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 combinations and arrangement of gravity and horizontal loads acting on floors, beams and structures are introduced. The design of reinforced and prestressed concrete elements, connections and structures will follow in later chapters.
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 combinations and arrangement of gravity and horizontal loads acting on floors, beams and structures are introduced. The design of reinforced and prestressed concrete elements, connections 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 function of a building than the structural design itself. The creation of a large ‘open plan’ accommodation 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 constructed 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 multistorey car parks, where the imposed loading (vehicle gross weight <30 kN according to the NA to BS EN 199111, 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 skeletal structure, in which case the spaces between the columns would be clad in brickwork, 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).
Figure 3.1 Precast skeletal structure showing large unobstructed spaces for the benefit of construction workers and the client.
Figure 3.2 Definitions in a precast skeletal structure.
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 loadbearing and they support oneway spanning floor slabs. There is less architectural freedom compared to the skeletal frame, for example walls should (preferably) be arranged on a rectangular grid and of fixed modular distance, usually 300 mm, which is quite important 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 very large inplane 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 breakdown in the weatherproof envelope and the eventual condemnation of buildings, which are structurally adequate.
Figure 3.3 Precast skeletal structure in Portugal. (Courtesy of Ergon, Belgium.)
Figure 3.4 Wall frames are best suited for apartments, hotels, schools, shopping units, as in this example at Rhodes, near Sydney, Australia.
Figure 3.5 Precast wall and slab frames in Kuala Lumpur, Malaysia.
Figure 3.6 Crosswall system in precast wall frames.
Figure 3.7 Precast wall and slab frame at Strijkijzer, Den Haag, the Netherlands.
The third category of precast building is the ‘portal frame’ used for industrial buildings and warehouses where clear spans of some 25–40 m Isection or Tsection prestressed rafters are necessary; Figures 3.8 and 3.9. Although portal frames are nearly always used for singlestorey buildings, they may actually be used to form the roof structure to a skeletal frame, and as this book is concerned with multistorey structures it gives us a reason to mention 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 flexible 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 momentresisting cantilevers, which might cause a problem in some structures as explained later in Section 3.6.2. A momentresisting 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 prestressed or reinforced beams of 6–8 m span supporting longspan 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.
Figure 3.8 Definition of a precast portal frame.
Figure 3.9 Precast portal frame. (Courtesy David FernandezOrdoñez, Escuela Técnica Superior de Ingeniería Civil, Madrid, Spain.)
Figure 3.10 Portal frame with folded plate roof units, the University of Sao Carlos, Brazil.
Use of building 
Number of storeys^{a} 
Interior spans (m) 
Skeletal frame 
Wall frame 
Portal frame 

Office 
2–0 
6–15 
✓ 



2–50 
6–15 

✓ 

Retail, shopping complex 
2–10 
6–10 
✓ 
✓ 

Cultural 
2–10 
6–10 
✓ 


Education 
2–5 
6–10 
✓ 
✓ 

Car parking 
2–10 
15–20 
✓ 


Stadia 
2–4 
6–8 
✓ 


Hotel 
2–30 
6–8 

✓ 

Hospital 
2–10 
6–10 

✓ 

Residential 
1–40 
4–6 

✓ 

Industrial 
1 
25–40 


✓ 
Warehouse with office 
2–3 
6–8 25–40 
✓ 

✓ 
One of the most frequently asked questions is … how is a precast concrete structure analysed 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 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 twodimensional intheplane simplification is appropriate in the first instance. This is defined in Figure 3.11 where there are no structural frame components, only simply supported floor units, connecting the 2D inplane frames together.
Figure 3.11 2D simplification of a 3D skeletal structure.
Figure 3.12 shows the approximate bending moments and deflected shape in a threestorey 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), contraflexure is at midspan, with the final location for combined loading at about 0.15 × span. In the column, contraflexure is always at midstorey 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 deflections 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 momentresisting (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 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 secondorder moments resulting from slenderness.
Figure 3.12 Deformation and bending moment distribution in a continuous structure subjected to (a) gravity loads and (b) horizontal sway load.
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 large spans so that this method is used for repetitious site casting. Therefore, a practical solution is to prefabricate a series of Lframes as shown in Figure 3.16b for longspan beams and smallstorey 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.
Figure 3.13 Beam halfjoints at 0.1× span close to points of contraflexure in a continuous beam.
The socalled Hframe is a variation on F3. Referring back to frame F1, if pinned connections are made at the points of column contraflexure, structural behaviour is similar to a continuous frame as explained in Figure 3.17. Connections between frames are made at midstorey height positions. Although in theory the connection is classed as pinned, in reality there will be some need for moment transfer, however small. Therefore, Hframe connections are designed with finite moment capacity, this also gives safety and stability to the Hframes, which by their nature tend to be massive. The foundation to halfstorey 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 Hframe has been used in a number of multistorey grandstands.
Figure 3.14 Deformation and bending moment distribution in a pinned jointed structure subjected to (a) gravity loads. (b) horizontal sway load.
Figure 3.15 Partially braced structures.
Figure 3.16 Structural systems for (a) portal Uframes and (b) portal Lframes.
The object of analysis of a structure is to determine bending moments, shear and axial forces throughout the structure. Monolithic twodimensional plane frames are analysed using either rigorous elastic analysis, for example moment distribution or stiffness method, either manually or using a computer program. Moment redistribution may be included in the analysis if appropriate. However, often it is only required to determine the moments and forces in one beam or one column, so codes of practice allow simplified substructuring 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% agreement with full frame analysis.
Figure 3.17 Hframes (a) structural system, (b) deformation and bending moments.
Substructuring is also carried out in precast frame analysis, except that, where pinned connections are used, no moment distribution or redistribution is permitted. Figure 3.19 shows 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 199211 and EN 199212’ (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 50year reference period.”
Figure 3.18 Substructuring method for internal beam in a continuous frame.
Figure 3.19 Substructuring methods for internal beam and columns in a pinned jointed frame (a) internal beam. (b) upper floor column and (c) ground floor column.
The primary Eurocodes used in the design of precast concrete structures are
Each panEuropean document is accompanied by national annexes (NAs) appropriate to national working practices, regional conditions and established/historical precedence, for example stability ties for robustness are the same as in the British code BS 8110: 1997. This book will refer to the UK NAs to Eurocodes EC0 (NA to BS EN 1990 2002), EC1 (NA to BS EN 199111 2002), EC2 (NA to BS EN 199211 2004) and briefly to the NA to Eurocode 3 where steelwork, inserts, welding, etc. is required (NA to BS EN 199311 2005). Appendix 3A (at the end of this chapter) summarises the content of Eurocodes EC2 Parts 11 and 12, together with the specific clauses related to precast and prestressed concrete elements in the NA to BS EN 199211. Reference will also be made to the UK's Published Document PD 66871 (PD 66871 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 γ_{f} (in BS EN 1990) for gravity and horizontal loads in frames and beams. Four conditions are considered, each with their own values of γ_{f} follows
However, each condition varies depending on the nature of the loads. These are called ‘actions’ in the Eurocodes, and those applicable to the superstructure are as follows:
Dead, live and wind loads are based on the 95% characteristic value for uniformly distributed load (UDL) known as g_{k}, q_{k} and w_{k} [kN/m^{2}] and for line/beam loads and point loads as G_{k}, Q_{k} and W_{k} [kN/m or kN].
The selfweight of plain concrete made with normalweight aggregates (approx. 2600 kg/m^{3}) is taken as 24 kN/m^{3}, according to BS EN 199111, Table A.1, unless it is shown by the manufacturer that the characteristic selfweight of elements is different. An additional 1 kN/m^{3} 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/m^{3}. The density of wet concrete is taken as 25 kN/m^{3}. The densities or selfweight of other building materials and stored materials in warehouses, etc. are given in BS EN 199111, Tables A.2 through A.12. Note that the selfweight of masonry units are given in BS EN 771 (BS EN 771 2011) and not in the masonry code (BS EN 199611 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 number and location of the variable loads. Statistically, it is improbable that all imposed loads will be acting at their characteristic value Q_{k}_{1}, Q_{k}_{2} … Q_{ki} and at the same time, that is full live loads will not act at all floor levels in a multistorey building, or live, wind and snow 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 taken for both elements.
Historically, the ‘characteristic’ imposed (live) load Q_{k} was used in all serviceability calculations of service stresses in prestressed concrete, crack spacing and widths, and short and longterm deflections using viscoelastic deformations due to creep and other effects such as the relative shrinkage between concrete cast at different times. The Eurocodes consider this too severe for longterm effects of cracking and deflection, and, with the exception of calculating service stresses in prestressed concrete in order to avoid sudden rupture after cracking, reduced values of Q_{k} are permitted as shown in Figure 3.20. This is an illustration of the representative values for the characteristic Q_{k}, combination ψ_{0}Q_{k}, frequent ψ_{1}Q_{k}, and quasipermanent ψ_{2}Q_{k} values of imposed loading over a period of time, which can of course be extended to the whole life of the structure. In fact, clause A1.4.2 of EN 1990 allows the serviceability criteria to be specified for each project and agreed with the client, but the actual circumstantial definitions recommended to be used with particular serviceability requirements in clause A1.4.2 of the NA to BS EN 1990 are
Figure 3.20 Illustration of variable actions.
Floor usage 
ψ_{0} 
ψ_{1} 
ψ_{2} 

