Part E Worked examples
E13 DESIGN OF A CONCRETE STEM WALL
E13.1 Data
Figure E13.1 shows a concrete stem wall supporting sloping ground. The characteristic values of the soil parameters are shown in the figure, together with some fixed dimensions. The design is to find the required length of the heel of the wall, and hence the footing width, B, together with the bending moments and shear forces in the wall for structural design. Both ultimate and serviceability limit states must be considered.
E13.2 Ultimate limit states
For ultimate limit state design, the steps of the calculation are shown in Table E13.1. Both Cases B and C must be considered. Initially, it is assumed that active earth pressures may be used for the design, and the implications of this are reviewed later.
Table E13.1 Calculations for concrete stem wall
Case C1 Case C2 Case B1 Case B2
Characteristic φ′(°) 32.5 32.5 32.5 32.5
Factor on tanφ′ 1.25 1.25 1.0 1.0
Design φ′(°) 27.0 27.0 32.5 32.5
γsoil(kN/m3) 19 19 19 19
γcct(kN/m3) 25 25 25 25
Characteristic surcharge p (kPa) 5 5 5 5
Factor on surcharge 1.3 1.3 1.1[b] 1.1[b]
Design surcharge p (kPa) 6.5[a] 6.5[a] 5.5[a] 5.5[a]
δ/ φ′ active 2/3 2/3
passive 2/3 2/3 2/3 2/3
base 1 1 1 1
Design δ° active 20 (= β)[c] 18 20 (= β)[c] 21.3
passive 18 18 21.3 21.3
base 27 27 32.5 32.5
Characteristic Ka 0.35 0.35
Factor on Ka 1.35 1.35
Design Ka[d] 0.48 0.49 0.47 0.47
Design Kp[d] 4.1 4.1 6.0 6.0
Design Nq[e] 13.2 24.6
Design Nc[e] 24.0 37.0
Design Nγ[e] 12.4 30.1
B (m) 5.3 ~ 4.3
tan–1(H / V) for base (°)[f] 22.7 22.2
< 27 √ < 32.5 √
Max bending moment in wall (kN m/m) 396 372
Max bending moment in toe (kN m/m) 79 78
Max bending moment in heel (kN m/m) 365 336
[a] The partial factor on surcharge, taken to be a variable load, is zero when it is beneficial (ie beneficial surcharge is ignored). Cases C1 and B1 consider the overall stability of the wall, so no surcharge is assumed between the wall and the virtual back. Cases C2 and B2 consider the bending moments in the wall, for which surcharge between the wall and virtual back is adverse and so is included
[b] Because the effect of the surcharge will later be factored by 1.35, the input value of this variable action is here factored by 1.5 / 1.35 = 1.1
[c] On the virtual back, δis set equal to the slope angle β; this is consistent with 8.5.2(2), though this paragraph refers to at rest states. Elsewhere, δis limited to 2/3φ′d, in accordance with 8.5.1(4) for ‘precast’ concrete (which really means concrete not cast directly against the soil). The use of δ= βis noted by Clayton et al (1993, p163)
[d] Values taken from Figures G2 and G3 in Annex G [e] Values calculated using Annex B
[f] H and V are the horizontal and vertical components of the force between the base and the ground
The surcharge is a variable action, which may occur, or be absent, randomly at any point. It is therefore necessary to consider two load cases, with and without a surcharge in the area between the wall and the ‘virtual back’. In checking the overall
equilibrium of the system, the surcharge in this area would be favourable (Cases C1 and B1), and so has an applied partial factor of 0.0, ie it is omitted. However, in checking the strength of the wall, the surcharge is unfavourable, and so is included (Cases C2 and B2). Surcharge beyond the virtual back is always unfavourable and so is always included.
It is reasonable to expect that Case C will govern the dimensioning of the heel so this case is checked first, giving a base width of 5.3 m. During the calculation for Case B, a rough calculation is performed to check that the dimension derived for Case C is adequate; a base width of about 4.3 m would be sufficient for Case B.
PART E WORKED EXAMPLES
155
β = 20°
Figure E13.1 Concrete stem wall supporting sloping ground
Figure 13.2 Results for Case C 1
Each column in the table represents a continuous calculation. Partial factors are applied to the soil strength (tanφ′in this case) and to the surcharge load p.
Values of the coefficients of earth pressure, K, and of the bearing capacity factors, N, are then derived directly from EC7 Annexes G and B, using the design values of φ′.
