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.     

Bidirectional evolutionary structural optimization for nonlinear structures under dynamic loads

This study aims to develop efficient numerical optimization methods for finding the optimal topology of nonlinear structures under dynamic loads. The numerical models are developed using the bidirectional evolutionary structural optimization method for stiffness maximization problems with mass constraints. The mathematical formulation of topology optimization approach is developed based on the element virtual strain energy as the design variable and minimization of compliance as the objective function. The suitability of the proposed method for topology optimization of nonlinear structures is demonstrated through a series of two- and three-dimensional benchmark designs. Several issues relating to the nonlinear structures subjected to dynamic loads such as material, geometric, and contact nonlinearities are addressed in the examples. It is shown that the proposed approach generates more reliable designs for nonlinear structures.     

Nonlinear dynamic analysis of suspension bridges under random wind loads by stochastic linearization

For the investigation of the geometric nonlinear effect of suspension bridges under random wind loads, a frequency domain method based on a stochastic linearization technique is presented. The equation of motion is formulated by including the coupling between the vertical and the torsional motions. In the linearization procedure, the nonlinear terms are approximated as a sum of linear and constant terms in order to take into account the nonzero mean components as well as the fluctuating components of structural responses. The verification has been made on a case with four modal degrees of freedom. Example analyses have been carried out on two structures for various wind speeds and wind force parameters. Numerical results indicate that, by including the nonlinearity into the analysis, the dynamic responses of the bridges, particularly in the vertical direction, change considerably.     

Procedure of optimization of the structure of composite plates under dynamic loads

We analyze the applicability of the methods of the theory of nonlinear programming to the development of the optimal design of laminated plates with maximum damping. Translated from Problemy Prochnosti, No. 6, pp. 91–99, November–December, 2008.     

Robust design optimization of nonlinear energy sink under random system parameters

Generally, in designing nonlinear energy sink (NES), only uncertainties in the ground motion parameters are considered and the unconditional expected mean of the performance metric is minimized. However, such an approach has two major limitations. First, ignoring the uncertainties in the system parameters can result in an inefficient design of the NES. Second, only minimizing the unconditional mean of the performance metric may result in large variance of the response because of the uncertainties in the system parameters. To address these issues, we focus on robust design optimization (RDO) of NES under uncertain system and hazard parameters. The RDO is solved as a bi-objective optimization problem where the mean and the standard deviation of the performance metric are simultaneously minimized. This bi-objective optimization problem has been converted into a single objective problem by using the weighted sum method. However, solving an RDO problem can be computationally expensive. We thus used a novel machine learning technique, referred to as the hybrid polynomial correlated function expansion (H-PCFE), for solving the RDO problem in an efficient manner. Moreover, we adopt an adaptive framework where H-PCFE models trained at previous iterations are reused and hence, the computational cost is less. We illustrate that H-PCFE is computationally efficient and accurate as compared to other similar methods available in the literature. A numerical study showcasing the importance of incorporating the uncertain system parameters into the optimization procedure is shown. Using the same example, we also illustrate the importance of solving an RDO problem for NES design. Overall, considering the uncertainties in the parameters have resulted in a more efficient design. Determining NES parameters by solving an RDO problem results in a less sensitive design.     

On fatigue reliability under random loads

We consider the problem of estimating the probability of survival (non-failure) and the probability of safe operation (strength greater than a limiting value) of structures subjected to random loads. These probabilities are formulated in terms of the probability distributions of the loads and the material strength. For the material strength, the Weibull distribution is assumed, the parameters of which are estimated by a statistical analysis of the experimental tensile strength of steel specimens subjected to different periods of random loads. The statistical analysis shows that, with the application of random loads, the initial homogeneous distribution of strength changes to a two-component distribution, reflecting the two-stage fatigue damage. In the crack initiation stage, the strength increases initially and then decreases, while an abrupt decrease of strength is seen in the crack propagation stage. The consequences of this behaviour on the fatigue reliability are discussed.     

