5.85 + 0.15; 5.85 + 0.46} = 6.0 m leff = min{5.85 + 0.15; 5.85 + 0.46} = 6.0 m
M = 110.1 × 6.02/8 = 495.5 kNm MEd = 100.7 × 6.02/8 = 453.1 kNm
{3.4.4.4} K = 495.5 × 106/40 × 300 × 5562 {3.1.7(3)} K = 453.1 × 106/32 × 300 × 5562
= 0.134 < 0.156 for x/d ≤ 0.5 = 0.153 < 0.206 for x/d ≤ 0.6
z/d = 0.5 + √0.25 – K/0.9 = 0.82 < 0.95 z/d = 0.5 + √0.25 – K/1.133 = 0.84 < 0.95
z = 0.82 × 556 = 455 mm z = 0.84 × 556 = 467 mm
As = 495.5 × 106/455 × 0.87 × 500 = 2503 mm2 As = 453.1 × 106/467 × 0.87 × 500 = 2230 mm2
Use 2 no. H32 ± 2 no. H25 bars (2,590) Use 3 no. H32 bars (2,412)
Spacing = (300 – 88 – 114)/3 = 33 mm Spacing = (300 – 88 – 96)/2 = 58 mm
{3.12.11.1} > 20 + hagg 5 = 25 mm OK {Table 7.3N} < 100 mm for any value of σs
{Table 7.1, NAD Table NA.4}wk = 0.3 mm
Although compression steel is not required
{Table 3.25} Min As′ = 0.2% = 360 mm2 {9.2.1.1(1)} Min As′ = 0.13% = 234 mm2
Use 2 no. H16 (402) at d ′ = 36 mm Use 2 no. H16 (402) at d ′ = 36 mm
Comments. EC2 requires 11% less area of rebar.
Shear design
{3.4.5.3} lv = 5.85 – 2 × 0.556 = 4.74 m {6.2.1(8)} lv = 5.85 – 2 × 0.556 = 4.74 m
V = 110.1 × 4.74/2 = 261.0 kN VEd = 100.7 × 4.74/2 = 238.7 kN
{3.4.5.3} v = 261.0 × 103/300 × 556 {6.3.2. Exp. 6.9}VRd,max = v1 b z fcd 0.5 sin 2θ
= 1.56 N/mm2 < 0.8√fcu = 5.06 N/mm2 v1 = 0.6 (1 − fck/250), z = 0.9d and fcd = fck/1.5
> vc = 0.79 × 1.51/3 × 1.17/1.25 = 0.84 N/mm2 θ = 0.5 sin−1 (238,700/(0.5 × 0.523 × 300 × 501
where 100 As/bvd = 1.5, (fcu/25)1/3 = 1.17 × (32/1.5)) = 8.2o < 22.5o ∴ use cot θ = 2.5
{Table 3.7} v > vc {6.3.2. Exp. 6.8}
Asv/sv = 300 × (1.56 – 0.84)/(0.87 × 500) Asw/s = 238,700/501 × 0.87 × 500 × 2.5
= 0.5 mm2/mm = 250 mm2/m/leg = 0.438 mm2/mm = 219 mm2/m/leg
{3.4.5.5} s < 0.75 × 556 = 417 mm {9.2.2(6)} s ≤ 0.75d = 417 mm
Use H8 links at 200 mm c/c (250) Use H8 links at 225 mm c/c (222)
Nominal where v = vc + 0.4 = 1.24 N/mm2 {Exp. 9.5N} Asw,min/s = 0.08 × √32 × 300/500
or where shear force = 0.272 mm2/m
∴Vnom = 1.24 × 300 × 556 × 10–3 = 207 kN ∴ VRd,c,min = 238.7 × (0.272/0.438) = 148.0 kN
at 1,530 mm from centre of support. at 1,120 mm from the centre of support
Comments.
EC2 requires 13% less area of links, and nominal links to EC2 start at 1.36 times the distance
for BS 8110.
