Analysis and design charts for 3D active earth pressure in cohesive soils with cracks

Abstract

The prediction of active earth pressure was generally implemented under the assumptions of two-dimensional conditions and cohesionless soils. However, in practice, the soils usually display a considerable level of cohesion, and the collapse of retained slopes exhibits a three-dimensional (3D) nature. Considering this fact, this paper intends to predict the 3D active earth pressure in cohesive soils based on the kinematic limit-analysis method and a 3D rotational collapse mechanism. The influence of cracks is considered, including a crack forming before the failure of retained soil masses (open crack) and a crack forming simultaneously with the failure (formation crack). The active earth pressure coefficient is estimated based on the work-energy balance principle. In order to facilitate direct application, several design charts are provided. It is shown that accounting for soil cohesion and 3D effects results in a notable decrease in the active earth pressure, whereas considering the existence of cracks would increase the pressure value. This paper develops the studies on active earth pressure, which considers the presence of cohesion, cracks, and 3D effects together for the first time. The results of this paper can offer references in designs of retaining structures for cohesive slopes.

Appendices

Appendix 1: Detailed expressions used in the calculations

This appendix presents the detailed expressions that are used in the derivation of active earth pressure coefficient, i.e., internal energy dissipation rate $D$, work rate due to gravity $W_{\gamma }$, work rate due to the resultant of active earth pressure $W_{{{\mathrm{P}}_{a} }}$, and equivalent width $B_{{\mathrm{e}}}$. Note that only the final expressions are reported below. For the detailed derivation process, one can refer to several relevant studies (e.g., [10, 15, 16, 20, 21, 35]).

The internal dissipation rate ($D$) can be obtained as algebraic summation of the contributions along surfaces CE ($D_{{{\mathrm{CE}}}}$) and BC ($D_{{{\mathrm{BC}}}}$), and $D_{{{\mathrm{CE}}}}$ can be calculated as: $D_{{{\mathrm{AE}}}} - D_{{{\mathrm{AC}}}}$, where $D_{{{\mathrm{AE}}}}$ and $D_{{{\mathrm{AC}}}}$ are the dissipation rates along surfaces AE and AC, respectively. As a consequence, $D$ can be recast as:

$$D = D_{{{\mathrm{AE}}}} - D_{{{\mathrm{AC}}}} + D_{{{\mathrm{BC}}}}$$

(A1)

The mechanism can be divided into two parts: the 3D part and the inserted part. Therefore, the terms in the right-hand side of Eq. (A1) can be calculated as the sum of corresponding rates in the two parts. The detailed expressions for the three terms in the right-hand side of Eq. (A1) are reported as follows:

$$D_{{{\mathrm{AE}}}} = \omega cr_{0}^{3} \left( {g_{1} + g_{2} } \right)$$

(A2)

$$D_{{{\mathrm{AC}}}} = \omega cr_{0}^{3} \left( {g_{3} + g_{4} } \right)$$

(A3)

$$D_{{{\mathrm{BC}}}} = \omega cr_{0}^{3} \left( {g_{5} + g_{6} } \right)$$

(A4)

where $\omega$ = angular velocity of the mechanism, and $g_{1}$ − $g_{6}$ are dimensionless functions, with $g_{1}$, $g_{3}$, and $g_{5}$ for the 3D part, and $g_{2}$, $g_{4}$, and $g_{6}$ for the interested part, which take the following forms:

$$g_{1} = 2\int_{{\theta_{0} }}^{{\theta_{{\mathrm{h}}} }} {\left[ {\left[ {2f_{1} f_{2} + \frac{1}{2}f_{2} \left( {f_{3} - f_{1} } \right)} \right]\sqrt {f_{2}^{2} - \left( {f_{3} - f_{1} } \right)^{2} } + \left( {f_{1}^{2} f_{2} + \frac{1}{2}f_{2}^{3} } \right)\arccos \frac{{f_{3} - f_{1} }}{{f_{2} }}} \right]} {\mathrm{d}}\theta$$

(A5)

$$g_{2} = \frac{{B_{{{\mathrm{insert}}}} }}{{r_{0} }}\frac{{{\mathrm{e}}^{{2\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} - 1}}{2\tan \varphi }$$

