Inverse reliability based structural design for system dependent critical earthquake loads

The problem of reliability based design of structures subjected to partially specified random earthquake loads is considered. A procedure for the determination of structure–excitation pair that maximizes a specified response variable and, at the same time, achieves a target reliability is outlined. The procedure combines concepts from inverse first order reliability methods and methods for determining random critical earthquake loads. The formulation is shown to lead to a problem in constrained non-linear optimization. Issues related to spatial variability of earthquake loads are also addressed. Illustrative examples on a singly supported multi-degree of freedom system and a doubly supported single degree of freedom system are included.     

基于车辆荷载效应截尾分布的桥梁限载分析方法

根据车辆荷载效应右截尾分布模型, 提出一种基于结构可靠度理论的中、小跨度桥梁限载分析方法。首先以规范规定的车辆荷载效应分布为原始分布, 构造右截尾的车辆荷载效应概率密度函数;然后, 假定抗力、恒载效应及车辆荷载效应为相互独立的随机变量, 建立了考虑车辆荷载效应右截尾分布特征的限载系数反演模型;继而, 通过分析车辆荷载效应均值变化对限载系数的影响规律, 提出了理想抗力桥梁条件限载系数的确定方法, 并讨论了条件限载系数对车辆荷载效应变异系数与分布类型的敏感程度;最后, 根据原桥梁规范受弯构件承载能力设计表达式, 计算了按原规范设计桥梁的条件限载系数。结果表明, 桥梁限载取值仅与设计荷载等级、容许失效概率以及设计采用的活恒载比值有关, 而与车辆荷载效应的统计参数关系不大。所提限载分析方法可为中、小跨径桥梁提供具有一致可靠度水平的限载取值。     

Uncertainty analysis and reliability estimation of clamp band joints

In this paper, a mathematical model is developed to estimate specifications of a clamp band joint by taking into account full effects of the uncertainty of the joint parameters with a reasonable amount of computational cost. In the model, the clamp band joint is partitioned into sectors which are substituted with equivalent springs. The mathematical formulation of the stiffness and load bearing capacity of the joint is derived under combination of the loads. Nonlinear finite-element analysis is conducted to verify the proposed formulation, where all parameters have their nominal value. Finally, a procedure is introduced, in which one can specify the uncertainty of the joint parameters and the safety factor to obtain a desired reliability level from a table.     

Mechanical reliability by design

With distributed loads and strengths the condition of minimum whole life cost defines uniquely the safety margin (or factor of safety) that must be used. Established empirical factors used in current deterministic design are not then relevant. It is postulated that computer-aided design packages should incorporate a simultaneous optimization of the safety margin and maintenance policy in place of these factors, since once the design is finalized the whole life cost is finalized. The full development of stochastic CAD methods is seen as the next essential step to advance design techniques.     

Optimization of laminated composites subject to uncertain buckling loads

Optimal design of composite laminates under buckling load uncertainty is presented. The laminates are subjected to biaxial compressive loads and the buckling load is maximized under worst case in-plane loading which is computed using an anti-optimization approach. The magnitudes of the in-plane loads are not known a priori resulting in load uncertainty subject to the only constraint that the loads belong to a given uncertainty domain. Results are given for continuous and discrete fibre orientations which constitute the optimization problem coupled to load anti-optimization problem leading to a nested solution method. It is observed that the stacking sequence of a laminate designed for a deterministic load case only differs considerably from that of a robust laminate designed taking load uncertainties into account. Consequently the buckling load carried by a deterministic design is considerably less than the one carried by a robust design when both are subjected to uncertain loads.     

