Pseudo-Static Bearing Capacity Analysis of Shallow Strip Footing over Two-Layered Soil Considering Punching Shear Failure

Abstract

In present analysis, ultimate bearing capacity of footings residing on sub soils comprising of two layers has been investigated for the event of a thin soil layer overlying a weak deposit. Pseudo-static approach has been applied to find out the ultimate bearing capacity of shallow strip footing resting on two layered soil. Ultimate bearing capacity equation was derived as a function of the properties of soils, the footing width and the top soil thickness. Particle swarm optimization algorithm is used to optimize the solution using MATLAB R2009. The paper acquaints a detailed parametric study of the design parameters including the effect of angle of friction, mobilization factor, the ratio of the depth of embedment to the footing width and the ratio of the cohesions of both the layers. Design charts were developed in dimensionless form for very wide range of design parameters. Comparative study of the present analysis has been applied with respect to different analytical, numerical and experimental studies.

Appendices

Appendix 1

$$\begin{aligned} & A_{1} = \frac{{\gamma_{1} k_{p} }}{{2B_{0} \left\{ {k_{h} + \left( {1 \pm k_{v} } \right)} \right\}}}\left\{ {(1 + m)\tan \delta_{1} + (m - 1)} \right\} \\ & A_{2} = \frac{{\gamma_{1} D_{f} k_{p} }}{{B_{0} \left\{ {k_{h} + \left( {1 \pm k_{v} } \right)} \right\}}}\left\{ {(1 + m)\tan \delta_{1} + (m - 1)} \right\} \\ & \quad - \frac{{c_{1} k_{p} }}{{B_{0} \left\{ {k_{h} + \left( {1 \pm k_{v} } \right)} \right\}}}\left\{ {(1 + m)\tan \delta_{1} + (m - 1)} \right\}z - \frac{{c_{1} (1 + m)}}{{B_{0} \left\{ {k_{h} + \left( {1 \pm k_{v} } \right)} \right\}}} - \gamma_{1} \\ & p_{p\gamma } = \frac{1}{2}\gamma_{2} \frac{{B_{0}^{2} \sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin^{3} \alpha_{p} \,k_{p(static)} - \frac{1}{8}\gamma_{2} \frac{{B_{0}^{2} \sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin 2\alpha_{p} \sin \alpha_{p} \;k_{h} \\ & \quad + \frac{1}{4}\gamma_{2} \frac{{B_{0}^{2} \sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin 2\alpha_{p} \cos \alpha_{p} (1 \pm k_{v} ) - \frac{1}{4}\gamma_{2} \frac{{B_{0}^{2} \sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin^{3} \alpha_{p} \,k_{p(Decrement)} \\ & \quad + \frac{1}{2}\frac{1}{{\cos \phi_{2} \left( {1 + 9\tan^{2} \phi_{2} } \right)}}\frac{{B_{0}^{2} \sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\gamma_{2} \left[ {e^{{3\beta \tan \phi_{2} }} \left( {3\tan \phi_{2} \sin \beta - \cos \beta } \right) + 1} \right] \\ & \left[ {\frac{{\sin \nu - \cos \left( {\nu - \vartheta } \right)\cos \alpha_{A2} }}{\sin \nu \cos \vartheta }} \right]\left[ {\left( {1 \pm k_{v} } \right) - \cot \left( {\nu + \alpha_{A2} - \pi /2} \right)k_{h} } \right] \\ & p_{pq} = \gamma_{1} \left( {D_{f} + h_{1} } \right)\frac{{B_{0} \sin \alpha_{A1} }}{{\sin \left( {\alpha_{A1} + \alpha_{A2} } \right)}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}k_{p(static)} \sin^{2} \alpha_{p} - \frac{2}{3}\gamma_{1} \left( {D_{f} + h_{1} } \right)\frac{{B_{0} \sin \alpha_{A1} }}{{\sin \left( {\alpha_{A1} + \alpha_{A2} } \right)}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }} \\ & k_{p(Decrement)} \sin^{2} \alpha_{p} + \left( {D_{f} + h_{1} } \right)\gamma_{1} \frac{{B_{0} \sin \alpha_{A1} }}{{\sin \left( {\alpha_{A1} + \alpha_{A2} } \right)}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\cos^{2} \alpha_{p} \left( {1 \pm k_{v} } \right) \\ & p_{pc} = 2c_{2} \frac{{B_{0} \sin \alpha_{A1} }}{{\sin (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin^{2} \alpha_{p} \sqrt {k_{p(static)} } + c_{2} \frac{{B_{0} \sin \alpha_{A1} }}{{\sin (\alpha_{A1} + \alpha_{A2} )\tan \phi_{2} }}\frac{{\left( {e^{{2\beta \tan \phi_{2} }} - 1} \right)}}{{\cos \phi_{2} }} \\ \end{aligned}$$

