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Seismic Bearing Capacity of Shallow Strip Footings on Sand Deposits with Weak Inter-layer

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Abstract

The main objective of this study is to evaluate the ultimate seismic bearing capacity of a shallow strip footing resting on a frictional soil stratum containing a weak intervening layer. The majority of the studies throughout the literature pertain to the static loading condition. The previous seismic analyses have also been devoted to the studies on the bearing capacity of shallow strip footings resting on a two-layered soil. The influence of weak middle layer on the pseudo-static seismic bearing capacity of shallow foundations is the main focus of the present study. To determine the seismic bearing capacity, the limit equilibrium method (LEM) was combined with the pseudo-static seismic loading approach. Bearing capacity was defined by a single equivalent coefficient which combines the contributions of cohesion, surcharge and soil weight. A two-wedge failure surface, known as the Coulomb failure mechanism, was adopted to model the slip lines in each layer to calculate the seismic bearing capacity of the overlying shallow strip footing. The Particle Swarm Optimization (PSO) algorithm was invoked to seek the optimal bearing capacity value under different strength and loading conditions. In order to verify the validity of the presented formulations, the results were compared with some Finite Elements Method (FEM) analyses available in literature. Furthermore, the influence of the embedment depth, thickness, and strength of the weak inter-layer on the seismic bearing capacity of the shallow footing is investigated in the presence of different seismic loading arrangements. The results of this study could be very helpful in the seismic analysis and design of shallow foundations overlying a soil medium containing a weak layer of various strengths.

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Authors and Affiliations

  1. Department of Civil Engineering, Faculty of Engineering, University of Guilan, Rasht, Iran

    Hamed Haghsheno & Reza Jamshidi Chenari

  2. Rocscience Inc., 54 Saint Patrick St, Toronto, ON, Canada

    Sina Javankhoshdel

  3. Central Tehran Branch, Islamic Azad University, Tehran, Iran

    Tooraj Banirostam

Authors
  1. Hamed Haghsheno
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  2. Reza Jamshidi Chenari
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  3. Sina Javankhoshdel
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  4. Tooraj Banirostam
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Correspondence to Reza Jamshidi Chenari.

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Appendix: Analytical Functions of

Appendix: Analytical Functions of

Eq. ( 6 )

