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.     

钢筋混凝土梁桥船舶撞击连续倒塌数值模拟

详细介绍了钢筋混凝土桥梁船撞倒塌模拟涉及的材料模型和单元失效指标等问题,建立某连续梁上部、下部结构、支座和船舶结构精细有限元模型,将弹塑性损伤帽盖模型用于混凝土,弹塑性随动强化模型用于钢筋,对船舶撞击钢筋混凝土连续梁引起的结构连续倒塌过程进行了数值模拟,揭示了连续梁桥结构的船撞倒塌机理。     

近断层钢筋混凝土框架结构地震倒塌机制分析

由于近断层地震动的特殊性,近断层混凝土结构的损伤倒塌较一般地震动下更为严重,其损伤过程和倒塌机制也更为复杂。采用改进的拆除构件法,对抗倒塌分析中传统的拆除构件法进行了改进,并结合近断层地震动下混凝土框架结构的反应特征,建立了典型10层钢筋混凝土框架模型,通过设置12种拆除工况,选取3条简谐波和3条实际近断层地震记录作为结构输入,进行动力弹塑性时程分析,研究了钢筋混凝土框架在近断层地震作用下的倒塌机制。研究表明,底层框架柱损伤对结构倒塌模式的影响较大,中、上部楼层框架柱损伤对结构倒塌模式的影响较小;框架柱不对称损伤对所在楼层层间位移的影响大于对称损伤的影响,角柱损伤的影响大于边柱损伤和中柱损伤的影响;即使结构的损伤部位不同,但也可能具有相同的倒塌模式;采用与结构基本周期相接近的简谐波作用能够从总体上模拟结构的整体倒塌模式。     

交叉模态对结构随机振动疲劳损伤的贡献

以小阻尼简支梁为例,讨论了交叉模态对随机振动疲劳损伤累积的贡献;结构的响应分析采用模态叠加方法,随机振动疲劳损伤累积分析采用M iner线性疲劳损伤准则。数值模拟结果表明,即使是小阻尼稀疏模态结构,交叉模态对结构振动疲劳损伤累积的贡献也是不容忽视的。     

Life estimation model of a cantilevered beam subjected to complex random vibration

An analytic methodology was developed to predict the fatigue life of a structure experiencing stationary, Gaussian random vibration excitation. This method allows the estimation of fatigue life using a frequency domain method, where only the input power spectral density and damping factor are required. The methodology uses linear elastic fracture mechanics for fatigue crack propagation and accounts for the frequency shifting that occurs due to fatigue crack evolution. Good results have been obtained comparing the analytic model to both finite element analysis (FEA) and experimental results, for mild‐steel cantilever beams.     

Extremes of random fields over arbitrary domains with application to concrete rupture stresses

To find the exact probability distribution of the global maximum or minimum of a random field within a bounded domain is a pending problem even for Gaussian fields. Except for very special examples of fields, recourse must be taken to approximate reasoning or asymptotic considerations to be judged with respect to accuracy by simulations. In this paper, the problem is addressed through a functional equation that leads to the definition of a class of distribution functions that depend solely on process or field characteristics and domain quantities that can be calculated explicitly. This distribution function class is studied for Gaussian processes in earlier works by the author and it has been obtained explicitly for Gaussian fields on rectangular domains in the plane. Simulation studies show that rather good predictions are obtained for sufficiently smooth wide band Gaussian processes and fields. In this paper, the distribution function is obtained in general for Gaussian fields over arbitrary bounded domains with piecewise continuous and differentiable boundaries, and as in earlier works the distribution function is tested against empirical distribution functions obtained by simulation of sample functions of a smooth approximately Gaussian field, herein called a broken line Hino field. For completeness this particular field type is defined in appendix a and appendix b. The paper concludes with a statistical application on data for plain concrete tensile strength.     

A damage gradient model for fatigue life prediction of notched metallic structures under multiaxial random vibrations

In the present paper, a damage gradient model combing the damage concept with the theory of critical distance (TCD) is established to estimate the fatigue lives of notched metallic structures under multiaxial random vibrations. Firstly, a kind of notched metallic structure is designed, and the biaxial random vibration fatigue tests of the notched metallic structures are carried out under different correlation coefficients and phase differences between two vibration axes. Then, the fatigue lives of the notched metallic structures are evaluated utilizing the proposed model with the numerical simulations. Finally, the proposed model is validated by the experiment results of the biaxial random vibration fatigue tests. The comparison results demonstrate that the proposed model can provide fatigue life estimation with high accuracy.     

