Displacement versus pseudo-static evaluation of the seismic performance of sliding retaining walls

Abstract

Pseudo-static seismic analysis of retaining walls requires the selection of an equivalent seismic coefficient synthetically representing the effects of the transient seismic actions on the soil-wall system. In this paper, a rational criterion for the selection of the equivalent seismic coefficient is proposed with reference to sliding retaining walls. In the proposed approach earthquake-induced permanent displacements are assumed as a suitable parameter to assess the seismic performance and an alternative definition of the wall safety factor is introduced comparing expected and limit values of permanent displacements. Using a simplified displacement prediction model it is shown that, for a given design earthquake, reliable values of the equivalent seismic coefficient should depend on all the factors affecting the stability condition of the soil-wall system and on a threshold value of permanent displacement related to a given ultimate or serviceability limit state. To achieve a match between the results of the pseudo-static and of the displacement-based analysis, the proposed procedure detects the value of the equivalent seismic coefficient for which the two approaches provide the same factor of safety. Thus, without necessarily carrying out a displacement analysis, a measure of the safety condition of a soil-wall system consistent with the actual seismic performance may be achieved through an equivalent pseudo-static analysis.

Appendices

Appendix

For the soil-wall systems shown in Figs. 1a, 2 this Appendix lists the equations derived by Biondi and Cascone (2014) and by Cascone and Biondi (2014) for the evaluation of the critical,$k_\mathrm{h,c}$, and of the equivalent, $k_\mathrm{h,eq}$, seismic coefficients and of the displacement factor $C_\mathrm{w}$ assuming $k_\mathrm{v}$ $=$ 0 ($\Omega$ $=$ 0).

Appendix 1: Horizontal component of the critical acceleration coefficient, $k_\mathrm{h,c}$: R–E procedure

$k_\mathrm{h,c}$ can be computed solving, iteratively, the following equation (Biondi and Cascone 2014):

$$\begin{aligned} k_{\mathrm{h,c}} =\frac{\frac{2\cdot c_\mathrm{b} \cdot B_\mathrm{b} }{\gamma \cdot H^{2}\cdot \Gamma _\mathrm{w} \cdot \cos \upalpha _\mathrm{b} }+E_\mathrm{c} }{\frac{\cos \left( {\upphi _\mathrm{b} -\upalpha _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }+\Omega \cdot E_\mathrm{c} } \end{aligned}$$

(31)

where $c_\mathrm{b}$ and $\upphi _\mathrm{b}$ are the shear strength parameters at the wall-foundation soil interface, $B_\mathrm{b}$ is the wall base width, $\upalpha _\mathrm{b}$ is the inclination of the wall base to the horizontal and $H$ is the height of the retained soil having a unit weight $\upgamma$;

$$\begin{aligned} E_\mathrm{c}&= \frac{\sin \left( {\upphi _\mathrm{b} -\upalpha _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }+\frac{K_{\mathrm{ae,c}} }{\Gamma _\mathrm{w} }\cdot \frac{\sin \left( {\updelta +\upbeta +\upphi _\mathrm{b} -\upalpha _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }\end{aligned}$$

(32)

$$\begin{aligned} \Omega&= \frac{k_{\mathrm{v,c}} }{k_{\mathrm{h,c}} } \end{aligned}$$

(33)

is the ratio of the vertical to the horizontal seismic coefficient at limit equilibrium;

$$\begin{aligned} \Gamma _\mathrm{w} =\frac{2\cdot W_\mathrm{w} }{\upgamma \cdot H^{2}} \end{aligned}$$

(34)

is the normalized wall weight ($W_\mathrm{w}$ is the wall weight);

$K_\mathrm{ae,c}$ is the value at limit equilibrium of the Mononobe–Okabe active earth-pressure coefficient $K_\mathrm{ae}$:

$$\begin{aligned} K_{\mathrm{ae}} =\frac{\cos ^{2}(\upbeta +\uptheta -\upvarphi ^{\prime })}{\cos ^{2}\upbeta \cdot \cos \uptheta \cdot \cos (\updelta +\upbeta +\uptheta )\cdot \left[ {1+\sqrt{\frac{\sin (\upvarphi ^{\prime }-i-\uptheta )\cdot \sin (\updelta +\upvarphi ^{\prime })}{\cos (i-\upbeta )\cdot \cos (\upbeta +\updelta +\uptheta )}}} \right] ^{2}} \end{aligned}$$

