Two-scale random field model for quasi-brittle materials

A new idea of two-scale random field is presented by taking the mechanical behaviors of quasi-brittle materials as an example. In this way, the random fluctuation of material damage at the micro-meso-scale and the spatial correlation of mechanical properties at the macro-scale are both well described by the random distribution of fracture strain. Moreover, it is proved that by introducing a transform matrix, the mesoscopic random fields can be described by a two-scale random field. Based on this, a two-step numerical implementation of the two-scale random field is proposed, and the mechanical behaviors of quasi-brittle materials can be analyzed by incorporating the probability density evolution theory. Finally, an experimental example is presented to demonstrate the capability of the proposed model. Ideal agreements are achieved against the experimental results. Particularly, the randomness of material properties at both scales can be well reproduced.     

Generalized F-discrepancy-based point selection strategy for dependent random variables in uncertainty quantification of nonlinear structures

In the performance evaluation of structures under disastrous actions, for example, earthquakes, it is important to take into account the randomness of structural parameters. Generally, these random parameters are treated either as independent or perfectly dependent, but practically they are partly dependent. This article aims at developing a point selection strategy for uncertainty quantification of nonlinear structures involving probabilistically dependent random parameters characterized by copula function. For this purpose, the point selection strategy for structures involving independent basic variables is first revisited. As an improvement, a generalized F-discrepancy diminishing oriented iterative screening algorithm is proposed. Then, combining with the conditional sampling method, a conditional point set rearrangement method and a conditional iterative screening-rearrangement method are proposed for probabilistically dependent variables. These new point selection strategies are readily incorporated into the probability density evolution method for uncertainty quantification of nonlinear structures involving dependent random parameters, which is characterized by copula function. The proposed methods are illustrated by two examples including a shear frame with hysteretic restoring forces and a reinforced concrete frame structure with the damage constitutive model of concrete, where the material parameters are probabilistically dependent. The results demonstrate the effectiveness of the proposed method. Problems to be studied are discussed.     

Identification of probabilistic distribution parameters for the mesoscopic stochastic fracture model

As a kind of multiphase composite material, the basic mechanical behaviors of concrete are randomness and nonlinearity. The mesoscopic stochastic fracture model (MSFM) which can reflect the coupling effect of randomness and nonlinearity, has been widely used for the nonlinear analysis of concrete structures. In this paper, we proposed a new stochastic modeling principle to identify the probabilistic distribution parameters of MSFM. In order to reduce the modeling works, a dimension-reduced algorithm is proposed as well. In this paper, an overview of MSFM is firstly presented to introduce the background of the research. Then the stochastic harmonic function (SHF) representation is introduced to express the random field mentioned in the MSFM, and the probability density evolution method (PDEM) is applied to obtain the theoretical probability density function (PDF) of the stress–strain relationships. Furthermore, a stochastic modeling principle is proposed, in which minimizing the Kullback–Leibler divergence (KLD) is taken as the optimization object. Based on the framework of genetic algorithm, a dimension-reduced algorithm is proposed to identify the parameters with reference to the data from tested complete curves of uniaxial compressive and uniaxial tensile stress–strain relationship of concrete. The results indicate that the proposed principle and algorithm can be used to identify the parameters of MSFM accurately and efficiently.     

基于概率密度演化理论的结构抗震可靠性分析

运用随机过程的正交展开方法,将地震动加速度过程表示为由10个左右的独立随机变量所调制的确定性函数的线性组合形式。结合概率密度演化方法和等价极值事件的基本思想,研究了非线性结构的抗震可靠度分析问题。以具有滞回特性的非线性结构为例,对某一多自由度的剪切型框架结构进行了抗震可靠性分析。结果表明：按照复杂失效准则计算的结构抗震可靠度较之结构各层抗震可靠度均低。这一研究为基于概率密度函数的、精细化的抗震可靠度计算提供了新的途径。     

