Advances in Civil Engineering

Volume 2017, Article ID 8643801, 10 pages

https://doi.org/10.1155/2017/8643801

## A Comparative Study of First-Order Reliability Method-Based Steepest Descent Search Directions for Reliability Analysis of Steel Structures

^{1}Department of Civil Engineering, Saravan Branch, Islamic Azad University, Saravan, Iran^{2}Department of Civil Engineering, University of Zabol, Zabol, Iran^{3}Department of Civil Engineering, Zahedan Branch, Islamic Azad University, Zahedan, Iran

Correspondence should be addressed to Hamed Makhduomi; moc.liamg@km.ymah

Received 28 January 2017; Accepted 11 May 2017; Published 7 September 2017

Academic Editor: Sertong Quek

Copyright © 2017 Hamed Makhduomi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Three algorithms of first-order reliability method (FORM) using steepest descent search direction are compared to evaluate the reliability index of structural steel problems which are designed by the Iranian National Building code. The FORM formula is modified based on a dynamic step size which is computed based on the merit functions named modified Hasofer-Lind and Rackwitz-Fiessler (MHL-RF) method. The efficiency of the gradient, HL-RF, and MHL-RF method was compared for a bar structure under tensile capacity, a multispan beam under bending capacity, a connection under tension load, and a column under axial force. The results illustrated that the MHL-RF method is more efficient than the HL-RF and gradient method. The designed steel components by the Iranian National Building code showed good confidence levels with the reliability index in the range from 2.5 to 3.0.

#### 1. Introduction

In real engineering systems, various uncertainties are produced in designing, constructing, and servicing phases of structural systems. The analytical modeling, poor acknowledgement from the analytical modeling, human factors, and data experiments from the input variables such as mechanical properties, load, failure mode, and geometrical dimensions of engineering systems are some kinds of uncertainties. These uncertainties can be considered based on probabilistic models and the reliability analysis can evaluate safety levels for engineering problems by using the failure probability. The analytical methods including first-order reliability method [1, 2], second-order reliability method [3], and moment methods [4] or simulation methods such as Monte Carlo simulation (MCS) [2], subset simulation [5], and weighted simulation [6] are generally used for estimating failure probabilities based on probabilistic model. The simulation methods are time-consuming approach for complicated real engineering structures [5–7]. However, the analytical methods are more efficient with appropriate accuracy for real structural problems in reliability analysis. The first-order reliability method is widely used in robust design, code calibration, load combination, and evaluating the confidence level of engineering systems due to its simplicity and efficiency [8, 9].

Hasofer and Lind (1974) proposed an iterative formula to search the most probable failure point (MPP) which is a point on the failure surface with minimum distance from origin in the standard normal space. Thus, the main effort of FORM is to search the MPP based on the following optimization model [10]: in which is the reliability index and is the limit state function or the performance function in normal standard space. Hasofer and Lind utilized the above model for normal variables where later Rackwitz and Flessler (1978) extended the iterative FORM formula based on the distribution information of the basic random variables; that is, it is briefly called HL-RF. The HL-RF method was improved by Liu and der Kiureghian (1991) using a merit function to enhance the convergence properties. Santosh et al. (2006) improved the HL-RF method based on Armijo rule. The convergence properties of HL-RF are improved based on the stability transformation method (STM) [8], the relaxed approach [10, 11], conjugate search direction [7, 12–14], and sufficient descent condition [13]. The STM [8] using small step size and relaxed HL-RF [11] using a dynamic relaxed factor less than 1 can provide stable results for highly nonlinear performance function but are formulated with complicated formulations [9]. The drawback of the STM is enhanced by the conjugate search direction with chaotic step size using logistic map [7, 13] and the directional search direction [15]. The new modified FORM formulas have been developed based on a complicated formulation to search MPP. Thus, efficiency, simplicity, and robustness can be applied to select an iterative FORM formula for reliability analysis of real engineering problems. Therefore, the present study is organized based on two main aims as follows:(1)Utilize the FORM formula for reliability analysis of steel structures designed by Iranian National Building code.(2)Develop a simple FORM formula to compute the reliability index of steel structures.This study is organized based on estimating the failure probabilities or reliability indexes of steel components using three FORM algorithms including the HL-RF, gradient method, and a modified HL-RF method. The modified HL-RF is developed based on a dynamic step size where the proposed step size is adjusted based on the merit function. The normal and nonnormal random variables with log-normal and Gumbel distributions are implemented to simulate the statistical properties of the basic random variables in reliability analysis of steel components. Thus, a MATLAB code is developed to consider the statistical characteristics of basic random variables in performance functions which are extracted from Iranian National Building code. The HL-RF, gradient, and modified HL-RF are compared using four steel examples as a bar, a multispan beam, a connection, and a column. The effects of the applied loads are evaluated to obtain a reliable level with reliability index of 3. Results indicate that the proposed MHL-RF method is as robust as the HL-RF and gradient method but is more efficient. The reliability indexes under the service load of these four steel components are obtained in range from 2.5 to 3.

