# Seismic Passive Earth Pressure on Walls with Bilinear Backface Using Pseudo-Dynamic Approach

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## Abstract

By using pseudo-dynamic approach, a method has been proposed in this paper to compute the seismic passive earth pressure behind a rigid cantilever retaining wall with bilinear backface. The wall has sudden change in inclination along its depth and a planar failure surface has been considered behind the retaining wall. The effects of a wide range of parameters like soil friction angle, wall inclination, wall friction angle, amplification of vibration, variation of shear modulus and horizontal and vertical seismic accelerations on the passive earth pressure have been explored in the present study. For the sake of illustration, the computations have been exclusively carried out for constant wall friction through out the depth. Unlike the Mononobe-Okabe method, which incorporates pseudo-static analysis, the present analysis predicts a nonlinear variation of passive earth pressure along the wall.

### Keywords

Bilinear backface Cantilever retaining wall Earthquakes Passive earth pressure Pseudo-dynamic analysis### Abbreviations

*a*_{h}(*z*,*t*)Horizontal acceleration at depth

*z*and time*t**a*_{v}(*z*,*t*)Vertical acceleration at depth

*z*and time*t**f*_{a}Amplification factor

*G*Shear modulus of the backfill soil

*G*_{0}Constant

*g*Acceleration due to gravity

*H*Height of retaining wall

*H*_{1}Height of the upper part of wall backface

*K*_{pe1}Passive thrust coefficient relative to the thrust

*P*_{pe1}(t)*K*_{pe2}Passive thrust coefficient relative to the thrust

*P*_{pe2}(t)*m*(*z*)Mass of small shaded part of thickness d

*z*in wedge ABDE*m*_{1}(*z*)Mass of small shaded part d

*z*in wedge ABC*m*_{21}(*z*)Mass of small shaded part d

*z*in wedge ABDE for*z*varying from 0 to*H*_{1}*m*_{22}(*z*)Mass of small shaded part d

*z*in wedge ABDE for*z*varying from*H*_{1}to*H**P*_{pe1}(t)Passive thrust on the upper part of wall backface

*P*_{pe2}(t)Passive thrust on the lower part of wall backface

*p*_{pe}(*z*,*t*)Passive earth pressure behind the wall at depth

*z*and time*t**p*_{pe1}(*z*,*t*)Passive earth pressure on the upper part of wall at depth

*z*and time*t**p*_{pe2}(*z*,*t*)Passive earth pressure on the lower part of wall at depth

*z*and time*t**Q*_{h}(t)Horizontal inertia force in the wedge ABDE due to horizontal seismic acceleration

*Q*_{v}(t)Vertical inertia force in the wedge ABDE due to vertical seismic acceleration

*Q*_{h1}(t)Horizontal inertia force in the wedge ABC due to horizontal seismic acceleration

*Q*_{v1}(t)Vertical inertia force in the wedge ABC due to vertical seismic acceleration

*R*Thrust acting on the internal failure plane limiting the thrust wedge

*T*Period of lateral shaking

*t*Time

- Δ
*t*_{p} Time increment for passage of primary wave from the base to a depth

*z*- Δ
*t*_{s} Time increment for passage of shear wave from the base to a depth

*z**V*_{p}Primary wave velocity

*V*_{pavg}Mean primary wave velocity

*V*_{s}Shear wave velocity

*V*_{savg}Mean shear wave velocity

*W*Weight of the failure wedge ABDE behind the wall

*W*_{1}Weight of the failure wedge ABC behind the upper part of wall

*z*Depth of a generic point below the wall top

*α*_{1}Inclination angle of the failure plane BC limiting the local thrust wedge acting on the upper part of wall backface

*α*_{2}Inclination angle of the failure plane DE limiting the thrust wedge behind the wall

*α*_{h}Seismic acceleration coefficient in the horizontal direction

*α*_{v}Seismic acceleration coefficient in the vertical direction

*β*Depth exponent causing shear modulus variation

*δ*_{1}Friction angle between soil and wall along the upper part of wall backface

*δ*_{2}Friction angle between soil and wall along the lower part of wall backface

*ϕ*Friction angle of the backfill

*γ*Unit weight of the soil

*ν*Poisson’s ratio

*θ*_{1}Angle of inclination of the upper part of wall backface

*θ*_{2}Angle of inclination of the lower part of wall backface

*ρ*Density of the soil

*ω*Angular frequency of base shaking