Slope Stability Analysis Based on Coupled Approach

Abstract

Landslides that can result from high-intensity rainfall can be detrimental to infrastructure and can contribute to loss of human lives. Shallow rainfall-caused landslides are represented as sliding that appears down a surface parallel to sloping ground.

Appendix

Applying the Laplace transform and replacing W by $\bar{W}$, Eq. (6.20) becomes

$$ \frac{{\partial^{2} \bar{W}}}{{\partial z^{2} }} + \frac{{\partial \bar{W}}}{\partial z} - s\bar{W} + W_{0} = 0 $$

(6.60)

The bottom boundary condition (Eq. (6.21b)) and surface top boundary condition (Eq. (6.21c)) for the soil slope are given by

$$ \bar{W}\left( 0 \right) = \frac{{e^{{\alpha \psi_{0} }} }}{s} $$

(6.61)

$$ \left. {\frac{{\partial \bar{W}}}{\partial z} - s\bar{W}} \right|_{z = L} = \frac{{q_{B} }}{s} $$

(6.62)

The general solution to Eq. (6.60) with the boundary conditions that are described by Eqs. (6.61) and (6.62) is expressed as

$$ \bar{W} = \frac{{W_{0} \left( z \right)}}{s} + \left( {q_{B} - q_{A} } \right)e^{{{{\left( {L - z} \right)} \mathord{\left/ {\vphantom {{\left( {L - z} \right)} 2}} \right. \kern-0pt} 2}}} G\left( s \right) $$

(6.63)

in which

$$ G\left( s \right) = \frac{1}{s}\frac{{\sinh \left[ {z\left( {s + 0.25} \right)^{0.5} } \right]}}{{\left( {s + 0.25} \right)^{0.5} \cosh \left[ {L\left( {s + 0.25} \right)^{0.5} } \right] + 0.5\sinh \left[ {L\left( {s + 0.25} \right)^{0.5} } \right]}} $$

(6.64)

The inversion of G(s) is achieved by using the residue theorem as the sum of residues of e^SlG(s) at the poles of G(s). The residue at s = 0 is e^–(L−z)/2 − e^−(L+z)/2. The other poles are acquired by assigning (s + 0.25)^1/2 as a complex number. All the poles are at pure imaginary values of (s + 0.25)^1/2 that is a function of i$\lambda$. We can obtain $\lambda$ that satisfies the positive roots of the characteristic equation as follows:

$$ \tan \left( {\lambda L} \right) + 2\lambda = 0, $$

(6.65)

and the residue at $\lambda_{n}$, which is the nth root of Eq. (6.65), can be written as

$$ = - \frac{{4\sin \left( {L\lambda_{n} } \right)\sin \left( {\lambda_{n} z} \right)e^{{ - \left( {0.25 + \lambda_{n}^{2} } \right)t}} }}{{1 + 0.5L + 2L\lambda_{n}^{2} }}. $$

(6.66)

The equation for W, i.e., Equation (6.22), is therefore obtained.