Domestic, residential, offices 
0.7 
0.5 
0.3 
Shopping, congregation 
0.7 
0.7 
0.6 
Storage 
1.0 
0.9 
0.8 
Traffic area < 3t vehicle weight 
0.7 
0.7 
0.6 
Traffic area > 3t vehicle weight 
0.7 
0.5 
0.3 
Roof 
0.7 
0 
0 
Snow at altitude > 1000 m 
0.7 
0.5 
0.2 
Snow at altitude < 1000 m 
0.5 
0.2 
0 
Wind pressure 
0.5 
0.2 
0 
Source: From NA to BS EN 1990, Table NA.A1.1.
ψ_{0} is used for load combinations of ultimate strength, ψ_{1} is used for checking crack widths and decompression stresses for certain durability requirements, ψ_{2} is used for calculating deflections.
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 V_{s} and end reaction F_{s} are based on the design service load = characteristic load × set of load factors ψ as follows:
Figure 3.21 Example of the characteristic, frequent and quasipermanent combinations of service stresses in a fictitious prestressed concrete section.
Continuing the example mentioned earlier would not be meaningful as the ‘quasipermanent’ combination is used for calculating deflections, but for completeness f_{b} = +3.4 N/mm^{2} (compression). 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.
This limit state is known as ‘structure STR’. The design ultimate moment M_{Ed}, shear force V_{Ed} and end reaction F_{Ed} are based on the design ultimate load E_{d} = characteristic load x set of PSFs γ. Loads are called ‘favourable’ or ‘unfavourable’ in creating the worst possible effects in an element, frame or subframe. The ultimate load combination is according to BS EN 1990, Exp. 6.10, or for STR limit state the least favourable (greater) of Exp. 6.10a and 6.10b, which will always be less than Exp. 6.10. The NA to BS EN 1990, Table NA.A1.2 (B) – Design values of actions (STR) (Set B) give the PSF values. The combinations are
with j ≥ 1 and i > 1
where γ_{G,j} = 1.35 unfavourable, γ_{G,j} = 1.0 favourable, γ_{Q,1} = 1.5, ξ = 0.925
for prestress γ_{P} = 0.9 (used for ultimate shear capacity)
NA to BS EN 1990, Table NA.A1.2 (B) notes: “Either expression 6.10, or expression 6.10a together with and 6.10b may be made, as desired. The characteristic values of all permanent actions from one source are multiplied by γ_{G,sup} = 1.35, if the total resulting action effect is unfavourable and γ_{G,inf} = 1.0, if the total resulting action effect is favourable. For example all actions originating from the selfweight of the structure may be considered as coming from one source; this also applies if different materials are involved. When variable actions are favourable Q_{k} should be taken as 0.” In other words, in frame or subframe analysis 1.35 G_{k,j} + 1.5 ψ_{0,} Q_{k} (6.10a or b) is carried on one element and 1.0 G_{k,j} on the adjacent element. Each live load is taken as the ‘dominant’ in turn, with all of the others as ‘accompanying’ in turn, until the maximum combination is found. Therefore, if there are two live loads present there will be three load combinations, that is 6.10a; 6.10b Q_{k}_{1} dominant; and 6.10b Q_{k}_{2} dominant.
If the building is domestic with ψ_{0} = 0.7, use the greater of
To satisfy the ULS design, the three load combinations must be used to determine the maximum end reactions, and bending moments and shear forces at all points along the span.
Finally, it is necessary to define the effective span of floor slabs (simply supported, cantilevers) and beams (on dry bearings or mechanical connectors). The clear span of floors l_{n} = distance between beam centres – sum of half breadth of beams. Referring to BS EN 199211,
where
For cantilevers, L_{b}_{2} is also the width of the support. For continuous elements after completion of the continuity (i.e. stage 2 loading), l_{eff} = distance between beam centres. If a bearing medium (pad, plate) is provided, l_{eff} is to the centre of the pad, and this is also the case for beams supported on steel inserts, cleats and plates, etc.
Calculate the maximum ultimate end reaction F_{Ed} in a simply supported beam of effective span 6.0 m subjected to G_{k} = 30 kN/m, Q_{k}_{1} = 20 kN/m, and Q_{k}_{2} = 40 kN point load at midspan. ψ_{0} = 0.7.
Solution
This limit state, known as ‘equilibrium EQU’, is used to check uplift in the backspan of cantilevers, and overturning of frames including the effect of horizontal wind pressure or other forces. The equilibrium load combination is according to BS EN 1990, Exp. 6.10. The PSFs are given in NA to BS EN 1990, Table NA.A1.2 (A) – Design values of actions (EQU) (Set A) Exp. 6.10 as follows:
with j ≥ 1 and i > 1
Calculate the minimum end reaction F_{Ed} in a simply supported beam of effective span 6.0 m with a 3.0 m span overhanging cantilever at one end subjected to G_{k} = 30 kN/m and Q_{k} = 20 kN/m.
Solution
The accidental load combination is according to BS EN 1990, Exp. 6.11. The PSFs are all γ = 1.0. ψ values are given in NA to BS EN 1990, Table NA.A1.3 – Design values of actions for use in accidental combinations of actions, Exp. 6.11 as follows:
with j ≥ 1 and i > 1
where A_{d} is the value of the accidental action. ψ_{1,1} is applied to the dominant action and ψ_{2,i} to the others. However, in accidental situations, it may not be obvious which is which and therefore ψ_{1} is applied to all.
In frame and subframe analysis without sway, the critical gravity load combinations with their associated PSFs γ_{G} and γ_{Q} are
For frame analysis with sway, horizontal loads W_{k} are combined with gravity G_{k} and Q_{k} load combinations 6.10a and 6.10b for three situations:
Exp. 
Load combination 
Permanent load 
Imposed load 


Adverse 
Beneficial 
Adverse 
Beneficial 
Wind 

6.10a 
Permanent + imposed 
1.35 
1.0 
ψ_{0} 1.5 
0 
N/A 

Permanent + wind 
1.35 
1.0 
N/A 
N/A 
ψ_{0} 1.5 = 0.75 

All 
1.35 
1.0 
ψ_{0} 1.5 
0 
ψ_{0} 1.5 = 0.75 
6.10b 
Permanent + imposed 
1.25 
1.0 
1.5 
0 
N/A 

Permanent + wind 
1.25 
1.0 
N/A 
N/A 
1.5 

All 
1.25 
1.0 
1.5 
0 
ψ_{0} 1.5 = 0.75 
Source: From NA to BS EN 1990, Table NA.A1.2 (B).
Alternative terminology to BS EN 1990: beneficial = favourable, adverse = unfavourable.
For wind load, ψ_{0} = 0.5 according to Table NA.A1.1 of NA to BS EN 1990 (see Table 3.2).
The fundamental PSF for wind load (notation used here γ_{W}) is as NA to BS EN 1990, Table NA.A1.2 (B) – Design values of actions (STR) (Set B) γ_{W} = 1.5, and is modified by ψ_{0} = 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.
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 G_{k} = 40 kN/m and Q_{k} = 30 kN/m, and the wind pressure equates to a horizontal load W_{k} = 12 kN. The carryover moment at the lower end of the column is equal to 50% of the upper end moment due to beam eccentricity. Let ψ_{0} (gravity load) = 0.7.
Solution
Eccentricity of beam reaction R from the centre of column e = 300/2 + 100 = 250 mm.
Figure 3.22 Detail to Example 3.3.
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.
Load combination 
Gravity beam loads 
Wind load 


w_{Ed} (kN/m) 
R_{Ed} (kN) 
M_{Ed} = 0.5 R_{Ed}e (kNm) 
W_{Ed} (kN) 
M_{Ed} = W_{Ed} h/2 (kNm) 
Total M_{Ed} (kNm) 

6.10a 
P + I 
85.5 
342.0 
42.75 
— 
— 
42.75 

P + W 
54.0 
216.0 
27.0 
9.0 
18.0 
45.0 

All 
85.5 
342.0 
42.75 
9.0 
18.0 
60.75 
6.10b 
P + I 
95.0 
380.0 
47.5 
— 
— 
47.5 

