Uncertain linear systems in dynamics: Retrospective and recent developments by stochastic approaches

This paper provides an overview along with a critical appraisal of available methods for uncertainty propagation of linear systems subjected to dynamic loading. All uncertain structural properties are treated as random quantities by employing a stochastic approach. The loading can be either of deterministic or stochastic nature, described by white noise, filtered white noise, and more generally, by a Gaussian stochastic process.The assessment of the variability of the uncertain response in terms of the mean and variance is described by reviewing the random eigenvalue problem and procedures to evaluate the first two moments of the stochastic (uncertain) response. Computational procedures which are efficiently applicable for general FE-models are the focus of this work.Most recent developments for the reliability assessment–besides a retrospective review–are summarized, with particular emphasis on numerical Monte Carlo Simulation approaches. This review comprises methods to assess the first excursion probability directly by efficient numerical methods. General “black box” procedures and approaches applicable only for linear systems are critically discussed. Specific procedures applicable to linear systems subjected to general Gaussian excitation are subsequently addressed. Methods applicable for deterministic structural systems are introduced first. Finally, a procedure to exploit the solutions for deterministic linear systems for stochastic uncertain systems in an efficient manner is described.     

Practical random field discretization in stochastic finite element analysis

Stochastic finite element-based reliability analysis is applied to structures with distributed parameters that can be modeled as random fields. In this method, reliability is estimated through analytical computation of the sensitivity of stochastic response to the basic random variables. The random fields are discretized into sets of correlated random variables using two methods of discretization. The sensitivity measures are further used to selectively consider only a few of the distributed parameters as random fields, to ensure computational efficiency. The issue of choosing the appropriate mesh for the discretization of the random field is addressed through mesh refinement studies. With the help of three numerical examples, the paper examines the effects of the correlation characteristics of the random field on discretization and reliability analysis, and develops guidelines for efficient application of stochastic finite element analysis to structures with distributed parameters.     

Random analysis of geometrically non-linear FE modelled structures under seismic actions

In the framework of the finite element (FE) method, by using the “total Lagrangian approach”, the stochastic analysis of geometrically non-linear structures subjected to seismic inputs is performed. For this purpose the equations of motion are written with the non-linear contribution in an explicit representation, as pseudo-forces, and with the ground motion modelled as a filtered non-stationary white noise Gaussian process, using a Tajimi-Kanai-like filter. Then equations for the moments of the response are obtained by extending the classical Itô's rule to vectors of random processes. The equations of motion, and the equations for moments, obtained here, show a perfect formal similarity. By using this similarity a very effective computational procedure for evaluating response moments of any order is proposed.

Within the framework of non-Gaussian closure schemes, a technique is here presented based on a truncated Gram-Charlie expansion. To achieve this the Hermite coefficients are evaluated for multi-degree-of-freedom systems, once the multi-dimensional Hermite polynomials have been obtained in compact form.     

Life-cycle cost optimal design of passive dissipative devices

The cost-effective performance of structures under natural hazards such as earthquakes and hurricanes has long been recognized to be an important topic in the design of civil engineering systems. A realistic comprehensive treatment of such a design requires proper integration of (i) methodologies for treating the uncertainties related to natural hazards and to the structural behavior over the entire life-cycle of the building, (ii) tools for evaluating the performance using socioeconomic criteria, as well as (iii) algorithms appropriate for stochastic analysis and optimization. A systematic probabilistic framework is presented here for detailed estimation and optimization of the life-cycle cost of engineering systems. This framework is a general one but the application of interest here is the design of passive dissipative devices for seismic risk mitigation. A comprehensive methodology is initially presented for earthquake loss estimation; this methodology uses the nonlinear time-history response of the structure under a given excitation to estimate the damage in a detailed, component level. A realistic probabilistic model is then presented for describing the ground motion time history for future earthquake excitations. In this setting, the life-cycle cost is uncertain and can be quantified by its expected value over the space of the uncertain parameters for the structural and excitation models. Because of the complexity of these models, calculation of this expected value is performed using stochastic simulation techniques. This approach, though, involves an unavoidable estimation error and significant computational cost, features which make efficient design optimization challenging. A highly efficient framework, consisting of two stages, is discussed for this stochastic optimization. An illustrative example is presented that shows the efficiency of the proposed methodology; it considers the seismic retrofitting of a four-story non-ductile reinforced-concrete building with viscous dampers.     