随机结构复合随机振动分析的概率密度演化方法

提出了随机结构复合随机振动分析的概率密度演化方法。通过引入扩展状态向量,构造具有随机初始条件的状态方程,导出了复合随机振动反应的概率密度演化方程。结合精细时程积分方法和Lax-Wendroff差分格式对概率密度演化方程提出了数值求解方法。进行了八层层间剪切框架结构复合随机振动反应的概率密度演化分析,证明提出的方法具有计算高效、收敛性稳定与精度高的特点。研究表明随着时间的增长,复合随机振动反应概率密度曲线趋于复杂,基于正态分布假定的二阶矩分析方法可能造成可靠度分析结果的显著偏差。与仅考虑结构参数随机性和仅考虑输入随机性时的结构反应相比,复合随机振动反应概率密度曲线峰值降低、分布变宽,且随机涨落显著增强。     

A physical approach to structural stochastic optimal controls

The generalized density evolution equation proposed in recent years profoundly reveals the intrinsic connection between deterministic systems and stochastic systems by introducing physical relationships into stochastic systems. On this basis, a physical stochastic optimal control scheme of structures is developed in this paper, which extends the classical stochastic optimal control methods, and can govern the evolution details of system performance, while the classical stochastic optimal control schemes, such as the LQG control, essentially hold the system statistics since there is still a lack of efficient methods to solve the response process of the stochastic systems with strong nonlinearities in the context of classical random mechanics. It is practically useful to general nonlinear systems driven by non-stationary and non-Gaussian stochastic processes. The celebrated Pontryagin’s maximum principles is employed to conduct the physical solutions of the state vector and the control force vector of stochastic optimal controls of closed-loop systems by synthesizing deterministic optimal control solutions of a collection of representative excitation driven systems using the generalized density evolution equation. Further, the selection strategy of weighting matrices of stochastic optimal controls is discussed to construct optimal control policies based on a control criterion of system second-order statistics assessment. The stochastic optimal control of an active tension control system is investigated, subjected to the random ground motion represented by a physical stochastic earthquake model. The investigation reveals that the structural seismic performance is significantly improved when the optimal control strategy is applied. A comparative study, meanwhile, between the advocated method and the LQG control is carried out, indicating that the LQG control using nominal Gaussian white noise as the external excitation cannot be used to design a reasonable control system for civil engineering structures, while the advocated method can reach the desirable objective performance. The optimal control strategy is then further employed in the investigation of the stochastic optimal control of an eight-storey shear frame. Numerical examples elucidate the validity and applicability of the developed physical stochastic optimal control methodology.     

Probability density evolution analysis of stochastic seismic response of structures with dependent random parameters

Performance evaluation and reliability assessment of real-world structures under earthquakes is of paramount importance. Generally, different mechanical property parameters of a structure are usually not independent, nor completely dependent, but partly dependent or correlated. Therefore, how to reasonably characterize such partial dependency and whether such partial dependency real matters in the stochastic response and reliability of structures under earthquakes are crucial issues. For this purpose, in the present paper, a novel physically-guided data-driven methodology of capturing the correlation configuration of basic random variables and the probability density evolution method are synthesized. The physically-guided data-driven methodology is firstly outlined. In this methodology, the underlying physical mechanism between dependent random variables is firstly involved to establish a random function model, and then the available observed data are adopted to identify the parameters in this model. What is more, physical constraints are also revealed for the initial modulus of elasticity and compressive strength of concrete. The probability density evolution method is then adopted, and the point selection by minimizing the GF-discrepancy is adjusted according to the correlation configuration and physical constraints. A reinforced concrete frame structure subjected to earthquake input is studied. It is found that when the structure is in the strongly nonlinear stage, the correlation configuration has considerable effects on the standard deviation of the stochastic responses, by a factor of nearly 2. In addition, whether the mechanical parameters in different floors are independent or not has great effects on the stochastic responses as well. Problems to be further studied are also outlined.     