Estimation of cable safety factors of suspension bridges using artificial neural network‐based inverse reliability method

The design of the main cables of suspension bridges is based on the verification of the rules defined by standard specifications, where cable safety factors are introduced to ensure safety. However, the current bridge design standards have been developed to ensure structural safety by defining a target reliability index. In other words, the structural reliability level is specified as a target to be satisfied by the designer. Thus, calibration of cable safety factors is needed to guarantee the specified reliability of main cables. This study proposes an efficient and accurate algorithm to solve the calibration problem of cable safety factors of suspension bridges. Uncertainties of the structure and load parameters are incorporated in the calculation model. The proposed algorithm integrates the concepts of the inverse reliability method, non‐linear finite element method, and artificial neural networks method. The accuracy and efficiency of this method with reference to an example long‐span suspension bridge are studied and numerical results have validated its superiority over the conventional deterministic method or inverse reliability method with Gimsing's simplified approach. Finally, some important parameters in the proposed method are also discussed. Copyright © 2006 John Wiley & Sons, Ltd.     

相关荷载效应组合及结构可靠度计算

相关性是孪生性荷载的基本特征之一。本文研究了相关荷载效应的组合问题,通过引入一个综合了相关荷载效应特性的等效荷载效应,提出了基于Ｔｕｒｋｓｔｒａ规则的荷载效应实用组合方法,并给出相关荷载效应组合和可靠度分析的算例。算例分析表明,荷载效应的相关性对结构的可靠度有一定的影响,在有条件的情况下应于以考虑。     

Reliability-based design optimization under stationary stochastic process loads

Time-dependent reliability-based design ensures the satisfaction of reliability requirements for a given period of time, but with a high computational cost. This work improves the computational efficiency by extending the sequential optimization and reliability analysis (SORA) method to time-dependent problems with both stationary stochastic process loads and random variables. The challenge of the extension is the identification of the most probable point (MPP) associated with time-dependent reliability targets. Since a direct relationship between the MPP and reliability target does not exist, this work defines the concept of equivalent MPP, which is identified by the extreme value analysis and the inverse saddlepoint approximation. With the equivalent MPP, the time-dependent reliability-based design optimization is decomposed into two decoupled loops: deterministic design optimization and reliability analysis, and both are performed sequentially. Two numerical examples are used to show the efficiency of the proposed method.     

Robust topology optimization under multiple independent uncertainties of loading positions

The topology optimization problem of a continuum structure is further investigated under the independent position uncertainties of multiple external loads, which are now described with an interval vector of uncertain-but-bounded variables. In this study, the structural compliance is formulated with the quadratic Taylor series expansion of multiple loading positions. As a result, the objective gradient information to the topological variables can be evaluated efficiently upon an explicit quadratic expression as the loads deviate from their ideal application points. Based on the minimum (largest absolute) value of design sensitivities, which corresponds to the most sensitive compliance to the load position variations, a two-level optimization algorithm within the non-probabilistic approach is developed upon a gradient-based optimization method. The proposed framework is then performed to achieve the robust optimal configurations of four benchmark examples, and the final designs are compared comprehensively with the traditional topology optimizations under the loading point fixation. It will be observed that the present methodology can provide a remarkably different structural layout with the auxiliary components in the design domain to counteract the load position uncertainties. The numerical results also show that the present robust topology optimization can effectively prevent the structural performance from a noticeable deterioration than the deterministic optimization in the presence of load position disturbances.     

Identification of stochastic loads applied to a non-linear dynamical system using an uncertain computational model and experimental responses

The paper is devoted to the identification of stochastic loads applied to a non-linear dynamical system for which experimental dynamical responses are available. The identification of the stochastic load is performed using a simplified computational non-linear dynamical model containing both model uncertainties and data uncertainties. Uncertainties are taken into account in the context of the probability theory. The stochastic load which has to be identified is modelled by a stationary non-Gaussian stochastic process for which the matrix-valued spectral density function is uncertain and is then modelled by a matrix-valued random function. The parameters to be identified are the mean value of the random matrix-valued spectral density function and its dispersion parameter. The identification problem is formulated as two optimization problems using the computational stochastic model and experimental responses. A validation of the theory proposed is presented in the context of tubes bundles in Pressurized Water Reactors.     

