An inverse problem analysis for the determination of probalistic parameters of concrete behavior modeled by a statistical approach

When the Monte Carlo method is used to simulate the heterogeneous behavior of concrete in the framework of a finite element probabilistic analysis, N samples of the vector of random variables (tensile strength, Young's modulus, etc.) are generated from a specific probability density function. If the uncertainties of these material parameters are assumed to vary spatially following a normal distribution, the N samples corresponding to a simulation are function of the mean and the standard deviation that define the Gauss density function. The problem is that these statistical moments are not known,a priori, for the characteristic volume of the finite elements for which the problem has been discretized. In this paper an algorithm is proposed to evaluate the parameters characterizing the statistical distribution (e. g. for a normal distribution: the mean and the standard deviation) for a given response of the structure (for instance, a mean load-displacement curve) following an inverse analysis procedure. A very simple mechanical system is used to verify the feasibility of the procedure. By way of example, it is shown that this kind of inverse problem for the identification of statistical parameters is suitable for concrete.

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Résumé Si la méthode de Monte-Carlo est utilisée pour simuler l'hétérogénéité du béton dans le cadre d'une analyse probabiliste par éléments finis, N échantillons du vecteur des variables aléatoires (résistance à la traction, module d'Young, etc.) sont générés à partir d'une fonction de densité de probabilité donnée. Si la dispersion de ce matériau peut être représentée par une distribution normale, les N échantillons correspondant à une simulation, sont fonction de la moyenne et de l'écart type qui définissent la fonction de densité de Gauss. Le problème c'est que ces moments statistiques ne sont pas connus,a priori, pour le volume caractéristique des éléments finis selon lesquels le problème a été discrétisé. Dans cet article, on propose un algorithme pour l'évaluation des paramètres qui caractérisent la distribution statistique (comme exemple, pour une distribution normale: la moyenne et l'écart type) correspondant à la réponse donnée d'une structure (par exemple: une courbe force-déplacement) suivant un processus d'analyse inverse. Un système mécanique très simple est utilisé pour vérifier la faisabilité du processus. Il est montré, à partir d'un exemple, que ce type d'analyse inverse peut être utilisé pour le béton.

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Editorial note COPPE/UFRJ and LCPC are RILEM Titular Members.     

Effective Elastic Property Estimation for Bi-continuous Heterogeneous Solids

In the present study we performed finite element simulation for bi-continuous heterogeneous solids via a random distribution of materials to predict effective elastic properties. With a random distributing scheme, a statistical analysis via finite element becomes feasible for the multi-phase heterogeneous solids. Using a two-phase bi-continuous material as example, the numerical prediction of the effective properties is obtained in terms of a mean value and standard deviation with a sample size of 30 for each of given volume fraction. The finite element simulation results fall within the analytical bounds proposed by Hashin and Shtrikman (1963) based on the principle of variation. Comparison between the effective modulus based on the present bio-continuous morphology with the matrix-fiber configuration shows big difference.     

A spectral stochastic element free Galerkin method for the problems with random material parameter

This paper presents a spectral stochastic element free Galerkin method (SSEFGM) for the problems involving a random material property. The random material property and resulting system response quantity are represented by a probabilistic spectral expansion techniques (Karhunen–Loeve expansion and Polynomical Chaos series, respectively) and implemented into the element free Galerkin (EFG) analysis. Numerical solutions in 1D linear elastic problem with random elastic modulus are introduced, and compared with those of Monte Carlo simulation (MCS) so as to provide the validation of the proposed approach. The present SSEFGM approach can produce a probabilistic density distribution as well as a first‐ and second‐order statistical moments (mean and variance) of response quantity by a single calculation, which is distinguished from an iterative MCS. Moreover, the method is based on an element free analysis so that there is no need of nodal connectivities, which usually require more time and labourious task than main calculations. Thus the proposed SSEFGM approach can provide an alternative analysis tool for the problems contains a stochastic material property, and demands complex mesh structures. Copyright © 2004 John Wiley & Sons, Ltd.     

Fatigue life assessment of a low pressure steam turbine blade during transient resonant conditions using a probabilistic approach

This paper presents a sequential approach used in fatigue life prediction of a low pressure steam turbine blade during resonance conditions encountered during a turbine start-up by incorporating probabilistic principles. Material fatigue properties are determined through experimental testing of used blade material X22CrMoV12-1 along with statistical modelling using regression analysis to interpret the stress-life diagram. A finite element model of a free-standing LP blade is developed using the principle of sub-structuring which enables the vibration characteristics and transient stress response of the blade to be determined for variations in blade damping. Random curve fitting routines are performed on the fatigue and FEM stress data to ensure that the selection of the random variables used in fatigue life calculations is stochastic in nature. The random vectors are selected from a multivariate normal distribution. The use of confidence intervals in the probabilistic fatigue life model works effectively in being able to account for uncertainty in the material fatigue strength parameters and varying stress in the blade root. The predicted fatigue life of the blade is shown to be in good agreement with discrete life modelling results.     

Comparison of finite element reliability methods

The spectral stochastic finite element method (SSFEM) aims at constructing a probabilistic representation of the response of a mechanical system, whose material properties are random fields. The response quantities, e.g. the nodal displacements, are represented by a polynomial series expansion in terms of standard normal random variables. This expansion is usually post-processed to obtain the second-order statistical moments of the response quantities. However, in the literature, the SSFEM has also been suggested as a method for reliability analysis. No careful examination of this potential has been made yet. In this paper, the SSFEM is considered in conjunction with the first-order reliability method (FORM) and with importance sampling for finite element reliability analysis. This approach is compared with the direct coupling of a FORM reliability code and a finite element code. The two procedures are applied to the reliability analysis of the settlement of a foundation lying on a randomly heterogeneous soil layer. The results are used to make a comprehensive comparison of the two methods in terms of their relative accuracies and efficiencies.     

Transverse failure of carbon fiber composites: Analytical sensitivity to the distribution of fiber/matrix interface properties

An analytic differentiation method is presented to calculate the sensitivity of the transverse failure response of carbon fiber composite laminates to the distribution parameters of the fiber/matrix interface properties. The method starts with the evaluation of the sensitivities of the transverse failure response with respect to the interface properties of each fiber, ie, the cohesive failure strength and the critical displacement jump. These individual sensitivities are then used to calculate the sensitivities with respect to the mean and standard deviation of the interface properties. The derived sensitivities are implemented in a nonlinear interface-enriched generalized finite element method solver specially developed for this application. The interface-enriched generalized finite element method solver combines a cohesive modeling of the fiber/matrix interface failure with finite element meshes that do not conform to the composite microstructure. The approach is first demonstrated on a model material involving a one-dimensional domain containing N cohesive interfaces described by randomly selected cohesive failure properties. The method is then applied to the more complex problem of a composite laminate involving a large number of fibers.     

Variability response functions for apparent material properties

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

Non-Deterministic Structural Response and Reliability Analysis Using a Hybrid Perturbation-Based Stochastic Finite Element and Quasi-Monte Carlo Method

The random interval response and probabilistic interval reliability of structures with a mixture of random and interval properties are studied in this paper. Structural stiffness matrix is a random interval matrix if some structural parameters and loads are modeled as random variables and the others are considered as interval variables. The perturbation-based stochastic finite element method and random interval moment method are employed to develop the expressions for the mean value and standard deviation of random interval structural displacement and stress responses. The lower bound and upper bound of the mean value and standard deviation of random interval structural responses are then determined by the quasi-Monte Carlo method. The structural reliability is not a deterministic value but an interval as the structural stress responses are random interval variables. Using a combination of the first order reliability method and interval approach, the lower and upper bounds of reliability for structural elements, series, parallel, parallel-series and series-parallel systems are investigated. Three numerical examples are used to demonstrate the effectiveness and efficiency of the proposed method.     