#### 2. First-Order Reliability Method (FORM)

Failure probability of steel components is computed through a probabilistic model that includes strength unreliability (elastic modulus, dimensions, Poisson index, ultimate tension, and yielding tension) and load (live or dead loads, load capacity). The failure probability can be approximated based on the reliability index in FORM as follows [8]:in which is failure probability, is the limit state function which separated design domain into safe and failure domains using the basic random variables as is failure regain and is safe regain. The main effort in the FORM is to search the maximum probable point (MPP, i.e., ) which is a point on the limit state surface with minimum distance to organ into normal standard space. This distance is defined as the reliability index. Consequently, [14]. In FORM, the random variables from original space should be transferred into normal standard space where these variables are independent with means of zero and standard deviations of one using Rosenblatt’s transformation; that is, as follows [11]:in which and are, respectively, equivalent mean and standard deviation of random variable . According to Rosenblatt’s transformation, equivalent mean and standard deviation for no-normal variables are calculable as follows [10]:where and are the normal probability distribution function and cumulative distribution function, respectively. and are, respectively, probability distribution function and cumulative distribution function of random variable at point . For searching the MPP, there are various FORM algorithms such as Hasofer-Lind method [16, 17], stability transformation method [8, 15], and conjugate gradient [9, 12], finite-step length [18], relaxed HL-RF method [10, 13], and chaotic conjugate search direction [7, 13]. The main effort to develop the FORM formula is to improve the efficiency and robustness of FORM. The HL-RF, gradient, and modified HL-RF methods which are formulated using the steepest descent search direction are applied to find MPP.

##### 2.1. HL-RF Method

The iterative formula of FORM can be described by the following relation:where is step size. In HL-RF method, the step size is considered as 1. is search direction vector, which is computed as follows [1]:in which is gradient vector of the limit state function at point . By replacing (6) with (5) and considering the step size equivalent to 1, the FORM formula-based HL-RF method is rewritten as follows:This approach can be moved from a point on the feasible design region to MPP on the limit state surface; thus the constraint of the probabilistic model in (1) is satisfied at the final iteration of the HL-RF method when stable results are obtained.

##### 2.2. Gradient Method

In gradient method, the probabilistic constraint of the optimization model in (1) should be satisfied at each iteration. Therefore, the iterative formula of the HL-RF is adjusted based on the following search direction in gradient method [1]: In this method, the limit state function should be considered equal to zero at each iteration as *. *Therefore, the probabilistic constraint of (1) can be satisfied using Newton method at each iteration by the following iterative formula:The gradient and HL-RF methods are formulated based on steepest descent search direction based on (6) and (8). However, it needs an inner loop in order to satisfy the limit state function equal to zero, that is, , based on the gradient approach. As seen, the amount of evaluating the limit state function in the gradient method is slightly increased compared to the HL-RF method with same evaluating of the gradient vectors. In addition, the HL-RF method is similar to the gradient method with step size of 1 for reliability analysis. These methods may provide unstable results for highly nonlinear reliability problems [7, 8, 15, 18]. However, the gradient and HL-RF method are FORM algorithms with a fast convergence rate because the step size in these approaches is selected equivalent to one, while the step size in the modified versions of FORM-based steepest descent search direction such as improved HL-RF [1, 19], RHL-RF [11], and STM [8] is given less than 1 to achieve the stabilization. Hence, the improved versions of HL-RF may enhance the robustness of FORM formula compared to HL-RF for highly nonlinear problems but these approaches are computationally inefficient for moderately nonlinear or linear performance functions [7]. The conjugate search direction can improve the robustness and efficiency of FORM compared to the improved HL-RF methods, but the FORM-based conjugate search direction was formulated with complex relations. In this study, a modified HL-RF is developed to improve the capability of FORM formula.

#### 3. Modified HL-RF Method

According to (5), the step size and search direction are two affective parameters in the iterative FORM formula. The iterative FORM formula can be controlled based on step size to search MPP. Therefore, the iterative formula of modified HL-RF (MHL-RF) can be obtained as follows:where is the adjusted step size. In this study, the step size of MHL-RF method in (10) can be dynamically adjusted in range from 1.5 to 0. It is supposed that the step size is adjusted by the following merit function:It is clear that the merit function is a positive value and it is computed based on the previous results as well as the HL-RF method. The second term of this merit function is a positive dimensionless value that should be decreased for sequence iterations of MHL-RF to satisfy the constraint of the probabilistic optimization model in (1) as . If is equivalent to zero, then a fixed point is obtained or the MHL-RF is converged. Thus and when (8); it is supposed that for then ; this means that ; consequently, ; therefore, is a fixed point and the proposed method is converged. It is assumed that thus for . Therefore, the step size is computed as follows:in which the initial step size is considered as 1.5 (i.e., ). According to the above adaptive step size in (12), it can be concluded that and also, as it is mentioned, if , then ; thus when . If , then according to (12) ; thus for , and for highly nonlinear problems. If , then the new points are located on the previous point based on the modified HL-RF method; thus robust results for reliability analysis are obtained. The step size may be obtained more than 1 when a linear or moderately concave limit state function is selected. It is clear that this method is as simple as the HL-RF method but the MHL-RF method may provide a higher convergence rate than the HL-RF for moderate performance functions. The algorithm of the modified HL-RF method to code in computer software can be presented using the following steps:

(1) Define the limit state function and the convergence criterion , statistical characteristics of basic random variables (mean, standard deviation, and distribution function for each random variable), and set , , and .

(2) Transfer the random variables from -space to -space according to (3) and (4).

(3) Compute the gradient vector and limit state function at point .

(4) Compute the merit function using (10) and the new adjusted step size based on (12), for .

(5) Compute the steepest descent search direction in terms of (6).

(6) Determine the new point according to (5).

(7) Transfer the basic random variables from -space to -space.

(8) Control the convergence criterion as ; if converged, stop and go to step (); else, and go to step ().

(9) Compute the reliability index and failure probability.

#### 4. Reliability Results and Comparative Examples

According to three algorithms of HL-RF, gradient, and MHL-RF, the reliability indexes of four moderately structural examples which were designed based on the Iranian National Building were evaluated to illustrate their performances in reliability analysis-based FORM. The numbers of gradient vector (iterations) for limit state functions which are computed using the central difference method as and reliability indexes were considered as comparative criteria to compare the FORM-based steepest descent search directions which were coded in MATLAB using the stopping criterion as .

##### 4.1. A Bar under Axial Tensile Force

A structure under a uniform load on the beam ED which is shown in Figure 1 is supported using a bar FC and bar AB is connected at point B with bar AB. The performance function for this structure is defined based on the tensile capacity of the bar AB for this example.