Date Published: March 8, 2019
Publisher: Public Library of Science
Author(s): Mahdi Shadab Far, Hongwei Huang, Eva O. L. Lantsoght.
An important segment of the reliability-based optimization problems is to get access to the sensitivity derivatives. However, since the failure probability is not a closed-form function of the input variables, the derivatives are not explicitly computable and rather require a full reliability analysis which is computationally expensive. In this paper, a step-by-step algorithm has been presented to calculate the derivatives of the probability of failure and safety index with respect to the input parameters based on the advanced first-order second-moment (AFOSM) reliability method. The proposed algorithm is then implemented in a spreadsheet using Visual Basic for Application (VBA) programming language. Two geotechnical and structural examples are then presented to examine the program and describe the modeling procedure. The robustness of the proposed method is examined using a Gaussian random perturbation. The capability of the proposed method in the calculation of the sensitivity derivatives of the model uncertainty is explained in a separate section. Finally, the proposed model has been compared to the forward finite difference (FFD) method and the results are validated.
Because of aleatory and/or epistemic uncertainties associated with loads and capacity of structures alike, deterministic models fail to provide a sufficiently reliable estimation of design variables . In order to consider the uncertainties, probabilistic methods are thus preferred [2–4]. That is, reliability-based design optimization is adopted for structural design process . More precisely, the philosophy of optimization tends to minimize the risk rather than focusing on the stress/strain level induced in structural elements. As such, the probability of failure should be limited to a certain extent [6, 7]. This can be achieved by considering the failure probability as a constraint in the optimization process. Mathematically speaking, this can be summarized as follows:
Consider a system with a performance function G(b, x) where b is the vector of design variables and x is the vector of random variables. When AFOSM is employed for reliability analysis, the probability of failure is calculated as follows :
where β is safety index and Φ(•) is cumulative density function of standard normal distribution. As such, the sensitivity of failure probability with respect to design variable (∂Pf/∂b) can be defined as follows:
where ∂β/∂b is the derivative of safety index with respect to design variable (b) and ϕ(•) is density function of standard normal distribution. The type of distribution function governing the variables is one of the important issues in the estimation of sensitivity derivatives. To address this issue in the proposed algorithm, the input variables are projected to the normal standard space using the change of variable. Then, the variables are entered into the algorithm as a vector of equivalent normal variables. This technique enables the algorithm to cover a wide range of probabilistic distribution functions. For this purpose, regardless of the type of distribution function governing the involved random variables, the standard normal form of variables (U), which is commonly used in reliability analysis, is calculated as follows :
where μx and σx are the mean and standard deviation of random variables, respectively. With this assumption, the performance function can be rewritten based on standard normal variables in the form of G(b, U). Kwak and Lee  showed that, as a general case, the derivative of β with respect to design variable (b) can be expressed as follows:
where λ, the Lagrange multiplier, can be calculated as follows:
A numerical example is presented herein to demonstrate the application of the described approach. The example is about a geotechnical project related to the damage induced in a single-hole rock explosion. Following the explosion, the resulting stress waves affect the surrounding environment severely, creating a highly damaged zone around the blast point, i.e. “crushed zone” [22, 23]. Many researchers have worked on the dimensional determination of the crushed zone. Djordjevic  presented the following equation to estimate the radius of the crushed zone:
where r0 is borehole radius (mm), T is tensile strength of the rock material (Pa), and Pb is borehole pressure (Pa). It is reported that Pb can be obtained as follows :
where ρ0 is unexploded explosive density (kg/m3) and DCJ is ideal detonation velocity (m/s). Substituting Pb from Eq (21) into Eq (20), rc can be rewritten as:
Given that the process through which the reliability sensitivity derivatives were calculated was provided in the form of a step-by-step algorithm, it could be easily implemented. In this paper, Visual Basic for Application (VBA) was used to implement the proposed algorithm in Microsoft Excel spreadsheet for any given performance function (provided that it is differentiable) and any number of random variables. The designed spreadsheet and relevant codes are available along with the paper (or, can be downloaded from this link: https://sourceforge.net/projects/rsa-v1/). In the following, the modeling and analysis of the problem in the spreadsheet are described.
In this paper, sensitivity analysis was performed for both the random and deterministic variables. For this purpose, after defining the performance function and the vector of involved variables, the problem was formulated and the required processes were established. The proposed algorithm was then implemented in Microsoft Excel using VBA programming language. Capability of the program for modeling and performing sensitivity analysis was then investigated. Next, two geotechnical and structural examples were presented to examine the proposed method. Finally, the proposed model was validated using the forward finite difference method. The main contributions of this paper are as follows: