# Infinite Slope Model

The infinite slope model is a 6-dimensional test function for reliability analysis methods. The model is used in geotechnical engineering to quantify the stability of a sloped soil under the influence of water infiltration (Phoon, 2008).

## Description

The limit state function of the infinite slope model compares the resisting force R due to friction with the loading S due to the soil and groundwater table.
The analytical expression of the function is given as:

g = \frac{R}{S} - 1 = \frac{(\gamma (H-h) + h (\gamma_{\text{sat}}) - \gamma_w) \cos{\theta} \tan{\phi}}{(\gamma (H-h) + h \gamma_{\text{sat}}) \sin{\theta}} - 1

where:

• H is the depth of soil above the bedrock [\text{m}];
• h is the height of the groundwater table above bedrock [\text{m}];
• \gamma is the moist unit weight of the surface soil [\text{kN.m}^{-3}]
• \gamma_{\text{sat}} is the saturated unit weight of the surface soil [\text{kN.m}^{-3}]
• \gamma_w is the unit weight of water (a constant with a value of 9.81 [\text{kN.m}^{^-3}])
• \phi is the effective stress friction angle between the rock and soil [-]
• \theta is the slope inclination [-]

The configuration of the model is illustrated in Figure 1.

Figure 1: Illustration for the infinite slope problem, adapted from Phoon (2008).

Notice that the height of the groundwater table bedrock h is limited by the depth of the soil above bedrock H. Furthermore, h can only take positive values. Therefore, h is modeled using a uniform random variable such that h = H U_h, where U_h is a uniform random variable.

The moist and saturated soil unit weights are not independent, because they are related to the specific gravity of the soil solids G_s and the void ratio e. The uncertainty in \gamma and \gamma_{\text{sat}} are characterized by modeling G_s and e as two independent uniform random variables. The moist and the saturated soil unit weights are computed using:

\gamma = \gamma_w \frac{G_s+ke}{1+e} \;\;\; \text{and} \;\;\; \gamma_{\text{sat}} = \gamma_w \frac{G_s + e}{1 + e},

respectively. k is the degree of saturation of moist soil with a constant value of 0.2.

All in all, the input variables are represented by six variables \mathbf{x} = \{H, U_h, \phi, \theta, G_s, e\}.

The failure event (i.e., unstable slopes) is defined as g(\mathbf{x}) \leq 0, that is if the resistance R is smaller than the loading S. The failure probability, in turn, is defined as P_f = \mathbb{P}[g(\mathbf{x}) \leq 0].

## Inputs

The parameters for the six independent random inputs are described in the table below.

No Variable Distribution Parameters Description
1 H Uniform H_{\min} = 2,
H_{\max} = 8
Depth of soil above bedrock [\text{m}]
2 U_h Uniform U_{h,\min} = 0,
U_{h,\max} = 1
Relative height of water table [-]
3 \phi Lognormal \mu_{\phi} = 0.6109,
\sigma_{\phi} = 0.0489
Effective stress friction angle [-]
4 \theta Lognormal \mu_{\theta} = 0.3491,
\sigma_{\theta} = 0.0175
Slope inclination [-]
5 G_s Uniform G_{s,\min} = 2.5,
G_{s,\max} = 2.7
Specific gravity of soil [-]
6 e Uniform e_{\min} = 0.3,
e_{\max} = 0.6
Void ratio of soil [-]

## Reference values

Some reference values for the failure probability P_f from the literature are shown in the table below.

Method N \hat{P}_f \text{CoV}[\hat{P}_f] Source
FORM - 7.64 \times 10^{-2} - Phoon (2008)
MCS - 5.82 \times 10^{-2} - Phoon (2008)
MCS 10^6 5.78 \times 10^{-2} 0.4\% UQLab v1.2.1

## Resources

The vectorized implementation of the limit state function of the infinite slope model in MATLAB as well as the script file with the model and probabilistic inputs definitions for the function in UQLAB can be downloaded below:

uq_infiniteSlope.zip (2.9 KB)

The contents of the file are:

Filename Description
uq_infiniteSlope.m vectorized implementation of the limit state function of the infinite slope model in MATLAB
uq_Example_infiniteSlope.m definitions for the model and probabilistic inputs in UQLab
LICENSE license for the function (BSD 3-Clause)

## References

• K.-K Phoon, “Numerical recipes for reliability analysis – a primer,” in Reliability-based Design in Geotechnical Engineering: Computations and Applications, K.-K Phoon, Ed. London: CRC Press, 2008, pp. 34–35. DOI:10.1201/9781482265811