A method for designing the longitudinal spacing of slope-stabilising shafts

Abstract

The work at hand deals with the design of the longitudinal spacing among rows of closely spaced large-diameter shafts used to stabilise a precarious slope. The problem under consideration is idealised through a conceptual framework where an unstable mass of an infinitely long slope pushes a stable portion of soil adjacent to shafts, leading to failure along a slip surface passing through the upper end of the reinforcement elements. By exploiting the upper bound theorem of plastic collapse, a closed-form solution is derived for the load required for the failure of the stable mass as a function of geometrical and mechanical parameters of the slope and the soil. Results are validated through physical model tests by means of geotechnical centrifuge. Given the satisfactory agreement between analytical and experimental results, the model is extended to evaluate the safety conditions of the reinforced slope.

Appendix

1.1 Derivation of Equation (4)

With reference to the triangle OCB (see Figs. 4 and 5), the law of sines assures that:

$$\frac{{\sin {\text{O}}\widehat{\text{C}}{\text{B}}}}{{\overline{\text{OB}} }}=\frac{{\sin {\text{C}}\widehat{\text{O}}{\text{B}}}}{{\overline{\text{CB}} }} \,$$

(16)

By making use of the law of cosines:

$$\sin \Delta_{2} = \sin {\text{O}}\widehat{\text{C}}{\text{B}} = \sin {\text{C}}\widehat{\text{O}}{\text{B}}\frac{{\overline{\text{OB}} }}{{\sqrt {\overline{\text{OB}}^{2} + \overline{\text{OC}}^{2} - 2\overline{{ \cdot {\text{OB}}}} \cdot \overline{\text{OC}} \cdot \cos {\text{C}}\widehat{\text{O}}{\text{B}}} }}$$

(17)

and therefore:

$$ \begin{aligned} \sin \Delta_{2} = \sin \left( {\theta_{s} - \theta_{0} } \right) \hfill \\ \frac{{r_{0} }}{{\sqrt {r_{0}^{2} + r_{0}^{2} e^{{2\left( {\theta_{s} - \theta_{0} } \right)\tan \varphi }} - 2r_{0}^{2} e^{{\left( {\theta_{s} - \theta_{0} } \right)\tan \varphi }} \cos \left( {\theta_{s} - \theta_{0} } \right)} }} \hfill \\ = \sin \left( {\theta_{s} - \theta_{0} } \right) \hfill \\ \left( {1 + e^{{2\left( {\theta_{s} - \theta_{0} } \right)\tan \varphi }} - 2e^{{\left( {\theta_{s} - \theta_{0} } \right)\tan \varphi }} \cos \left( {\theta_{s} - \theta_{0} } \right)} \right)^{{ - \frac{1}{2}}} \hfill \\ \end{aligned} $$

(18)

from which we obtain Eq. (4).

1.2 Derivation of Equation (11)

Solving Eq. (10) for $\dot{Q}$ and dividing both sides of the equation by $\dot{\omega }\,\gamma H^{2} \left( {r_{0} \cos \varphi - h} \right)$:

$$ \frac{{q_{\lim } }}{\gamma H} = \frac{\begin{aligned} \dot{\omega }\,\gamma \left[ { - \frac{1}{6}Hr_{0} \left( {H\sin \alpha - 2r_{0} \cos \theta_{0} } \right)\sin \varphi } \right]\, \hfill \\ + \dot{\omega }\,\gamma \,\left[ { - \frac{1}{6}r_{s} \xi_{s} \left( \begin{aligned} 2H\sin \alpha - 2r_{0} \cos \theta_{0} \hfill \\ + \xi_{s} \cos \alpha \hfill \\ \end{aligned} \right)\sin \left( {\theta_{s} + \alpha } \right)} \right] \hfill \\ - \dot{\omega }\,\gamma \,\frac{\begin{aligned} \left( {3\tan\varphi \cos \theta_{s} + \sin \theta_{s} } \right)r_{s}^{3} \hfill \\ - \left( {3\tan\varphi \cos \theta_{0} + \sin \theta_{0} } \right)r_{0}^{3} \hfill \\ \end{aligned} }{{3\left( {1 + 9\tan^{2} \varphi } \right)}} \hfill \\ + \dot{\omega }\frac{c}{2\tan \varphi }\left( {r_{s}^{2} - r_{0}^{2} } \right) \hfill \\ \end{aligned} }{{\dot{\omega }\,\gamma H^{2} \left( {r_{0} \cos \varphi - h} \right)}} $$

(19)

Multiplying both numerator and denominator of the right side of the above equation by (6/H³) one obtains Eq. (11).