A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.     

Probabilistic analysis of truss structures with uncertain parameters (virtual distortion method approach)

A new approach for probabilistic characterization of linear elastic redundant trusses with uncertainty on the various members subjected to deterministic loads acting on the nodes of the structure is presented. The method is based on the simple observation that variations of structural parameters are equivalent to superimposed strains on a reference structure depending on the axial forces on the elastic modulus of the original structure as well as on the uncertainty (virtual distortion method approach). Superposition principle may be applied to separate contribution to mechanical response due to external loads and parameter variations. Statically determinate trusses dealt with the proposed method yields explicit analytic solution in terms of displacements while redundant trusses have been studied by means of an asymptotic expansion exhibiting explicit dependence on parameter fluctuations. Probabilistic characterization of the response may then be obtained both for statically determinate and statically indeterminate stochastic trusses.     

Simulation of stationary Gaussian/non-Gaussian stochastic processes based on stochastic harmonic functions

A new model is proposed to represent and simulate Gaussian/non-Gaussian stochastic processes. In the proposed model, stochastic harmonic function (SHF) is extended to represent multivariate Gaussian process firstly. Compared with the conventional spectral representation method (SRM), the SHF based model requires much fewer variables and Cholesky decompositions. Then, SHF based model is further extended to univariate/multivariate non-Gaussian stochastic process simulation. The target non-Gaussian process can be obtained from the corresponding underlying Gaussian processes by memoryless nonlinear transformation. For arbitrarily given marginal probability distribution function (PDF), the covariance function of the underlying multivariate Gaussian process can be determined easily by introducing the Mehler’s formula. And when the incompatibility between the target non-Gaussian power spectral density (PSD) or PSD matrix and marginal PDF exists, the calibration of the target non-Gaussian spectrum will be required. Hence, the proposed model can be regarded as SRM to efficiently generate Gaussian/non-Gaussian processes. Finally, several numerical examples are addressed to show the effectiveness of the proposed method.     

Spectral stochastic analysis of structures with uncertain parameters

We present a probabilistic analysis of a structure with uncertain parameters subject to arbitrary stochastic excitations in a frequency domain. The problem of stochastic dynamic analysis of a linear system in a frequency domain is formulated by taking into consideration the uncertainty of structural parameters. The solution is based on the idea of a random frequency response vector for stationary input excitation and a transient random frequency response vector for nonstationary one which are used in the context of spectral analysis in order to determine the influence of structural uncertainty on the random response of structure. The numerical spectral analysis of the building structure under wind and earthquake excitation is provided to demonstrate the described algorithms in the context of computer implementation.     

Response analysis of stochastic parameter structures under non-stationary random excitation

 The stochastic orthogonal polynomial expansion method is extended with the pseudo-excitation method in this paper. This extension enables the stochastic orthogonal polynomial method to be readily used in the analysis of stochastic parameter structures under non-stationary random excitation. The probabilistic information of structural response, such as the power spectral density, standard deviation function, etc. can be obtained directly with this method. A dynamic condensation algorithm for order-expanded equation resulting from the orthogonal polynomial expansion method is also presented in this paper. The applicability of the proposed methodology is demonstrated by numerical examples. Received 21 July 2000     

The two-level probabilistic description of the compressive constitutive parameters of concrete

The constitutive model of concrete is the critical basis for the nonlinear analyses of concrete structures. Due to the fact that the mechanical behavior of concrete exhibits remarkable randomness, the probabilistic modeling of the key parameters of the concrete constitutive model is of paramount significance. In the present study, a two-level probabilistic model is proposed to describe the dependent random constitutive parameters based on the compressive test results of several batches of concrete specimens of different strength grades. Both the local probabilistic dependence, i.e., the dependence of the parameters of intra-batch specimens, and the global probabilistic dependence, i.e., the dependence of the parameters of inter-batch specimens, are captured by the proposed method. To this end, the constitutive parameters of concrete are first standardized by their means and coefficients of variation (COVs). In this way, the local dependence can be represented by the same copula model for the concrete of different strength grades. The global dependence is expressed as the empirical formulae in terms of the means and COVs of the parameters. The full probabilistic model can then be obtained by synthesizing the local and global dependence models. The effectiveness of the proposed approach is demonstrated by comparing the generated samples with the test results. To illustrate the effect of the probabilistic dependence of the compressive constitutive parameters on the structural evaluation result, the nonlinear stochastic dynamic response analysis of a reinforced concrete frame is carried out. The results indicate that the probabilistic dependence of the constitutive parameters of concrete has a non-negligible effect on the structural response and reliability, and should be reasonably considered in practice.     

