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Pseudo Static Seismic Stability Analysis of Reinforced Soil Structures

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Abstract

The paper pertains to the pseudo-static seismic stability analysis of reinforced soil structures. Using limit equilibrium method and assuming the failure surface to be logarithmic spiral, analysis has been conducted to maintain internal stability against both tensile and pullout failure of the reinforcements. The external stability of the reinforced earth wall is also assessed in terms of its sliding, overturning, eccentricity and bearing modes of failure. The influence of the intensity of the surcharge load placed on the backfill is also considered in the analysis. The obtained results are validated by comparing the same with those reported in literature. Studies have also been made regarding the influence of backfill soil friction angle, horizontal and vertical seismic accelerations, surcharge load, the tensile strength of reinforcement, pullout length of the reinforcement and number of reinforcement layers on the seismic stability against various failure modes as mentioned earlier.

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Acknowledgments

The authors thank the anonymous reviewers for their constructive comments and useful suggestions which have been of immense help in revising the manuscript.

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Correspondence to B. Munwar Basha.

Appendix

Appendix

With reference to Fig. 1, the following equations can be derived.

$$ H = KC = EG = r_{1} \sin \left( {\theta_{1} + \theta_{2} } \right) - r_{o} \sin \theta_{2} = ar_{0} $$
(A1)
$$ a = \left[ {e^{{\theta_{1} \tan \phi }} \sin \left( {\theta_{1} + \theta_{2} } \right) - \sin \theta_{2} } \right] $$
(A2)
$$ BC = DG = r_{1} \sin \left( {\theta_{1} + \theta_{2} } \right) - r_{1} \sin \left( {\theta_{2} + \theta } \right) $$
(A3)
$$ EH_{1} = L_{g} = r_{o} \cos \theta_{2} - r_{1} \cos \left({\theta_{1} + \theta_{2} } \right) = br_{o}$$
(A4)
$$ b = \left[ {\cos \theta_{2} - e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right] $$
(A5)

1.1 A. 1 Weight of Log Spiral Wedge SH 1 G (\( W_{{SH_{1} G}} \))

Weight of log spiral wedge AH 1 G can be written as

$$ W_{{AH_{1} G}} = \gamma \left[ {{\frac{{r_{1}^{2} - r_{0}^{2} }}{4\tan \phi }}} \right] = \gamma r_{0}^{2} \left[ {{\frac{{e^{{2\theta_{1} \tan \phi }} - 1}}{4\tan \phi }}} \right] $$
(A6)

Weight of triangular wedge ESGcan be expressed as

$$ W_{ESG} = \frac{1}{2}\gamma H^{2} \cot \alpha = \frac{1}{2}\gamma a^{2} r_{o}^{2} \cot \alpha $$
(A7)

Weight of triangular wedge AH 1 EK can be expresses as

$$ W_{{AH_{1} EK}} = \gamma \frac{1}{2}\left( {r_{0} \cos \theta_{2} } \right)\left( {r_{0} \sin \theta_{2} } \right) = \frac{1}{2}\gamma r_{0}^{2} \cos \theta_{2} \sin \theta_{2} $$
(A8)

Weight of rectangular wedge KEGC can be expresses as

$$ W_{KEGC} = \gamma r_{1} \cos \left( {\theta_{1} + \theta_{2} } \right)H = \gamma r_{1} \cos \left( {\theta_{1} + \theta_{2} } \right)\left( {ar_{0} } \right) = a\gamma r_{0}^{2} e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right) $$
(A9)

Weight of triangular wedge AGCcan be written as

$$ W_{AGC} = \gamma \frac{1}{2}r_{1} \cos \left( {\theta_{1} + \theta_{2} } \right)r_{1} \sin \left( {\theta_{1} + \theta_{2} } \right) = \frac{1}{4}\gamma r_{0}^{2} e^{{2\theta_{1} \tan \phi }} \sin 2\left( {\theta_{1} + \theta_{2} } \right) $$
(A10)
$$ W_{{SH_{1} G}} = \frac{1}{2}\gamma r_{0}^{2} \left\{ {\left[ {{\frac{{e^{{2\theta_{1} \tan \phi }} - 1}}{2\tan \phi }}} \right] - \left( {\begin{array}{*{20}c} {\left[ {a^{2} \cot \alpha } \right] + \left[ {\cos \theta_{2} \sin \theta_{2} } \right] + \left[ {2ae^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right]} \\ { - \left[ {\frac{1}{2}e^{{2\theta_{1} \tan \phi }} \sin 2\left( {\theta_{1} + \theta_{2} } \right)} \right]} \\ \end{array} } \right)} \right\} $$
(A11)

1.2 A. 2 Derivation for Active Length of the Reinforcement

$$ L_{a} = br_{o} - L_{s} - L_{b} $$
(A12)

where L b  = r0 cos θ2 − r cos (θ2 + θ) and L s  = (H − z) cot α.