(A6)

$$g_{3} = 2\int_{{\theta_{0} }}^{{\theta_{{\mathrm{C}}} }} {\left[ {\left[ {2f_{1} f_{2} + \frac{1}{2}f_{2} \left( {f_{4} - f_{1} } \right)} \right]\sqrt {f_{2}^{2} - \left( {f_{4} - f_{1} } \right)^{2} } + \left( {f_{1}^{2} f_{2} + \frac{1}{2}f_{2}^{3} } \right)\arccos \frac{{f_{4} - f_{1} }}{{f_{2} }}} \right]} {\mathrm{d}}\theta$$

(A7)

$$g_{4} = \frac{{B_{{{\mathrm{insert}}}} }}{{r_{0} }}\frac{{{\mathrm{e}}^{{2\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} - 1}}{2\tan \varphi }$$

(A8)

$$g_{5} = \left\{ \begin{gathered} 0,\;\;\;\;\;\;\;{\mathrm{for}}\;{\mathrm{pre - existing}}\;{\mathrm{cracks}} \hfill \\ 2\frac{\cos \varphi }{{1 - \sin \varphi }}\int_{{\theta_{{{\mathrm{B}}}} }}^{{\theta_{{{\mathrm{C}}}} }} {f_{4}^{2} \sqrt {f_{2}^{2} - \left( {f_{4} - f_{1} } \right)^{2} } \frac{1 - \sin \theta }{{\cos \theta }}} \;{\mathrm{d}}\theta ,\;\;\;{\mathrm{for}}\;{\mathrm{formation}}\;{\mathrm{cracks}}\;\;\; \hfill \\ \end{gathered} \right.$$

(A9)

$$g_{6} = \left\{ \begin{gathered} 0,\;\;\;\;\;\;\;{\mathrm{for}}\;{\mathrm{pre - existing}}\;{\mathrm{cracks}} \hfill \\ \frac{{B_{{{\mathrm{insert}}}} }}{{r_{0} }}\frac{\cos \varphi }{{1 - \sin \varphi }}\int_{{\theta_{{{\mathrm{B}}}} }}^{{\theta_{{{\mathrm{C}}}} }} {f_{4}^{2} \frac{1 - \sin \theta }{{\cos \theta }}} \;{\mathrm{d}}\theta ,\;\;\;{\mathrm{for}}\;{\mathrm{formation}}\;{\mathrm{cracks}}\;\;\; \hfill \\ \end{gathered} \right.$$

(A10)

where:

$$f_{1} = \frac{{r_{{\mathrm{m}}} }}{{r_{0} }} = \frac{1}{2}\left[ {{\mathrm{e}}^{{\left( {\theta - \theta_{0} } \right)\tan \varphi }} + \frac{{r^{\prime}_{0} }}{{r_{0} }}{\mathrm{e}}^{{ - \left( {\theta - \theta_{0} } \right)\tan \varphi }} } \right]$$

(A11)

$$f_{2} = \frac{R}{{r_{0} }} = \frac{1}{2}\left[ {{\mathrm{e}}^{{\left( {\theta - \theta_{0} } \right)\tan \varphi }} - \frac{{r^{\prime}_{0} }}{{r_{0} }}{\mathrm{e}}^{{ - \left( {\theta - \theta_{0} } \right)\tan \varphi }} } \right]$$

(A12)

$$f_{3} = \frac{{s_{1} }}{{r_{0} }} = \left\{ \begin{gathered} \frac{{\sin \theta_{0} }}{\sin \theta },\;\;\;\;\;\;\theta_{0} < \theta \le \theta_{{\mathrm{D}}} \hfill \\ \frac{{\sin \left( {\theta_{{\mathrm{h}}} + \beta } \right)}}{{\sin \left( {\theta + \beta } \right)}}{\mathrm{e}}^{{\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} ,\;\;\;\;\;\;\theta_{{\mathrm{D}}} < \theta \le \theta_{{\mathrm{h}}} \hfill \\ \end{gathered} \right.$$

(A13)

$$f_{4} = \frac{{s_{2} }}{{r_{0} }} = \left\{ \begin{gathered} \frac{{\sin \theta_{0} }}{\sin \theta },\;\;\;\;\;\;\theta_{0} < \theta \le \theta_{{\mathrm{B}}} \hfill \\ \frac{{\cos \theta_{C} }}{\cos \theta }{\mathrm{e}}^{{\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} ,\;\;\;\;\;\;\theta_{{\mathrm{B}}} < \theta \le \theta_{{\mathrm{C}}} \hfill \\ \end{gathered} \right.$$

(A14)

where $s_{1}$ = distance from the rotation center O to any generic point on surface AD or DE, and $s_{2}$ = distance from the rotation center O to any generic point on surface AB and BC.