P + W 
50.0 
200.0 
25.0 
18.0 
36.0 
61.0 

All 
95.0 
380.0 
47.5 
9.0 
18.0 
65.5 
P, permanent (dead load); I, imposed (live load), W, wind load.
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, w_{ult} = 1.2 × 70 = 84 kN/m, M_{u} (gravity) = 42 kNm, W_{u} = 1.2 × 12 = 14.4 kN, M_{u} (wind) = 28.8 kNm. Total M_{u} = 70.8 kNm).
All buildings, including precast concrete structures built with the greatest practical accuracy, will contain imperfections due to construction methods, errors or natural effects. Some of these are unavoidable, for example overturning moments due to balconies, hanging façade panels, etc. resulting in horizontal deflection and curvature in columns and walls. The reactions to the precast frame from the inclined staircase shown in Figure 3.23 are not exactly imperfections but demonstrate the point of transferring inclined gravity loads into horizontal forces.
Figure 3.23 Inclined staircase imposing horizontal forces to the structure at Bella Sky Hotel, Denmark. (Courtesy Ramboll, Denmark.)
BS EN 199211 defines imperfections as ‘possible deviations’ in geometry and ‘positions’ of loads in Section 5.2 and as quantified by code Exp. 5.1 through 5.4 as an ultimate horizontal force H_{i} = Nθ_{i} due the inclination θ_{i} 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 θ_{i} 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 firstorder bending moments and shear forces, but not when checking deflections.
BS EN 199211 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:
Figure 3.24 Examples of the effect of geometric imperfections. (a) Bracing system and (b) isolated column in unbraced structure. (Adapted from BS EN 199211. 2004, Eurocode 2: Design of Concrete Structures – Part 11: General rules and rules for buildings, BSI, London, February 2014, Fig. 5.1a1 and b.)
Calculate the horizontal forces at each roof and floor level and the overturning 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.
Figure 3.25 Detail to Example 3.4 (braced) and Example 3.5 (unbraced).
Solution
Calculate the horizontal forces at each roof and floor level and the overturning 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
(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).
Figure 3.19a. The subframe consists of the beam to be designed of span L_{1}, and half of the adjacent beams of span L_{2} and L_{3}. The eccentricity of the beam end reaction from the centroidal axis of the column is e. Alternate pattern loading is used. The height of the column above and below the beam is actually of no consequence to beam. It is assumed that the cross section and flexural stiffness of the column is constant.
(Note the shear force in the beam is V_{Ed} = w_{Ed,}_{max} (L_{2} – 2e)/2).
The resulting maximum bending moment in the column is given by
assuming that R_{1} < R_{3} and h_{1} > h_{3}. Figure 3.26a shows the final moments.
Figure 3.26 Bending moments in a pinned jointed frame for (a) internal beams, (b) upper floor columns. (c) ground floor columns.
Figure 3.19b. The subframe consists of the column to be designed of height (distance between centres of beam bearing) h_{2}, and half the adjacent columns of heights h_{1} and h_{3}. Because the column is continuous, the cross section and flexural stiffness EI of each part of the column is considered as shown in the figure. The beams are pattern loaded as mentioned earlier, of span L_{4}/2 and L_{5}/2, and the eccentricity of each beam end reaction from the centroidal axis of the column is e_{4} and e_{5}, respectively. The moment at the upper end of the designed column is given by
and at the lower end is
where R_{4} 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.
Figure 3.19c The subframe consists of the column to be designed of height (distance between the centre of first floor beam bearing and 50 mm below top of foundation (see section 9.4)) h_{1}, and half the adjacent column of height h_{2}. All other details are as before. If the foundation is rigid (moment resisting), the moment at the upper end of the designed column is given by Equation 3.5 with appropriate notation. The carryover 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
and the lower end moment is zero. Figure 3.26c shows the final moments. Patch loads produce single curvature in the columns.
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 G_{k} = 40 kN/m and Q_{k} = 30 kN/m.
Solution
w_{Ed,}_{max} = max{1.35 × 40 + 1.05 × 30; 1.25 × 40 + 1.5 × 30} = max {85.5; 95.0} = 95.0 kN/m; w_{Ed,}_{min} = 40 kN/m.
Beam subframe
e = 450/2 + 100 = 325 mm
Column Y subframe
Beam end reactions R_{1} = 95.0 × 8.000/2 = 380.0 kN; R_{2} = 40 × 6.000/2 = 240.0 kN
Figure 3.27 Detail to Example 3.6.
Column Z subframe
Beam end reactions as before. e_{1} = e_{2} = 450/2 + 100 = 325 mm
Given that E is constant
Equation 3.5. At upper end, M_{col,upper} = (380 – 240) × 0.325 × 451/(451 + 211) = 31.0 kNm
At lower end, M_{col,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 loadbearing 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 versus 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 similar things. Connections are sometimes called ‘joints’ – the terminology is loose and often interposed. The definition adopted in this book is as follows:
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 values. The effects of localised stress concentrations near to inserts and geometric discontinuities 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 manufacturing tolerances, erection methods and workmanship.
Figure 3.28 Moment and shear transfer at a bearing corbel.
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 concrete and steel, handling and selfweight 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 connections. 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 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.
Figure 3.29 Force paths in beam to column corbel connection.
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 semirigid manner, especially after the onset of flexural cracking. Using a ‘beamline’ analysis, Figure 3.33, we can assess the structural classification of a connection. Although the beamline 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 (Mθ) 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 M_{Rd} > wL^{2}/12, and the rotation of a pin ended beam subjected to a UDL of w is θ = wL^{3}/24EI. The gradient of the beam line is 2EI/L. The Mθ 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 Mθ plot for the connection fails to pass through the beamline, that is plot 5, the connection is deemed not to possess sufficient ductility and should be considered in design as ‘pinned’. Furthermore, its inherent stiffness (given by the gradient of the Mθ plot) is ignored. Conversely, if the Mθ 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%.
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.
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 maturity of cast in situ concrete/grout joints, and several days if structural cast in situ toppings are used to transfer horizontal forces. A stabilising system must comprise two things as shown in Figure 3.34:
Figure 3.31 Precast column to pocket connection.
Figure 3.32 Types of connections in a precast structure.
Connection type 
Location in Figure 3.32 
Classification 
Method of jointing 

Beam – column head 
1 
Pinned 
Dowel 
Beam – column head 
2 
Rigid 
Dowel plus continuity top steel 
Rafter – column head 
3 
Pinned 
Dowel 
Rafter – column head 
4 
Rigid 
Bolts (couple) 
Column splice 
5 
Pinned 
Bolts/dowel 


Rigid 
Bars in grouted sleeve (couple) 



Threaded couplers 



Steel shoes 
Beam – column face 
6 
Pinned 
Bolts 



Welded plates 



Notched plates 



Dowels 
Beam – column corbel 
7 
Pinned 
Dowel 
Beam – column corbel 
8 
Rigid 
Dowel plus continuity top steel 
Beam – beam 
9 
Pinned 
Bolts 



Dowels 
Slab – beam 
10 
Pinned 
Tie bars 
Slab – wall 
11 
Pinned 
Tie bars 
Column – foundation 
12 
Pinned 
Bolts 
Column – cast in situ beam or retaining wall 
13 
Pinned or rigid^{a} 
Bolts Rebars in grouted sleeve 
Figure 3.33 Definition of moment–rotation characteristics.
Figure 3.34 Stabilising systems in braced frames.
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 lackofplumb forces, the floor slab acts as a deep beam and is subjected to bending moments M_{h} and shear forces V_{h} (h being the subscript used for horizontal diaphragms). The basic design method is shown in Figure 3.35. The design is a threestage approach:
where
High tensile rebar with f_{yk} = 500 N/mm^{2} or standard helical strand with f_{pk} = 1770 N/mm^{2} 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 199211, clause 6.2.5(2). Hollow core slabs are considered as being untreated and smooth, then µ = 0.6 with no special, that is exfactory, edge preparation (see Section 8.2). For exsteel mould with smooth surfaces, µ = 0.5.
Figure 3.35 Diaphragm floor action. (a) Deep beam analogy. (b) Reinforced structural topping in doubletee floors. (c) Perimeter reinforcement in hollow core floors.
Diaphragms may be reinforced in several ways. In Figure 3.35b, a reinforced cast in situ topping 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 concretefilled longitudinal joints between the units.
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 exfactory edge finish. The characteristic wind pressure on the floor w_{k} = 3 kN/m. Tie steel is high tensile ribbed bar f_{yk} = 500 N/mm^{2}. Suggest some reinforcement details.
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) γ_{W} = 1.5
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 direction of the bracing. However, there is a broad classification as the structure is
Figure 3.37 Alternative sway mechanisms and resulting column effective length factors.
Figure 3.38 Alternative fullheight bracing mechanisms and resulting column effective length factors.
The type of stabilising system may be different in other directions. The floor plan arrangement 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 of three or more storeys in height are mostly braced or partially braced. This is to avoid having to use deep columns to cater for sway deflections, which give rise to large secondorder bending moments. Section 6.2.6 refers in more detail.
Figure 3.39 Alternative partial height bracing mechanisms and resulting column effective length factors.
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. (c) bracing methods and positions in partially braced irregular or nonsymmetrical buildings.
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 component of thickness t and length L is given by E_{cm,long} I, where E_{cm,long} = longterm 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.
Propose stabilising systems for the fivestorey 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 uniform 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 onestorey 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 external pressure, that is at x ≈ 24 m and y ≈ 16 m. It is necessary to first consider the two orthogonal directions.
Stability in ydirection
The centroid of the stability walls x′ ≈ 24 m.
Figure 3.41 Detail to Example 3.8 (dimensions in metres). (a) Plan view and crosssection of framed structure, (b) Plan view showing positions of shear walls.
Select walls as shown in Figure 3.41b. On the assumption that the material and construction 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.
Centroid of stiffness ${x}^{\prime}=\frac{({4}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}0)\text{\hspace{0.17em}}+({4}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}3)\text{\hspace{0.17em}}+({3}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}33)\text{\hspace{0.17em}}+({3}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}36)\text{\hspace{0.17em}}+({4}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}45)\text{\hspace{0.17em}}+({4}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}48)}{(4\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{4}^{3})\text{\hspace{0.17em}}+\text{\hspace{0.17em}}(2\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{3}^{3})}=25.0\text{\hspace{0.17em}m}$
, which is sufficiently close to the required point to eliminate significant torsional effects.Stability in xdirection
The centroid of the stability walls y′ ≈ 32/2 = 16 m.
Centroid of stiffness ${y}^{\prime}=\frac{({3}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}0)\text{\hspace{0.17em}}+({3}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}\hspace{0.17em}}16)\text{\hspace{0.17em}}+({3}^{3}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}\hspace{0.17em}}32)}{3\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}{3}^{3}}=16.0\text{\hspace{0.17em}m}$
, which is at the correct point.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:
Clauses and tables in the codes are indicated within ‘{}’.
A 600 mm deep × 300 mm wide r.c. beam carries uniformly distributed dead and live loading of 40 and 30 kN/m over a simply supported clear span of 5.85 m with bearing lengths of 150 mm. The beam carries office loading. The exposure is internal, and the fire resistance is 60 min. The design life is 50 years.
Design the main reinforcement at midspan and the shear reinforcement at the support and position where nominal links are required. Calculate the crack width and the longterm deflection using the appropriate creep factor and check the span/depth ratio for the area of rebars designed. Use f_{c}_{k}/f_{c}_{u} = 32/40, high tensile main bars and links grade f_{y}_{k}/f_{y} = 500 N/mm^{2}, and normalweight concrete with a 20 mm coarse aggregate.
BS 8110 solution 
Eurocodes solution 

Durability. BS 85001. Table A.4 for 50 years 

Exposure XC1. Cover c = 15 + Δc 
Exposure XC1. C_{nom} = 15 + ΔC_{dev} 
{7.3} Δc = 5 mm 
{4.4.1.3(3)} ΔC_{dev} = 5 mm 
Cover to links c = 20 mm 
Cover to links = 20 mm 
Fire. 1 h 
Fire. R60. BS EN 199212 
{Table 3.4} c = 20 mm 
{Table 5.5} for b = 300 mm, axis a = 25 mm 
Section properties. Let links φ = 8 mm 
∴a = max{20 + 8 = 28 mm; 25 mm} 
Assume main bars φ = 32 mm 

b = 300 mm; d = 600 – 20 – 8 – 16 = 556 mm 
b = 300 mm; d = 600 – 28 – 16 = 556 mm 
Flexural design 