A method for the preliminary design of gears using a reduced number of American Gear Manufacturers Association (AGMA) correction factors

Typically, the preliminary design of mechanical components such as gears is carried out using standardized design processes such as those developed by the American Gear Manufacturers Association (AGMA). These design standards include a large number of ‘correction factors’ to account for various uncertainties. As the knowledge about these uncertainties increases, it becomes possible to include them systematically in the design procedure, thereby reducing the number of empirical correction factors. Robust design provides a way to design in the presence of various uncertainties. In this article, a design method is proposed to eliminate empirical correction factors and is demonstrated by eliminating two correction factors from the AGMA design standards for a spur gear, namely, the factor of safety in contact and the reliability factor by the formal introduction of uncertainty in the magnitude of load and material properties. The proposed method is illustrated with the design of an automotive gear with desired reliability, cost and robustness.     

Decision analysis for deteriorating structures

Measures that improve durability of a structure usually increase its initial cost. Thus, in order to make a decision about a cost-effective solution the life-cycle cost of a structure including cost of structural failure needs to be considered. Due to uncertainties associated with structural properties, loads and environmental conditions the cost of structural failure is a random variable. The paper derives probability distributions of the cost of failure of a single structure and a group of identical structures when single or multiple failures are possible during the service life of a structure. The probability distributions are based on cumulative probabilities of failure of a single structure over its service life. It is assumed that failures occur at discrete points in time, the cost of failure set at the time of decision making remains constant for a particular design solution and the discount rate is a deterministic parameter not changing with time. The probability distributions can be employed to evaluate the expected life-cycle cost or the expected utility, which is then used in decision making. An example, which considers the selection of durability specifications for a reinforced concrete structure built on the coast, illustrates the use of the derived probability distributions.     

Comparison of robust,reliability-based and non-probabilistic topology optimization under uncertain loads and stress constraints

It is nowadays widely acknowledged that optimal structural design should be robust with respect to the uncertainties in loads and material parameters. However, there are several alternatives to consider such uncertainties in structural optimization problems. This paper presents a comprehensive comparison between the results of three different approaches to topology optimization under uncertain loading, considering stress constraints: (1) the robust formulation, which requires only the mean and standard deviation of stresses at each element; (2) the reliability-based formulation, which imposes a reliability constraint on computed stresses; (3) the non-probabilistic formulation, which considers a worst-case scenario for the stresses caused by uncertain loads. The information required by each method, regarding the uncertain loads, and the uncertainty propagation approach used in each case is quite different. The robust formulation requires only mean and standard deviation of uncertain loads; stresses are computed via a first-order perturbation approach. The reliability-based formulation requires full probability distributions of random loads, reliability constraints are computed via a first-order performance measure approach. The non-probabilistic formulation is applicable for bounded uncertain loads; only lower and upper bounds are used, and worst-case stresses are computed via a nested optimization with anti-optimization. The three approaches are quite different in the handling of uncertainties; however, the basic topology optimization framework is the same: the traditional density approach is employed for material parameterization, while the augmented Lagrangian method is employed to solve the resulting problem, in order to handle the large number of stress constraints. Results are computed for two reference problems: similarities and differences between optimized topologies obtained with the three formulations are exploited and discussed.     

不确定视角下产品结构性能优化设计综述与展望

目的 复杂产品在工程装备、航空航天等我国优先发展的战略领域中扮演着不可替代的重要角色,在其结构设计过程中普遍存在着各种不确定性,导致产品结构性能很难实现最优,甚至出现重大故障。方法 针对复杂产品设计过程中的多粒度模糊不确定、随机不确定、不完备区间不确定和高维混合不确定等特点,将不确定性理论与产品结构设计过程相结合,系统地构建了不确定视角下产品结构性能优化设计理论体系,提出了多粒度模糊不确定下产品质量特性精准提取、随机不确定下产品功能结构模块化求解、不完备区间不确定下产品结构方案多属性决策、高维混合不确定下产品关键结构可靠性优化等关键技术,并指出了产品结构性能优化设计的未来发展方向。结论 此设计理论能够充分适应和利用产品设计过程的多种不确定信息,为在结构设计环节切实提升产品性能提供了有力参考。     