Variability response functions for stochastic plate bending problems

In this paper, the weighted integral method and the concept of variability response function are successfully extended to plate bending problems where the elastic modulus of the structure is considered to be a two-dimensional, homogeneous stochastic field, overcoming earlier computational problems associated with the large number of terms in the expression for the variability response function. The concept of the variability response function is used to compute spectral-distribution-free upper bounds of the response variability. Such bounds are of paramount importance for the majority of real-life problems where only first and second moments of the stochastic material properties can be estimated with reasonable accuracy. Under the assumption of a prespecified power spectral density function of the stochastic field describing the elastic modulus, it is also possible to compute the response variability (in terms of second moments of response quantities) and the reliability (in terms of the safety index) of the stochastic plate. The use of a variability response function based on the local averaging method reduces the computational effort associated with the weighted integral method, with only a small loss of accuracy in most cases. Numerical examples are provided to demonstrate all of the above capabilities. One of the conclusions is that the coefficient of variation of certain response quantities can become larger than the coefficient of variation of the elastic modulus (the input quantity).     

On the propagation of uncertainty in inflow turbulence to wind turbine loads

When stochastic simulation of inflow turbulence random fields is employed in the analysis or design of wind turbines in normal operating states, it is common to use well-established standard spectral models represented in terms of parameters that are usually treated as fixed or deterministic values. Studies have suggested, though, that many of these spectral parameters can exhibit some degree of variability. It is not unreasonable to expect, then, that derived flow fields based on simulation with such spectral models can be in turn highly variable for different realizations. Turbine load and performance variability would also be expected to result if response simulations are carried out with these variable flow fields. The aim here is to assess the extent of variability in derived inflow turbulence fields that arises from the noted variability in spectral model parameters. Simulation of these parameters as random variables forms the basis of this study. A commercial-sized 1.5 MW concept wind turbine is considered in the numerical studies. Variability in turbulence power spectra at field points on the rotor plane and in turbulence coherence functions for separations on the order of a rotor diameter and smaller is studied. Using time domain simulations, variability in various wind turbine response measures is also studied where the focus is on statistics such as response root-mean-square and 10-min extreme estimates. It is seen that while variability in inflow turbulence spectra can be great, the variability in turbine loads is generally considerably lower. One exception is for turbine yaw loads whose larger variability arises due to sensitivity to a coherence decay parameter that is itself highly variable. Finally, because reduced-order representations of turbulence random fields using empirical orthogonal decomposition techniques allow useful physical insights into spatial patterns of flow, variability in the energy distribution and the shapes of such empirical eigenmodes is studied and a simplified model is proposed that retains key variability sources in a limited number of modes and that accurately preserves overall inflow turbulence field uncertainty.     

Simplified multi‐degree‐of‐freedom model for estimation of seismic response of regular wall‐frame structures

A simplified multi‐degree‐of‐freedom (MDOF) model is developed for estimation of seismic response of tall wall‐frame structures. By using the continuum technique for the structure and adopting the bilinear hysteretic model for material properties, procedure for the development of the simplified MDOF model is derived. The numerical study for a 20‐storey reinforced concrete (RC) wall‐frame structure is conducted to investigate the accuracy of seismic response predicted by the proposed model. Results from the nonlinear response history analyses based on the proposed MDOF model and the detailed structural model with member‐by‐member representation are compared and show very good agreement. The proposed simplified MDOF model is shown to provide a simple, efficient and accurate method for estimation of seismic performance of tall wall‐frame structures. Copyright © 2009 John Wiley & Sons, Ltd.     