$$\begin{gathered} a = \left[ {\frac{{\left( {1 - k_{h} } \right)\sin \left( {\alpha_{A1} - \varphi } \right) + k_{h} \cos \left( {\alpha_{A1} - \varphi } \right)}}{{\cos \left( {\alpha_{A1} - \varphi - \delta } \right)}}} \right] - \frac{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} } \right)}}{{\left( {1 + \frac{{h_{1} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{A1} - \varphi } \right) + k_{h} \cos \left( {\alpha_{A1} - \varphi } \right)tan\varphi }}{{\cos \left( {\alpha_{A1} - \varphi - \delta } \right)}}} \right] \hfill \\ + \frac{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} } \right)}}{{\left( {1 + \frac{{h_{1} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{A2} } \right) + k_{h} \cos \left( {\alpha_{A2} } \right)tan\varphi }}{{\cos \left( {\alpha_{A2} } \right)}}} \right] \hfill \\ - \frac{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} } \right)\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} - \frac{{h_{2} }}{B}\cot \alpha_{A2} } \right)}}{{\left( {1 + \frac{{h_{1} }}{B}} \right)\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} + \frac{{h_{2} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{A2} } \right) + k_{h} \cos \left( {\alpha_{A2} } \right)tan\varphi }}{{\cos \left( {\alpha_{A2} } \right)}}} \right] \hfill \\ + \frac{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} } \right)\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} - \frac{{h_{2} }}{B}\cot \alpha_{A2} } \right)}}{{\left( {1 + \frac{{h_{1} }}{B}} \right)\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} + \frac{{h_{2} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{A3} - \varphi } \right) + k_{h} \cos \left( {\alpha_{A3} - \varphi } \right)tan\varphi }}{{\cos \left( {\alpha_{A3} - \varphi - \delta } \right)}}} \right] \hfill \\ \end{gathered}$$
$$\begin{gathered} d = 2\frac{{D_{f} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{B1} + \varphi } \right) - k_{h} \cos \left( {\alpha_{B1} + \varphi } \right)}}{{\cos \left( {\alpha_{B1} + \varphi + \delta } \right)}}} \right] \hfill \\ - \frac{{2\frac{{D_{f} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} } \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} } \right)}}{{\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} + \frac{{h_{1} }}{B}}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{B1} + \varphi } \right)}}{{\cos \left( {\alpha_{B1} + \varphi + \delta } \right)}} + \frac{{k_{h} \cos \left( {\alpha_{B1} + \varphi } \right)tan\varphi }}{{\cos \left( {\alpha_{B1} + \varphi + \delta } \right)}}} \right] \hfill \\ + \frac{{2\frac{{D_{f} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} } \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} } \right)}}{{\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} + \frac{{h_{1} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha_{B2} + k_{h} \cos \alpha_{B2} tan\varphi }}{{\cos \alpha_{B2} }}} \right] \hfill \\ - \frac{{2\frac{{D_{f} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} } \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} } \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} } \right)}}{{\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} + \frac{{h_{1} }}{B}} \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{2} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha_{B2} + k_{h} \cos \alpha_{B2} }}{{\cos \alpha_{B2} }}} \right] \hfill \\ + \frac{{2\frac{{D_{f} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + h_{2} \cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} } \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} } \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} } \right)}}{{\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{1} }}{B}\cot \alpha_{B1} + \frac{{h_{1} }}{B}} \right)\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} + \frac{{h_{2} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha_{B3} + \varphi } \right) + k_{h} \cos \left( {\alpha_{B3} + \varphi } \right)tan\varphi }}{{\cos \left( {\alpha_{B3} + \varphi + \delta } \right)}}} \right] \hfill \\ \end{gathered}$$
$$\begin{gathered} e = 2\frac{{h_{2} }}{B}\tan \alpha_{A2} + 2\frac{{h_{2} }}{B}\tan \alpha_{B2} + 2\frac{{h_{2} }}{B}cot\alpha_{B2} + \frac{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} } \right)\cos \left( {\alpha_{A1} - \varphi } \right)}}{{\cos \left( {\alpha_{A1} - \varphi - \delta } \right)}} \hfill \\ - \frac{{\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} + \frac{{h_{2} }}{B}\cot \alpha_{B2} - \frac{{h_{1} }}{B}cot\alpha_{B1} } \right)\cos \left( {\alpha_{B1} + \varphi } \right)}}{{\cos \left( {\alpha_{B1} + \varphi + \delta } \right)}} - \frac{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha_{A1} - \frac{{h_{2} }}{B}\cot \alpha_{A2} } \right)\cos \left( {\alpha_{A3} - \varphi } \right)}}{{\cos \left( {\alpha_{A3} - \varphi - \delta } \right)}} \hfill \\ + \frac{{\left( {\frac{{h_{3} }}{B}\cot \alpha_{B3} } \right)\cos \left( {\alpha_{B3} + \varphi } \right)}}{{\cos \left( {\alpha_{B3} + \varphi + \delta } \right)}} \hfill \\ \end{gathered}$$
$$\begin{aligned} b = & 2\frac{{h_{1} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} + 0.5\frac{{h_{1} }}{B}\cot \alpha _{{B1}} } \right)\left[ {\frac{{\left( {1 - k_{h} } \right)\sin \left( {\alpha _{{B1}} + \varphi } \right) - k_{h} \cos \left( {\alpha _{{B1}} + \varphi } \right)}}{{\cos \left( {\alpha _{{B1}} + \varphi + \delta } \right)}}} \right] \\ & \quad - 2\frac{{h_{1} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{B1}} + \varphi _{1} } \right) + k_{h} \cos \left( {\alpha _{{B1}} + \varphi } \right)\tan \varphi }}{{\cos \left( {\alpha _{{B1}} + \varphi + \delta } \right)}}} \right] \\ & \quad - \frac{{2\frac{{h_{1} }}{B}\frac{{h_{3} }}{B}\cot \alpha _{{B3}} \left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} } \right)}}{{\left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} + \frac{{h_{2} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{B2}} + k_{h} \cos \alpha _{{B2}} }}{{\cos \alpha _{{B2}} }}} \right] \\ & \quad + 2\frac{{h_{1} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{B2}} + k_{h} \cos \alpha _{{B2}} \tan \varphi }}{{\cos \alpha _{{B2}} }}} \right] \\ & \quad + 2\frac{{\gamma _{w} }}{\gamma }\frac{{h_{2} }}{B}\left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + 0.5\frac{{h_{2} }}{B}\cot \alpha _{{B2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{B2}} - k_{h} \cos \alpha _{{B2}} }}{{\cos \alpha _{{B2}} }}} \right] \\ & \quad - 2\frac{{\gamma _{w} }}{\gamma }\frac{{h_{2} }}{B}\frac{{h_{3} }}{B}\cot \alpha _{{B3}} \left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{B2}} + k_{h} \cos \alpha _{{B2}} }}{{\cos \alpha _{{B2}} }}} \right] \\ & \quad + \frac{{2\frac{{h_{1} }}{B}\frac{{h_{3} }}{B}\cot \alpha _{{B3}} \left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} } \right)}}{{\left( {\frac{{h_{3} }}{B}\cot \alpha _{{B3}} + \frac{{h_{2} }}{B}\cot \alpha _{{B2}} + \frac{{h_{2} }}{B}} \right)}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{B3}} + \varphi } \right) + k_{h} \cos \left( {\alpha _{{B3}} + \varphi } \right)\tan \varphi }}{{\cos \left( {\alpha _{{B3}} + \varphi + \delta } \right)}}} \right] \\ & \quad + 2\frac{{\gamma _{w} }}{\gamma }\frac{{h_{2} }}{B}\frac{{h_{3} }}{B}\cot \alpha _{{B3}} \left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{B3}} + \varphi } \right) + k_{h} \cos \left( {\alpha _{{B3}} + \varphi } \right)\tan \varphi }}{{\cos \left( {\alpha _{{B3}} + \varphi + \delta } \right)}}} \right] \\ & \quad + \frac{{2\frac{{h_{1} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} } \right)\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} - \frac{{h_{2} }}{B}\cot \alpha _{{A2}} } \right)}}{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} } \right) + \frac{{h_{2} }}{B}}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{A2}} + k_{h} \cos \alpha _{{A2}} \tan \varphi }}{{\cos \alpha _{{A2}} }}} \right] \\ & \quad - \frac{{2\frac{{h_{1} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} - \frac{{h_{2} }}{B}\cot \alpha _{{A2}} } \right)}}{{\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} } \right) + \frac{{h_{2} }}{B}}}\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{A3}} - \varphi } \right) + k_{h} \cos \left( {\alpha _{{A3}} - \varphi } \right)\tan \varphi }}{{\cos \left( {\alpha _{{A3}} - \varphi - \delta } \right)}}} \right] \\ & \quad + 2\frac{{h_{1} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{A1}} - \varphi } \right) + k_{h} \cos \left( {\alpha _{{A1}} - \varphi } \right)\tan \varphi }}{{\cos \left( {\alpha _{{A1}} - \varphi - \delta } \right)}}} \right] \\ & \quad - 2\frac{{h_{1} }}{B}\left( {1 - 0.5\frac{{h_{1} }}{B}\cot \alpha _{{A1}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{A1}} - \varphi } \right) + k_{h} \cos \left( {\alpha _{{A1}} - \varphi } \right)}}{{\cos \left( {\alpha _{{A1}} - \varphi - \delta } \right)}}} \right] \\ & \quad - 2\frac{{h_{1} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} } \right)\left[ {\frac{{\left( {1 - K_{v} } \right)\sin \alpha _{{A2}} + K_{h} \cos \alpha _{{A2}} \tan \varphi _{{ave}} }}{{\cos \alpha _{{A2}} }}} \right] \\ & \quad + 2\frac{{\gamma _{w} }}{\gamma }\frac{{h_{2} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} - \frac{{h_{2} }}{B}\cot \alpha _{{A2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{A2}} + k_{h} \cos \alpha _{{A2}} \tan \varphi }}{{\cos \alpha _{{A2}} }}} \right] \\ & \quad - 2\frac{{\gamma _{w} }}{{\bar{\gamma }}}\frac{{h_{2} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} - 0.5\frac{{h_{2} }}{B}\cot \alpha _{{A2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \alpha _{{A2}} + k_{h} \cos \alpha _{{A2}} }}{{\cos \alpha _{{A2}} }}} \right] \\ & \quad - 2\frac{{\gamma _{w} }}{\gamma }\frac{{h_{2} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} - \frac{{h_{2} }}{B}\cot \alpha _{{A2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{A3}} - \varphi } \right) + k_{h} \cos \left( {\alpha _{{A3}} - \varphi } \right)\tan \varphi }}{{\cos \left( {\alpha _{{A3}} - \varphi - \delta } \right)}}} \right] \\ & \quad + \left( {\frac{{h_{3} }}{B}} \right)^{2} \cot \alpha _{{B3}} \left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{B3}} + \varphi } \right) - k_{h} \cos \left( {\alpha _{{B3}} + \varphi } \right)}}{{\cos \left( {\alpha _{{B3}} + \varphi + \delta } \right)}}} \right] \\ & \quad - \frac{{h_{3} }}{B}\left( {1 - \frac{{h_{1} }}{B}\cot \alpha _{{A1}} - \frac{{h_{2} }}{B}\cot \alpha _{{A2}} } \right)\left[ {\frac{{\left( {1 - k_{v} } \right)\sin \left( {\alpha _{{A3}} - \varphi } \right) + k_{h} \cos \left( {\alpha _{{A3}} - \varphi } \right)}}{{\cos \left( {\alpha _{{A3}} - \varphi - \delta } \right)}}} \right] \\ \end{aligned}$$

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Haghsheno, H., Jamshidi Chenari, R., Javankhoshdel, S. et al. Seismic Bearing Capacity of Shallow Strip Footings on Sand Deposits with Weak Inter-layer. Geotech Geol Eng 38, 6741–6754 (2020). https://doi.org/10.1007/s10706-020-01466-4

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