(35)

with:

$$\begin{aligned} \tan \uptheta =\tan \mathop {\uptheta }\nolimits _\mathrm{c} =\frac{k_{\mathrm{h,c}} }{1-k_{\mathrm{v,c}} }=\frac{1}{1/{k_{\mathrm{h,c}} }-\Omega _\mathrm{c} } \end{aligned}$$

(36)

In the case of horizontal wall base ($\upalpha _\mathrm{b}$ $=$ 0; Fig. 1a) Eqs. 31 and 32 reduce to:

$$\begin{aligned} k_{\mathrm{h,c}}&= \frac{\frac{2\cdot c_\mathrm{b} \cdot B_\mathrm{b} }{\upgamma \cdot H^{2}\cdot \Gamma _\mathrm{w} }+E_\mathrm{c} }{1+\Omega \cdot E_\mathrm{c} }\end{aligned}$$

(37)

$$\begin{aligned} E_\mathrm{c}&= \tan \upphi _\mathrm{b} +\frac{K_{\mathrm{ae,c}} }{\Gamma _\mathrm{w} }\cdot \frac{\sin \left( {\updelta +\upbeta +\upphi _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b}} \end{aligned}$$

(38)

If the vertical component of the ground acceleration is neglected ($k_\mathrm{v}$ $=$ 0; $\Omega$ $=$ 0), it is:

$$\begin{aligned} k_{\mathrm{h,c}} =\frac{2\cdot c_\mathrm{b} \cdot B_\mathrm{b} }{\upgamma \cdot H^{2}\cdot \Gamma _\mathrm{w} }+\tan \upphi _\mathrm{b} +\frac{K_{\mathrm{ae,c}} }{\Gamma _\mathrm{w} }\cdot \frac{\sin \left( {\updelta +\upbeta +\upphi _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b}} \end{aligned}$$

(39)

Appendix 2: Horizontal component of the critical acceleration coefficient: 2-$W$ procedure

For the case $c_\mathrm{b}$ $=$ 0 and $\upalpha _\mathrm{b}$ $=$ 0, the horizontal component of the critical acceleration coefficient $k_\mathrm{h,c}$ can be computed through the following equation (Biondi and Cascone 2014):

$$\begin{aligned} \frac{k_{\mathrm{h,c}} }{1-\Omega \cdot k_{\mathrm{h,c}} }=\frac{b+\sqrt{b^{2}-a\cdot c}}{a} \end{aligned}$$

(40)

where:

$$\begin{aligned} a=B_4^2 -4\cdot A_4 \cdot C_4 \quad b=2\cdot A_3 \cdot C_4 -2\cdot C_3 \cdot A_4 -B_3 \cdot B_4 \quad c=B_3^2 +4\cdot A_3 \cdot C_3\nonumber \\ \end{aligned}$$

(41)

being:

$$\begin{aligned} A_3&= \frac{\hbox {cos}\left( \upbeta +\updelta +\upphi _\mathrm{b} \right) }{\cos \upphi _\mathrm{b}}\cdot \left( {1+\tan i\cdot \tan \upbeta } \right) \cdot \tan \upbeta \nonumber \\&\quad -\frac{\sin \left( {\upbeta +\updelta +\upvarphi ^{\prime }} \right) }{\cos \upvarphi ^{\prime }}\cdot \tan {\upphi }_\mathrm{b} \cdot \Gamma _\mathrm{w}\end{aligned}$$

(42)

$$\begin{aligned} A_4&= \frac{\cos \left( {\upbeta +\updelta +\upphi _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }\cdot \left( {1+\tan i\cdot \tan \upbeta } \right) \cdot \tan \upbeta \cdot \tan {\upvarphi }^{\prime }\nonumber \\&\quad +\frac{\sin \left( {\upbeta +\updelta +\upvarphi ^{\prime }} \right) }{\cos \upvarphi ^{\prime }}\cdot \Gamma _\mathrm{w}\end{aligned}$$

(43)

$$\begin{aligned} B_3&= \frac{\cos \left( {\upbeta +\updelta +\upphi _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }\cdot \left( {1+\tan i\cdot \tan \upbeta } \right) \cdot \left( {1-\tan \upbeta \cdot \tan {\upvarphi }^{\prime }} \right) \nonumber \\&\quad -\frac{\cos \left( {\upbeta +\updelta +\upvarphi ^{\prime }+i} \right) }{\cos \upvarphi ^{\prime }\cdot \cos i}\cdot \Gamma _\mathrm{w} \cdot \tan {\upphi }_\mathrm{b}\end{aligned}$$