Deflection
Short-term Young’s modulus
{2.5.4} Es = 200 kN/mm2 {3.2.7(4)} Es = 200 kN/mm2
{Part 2, 7.2} Ec = 20 + 0.2 × 40 = 28 kN/mm2 {Table 3.1} Ecm = 22 (40/10)0.3 = 33.34 kN/mm2
α = 200/28 = 7.14 α = 200/33.34 = 6.00
Long-term Young’s modulus
Bottom and sides exposed ho = 2Ac/u = 360,000/
(300 + 2 × 600) = 240 mm
(Continued)
110 Precast Concrete Structures
Wk = 7 kN
Floor beams
3000
designed in 3.6.1
Clear 6100
Wk = 14 kN
3500
Clear 2900
(a)
e Vroof
Vfloor
(b) First order frame moments Due to wind loads Second order deflection Second order moments
Figure 3.42 Detail to code comparison for reinforced concrete column. (a) Frame arrangement, (b) bending moments
due to beam eccentricity, horizontal loads (wind and imperfections) and second-order deflections.
1.4
1.3 d/h = 0.87
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K = 0.1 d
1.2
1.1 0.2
As fyk/bh fcu
b
1.0 0.3
0.9 0.4 h
1.0 As = total steel area
0.8 0.5
N/fcu bh
0.8
0.7
0.6 0.6
0.6
0.4 0.7
0.5
0.2 0.8
0.4
0.9
0.3
0.2 1.0
0.1
0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
(a) M/fcu bh2
1.5
1.4 d/h = 0.87
Kr = 0.1
1.3 d
0.2
1.2
b
1.1 0.3 As fyk/bh fck
1.0 0.4 h
0.9 As = total steel area
0.5
N/fck bh
0.8 1.0
0.8 0.6
0.7
0.6 0.7
0.6
0.4
0.5 0.2 0.8
0.4 0.9
0.3 1.0
0.2
0.1
0.0
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
(b) M/fck bh2
Figure 3.43 (a) Reinforced concrete column design chart to BS 8110. (b) Reinforced concrete column design
chart to BS EN 1992-1-1.
Use fck /fcu = 45/55, at transfer fck(t)/fci = 30/35, f pk /f pu = 1770 N/mm 2 , cement CEM I class
52.5R and a 10 mm coarse gravel aggregate. The self-weight of the precast concrete is
determined by the manufacturer as 24.5 kN/m3 giving a self-weight of 4.90 kN/m 2 . In this
exercise, ignore the reduced compression acting at the level of the strands due to self-weight
and dead loads in the calculation of creep losses.
Precast frame analysis 117
Durability. BS 8500-1. Table A.4 for 50 years Durability. BS 8500-1. Table A.4 for 50 years
Exposure XC1. Cover c = 15 + Δc Exposure XC1. Cnom = 15 + ΔCdev
{7.3} Δc = 5 mm {4.4.1.3(3)}Cover controlled ΔCdev = 5 mm
Cover to tendons c ≥ 20 mm Cover to tendons Cnom ≥ 20 mm
Fire. 1 h Fire. R60. BS EN 1992-1-2
{Table 3.4} c = 20 mm {Table 5.8} Depth h ≥ 80 mm < 200 mm
{5.2(5)}Axis a = 25 + 15 − Δa = 29 mm
{Exp. 5.3} Δa = 0.1 (500 – θcr) = 10.7 mm
{Fig. 5.1, curve 3}θcr = 390°C for
{Exp. 5.2} kp(θcr) = 805/1,770 = 0.455
σp,fi = 0.523 × 1,770/1.15 = 805 N/mm2
where Ed,fi/Ed = (7.0 + 0.3 × 5.0)/(1.25 × 7.0 +
1.5 × 5.0) = 0.523
{4.3}using ψfi = quasi-permanent ψ2 = 0.3
with Gk = 4.9 + 1.5 + 0.6 = 7.0 kN/m2 and
Qk = 5.0 kN/m2
Centroid to steel tendons in fire zone, a = yT =
35.7 mm > 29 mm
Young’s modulus