受宽带随机激励的最优反馈控制

文章提出了基于平稳响应的随机最优反馈控制策略，研究了受宽带随机激励的Hamilton系统的最优反馈控制。首先应用拟Hamilton系统随机平均法建立平均后的Ito随机微分方程；接着求解相应的Fokker—Plank—Kol—mogorov（FPK）方程得到系统精确平稳解；然后将控制力的均方值和Hamilton能量均值结合作为系统控制指标，得出最优反馈控制规律；最后用两个例子详细说明了这一控制方法的实施过程的有效性，数字模拟结果与理论结果完全吻合。     

Corrugated panels under dynamic loads

Corrugated panels are commonly used as blast walls on offshore installations, and an assessment of their response to dynamic pressure loads needs to be made during design. Although detailed numerical models are commonly used, simplified models based on spring-mass idealisations, such as Biggs’ method, are still important for preliminary design. Based on an earlier static formulation, the dynamic response of a corrugated panel (as characterised by its central deflection) subjected to a uniformly distributed pressure pulse is analysed. A simplified theoretical method is built up and new transformation factors for panels with straight endplates are derived. The method is calibrated against corresponding finite element and experimental results. Good agreement is achieved.     

典型结构在单、多轴随机振动下的动力学特性对比研究

以某典型线性结构和非线性结构为例,通过理论分析和试验研究,对比结构在单、多轴随机振动载荷下动力学特性的差别.研究结果表明:对于线性结构,单轴和多轴加载时结构的自振频率不发生变化,但由于单轴和多轴加载对结构模态有不同激发效果.振动响应谱会出现不同数量的共振峰;对于刚度非线性结构,由于多轴加载时的变形比单轴加载时大.系统刚性发生变化,自振频率也随之变化.     

Structural shape optimization using equivalent static loads transformed from dynamic loads

In structural optimization, static loads are generally utilized although real external forces are dynamic. Dynamic loads have been considered only in small‐scale problems. Recently, an algorithm for dynamic response optimization using transformation of dynamic loads into equivalent static loads has been proposed. The transformation is conducted to match the displacement fields from dynamic and static analyses. This algorithm can be applied to large‐scale problems. However, the application has been limited to size optimization. The present study applies the algorithm to shape optimization. Because the number of degrees of freedom of finite element models is usually very large in shape optimization, it is difficult to conduct dynamic response optimization with conventional methods that directly treat dynamic response in the time domain. The optimization process is carried out by interfacing an optimization system and an analysis system for structural dynamics. Various examples are solved to verify the algorithm. The results are compared to the results from static loads. It is found that the algorithm using static loads transformed from dynamic loads based on displacement is valid for very large‐scale shape optimization problems. Copyright © 2005 John Wiley & Sons, Ltd.     

Reliability-based design optimization with progressive surrogate models

Reliability-based design optimization (RBDO) has traditionally been solved as a nested (bilevel) optimization problem, which is a computationally expensive approach. Unilevel and decoupled approaches for solving the RBDO problem have also been suggested in the past to improve the computational efficiency. However, these approaches also require a large number of response evaluations during optimization. To alleviate the computational burden, surrogate models have been used for reliability evaluation. These approaches involve construction of surrogate models for the reliability computation at each point visited by the optimizer in the design variable space. In this article, a novel approach to solving the RBDO problem is proposed based on a progressive sensitivity surrogate model. The sensitivity surrogate models are built in the design variable space outside the optimization loop using the kriging method or the moving least squares (MLS) method based on sample points generated from low-discrepancy sampling (LDS) to estimate the most probable point of failure (MPP). During the iterative deterministic optimization, the MPP is estimated from the surrogate model for each design point visited by the optimizer. The surrogate sensitivity model is also progressively updated for each new iteration of deterministic optimization by adding new points and their responses. Four example problems are presented showing the relative merits of the kriging and MLS approaches and the overall accuracy and improved efficiency of the proposed approach.     

Optimal structural design under dynamic loads

This paper presents an algorithm for optimal design of elastic structures, subjected to dynamic loads. Finite element, modal analysis and a generalized steepest descent method are employed in developing a computational algorithm. Structural weight is minimized subject to constraints on displacement, stress, structural frequency, and member size. Optimum results for several example problems are presented and compared with those available in the literature.     

随机激励下重载车辆空气悬架参数多目标优化

为提高车辆行驶平顺性、减小轮胎对路面的动载荷,以某重载车辆空气悬架系统为研究对象,建立四自由度1/2车辆多目标优化模型,提出改进的多目标自适应遗传算法对悬架参数进行优化。与一般遗传算法相比,车身垂向加速度、前后轮动载荷有效值约减小10%,目标函数值改善度降低57.03%。该方法不仅能提高车辆行驶稳定性,且可减小轮胎对路面的动载荷,已进一步证明该方法的有效性及可行性。     

Behaviour of shear connectors under dynamic loads

The details of an experimental investigation of the fatigue strength of stud type shear connectors are described. A new type of test set-up called ‘New Standard Test Method’ has been employed to assess the fatigue strength of an individual shear connector. The results of the fatigue tests are discussed and a simple relationship between the stress range and the number of cycles has been obtained. This relationship is compared to the one obtained by Slutter on the basis of his experiments.     

