A Simplified Analytical Method for Stabilizing Micropile Groups in Slope Engineering

Abstract

Stabilizing micropile groups is a light retaining structure constructed quickly and safely for slope reinforcement in practice. To carry out engineering design of any structure, a simplified analytical procedure for a micropile group with consideration of stability of the piled slope is presented. According to the upper bound theorem of kinematical limit analysis, an analytical method is proposed to evaluate the net thrust force on a micropile group with 3 × 3 layout of piles and the slip surface of the piled slope for a specified factor of safety. Then, internal forces of the micropile group can be computed using plane rigid frame model for the part of the structure above the slip surface under the net thrust force and beam-on-elastic-foundation model for the rest part. A laboratory model test and corresponding 3D-numerical simulation are conducted to verify the proposed method. Moreover, analysis of a practical slope shows that flexural rigidity of a micropile and micropile numbers of a group have a great effect on internal forces of micropiles. In particular, the internal forces are relatively sensitive to pile numbers in a group. However, micropile length and spacing in plane in a group have little effect on the internal forces, which is rather different from traditional stabilizing piles with a large cross section.

Appendix

In Fig. 2, there are following geometric relationships:

$$\frac{X}{{r_{0} }} = \frac{H}{{r_{0} }} \cdot \frac{{\sin \left( {\beta - \beta^{\prime}} \right)}}{{\sin \beta \sin \beta^{\prime}}},$$

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where X is the horizontal distance from the slope toe to the intersection between the log-spiral slip line and the ground outside the toe. H is the height of the slope. β and β′ are dip angles of line JN and JN′ in Fig. 2, respectively.

$$H\cot \beta = \kappa_{1} H\cot \beta_{1} + \kappa_{2} H\cot \beta_{2} + Y,$$

(20)

where κ and β are the ratio of local slope height over the whole slope height and the dip angle of the slope face, respectively. And the subscripts 1 and 2 denote the upslope and downslope in Fig. 2, respectively. Y is the width of bench of the slope.

$$\kappa_{1} + \kappa_{2} { = }1,$$

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$$\frac{L}{{r_{0} }} = \frac{{\sin \left( {\theta_{\text{h}} - \theta_{0} } \right)}}{{\sin \left( {\theta_{\text{h}} + \delta } \right)}} - \frac{{\sin \left( {\theta_{\text{h}} + \beta^{\prime}} \right)}}{{\sin \left( {\theta_{\text{h}} + \delta } \right)\sin \left( {\beta^{\prime} - \delta } \right)}}\left[ {{\text{e}}^{{\left( {\theta_{\text{h}} - \theta_{0} } \right)\tan \varphi_{\text{f}} }} \cdot \sin \left( {\theta_{\text{h}} + \delta } \right) - \sin \left( {\theta_{0} + \delta } \right)} \right].$$

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According to the concept of the gravity work rate [26], the coefficients of the gravity work rate can be derived and expressed as follows:

$$f_{1} = \frac{{\left( {3\tan \varphi_{\text{f}} \cos \theta_{\text{h}} + \sin \theta_{\text{h}} } \right) \cdot {\text{e}}^{{3\left( {\theta_{\text{h}} - \theta_{0} } \right)\tan \varphi_{\text{f}} }} - \left( {3\tan \varphi_{\text{f}} \cdot \cos \theta_{0} + \sin \theta_{0} } \right)}}{{3\left( {1 + 9\tan^{2} \varphi_{\text{f}} } \right)}},$$

(23)

$$f_{2} = \frac{1}{6}\frac{L}{{r_{0} }}\left[ {2\cos \theta_{0} - \frac{L}{{r_{0} }}\cos \delta } \right]\sin \left( {\theta_{0} + \delta } \right){\kern 1pt} ,$$

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$$\begin{aligned} f_{3} &= \frac{1}{6}\kappa_{1} \frac{H}{{r_{0} }}\left( {2\left( {\cos \theta_{0} - \frac{L}{{r_{0} }}\cos \delta } \right) - \kappa_{1} \frac{H}{{r_{0} }}\cot \beta_{1} } \right) \hfill \\ &\quad \cdot \left[ {\left( {\cos \theta_{0} - \frac{L}{{r_{0} }}\cos \delta } \right) + \cot \beta_{1} \left( {\sin \theta_{0} + \frac{L}{{r_{0} }}\sin \delta } \right)} \right], \hfill \\ \end{aligned}$$

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$$f_{4} = \frac{1}{6} \cdot \frac{Y}{{r_{0} }} \cdot \left( {\kappa_{1} \frac{H}{{r_{0} }} + \frac{L}{{r_{0} }}\sin \delta + \sin \theta_{0} } \right) \cdot \left( \begin{aligned} &2\cos \theta_{0} - 2\frac{L}{{r_{0} }}\cos \delta \hfill \\ &\quad - 2\kappa_{1} \frac{H}{{r_{0} }}\cot \beta_{1} - \frac{Y}{{r_{0} }} \hfill \\ \end{aligned} \right),$$

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$$f_{5} = \frac{1}{6}\kappa_{2} \frac{H}{{r_{0} }}\left[ \begin{aligned} &\left( {{\text{e}}^{{\left( {\theta_{\text{h}} - \theta_{0} } \right)\tan \varphi_{\text{f}} }} \cdot \cos \theta_{\text{h}} + \frac{X}{{r_{0} }}} \right) \hfill \\ &\quad + \cot \beta_{2} \cdot \left( {\frac{H}{{r_{0} }} + \frac{L}{{r_{0} }}\sin \delta + \sin \theta_{0} } \right) \hfill \\ \end{aligned} \right] \cdot \left[ \begin{aligned} &2\left( {{\text{e}}^{{\left( {\theta_{\text{h}} - \theta_{0} } \right)\tan \varphi_{\text{f}} }} \cdot \cos \theta_{\text{h}} + \frac{X}{{r_{0} }}} \right) \hfill \\ &\quad + \kappa_{2} \frac{H}{{r_{0} }}\cot \beta_{2} \hfill \\ \end{aligned} \right],$$

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$$f_{6} = \frac{1}{6}\frac{X}{{r_{0} }}\left( {2{\text{e}}^{{\left( {\theta_{\text{h}} - \theta_{0} } \right)\tan \varphi_{\text{f}} }} \cdot \cos \theta_{\text{h}} + \frac{X}{{r_{0} }}} \right) \cdot \left( {\frac{H}{{r_{0} }} + \frac{L}{{r_{0} }}\sin \delta + \sin \theta_{0} } \right).$$

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