非平稳随机地震作用的结构整体可靠度分析

应用随机过程的正交展开-随机函数方法,建立了非平稳地震动过程的概率模型,实现了用一个基本随机变量来表达地震动过程的目的。通过选取基本随机变量的代表性离散点集,可以直接获取地震动过程的代表性样本集合。结合概率密度演化理论,进行了多自由度Duffing系统的随机地震反应分析与抗震可靠度计算。研究表明,非平稳地震动过程的概率模型与概率密度演化理论有机结合,可以实现复杂工程结构整体抗震可靠度的精确计算。     

Probabilistic analysis of concrete cracking using neural networks and random fields

In this paper an algorithm for the probabilistic analysis of concrete structures is proposed which considers material uncertainties and failure due to cracking. The fluctuations of the material parameters are modeled by means of random fields and the cracking process is represented by a discrete approach using a coupled meshless and finite element discretization. In order to analyze the complex behavior of these nonlinear systems with low numerical costs a neural network approximation of the performance functions is realized. As neural network input parameters the important random variables of the random field in the uncorrelated Gaussian space are used and the output values are the interesting response quantities such as deformation and load capacities. The neural network approximation is based on a stochastic training which uses wide spanned Latin hypercube sampling to generate the training samples. This ensures a high quality approximation over the whole domain investigated, even in regions with very small probability.     

A stochastic damage model for evaluating the internal deterioration of concrete due to freeze–thaw action

This paper presents a stochastic damage model for evaluating the internal deterioration of concrete due to freeze–thaw action, which involves great uncertainty and randomness. In this model, the structural element of concrete is discretized into infinite microelements, whose lifetimes are assumed to be independent random variables. Then expressions for the mean and variance of the damage of concrete are analytically derived. To calibrate the model parameters, a series of freeze–thaw tests in water on non-air-entrained concrete were conducted and back-calculation analyses were performed on the test results of dynamic modulus. The reliability of the proposed stochastic damage model is further validated through comparisons with the results of 80 other existing test specimens. The present model offers a theoretical basis for exploring the statistical aspect of concrete behavior during freeze–thaw.     

Numerical investigation of statistical variation of concrete damage properties between scales

Concrete is typically treated as a homogeneous material at the continuum scale. However, the randomness in micro-structures has profound influence on its mechanical behavior. In this work, the relationship of the statistical variation of macro-scale concrete properties and micro-scale statistical variations is investigated. Micro-structures from CT scans are used to quantify the stochastic properties of a high strength concrete at the micro-scale. Crack propagation is then simulated in representative micro-structures subjected to tensile and shear tractions, and damage evolution functions in the homogenized continuum are extracted using a Helmholtz free energy correlation. A generalized density evolution equation is employed to represent the statistical variations in the concrete micro-structures as well as in the associated damage evolution functions of the continuum. This study quantifies how the statistical variations in void size and distribution in the concrete microstructure affect the statistical variation of material parameters representing tensile and shear damage evolutions at the continuum scale. The simulation results show (1) the random variation decreases from micro-scale to macro-scale, and (2) the coefficient of variation in shear damage is larger than that in the tensile damage.     

A solution method for linear and geometrically nonlinear MDOF systems with random properties subject to random excitation

A method for computing the lower-order moments of response of randomly excited multi-degree-of-freedom (MDOF) systems with random structural properties is proposed. The method is grounded in the techniques of stochastic calculus, utilizing a Markov diffusion process to model the structural system with random structural properties. The resulting state-space formulation is a system of ordinary stochastic differential equations with random coefficients and deterministic initial conditions which are subsequently transformed into ordinary stochastic differential equations with deterministic coefficients and random initial conditions. This transformation facilitates the derivation of differential equations which govern the evolution of the unconditional statistical moments of response. Primary consideration is given to linear systems and systems with odd polynomial nonlinearities, for in these cases there is a significant reduction in the number of equations to be solved. The method is illustrated for a five-story shear-frame structure with nonlinear interstory restoring forces and random damping and stiffness properties. The results of the proposed method are compared to those estimated by extensive Monte-Carlo simulation.     

