یک روش جایگزین برای مقابله با شکست روش عامل احتمال ایمنی همراه با تجزیه و تحلیل حساسیت

The paper introduces a method for solving the failure probability-safety factor problem for designing engineering works proposed by Castillo et al. that optimizes an objective function subject to the standard geometric and code constraints, and two more sets of constraints that simultaneously guarantee given safety factors and failure probability bounds associated with a given set of failure modes. The method uses the dual variables and is especially convenient to perform a sensitivity analysis, because sensitivities of the objective function and the reliability indices can be obtained with respect to all data values. To this end, the optimization problems are transformed into other equivalent ones, in which the data parameters are converted into artificial variables, and locked to their actual values. In this way, some variables of the associated dual problems become the desired sensitivities. In addition, using the proposed methodology, calibration of codes based on partial safety factors can be done. The method is illustrated by its application to the design of a simple rubble mound breakwater and a bridge crane.

Engineering design of structural elements is a complicated and highly iterative process that usually requires a long experience. Iterations consists of a trial-and-error selection of the design variables or parameters, together with a check of the safety and functionality constraints, until reasonable structures, in terms of cost and safety, are obtained. Optimization procedures are a good solution to free the engineer from the above-mentioned cumbersome iterative process, i.e. to automate the design process [1], [18], [19], [2], [5] and [13]. Safety of structures is the fundamental criterion for design [3], [8], [9], [16], [10], [14], [15], [20], [21] and [22]. To this end, the engineer first identifies all failure modes of the work being designed and then establishes the safety constraints to be satisfied by the design variables. To ensure satisfaction of the safety constraints, two approaches are normally used: (a) the classical safety factor approach and (b) the probability-based approach. With the purpose of illustration, consider the case of designing a breakwater (see Fig. 1) fixing its geometry and dimensions, and checking its behavior with respect to the most important failure modes, as overtopping, overturning and sliding. This check can be done using safety factors, failure probabilities or both. Each failure mode has a probability of occurrence that depends on the selected geometry. A given design must guarantee that the failure probabilities associated with all failure modes are smaller than the values required by the engineering codes. In addition, it is fundamental to choose a design that minimizes the cost. Full-size image (4 K) Fig. 1. Parameterized rubblemound breakwater used in the example. Figure options Classic engineers criticize the probabilistic approach because of its sensitivity to statistical hypotheses, especially tail assumptions [4] and [11]. Similarly, probability-based engineers question classical designs because it is not clear how far are their designs from failure. To avoid the lack of agreement between defenders of both approaches, and to obtain a more reliable design, Castillo et al. [6] and [7] proposed a mixed method, the failure probability-safety factor method (FPSF) that combines safety factors and failure probability constraints. Since the failure probability bounds cannot be directly imposed in the form of standard constraints, optimization packages cannot deal directly with problems involving them. In fact, failure probability constraints require themselves the solution of other optimization problems. Fortunately, there are some iterative methods for solving this problem that converge in a few iterations to the optimal solution (see for example Refs. [6] and [7]). However, since the proposed method consists of a bilevel minimization process, one that minimizes cost and others that calculate the reliability indices, and not all variables are involved in both problems, the final result is that only some sensitivities are obtained. In addition, the method requires the use of a relaxation factor that has to be fixed experimentally. In this paper, an alternative procedure that avoids the relaxation factor and allows to perform a complete sensitivity analysis is presented. The remaining of this paper is structured as follows. In Section 2 the FPSF method for designing engineering works is presented and the methods proposed by Castillo et al. [6] and [7] for performing a sensitivity analysis are reviewed. In Section 3 an alternative method optimized to perform a complete sensitivity analysis is presented. In 4 and 5, examples of a breakwater and a bridge girder are given to illustrate the new proposals. Finally, in Section 6 some conclusions are drawn.

The main conclusions that can be derived from the previous sections are: 1. The FPSF method for engineering design gives a dual information on the safety of the structures being designed: safety factors and failure probabilities, giving a double way of safety control, and interesting calibration possibilities for the classic and probability-based designs. Errors in the safety factor assumptions approach can be detected by the failure probability approach and vice versa. 2. The proposed alternative method for solving the failure-probability-safety-factor method converges in a few iterations, has a robust computational behavior, and is faster than the method proposed in Castillo et al. [6] and [7], that uses a relaxation factor. 3. Since the alternative proposed method involves all the adequate variables in the right hand side of the constraints, a complete sensitivity analysis of the cost function and the β reliabilities associated with all modes of failure with respect to all data values can be easily performed.