{Part 2, 7.2} Ec = 20 + 0.2 × 55 = 31 kN/mm2 {Table 3.1} Ecm = 22 (53/10)0.3 = 36.3 kN/mm2
{4.8.3.1} Eci = 20 + 0.2 × 35 = 27 kN/mm2 Ecm(t) = 22 (38/10)0.3 = 32.8 kN/mm2
{BS 5896, Table 6}Es = 195 kN/mm2 strand {3.3.6(3)} Ep = 195,000 N/mm2 strand
Although Es for wire = 205 kN/mm2 use same as for
strand
mi = 195/27 = 7.22. m = 195/31 = 6.29 m(t) = 195/32.8 = 5.94. m = 195/36.3 = 5.37
Section properties
Zb = 779 × 106/99.4 = 7.836 × 106 mm3
Zt = 779 × 106/101.6 = 7.744 × 106 mm3
e = 99.4 – 47.0 = 52.4 mm
Compound (subscript co) section properties using concrete + (m − 1) Aps
m − 1 = 5.29 m-1 = 4.37
yb,co = 98.3 mm; Ix-x,co = 799.6 × 106 mm4 yb,co = 98.5 mm; Ix-x,co = 796.0 × 106 mm4
Zb,co = 8.134 × 106 mm3; Zt,co = 7.862 × 106 mm3 Zb,co = 8.082 × 106 mm3; Zt,co = 7.842 × 106 mm3
Flexural capacity – service limit of stress
Aps = 6 × 52 + 6 × 93 + 4 × 19.6 = 948.5 mm2
Initial σi = 0.7 × 1770 = 1239 N/mm2 σpi = 0.7 × 1770 = 1,239 N/mm2
Pi = 1239 × 948.5 = 1,175,241 N Ppi = 1239 × 948.5 = 1,175,241 N
{4.8.2.1} relaxation loss = 1.2 × 2% = 0.024 {3.3.2(7)} Relaxation at t = 20 h
{Exp. 3.29} for Class 2. μ = 0.7; ρ1000 = 2.5%
Loss Δσpr = 1239 × 0.66 × 2.5 × e(9.1 × 0.7)
(20/1000)0.75(1 – 0.7) = 4.95 N/mm2
fcc after relaxation loss = +8.99 N/mm2 σc after relaxation loss = +9.17 N/mm2
Shortening loss = 8.99 × 7.22/1,239 = 0.0524 {Exp. 3.29} Δσel = 9.17 × 5.94 = 54.46 N/mm2
σpm0 = 1,239.0 – 4.95 – 54.46 = 1,179.6 N/mm2
Transfer Rtr = 1 – 0.024 – 0.0524 = 0.924 Transfer Rtr = 1,179.6/1,239 = 0.952
Transfer Pt = 0.924 × 1,175,241 = 1,085,467 N Ppm0 = 0.924 × 1,175,241 = 1,085,467 N
(Continued)
118 Precast Concrete Structures
936
195000
{4.3.8.1} Ultimate shear capacity Vco {6.2.2(2)} Ultimate shear capacity VRd,c
{4.3.8.4} x = 100 + 99.4 = 199.4 mm {6.2.2(2)} lx = 100 + 99.4 = 199.4 mm
Mean diameter of strands = 10.9 mm {Exp. 8.16 and 8.18} lpt2 = 1.2 × 0.19 × 979.5 ×
lp = 10.9 × 240/√35 = 442 mm 10.9/4.06 = 721 mm
{Exp. 8.15} fpbt = 3.2 × 1.0 × 1.27 = 4.06 N/mm2
{8.10.2.2} fctd(t) = 0.7 × 2.72/1.5 = 1.27 N/mm2
x/lp = 199.4/442 = 0.451 αl = lx/lpt2 = 199.4/721 = 0.276
fcx = 927,621/232,040 = 4.00 N/mm2 σcp = 0.9 × 929,076/232,040 = 3.60 N/mm2
fcpx = 4.00 [0.451 × (2 − 0.451)] = 2.79 N/mm2 where {2.4.2.2(1)}γp,fav = 0.9
ft = 0.24√55 = 1.78 N/mm2 {3.1.6.2(P)} fctd =0.3 × 452/3 × 0.7/1.5 = 1.77 N/mm2
first m.o.a. Sx-x = 5.8326 × 106 mm3
{Eq. 54} Vco = 0.67 × 1134 × 200 × {Exp. 6.4} VRd,c = (779 × 1,134/5.8326) ×
√(1.782 + 0.8 × 2.79 × 1.78) = 406.1 kN √(1.772 + 0.276 × 3.60 × 1.77) = 335.3 kN
Comments
1. MsR to the Eurocodes is greater than Class 2 BS 8110 because fctm is 13% greater than fct. However, there
is no greater stress than fctm allowed in the Eurocodes, such as Class 3 (0.2) in BS 8110.