Performance-based seismic design of steel frames using ant colony optimization

In this paper, a performance-based optimal seismic design of frame structures is presented using the ant colony optimization (ACO) method. This discrete metaheuristic algorithm leads to a significant improvement in consistency and computational efficiency compared to other evolutionary methods. A nonlinear analysis is utilized to arrive at the structural response at various seismic performance levels, employing a simple computer-based method for push-over analysis which accounts for first-order elastic and second-order geometric stiffness properties. Two examples are presented to illustrate the capabilities of ACO in designing lightweight frames, satisfying multiple performance levels of seismic design constraints for steel moment frame buildings, and a comparison is made with a standard genetic algorithm (GA) implementation to show the superiority of ACO for the discussed optimization problem.     

装配式RCS组合节点减震框架结构地震响应研究

为研究布置扇形铅黏弹性阻尼器的装配式RCS组合节点框架结构地震响应,选取不可替换的装配式RCS组合节点JD-1、可替换的装配式RCS组合节点JD-2、JD-3、JD-4连接的减震框架结构与普通刚接减震框架结构,对其进行地震响应分析并将计算结果与未设置阻尼器的RCS框架结构结果进行对比分析,研究其减震效果.结果表明:在布...     

Performance-based seismic fragility analysis of retaining walls based on the probability density evolution method

Seismic fragility analysis is an efficient way to study the seismic behaviour and performance of structures under the excitation of earthquakes of varying intensity, and an essential part of the seismic risk assessment of structures. A recently developed dynamic reliability methodology, the probability density evolution method (PDEM), is proposed for the dynamic reliability and seismic fragility analysis of a retaining wall. The PDEM can obtain an instantaneous probability density function of the seismic responses and easily acquire the seismic reliability of the structural system. An important advantage of the PDEM is its high efficiency relative to that of the Monte Carlo simulation method, which is often used in the reliability and fragility analysis of structures. The present study uses a typical gravity retaining wall to illustrate stochastic seismic responses and fragility curves that can be obtained by the PDEM. The combined uncertainties of the seismic force and soil properties are explicitly and systematically modelled by stochastic ground motions and random variables respectively. The performance of the retaining wall is analysed for different acceptable levels of backfill settlement. Additionally, seismic fragility curves are constructed without assuming the distribution of the seismic response.     

既有RC框架结构在地震作用下弹性反应的简化分析方法

提出了采用连续体模型确定既有钢筋混凝土框架结构地震弹性反应的简化分析方法.该模型参数:剪切刚度系数K由实测的第一自振周期反推而得,结构的自重系数M由经验估算而得.利用振型分解反应谱法求得结构的最大层间位移;并借助opensees计算平台对该结构进行大量的时程分析计算,得到的最大层间位移与连续体模型计算结果符合度较好,证明了此简化方法的可行性.     

钢筋混凝土框架结构的多尺度变异性分析

混凝土材料的变异性是造成钢筋混凝土结构非线性受力行为不可精确预测和不可准确控制的重要原因之一。考虑混凝土材料抗压强度的变异性,将具有确定性和高精度特征的拟蒙特卡洛法引入到钢筋混凝土框架结构的非线性静、动力分析中,结合增量变异性分析方法,定量分析了钢筋混凝土框架结构材料的变异性对构件和结构尺度非线性静、动力响应变异性的影响规律。研究结果表明：混凝土材料的变异性在构件尺度和结构尺度具有不同程度的衰减;混凝土材料的变异性在构件尺度呈现出复杂的随机演化特征,导致构件破坏模式和内力重分布路径的多样;在结构尺度,尤其是结构承载力和自振周期上具有相对的稳定性,这为结构整体失效模式的控制提供了可能。     

Stochastic wave models for stationary and homogeneous seismic ground motion

A fundamental theory of stationary, homogeneous stochastic waves is developed and a technique for digitally generating samples of such waves is introduced as a further extension of the spectral representation method. This is done primarily for the purpose of developing an analytical model for propagating seismic waves that can account for their stochastic characteristics in the time and space domain. From this model, the corresponding sample seismic waves can be digitally generated with great computational efficiency. The efficacy of this technique is demonstrated with the aid of a numerical example in which a sample of a stationary, homogeneous, two-dimensional, dispersive Rayleigh wave consistent, in its analytical form of spatial variability, with Lotung, Taiwan dense-array data is digitally generated. Although the stochastic wave model considered in this work is stationary and homogeneous, it is a straightforward task to extend the methodology introduced to nonstationary and / or non-homogeneous stochastic waves characterized by an evolutionary power spectrum.     