The base width B is required to provide equilibrium between the lateral earth pressures and available bearing resistance. This may be achieved by hand calculation or using suitable software; iteration may be required to find the minimum base width which provides equilibrium. In this case, the Oasys program GRETA was used for the wall equilibrium calculations. The shear on the base, represented by the ratio of forces H / V must be checked to be less than the available base friction, δ.
The earth pressures for the critical Case C1 are shown in Figure E13.2. On the passive side of the wall, the ground level shown in Figure E13.1 should be the worst that is foreseeable, allowing for excavation of service trenches, etc.
Even then, the upper 0.5 m of passive soil is neglected (EC7, 8.3.2.1(2));
although the remaining passive resistance is included, it plays a minor role in the calculation.
The calculations for Case B are applied to the same wall geometry as Case C, ie that which will finally be built. The method of applying partial
factors in Case B to designs of this type is slightly uncertain. The approach adopted here is to increase the coefficient of active earth pressure by a factor of 1.35. This is considered to be consistent with 2.4.2(17), at least in spirit. Because a larger factor of 1.5 should be applied to the surcharge (a variable action), the surcharge is multiplied at source by 1.1 (= 1.5 / 1.35), in the
knowledge that the earth pressures it causes will be factored by 1.35. The vertical earth pressures and passive earth pressure have not been increased by 1.35. Such a factor seems unreasonable; although it would affect parts of the Case B calculation, it would have very little effect on the final design.
Bending moments from Case C2 are shown in Figure E13.3. Following 6.8(2), the bearing pressure is assumed to be distributed linearly beneath the base for calculation of bending moments in the base. For this problem, there is little difference between the results of the two cases, Case C proving to be critical. These bending moments may be taken directly into Eurocode 2 as ultimate limit state design values; no further factors are applied to them. More typically, for level ground supported behind the wall, Case B normally gives the more severe structural action effects, especially if water pressure is present.
Figure E13.3 Results for Case C2
E13.3 Serviceability limit states
Serviceability considerations apply to the displacement of the structure and surrounding ground, and to the performance of the concrete, especially with regard to cracking.
EC7, 8.7.2 notes that it is often possible to avoid detailed analysis of displacement by noting comparable examples. It would not normally be necessary to calculate the displacement of a wall of this type on a sand foundation. If it were, the methods of calculating settlements of footings, noted in E5, could be adopted. For economy of effort, the loads on the footing could be taken from the ULS Case B calculation, but if these lead to a marginal situation a more accurate calculation of service loads should be undertaken.
EC7, 8.5.1(6) notes that the earth pressures for ultimate and serviceability limit states are, in principle, derived from different calculations. EC7, 8.5.4 considers the relationship of earth pressure to movement, and 8.5.5 considers compaction effects. In considering ‘Structural serviceability limit states’, subclause 8.7.4 notes that all these factors are relevant and that design earth pressures for serviceability limit states will not necessarily be limiting values.
EC7 is unable to be more specific because earth pressures existing in the service state are very dependent on individual circumstances.
A minimum earth pressure sometimes used for the serviceability check of this type of wall may be obtained using a coefficient of earth pressure, KSLS, given by:
KSLS= 1/2(Ka+ Knc)
where Kais the coefficient of active pressure and Knc= (1 – sinφ′). For sloping ground, Kncwould logically be replaced by Knc(1 + sinβ), as in 8.5.2(2). Where it is known that heavy compaction will be used, a larger value should be adopted, however. (Currently, for design of backfilled retaining walls and bridge abutments on the trunk road network in the United Kingdom, BD30/87 has much more severe requirements – see Carder (1998).)
For this calculation, characteristic values of φ′should be used, so:
KSLS= 1/2(Ka+ Knc,k).
Typically, if δ= 2/3φ′is used in deriving Ka, this gives a value for KSLS roughly equal to the value used for ULS design, which cannot be less than 1.35 Kak, used in Case B. So it is likely that serviceability will dominate the structural design. This situation may be compared with that of BS 8002, which proposes that both SLS and ULS design of the structural sections should be based on the same earth pressures. In this problem, they would be equivalent to about 1.25 Kak, slightly smaller than the value of KSLSsuggested here. Using either code, structural designers may prefer to carry out the ULS design for a larger bending moment in order to give simple SLS calculations, as discussed in C8.6.6.
PART E WORKED EXAMPLES