An alternative approach for addressing the failure probability-safety factor method with sensitivity analysis

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.     

A comparison of deterministic,reliability-based and risk-based structural optimization under uncertainty

In this paper, the effects of uncertainty and expected costs of failure on optimum structural design are investigated, by comparing three distinct formulations of structural optimization problems. Deterministic Design Optimization (DDO) allows one the find the shape or configuration of a structure that is optimum in terms of mechanics, but the formulation grossly neglects parameter uncertainty and its effects on structural safety. Reliability-based Design Optimization (RBDO) has emerged as an alternative to properly model the safety-under-uncertainty part of the problem. With RBDO, one can ensure that a minimum (and measurable) level of safety is achieved by the optimum structure. However, results are dependent on the failure probabilities used as constraints in the analysis. Risk optimization (RO) increases the scope of the problem by addressing the compromising goals of economy and safety. This is accomplished by quantifying the monetary consequences of failure, as well as the costs associated with construction, operation and maintenance. RO yields the optimum topology and the optimum point of balance between economy and safety. Results are compared for some example problems. The broader RO solution is found first, and optimum results are used as constraints in DDO and RBDO. Results show that even when optimum safety coefficients are used as constraints in DDO, the formulation leads to configurations which respect these design constraints, reduce manufacturing costs but increase total expected costs (including expected costs of failure). When (optimum) system failure probability is used as a constraint in RBDO, this solution also reduces manufacturing costs but by increasing total expected costs. This happens when the costs associated with different failure modes are distinct. Hence, a general equivalence between the formulations cannot be established. Optimum structural design considering expected costs of failure cannot be controlled solely by safety factors nor by failure probability constraints, but will depend on actual structural configuration.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

预应力混凝土梁桥系统失效树分析

针对预应力混凝土梁桥,采用失效树的方法进行系统可靠性分析。运用全局临界强度分枝-约界准则识别结构系统主要失效模式,以JC法计算各主要失效模式的可靠指标,采用Ditlevsen上下界公式分析结构体系的失效概率。以某汉江公路大桥为例,分析了失效树图形,研究了主要失效模式的可靠指标、失效概率及桥梁结构体系可靠性。与实桥运营状态的对比分析表明,上述理论分析较好地预测了实际的结构行为,能用于桥梁结构安全性及可靠性分析评估。     

A discrete stress-strength interference model for unreliability bounds

This study presents a multiple-line-segment method of implementing the stress-strength interference model when only discrete interval probabilities of stress and strength inside an interference region are available. Based on the discrete interval probabilities, the probability density functions are represented by line segments, which are in turn expressed by piecewise linear polynomials. Thus, unlike in existing empirical methods, when both stress and strength fall into the same subinterval, one can calculate the probability of failure by directly integrating the joint probability of stress and strength over the subinterval. Quadratic programming optimization is implemented to determine the upper and lower bounds on the failure probability, and the average value of the bounds is treated as the point estimate of the failure probability. Finally, numerical examples are carried out to demonstrate the proposed method. Three quantities are used to evaluate the accuracy and effectiveness of the proposed method—the relative error of the point estimate of the unreliability, the width of the bounds on the unreliability, and the minimum number of subintervals needed to generate the bounds that contain the exact unreliability. These three quantities are compared with results in the literature.     

The Design Space Root Finding method for efficient risk optimization by simulation

Reliability-Based Design Optimization (RBDO) is computationally expensive due to the nested optimization and reliability loops. Several shortcuts have been proposed in the literature to solve RBDO problems. However, these shortcuts only apply when failure probability is a design constraint. When failure probabilities are incorporated in the objective function, such as in total life-cycle cost or risk optimization, no shortcuts were available to this date, to the best of the authors knowledge. In this paper, a novel method is proposed for the solution of risk optimization problems. Risk optimization allows one to address the apparently conflicting goals of safety and economy in structural design. In the conventional solution of risk optimization by Monte Carlo simulation, information concerning limit state function behavior over the design space is usually disregarded. The method proposed herein consists in finding the roots of the limit state function in the design space, for all Monte Carlo samples of random variables. The proposed method is compared to the usual method in application to one and n-dimensional optimization problems, considering various degrees of limit state and cost function nonlinearities. Results show that the proposed method is almost twenty times more efficient than the usual method, when applied to one-dimensional problems. Efficiency is reduced for higher dimensional problems, but the proposed method is still at least two times more efficient than the usual method for twenty design variables. As the efficiency of the proposed method for higher-dimensional problems is directly related to derivative evaluations, further investigation is necessary to improve its efficiency in application to multi-dimensional problems.     

