An algorithm for finding a sequence of design points in reliability analysis Ziqi Wang a,⇑ , Marco Broccardo b , Armen Der Kiureghian b a Department of Bridge Engineering, Southwest Jiaotong University, Chengdu, China b Department of Civil and Environmental Engineering, University of California, Berkeley, CA, USA article info Article history: Received 2 March 2015 Received in revised form 31 July 2015 Accepted 17 September 2015 Keywords: Design point Nonlinear analysis Optimization algorithm Quasi-Newton Reliability analysis abstract In the analysis of structural reliability, often a sequence of design points associated with a set of thresh- olds are sought in order to determine the tail distribution of a response quantity. In this paper, after a brief review of methods for determining the design point, an inverse reliability method named the k- method is introduced for efficiently determining the sequence of design points. The k-method uses Broyden’s ‘‘good” method to solve a set of nonlinear simultaneous equations to find the design points for the values of an implicitly defined threshold that is associated with parameter k. In a special param- eter setting, the k parameter equals the reliability index, thus allowing convenient implementation of the method. Three numerical examples illustrate the accuracy and efficiency of the proposed method. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction In analysis of structural reliability, there is often interest in solv- ing the reliability problem for a range of threshold values. Three specific examples include: (a) determining the probability distri- bution of a structural response quantity by, e.g., probabilistic finite-element analysis; (b) determining the fragility (conditional probability of failure) of a structure for a given range of demand thresholds; and (c) in stochastic dynamic analysis for determining such statistics of the response, as up-crossing rates of various thresholds or probabilities of exceeding a range of thresholds. In general, these problems can be formulated in terms of a limit- state function of the form gðx; rÞ¼ r  RðxÞ, where x denotes a vec- tor of random variables representing uncertain structural or load values, RðxÞ denotes the response or capacity quantity of interest, and r is the threshold. The objective is to solve the reliability prob- lem for a range of values of r. For the specific examples above, the corresponding probability expressions are as follows: (a) the cumulative distribution function (CDF) of response RðxÞ is given by F R ðrÞ¼ Pr½RðxÞ 6 r¼ 1  Pr½gðx; rÞ 6 0; (b) the fragility func- tion for capacity RðxÞ is given by Pr½RðxÞ 6 DjD ¼ r ¼ 1  Pr½gðx; rÞ 6 0, where r now denotes the threshold of a demand D; and (c) for a nonlinear stochastic response process Rðx; tÞ, the first-order solution of the problem Pr½r 6 Rðx; tÞ ¼ Pr½gðx; t; rÞ 6 0 leads to a tail-equivalent linear system for which all statistics of interest for threshold r can be determined by linear random vibration analysis (see Fujimura and Der Kiureghian [16] and Section 7 in this paper). For the sake of convenience, hereafter we call the above class of problems threshold-reliability problems. Since analytical solutions of reliability problems are often unavailable, approximate techniques, such as the first- and second-order reliability methods (FORM and SORM) [1,2], response surface methods (RSM) [3,4], various sampling techniques [5–7], and expansion methods [8] are used. Several popular methods among these, including FORM, SORM and importance sampling (IS) [5], require determination of the so-called ‘‘design point” for each limit-state function. This is the point on the limit-state sur- face in a transformed standard normal space, which has minimum distance from the origin [1]. This point has the property of having the highest probability density among all failure points in the stan- dard normal space. Hence, it is an optimal point for constructing approximations of the surface (first-order in FORM, second-order in SORM) or as a center of sampling in IS. Furthermore, in FORM, the distance from the origin to the design point, known as the reli- ability index, is directly related to the first-order approximation of the failure probability. The design point is usually obtained by solving a constrained optimization problem by a gradient-based algorithm (see, e.g., [9–13,20]). When the reliability problem is to be solved for a range of thresholds, a sequence of design points must be computed. This can be a costly computation when evalu- ation of the limit-state function or its gradient involves extensive numerical calculations. http://dx.doi.org/10.1016/j.strusafe.2015.09.004 0167-4730/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. Structural Safety 58 (2016) 52–59 Contents lists available at ScienceDirect Structural Safety journal homepage: www.elsevier.com/locate/strusafe