Similarly, $W_{\gamma }$ can be derived as the gravity work rate of block A–D–E ($W_{{\gamma - {\mathrm{ADE}}}}$) minus that of block A–B–C ($W_{{\gamma - {\mathrm{ABC}}}}$), namely:

$$W_{\gamma } = W_{{\gamma - {\mathrm{ADE}}}} - W_{{\gamma - {\mathrm{ABC}}}}$$

(A15)

The two terms in the right-hand side of the above equation take the following forms:

$$W_{{\gamma - {\mathrm{ADE}}}} = \omega \gamma r_{0}^{4} \left( {g_{7} + g_{8} } \right)$$

(A16)

$$W_{{\gamma - {\mathrm{ABC}}}} = \omega \gamma r_{0}^{4} \left( {g_{9} + g_{10} } \right)$$

(A17)

where $g_{7}$—$g_{10}$ are dimensionless functions, with $g_{7}$ and $g_{9}$ for the 3D part, and $g_{8}$ and $g_{10}$ for the inserted part. Their detailed expressions are given by:

$$\begin{gathered} g_{7} = 2\int_{{\theta_{0} }}^{{\theta_{{\mathrm{h}}} }} {\left\{ {\left[ {\frac{1}{8}f_{2}^{2} \left( {f_{3} - f_{1} } \right) - \frac{1}{4}\left( {f_{3} - f_{1} } \right)^{3} - \frac{2}{3}f_{1} \left( {f_{3} - f_{1} } \right)^{2} - \frac{1}{2}f_{1}^{2} \left( {f_{3} - f_{1} } \right) + \frac{2}{3}f_{1} f_{2}^{2} } \right]} \right.} \sqrt {f_{2}^{2} - \left( {f_{3} - f_{1} } \right)^{2} } \\ \left. { + \left( {\frac{1}{8}f_{2}^{4} + \frac{1}{2}f_{1}^{2} f_{2}^{2} } \right)\arccos \frac{{f_{3} - f_{1} }}{{f_{2} }}} \right\}\cos \theta {\mathrm{d}}\theta \\ \end{gathered}$$

(A18)

$$g_{8} = \frac{{B_{{{\mathrm{insert}}}} }}{{r_{0} }}\left( {t_{1} - t_{2} - t_{3} } \right)$$

(A19)

where:

$$t_{1} = \frac{1}{{3\left( {1 + 9\tan^{2} \varphi } \right)}}\left[ {\left( {3\tan \varphi \cos \theta_{{\mathrm{h}}} + \sin \theta_{{\mathrm{h}}} } \right){\mathrm{e}}^{{3\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} - 3\tan \varphi \cos \theta_{0} - \sin \theta_{0} } \right]$$

(A20)

$$t_{2} = \frac{1}{6}\frac{L}{{r_{0} }}\left( {2\cos \theta_{0} - \frac{L}{{r_{0} }}} \right)\sin \theta_{0}$$

(A21)

$$t_{3} = \frac{1}{6}{\mathrm{e}}^{{\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} \left[ {\sin \left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right) - \frac{L}{{r_{0} }}\sin \theta_{{\mathrm{h}}} } \right]\left[ {\cos \theta_{0} - \frac{L}{{r_{0} }} + \cos \theta_{{\mathrm{h}}} {\mathrm{e}}^{{\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} } \right]$$

(A22)

$$\begin{gathered} g_{9} = 2\int_{{\theta_{0} }}^{{\theta_{{\mathrm{C}}} }} {\left\{ {\left[ {\frac{1}{8}f_{2}^{2} \left( {f_{4} - f_{1} } \right) - \frac{1}{4}\left( {f_{4} - f_{1} } \right)^{3} - \frac{2}{3}f_{1} \left( {f_{4} - f_{1} } \right)^{2} - \frac{1}{2}f_{1}^{2} \left( {f_{4} - f_{1} } \right) + \frac{2}{3}f_{1} f_{2}^{2} } \right]} \right.} \sqrt {f_{2}^{2} - \left( {f_{4} - f_{1} } \right)^{2} } \\ \left. { + \left( {\frac{1}{8}f_{2}^{4} + \frac{1}{2}f_{1}^{2} f_{2}^{2} } \right)\arccos \frac{{f_{4} - f_{1} }}{{f_{2} }}} \right\}\cos \theta {\mathrm{d}}\theta \\ \end{gathered}$$