基于贝叶斯理论的非线性结构模型修正及其动力可靠度分析EI北大核心CSCD

提出一种基于贝叶斯推理的非线性结构模型修正方法,同时考虑激励的随机性,建立了复合随机振动系统的动力可靠度分析方法。利用实测结构动力响应主分量的瞬时特征参数作为非线性指标构建似然函数,结合拒绝延缓自适应(Delayed Rejection and Adaptive Metropolis,DRAM)算法和高斯过程替代模型实现了非线性结构模型修正及其参数的不确定性量化。根据首次超越破坏准则,利用广义概率密度演化方法,分别对仅考虑激励随机性的确定性模型和同时考虑结构参数与激励不确定性的复合随机振动模型进行动力可靠度分析,并利用蒙特卡洛随机抽样方法验证了计算结果的准确性。研究结果表明:基于振动响应瞬时特征参数的贝叶斯推理方法能够快速、准确地实现结构的非线性模型修正及其参数的不确定性量化。与具有初始设计参数名义值的确定性模型相比,考虑参数不确定性的复合随机模型的动力可靠度总体偏低,因此,在结构安全评估中应考虑非线性模型参数不确定性的影响,使评估结果更加安全、可靠。     

Integrated system identification and reliability evaluation of stochastic building structures

System identification and reliability evaluation play a significant role in structural health monitoring to ensure the serviceability and safety of existing structures. Although the development of system identification methods has attained much attention and some degree of maturity, reliability evaluation of existing structures still remains a challenging problem especially when uncertainties in measurement data and inherent randomness, which are inevitably involved in civil structures, are considered. In this regard, this paper presents a framework for integrated system identification and reliability evaluation of stochastic building structures. Two algorithms are proposed to respectively evaluate component reliability and system reliability of stochastic building structures by combining a statistical moment-based system identification method and a probability density evolution equation-based reliability evaluation method. System identification is embedded in the procedure of reliability evaluation of a stochastic building structure. The uncertainties in both the structure and the external excitation are considered. Numerical examples show that the structural component and system reliabilities of a three-story shear building structure with three damage scenarios can be effectively evaluated by the proposed methods.     

Integrated system identification and reliability evaluation of stochastic building structures

System identification and reliability evaluation play a significant role in structural health monitoring to ensure the serviceability and safety of existing structures. Although the development of system identification methods has attained much attention and some degree of maturity, reliability evaluation of existing structures still remains a challenging problem especially when uncertainties in measurement data and inherent randomness, which are inevitably involved in civil structures, are considered. In this regard, this paper presents a framework for integrated system identification and reliability evaluation of stochastic building structures. Two algorithms are proposed to respectively evaluate component reliability and system reliability of stochastic building structures by combining a statistical moment-based system identification method and a probability density evolution equation-based reliability evaluation method. System identification is embedded in the procedure of reliability evaluation of a stochastic building structure. The uncertainties in both the structure and the external excitation are considered. Numerical examples show that the structural component and system reliabilities of a three-story shear building structure with three damage scenarios can be effectively evaluated by the proposed methods.     

土木结构损伤多尺度并发计算方法及其应用

大型土木结构的损伤破坏是跨尺度演化的结果, 因此单一尺度下的结构分析难以正确地反映结构的非线性损伤失效过程。该文根据结构损伤在宏观、细观尺度下的不同特征建立结构一致多尺度模型, 并通过多点约束法进行跨尺度关联, 实现了结构整体线弹性响应分析和局部细节易损部位的细观层次上弹塑性损伤分析的并发进行。计算结果表明:该文提出的结构损伤多尺度并发计算方法能够兼顾结构整体上的线弹性响应和局部易损部位在细观层次上的塑性损伤特征, 在对结构多尺度响应与损伤特征进行准确描述的基础上, 能够获得结构易损局部的细观损伤状态、演化过程及其对结构宏观响应与失效的影响。     

A contribution to the SFE-based reliability assessment of nonlinear structures under dynamic loading

This paper presents a procedure which allows for a stochastic finite element (SFE)-based reliability analysis of large nonlinear structures under dynamic loading involving both structural and loading randomness with relatively little computational effort when compared to traditional Monte Carlo methods. The analysis is based on the identification of important random variables by means of a transformation of the vector of original random variables to the uncorrelated space and subsequent sensitivity analyses. Only few nonlinear computations using the most important identified random variables are performed to determine points on the limit state surface. Subsequently, the response surface method (RSM) is employed to estimate the reliability of the structure.