A closed-form solution for seismic passive earth pressure behind a retaining wall supporting cohesive–frictional backfill

Abstract

The evaluation of seismic passive earth pressure is an important aspect in designing safe retaining walls. In this paper, a slice analysis method is adopted to study the nonlinear distribution of seismic passive earth pressure while considering most of the possible parameters. The closed-form expressions for the resultant force of seismic passive earth pressure, earth pressure distribution, and its application position are obtained. The explicit solution for the critical failure angle of seismic passive earth pressure is proposed by simplifying the relation between the resultant force and the failure angle based on a graphical analysis method. Under certain conditions, the present method correlates with classical passive earth pressure theories. By comparing the present method with test results and previously published solutions, the results are found to be consistent. The influence of the seismic coefficients on seismic passive earth pressure is also studied.

Appendices

Appendix 1: Expressions for the symbols in text

$$k_{1} = \frac{{ - \cos \left( {\alpha + \theta - \varphi } \right)\left[ {\cos (\alpha + \beta )\sin (i - \theta ) - \cos (i - \beta )\sin (\alpha + \theta )} \right]}}{{\cos (\alpha + \beta )\left[ {\cos (\alpha + \theta - \varphi )\sin (\beta + \delta - i) + \cos (\delta + \theta - i)\sin (\alpha + \beta - \varphi )} \right]}}$$

(19)

$$k_{2} = - \frac{\cos (\alpha + \theta - \varphi )\cos (i - \beta ) + \sin (i - \theta )\sin (\beta + \alpha - \varphi )}{\cos (\alpha + \theta - \varphi )\sin (\beta + \delta - i) + \cos (\delta + \theta - i)\sin (\alpha + \beta - \varphi )}$$

(20)

$$k_{3} = \frac{\cos (i - \beta )\cos (\beta - \theta )\cos \varphi }{{\cos (\beta + \alpha )\left[ {\cos (\alpha + \theta - \varphi )\sin (\beta + \delta - i) + \cos (\delta + \theta - i)\sin (\alpha + \beta - \varphi )} \right]}}$$

(21)

$$l_{1} = 1 - k_{1} \frac{\cos (\alpha + \beta )\sin (\alpha + i - \varphi - \delta )}{\cos (\alpha + \theta - \varphi )\sin (i + \alpha )}$$

(22)

$$l_{2} = \frac{{2 \times \left[ {\cos (\alpha + \beta )\sin (\alpha + \beta - \varphi )\cos \delta \cdot c_{\text{w}} + \sin (\delta + \beta - i)\cos (i - \beta )\cos \varphi \cdot c} \right]}}{{\sin (i + \alpha )\left[ {\cos (\alpha + \theta - \varphi )\sin (\beta + \delta - i) + \cos (\delta + \theta - i)\sin (\alpha + \beta - \varphi )} \right]}}$$

(23)

$$\begin{aligned} T_{1} & = \frac{\cos (i - \delta )}{{\cos i\cos \theta \cos^{2} (\beta + \delta + \varphi - i)}} \\ & \quad \times\, \left[ \begin{array}{c} \frac{1}{2}\gamma H(1 - k_{\text{v}} )\cos (i - \beta )\sin (\varphi + \beta - \theta ) \hfill \\ +\,q_{0} (1 - k_{\text{v}} )\cos i\sin (\varphi + \beta - \theta ) + c \cdot \cos \theta \cos \varphi \cos i \hfill \\ \end{array} \right] \\ \end{aligned}$$

(24)

$$\begin{aligned} T_{2} & = \frac{{\cos (i - \beta )H^{2} }}{{\cos^{3} i\cos \theta \cos (i - \delta )\cos^{2} (\beta + \varphi + \delta - i)}} \\ & \quad \times\, \left[ \begin{array}{c} \frac{1}{2}\gamma H(1 - k_{\text{v}} )\cos (i - \beta )\sin (\delta + \varphi )\cos (\delta + \theta - i) + q_{0} (1 - k_{\text{v}} )\cos i\sin (\delta + \varphi )\cos (\delta + \theta - i) \hfill \\ +\,c \cdot \cos \theta \cos \varphi \cos i\cos (i - \beta ) + c_{\text{w}} \cdot \cos i\cos \theta \cos \delta \cos (\beta + \delta + \varphi - i) \hfill \\ \end{array} \right] \\ \end{aligned}$$

