Probabilistic analysis of landslide potential of an inclined uniform soil layer of infinite length: application

Abstract

An extended probabilistic model that is a modification of the Chen et al. (2007,) model for evaluating the failure probability of an inclined soil layer with an infinite length was developed in the present paper, and then applied to evaluate the occurrence probability of landslide-related debris flow in Tungmen gully located in the eastern Taiwan, which occurred a devastating debris flow in 1990. The statistical properties of hydrogeological parameters were collected and summarized, and then used to evaluate the landslide-related debris-flow probabilities at various relative water depths for Tungmen gully by using the probabilistic model. Under the assumption that the soil is saturated, the soil’s cohesion is negligible and the specific gravity of the solid particles of soils is a constant, a simplified probabilistic critical slope equation for the stability of an infinite slope of soils was also developed, and used to estimate the occurrence probability of debris flow. The result shows that probabilistic landslide analysis for an infinite slope could provide a suitable approximation for the risk analysis of debris flow mobilization at a given gully.

Appendices

Appendix I: coefficients in Eq. (6)

$$ \begin{aligned} a_{1} &= \mu_{n} {\left[ {(G_{s} - \mu_{d}) - \mu_{s} (1 - \mu_{d})} \right]}\\ b_{1} &= - a_{1} \tan \mu_{\phi}\\ a_{2} &= 0\\ b_{2} &= - \mu_{\phi} {\left[ {(1 - \mu_{n})(G_{s} - \mu_{d}) + \mu_{s} \mu_{n} (1 - \mu_{d})} \right]}(1 + \tan^{2} \mu_{\phi})\\ a_{3} &= \left\{\begin{aligned}\,& - \mu_{m} \mu_{n} (1 - \mu_{s}),\,\hbox{if}\,\mu_{m} < 1 \\\,& \\\,& - \mu_{m}, \,\hbox{if}\,\mu_{m} \geqslant 1 \\ \end{aligned} \right.\\ b_{3} &= \left\{\begin{aligned}\,& \mu_{m} {\left[ {(1 - \mu_{s})\mu_{n} - 1} \right]}\tan \mu_{\phi}, \,\hbox{if}\,\mu_{m} < 1 \\\,& \\\,& 0,\,\hbox{if}\,\mu_{m} \geqslant 1 \\ \end{aligned} \right.\\ a_{4} &= \left\{\begin{aligned}\,& - {\left[ {(1 - \mu_{n})G_{s} + \mu_{s} \mu_{n}} \right]},\,\hbox{if}\,\mu_{m} < 1 \\\,& \\\,& - (1 - \mu_{n})(G_{s} - 1),\,\hbox{if}\,\mu_{m} \geqslant 1 \\ \end{aligned} \right.\\ b_{4} &= - a_{4} \tan \mu_{\phi}\\ a_{5} &= - \mu_{\theta} {\left[ {(1 - \mu_{n})(G_{s} - \mu_{d}) + \mu_{s} \mu_{n} (1 - \mu_{d})} \right]}\tan \mu_{\phi}\\ b_{5} &= - \mu_{\theta} {\left[ {(1 - \mu_{n})(G_{s} - \mu_{d}) + \mu_{s} \mu_{n} (1 - \mu_{d}) + \mu_{m}} \right]}\\ a_{6} &= \mu_{c} /(\gamma_{w} \mu_{H} \cos \mu_{\theta})\\ b_{6} &= 0\\ a_{7} &= - (1 - \mu_{d})\mu_{s} \mu_{n}\\ b_{7} &= - a_{7} \tan \mu_{\phi} \end{aligned} $$

Appendix II: notations

The following symbols are used in this paper:

a _i , b _i :

= coefficients relating to Eq. (6)

B :

= coefficient relating to Eq. (3)

c :

= cohesion of soil

CV _xi :

= coefficients of variation of parameter x _i

G :

= state function(=τ _R − τ _D )

G _s :

= specific gravity of soil solids

h :

= water depth

H :

= thickness of a soil layer

k :

= coefficient relating to Eq. (12)

m :

= relative water depth (=h/H)

n :

= porosity of soil

N :

= sample size

p _f :

= failure probability of a soil layer

s :

= degree of soil’s saturation

S ^* :

 = tan μ_θ /μ _A

x _i :

= variables

x _li :

= lower values of variables

x _ui :

= upper values of variables

V :

= parameter relating to Eq. (3)

V ^* :

= parameter relating to Eq. (7)

ϕ:

= soil’s friction angle

γ _w :

= specific gravity of water

θ:

= inclined angle of a soil layer

τ _D :

= driving stress

τ _R :

= resistance stress

μ _A :

= reference slope, which is defined as Eq. (4)

μ _c :

= mean value of soil cohesion c

μ _d :

= parameter relating to μ _m

μ _n :

= mean value of soil porosity n

μ _s :

= mean value of degree of saturation s

μ_ϕ :

= mean value of friction angle ϕ

μ_θ :

= mean value of inclined angle θ

μ _h :

= mean value of water depth h

μ _H :

= mean value of soil thickness H

μ _m :

= mean values of relative water depth m

μ _xi :

= mean values of variables x _i

σ _xi :

= standard deviation of variables x _i

κ:

= Takahashi’s parameter (κ = 0.7–0.75)