Reliability-based robust geotechnical design using Monte Carlo simulation

Abstract

The reliability-based robust geotechnical design (RGD) approach provides an effective tool to deal with the uncertainty in the estimated statistics of geotechnical parameters in a reliability-based design. However, the existing reliability-based RGD approach is not straightforward to apply as it involves multiple concepts. In this paper, the applicability of the existing reliability-based RGD approach is improved by utilizing Monte Carlo simulation (MCS). Here, an MCS-based weighted technique is used to evaluate the failure probability of a geotechnical system. With the aid of this weighted technique, the variation in the failure probability, caused by the uncertainty in the estimated statistics of geotechnical parameters, is computed using MCS. To further improve the efficiency of the RGD method, a series of single-objective optimizations are used in lieu of a multi-objective optimization in the robust design optimization process. The proposed MCS-based RGD approach is illustrated through an example of rock slope design. Compared with the existing reliability-based RGD approach, the MCS-based RGD approach is not only more intuitive and easier to follow, but also more computationally efficient.

Appendix: Solution model for rock slope stability analysis

In reference to Fig. 4, the factor of safety (FS) for the rock slope is expressed as the ratio of the sum of all resisting forces to the sum of all driving forces (Hoek and Bray 1981; Hoek 2006):

$$ FS = \frac{{cA + [W(\text{cos}\psi_{p} - \alpha \text{sin}\psi_{p} ) - U - V\sin \psi_{p} ]\tan \varphi }}{{W(\sin \psi_{p} + \alpha \cos \psi_{p} ) + V\cos \psi_{p} }} $$

(9)

where c is the cohesive strength along the sliding surface, A is the base area of the wedge, W is the weight of rock wedge resting on the failure surface, ψ _p is the angle of failure surface measured from the horizontal plane, α is the gravitational acceleration coefficient defined by the ratio of horizontal to the gravitational acceleration, U is the uplift force due to the water pressure on the failure surface, V is the horizontal force due to the water in the tension crack, and φ is the friction angle of the sliding surface.

The following intermediate terms for computing FS are obtained from the basic slope geometry and rock properties (Hoek 2006):

$$ A = (H - z)/{\text{sin}}\psi_{p} $$

(10)

$$ W = 0.5\gamma H^{2} \{ [1 - (z/H)^{2} ]\cot \psi_{p} - \cot \psi_{f} \} $$

(11)

$$ U = 0.5\gamma_{w} z_{w} A $$

(12)

$$ V = 0.5\gamma_{w} z_{w}^{2} $$

(13)

$$ i_{w} = z_{w} /z $$

(14)

where H is the height of the overall slope, z is the depth of tension crack, z _w is the depth of the water in the tension crack, ψ _f is the overall slope angle measured from the horizontal, and i _w is the percentage of the tension crack depth filled with water.

Based on Eq. (9), the corresponding performance function can be expressed as:

$$ G(\varvec{X}) = FS - 1. $$

(15)