A Reliability-Based Approach for the Design of Rockfall Protection Fences

Abstract

This paper proposes a method for improving the design of rockfall protection fences and accounting for the variability of loading cases. It is based on a probabilistic reliability analysis and can combine loading cases from rockfall propagation simulations with numerical simulations of the structure response to the block impact. The advantage of such a reliability-based approach is that statistically relevant results can be obtained concerning the fence’s efficiency in stopping the block with a limited number of simulations. This method was employed in a study case, involving a low-energy tree-supported fence placed on a forested slope. The trajectory simulations were conducted using Rockyfor3D and the fence was modelled using a three-dimensional discrete element method model. For demonstration purposes, two parameters were considered: block velocity and the block’s angle of incidence before impact. The probability of the fence stopping the block was evaluated accounting for the variability of these two parameters separately and together, either considering these variables as non-correlated, or as correlated. The value of this approach is demonstrated in terms of computation cost. In addition, the results revealed the importance of accounting for both these parameters in designing the structure as well as in estimating the residual hazard downslope from the protective structure.

Appendices

Appendix 1: Roots of Hermite Polynomials

The values of x _i , corresponding to the roots of Nth-degree Hermite polynomials, denoted H _N , should be computed numerically or found in tables (Press et al. 1994). For instance, Table 6 provides points x _i for N = 3, 4, 5, where H ₃(x) = x ³ − 3x, H ₄(x) = x ⁴ − 6x ² + 3, H ₅(x) = x ⁵ − 10x ² + 15x. Such values are classically deduced from quadrature formulas (see also Baroth et al. 2007).

Table 6 Points x _i for N = 3, 4, 5

Appendix 2: Gaussian Standardization of Two Correlated Log-Normal Random Variables

Let $\mathbf{Y} =(Y_1, Y_2)$ be a 2D lognormal random variable with given mean μ _Y  = (μ _{Y_1}, μ _{Y_2}) and standard deviation σ _Y  = (σ _{Y_1}, σ _{Y_2}) and coefficient of correlation ρ _{Y_1 Y_2}. In this case, the Gaussian standardization of $\mathbf{Y}$ is written (Baroth et al. 2006):

$${\mathbf{Y}}=T({\mathbf{X}}) \Leftrightarrow \left\lbrace \begin{array}{ll} Y_1 = \frac{\mu_{Y_1}}{\sqrt{1+ {\rm Cv}_{Y_1}^2}} {\rm exp} \lbrace L_{11} X_1\rbrace \\ Y_2 = \frac{\mu_{Y_2}}{\sqrt{1+ {\rm Cv}_{Y_2}^2}} {\rm exp} \lbrace L_{21} X_1 + L_{22} X_2 \rbrace \end{array} \right.,$$

(4)

with:

$$L_{11} = \sqrt{\ln(1+ {\rm Cv}_{Y_1}^2)},$$

(5)

$$L_{21} = \frac{\ln(1 + \rho_{Y_1 Y_2} {\rm Cv}_{Y_1} {\rm Cv}_{Y_2})} {L_{11}},$$

(6)

$$L_{22} = \sqrt{ \frac{ \ln(1 + {\rm Cv}_{Y_1}^2) \ln(1 + {\rm Cv}_{Y_2}^2) - \ln^2(1+ \rho_{Y_1 Y_2} {\rm Cv}_{Y_1} {\rm Cv}_{Y_2})} {L_{11}^2}},$$

(7)

where ${\rm Cv}_{Y_i}=\frac{\sigma_{Y_i}}{\mu_{Y_i}}, i=1,2$ are the coefficients of variation of Y ₁ and Y ₂.