Determination of damage evolution in CFRP subjected to cyclic loading using DIC

Damage evolution in carbon fibre reinforced plastic subjected to fatigue loading (R = 0.5, ?1, 2) has been studied using digital image correlation to obtain full‐field surface strains. Damage initiation being a local phenomenon, its effect on global parameters is not significant. The local transverse strain is a better indicator of delamination which affects transverse strain more than the longitudinal strain. Variation of normalized local transverse strain (ratio of local transverse strain to applied stress) near the initiated delamination indicates that the damage evolution occurs over 2 to 3 stages. Each stage has a stable damage growth with a drastic increase between the stages. The stages correspond to different damage mechanisms (matrix cracking, fibre‐matrix debonding, delamination, and fibre breakage) dominating at different periods during the fatigue life. Scatter in normalized local transverse strain plots due to large relative displacements was eliminated using different reference images for DIC. Waviness due to shift in the time at which the images are captured during the loading cycles was avoided using a sine curve fit to obtain maximum transverse strain in a cycle. Normalized local transverse strain plots were found to qualitatively reflect the physical extent of damage, thereby providing confidence in the methodology. Fatigue life curves were generated and run‐out lives were determined.     

An improved sieve point method for the reliability analysis of structures

The efficiency of existing stochastic analysis method depends on the discretization of the random variables domain. The number theoretical method has been proposed to discretize the random variable space and solve the generalized density evolution equation via sampling strategy. This method traditionally involves hyper-ball sieving (HS) algorithm to sample the representative point set. However, the sieving radius of the hyper-ball is determined subjectively, and the efficiency and accuracy of the analysis depend on the selected radius. To avoid this subjective selection, an equal volume hyper-ball sieving method is presented in this paper. By transforming the hypercube spatial volume of random variables into that of an equivalent hyper-ball, the radius of the equal volume hyper-ball is obtained analytically. This radius is further optimized with a minimum star discrepancy in the representative point set. The performance and accuracy of the proposed method are checked in four numerical examples, and the representative point set such obtained is more uniform with smaller NRP leading to more accurate and efficient subsequent stochastic analysis than the HS method.     

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.     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.     

Lifecycle operational reliability assessment of water distribution networks based on the probability density evolution method

In this study, a lifecycle operational reliability assessment framework for water distribution networks (WDNs) is proposed on the basis of the probability density evolution method (PDEM). The occurrence models of daily accidents are fitted using the maintenance data provided by a local water administration sector. For a given accident, two types of accidents (e.g., leaks and bursts) are distinguished in different occurrence probabilities and simulated in various ways. The pipe deterioration process in the lifecycle is reflected by incorporating the time-dependent pipe roughness model. Considering various randomness in the model, PDEM, a newly proposed and developed method for a stochastic system, is used to evaluate the lifecycle operational reliability of WDNs. The framework is demonstrated using an actual WDN, and the nodal reliabilities in the lifecycle are obtained. Comparisons of the operational reliabilities of all nodes calculated via the PDEM and Monte Carlo simulations prove that PDEM is an accurate and highly efficient method.     

An updated continuum damage model to investigate fracture process of structures in DBTT region

In the Ductile–Brittle Transition Temperature (DBTT) region, it is not realistic to take unique fracture stress or fracture strain as the fracture criterion to investigate the fracture properties. In this paper, an updated continuum damage model was proposed, in which the fracture energy density, a function of the stress triaxiality, temperature and strain rate in the transition region was taken as the critical damage factor. Uniaxial tension tests were carried out to get the basic material properties at different temperatures, to calibrate the fracture model constants and verify the validity of the damage model. The fracture behaviour of pipes with penetrating cracks under the internal pressure was experimentally investigated with the load–deflection curves and the crack propagation length captured from tests. The J–R curves were obtained from the testing results for different temperatures. Based on the Finite Element Analyses (FEA) with the proposed fracture criterion of the updated continuum damage model, the loading level of pipes with penetrating cracks were estimated and compared with the experimental results. Meanwhile the fracture processes of the pipeline structures in the transition region were reproduced. The experimental and numerical results agreed very well in present calculations. It has been shown that the fracture process in the transition region strongly depends on both the stress and strain states, and could be effectively predicted using the continuum damage model.     

Constitutive and damage modelling of H11 subjected to low‐cycle fatigue at high temperature

Hot‐work tool steel H11 is extensively applied in extrusion industries as extrusion tools. The understanding of its mechanical properties and damage evolution as well as failure is crucial for its implementation. In this paper, a finite element (FE) model employing Chaboche unified constitutive model and ductile damage rule is proposed to simulate the mechanical responses of H11 subjected to low‐cycle fatigue (LCF). Accumulated inelastic hysteresis energy is adopted to demonstrate the impact on damage initiation and evolution rules. A series of tension and LCF experiments were conducted to investigate H11's mechanical properties and its deterioration processes. In addition, to deeply understand the deformation and damage mechanism, scanning electron microscope (SEM) investigations were performed on the fracture section of gauge‐length part of the specimen after failure. Furthermore, the parameters in both constitutive model and damage rule are identified based on experimental data. The comparison of the hysteresis loop of the first cycle and stable cycle with different strain amplitudes demonstrates that the Chaboche constitutive model provides high precision to predict the evolution of mechanical properties. Based on the reliable achieved constitutive model, LCF behaviour prediction with damage rule was executed successfully using FE model and gains a good agreement with the experiments. It is believed that the proposed FE method lays the foundation of structure analysis and rapid design optimization in further applications.