Selfweight 
Selfweight {BS EN 199111, Table A1.1} 
= 0.6 × 0.3 × 24 = 4.32 kN/m 
= 0.6 × 0.3 × 25 = 4.5 kN/m 
Ultimate load 
{BS EN 1990, Table A1.2(B) and 
Table A1.1} ψ_{0} = 0.7 

w_{u} = 1.4 × 44.32 + 1.6 × 30 = 110.1 kN/m 
{Exp. 6.10a; 6.10b} w_{Ed} = 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} 
l_{o} = min{5.85 + 0.15; 5.85 + 0.46} = 6.0 m 
l_{eff} = min{5.85 + 0.15; 5.85 + 0.46} = 6.0 m 
M = 110.1 × 6.0^{2}/8 = 495.5 kNm 
M_{Ed} = 100.7 × 6.0^{2}/8 = 453.1 kNm 
{3.4.4.4} K = 495.5 × 10^{6}/40 × 300 × 556^{2} 
{3.1.7(3)} K = 453.1 × 10^{6}/32 × 300 × 556^{2} 
= 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 
A_{s} = 495.5 × 10^{6}/455 × 0.87 × 500 = 2503 mm^{2} 
A_{s} = 453.1 × 10^{6}/467 × 0.87 × 500 = 2230 mm^{2} 
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 + h_{agg} 5 = 25 mm OK 
{Table 7.3N} < 100 mm for any value of σ_{s} 

{Table 7.1, NAD Table NA.4}w_{k} = 0.3 mm 
Although compression steel is not required 

{Table 3.25} Min A_{s}_{′} = 0.2% = 360 mm^{2} 
{9.2.1.1(1)} Min A_{s}_{′} = 0.13% = 234 mm^{2} 
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} l_{v} = 5.85 – 2 × 0.556 = 4.74 m 
{6.2.1(8)} l_{v} = 5.85 – 2 × 0.556 = 4.74 m 
V = 110.1 × 4.74/2 = 261.0 kN 
V_{Ed} = 100.7 × 4.74/2 = 238.7 kN 
{3.4.5.3} v = 261.0 × 10^{3}/300 × 556 
{6.3.2. Exp. 6.9}V_{Rd,}_{max} = v_{1} b z f_{cd} 0.5 sin 2θ 
= 1.56 N/mm^{2} < 0.8√f_{cu} = 5.06 N/mm^{2} 
v_{1} = 0.6 (1 − f_{ck}/250), z = 0.9d and f_{cd} = f_{ck}/1.5 
> v_{c} = 0.79 × 1.5^{1/3} × 1.17/1.25 = 0.84 N/mm^{2} 
θ = 0.5 sin^{−1} (238,700/(0.5 × 0.523 × 300 × 501 
where 100 A_{s}/b_{v}d = 1.5, (f_{cu}/25)^{1/3} = 1.17 
× (32/1.5)) = 8.2° < 22.5° ∴ use cot θ = 2.5 
{Table 3.7} v > v_{c} 
{6.3.2. Exp. 6.8} 
A_{sv}/s_{v} = 300 × (1.56 – 0.84)/(0.87 × 500) 
A_{sw}/s = 238,700/501 × 0.87 × 500 × 2.5 
= 0.5 mm^{2}/mm = 250 mm^{2}/m/leg 
= 0.438 mm^{2}/mm = 219 mm^{2}/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 = v_{c} + 0.4 = 1.24 N/mm^{2} 
{Exp. 9.5N} A_{sw,}_{min}/s = 0.08 × √32 × 300/500 
or where shear force 
= 0.272 mm^{2}/m 
∴V_{nom} = 1.24 × 300 × 556 × 10^{–3} = 207 kN 
∴ V_{Rd,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 

Shortterm Young's modulus 

{2.5.4} E_{s} = 200 kN/mm^{2} 
{3.2.7(4)} E_{s} = 200 kN/mm^{2} 
{Part 2, 7.2} E_{c} = 20 + 0.2 × 40 = 28 kN/mm^{2} 
{Table 3.1} E_{cm} = 22 (40/10)^{0.3} = 33.34 kN/mm^{2} 
α = 200/28 = 7.14 
α = 200/33.34 = 6.00 
Longterm Young's modulus 

Bottom and sides exposed h_{o} = 2A_{c}/u = 360,000/(300 + 2 × 600) = 240 mm 

Age at loading = 28 days. Indoor exposure RH = 50% 

{Part 2, Fig. 7.1} φ = 2.45 
{Fig. 3.1a} φ_{(∞,to)} = 2.4 
{Part 2, 3.6} E_{c,long} = 28/3.45 = 8.11 kN/mm^{2} 
{7.20} E_{c,ef} = 33.34/3.4 = 9.80 kN/mm^{2} 
∴α_{e} = 200/8.11 = 24.64 
∴ α_{e} = 200/9.80 = 20.39 
α_{e} − 1 (uncracked concrete) = 23.64 
α_{e} − 1 = 19.39 
Uncracked section properties 
Uncracked transformed section 
Not required in BS 8110 
bh + Σ(α_{e} – 1) A_{s} = 234,587 mm^{2} 

x_{u} = 300 × 600^{2}/2 + 19.39 × (2,590 × 566 + 402 × 36)/234,587 = 342.3 mm from top 

I_{u} = 8,590 × 10^{6} mm^{4} 

Z_{b} = 8,590/257.5 = 33.36 × 10^{6} mm^{4} 

{7.4.3} M_{cr} = 33.36 × 3.02 = 100.8 kNm 

{Table 3.1} f_{ctm} = 0.3 × 32^{2/3} = 3.02 N/mm^{2} 

M_{s,QP} = (44.5 + 0.3 × 30) × 6.0^{2}/8 = 240.7 kNm 

> M_{cr} use partially cracked I_{ef} 
Cracked section properties 

{Part 2, 3.6} Instantaneous value 

Solving first m.o.a. b x_{c}^{2}/2 + (α − 1) A_{s}′(x_{c} – d′) = α A_{s} (d – x_{c}) 

x_{c} = 202.2 mm 

I_{xx,c} = 3211 × 10^{6} mm^{4} 

Longterm value 

Solving first m.o.a. b x_{c}^{2}/2 + (α_{e}×1) A_{s}′(x_{c} – d′) = α_{e} A_{s} (d – x_{c}) 

x_{c} = 302.0 mm 
x_{c} = 279.4 mm 
I_{xx,c} = 7545 × 10^{6} mm^{4} 
I_{c} = 6,407 × 10^{6} mm^{4} 
Curvature 1/r_{b} = M/EI 
Effective I_{ef} 
{Part 2, 3.3.3) ψ = 0.25 offices 
{Exp. 7.18 and 7.19} better presented as 
Instantaneous curvature (×10^{–6} mm^{−1} units) 
I_{ef} = I_{c} + [(I_{u} – I_{c}) 0.5 (M_{cr}/M_{s,QP})^{2}] 
w_{δ} = 44.32 + 0.25 × 30 = 51.82 kN/m 
I_{ef} = 6407 + [(8,590 – 6,407) × 0.5 × (100.8/240.7)^{2}] = 6599 × 10^{6} mm^{4} 
M_{s,total} = 51.82 × 6.0^{2}/8 = 233.2 kNm 

1/r_{b,total} = 233.2/(28,000 × 3,211) = 2.59 

M_{s,Gk} = 44.32 × 6.0^{2}/8 = 199.4 kNm (dead) 

1/r_{b,Gk} = 199.4/(28,000 × 3,211) = 2.22 

Longterm curvature 
Longterm curvature (×10^{–6} mm^{−1} units) 
1/r_{b,Gk} = 199.4/(8,116 × 7,545) = 3.26 
1/r_{b} = M_{s,QP}/E_{c,ef} I_{ef} 
1/r_{b} = 3.26 + (2.59 − 2.22) = 3.63 
1/r_{b} = 240.7/9,806 × 6,599 = 3.72 
{Part 2, 3.7.2} Deflection δ = K l_{o}^{2} (1/r_{b}) 
δ = K l_{eff}^{2} (1/r_{b}) 
{Part 2, Table 3.1} K = 0.104 

δ = 0.104 × 6000^{2} × 3.63 × 10^{–6} = 13.6 mm 
δ = 0.104 × 6,000^{2} × 3.72 × 10^{–6} = 13.9 mm 
{3.4.6.3} l_{o} /δ = 440 > 250 OK 
{7.4.1(4)} l_{eff} /δ = 431 > 250 OK 
Span/depth ratio 

{3.4.6.1} l_{o}/d = 20 × 0.744 = 14.87 
{7.4.2(2)} l/d = 11 + 1.5√32 × 0.432 = 14.67 
{Table 3.9} Basic l_{o}/d = 20 
where ρ_{o}/ρ = 0.432; ρ’ = 0 
{3.4.6.5} MF = 0.55 + [(477 – 332) 
ρ_{o} = 10^{–3} √32 = 0.00566 
/(120 (0.9 + 5.34))] = 0.744 
ρ = 2,230/300 ×566 = 0.0131 > ρ_{o} {Exp. 7.16b} 
{Table 3.10} f_{s} = (2/3) × 500 × (2,503/2,590) 
(Table 7.4N} K = 1 
= 332 N/mm^{2} 

M/bd^{2} = 495.5 × 10^{6}/300 × 556^{2} = 5.34 

∴d > 6,000/14.87 = 404 mm < 556 OK 
∴ d > 6,000/14.67 = 409 mm < 556 OK 
Comments. 

Deflections and span/depth ratios are the same in spite of different approaches in the two codes and that EC2’s E_{cm} = 1.19 E_{c}. Changing the creep coefficient φ from 2.4 to 1.5 results in only a 1.3 mm reduction in deflection, showing that φ is not a sensitive parameter. 

