Optimal location of piles in slope stabilization by limit analysis

Abstract

Many studies have been conducted to establish the optimal location of a row of piles to reinforce and stabilize slopes. However, the results obtained are very different, and in some cases even inconsistent and contradictory. The factor of safety of piled slopes is determined by the magnitude of resistive forces exerted by the piles on the slope. At the same time, the maximum retaining forces provided by the piles are also affected by the pile position. In this paper, the problem of the optimal location of piles used to stabilize slopes is analyzed using a combination of limit slope stability analysis and the theory of Ito and Matsui (Soils Found 15:43–59, 8) to calculate limit lateral loads on piles. Using an illustrative example slope, some of the issues including the most effective position, the most suitable position, and the position with the largest safety factor are discussed. The results show that the most effective pile position, the most suitable pile position, and pile position where the factor of safety can take maximum value are different from each other for a given slope.

Appendix

$$ \begin{aligned} f_{1} & = \frac{{(3\tan \varphi_{\text{t}} \cos \theta_{\text{h}} + \sin \theta_{\text{h}} )\exp [3(\theta_{\text{h}} - \theta_{ 0} )\tan \varphi_{\text{t}} ]}}{{3(1 + 9\tan^{2} \varphi_{\text{t}} )}} \\ & \quad - \frac{{3\tan \varphi_{\text{t}} \cos \theta_{ 0} + \sin \theta_{ 0} }}{{3(1 + 9\tan^{2} \varphi_{\text{t}} )}} \\ f_{2} & = \frac{1}{6}\frac{L}{{r_{0} }}\left( {2\cos \theta_{0} - \frac{L}{{r_{0} }}\cos \alpha } \right)\sin (\theta_{0} + \alpha ) \\ f_{3} & = \frac{{\exp [(\theta_{\text{h}} - \theta_{ 0} )\tan \varphi_{\text{t}} ]}}{6}\left[ {\sin (\theta_{\text{h}} - \theta_{ 0} ) - \frac{L}{{r_{0} }}\sin (\theta_{\text{h}} + \alpha )} \right] \\ & \quad \times \left\{ {\cos \theta_{ 0} - \frac{L}{{r_{0} }}\cos \alpha + \cos \theta_{\text{h}} \cdot \exp [(\theta_{\text{h}} - \theta_{ 0} )\tan \varphi_{\text{t}} ]} \right\} \\ f_{4} & = \left( {\frac{H}{{r_{0} }}} \right)^{2} \frac{{\sin (\beta - \beta^{\prime } )}}{{2\sin \beta \sin \beta^{\prime } }} \\ & \quad \times \left( {\cos \theta_{0} - \frac{L}{{r_{0} }}\cos \alpha - \frac{H}{{3r_{0} }}[\cot \beta + \cot \beta^{\prime } ]} \right) \\ f_{5} & = \frac{1}{2\tan \varphi }\{ \exp [2(\theta_{\text{h}} - \theta_{ 0} )\tan \varphi_{\text{t}} ] - 1\} \\ \end{aligned} $$

where β is slope angle, tan φ _t = tan φ/F, and L is the distance between the failure surface at the top of the slope and the edge of the slope. It is given by

$$ L = \frac{{r_{0} \sin (\theta_{\text{h}} - \theta_{ 0} )}}{{\sin (\theta_{\text{h}} + \beta^{\prime \prime } )}} - \frac{{r_{0} \sin (\theta_{\text{h}} + \beta^{\prime \prime } )}}{{\sin (\theta_{\text{h}} + \beta '')\sin (\beta^{\prime } - \beta^{\prime \prime } )}}\{ \sin (\theta_{\text{h}} + \beta^{\prime \prime } )\exp [(\theta_{\text{h}} - \theta_{ 0} )\tan \phi_{\text{t}} ] - \sin (\theta_{ 0} + \beta^{\prime \prime } )\} $$

$$ \frac{H}{{r_{0} }} = \frac{{\sin \beta^{\prime } }}{{\sin (\beta^{\prime } - \beta^{\prime \prime } )}} \times \left\{ {\sin (\theta_{\text{h}} + \beta^{\prime \prime } )\exp [(\theta_{\text{h}} - \theta_{ 0} )\tan \varphi_{\text{t}} ] - \sin (\theta_{ 0} + \beta^{\prime \prime } )} \right\} $$