结构力学中的广义力法

本文根据目前结构力学的力法原理、线性空间理论和最小应变能原理,对杆系结构的力法作了进一步的探讨,提出了静不定结构的解空间的概念,并建立了广义力法典型方程。由于可以灵活地选择不同的基本结构计算系数δ_(ij)和自由项Δ_(ip),从而使线性方程组的建立在一定程度上得以简化,同时本文进一步完善了力法原则,有助于理解超静定结构内力解空间的分析构造。     

结构随机延性需求谱的应用研究

结构的地震反应危险性曲线体现了结构的地震反应与地震动强度之间的关系,采用随机延性需求谱可以很方便地建立结构的地震反应危险性曲线。非线性结构在随机地震作用下的位移反应分析属于非线性随机振动问题,采用随机延性需求谱可以简单而有效的获得非弹性单自由度体系的随机位移反应的统计量,结合非线性静力分析,还可以进行多自由度结构的随机地震反应分析。此外,随机延性需求谱还可以用于结构的抗震可靠度分析。     

Simulation of wind velocity time histories on long span structures modeled as non-Gaussian stochastic waves

A methodology is proposed for efficient and accurate modeling and simulation of correlated non-Gaussian wind velocity time histories along long-span structures at an arbitrarily large number of points. Currently, the most common approach is to model wind velocities as discrete components of a stochastic vector process, characterized by a Cross-Spectral Density Matrix (CSDM). To generate sample functions of the vector process, the Spectral Representation Method is one of the most commonly used, involving a Cholesky decomposition of the CSDM. However, it is a well-documented problem that as the length of the structure – and consequently the size of the vector process – increases, this Cholesky decomposition breaks down numerically. This paper extends a methodology introduced by the second and fourth authors to model wind velocities as a Gaussian stochastic wave (continuous in both space and time) by considering the stochastic wave to be non-Gaussian. The non-Gaussian wave is characterized by its frequency–wavenumber (FK) spectrum and marginal probability density function (PDF). This allows the non-Gaussian wind velocities to be modeled at a virtually infinite number of points along the length of the structure. The compatibility of the FK spectrum and marginal PDF according to translation process theory is secured using an extension of the Iterative Translation Approximation Method introduced by the second and third authors, where the underlying Gaussian FK spectrum is upgraded iteratively using the directly computed (through translation process theory) non-Gaussian FK spectrum. After a small number of computationally extremely efficient iterations, the underlying Gaussian FK spectrum is established and generation of non-Gaussian sample functions of the stochastic wave is straightforward without the need of iterations. Numerical examples are provided demonstrating that the simulated non-Gaussian wave samples exhibit the desired spectral and marginal PDF characteristics.     

Dynamic response and reliability analysis of non-linear stochastic structures

A new probability density evolution method is proposed for dynamic response analysis and reliability assessment of non-linear stochastic structures. In the method, a completely uncoupled one-dimensional governing partial differential equation is derived first with regard to evolutionary probability density function (PDF) of the stochastic structural responses. This equation holds for any response or index of the structure. The solution will put out the instantaneous PDF. From the standpoint of the probability transition process, the reliability of the structure is evaluated in a straightforward way by imposing an absorbing boundary condition on the governing PDF equation. However, this does not induce additional computational efforts compared with the dynamic response analysis. The computational algorithm to solve the PDF equation is studied. A deterministic dynamic response analysis procedure is embedded to compute coefficient of the evolutionary PDF equation, which is then numerically solved by the finite difference method with total variation diminishing scheme. It is found that the proposed hybrid algorithm may deal with non-linear stochastic response analysis problem with high accuracy. Numerical examples are investigated. Parts of the results are illustrated. Some features of the probabilistic information of the response and the reliability are observed and discussed. The comparisons with the Monte Carlo simulations demonstrate the accuracy and efficiency of the proposed method.     

A generalized stochastic perturbation technique for plasticity problems

The main aim of this paper is to present an algorithm and the solution to the nonlinear plasticity problems with random parameters. This methodology is based on the finite element method covering physical and geometrical nonlinearities and, on the other hand, on the generalized nth order stochastic perturbation method. The perturbation approach resulting from the Taylor series expansion with uncertain parameters is provided in two different ways: (i) via the straightforward differentiation of the initial incremental equation and (ii) using the modified response surface method. This methodology is illustrated with the analysis of the elasto-plastic plane truss with random Young’s modulus leading to the determination of the probabilistic moments by the hybrid stochastic symbolic-finite element method computations.     

Variability response functions for stochastic systems under dynamic excitations

The concept of variability response functions (VRFs) is extended in this work to linear stochastic systems under dynamic excitations. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of linear stochastic systems under static loads, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is assumed. This assumption is here validated with brute-force Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). The same integral expression can be used to calculate the mean response of a dynamic system using a Dynamic Mean Response Function (DMRF) which is a function similar to the DVRF. These integral forms can be used to efficiently compute the mean and variance of the transient system response together with time dependent spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability system response.     

Probabilistic nonconvex constrained optimization with fixed number of function evaluations

A methodology is proposed for the efficient solution of probabilistic nonconvex constrained optimization problems with uncertain. Statistical properties of the underlying stochastic generator are characterized from an initial statistical sample of function evaluations. A diffusion manifold over the initial set of data points is first identified and an associated basis computed. The joint probability density function of this initial set is estimated using a kernel density model and an Itô stochastic differential equation (ISDE) constructed with this model as its invariant measure. This ISDE is adapted to fluctuate around the manifold yielding additional joint realizations of the uncertain parameters, design variables, and function values, which are obtained as solutions of the ISDE. The expectations in the objective function and constraints are then accurately evaluated without performing additional function evaluations. The methodology brings together novel ideas from manifold learning and stochastic Hamiltonian dynamics to tackle an outstanding challenge in stochastic optimization. Three examples are presented to highlight different aspects of the proposed methodology.     