Crack width 

{Part 2, 3.8.3} E_{c}/2 = 14 kN/mm^{2} 
{7.3.4(1)} w_{k} = s_{r,}_{max} (∊_{sm} − ∊_{cm}) 
Spacing = 3 × 28 = 84 mm 
{7.3.4(3)} s_{r,}_{max} = 3.4 × 28 + 0.8 × 0.5 × 0.425 × 32/0.075 = 168 mm 
c_{w} = 84 × 736 × 10^{–6} = 0.06 mm 
w_{k} = 168 × 913 × 10^{–6} = 0.15 mm 
where ∊_{m} = ∊_{1} − ∊′ = 804 – 68 = 736 × 10^{–6} 
where ∊_{sm} − ∊_{cm} = [206 – (0.4 × 3.02/0.075) × 
∊_{1} = M_{s} (d – x_{c})/E_{c}/2 I_{xx,c} = 334.4 × 
(1 + 6.0 × 0.075)]/200,000 = 913 × 10^{–6} 
(556 – 302)/(14,000 × 7545) = 804 x 10^{–6} 
but > 0.6 × 206/200,000 = 618 × 10^{–6} 
where M_{s} = 74.32 × 6.0^{2}/8 = 334.4 kNm 
{7.3.4(2)}ρ_{eff} = 2412/300 × 107 = 0.075 
w_{s} = 44.32 + 30 = 74.32 kN/m 
{7.3.2(3)}Minimum is A_{c,eff} = (h – x_{c})/3 
{Part 2, Eq. 13} ∊′ = 300 × 298 × 302/(3 × 
= (600 – 279.4)/3 = 107 mm 
200,000 × 2,590 × 254) = 68 x 10^{–6} 
{7.1(2)} f_{ct,eff} = f_{ctm} = 0.3 × 32^{2/3} = 3.02 N/mm^{2} 

σ_{s} = α_{e} M_{s,QP} (d – x_{c})/I_{ef} = 20.39 × 240.7 × (556 − 279.4)/6599 = 206 N/mm^{2} 
{3.2.4} c_{w} < c_{w} max = 0.3 mm 
{Table NA.4} w_{k} < w_{max} = 0.3 mm for XC1 
Comments. 

Crack width is greater in EC2 due to increased crack spacing, that is 84 and 168 mm in the two codes. The final effective strains are also greater in EC2, possibly because of the assumption in BS 8110 using E_{c}/2 = 14 kN/mm^{2} compared to the longterm value in EC2. 
A twostorey 300 mm × 300 mm edge column supports beams on one side only in an unbraced sway frame as shown in Figure 3.42a. The exposure, fire resistance, design life and beam end reactions are as given in 3.6.1, except that the roof beam may be taken as 60% of the floor beam reactions. The beam reactions act at 80 mm from the face of the column. The flexural stiffness of the beamtocolumn connection may (in this exercise) be taken as^{1}/_{10} of that of the column. The characteristic horizontal wind load is shown in Figure 3.42a.
Design the main reinforcement and specify the shear links. Use f_{c}_{k}/f_{c}_{u} = 40/50, f_{y}_{k}/f_{y} = 500 N/mm^{2}, and normalweight concrete. Effective creep factor φ_{ef} = 1 (used in EC2). Moment distribution factors at the first floor (upper end of column is pinned, lower end at foundation is fixed) = 4EI/3.5 /(4EI/3.5 + 3EI/3.0) = 54% with 50% carryover (c/o) to the foundation.
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 secondorder deflections.
BS 8110 solution 
Eurocodes solution 

Durability is the same as for the beam, ∴ cover to links 20 mm 

Fire. 1 h 
Fire. R60. BS EN 199212 
{Table 3.4} c = 20 mm 
{BS EN 1990, Table A1.1} In fire ψ_{2} = 0.3 for offices beam load and ψ_{2} = 0 for wind load 

G_{k} = 44.5 × 6.0/2 = 133.5 kN per beam 

Q_{k} = 30 × 6.0/2 = 90 kN per beam 

{2.4.2} V_{Ed,fi,}_{max} = 133.5 + 0.3 × 90 = 160.5 kN 

Per floor plus 60% at roof = 96.3 kN 

Selfweight = 0.3 × 0.3 × 25 × 3 = 6.75 kN 

N_{Ed,fi} at first floor level = 424 kN 

Eccentricity of reaction = 150 + 80 = 230 mm 

M_{Ed,fi} = 160.5 × 0.23 = 36.9 kNm × 54% as distributed = 19.9 kNm 

{5.3.2} e_{fi} = 19.9/424 = 0.047 m = 47 mm 

{5.3.2. Method B, Table 5.2b}as l_{0} > 3.0 m 

e/h = 47/300 = 0.16 < 0.25 

Then n = N_{Ed,fi}/(0.7(A_{c} f_{cd} + A_{s} f_{yd})) 

Try 4 H25 bars = 1963 mm^{2} 

n = 424/(0.7 (300^{2} × 26.67 +1963 × 435) x 10^{–3} 

= 0.19 and ω = 1963 × 435/300^{2} × 26.67 = 0.36 

{Table 5.2b} R60 requires b_{min}/a = 300/25 

But 25 mm < c + φ_{link} + φ_{bar}/2 = 20 + 8 + 12 = 40 mm ∴ not critical 
Section properties. 

Assume main bars φ = 25 mm 
Assume main bars φ = 20 mm 
{3.12.7.1} Links ≥ φ/4 use 8 mm 
{9.5.3(1)} Links ≥ φ/4 use 8 mm 
b = 300 mm; d = 300 − 20 − 8 − 12 = 260 mm 
b = 300 mm; d = 300 − 40 = 260 mm 
d/h = 0.87. Use design chart in Figure 3.43a 
d/h = 0.87. Use design chart in Figure 3.43b 
Selfweight = 0.3 × 0.3 × 24 × 6.5 = 14.0 kN 
Selfweight = 0.3 × 0.3 × 25 × 6.5 = 14.6 kN 
Effective height factors 

{Part 2, clause 2.5} β = 2 + 0.3 α_{c,min} where 
{5.8.3.2, Exp. 5.16} 
α_{c} = 1.0 for foundation and 10 given for beam 
k_{1} = 0.1 and k_{2} = 1/(^{1}/_{10}) = 10 
β = 2 + 0.3 × 1.0 = 2.3 
β = (1 + 0.1/1.1) × (1 + 10/11) = 2.08 
Clear first floor l_{0} = 3500 – 600 = 2,900 mm 

l_{e} = 2.3 × 2,900 = 6,670 mm. l_{e}/h = 22.2 
l_{0} = 2.08 × 2,900 = 6032 mm. λ = l_{0}/i = 69.6 
{3.8.1.3} For unbraced l_{e}/h > 10 ∴ slender 
where i = 300/√12 = 87 mm 
{3.8.1.7} l_{e}/h < 60 limit 
{5.8.3.1(1)} ${\lambda}_{\mathrm{lim}}=20\text{ABC/}\sqrt{n}=33.4$ 
{3.8.3.1} a_{u} = 22.2^{2} × 300 K/2,000 = 73.9K 
where A = 1/(1 + 0.2φ_{ef}) = 1/1.2 = 0.83 

$B=\sqrt{1\text{\hspace{0.17em}}+2\omega}=1.31$ where ω = 0.36 

C = 1.7 − r_{m} = 0.7 where r_{m} = +1 for unbraced 

n = 502 × 10^{3*}/300^{2} × 26.67 = 0.209 

* N_{Ed} = 502 kN (see case 1 in the following) 

λ >λ_{lim} 69.6 > 33.4 ∴ slender 

{5.8.8.2(3)} e_{2} = 19.4 × 10^{–6} × 6,032^{2}/10 = 71 mm 

{5.8.8.3(1)} where 1/r = K_{r} K_{φ} f_{yd}/0.45 d E_{s} = 1.05 × 435/0.45 × 260 × 200,000= 19.4 × 10^{–6} mm^{−1} 

{5.8.8.3(3)} K_{r} Figure 3.43b = 1 (because N_{Ed}/f_{ck} bh < 0.25 see the following) 

{5.8.8.3(4)} K_{φ} = 1 + (0.35 + f_{ck}/250 − λ/150) φ_{ef} 

= 1 + (0.35 + 0.16 − 69.6/150) × 1 = 1.05 
Clear height to roof l_{0} = 6,500 – 400 = 6,100 m 

l_{e} = 2.3 × 6,100 = 14,030 mm. l_{e}/h = 46.7 < 60 
l_{0} = 2.08 × 6,100 = 12,688 mm. λ = l_{0}/i = 146 
{3.8.3.1} a_{u} = 46.7^{2} × 300 K/2,000 = 327K 
e_{2} = 18.6 × 10^{–6} × 12,688^{2}/10 = 299 mm 
{3.8.3.8}All columns have equal stiffness 
where K_{φ} = 1 and 1/r = 18.6 × 10^{–6} mm^{−1} 
{3.8.5} Column deflection checked as l_{e}/h > 30 

Horizontal loads and moments per column 

Refer to Figure 3.42b 

M_{wind} = (7.0 × 6.5 + 14.0 × 3.5)/3 = 31.5 kNm 
M_{wk} = 31.5 kNm 
{3.1.4.2} Total G_{k} for floor beams + selfweight 
{5.2(5)} Imperfection at first floor 
G_{k,floor} = 44.32 × 12 + 3 × 14 = 574 kN 
α_{h} = 2/√3.5 = 1.069 use 1, α_{m} = 1 
H_{floor} = 1.5% G_{k} = 0.015 × 574 = 8.6 kN 
{5.2(7)} e_{i} = (1/200) × 1 × 6.032/2 = 0.015 m 
<14 kN wind load 
M_{i,floor} = 302.1* × 0.015 = 4.6 kNm *see the following 
At roof H_{roof} = 0.6 × 8.6 = 5.2 kN 
At roof α_{h} = 2 /√6.5 = 0.784, α_{m} = 1 
<7 kN wind load 
e_{i} = (1/200) × 0.784 × 12.688/2 = 0.025 m 

