# General Solution for Active Earth Pressure on Rigid Walls Under Strip Surcharge on Retained Soils Using Variational Method

## Abstract

Hypothesis of the slip surface shape in retained soils makes most of the existing methods for active earth pressure on rigid walls be of no logical basis. The present work aims to rigorously calculate the earth pressure under general conditions. Based on the limit equilibrium conditions of a soil mass retained by rigid walls under a distanced backfill strip surcharge, a variational calculus method was developed to compute the active earth pressure on the walls. A functional relationship between the earth thrust and critical slip surface of the retained soil, and the normal stress acting on it without considering the hypothesis of the slip surface shape, were established. A unified closed-form solution for the earth thrust was derived and could be obtained using an implicit solution technique via MATLAB. The analysis model is of high generality, and the effects of 11 basic parameters on the earth thrust and the corresponding slip surface were analyzed. Some of the considered parameters were the soil properties, soil–wall interface friction angle, wall height, strip surcharge, dip angle between the wall back and soil top surface, horizontal net distance from the wall back to the surcharge, and distribution width of the surcharge. The examples showed that the earth thrusts calculated using the proposed method were approximately 4–16% higher than Coulomb’s solutions but 3–11% lower than traditional stress distribution solutions based on the elastic theory. The proposed method in which the critical slip surface shape need not be subjectively assumed has comparatively more extensive applicability.

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## Abbreviations

a :

Horizontal net distance from the wall back to the strip surcharge

b :

Distribution width of the strip surcharge on the top surface of the retained soil

b e :

Distribution width of the surcharge on a potential slide mass of the retained soil

c :

Cohesion of the soil

E a :

Active earth pressure on the retaining wall

f(x):

Function of the potential slip surface OB

g(x):

Function of the top surface AB of the retained soil

H :

Height of the wall

$$\bar{H}$$ :

Resultant force on the potential slide mass of the retained soil in the horizontal direction

K 1, K 2 :

Integral constant

l :

Curve length of the potential slip surface OB

m :

Intermediate variable

$$\bar{M}$$ :

Resultant moment on the potential slide mass of the retained soil on the plane

q :

Surcharge on the top surface of the retained soil

r :

Radius with respect to pole Oc of the polar coordinate system

$$\bar{V}$$ :

Resultant force acting on the potential slide mass of the retained soil in the vertical direction

x :

Horizontal coordinates with respect to origin O of the rectangular coordinate system, where subscripts B and c denote points B and Oc, respectively

y :

Vertical coordinate with respect to origin O of the rectangular coordinate system, where subscript c denotes point Oc

y′:

First-order derivative of f(x)

z a :

Height of action point from the earth thrust to the retaining wall heel

α :

Dip angle of the wall back

β :

Dip angle of the top surface of the retained soil

γ :

Unit weight of soil

δ :

Soil–wall interface friction angle

θ :

Angular coordinate rotating counterclockwise around pole Oc of the polar coordinate system

η :

Dip angle of tangent at a point on the potential slip surface

λ 1, λ 2 :

Lagrange multiplier

ξ :

Ratio of za to H

σ :

Normal stress on potential slip surface OB

τ :

Shear stress on potential slip surface OB

ϕ :

Angle of internal friction of soil

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## Acknowledgements

This study was supported by the National Natural Science Foundation of China (Grant no. 51578466).

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Correspondence to Shiguo Xiao.

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Xiao, S., Yan, Y. & Xia, P. General Solution for Active Earth Pressure on Rigid Walls Under Strip Surcharge on Retained Soils Using Variational Method. Int J Civ Eng 19, 881–896 (2021). https://doi.org/10.1007/s40999-020-00579-4

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### Keywords

• Retaining walls
• Active earth pressure
• Variational calculus method
• Limit equilibrium
• Strip surcharge