非概率不确定性及其对船舶坐墩配墩优化的影响

讨论了船舶坐墩配墩设计中存在的非概率不确定性及其描述。提出了一个非概率不确定性条件下船舶坐墩支墩配置优化设计的数学模型。着重分析了船体梁载荷和支墩组合刚度不确定性对设计结果的影响。计算结果表明,考虑参数不确定性后,配墩方案发生了变化,支墩结构重量也有所增加。增加的支墩材料是用于提高结构物抵抗不确定性参数波动变化的能力。相对于船体梁载荷不确定性,支墩组合刚度不确定性对设计结果的影响较大。     

Inversion of loading time history using displacement response of composite laminates: Three-dimensional cases

Summary An inverse procedure is proposed to reconstruct the time history of transient loads on the surface of composite laminates from the knowledge of dynamic displacement response at only one receiving point. A hybrid numerical method (HNM) is adopted as the forward solver to compute the dynamic displacement response of composite laminates subjected to arbitrary loads. By introducing a kernel displacement function — the dynamic displacement response of composite laminates excited by a point stepimpact load and the displacement response subjected to a load with an arbitrary force function are expressed in a form of convolution integral. The force history is reconstructed by employing an inversion algorithm, in which the least-squares optimization method being adopted to deconvolute the integral. Both point loads and loads with small spatial distribution are investigated and numerical verifications are given. The robustness of the procedure in the presence of noise is also investigated. Good agreements between the identified and true functions for all cases demonstrate the effectiveness of the present inverse procedure. The present inverse procedure is useful for determining impact loads on material surface using response on a point remote to the impact point.     

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.     

复合材料点阵夹芯结构平压性能不确定分析与优化

考虑一体化成型工艺制备的复合材料点阵夹芯结构及其不确定性, 采用区间向量实现不确定参数定量化, 建立复合材料点阵夹芯结构平压性能区间分析模型。考虑结构功能状态判断的模糊性, 分别在不考虑设计容差与考虑设计容差情形下, 建立了不确定平压载荷作用下含区间参数模糊可靠性分析与优化模型。研究结果表明: 材料参数及结构参数不确定性, 特别是设计容差对复合材料点阵夹芯结构平压性能影响明显, 因此在工程优化中不仅需要充分考虑材料参数与外部载荷等不确定性, 而且需要充分重视传统不确定设计方法中未计及的设计容差的影响。本研究实现了理论成果与工程应用的有机结合, 为工程领域复合材料点阵夹芯结构平压性能分析与优化提供有效理论方法。     

Gearbox design for uncertain load requirements using active robust optimization

Design and optimization of gear transmissions have been intensively studied, but surprisingly the robustness of the resulting optimal design to uncertain loads has never been considered. Active Robust (AR) optimization is a methodology to design products that attain robustness to uncertain or changing environmental conditions through adaptation. In this study the AR methodology is utilized to optimize the number of transmissions, as well as their gearing ratios, for an uncertain load demand. The problem is formulated as a bi-objective optimization problem where the objectives are to satisfy the load demand in the most energy efficient manner and to minimize production cost. The results show that this approach can find a set of robust designs, revealing a trade-off between energy efficiency and production cost. This can serve as a useful decision-making tool for the gearbox design process, as well as for other applications.     

Structural reliability optimization using an efficient safety index calculation procedure

The objective of this paper is to conduct reliability-based structural optimization in a multidisciplinary environment. An efficient reliability analysis is developed by expanding the limit functions in terms of intermediate design variables. The design constraints are approximated using multivariate splines in searching for the optimum. The reduction in computational cost realized in safety index calculation and optimization are demonstrated through several structural problems. This paper presents safety index computation, analytical sensitivity analysis of reliability constraints and optimization using truss, frame and plate examples.