M_{i,roof} = 181.2* × 0.025 = 4.5 kNm *see the following 

M_{Ed,i} = 4.6 + 4.5 = 9.1 kNm (added to γ_{w}M_{wind}) 
Ultimate loads and moments at foundation 

Case 1. Gravity dead + live 

V_{floor} = 110.1 × 6.0/2 = 330.3 kN 
F_{Ed,floor} = 100.7 × 6.0/2 = 302.1 kN 
V_{roof} = 0.6 × 330.3 = 198.2 kN 
F_{Ed,roof} = 0.6 × 302.1 = 181.2 kN 
N = 330.3 + 198.2 + 1.4 × 14.0 = 548 kN 
N_{Ed} = 302.1 + 181.3 + 1.25* × 14.6 = 502 kN 
Refer to Figure 3.42b 

Node moment M_{0} = Ve, where e = 150 + 80 
*γ_{G} = 1.25 is for critical beam load Exp. 6.10b 
= 230 mm from the centre of column 
e = 230 mm 
M_{0} = 330.3 × 0.23 = 76.0 kNm × 54% as 
M_{0Ed} = 302.1 × 0.23 × 54% × 50% c/o 
distributed x 50% c/o = 20.5 kNm 
= 18.8 kNm 
{Eq. 35} M_{add} = N a_{u} = 330.3 × 0.0739K 
{5.8.8.2(3)} M_{2} = N_{Ed} e_{2} = 302.1 × 0.071 
+ 198.2 × 0.327K = 89.2K kNm 
+ 181.3 × 0.299 = 75.7 kNm 
M = 20.5 + 89.2 = 109.7 kNm 
M_{Ed} = 18.8 + 75.7 + 9.1 = 103.6 kNm 
{3.8.2.4} M_{min} = 548 × 0.020 = 11.0 kNm 
{6.1(4)} M_{Ed,}_{min} = 502 × 0.020 = 10.0 kNm 
N/f_{cu} bh = 548 × 10^{3}/50 × 300^{2} = 0.12 
N_{Ed}/f_{ck} bh = 502 × 10^{3}/40 × 300^{2} = 0.14 
M/f_{cu} bh^{2} = 109.7 × 10^{6}/50 × 300^{3} = 0.081 
M_{Ed}/f_{ck} bh^{2} = 103.6 × 10^{6}/40 × 300^{3} = 0.096 
Figure 3.43a gives K = 1 
Figure 3.43b confirms K = 1 
∴A_{sc} = 0.12 × 50 × 300 × 300/500 = 1080 mm^{2} 
∴ A_{s} = 0.15 × 40 × 300 × 300/500 = 1080 mm^{2} 
Case 2. Dead + live + wind 

γ_{f} = all 1.2 
Table 3.3. Exp. 6.10b is critical over 6.10a 

γ_{G} = 1.25, γ_{Q} = 1.5, γ_{W} = 0.75 
V_{floor} = 1.2 × 74.32 × 6.0/2 = 267.5 kN 
N_{Ed} as case 1. M_{Ed} as case 1 + wind M_{Ed,w} 
V_{roof} = 0.6 × 267.5 = 160.5 kN 

N = 267.5 + 160.5 + 1.2 × 14.0 = 445 kN 

M_{0} = 267.5 × 0.23 × 54% × 50% = 16.6 kNm 

M_{add} = 267.5 × 0.0739K + 160.5 × 0.327K = 72.2K kNm 

M_{wind} = 1.2 × 31.5 = 37.8 kNm 
M_{Ed,w} = 0.75 × 31.5 = 23.6 kNm 
M = 16.6 + 72.2 + 37.8 = 126.6 kNm 
M_{Ed} = 103.6 + 23.6 = 127.2 kNm 
N/f_{cu} bh = 0.10 and M/f_{cu} bh^{2} = 0.094 
N_{Ed}/f_{ck} bh = 0.14 and M_{Ed}/f_{ck} bh^{2} = 0.118 
∴ A_{sc} = 0.19 × 50 × 300 × 300/500 = 1710 mm^{2} 
∴ A_{s} = 0.21 × 40 × 300 × 300/500 = 1,512 mm^{2} 
Case 3. Dead + wind 

γ_{f} = 1.0 and 1.4 
γ_{G} = 1.25, γ_{W} = 1.5 
V_{floor} = 1.0 × 44.32 × 6.0/2 = 133.0 kN 
F_{Ed,floor} = 44.5 × 6.0/2 = 133.5 kN 
V_{roof} = 0.6 × 133.0 = 79.8 kN 
F_{Ed,roof} = 0.6 × 133.5 = 80.1 kN 
N = 133.0 + 79.8 + 14.0 = 226.8 kN 
N_{Ed} = 133.5 + 80.1 + 14.6 = 228.2 kN 
M_{0} = 133.0 × 0.23 × 54% × 50% = 8.3 kNm 
M_{0Ed} = 133.5 × 0.23 × 54% × 50% = 8.3 kNm 
M_{add} = 133.0 × 0.0739K + 79.8 × 0.327K 
M_{2} = 133.5 × 0.071 + 80.1 × 0.299 = 33.4 kNm 
= 35.9K kNm 
M_{Ed,i} = 133.5 × 0.015 + 80.1 × 0.025 = 4.0 kNm 
M_{wind} = 1.4 × 31.5 = 44.1 kNm 
M_{Ed,w} = 1.5 × 31.5 = 47.2 kNm 
M = 8.3 + 35.9 + 44.1 = 88.3 kNm 
M_{Ed} = 8.3 + 33.4 + 4.0 + 47.2 = 92.9 kNm 
N/f_{cu} bh = 0.05 and M/f_{cu} bh^{2} = 0.065 
N_{Ed}/f_{ck} bh = 0.063 and M_{Ed}/f_{ck} bh^{2} = 0.086 
∴ A_{sc} = 0.13 × 50 × 300 × 300/500 = 1170 mm^{2} 
∴ A_{s} = 0.19 × 40 × 300 × 300/500 = 1368 mm^{2} 
Maximum A_{sc} = 1710 mm^{2} (Case 2) 
Maximum A_{s} = 1,512 mm^{2} (Case 2) 
Use 4 no. H25 bars (1963) 
Use 4 no. H25 bars (1963) 
{3.12.7.1} Links ≥ φ/4 use 8 mm 
{9.5.3(1)} Links ≥ φ/4 use 8 mm 
Spacing s = 12φ = 300 mm 
{9.5.3(3)} s = min {20φ, b, h, 300} = 300 mm 

{9.5.3(4)} s = 0.6s within h at top and bottom 
{3.12.6.2} A_{sc,}_{max} = 0.08 × 300^{2} = 7,200 mm^{2} 
{9.5.2(3)} A_{s,}_{max} = 0.04 × 300^{2} = 3,600 mm^{2} 
A_{sc,}_{min} = 0.004 × 300^{2} = 360 mm^{2} 
{9.5.2(2)} A_{s,}_{min} = 0.002 × 300^{2} = 180 mm^{2} or 0.1 × 502 × 10^{3}/435 = 115 mm^{2} 
Comments 

The Eurocodes require 12% less A_{sc}, due mainly to reduced γ_{f} (1.25 & 1.5 versus 1.4 & 1.6) which reduces N by 9%. First and secondorder bending moments are roughly equal. 
Calculate the service and ultimate moment of resistance, and the ultimate shear capacity of a 1200 mm wide × 200 mm deep prestressed concrete solid floor slab. The slab is pretensioned using 6 no. 12.5 mm plus 6 no. 9.3 mm diameter standard 7wire helical strands at 30 mm bottom cover, plus 4 no. 5 mm diameter wires at 25 mm top cover. The tendons are Class 2 relaxation and stressed initially at 70%. The exposure is internal with relative humidity = 50%, fire resistance is 60 minutes, and design life is 50 years. Bearing length is 100 mm. The slab is designed as a Class 2 member for permissible tension to BS 8110. The floor carries office loading for characteristic dead loads of 1.5 kN/m^{2} finishes and 0.6 kN/m^{2} services/ceiling, a superimposed live load of 4.0 kN/m^{2} and demountable partitions of 1.0 kN/m^{2}.
The section properties are crosssectional area A_{c} = 232,040 mm^{2}; second m.o.a I_{xx} = 779 × 10^{6} mm^{4}; depth = 200 mm; centroid from bottom y_{b} = 99.4 mm; breadth at top, bottom and at centroid = 1154, 1197 and 1134 mm, respectively. Height to the centroid of all tendons y_{s} = 47.0 mm. Height to the centroid of tendons in tension zone y_{T} = 35.7 mm.
Figure 3.43 (a) Reinforced concrete column design chart to BS 8110. (b) Reinforced concrete column design chart to BS EN 199211.
Use f_{c}_{k}/f_{c}_{u} = 45/55, at transfer f_{c}_{k}(t)/f_{c}_{i} = 30/35, f_{p}_{k}/f_{p}_{u} = 1770 N/mm^{2}, cement CEM I class 52.5R and a 10 mm coarse gravel aggregate. The selfweight of the precast concrete is determined by the manufacturer as 24.5 kN/m^{3} giving a selfweight of 4.90 kN/m^{2}. In this exercise, ignore the reduced compression acting at the level of the strands due to selfweight and dead loads in the calculation of creep losses.
BS 8110 solution 
Eurocodes solution 

Durability. BS 85001. Table A.4 for 50 years 
Durability. BS 85001. Table A.4 for 50 years 
Exposure XC1. Cover c = 15 + Δc 
Exposure XC1. C_{nom} = 15 + ΔC_{dev} 
{7.3} Δc = 5 mm 
{4.4.1.3(3)}Cover controlled ΔC_{dev} = 5 mm 
Cover to tendons c ≥ 20 mm 
Cover to tendons C_{nom} ≥ 20 mm 
Fire. 1 h 
Fire. R60. BS EN 199212 
{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 – θ_{cr}) = 10.7 mm 