(44)

$$\begin{aligned} B_4&= \frac{\cos \left( {\upbeta +\updelta +\upphi _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }\cdot \left( {1+\tan i\cdot \tan \upbeta } \right) \cdot \left( \tan \upbeta +\tan {\upvarphi }^{\prime } \right) \nonumber \\&\quad +\frac{\cos \left( {\upbeta +\updelta +\upvarphi ^{\prime }+i} \right) }{\cos \upvarphi ^{\prime }\cdot \cos i}\cdot \Gamma _\mathrm{w}\end{aligned}$$

(45)

$$\begin{aligned} C_3&= \frac{\cos \left( {\upbeta +\updelta +\upphi _\mathrm{b}} \right) }{\cos \upphi _\mathrm{b} }\cdot \left( {1+\tan i\cdot \tan \upbeta } \right) \cdot \tan {\upvarphi }^{\prime }\nonumber \\&\quad -\frac{\cos \left( {\upbeta +\updelta +\upvarphi ^{\prime }} \right) }{\cos \upvarphi ^{\prime }}\cdot \Gamma _\mathrm{w} \cdot \tan i\cdot \tan {\upphi } _\mathrm{b}\end{aligned}$$

(46)

$$\begin{aligned} C_4&= \frac{\cos \left( {\upbeta +\updelta +\upphi _\mathrm{b} } \right) }{\cos \upphi _\mathrm{b} }\cdot \left( {1+\tan i\cdot \tan \upbeta } \right) \nonumber \\&\quad -\frac{\cos \left( {\upbeta +\updelta +\upvarphi ^{\prime }} \right) }{\cos \upvarphi ^{\prime }}\cdot \tan i\cdot \Gamma _\mathrm{w} \end{aligned}$$

(47)

Appendix 3: Wall displacement factor $C_\mathrm{w}$

For the case $c_\mathrm{b}$ $=$ 0 the wall displacement factor $C_\mathrm{w}$ is (Biondi and Cascone 2014):

$$\begin{aligned} C_\mathrm{w} =\frac{\Gamma _\mathrm{w} \cdot A_5 +B_5 }{\Gamma _\mathrm{w} \cdot A_6 +B_6 } \end{aligned}$$

(48)

where:

$$\begin{aligned} A_5&= \cos \left( {\upalpha _\mathrm{c} -\updelta -\upbeta -\upvarphi _{}^{\prime }} \right) \cdot \cos \left( {\upalpha _\mathrm{c} -\upbeta } \right) \cdot \cos \left( {\upphi _\mathrm{b} -\upalpha _\mathrm{b} } \right) \end{aligned}$$

(49)

$$\begin{aligned} \hbox {A}_\mathrm{6}&= \cos \left( {\upalpha _\mathrm{c} -\updelta -\upbeta -\upvarphi _{}^{\prime }} \right) \cdot \cos \left( {\upalpha _\mathrm{c} -\upbeta } \right) \cdot \cos \upphi _\mathrm{b}\end{aligned}$$

(50)

$$\begin{aligned} B_5&= \frac{\cos \left( {\upbeta -i} \right) \cdot \cos \left( {\upalpha _\mathrm{c} -\upvarphi ^{\prime }} \right) \cdot \cos \left( {\upphi _\mathrm{b} -\upalpha _\mathrm{b} +\updelta +\upbeta } \right) }{\cos \left( {\upalpha _\mathrm{c} -i} \right) }\cdot \left[ {\frac{\cos \left( {\upalpha _\mathrm{c} -\upbeta } \right) }{\cos \upbeta }} \right] ^{2}\end{aligned}$$

(51)

$$\begin{aligned} B_6&= \frac{\cos \upvarphi ^{\prime }\cdot \cos \left( {\upphi _\mathrm{b} -\upalpha _\mathrm{w} +\upbeta +\updelta } \right) \cdot \cos \left( {\upbeta -\upalpha _\mathrm{w} } \right) \cdot \left( {\cot \upalpha _\mathrm{c} +\tan \upbeta } \right) \cdot \left( {1+\tan i\cdot \tan \upbeta } \right) }{1-\tan i\cdot \cot \upalpha _\mathrm{c} }\nonumber \\ \end{aligned}$$

(52)