(25)

$$\begin{aligned} T_{3} & = \frac{H}{{\cos^{2} i\cos \theta \cos^{2} (\beta + \varphi + \delta - i)}} \\ & \quad \times\, \left\{ \begin{array}{c} \left[ {\frac{1}{2}\gamma H(1 - k_{\text{v}} )\cos (i - \beta ) + q_{0} (1 - k_{\text{v}} )\cos i} \right] \hfill \\ \times\, \left[ {\cos (\delta + \theta - i)\cos (i - \beta ) + \sin (\varphi + \delta )\sin (\varphi + \beta - \theta )} \right] \hfill \\ + 2c \cdot \cos i\cos \theta \cos \varphi \cos (i - \beta )\sin (\beta + \varphi + \delta - i) \hfill \\ + c_{\text{w}} \cdot \cos i\cos \theta \sin (\beta + \varphi - i)\cos (\beta + \varphi + \delta - i) \hfill \\ \end{array} \right\} \\ \end{aligned}$$

(26)

Appendix 2: Transformation on the relation between resultant force and failure angle

According to Fig. 3, $\overline{AB}$, $\overline{BF}$, $\overline{AF}$, $\overline{BD}$, $\overline{AD}$, and $\overline{DF}$ do not change when the point C changes, and they are independent of the failure angle α. Based on the sine theorem in ΔABD, ΔABF, and ΔADF, they can be expressed as follows:

$$\left. \begin{array}{l} \overline{AD} = \frac{\cos (i + \varphi )H}{\cos i\sin (\varphi + \beta )},\quad \overline{BD} = \frac{\cos (i - \beta )H}{\cos i\sin (\varphi + \beta )},\quad \overline{AF} = \frac{\cos (i + \varphi )H}{\cos i\cos (i - \delta )}, \hfill \\ \overline{BF} = \frac{\sin (\delta + \varphi )H}{\cos i\cos (i - \delta )},\quad \overline{DF} = \frac{\cos (i + \varphi )\cos (\beta + \varphi + \delta - i)H}{\cos i\cos (i - \delta )\sin (\varphi + \beta )} \hfill \\ \end{array} \right\}$$

(27)

Based on the sine theorem in ΔABC and ΔBCG:

$$\left. \begin{array}{l} \sin (\alpha + i) = \frac{{\overline{AC} }}{{\overline{BC} }}\cos (i - \beta ),\quad\cos (\alpha + \beta ) = \frac{{\overline{AB} }}{{\overline{BC} }}\cos (i - \beta ) \hfill \\ \cos (\alpha - \varphi ) = \frac{{\overline{CG} }}{{\overline{BC} }}\cos (i - \delta ),\quad\sin (i + \alpha - \varphi - \delta ) = \frac{{\overline{BG} }}{{\overline{BC} }}\cos (i - \delta ) \hfill \\ \end{array} \right\}$$

(28)

Based on the cosine theorem in ΔBCG:

$$\overline{BC}^{2} = \overline{BG}^{2} + \overline{CG}^{2} - 2\overline{BG} \cdot \overline{CG} \cos (90^\circ - i + \delta )$$

(29)

Consequently, the following equation can be derived:

$$\sin (\alpha - \varphi ) = \sqrt {1 - \cos^{2} (\alpha - \varphi )} = \frac{{\overline{BG} - \overline{CG} \cdot \sin (i - \delta )}}{{\overline{BC} }}$$

(30)

According to ΔCDG–ΔADF:

$$\left. {\overline{CG} = \frac{{\overline{AF} }}{{\overline{DF} }}(\overline{BD} + \overline{BG} ),\quad \overline{AC} = \frac{{\overline{AD} }}{{\overline{DF} }}(\overline{BG} + \overline{BF} )} \right\}$$

(31)

