Effects of seismic force and pore water pressure on stability of 3D unsaturated hillslopes

Abstract

Three-dimensional (3D) kinematic limit analysis of unsaturated hillslopes is presented in this paper. Different from the traditional two-dimensional (2D) mechanism based on the infinite slope model, the 3D failure mechanism is more rational for its advantage in taking the out-of-plane geometries and soil properties into consideration. The soils in engineering practice are mostly unsaturated in nature and are commonly characterized with an arbitrary distribution of the moisture content, i.e., the matric suction, and therefore the formation of the work balance equation becomes much more elaborated using traditional methods. To tackle the nonlinear features of matric suction, a semi-analytical method is presented and is validated through comparisons with the benchmark solutions. The hillslope stability under seismic and pore water pressure conditions are both numerically studied. The results are presented in forms of graphs for a practical range of parameters, indicating the significance in accounting for the influences of 3D constraints, seismic loads and pore water pressures in hillslope stability assessments.

Appendices

Appendix 1: Details about the ILSM

The work rates provided by soil gravity and seismic forces in saturated zone are given as follows:

$$\begin{gathered} W_{{\gamma_{{{\text{sat}}}} }}^{{}} + W_{{\text{s}}}^{{}} = \frac{1}{4}br_{0}^{3} \omega \gamma_{{{\text{sat}}}} \int_{{\theta_{C} }}^{{\theta_{{C^{\prime}}} }} {\left( {\cos \theta + k_{{{\text{hE}}}} \sin \theta } \right)} \\ \times \left\{ {\exp \left[ {\left( {\theta - \theta_{0} } \right)\tan \phi^{\prime}} \right] + \frac{{\sin \left( {\theta_{0} + \beta } \right)}}{{\sin \left( {\theta + \beta } \right)}}} \right\}^{2} \left\{ {\exp \left[ {\left( {\theta - \theta_{0} } \right)\tan \phi^{\prime}} \right] - \frac{{\sin \left( {\theta_{0} + \beta } \right)}}{{\sin \left( {\theta + \beta } \right)}}} \right\}d\theta \\ + 2\omega \gamma_{{{\text{sat}}}} \int_{{\theta_{C} }}^{{\theta_{{C^{\prime}}} }} {\int_{0}^{{\sqrt {R^{2} - a^{2} } }} {\int_{a}^{{\sqrt {R^{2} - x^{2} } }} {\left( {\cos \theta + k_{{{\text{hE}}}} \sin \theta } \right)\left( {r_{{\text{m}}} + y} \right)^{2} dydx} } d\theta } \\ \end{gathered}$$

(27)

The variables θ_C, θ_C′, R, a and r_m are given as follows:

$$\left( {r_{0} \exp \left[ {\left( {\theta_{C} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{C} - r_{0} \sin \theta_{0} } \right) - \left( {r_{0} \cos \theta_{0} - r_{0} \exp \left[ {\left( {\theta_{C} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{C} } \right)\tan \beta = H$$

(28)

$$\frac{{\exp \left[ {\left( {\theta_{{C^{\prime}}} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{{C^{\prime}}} - \exp \left[ {\left( {\theta_{C} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{C} }}{{\exp \left[ {\left( {\theta_{C} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{C} - \exp \left[ {\left( {\theta_{{C^{\prime}}} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{{C^{\prime}}} }} = \tan \beta$$

(29)

$$R = \frac{1}{2}\left( {r - r^{\prime}} \right)$$

(30)

$$r_{m} = \frac{1}{2}\left( {r + r^{\prime}} \right)$$

(31)

$$a = \frac{{r_{0} \sin \left( {\theta_{0} + \beta } \right)}}{{\sin \left( {\theta + \beta } \right)}} - r_{m}$$

(32)

For cohesive and frictional soils, the expression of $D_{{c^{\prime } }}^{3D}$ is

$$D_{{c^{\prime}}}^{{}} = - \frac{{2c^{\prime}\omega r_{0}^{2} \sin^{2} \left( {\theta_{0} + \beta } \right)}}{{\tan \phi^{\prime}}}\int_{{\theta_{0} }}^{{\theta_{h} }} {\frac{{\cos \left( {\theta + \beta } \right)}}{{\sin^{3} \left( {\theta + \beta } \right)}}\left( {\sqrt {R^{2} - a^{2} } + b} \right)d\theta }$$

(33)