2. M Rd < Mur because fp is not allowed to reach maximum design stress in the Eurocodes stress versus
strain idealisation.
3. VRd,c < Vco because lpt2 is 63% greater than lp, and the build-up of prestress is linear in the Eurocodes
rather than parabolic in BS 8110.
* See Section 4.3.3 for the application of MsR in the top surface in design.
120 Precast Concrete Structures
This code effectively replaces BS 8110, Parts 1 to 3, although the execution of work (toler-
ances, setting out, etc.) is found in BS EN 13670:2009, Execution of concrete structures.
The division between commonplace and special design work separated in BS 8110 Parts 1
and 2 no longer exists, and there are no N-M interaction charts for column design. The last
point reflects the fact that EC2 is a limit state code of principles rather than methods. The
current amendment was published in February 2014. The UK Technical Committee B/525
(sub-committee 2) is currently engaged in a revision of the code.
Precast concrete is not treated as a separate design and construction method although, as
with BS 8110, there are certain aspects of design, such as bearings, anchorage at supports,
bursting, floor systems, compression/tension/shear joints, connections, pocket foundations,
and corbels, collected in a separate section, in this case Section 10.
The format of BS EN 1992-1-1, as with all material based on the Eurocodes, is as
follows:
Section 1 Scope – references; assumptions; definitions; symbols. Note that symbols are often only
defined here and not in the text
Section 2 Basis of design – requirement related to BS EN 1990, Annex B; requirements related to BS EN
1991-1; material properties; PSFs γc and γs, load combinations and equilibrium
Section 3 Materials – (concrete, rebar, tendons) strength, stress – strain models, deformation, shrinkage
and creep; fatigue; anchorage; prestressing
Section 4 Durability – environmental and exposure classes; cover to reinforcement
Section 5 Structural analysis – load cases; imperfections, sway; structural models; linear elastic, plastic and
non-linear analysis; redistribution; second-order effects with axial load (columns, walls);
prestressing – stressing; forces; losses; service and ultimate; fatigue
Section 6 ULS – bending, shear, torsion and punching shear; strut-and-tie models; anchorage and laps;
partially loaded areas (localised bearings); fatigue
Section 7 Serviceability limit state – crack control, spacing and crack width, deflections
Section 8 Detailing in general – rebars – bar spacing, anchorage, laps, links details; prestressing tendons
– anchorage, transmission length, development length
Section 9 Detailing in particular – maximum and minimum areas; anchorage at supports; shear, torsion
and surface reinforcement; solid and flat slabs, columns and walls, deep beams and stability ties
Section 10 Precast concrete elements and structures – materials; losses of prestress; bearings; anchorage
at supports; bursting; floor systems; compression/tension/shear joints; half joints; pocket
foundations; corbels
Section 12 Plain and lightly reinforced concrete – reduction factors for strength; precast walls and infill
shear walls, construction joints, strip and pad footings
Informative annexes – (A) improved PSFs; (B) creep and shrinkage strains in detail; (C)
reinforcement properties; (D) prestressing tendons losses; (E) strength classes for durability;
(F) tensile stresses in rebars in biaxial and shear stress fields; (G) soil-structure; (H) second-
order effects; (I) flat slab and shear walls; (J) regions of discontinuity
The code is not prescriptive, and it is necessary to turn to calculation methodology given in
documents published for example by The Concrete Centre, for example calculation of area
of flexural and shear reinforcement in beams and N–M charts for r.c. columns.
Precast frame analysis 121
The design procedure gives an analytical procedure taking into account the behaviour
of the structural system at elevated temperatures, the potential heat exposure and the
beneficial effects of active and passive fire protection systems, together with the conse-
quences of failure. The main text, together with informative annexes A to E, includes
most of the principal concepts and rules necessary for structural fire design of concrete
structures.