Reliability Based Design Optimization Using Design Explorer

Experimental design methods can be applied to engineering design activities to understand which variables affect the system under consideration, how these variables affect the system, and how to select variable settings that will give uniformly long life to the system. The objective of this paper is to demonstrate the use of Design and Analysis of Computer Experiment (DACE) methods (Sacks, J. et al., 1989) and design optimization via the Surrogate Management Framework (Booker, A. J. et al., 1999; Audet, C. et al., 2000) on reliability optimization problems. Reliabilities are calculated using the Probabilistic Structural Analysis Method (Palle Thoft–Christensen and Baker, 1982; Achintya Haldar and Sankaran Mahadevan, 2000), a method for estimation of reliabilities and reliability indices for a structural model given probability distributions for design variables and “environmental” variables such as loads. By maximizing reliability, or minimizing the probability of failure, we attempt to achieve a minimum cost design that is affected minimally by the variability in the design variables.     

Dynamic design of structures under random excitation

An optimum design method to minimize the weight of a linear elastic structural system subjected to random excitations is presented. It is focused on the constraints of the first passage failure and displacement response mean square (RMS) at certain degree of freedoms. Constraints on natural frequencies and bounds of design variables are also considered in the optimization. Both correlated and un-correlated generalized random excitations are considered in the present formulation. The sensitivities of the expected number of crossings as well as the displacement RMS with respect to the design variables are also derived. The present method is applicable to stationary Guassian random excitations. Computational examples show the feasibility and efficiency of the proposed method.     

Fault tree truncation error bounds

The solution of large fault trees can only performed if cut sets of a high order or low probability are neglected. This procedure is non-pessimistic since possible contributions to the top gate failure probability are ignored. Since the number of cut sets neglected is generally large, it is possible that their total contribution is significant compared to the total probability of those cut sets included in the evaluation of the top gate failure probability.In this paper a practical method for the estimation of upper bounds on the total failure probability of cuts sets of a given order is presented. This allows bounds on the contribution of all the cut sets neglected through cut-off procedures in fault tree solutions to be calculated, and so validate the top event failure probability. The method given here is much superior to that suggested by Modarres and Dezfuli in that it produces much lower and hence more useful bounds. The method of Modarres and Dezfuli can be refined, but for realistic fault tree examples the method given here always gives the least pessimistic and most practical bounds.     

Reliability analysis and reliability-based design optimization of roadway horizontal curves using a first-order reliability method

This article presents reliability analysis and reliability-based optimization of roadway minimum radius design based on vehicle dynamics, mainly focusing on exit ramps and interchanges. The performance functions are formulated as failure modes of vehicle rollover and sideslip. To accurately describe the failure modes, analytical models for rollover and sideslip are derived considering nonlinear characteristics of vehicle behaviour using the commercial software TruckSim. The probability of an accident is evaluated using the first-order reliability method and numerical studies are conducted using a single-unit truck model. To propose a practical application for the study, the reliability analysis for the minimum radius recommended by American Association of State Highway and Transportation Officials is conducted. The results show that, even if there are deviations from assumed design conditions of the current design guideline, the proposed design method can guarantee given target margins of safety against rollover and sideslip. Based on the reliability analysis, reliability-based design optimization is carried out and the results indicate new recommendations for minimum radii satisfying given target reliability levels.     

Matrix-based system reliability method and applications to bridge networks

Using a matrix-based system reliability (MSR) method, one can estimate the probabilities of complex system events by simple matrix calculations. Unlike existing system reliability methods whose complexity depends highly on that of the system event, the MSR method describes any general system event in a simple matrix form and therefore provides a more convenient way of handling the system event and estimating its probability. Even in the case where one has incomplete information on the component probabilities and/or the statistical dependence thereof, the matrix-based framework enables us to estimate the narrowest bounds on the system failure probability by linear programming. This paper presents the MSR method and applies it to a transportation network consisting of bridge structures. The seismic failure probabilities of bridges are estimated by use of the predictive fragility curves developed by a Bayesian methodology based on experimental data and existing deterministic models of the seismic capacity and demand. Using the MSR method, the probability of disconnection between each city/county and a critical facility is estimated. The probability mass function of the number of failed bridges is computed as well. In order to quantify the relative importance of bridges, the MSR method is used to compute the conditional probabilities of bridge failures given that there is at least one city disconnected from the critical facility. The bounds on the probability of disconnection are also obtained for cases with incomplete information.     