{Fig. 5.1, curve 3}θ_{cr} = 390°C for 

{Exp. 5.2} k_{p}(θ_{cr}) = 805/1,770 = 0.455 

σ_{p,fi} = 0.523 × 1,770/1.15 = 805 N/mm^{2} 

where E_{d,fi}/E_{d} = (7.0 + 0.3 × 5.0)/(1.25 × 7.0 + 1.5 × 5.0) = 0.523 

{4.3}using ψ_{fi} = quasipermanent ψ_{2} = 0.3 

with G_{k} = 4.9 + 1.5 + 0.6 = 7.0 kN/m^{2} and Q_{k} = 5.0 kN/m^{2} 

Centroid to steel tendons in fire zone, a = y_{T} = 35.7 mm > 29 mm 
Young's modulus 

{Part 2, 7.2} E_{c} = 20 + 0.2 × 55 = 31 kN/mm^{2} 
{Table 3.1} E_{cm} = 22 (53/10)^{0.3} = 36.3 kN/mm^{2} 
{4.8.3.1} E_{ci} = 20 + 0.2 × 35 = 27 kN/mm^{2} 
E_{cm}(t) = 22 (38/10)^{0.3} = 32.8 kN/mm^{2} 
{BS 5896, Table 6}E_{s} = 195 kN/mm^{2} strand 
{3.3.6(3)} E_{p} = 195,000 N/mm^{2} strand 
Although E_{s} for wire = 205 kN/mm^{2} use same as for strand 

m_{i} = 195/27 = 7.22. m = 195/31 = 6.29 
m(t) = 195/32.8 = 5.94. m = 195/36.3 = 5.37 
Section properties 

Z_{b} = 779 × 10^{6}/99.4 = 7.836 × 10^{6} mm^{3} 

Z_{t} = 779 × 10^{6}/101.6 = 7.744 × 10^{6} mm^{3} 

e = 99.4 – 47.0 = 52.4 mm 

Compound (subscript co) section properties using concrete + (m − 1) A_{ps} 

m − 1 = 5.29 
m1 = 4.37 
y_{b,co} = 98.3 mm; I_{xx,co} = 799.6 × 10^{6} mm^{4} 
y_{b,co} = 98.5 mm; I_{xx,co} = 796.0 × 10^{6} mm^{4} 
Z_{b,co} = 8.134 × 10^{6} mm^{3}; Z_{t,co} = 7.862 × 10^{6} mm^{3} 
Z_{b,co} = 8.082 × 10^{6} mm^{3}; Z_{t,co} = 7.842 × 10^{6} mm^{3} 
Flexural capacity – service limit of stress 

A_{ps} = 6 × 52 + 6 × 93 + 4 × 19.6 = 948.5 mm^{2} 

Initial σ_{i} = 0.7 × 1770 = 1239 N/mm^{2} 
σ_{pi} = 0.7 × 1770 = 1,239 N/mm^{2} 
P_{i} = 1239 × 948.5 = 1,175,241 N 
P_{pi} = 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; ρ_{1000} = 2.5% 

Loss Δσ_{pr} = 1239 × 0.66 × 2.5 × e^{(9.1 × 0.7)} 

(20/1000)^{0.75(1 – 0.7)} = 4.95 N/mm^{2} 
f_{cc} after relaxation loss = +8.99 N/mm^{2} 
σ_{c} after relaxation loss = +9.17 N/mm^{2} 
Shortening loss = 8.99 × 7.22/1,239 = 0.0524 
{Exp. 3.29} Δσ_{el} = 9.17 × 5.94 = 54.46 N/mm^{2} 

σ_{pm0} = 1,239.0 – 4.95 – 54.46 = 1,179.6 N/mm^{2} 
Transfer R_{tr} = 1 – 0.024 – 0.0524 = 0.924 
Transfer R_{tr} = 1,179.6/1,239 = 0.952 
Transfer P_{t} = 0.924 × 1,175,241 = 1,085,467 N 
P_{pm}_{0} = 0.924 × 1,175,241 = 1,085,467 N 
Prestress at transfer 
{Table 3.1} f_{ctm} = 0.3 × 45^{2/3} = 3.80 N/mm^{2} 
{4.3.5.2} f_{cti} = 0.45√35 = 2.66 N/mm^{2} 
{Exp. 3.4} f_{ctm}(t) = (38/53) × 3.80 = 2.72 N/mm^{2} 
f_{bci} = +11.94 N/mm^{2} < 0.5 × 35 = 17.5 OK 
σ_{b}(t) = +12.31 N/mm^{2} < 0.6 × 30 = 18.0 OK 
f_{bti} = −2.67 N/mm^{2} > −2.66 say OK 
σ_{t}(t) = −2.75 N/mm^{2} > −2.72 say OK 
{4.8.5.2} Creep φ = 1.8 
{Exp. B.2} φ(t,t_{o}) = 2.31 × 1.39 × 0.62 × 0.99 

=1.98 

where {Exp. B.4} β (f_{cm}) = 2.31 

{Exp. B.3b/B.8c} φ_{RH} = 1.39, with h_{o} = 400 mm 

{Exp. B.5}β (t_{i}) = 0.62, where t_{i} = 7.6 days at 

50°C curing for 20 h 

{Exp. B.7}β_{c} (t,t_{i}) = 0.99 for t = 20,833 days and {Exp. B.8b}β_{H} (days) = 803 

σ_{c} after initial loss = +8.77 N/mm^{2} 
{4.8.2.1} Loss = 1.8 × 0.924 × 0.0524 = 0.087 
{Exp. 5.46} Δσ_{p,c} = 1.98 × 8.77 × 5.37/1.103 = 

84.40 N/mm^{2} 

where denominator in code Exp. 5.46 = 1.103 
{4.8.4} Shrinkage strain e_{sh} = 300 × 10^{–6} 
{Exp. 3.8} ∊_{cs} = ∊_{cd} + ∊_{ca} = 404 +88= 492 × 10^{–6} 

where {Exp. 3.9} ∊_{cd} = 0.985 × 0.73 × 566 × 10^{–6} 

= 404 × 10^{–6} 

β_{ds} (t,t_{s}) = 0.985 with t as earlier, t_{s} = 1 day 

{Table 3.3} k_{n} = 0.73 for h_{o} = 400 mm 

{Exp. B.11} ∊_{cd,o} = 0.85 × (220 + 110 × 6) × 

e^{(0.11 × 53/10)} × 1.356 = 566 × 10^{–6} 

β_{RH} = 1.55 × [1 − (50/100)^{3}] = 1.356 

{Exp. 3.12} ∊_{ca} = 2.5 × (45 − 10) × 10^{–6} = 88 × 10^{–6} 
Loss = 300 × 10^{–6} × 195,000/1,239 = 0.0472 
{Exp. 5.46} Δσ_{p,sh} = 492 × 10^{–6} × 195,000/1.103 

= 87.0 N/mm^{2} 

{5.10.6(1b)} Longterm relaxation 

{Exp. 3.29} µ = 1,179/1,770 = 0.666 

{Exp. 3.29} σ_{pr} = 1,179 × 0.66 × 2.5 × e^{(9.1 × 0.666)} 

(500,000/1,000)^{0.75(1 − 0.666)} = 39.65 N/mm^{2} 

{Exp. 5.46} Δσ_{p,r} = 0.8 × 39.65/1.103 

= 28.7 N/mm^{2} 

σ_{po} = 1,179.6 − 84.4 − 87.0 − 28.7 = 979.5 N/mm^{2} 
R_{wk} = 0.924 − 0.087 − 0.0472 = 0.789 (21.1%) 
R_{wk} = 979.5/1,239 = 0.7905 (20.95%) 
P_{f} = 0.789 × 1,175,241 = 927,621 N 
P_{po} = 0.791 × 1,175,241 = 929,076 N 
Prestress in service 

f_{bc} = +10.20 N/mm^{2} < 0.33 × 55 = 18.15 OK 
σ_{b} = +10.22 N/mm^{2} < 0.45 × 45 = 20.25 OK 
f_{bt} = −2.28 N/mm^{2} > 0.45 √55 = −3.34 OK 
σ_{t} = −2.28 N/mm^{2} > −3.80 OK 
Service moment of resistance M_{sr} 

Btm M_{sr} = (10.20 + 3.34) × 8.134 = 110.1 kNm 
M_{sR} = (10.22 + 3.80) × 8.082 = 113.3 kNm 
Top M_{sr} = (2.28 + 18.15) × 7.862 = 160.6 kNm 
M_{sR} = (2.28 + 20.25) × 7.842 = 176.7 kNm^{*} 
{4.3.7.1} Ultimate moment of resistance M_{ur} 
{3.3.6(7)} Ultimate moment of resistance M_{Rd} 
Refer to this book first ed. for full analysis 
Refer to Section 4.3.4 for full analysis 
f_{c} = 0.45 f_{cu} = 24.75 N/mm^{2} and λ = 0.9 
f_{cd} = 0.567 f_{ck} = 25.5 N/mm^{2} and λ = 0.8 
A_{ps} in tension zone = 870 mm^{2}; d = 164.3 mm 

Prestrain after losses ∊_{po} = 0.005015 
{5.10.9} ∊_{po} = 0.004946 
Refer to stress versus strain diagrams in the following 

Strain ∊_{p} = 0.012970; stress f_{p} = 1539 N/mm^{2} 
∊_{p} = 0.012700; stress f_{p} = 1,442 N/mm^{2} 
{4.3.7.3} X = 50.1 mm; z = 141.8 mm 
X = 51.2 mm; z = 143.8 mm 
M_{ur} = 870 × 1539 × 141.8 × 10^{–6} = 189.9 kNm 
M_{Rd} = 870 × 1442 × 143.8 × 10^{–6} = 180.5 kNm 
{4.3.8.1} Ultimate shear capacity V_{co} 
{6.2.2(2)} Ultimate shear capacity V_{Rd,c} 
{4.3.8.4} x = 100 + 99.4 = 199.4 mm 
{6.2.2(2)} l_{x} = 100 + 99.4 = 199.4 mm 
Mean diameter of strands = 10.9 mm 
{Exp. 8.16 and 8.18} l_{pt}_{2} = 1.2 × 0.19 × 979.5 × 
l_{p} = 10.9 × 240/√35 = 442 mm 
10.9/4.06 = 721 mm 