Appendix 2

$$\begin{aligned} & x = \left( {\frac{{\gamma_{2} }}{{\bar{\gamma }}}} \right)\frac{{\sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin^{3} \alpha_{p} k_{p(static)} - \frac{1}{4}\left( {\frac{{\gamma_{2} }}{{\bar{\gamma }}}} \right)\frac{{\sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin 2\alpha_{p} \sin \alpha_{p} k_{h} \\ & \quad + \frac{1}{2}\left( {\frac{{\gamma_{2} }}{{\bar{\gamma }}}} \right)\frac{{\sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin 2\alpha_{p} \cos \alpha_{p} (1 \pm k_{v} ) - \frac{1}{2}\left( {\frac{{\gamma_{2} }}{{\bar{\gamma }}}} \right)\frac{{\sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{3\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin^{3} \alpha_{p} k_{p(Decrement)} \\ & \quad + \frac{1}{{\left( {1 + 9\tan^{2} \phi_{2} } \right)}}\frac{{\sin^{2} \alpha_{A1} }}{{\sin^{2} (\alpha_{A1} + \alpha_{A2} )}}\frac{{\gamma_{2} }}{{\bar{\gamma }}}\frac{1}{{\cos \phi_{2} }}\left[ {e^{{3\beta \tan \phi_{2} }} \left( {3\tan \phi_{2} \sin \beta - \cos \beta } \right) + 1} \right]\,\left[ {\frac{{\sin \nu - \cos \left( {\nu - \vartheta } \right)\cos \alpha_{A2} }}{\sin \nu \,\cos \vartheta }} \right] \\ & \quad \left[ {\left( {1 \pm k_{v} } \right) - k_{h} \tan \left( {\nu + \alpha_{A2} - \frac{\pi }{2}} \right)} \right] \\ & y = 2\frac{{\gamma_{1} }}{{\bar{\gamma }}}\frac{{\left( {D_{f} + h_{1} } \right)}}{{B_{0} }}\frac{{\sin \alpha_{A1} }}{{\sin (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}k_{p(static)} \sin^{2} \alpha_{p} - \frac{4}{3}\left( {\frac{{\gamma_{1} }}{{\bar{\gamma }}}} \right)\left( {\frac{{\left( {D_{f} + h_{1} } \right)}}{{B_{0} }}} \right)\frac{{\sin \alpha_{A1} }}{{\sin (\alpha_{A1} + \alpha_{A2} )}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }} \\ & \quad k_{p(Decrement)} \sin^{2} \alpha_{p} + 2\frac{{h_{1} + D_{f} }}{{B_{0} }}\frac{{\gamma_{1} }}{{\bar{\gamma }}}\frac{{\sin \alpha_{A1} }}{{\sin \left( {\alpha_{A1} + \alpha_{A2} } \right)}}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\cos^{2} \alpha_{p} \left( {1 \pm k_{v} } \right) \\ & z = \frac{{2\bar{c}}}{{\bar{\gamma }B_{0} }}\frac{{c_{2} }}{{\bar{c}}}\left\{ {\frac{{2\sin \alpha_{A1} }}{{\sin \left( {\alpha_{A1} + \alpha_{A2} } \right)}}} \right\}\frac{{e^{{2\beta \tan \phi_{2} }} }}{{\cos \phi_{2} }}\sin^{2} \alpha {}_{p}\sqrt {k_{p(static)} } + \frac{{2\bar{c}}}{{\bar{\gamma }B_{0} }}\frac{{c_{2} }}{{\bar{c}}}\frac{{\sin \alpha_{A1} }}{{\sin \left( {\alpha_{A1} + \alpha_{A2} } \right)\tan \phi_{2} }}\frac{{\left( {e^{{2\beta \tan \phi_{2} }} - 1} \right)}}{{\cos \phi_{2} }} \\ \end{aligned}$$

$x_{1}$, $y_{1}$ and $z_{1}$ can be obtained by substituting the angle of internal friction $\phi_{2}$ by $\phi_{m2}$ and changing the wedge angle $\alpha_{A1}$ to $\alpha_{A2}$ and $\alpha_{A2}$ to $\alpha_{A1}$.