Performance optimization of multidisciplinary mechanical systems subject to uncertainties

This paper presents a methodology to solve a new class of stochastic optimization problems for multidisciplinary systems (multidisciplinary stochastic optimization or MSO) wherein the objective is to maximize system mechanical performance (e.g. aerodynamic efficiency) while satisfying reliability-based constraints (e.g. structural safety). Multidisciplinary problems require a different solution approach than those solved in earlier research in reliability-based structural optimization (single discipline) wherein the goal is usually to minimize weight (or cost) for a structural configuration subject to a limiting probability of failure or to minimize probability of failure subject to a limiting weight (or cost). For the problems solved herein, the objective is to maximize performance over the range of operating conditions, while satisfying constraints that ensure safe and reliable operation. Because the objective is performance based and because the constraints are reliability based, the random variables used in the objective must model variability in operating conditions, while the random variables used in the constraints must model uncertainty in extreme values (to ensure safety). Thus, the problem must be formulated to treat these two different types of variables at the same time, including the case when the same physical quantity (e.g. a particular load) appears in both the objective function and the constraints. In addition, the problem must be formulated to treat multiple load cases, which can again require modeling the same physical quantity with different random variables. Deterministic multidisciplinary optimization (MDO) problems have advanced to the stage where they are now commonly formulated with multiple load cases and multiple disciplines governing the objective and constraints. This advancement has enabled MDO to solve more realistic problems of much more practical interest. The formulation used herein solves stochastic optimization problems that are posed in this same way, enabling similar practical benefits but, in addition, producing optimum designs that are more robust than the deterministic optimum designs (since uncertainties are accounted for during the optimization process). The methodology has been implemented in the form of a baseline MSO shell that executes on both a massively parallel computer and a network of workstations. The MSO shell is demonstrated herein by a stochastic shape optimization of an axial compressor blade involving fully coupled aero-structural analysis.     

Reliability-based design optimization with frequency constraints using a new safest point approach

In the reliability-based design optimization (RBDO) model, the mean values of uncertain variables are usually applied as design variables, and the cost is optimized subject to prescribed probabilistic constraints as defined by a nonlinear mathematical programming problem. Therefore, an RBDO solution that reduces the structural weight in non-critical regions provides not only an improved design, but also a higher level of confidence in the design. Solving such nested optimization problems is extremely expensive for large-scale multidisciplinary systems that are likewise computationally intensive. This article focuses on the study of a particular problem representing the failure mode of structural vibration analysis. A new method is proposed, called safest point, that can efficiently give the reliability-based optimum solution under frequency constraints, and then several probability distributions are developed, which are mathematically nonlinear functions, for the proposed method. Finally, the efficiency of the extended approach is demonstrated for probability distributions such as log-normal and uniform distributions, and its applicability to the design of structures undergoing fluid–structure interaction phenomena, especially the design process of aeroelastic structures, is also demonstrated.     

Reliability-based design optimization using a generalized subset simulation method and posterior approximation

The evaluation of the probabilistic constraints in reliability-based design optimization (RBDO) problems has always been significant and challenging work, which strongly affects the performance of RBDO methods. This article deals with RBDO problems using a recently developed generalized subset simulation (GSS) method and a posterior approximation approach. The posterior approximation approach is used to transform all the probabilistic constraints into ordinary constraints as in deterministic optimization. The assessment of multiple failure probabilities required by the posterior approximation approach is achieved by GSS in a single run at all supporting points, which are selected by a proper experimental design scheme combining Sobol’ sequences and Bucher’s design. Sequentially, the transformed deterministic design optimization problem can be solved by optimization algorithms, for example, the sequential quadratic programming method. Three optimization problems are used to demonstrate the efficiency and accuracy of the proposed method.     

Rapid risk assessment using probability of fracture nomographs

Traditional risk-based design process involves designing the structure based on risk estimates obtained during several iterations of an optimization routine. This approach is computationally expensive for large-scale aircraft structural systems. Therefore, this paper introduces the concept of risk-based design plots that can be used for both structural sizing and risk assessment for fracture strength when maximum allowable crack length is available. In situations when crack length is defined as a probability distribution the presented approach can only be applied for various percentiles of crack lengths. These plots are obtained using normalized probability density models of load and material properties and are applicable for any arbitrary load and strength values. Risk-based design plots serve as a tool for failure probability assessment given geometry and applied load or they can determine geometric constraints to be used in sizing given allowable failure probability. This approach would transform a reliability-based optimization problem into a deterministic optimization problem with geometric constraints that implicitly incorporate risk into the design. In this paper, cracked flat plate and stiffened plate are used to demonstrate the methodology and its applicability.     

Set theoretic formulation of performance reliability of multiple response time‐variant systems due to degradations in system components

This paper presents a design stage method for assessing performance reliability of systems with multiple time‐variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time‐dependent limit‐state functions. System failure is defined as the non‐conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second‐order bounds. Further, for numerical efficiency probabilities are evaluated by first‐order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd.     

