3D Seismic Stability Analysis of Bench Slope with Pile Reinforcement

Abstract

Pile reinforcement has been widely applied to ensure the stability of slopes. However, few investigations have been conducted on multi-stage or bench slopes reinforced with piles. To give a brief stability description of such slopes, a three-dimensional kinematic analysis of pile reinforced bench slopes under static and seismic effects is carried out in this paper. The analytical expression of the seismic safety factor is derived by combining the pseudo-static method and the strength reduction method. Then comparisons were carried out to verify the obtained solutions. A detailed parametric study was also implemented to explore the variation rule of the safety factor. The effect of several factors, such as the pile location, friction angle, horizontal seismic force coefficient, slope width, bench width, were discussed. The results indicate that the pile location near/on the bench can raise the safety factor by nearly 30% than those on the slope toe. Furthermore, a 20% increase in safety factor can be achieved as the length of the bench increases from 0 to 4 m. Therefore, ignoring the benches in the calculation process will lead to a rather conservative solution.

Appendix 1

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

$$f_{2} (\theta_{h} ,\theta_{0} ) = \frac{1}{6}\frac{L}{{r_{0} }}(2\cos \theta_{0} - \frac{L}{{r_{0} }})\sin (\theta_{0} )$$

$$f_{3} = \frac{{\alpha_{2} }}{3}\frac{H}{{r_{0} }}[(\cos \theta_{0} )^{2} + (\frac{{L_{1} }}{{r_{0} }})^{2} - 2\frac{{L_{1} }}{{r_{0} }}\cos \theta_{0} + \sin \theta_{0} \cot \beta_{2} (\cos \theta_{0} - \frac{{L_{1} }}{{r_{0} }}) \, - \frac{{\alpha_{2} }}{2}\frac{H}{{r_{0} }}\cot \beta_{2} (\cos \theta_{0} - \frac{{L_{1} }}{{r_{0} }} + \sin \theta_{0} \cot \beta_{2} )]$$

$$f_{4} = \frac{{L_{2} }}{{r_{0} }}\{ \alpha_{2} \frac{H}{{r_{0} }}[\cos \theta_{0} + \frac{{L_{1} }}{{r_{0} }} - \alpha_{2} \frac{H}{{r_{0} }}\cot \beta_{2} - \sin \theta_{0} \cot \beta_{2} - \frac{1}{2}\frac{{L_{2} }}{{r_{0} }}] - \frac{{L_{1} }}{{r_{0} }}\sin \theta_{0} + \sin \theta_{0} \cos \theta_{0} - \frac{1}{2}\frac{{L_{2} }}{{r_{0} }}\sin \theta_{0} \}$$

$$f_{5} (\theta_{h} ,\theta_{0} ) = \frac{1}{3}\frac{H}{{r_{0} }}\{ \cot \beta_{1} [\cos \theta_{h} + \sin \theta_{h} \cot \beta_{1} ] + [(\cos \theta_{h} )^{2} + \sin \theta_{h} \cos \theta_{h} \cot \beta_{1} ]e^{{(\theta_{h} - \theta_{0} )\tan \varphi }} \} e^{{(\theta_{h} - \theta_{0} )\tan \phi }}$$

$$f_{6} (\theta_{h} ,\theta_{0} ) = \frac{1}{{3(1 + 9\tan^{2} \varphi )}}\left\{ {(3\tan \varphi \sin \theta_{h} - \cos \theta_{h} )\exp \left[ {3(\theta_{h} - \theta_{0} )\tan \varphi } \right] - (3\tan \varphi \sin \theta_{0} - \cos \theta_{0} )} \right\}$$

$$f_{7} (\theta_{h} ,\theta_{0} ) = \frac{1}{3}\frac{{L_{1} }}{{r_{0} }}(\sin \theta_{0} )^{2}$$

$$f_{8} (\theta_{h} ,\theta_{0} ) = \frac{{\alpha_{2} }}{3}\frac{H}{{r_{0} }}[\cot \beta_{2} (\sin \theta_{0} )^{2} + \sin \theta_{0} \cos \theta_{0} - \frac{{L_{1} }}{{r_{0} }}\sin \theta_{0} + \frac{{\alpha_{2} }}{2}\frac{H}{{r_{0} }}\cot \beta_{2} \sin \theta_{0} + \frac{{\alpha_{2} }}{2}\frac{H}{{r_{0} }}\cos \theta_{0} - \frac{{\alpha_{2} }}{2}\frac{H}{{r_{0} }}\frac{{L_{1} }}{{r_{0} }}]$$

$$f_{9} (\theta_{h} ,\theta_{0} ) = \frac{1}{3}\frac{{L_{2} }}{{r_{0} }}(\alpha_{2} \frac{H}{{r_{0} }} + \sin \theta_{0} )^{2}$$

$$f_{10} (\theta_{h} ,\theta_{0} ) = \frac{{\alpha_{1} }}{3}\frac{H}{{r_{0} }}(e^{{(\theta_{h} - \theta_{0} )\tan \varphi }} \sin \theta_{h} - \frac{{\alpha_{1} }}{2}\frac{H}{{r_{0} }})(\cos \theta_{h} + \sin \theta_{h} \cot \beta_{1} )e^{{(\theta_{h} - \theta_{0} )\tan \varphi }}$$