(A23)

$$g_{10} = \frac{{B_{{{\mathrm{insert}}}} }}{{r_{0} }}\left( {t_{4} - t_{5} - t_{6} } \right)$$

(A24)

where:

$$t_{4} = \frac{1}{{3\left( {1 + 9\tan^{2} \varphi } \right)}}\left[ {\left( {3\tan \varphi \cos \theta_{{\mathrm{C}}} + \sin \theta_{{\mathrm{C}}} } \right){\mathrm{e}}^{{3\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} - 3\tan \varphi \cos \theta_{0} - \sin \theta_{0} } \right]$$

(A25)

$$t_{5} = \frac{1}{6}\left[ {\cos^{2} \theta_{0} - \cos^{2} \theta_{{\mathrm{C}}} {\mathrm{e}}^{{2\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} } \right]\sin \theta_{0}$$

(A26)

$$t_{6} = \frac{1}{3}\cos^{2} \theta_{{\mathrm{C}}} {\mathrm{e}}^{{2\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} \left[ {\sin \theta_{{\mathrm{C}}} {\mathrm{e}}^{{\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} - \sin \theta_{0} } \right]$$

(A27)

The work rate due to force $P_{a}$ can be formulated as a dot product of force $P_{a}$ and the velocity at its action point. Consequently, $W_{{P_{a} }}$ can be calculated by:

$$W_{{P_{a} }} = - \omega P_{a} r_{0} g_{11}$$

(A28)

where $g_{11}$ is a dimensionless function given by:

$$g_{11} = \sin \left( {\beta + \delta } \right)\left[ {\sin \theta_{0} + \frac{2}{3}\frac{H}{{r_{0} }}} \right] - \cos \left( {\beta + \delta } \right)\left[ {\cos \theta_{{\mathrm{h}}} {\mathrm{e}}^{{\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} + \frac{1}{3}\frac{H}{{r_{0} }}\cot \beta } \right]$$

(A29)

The equivalent width $B_{{\mathrm{e}}}$ can be calculated by the following equation:

$$B_{{\mathrm{e}}} = r_{0} g_{12}$$

(A30)

where $g_{12}$ is a dimensionless equation calculated by:

$$g_{12} = 2\sin \left( {\theta_{{\mathrm{h}}} + \beta } \right)\sin \beta {\mathrm{e}}^{{\left( {\theta_{{\mathrm{h}}} - \theta_{0} } \right)\tan \varphi }} \frac{{r_{0} }}{H}\int_{{\theta_{{\mathrm{D}}} }}^{{\theta_{{\mathrm{h}}} }} {\frac{{\sqrt {f_{2}^{2} - f_{4}^{2} } }}{{\sin^{2} \left( {\theta + \beta } \right)}}} {\mathrm{d}}\theta + \frac{{B_{{{\mathrm{insert}}}} }}{{r_{0} }}$$

(A31)

Appendix 2: Solution to the maximum crack depth

As elaborated earlier, the maximum crack depth can be determined using the kinematic limit-analysis method. The horn-like mechanism is used to capture the collapse of the soil mass behind the crack boundary. The solution is still derived from the work rate balance equation, which takes the form as:

$$D^{\prime} = W^{\prime}_{\gamma }$$

(B1)

where $D^{\prime}$ = rate of energy dissipation along surface AC, and $W^{\prime}_{\gamma }$ = rate of work done by the soil self-weight in block A–B–C. The expressions for $D^{\prime}$ and $W^{\prime}_{\gamma }$ are the same as those for $D_{{{\mathrm{AC}}}}$ and $W_{{\gamma - {\mathrm{ABC}}}}$, as shown in Eqs. (A3) and (A17), respectively, namely:

$$D^{\prime} = \omega cr_{0}^{3} \left( {g_{3} + g_{4} } \right)$$

(B2)

$$W^{\prime}_{\gamma } = \omega \gamma r_{0}^{4} \left( {g_{9} + g_{10} } \right)$$

(B3)