3.2.3(5) Values for the parameters of the stress–strain relationship of reinforcing steel at elevated
temperatures. Use Class N (Table 3.2a).
3.2.4(2) Ditto cold worked (wires and strands) prestressing steel at elevated temperatures.
Use Class A.
5.6.1(1) Web thickness. Use dimensions for Class WA.
This PD gives guidance on some specific items that were not published in the concrete
Eurocodes or were in need of additional or noncontradictory additional information.
Background research is cited in many cases. It is not to be regarded as a British standard.
This PD gives noncontradictory complementary information for use with BS BS EN 1992
Parts 1-1 and 1-2 and their UK NAs.
Precast frame analysis 123
The main items in the PD relating to the design of precast concrete structures are
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2.5 Bond stress for mild steel cl. 8.4.2(2). The PD gives fbd = η1η2 (0,36√fck)/γc.
2.11 Calculation of effective length of columns, cl. 5.8.3.1, 5.8.3.2 (4) and (5). In the
calculation of the flexibility at the ends of the column, the stiffness of the
beam(s) attached to the column is taken as 2 (EI/L)beam to allow for the effects of
cracking.
2.11.3 Calculation of limiting slenderness ratio, λlim. where adjacent spans of beams do not
differ by more than 15%, columns may be assumed to be in double curvature
bending for the calculation of λlim (i.e. value of moment ratio rm < 0).
2.12 Design moment in columns, cl. 5.8.7.3 and 5.8.8.2. For braced structures,
MEd = maximum of (M0e + M2), (M02) or (M01+ 0.5 M2).
2.14 Design shear – point loads close to support. cl. 6.2.2 (6). Point loads close to
support will need to be considered in conjunction with other loads on the
member. Design shear VEd between the point load and the support is
VEd = VEd,other + β VEd,point-load, and therefore the reduction factor β cannot be applied
to the total VEd. The clause explains how to deal with this situation.
2.20 Stress limitation in serviceability limit state. cl. 7.2(5). A modular ratio of 15 should be
used when calculating tensile stresses in rebars (≤ 0.8 fyk and tendons ≤ 0.75 fpk)
under the characteristic combination of loads.
2.21.1 Control of cracking without direct calculation. cl. 7.3.3. Where the assumptions
relating to Table 7.2N and Table 7.3N are not met, crack width is verified using
the calculation procedure.
2.21.2 Calculation of crack widths, cl. 7.3.4.Values for hc,eff from Fig. 6 of the PD are
proposed. It is unsure how Fig. 6a is interpreted.
2.22 Crack widths for non-rectangular tension zones and irregular bar layouts. Based
on BS 8110, the recommendation is wk = 3acr εm/[1+ 2(acr – c)/(h – x)].
2.23.2 Span/depth ratio. Exp. 7.17 in cl. 7.4.2(2).Values of (As,prov/As,req) or (310/σs) should
be limited to 1.5.
2.23.3 The value for ζ in Exp. 7.18 in cl. 7.4.3, and hence the value of σs or Ms, should be
based on the frequent (not quasi-permanent) combination of loading.
2.26.1 Vertical ties, cl. 9.10.2. For notes that vertical ties are required in framed as well as
load-bearing structures, see Chapter 11 of this book.
2.26.2 Anchorage of precast floor and roof units and stair members. BS EN 1992-1-
1:2004 or BS EN 1991-1-7 does not cover this. All precast floor, roof and stair
members should be effectively anchored whether or not such members are used
to provide other ties required in BS EN 1992-1-1:2004, cl. 9.10.2. The anchorage
should be capable of carrying the dead weight of the member to that part of the
structure that contains the ties.
2.28 Detailing rules for particular situations, Annex J. NA to BS EN 1992-1-1:2004
declares that this is not applicable in the United Kingdom. Alternative versions for
frame corners and corbels are given in Annex B of this PD.
3 BS EN 1992-1-2:2004, Structural fire design. The tabular methods for assessing
the fire resistance of columns are limited to braced structures. However, at the
discretion of the designer, the methods given in BS EN 1992-1-2:2004 for
columns may be used for the initial design of unbraced structures. In critical
cases, the chosen column sizes should be verified using BS EN 1992-1-2:2004,
Annex B.
124 Precast Concrete Structures
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