On fatigue life under stationary Gaussian random loads

Load parameters for a stationary Gaussian random load are taken as the location, scale and shape parameters of its power spectrum. The centre frequency of a power spectrum is proposed as a measure of fatigue life. A fatigue life function, formulated in terms of the load parameters, is evaluated from the test results obtained by fatigue testing a structural steel under six different power spectral shapes. The concept of a shape operator is employed to correlate fatigue lives under different power spectral shapes.     

Reliability-based optimization of fibrous laminated composites

This paper is concerned with the optimum design of multiaxial fiber reinforced laminate systems under probabilistic conditions of loads and material properties. A multiaxially laminated composite is treated as a structural system with each ply contained in the composite as one element. The Tsai-Wu failure criterion is adopted as the limit state function of a unidirectional ply. It is assumed that the system failure occurs when any one of the plies in a laminate system fails. The multiple-check-point method is successfully applied to evaluate the system reliabilities of multiaxial laminates under probabilistic in-plane stresses. An optimization problem is defined to find the optimal number of fiber orientation axes, optimum orientation angles, and optimum ply ratios which yield the highest system reliability.     

动力荷载作用下水-桩-土相互作用简化分析研究EI北大核心CSCD

针对近海结构单桩基础在动力荷载作用下发生复杂的水-桩-土相互作用问题,建立了三维水-桩-土全耦合动力有限元分析模型。土体、桩和水体分别采用实体单元和声学单元模拟,土体截断边界采用滚轴边界条件、水体截断边界采用无反射吸收边界条件,并确定了合理的截断边界位置;以全耦合分析模型计算结果为参考解,系统研究了四种动荷载作用下水-桩不耦合(两者界面自由)和水-土不耦合(两者界面自由)对桩体和海床表面位移和动水压力响应的影响,揭示了不同水深和桩体半径变化下不考虑两种相互作用的影响规律。     

Optimum design criteria for elastic structures subject to random dynamic loads

A new stochastic optimization method, which makes use of a constraint on structural reliability, is proposed for structures subject to dynamic random loads. A minimum weight problem is posed, in which a constraint condition imposes that the failure probability must be smaller than a given admissible level. The failure is determined by the first crossing outside the safe domain of a suitable structural response vector. The method is used to find the optimal shape of an elastic vertical column supporting a fixed mass positioned on the top, subject to a Gaussian filtered stationary stochastic horizontal acceleration process. The column, with variable annular cross-section, is described by a deterministic elastic multi-degree-of-freedom system. It is assumed that failure is reached when its lateral displacement exceeds an acceptable threshold value. Under this constraint, the structural weight is minimized and the optimal shape is determined for different structural conditions.     

Iterative univariate optimization of nonlinear trusses with fuzzy strain constraints

A univariate iterative method for the optimization of nonlinear trusses with fuzzy constraints is proposed. The method is based on the control of elemental strains rather than stresses and hence it is called the desirable strain design criteria method. The objective of the fuzzy optimization is to find a design so that the weight or volume of the truss as well as all of the constraints on the elemental strains and joint deflections are desirable and satisfactory according to the fuzzy designer expressions. Such expressions are made in the form of fuzzy membership functions. Two illustrative examples, a 3- and a 10-bar truss, are optimized by the proposed method, where it is shown that in both cases the method monotonically converges to the highest desirability.     

Design optimisation of linear structures subjected to dynamic random loads with respect to fatigue life

This work concerns the optimisation of linear two-dimensional planar metallic structures subjected to stationary Gaussian random loads. A strategy intended to improve the design of this category of structures is proposed according to the multiaxial high-cycle fatigue, by varying the thicknesses of the zones where the structure is assumed to be divided. To achieve this goal, a computationally efficient framework for the determination of the fatigue life is firstly proposed. It is based on a frequency formulation of the Sines’ fatigue criterion, adapted according to Pitoiset and Preumont works [1]. Two examples are presented to demonstrate the ability of the methodology to provide better structural topologies.