A heuristic for optimizing stochastic activity networks with applications to statistical digital circuit sizing

A deterministic activity network (DAN) is a collection of activities, each with some duration, along with a set of precedence constraints, which specify that activities begin only when certain others have finished. One critical performance measure for an activity network is its makespan, which is the minimum time required to complete all activities. In a stochastic activity network (SAN), the durations of the activities and the makespan are random variables. The analysis of SANs is quite involved, but can be carried out numerically by Monte Carlo analysis. This paper concerns the optimization of a SAN, i.e., the choice of some design variables that affect the probability distributions of the activity durations. We concentrate on the problem of minimizing a quantile (e.g., 95%) of the makespan, subject to constraints on the variables. This problem has many applications, ranging from project management to digital integrated circuit (IC) sizing (the latter being our motivation). While there are effective methods for optimizing DANs, the SAN optimization problem is much more difficult; the few existing methods cannot handle large-scale problems. In this paper we introduce a heuristic method for approximately optimizing a SAN, by forming a related DAN optimization problem which includes extra margins in each of the activity durations to account for the variation. Since the method is based on optimizing a DAN, it readily handles large-scale problems. To assess the quality of the resulting suboptimal designs, we describe two widely applicable lower bounds on achievable performance in optimal SAN design. We demonstrate the method on a simplified statistical digital circuit sizing problem, in which the device widths affect both the mean and variance of the gate delays. Numerical experiments show that the resulting design is often substantially better than one in which the variation in delay is ignored, and is often quite close to the global optimum (as verified by the lower bounds).     

General principles of the use of safety factors in design and assessment

Any structure or component can be made to fail if it is subjected to loadings in excess of its strength. Structural integrity is achieved by ensuring that there is an adequate safety margin or reserve factor between strength and loading effects. The basic principles of ‘allowable stress’ and ‘limit state’ design methods to avoid failure in structural and pressure vessel components are summarised. The use of risk as a means of defining adequate safety is introduced where risk is defined as the product of probability of failure multiplied by consequences of failure. The concept of acceptable ‘target’ levels of risk is discussed. The use of structural reliability theory to determine estimates of probability of failure and the use of the reliability index β are described. The need to consider the effects of uncertainties in loading information, calculation of stresses, input data and material properties is emphasised. The way in which the effect of different levels of uncertainty can be dealt with by use of partial safety factors in limit state design is explained. The need to consider all potential modes of failure, including the unexpected, is emphasised and an outline given of safety factor treatments for crack tip dependent and time dependent modes. The relationship between safety factors appropriate for the design stage and for assessment of structural integrity at a later stage is considered. The effects of redundancy and system behaviour on appropriate levels of safety factors are discussed.     

Structural optimization of uncertain dynamical systems considering mixed-design variables

In this paper attention is directed to the reliability-based optimization of uncertain structural systems under stochastic excitation involving discrete-continuous sizing type of design variables. The reliability-based optimization problem is formulated as the minimization of an objective function subject to multiple reliability constraints. The probability that design conditions are satisfied within a given time interval is used as a measure of system reliability. The problem is solved by a sequential approximate optimization strategy cast into the framework of conservative convex and separable approximations. To this end, the objective function and the reliability constraints are approximated by using a hybrid form of linear, reciprocal and quadratic approximations. The approximations are combined with an effective sensitivity analysis of the reliability constraints in order to generate explicit expressions of the constraints in terms of the design variables. The explicit approximate sub-optimization problems are solved by an appropriate discrete optimization technique. The optimization scheme exhibits monotonic convergence properties. Two numerical examples showing the effectiveness of the approach reported herein are presented.     

Unified reliability analysis by active learning Kriging model combining with random‐set based Monte Carlo simulation method

Reliability analysis with both aleatory and epistemic uncertainties is investigated in this paper. The aleatory uncertainties are described with random variables, and epistemic uncertainties are tackled with evidence theory. To estimate the bounds of failure probability, several methods have been proposed. However, the existing methods suffer the dimensionality challenge of epistemic variables. To get rid of this challenge, a so‐called random‐set based Monte Carlo simulation (RS‐MCS) method derived from the theory of random sets is offered. Nevertheless, RS‐MCS is also computational expensive. So an active learning Kriging (ALK) model that only rightly predicts the sign of performance function is introduced and closely integrated with RS‐MCS. The proposed method is termed as ALK‐RS‐MCS. ALK‐RS‐MCS accurately predicts the bounds of failure probability using as few function calls as possible. Moreover, in ALK‐RS‐MCS, an optimization method based on Karush–Kuhn–Tucker conditions is proposed to make the estimation of failure probability interval more efficient based on the Kriging model. The efficiency and accuracy of the proposed approach are demonstrated with four examples. Copyright © 2016 John Wiley & Sons, Ltd.