{Exp. 8.15} f_{pbt} = 3.2 × 1.0 × 1.27 = 4.06 N/mm^{2} 

{8.10.2.2} f_{ctd}(t) = 0.7 × 2.72/1.5 = 1.27 N/mm^{2} 
x/l_{p} = 199.4/442 = 0.451 
α_{l} = l_{x}/l_{pt2} = 199.4/721 = 0.276 
f_{cx} = 927,621/232,040 = 4.00 N/mm^{2} 
σ_{cp} = 0.9 × 929,076/232,040 = 3.60 N/mm^{2} 
f_{cpx} = 4.00 [0.451 × (2 − 0.451)] = 2.79 N/mm^{2} 
where {2.4.2.2(1)}γ_{p,fav} = 0.9 
f_{t} = 0.24√55 = 1.78 N/mm^{2} 
{3.1.6.2(P)} f_{ctd} =0.3 × 45^{2/3} × 0.7/1.5 = 1.77 N/mm^{2} 

first m.o.a. S_{xx} = 5.8326 × 10^{6} mm^{3} 
{Eq. 54} V_{co} = 0.67 × 1134 × 200 × 
{Exp. 6.4} V_{Rd,c} = (779 × 1,134/5.8326) × 
√(1.78^{2} + 0.8 × 2.79 × 1.78) = 406.1 kN 
√(1.77^{2} + 0.276 × 3.60 × 1.77) = 335.3 kN 
Comments

This code effectively replaces BS 8110, Parts 1 to 3, although the execution of work (tolerances, 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 NM 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 (subcommittee 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 199211, 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 19911; material properties; PSFs γ_{c} and γ_{s}, 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 nonlinear analysis; redistribution; secondorder effects with axial load (columns, walls); prestressing – stressing; forces; losses; service and ultimate; fatigue 
Section 6 
ULS – bending, shear, torsion and punching shear; strutandtie 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) soilstructure; (H) secondorder 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.
The main issues relating to the design of precast concrete structures in NA to BS EN 199211 are
2.4.2.2(1) 
Partial factor for prestress at ULS γ_{P,fav} = 0.9. 
3.1.2(2)P 
Value of C_{max}. Shear strength of concrete classes higher than C50/60 should be determined by tests, etc. 
3.1.6(1)P 
Value of α_{cc} = 0.85 for compression in flexure and axial loading and 1.0 for other phenomena, that is in bending f_{cd} = 0.85 f_{ck}/1.5 = 0.567 f_{ck} but in shear f_{cd} = 0.667 f_{ck}. 
4.4.1.3(3) 
Δc_{dev} under controlled conditions, such as steel mounts known as soldiers in front of hollow core slabs machines, may be reduced to 10 mm > Δc_{dev} > 5 mm. 
5.1.3(1)P 
Simplified load arrangements. Consider the two following arrangements for ‘all spans’ and alternate spans: (i) all spans carrying γ_{G}G_{k} + γ_{Q}Q_{k} + P_{k}; and (ii) alternate spans carrying γ_{G}G_{k} + γ_{Q}Q_{k} + P_{k}, other spans carrying only γ_{G}G_{k} + P_{k}; the same value of γ_{G} should be used throughout the structure. For oneway spanning slabs, use the ‘all spans’ loaded if (i) area of each bay > 30 m^{2}; (ii) Q_{k}/G_{k} ≤ 1.25; and (iii) Q_{k} < 5 kN/m^{2} excluding partitions. 
5.5(4) 
Moment redistribution formula: values for steels with f_{yk} ≤ 500 N/mm^{2}, k_{1} = 0.4, k_{2} = 0.6 + 0.0014/∊_{cu2}. Then if ∊_{cu2} = 0.0035, k_{2} = 1. Code Exp. 5.10a for f_{ck} ≤ 50 N/mm^{2}, δ = k_{1} + k_{2} x_{u}/d for zero moment redistribution δ = 1 = 0.4 + x_{u}/d, then ≤ 0.6, that is for the balanced section, the limiting depth of the neutral axis is 0.6d. 
5.10.9(1)P 
For pretensioning, r_{sup} = 1.0 and r_{inf} = 1.0, that is there are no modifications to the action of prestress. 
6.2.3(3) 
Values of ν_{1} = ν unless the design stress of the shear reinforcement < 0.8 f_{yk}, ν_{1} is modified. 
7.2 
The different limits of compressive stress in service depending on durability requirements and the avoidance of nonlinear creep in prestressed sections in flexure. 
7.3.1(5) 
Limitations of crack width w_{max}. Use Table National Annex NA4. This reduces w_{max} in r.c. sections to 0.3 mm and, in prestressed sections, the limiting permissible tension in service to zero for exposure class greater than XC1, although the value of the imposed live load may be reduced. 
7.4.2(2) 
Values of basic span/depth ratios. Use Table NA5 which gives additional information and limits. 
8.3(2) 
Minimum mandrel diameter Φ_{m,min}. Use in Table NA6a and Table NA6b, which contain additional information regarding scheduling reinforcement. 
8.8(1) 
Additional rules for large diameter bars Φ_{large} > 40 mm. 
9.5.2(1) 
Minimum diameter of longitudinal reinforcement in columns Φ_{min} = 10 mm. 
9.5.2(3) 
Maximum area of longitudinal reinforcement in columns. The designer should consider the practical upper limit taking into account the ability to place the concrete around the rebar, that is when casting columns horizontally mould, the maximum area is often around 8 to 10% of the area of concrete. This issue is considered further in the PD 66871:2010 (PD 66871 2010). 
9.7(1) 
Minimum area of distribution reinforcement in deep beams = 0.2 % in each face. 
9.10.2.2(2) 
Force to be resisted by peripheral tie. q_{1} = (20 + 4n_{0}) where n_{0} is the number of storeys; q_{2} = 60 kN. 
9.10.2.3(3) 
Minimum tensile force internal tie. F_{tie,int} = [(q_{k} + g_{k})/7.5](l_{r}/5)(F_{t}) ≥ F_{t} kN/m, with full definitions. Maximum spacing of internal ties = 1.5l_{r}. 
9.10.2.3(4) 
Internal ties on floors without screed. F_{tie} = [(g_{k} + q_{k})/7.5](l_{r}/5)F_{t} ≥ F_{t} kN/m. 
9.10.2.4(2) 
Horizontal ties to external columns and/or walls at each floor level: F_{tie,col} = the greater of 2F_{t} ≤ l_{s}/2.5F_{t} and 3% N_{Ed} at that level. 
12.3.1(1) 
Values of α_{cc,pl} = 0.6 and α_{ct,pl} = 0.6 (plain concrete). 
Annex J 
Design of corbels use PD 66871:2010. 
This code provides the following two alternatives for designing r.c. and prestressed concrete elements and structures for the actions of fire:
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 consequences of 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.
Section 1 
Scope – references; assumptions; definitions; symbols 
Section 2 
Basis of design – requirements; actions; material properties; verification methods 
Section 3 
Material properties – at elevated temperatures; concrete with siliceous and calcareous aggregates; thermal elongation of bars and tendons 
Section 4 
Design procedures – simplified and advanced calculation methods; shear, torsion and anchorage; spalling; joints; protective layers 
Section 5 
Tabulated data – columns, walls, beams and slabs; fire thickness and axis distance to bars 
Section 6 
Highstrength concrete – calculation models and tabulated data for columns, walls, beams and slabs 
Informative annexes 
(A) Temperature profiles; (B) Simplified calculation methods; (C) Buckling of columns under fire conditions; (D) Calculations for shear, torsion and anchorage; (E) Simplified calculations for beams and slabs 
Section 1 Scope – references; assumptions; definitions; symbols.
The main issues relating to the design of precast concrete structures in NA to BS EN 199212 are
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 11 and 12 and their UK NAs.
The main items in the PD relating to the design of precast concrete structures are
2.5 
Bond stress for mild steel cl. 8.4.2(2). The PD gives f_{bd} = η_{1}η_{2} (0,36√f_{ck})/γ_{c}. 
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, λ_{lim}. 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 λ_{lim} (i.e. value of moment ratio r_{m} < 0). 
2.12 
Design moment in columns, cl. 5.8.7.3 and 5.8.8.2. For braced structures, M_{Ed} = maximum of (M_{0e} + M_{2}), (M_{02}) or (M_{01}+ 0.5 M_{2}). 
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 V_{Ed} between the point load and the support is V_{Ed} = V_{Ed}_{,other} + β V_{Ed}_{,pointload}, and therefore the reduction factor β cannot be applied to the total V_{Ed}. 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 f_{yk} and tendons ≤ 0.75 f_{p}_{k}) 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 h_{c,eff} from Fig. 6 of the PD are proposed. It is unsure how Fig. 6a is interpreted. 
2.22 
Crack widths for nonrectangular tension zones and irregular bar layouts. Based on BS 8110, the recommendation is w_{k} = 3a_{cr} ∊_{m/}[1+ 2(a_{cr} – c)/(h – x)]. 
2.23.2 
Span/depth ratio. Exp. 7.17 in cl. 7.4.2(2). Values of (A_{s,prov}/A_{s,req}) or (310/σ_{s}) 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 σ_{s} or M_{s}, should be based on the frequent (not quasipermanent) combination of loading. 
2.26.1 
Vertical ties, cl. 9.10.2. For notes that vertical ties are required in framed as well as loadbearing structures, see Chapter 11 of this book. 
2.26.2 
Anchorage of precast floor and roof units and stair members. BS EN 199211:2004 or BS EN 199117 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 199211:2004, 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 199211:2004 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 199212:2004, 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 199212:2004 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 199212:2004, Annex B. 