Optimization design of stabilizing piles in slopes considering spatial variability

Abstract

Although advances in piling equipment and technologies have extended the global use of stabilizing piles (to stabilize slope or landslide), the design of stabilizing piles remains a challenge. Specifically, the installation of stabilizing piles can alter the behavior of the slope; and the spatial variability of the geotechnical parameters required in the design is difficult to characterize with certainty, which can degrade the design performance. This paper presents an optimization-based design framework for stabilizing piles. The authors explicitly consider the coupling between the stabilizing piles and the slope, and the robustness of the stability of the reinforced slope against the spatial variability of the geotechnical parameters. The proposed design framework is implemented as a multiobjective optimization problem considering the design robustness as an objective, in addition to safety and cost efficiency, two objectives considered in the conventional design approaches. The design of stabilizing piles in an earth slope is studied as an example to illustrate the effectiveness of this new design framework. A comparison study is also undertaken to demonstrate the superiority of this new framework over the conventional design approaches.

Appendix 1. Subdomain sampling method (SSM) for estimating the statistics of system behavior

The essence of the SSM is to partition the possible domain of uncertain variables into a set of subdomains and then to generate samples of uncertain variables in each and every subdomain separately [29]. In which, a distance index (d) based upon Hasofer–Lind reliability index is adopted to locate the possible domain and to partition this domain.

$$ d = \sqrt {\left[ \varvec{n} \right]^{\text{T}} \left[ {\varvec{R}_{\varvec{n}} } \right]^{ - 1} \left[ \varvec{n} \right]} $$

(9)

where R_n is the correlation matrix among the equivalent standard normal variables n = [n₁, n₂, …, $n_{{n_{x} }}$]^T, where n_x is the number of uncertain variables. The standard normal variable n_i in n is related to the uncertain variable x_i in x.

$$ n_{i} = \varPhi^{ - 1} \left[ {F(x_{i} )} \right] $$

(10)

where F(x_i) is the cumulative distribution function (CDF) of uncertain variable x_i, and Φ(·) is the CDF of the standard normal variable. With the distance index formulated in Eq. (9), the possible domain of uncertain variables x, denoted as [0, d_max), can be located.

$$ \chi_{{n_{x} }}^{2} (d_{\hbox{max} }^{2} ) = \varepsilon $$

(11)

where $\chi_{{n_{x} }}^{2} ( \cdot )$ is the Chi square CDF with n_x degrees of freedom, and ε is a probability which is relatively low. The located possible domain of uncertain variables x, in terms of [0, d_max), is readily partitioned into a set of subdomains, in terms of [d₀, d₁), [d₁, d₂), [d₂, d₃), etc. The likelihoods of the uncertain variables x being located in these subdomains could be taken as a decreasing sequence for the purpose of being computationally efficient.

$$ p_{di} = \Pr \left[ {d_{i - 1} \le \sqrt {\left[ \varvec{n} \right]^{\text{T}} \left[ \varvec{R} \right]^{ - 1} \left[ \varvec{n} \right]} < d_{i} } \right] = \Pr \left[ {d_{i - 1}^{2} \le d^{2} < d_{i}^{2} } \right] = \chi_{{n_{x} }}^{2} (d_{i}^{2} ) - \chi_{{n_{x} }}^{2} (d_{i - 1}^{2} ) $$

(12)

where p_di is the likelihood of the uncertain variables x being located in the subdomain [d_i−1, d_i). Then, the samples of uncertain variables x are generated in each subdomain. The procedures for generating a target number of samples in the subdomain [d_i−1, d_i) are given in Gong et al. [19].

For ease of programming, a same target number of samples, denoted as t₁, is adopted in all these subdomains and this target number is taken as: t₁ = 10p_di/p_d(i−1). With the generated samples of uncertain variables, the deterministic analysis of the system behavior can readily be undertaken, from which the statistics of the system behavior, in terms of the mean E[g],the standard deviation σ[g], the skewness α₃[g] and the kurtosis α₄[g],can be approximated as:

$$ E[g] \approx \sum\limits_{i = 1}^{{i = n_{s} }} {\sum\limits_{j = 1}^{{j = t_{1} }} {p_{ij} \cdot g_{ij} } } $$

(13)

$$ \sigma [g] \approx \left[ {\sum\limits_{i = 1}^{{i = n_{s} }} {\sum\limits_{j = 1}^{{j = t_{1} }} {p_{ij} \cdot \left( {g_{ij} - E[g]} \right)^{2} } } } \right]^{0.5} $$

(14)

$$ \alpha_{3} [g] \approx \sum\limits_{i = 1}^{{i = n_{s} }} {\sum\limits_{j = 1}^{{j = t_{1} }} {p_{ij} \cdot \left( {\frac{{g_{ij} - E[g]}}{\sigma [g]}} \right)^{3} } } $$

(15)

$$ \alpha_{4} [g] \approx \sum\limits_{i = 1}^{{i = n_{s} }} {\sum\limits_{j = 1}^{{j = t_{1} }} {p_{ij} \cdot \left( {\frac{{g_{ij} - E[g]}}{\sigma [g]}} \right)^{4} } } $$

(16)

where g_ij is the system behavior evaluated with the jth sample in the ith subdomain, denoted as x_ij; n_s is the number of subdomains; and, p_ij is the likelihood or probability of the sample x_ij being generated in the domain of uncertain variables, which could be expressed as:

$$ p_{ij} = \frac{{p_{di} }}{{t_{1} }} = \frac{{\chi_{{n_{x} }}^{2} \left( {d_{i}^{2} } \right) - \chi_{{n_{x} }}^{2} \left( {d_{i - 1}^{2} } \right)}}{{t_{1} }}. $$

(17)