In addition, by substituting Eqs. (19)–(23) into Eq. (11):

$$\begin{aligned} E_{\text{p}} & = \frac{1}{2}\gamma H^{2} \frac{{1 - k_{\text{v}} }}{\cos \theta } \cdot \frac{{\cos (i - \beta )\sin (i + \alpha )\left[ {\cos (\alpha - \varphi )\cos \theta - \sin (\alpha - \varphi )\sin \theta } \right]}}{{\cos^{2} i\cos (\alpha + \beta )\sin (\alpha + i - \varphi - \delta )}} \\ & \quad +\,q_{0} H\frac{{1 - k_{\text{v}} }}{\cos \theta } \cdot \frac{{\sin (i + \alpha )\left[ {\cos (\alpha - \varphi )\cos \theta - \sin (\alpha - \varphi )\sin \theta } \right]}}{\cos i\cos (\alpha + \beta )\sin (\alpha + i - \varphi - \delta )} \\ & \quad +\, cH \cdot \frac{\cos (i - \beta )\cos \varphi }{\cos i\cos (\alpha + \beta )\sin (\alpha + i - \varphi - \delta )} \\ & \quad +\, c_{\text{w}} H \cdot \frac{\cos (\alpha - \varphi )\cos i - \sin (\alpha - \varphi )\sin i}{\cos i\sin (\alpha + i - \varphi - \delta )} \\ \end{aligned}$$

(32)

By substituting Eqs. (27)–(31) into Eq. (32), the relation between the resultant force and the failure angle can be simplified, as shown in Eq. (13).

Appendix 3: Special cases

3.1 Case of Mononobe–Okabe theory

When c = 0, c _w = 0, and q ₀ = 0, Eq. (16) can be rewritten as:

$$\begin{aligned} E_{\text{p}} & = 2\sqrt {T_{1} \cdot T_{2} } + T_{3} \\ & = \frac{1}{2}\gamma H^{2} \frac{{\cos (i - \beta )(1 - k_{\text{v}} )}}{{\cos^{2} i\cos \theta \cos^{2} (\beta + \varphi + \delta - i)}} \\ & \quad \times\, \left[ \begin{aligned} \cos (\delta + \theta - i)\cos (i - \beta ) + \sin (\varphi + \delta )\sin (\varphi + \beta - \theta ) \hfill \\ + 2\sqrt {\cos (\delta + \theta - i)\cos (i - \beta )\sin (\varphi + \delta )\sin (\varphi + \beta - \theta )} \hfill \\ \end{aligned} \right] \\ & = \frac{1}{2}\gamma H^{2} \frac{{\cos (i - \beta )(1 - k_{\text{v}} )}}{{\cos^{2} i\cos \theta \cos^{2} (\beta + \varphi + \delta - i)}} \\ & \quad \times\, \left[ {\frac{\cos (\varphi + \delta + \beta - i)\cos (i + \varphi - \theta )}{{\sqrt {\cos (\delta + \theta - i)\cos (i - \beta )} - \sqrt {\sin (\varphi + \delta )\sin (\varphi + \beta - \theta )} }}} \right]^{2} \\ & = \frac{1}{2}\gamma H^{2} \frac{{(1 - k_{\text{v}} )\cos^{2} (i + \varphi - \theta )}}{{\cos^{2} i\cos \theta \cos (\delta + \theta - i)\left[ {1 - \sqrt {\frac{\sin (\varphi + \delta )\sin (\varphi + \beta - \theta )}{\cos (\delta + \theta - i)\cos (i - \beta )}} } \right]^{2} }} \\ \end{aligned}$$

(33)

Equation (33) is the same as the Mononobe–Okabe passive earth pressure.

3.2 Case of Rankine theory

When i = 0, β = 0, δ = 0, c _w = 0, q ₀ = 0, k _h = k _v = 0, and θ = 0, Eq. (16) can be rewritten as:

$$\begin{aligned} E_{\text{p}} & = 2\sqrt {T_{1} \cdot T_{2} } + T_{3} = \frac{1}{2}\gamma H^{2} \left( {\frac{1 + \sin \varphi }{\cos \varphi }} \right)^{2} + 2cH\left( {\frac{1 + \sin \varphi }{\cos \varphi }} \right) \\ & = \frac{1}{2}\gamma H^{2} \tan^{2} \left( {45^\circ + \frac{\varphi }{2}} \right) + 2cH\tan \left( {45^\circ + \frac{\varphi }{2}} \right) \\ \end{aligned}$$

(34)

and Eq. (34) is the same as the Rankine passive earth pressure.

Besides, $\overline{BG}$ can be solved as $\overline{BG} = H$ according to Eq. (15). Thus, the expression for the critical failure angle (Eq. 18) can be simplified as:

$$\alpha_{\text{cr}} = \tan^{ - 1} \left( {\frac{1 + \sin \varphi }{\cos \varphi }} \right) = 45^\circ + \frac{\varphi }{2}$$

(35)

Equation (35) is known as the critical failure angle of passive soil wedge in Rankine theory.