For purely cohesive soils, the expression of $D_{{c^{\prime } }}$ is given as

$$D_{{c^{\prime}}}^{{}} = 2\omega c^{\prime}R\int_{{\theta_{0} }}^{{\theta_{h} }} {\int_{a}^{R} {\left( {r_{m} + y} \right)^{2} dyd\theta } } + \frac{{bc^{\prime}\omega r_{0}^{2} }}{{2\tan \phi^{\prime}}}\left( {\exp \left[ {2\left( {\theta_{h} - \theta_{0} } \right)\tan \phi^{\prime}} \right] - 1} \right)$$

(34)

The variables θ_j, θ_j′, θ_j-1, θ_j-1′ and a_j in Fig. 5 can be found from the geometrical relation as

$$\left\{ {r_{0} \exp \left[ {\left( {\theta_{j} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{j} - r_{0} \sin \theta_{0} } \right\} - \left\{ {r_{0} \cos \theta_{0} - r_{0} \exp \left[ {\left( {\theta_{j} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{j} } \right\}\tan \beta = \left( {m - j} \right)h$$

(35)

$$\frac{{\exp \left[ {\left( {\theta_{{j^{\prime}}} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{{j^{\prime}}} - \exp \left[ {\left( {\theta_{j} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{j} }}{{\exp \left[ {\left( {\theta_{j} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{j} - \exp \left[ {\left( {\theta_{{j^{\prime}}} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{{j^{\prime}}} }} = \tan \beta$$

(36)

$$\begin{gathered} \left\{ {r_{0} \exp \left[ {\left( {\theta_{j - 1} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{j - 1} - r_{0} \sin \theta_{0} } \right\} \hfill \\ - \left\{ {r_{0} \cos \theta_{0} - r_{0} \exp \left[ {\left( {\theta_{j - 1} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{j - 1} } \right\}\tan \beta = \left[ {m - \left( {j - 1} \right)} \right]h \hfill \\ \end{gathered}$$

(37)

$$\frac{{\exp \left[ {\left( {\theta_{{j - 1^{\prime}}} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{{j - 1^{\prime}}} - \exp \left[ {\left( {\theta_{j - 1} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \theta_{j - 1} }}{{\exp \left[ {\left( {\theta_{j - 1} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{j - 1} - \exp \left[ {\left( {\theta_{{j - 1^{\prime}}} - \theta_{0} } \right)\tan \phi^{\prime}} \right]\cos \theta_{{j - 1^{\prime}}} }} = \tan \beta$$

(38)

$$a_{j} = \frac{{r_{0} \sin \left( {\theta_{0} + \beta } \right) + \left( {m - j} \right)h\cos \beta }}{{\sin \left( {\theta + \beta } \right)}} - r_{m}$$

(39)

The variable z₁ in Eq. (21) is written as

$$z_{1} = H - r_{0} \left\{ {\frac{{\exp \left[ {\left( {\theta - \theta_{0} } \right)\tan \phi^{\prime}} \right]\sin \left( {\theta + \beta } \right) - \sin \left( {\theta_{0} + \beta } \right)}}{\cos \beta }} \right\}$$

(40)

Appendix 2: Details about the ILSM with the spherical cap failure mechanism

The volume of each subsoil layer is

$$V_{j} = 0.5\left( {S_{j} + S_{j - 1} } \right)h$$

(41)

where S_j and S_j-1 are the areas of upper and bottom surfaces of subsoil layer j, respectively; and h is the thickness of each subsoil layer. The expressions of S_j, S_j-1 and h can be expressed as

$$S_{j} = \pi r_{j}^{2} = \pi R^{{\prime}{2}} \cos^{2} \left( {\beta + \theta_{j} } \right)$$

(42)

$$S_{j - 1} = \pi r_{j - 1}^{2} = \pi R^{{\prime}{2}} \cos^{2} \left( {\beta + \theta_{j - 1} } \right)$$

(43)

$$h = \frac{{R^{\prime} - R^{\prime}\cos \delta }}{m}$$

(44)

where θ_j and θ_j-1 are angles shown in Fig. 14 and can be found from the following trigonometric relation:

$$\left[ {\sin \theta_{j} - \cos \left( {\delta + \beta } \right)} \right] - \left[ {\sin \left( {\delta + \beta } \right) - \cos \theta_{j} } \right]\tan \beta = \frac{{\left( {m - j} \right)\left( {1 - \cos \delta } \right)}}{m\cos \beta }$$

(45)

$$\left[ {\sin \theta_{j - 1} - \cos \left( {\delta + \beta } \right)} \right] - \left[ {\sin \left( {\delta + \beta } \right) - \cos \theta_{j - 1} } \right]\tan \beta = \frac{{\left( {m + 1 - j} \right)\left( {1 - \cos \delta } \right)}}{m\cos \beta }$$

(46)

$$l_{c} = R^{\prime}\cos \delta + 0.5h\left( {2m - 2j + 1} \right)$$

(47)