The expressions for $g_{3}$, $g_{4}$, $g_{9}$, and $g_{10}$ are provided in Eqs. (A7), (A8), (A23), and (A24), respectively. Substitution of the above two equations into Eq. (B1) leads to:

$$r_{0} = \frac{c}{\gamma }\frac{{g_{3} + g_{4} }}{{g_{9} + g_{10} }}$$

(B4)

The detailed expression for the maximum crack depth can then be derived as follows:

$$\begin{gathered} h_{{{\mathrm{c}} - \max }} = r_{0} \left[ {\sin \theta_{{\mathrm{C}}} {\mathrm{e}}^{{\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} - \sin \theta_{0} } \right] \\ { = }\frac{c}{\gamma }\frac{{g_{3} + g_{4} }}{{g_{9} + g_{10} }}\left[ {\sin \theta_{{\mathrm{C}}} {\mathrm{e}}^{{\left( {\theta_{{\mathrm{C}}} - \theta_{0} } \right)\tan \varphi }} - \sin \theta_{0} } \right] \\ \end{gathered}$$

(B5)

By trying different values of $\theta_{0}$, $\theta_{{\mathrm{C}}}$, and ${{r^{\prime}_{0} } \mathord{\left/ {\vphantom {{r^{\prime}_{0} } {r_{0} }}} \right. \kern-\nulldelimiterspace} {r_{0} }}$, different outcomes can be calculated from Eq. (B5). According to the kinematic approach, the results calculated from Eq. (B5) are not smaller than the actual solution. Therefore, the minimum value is needed and deemed as the most critical solution here. The critical solution can be determined using an optimization scheme under the following constrain conditions:

$$\left\{ \begin{gathered} 0 < \theta_{0} < \theta_{{\mathrm{B}}} < \theta_{{\mathrm{C}}} < \pi \hfill \\ 0 < {{r^{\prime}_{0} } \mathord{\left/ {\vphantom {{r^{\prime}_{0} } {r_{0} < 1}}} \right. \kern-\nulldelimiterspace} {r_{0} < 1}} \hfill \\ B_{{{\mathrm{insert}}}} \ge 0 \hfill \\ \end{gathered} \right.$$

(B6)

It should be noted here that the values of $\theta_{0}$, $\theta_{{\mathrm{C}}}$, and ${{r^{\prime}_{0} } \mathord{\left/ {\vphantom {{r^{\prime}_{0} } {r_{0} }}} \right. \kern-\nulldelimiterspace} {r_{0} }}$ determined in the calculation of the maximum crack depth may be not the same as those in computing the active earth pressure coefficient. In other words, in the calculation of the maximum crack depth, a set of $\theta_{0}$, $\theta_{{\mathrm{C}}}$, and ratio ${{r^{\prime}_{0} } \mathord{\left/ {\vphantom {{r^{\prime}_{0} } {r_{0} }}} \right. \kern-\nulldelimiterspace} {r_{0} }}$ is determined, and another set of $\theta_{0}$ and $\theta_{{\mathrm{C}}}$, and ratio ${{r^{\prime}_{0} } \mathord{\left/ {\vphantom {{r^{\prime}_{0} } {r_{0} }}} \right. \kern-\nulldelimiterspace} {r_{0} }}$ is determined when computing the active earth pressure coefficient. These two sets may be not the same, because the maximum active earth pressure coefficient may be obtained when the crack is not at its maximum.

As stated previously, the 2D case can be deemed as a special 3D one, where $B_{{{\mathrm{insert}}}} \to \infty$. To show the validity of the present solution for the maximum crack depth, the results calculated by the proposed approach under 2D conditions are compared with those calculated using the formula in Eq. (13). For convenience, a dimensionless coefficient is introduced as:

$$N = \frac{{\gamma h_{{{\mathrm{c}} - \max }} }}{c}$$

(B7)

Then, the coefficient corresponding to Eq. (13) takes the form as:

$$N = 3.83\tan \left( {\frac{\pi }{4} + \frac{\varphi }{2}} \right)$$

(B8)

The results calculated by the proposed approach are compared with those calculated by Eq. (3), with $\varphi$ varying from 10° to 45°, as shown in Table

Table 3 Comparison of the outcomes of the maximum crack depth obtained by the proposed approach with those calculated using Eq. (B8)

3. From the table, it can be observed that the results obtained by the proposed approach show little difference with those calculated from Eq. (3), with a maximum difference of 0.97%. The difference in the results may be due to the following reasons. Equation (13) or (B8) is indeed a fitting formula. Therefore, the results calculated by this formula may be slightly different from the actual solutions. Besides, the derivation of Eq. (13) is based on a planar mechanism, which is less critical than the log-spiral mechanism adopted in this work. Overall, the discrepancy is acceptable, suggesting that the present solutions are valid.