Application of stochastic differential equations to seismic reliability analysis of hysteretic structures

An analytical method of stochastic seismic response and reliability analysis of hysteretic structures based on the theory of Markov vector process is presented, especially from the methodological aspect. To formulate the above analysis in the form of stochastic differential equations, the differential formulations of general constitutive laws for a class of hysteretic characteristics are derived. The differential forms of the seismic safety measures such as the maximum ductility ratio, cumulative plastic deformation, low-cycle fatigue damage are also derived. The state equation governing the whole nonlinear dynamical system which is composed of the shaping filter generating seismic excitations, hysteretic structural system and safety measures is determined as the Itô stochastic differential equations. By introducing an appropriate non-Gaussian joint probability density function, the statistics and joint probability density function of the state variables can be evaluated numerically under nonstationary state. The merit of the proposed method is in systematically unifying the conventional response and reliability analyses into an analysis which requires knowledge of only first order (single-time) statistics or probability distributions.     

Approximate analysis of response variability of uncertain linear systems

A probabilistic methodology is presented for obtaining the variability and statistics of the dynamic response of multi-degree-of-freedom linear structures with uncertain properties. Complex mode analysis is employed and the variability of each contributing mode is analyzed separately. Low-order polynomial approximations are first used to express modal frequencies, damping ratios and participation factors with respect to the uncertain structural parameters. Each modal response is then expanded in a series of orthogonal polynomials in these parameters. Using the weighted residual method, a system of linear ordinary differential equations for the coefficients of each series expansion is derived. A procedure is then presented to calculate the variability and statistics of the uncertain response. The technique is extended to the stochastic excitation case for obtaining the variability of the response moments due to the variability of the system parameters. The methodology can treat a variety of probability distributions assumed for the structural parameters. Compared to existing analytical techniques, the proposed method drastically reduces the computational effort and computer storage required to solve for the response variability and statistics. The performance and accuracy of the method are illustrated by examples.     

Non-linear stochastic control via stationary response design

This paper presents a strategy for controlling externally excited stochastic systems with uncertain parameters. The control objective is to drive the system response from an arbitrary initial distribution to a prescribed stationary probability density function (PDF). This problem can be interpreted as the stochastic stabilization about a PDF. The control consists of a non-linear feedback part and a switching term inspired by robust sliding control concepts. The control of a one-dimensional stochastic process with a Gaussian target-PDF is used to illustrate the approach. The control performance is evaluated by studying the time evolution of the first and second order moments of the response. The dependence of the response on the number of feedback terms and the rate of convergence to the stationary PDF are studied numerically. Motivated by the fast convergence observed, a feedback controller with time-varying gains is applied to the problem of tracking a moving PDF. Monte Carlo simulations validate very well the control for both stabilization and tracking problems.     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

4X框架木箱结构内力计算方法研究

目的 以重型包装运输用木质框架结构为研究对象,解决木质框架结构内力求解困难的问题。方法 利用解析法,以4X框架结构为例,分别以经验简化静定桁架与无简化超静定桁架进行求解;基于ANSYS软件建立经验简化静定桁架、无简化超静定桁架、梁模型、梁杆组合模型及实体模型,并进行对比分析。结果 有限元求解结果与解析法结果具有高度的一致性;经验简化静定桁架的最大轴力超过无简化超静定桁架的最大轴力15%;梁模型、梁杆组合模型及实体模型的计算结果与无简化超静定桁架的计算结果一致。结论 经验法对框架结构的简化往往会导致比较大的误差,可能导致过包装设计或欠包装设计。梁杆组合模型和实体模型建模过程比较复杂,选用梁单元对木箱用框架结构进行数值计算分析,不仅计算精度高,分析处理快捷,而且对复杂工况具有较好的适应性。     

Optimal damper placement for critical excitation

Since earthquake ground motions are very uncertain even with the present knowledge, it is desirable to develop a robust structural design method taking into account these uncertainties. Critical excitation approaches are promising and a new probabilistic critical excitation method is proposed. Different from the conventional critical excitation methods, a stochastic response index is treated as the objective function to be maximized. The energy (area of power spectral density function) and the intensity (magnitude of power spectral density function) are fixed and the critical excitation is found under these restrictions. It is shown that the resonant characteristic of ground motions can be well represented by the proposed critical excitation. An original steepest direction search algorithm due to the present author is applied to the problem of optimal damper placement in structures subjected to the critical excitation. Closed-form expressions of the inverse of the coefficient matrix (tri-diagonal matrix) enable one to compute the transfer function and its derivative with respect to design variables very efficiently. A numerical example of a 6-DOF shear building model is presented to demonstrate the effectiveness and validity of the present method.