Substituting \( b = \cos \theta_{2} - e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right) \) in Eq (A12), we get

$$ L_{a} = r_{o} \left[ {\cos \theta_{2} - e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right] - \left( {H - z} \right)\cot \alpha - \left[ {r_{0} \cos \theta_{2} - r\cos \left( {\theta_{2} + \theta } \right)} \right] $$
(A13)
$$ L_{a} = \left[ { - r_{o} e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right) + r\cos \left( {\theta_{2} + \theta } \right)} \right] - \left( {H - z} \right)\cot \alpha $$
(A14)
$$ L_{a} = \left[ {r\cos \left( {\theta_{2} + \theta } \right) - r_{o} e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right] - \left( {H - z} \right)\cot \alpha $$
(A15)

Substituting \( r = r_{o} e^{\theta \tan \phi } \) in Eq (A15), we get

$$ L_{a} = {\frac{{ar_{o} }}{a}}\left[ {e^{\theta \tan \phi } \cos \left( {\theta_{2} + \theta } \right) - e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right] - \left( {H - z} \right)\cot \alpha $$
(A16)

Substituting ar o  = H in Eq (A16), we get

$$ L_{a} = \frac{H}{a}\left[ {e^{\theta \tan \phi } \cos \left( {\theta_{2} + \theta } \right) - e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right] - \left( {H - z} \right)\cot \alpha $$
(A17)
$$ {\frac{{L_{a} }}{H}} = \frac{1}{a}\left[ {e^{\theta \tan \phi } \cos \left( {\theta_{2} + \theta } \right) - e^{{\theta_{1} \tan \phi }} \cos \left( {\theta_{1} + \theta_{2} } \right)} \right] - \left( {1 - \frac{z}{H}} \right)\cot \alpha $$
(A18)

1.3 A. 3 Expressions for Bearing Capacity Factors

$$ N_{q} = \tan^{2} \left( {45 + \phi_{b} /2} \right)e^{{\pi \tan \phi_{b} }} ;\,N_{c} = \left( {N_{q} - 1} \right)\cot \phi_{b} ;\,N_{\gamma } = 2\left( {N_{q} + 1} \right)\tan \phi_{b} $$
(A19)
$$ F_{ci} = F_{qi} = \left[ {1 - {\frac{{\psi^{o} }}{90}}} \right]^{2} ;\,F_{\gamma i} = \left[ {1 - {\frac{{\psi^{o} }}{{\phi_{b} }}}} \right]^{2} \, \left( {\text{Das\,1999}} \right) $$
(A20)
$$ F_{cd} = \left( {1 + 0.4{\frac{{h_{e} }}{L - 2e}}} \right) = 1\, {\text{for}}\,{\text{no}}\,{\text{soil}}\,{\text{at}}\,{\text{the}}\,{\text{toe}}\,{\text{side}}\,{\text{of}}\,{\text{wall }} $$
(A21)
$$ F_{qd} = \left( {1 + 2\tan \phi_{b} \left( {1 - \sin \phi_{b} } \right)^{2} {\frac{h}{L - 2e}}} \right) = 1\,{\text{for}}\,{\text{no}}\,{\text{soil}}\,{\text{at}}\,{\text{the}}\,{\text{toe}}\,{\text{side}}\,{\text{of}}\,{\text{wall}} $$
(A22)
$$ \psi^{o} = \tan^{ - 1} \left( {{\frac{\sum H }{\sum V }}} \right) $$
(A23)
$$ \sum H = P_{aet} \cos \delta + k_{h} \left( {W_{ABFE} + qL} \right) $$
(A24)
$$ q_{u} = cN_{c} F_{ci} + 0.5\gamma_{b} \left( {L - 2e} \right)N_{\gamma } F_{\gamma i} $$
(A25)

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Basha, B.M., Basudhar, P.K. Pseudo Static Seismic Stability Analysis of Reinforced Soil Structures. Geotech Geol Eng 28, 745–762 (2010). https://doi.org/10.1007/s10706-010-9336-2

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