Appendix 3: An illustrative example

This paper provides a series of design charts for simple use. The aim of this appendix is to introduce how to use the charts provided by this work to design a retaining wall. To do this, an illustrative example is presented herein. As shown in Fig. 6, the wall has a top width of a and a bottom width of b. The wall is constructed using concrete with a unit weight of $\gamma_{{{\mathrm{wall}}}}$. The base soil is stiff clay with a base adhesion of $c_{{\mathrm{a}}}$. Two types of possible failure modes are considered herein, i.e., overturning about the wall toe and sliding along the base of the wall. The lumped safety factors for these failure types are defined below. Note that only per unit width of the wall is involved in the analysis.

The lumped safety factor against overturning failure is formulated as:

$$FS_{{{\mathrm{ot}}}} = \frac{{W_{1} Arm1 + W_{2} Arm2 + p_{{a{\mathrm{V}}}} Arm_{{a{\mathrm{V}}}} }}{{p_{{a{\mathrm{H}}}} Arm_{{a{\mathrm{H}}}} }}$$

(C1)

where $W_{1}$ and $W_{2}$ = two components of the retaining wall self-weight, with horizontal lever distances $Arm_{1}$ and $Arm_{2}$ measured from the wall toe; $p_{{a{\mathrm{V}}}}$ and $p_{{a{\mathrm{H}}}}$ = vertical and horizontal components of the active earth pressure resultant per unit width $p_{a}$, with lever distances $Arm_{{a{\mathrm{V}}}}$ and $Arm_{{a{\mathrm{H}}}}$, respectively. These parameters can be calculated as:

$$\begin{aligned} W_{1} & = \frac{1}{2}\gamma_{{{\mathrm{wall}}}} aH,\,\,Arm1 = \frac{2}{3}b + \frac{2}{3}H\cot \beta - \frac{1}{3}a,\,\,W_{2} = \frac{1}{2}\gamma_{{{\mathrm{wall}}}} bH,\,\,Arm2 = \frac{2}{3}b + \frac{1}{3}H\cot \beta , \\ p_{{a{\mathrm{V}}}} & = p_{a} \sin \left( {\delta + \beta - 90^\circ } \right),\,\,Arm_{{a{\mathrm{V}}}} = b + \frac{1}{3}H\cot \beta ,\,\,p_{{a{\mathrm{H}}}} = p_{a} \cos \left( {\delta + \beta - 90^\circ } \right),\,\,Arm_{{a{\mathrm{H}}}} = \frac{1}{3}H, \\ p_{a} & = \frac{1}{2}\gamma H^{2} K_{a} . \\ \end{aligned}$$

The lumped safety factor against the sliding failure takes the following form:

$$FS_{{{\mathrm{sl}}}} = \frac{{bc_{{\mathrm{a}}} }}{{p_{{a{\mathrm{H}}}} }}$$

(C2)

An illustrative example with the following parameters is considered: $H = 5{\mathrm{m}}$, $\beta = 90^\circ$, $B = 25{\mathrm{m}}$, $\varphi = 30^\circ$, $c = 5{\mathrm{kPa}}$, $\gamma = 20{{{\mathrm{kN}}} \mathord{\left/ {\vphantom {{{\mathrm{kN}}} {{\mathrm{m}}^{3} }}} \right. \kern-\nulldelimiterspace} {{\mathrm{m}}^{3} }}$, $\delta = 20^\circ$, $a = 0.4{\mathrm{m}}$, $b = 1.5{\mathrm{m}}$, $\gamma_{{{\mathrm{wall}}}} = 25{{{\mathrm{kN}}} \mathord{\left/ {\vphantom {{{\mathrm{kN}}} {{\mathrm{m}}^{3} }}} \right. \kern-\nulldelimiterspace} {{\mathrm{m}}^{3} }}$, and $c_{{\mathrm{a}}} = 60{\mathrm{kPa}}$.

