Design example of a sheet pile retaining wall using the global factor of safety and the Eurocode 7 approaches. Problem: A cantilever sheet pile wall supporting a 3.5m deep excavation in dry sand (dry= 15.3kNm-3, = 35, c= 0) is to be constructed. Determine the required depth of embedment 1. By using the global (overall) factor of safety method. 2. By using the Eurocode method. For both methods the same mechanism of failure is assumed (see the diagram below). The sheet pile rotates about a point O above the end of the sheet piling. Above point O an active stress state is assumed on the unexcavated side, and passive on the other. The stress state above point O is analysed with the forces below O being represented by a horizontal force R acting through O. The sheet pile wall above O is analysed by taking moments about O, hence R can be ignored. This enables the depth from the excavated surface to O (d0) to be found and then the full embedment depth is d0x1.2. Next the embedment depth can be checked to ensure that sufficient forces can be mobilised below O to ensure equilibrium. To carry out the analysis the first stage is to find the horizontal stress profiles, then the horizontal forces, then the moments to find d0. In the examples below the final horizontal force check is not undertaken but generally would be carried out. 1. Global Factor of Safety Approach.
de
d0
1 2 R O
Area 1 is active; area 2 is passive. Assume the material parameters given are typical design values. de = 3.5m. Ka =(1 - sin)/(1+sin) = (1-sin35)/(1+sin35) = 0.271 Kp = 1/Ka = 3.690
Horizontal active effective stress at O h = Ka.dry.z = (3.5 + d0)x15.3x0.271 Horizontal passive effective stress at O h = Kp.dry.z = d0x15.3x3.690 Area
Force
Lever arm above O
1 2
0.5x(3.5 + d0)2x15.3x0.271 0.5x d0 2x15.3x3.690/1.5*
(3.5 + d0)/3 d0/3
*Note: the 1.5 in the force for area 2 is the applied factor of safety used in this method of analysis. Equate moments about O Active moment = Passive moment (0.5x(3.5 + d0)2x15.3x0.271)x((3.5 + d0)/3) = (0.5x d0 2x15.3x3.690/1.5)x(d0/3) Hence d0 = 3.22m Therefore depth of embedment = d = 1.2x3.22 = 3.86m Length of sheet piling = 3.86 + 3.50 = 7.36m 2. Eurocode 7 Approach. Use Design Approach 1 and Combination 1 and 2 Combination 1 Combination 2
A1 + M1 + R1 A2 + M2 + R1
For the EC7 method it is usual to assume an over-excavation of 10% of the excavated depth to a maximum of 0.5m. Partial factors: Permanent Action, G,dst
A1 1.35
A2 1.0
tan, (unit weight),
M1 1.0 1.0
M2 1.25 1.0
Re(earth resistance), Re
R1 1.0
Assume the material parameters given are characteristic values (k).
Assume excavation depth = 3.5 + 0.1x3.5 = 3.85m Combination 1. des = tan-1((tank)/ ) = tan-1((tan35)/1) = 35 dry,des = dry,k/ = 15.3kNm-3 Therefore Ka and Kp as before. General stress distribution as above, but the values will change as the excavation depth is different and the partial factors are used: Horizontal active effective stress at O h = Ka.z.dry = (3.85 + d0)x15.3x0.271 Horizontal passive effective stress at O h = Kp.z.dry = d0x15.3x3.690 Area
Force
Lever arm above O
1 2
0.5x(3.85 + d0)2x15.3x0.271x G,dst 0.5x d0 2x15.3x3.690/Re
(3.85 + d0)/3 d0/3
Equate moments about O Passive moment = Active moment (0.5x(3.85 + d0)2x15.3x0.271 x1.35)x((3.85 + d0)/3) = (0.5x d0 2x15.3x3.690/1.0)x(d0/3) /1 Hence d0 = 3.32m Therefore depth of embedment = d = 1.2x3.32 = 3.98m Length of sheet piling = 3.98 + 3.85 = 7.83m Combination 2. des = tan-1((tank)/ = tan-1((tan35)/1.25) = 29 dry,des = dry,k/ = 15.3kNm-3 Therefore Ka = 0.347 and Kp = 2.882. General stress distribution as above, but the values will change as the excavation depth is different and the partial factors are used: Horizontal active effective stress at O h = Ka.z.dry = (3.85 + d0)x15.3x0.347
Horizontal passive effective stress at O h = Kp.z.dry = d0x15.3x2.882 Area
Force
Lever arm above O
1 2
0.5x(3.85 + d0)2x15.3x0.347x G,dst 0.5x d0 2x15.3x2.882/Re
(3.85 + d0)/3 d0/3
Equate moments about O Passive moment = Active moment (0.5x(3.85 + d0)2x15.3x0.347 x1)x((3.85 + d0)/3) = (0.5x d0 2x15.3x2.882/1.0)x(d0/3) /1 Hence d0 = 3.76m Therefore depth of embedment = d = 1.2x3.76 = 4.51m Length of sheet piling = 4.51 + 3.85 = 8.36m Therefore, by the EC7 method the critical length is 8.36m. _____________________________________________________________________ Notes: For anchored/propped sheet pile walls similar methods are used and the lateral force check is undertaken to give the tie/prop force. In the case above with a dry sand the stress distributions are simple. In the case of layered soils, water tables, surface surcharges etc. the same general method is used but the stress profile will be more complicated. If the water level is the same on each side of the wall, then the pore water pressure will need to be considered in calculating the effective stresses, but the effect of the water force moments can be ignored as they are assumed to cancel out. The method used for the cantilever wall analysis above is a simple one and an alternative method is to consider moment and horizontal equilibrium simultaneously with two unknowns, i.e. the depth from the excavated surface to the point of rotation and the depth from the point of rotation to the base of the pile. The resulting cubic equations can be solved for the two unknowns.