Probability density evolution analysis of stochastic nonlinear structure under non-stationary ground motions

In the present paper, a framework of dimension-reduction modeling method is developed for a dual stochastic dynamic structural system of spectrum-compatible non-stationary stochastic ground motion processes and stochastic structures. With the aid of the proposed method, the random variables used to describe the stochastic characteristics of the non-stationary ground motion processes and the structural parameters are respectively represented by the one-elementary-random-variable functions, contributing to an entire stochastic dynamic structural system readily described by merely two elementary random variables. Owing to the fact that the number of the elementary random variables needed is extremely small, the set of the representative points associated with the elementary random variables can thus be selected by the widely-used number theoretical method, and then the probability density evolution method can be conveniently combined to conduct the dynamic response analysis and dynamic reliability evaluation of nonlinear stochastic structures. In the numerical examples, the probability density evolution analysis of an eight-storey reinforced concrete frame structure with random parameters subjected to spectrum-compatible non-stationary stochastic ground motion processes is investigated as a case study. Numerical results fully demonstrated the effectiveness and robustness of the proposed method.     

On the reliability-based design of structures including passive energy dissipation systems

This contribution presents a methodology for stochastic design of structures including vibration protection systems. The approach is then used to investigate the effect of uncertain model parameters on the reliability-based optimal design of structures with a class of passive energy dissipation systems. The uncertainty of structural parameters as well as the variability of future excitations are characterized in a probabilistic manner. The optimal design problem is formulated as a non-linear constrained minimization problem involving multiple design requirements, including reliability constraints related to the structural performance. Failure events defined by a large number of random variables are used to characterize the reliability measures. A sequential optimization approach based on global conservative, convex and separable approximations is implemented for solving the optimization problem. The effects of uncertain model parameters on the performance, robustness and reliability of protected systems is illustrated by two example problems that consider multi-story buildings under stochastic ground excitation.     

高温后型钢混凝土框架结构抗震性能及其有限元分析

考虑受火时间、柱轴压比的影响,进行了考虑火灾升降温作用影响的高温后型钢混凝土框架结构抗震性能试验,升降温过程中和高温后阶段均保持柱顶竖向荷载的恒定。通过试验对高温后型钢混凝土框架结构在反复荷载作用下的破坏形态及滞回性能进行了研究。在此基础上,考虑升温阶段、降温阶段以及火灾后三阶段材料本构关系的转化以及型钢与混凝土之间、钢筋与混凝土之间界面的黏结滑移特性,建立了型钢混凝土框架结构温度场以及火灾后抗震性能的有限元分析模型,所得温度场以及试件的破坏形态、滞回曲线的分析结果与试验结果基本吻合,验证了所提出模型的合理性。     

An efficient stochastic dynamic analysis of soil media using radial basis function artificial neural network

Since a lot of engineering problems are along with uncertain parameters, stochastic methods are of great importance for incorporating random nature of a system property or random nature of a system input. In this study, the stochastic dynamic analysis of soil mass is performed by finite element method in the frequency domain. Two methods are used for stochastic analysis of soil media which are spectral decomposition and Monte Carlo methods. Shear modulus of soil is considered as a random field and the seismic excitation is also imposed as a random process. In this research, artificial neural network is proposed and added to Monte Carlo method for sake of reducing computational effort of the random analysis. Then, the effects of the proposed artificial neural network are illustrated on decreasing computational time of Monte Carlo simulations in comparison with standard Monte Carlo and spectral decomposition methods. Numerical verifications are provided to indicate capabilities, accuracy and efficiency of the proposed strategy compared to the other techniques.