For conservative and safe design, the active earth pressure coefficient for the case with open cracks is adopted. Given the above parameters, the value of $K_{a}$ can be found in Fig. 5g at $\varphi = 30^\circ$ and ${c \mathord{\left/ {\vphantom {c {\gamma H = 0.05}}} \right. \kern-\nulldelimiterspace} {\gamma H = 0.05}}$, where $K_{a} =$ 0.187. Then, the factors of safety against the overturning failure and sliding failure can be calculated using Eqs. (C1) and (C2). The calculated safety factors are $FS_{{{\mathrm{ot}}}} = 1.84$ and $FS_{{{\mathrm{sl}}}} = 2.05$, respectively, which both exceed the required minimum safety factors in many design guidelines. This suggests that the size of the retaining wall meets the requirement.

It should be noted that the active earth pressure coefficients may not be clearly shown in the design charts at some certain parameters. For example, the curves for $\varphi = 25^\circ$ are not included in the graphs. In this case, the corresponding active earth pressure coefficient can be determined using interpolation techniques based on the existing results in the design charts.

Glossary

a, b

Top and bottom width of the retaining wall

A

Area of the soil–wall interface covered by the 3D portion of the mechanism

$Arm_{1}$, $Arm_{2}$, $Arm_{{a{\mathrm{H}}}}$ $Arm_{{a{\mathrm{V}}}}$

Lever distances

B

Wall width

$B_{{\mathrm{3D - max}}}$

Maximum width of the 3D portion

$B_{{\mathrm{e}}}$

Equivalent width

$B_{{{\mathrm{insert}}}}$

Width of the insert

c

Soil cohesion

$c_{{\mathrm{a}}}$

Base adhesion

$D$

Energy dissipation rate owing to soil resistance

$D^{\prime}$

Energy dissipation rate along surface AC

$D_{{{\mathrm{CE}}}}$, $D_{{{\mathrm{AE}}}}$, $D_{{{\mathrm{AC}}}}$

Energy dissipation rates along surfaces CE, AE, and AC

$\kappa_{1}$, $f_{i}$, $g_{i}$, $t_{i}$ (i = 1, 2…,)

Dimensionless functions

$FS_{{{\mathrm{ot}}}}$, $FS_{{{\mathrm{sl}}}}$

Safety factors against overturning failure and sliding failure

$h_{{\mathrm{c}}}$

Crack depth

$h_{{\mathrm{c - max}}}$

Maximum crack depth

$K_{a}$

Active earth pressure coefficient

L, $L^{\prime}$

$\overline{{{\mathrm{AD}}}}$ and $\overline{{{\mathrm{BD}}}}$

$P_{a}$

Resultant force of active earth pressure

$p_{a}$

Resultant force of active earth pressure per unit width

$p_{{a{\mathrm{H}}}}$, $p_{{a{\mathrm{V}}}}$

Horizontal and vertical components of $p_{a}$

r, $r^{\prime}$

Radii of the lower and upper log-spirals

$r_{0}$, $r^{\prime}_{0}$

$\overline{{{\mathrm{OA}}}}$ and $\overline{{{\mathrm{OA}^{\prime}}}}$

$r_{{\mathrm{m}}}$

Distance from the rotation center O to the circle center

R

Circle radius

$s_{1}$

Distance from the rotation center O to a generic point on surface AD or DE

$s_{2}$

Distance from the rotation center O to a generic point on surface AB and BC

$W_{1}$, $W_{2}$

Components of the wall weight

$W_{\gamma }$, $W_{{P_{a} }}$

External work rates due to the self-weight of soil masses and the resultant active earth pressure

$W^{\prime}_{\gamma }$

Work rate done by the soil self-weight in block A–B–C

$W_{{\gamma - {\mathrm{ADE}}}}$, $W_{{\gamma - {\mathrm{ABC}}}}$

Rates of gravity work done by regions A–D–E and A–B–C

$\beta$

Inclination angle

$\delta$

Friction angle of the soil–wall interface

$\gamma$

Soil unit weight

$\gamma_{{{\mathrm{wall}}}}$

Wall unit weight

$\omega$

Angular velocity

$\varphi$

Soil internal friction angle

$\theta$, $\rho$

Polar coordinates

$\theta_{0}$, $\theta_{{\mathrm{B}}}$, $\theta_{{\mathrm{C}}}$, $\theta_{{\mathrm{D}}}$, $\theta_{{\mathrm{h}}}$

Polar angles of points A, B, C, D, and E