Statistical investigation of the fracture behaviour of inhomogeneous materials in tension and three-point bending

Development of damage in heterogeneous materials submitted to tensile tests and flexural tests is analysed using finite element analysis and considering statistical distribution of material strength. Materials are assumed to have a brittle local behaviour and fracture stresses are distributed randomly through test specimens. Also, the analysis considers that it exists a dimension which is characteristic of damage growing, depending on the fracture processes induced. A simulation procedure for evaluating damage development through test specimens is next implemented and the influence of the scattering width of the fracture stress distributions is analysed.     

Effect of amount, size, and spatial distribution of intermetallics on incipient localization during plastic deformation

Virtual microstructures having a systematic variation of amount, mean size, standard deviation of size, and spatial arrangement of intermetallics have been synthesized, and their deformation behavior in uniaxial tension has been evaluated using finite element analysis. Four spatial arrangements of intermetallics have been considered in this work, namely: random, clustered, and two-ordered structures. Various mathematical quantities have been developed to quantify the severity of deformation including plastic work density distribution (PWDD), percentile work-density volume criterion (PWC), and percentile stress volume criterion (PSC). This approach eliminates the need for an external trigger in FEA to achieve localization. The method developed has led to a better understanding of the effect of different microstructural attributes on the process of deformation. This has resulted in guidelines for optimizing the microstructure to minimize material damage and thereby maximize ductility.     

Effect of statistical yarn tensile strength on the probabilistic impact response of woven fabrics

The probabilistic impact response of flexible woven fabrics can be described through the V₀–V₁₀₀ or probabilistic velocity response (PVR) curve which describes the probability of fabric penetration as a function of projectile impact velocity. One source of variability that affects the probabilistic nature of fabric impact performance is the statistical distribution of yarn tensile strengths. In this paper the effects of the statistical yarn strength distribution characteristics on the probabilistic fabric impact response are computationally studied using five different strength distributions with differing mean strengths and distribution widths. Corresponding fabric PVR curves are generated for each strength distribution using a probabilistic computational framework that involves randomly mapping yarn strengths onto the individual woven yarns of a fabric finite element model and then running a series of impact simulations for the case of a four-sided clamped fabric impacted at the center by a spherical projectile.     

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.     

Robust topology optimization of continuum structures with loading uncertainty using a perturbation method

An approach for the robust topology optimization (RTO) of continuum structures with loading uncertainty is investigated. The loading uncertainties are quantified using the second order Taylor series expansion of uncertain loading magnitudes and directions, and then the response statistic mean and standard deviation of compliance are calculated using the uncertain perturbation propagation method. A robust design Lagrange function considering the compliance objective and finite element constraints is developed, and a sensitivity analysis is performed to calculate the Lagrange coefficients. The Lagrange objective function is optimized using the modified solid isotropic material with penalization (SIMP) algorithm; thus, the optimum material distribution under loading uncertainty is acquired. The proposed methodology is used for the RTO of two examples, revealing its efficiency under both concentrated and distributed uncertain loadings. The accuracy of the results is verified by comparison with similar cases found in the literature where a different modelling approach was used.     

结构随机延性需求谱的理论研究

引入结构屈服水平系数,建立了结构的延性需求谱,使地震动的幅值特性和频谱特性可以分开考虑.利用双线性单自由度体系的时程反应分析结果,对延性需求系数的条件分布类型进行了假设检验,通过回归得到了Ⅰ类和Ⅱ类场地上延性系数的条件均值和条件标准差的拟合公式.在此基础上,提出了延性系数的统计参数的计算方法,建立了随机的延性需求谱,进而可获得随机的弹塑性位移反应谱.     

基于薄层单元的螺栓连接结构接触面不确定性参数识别

提出了一种螺栓连接接触面不确定性参数识别方法,首先采用薄层单元对接触面进行参数化,然后根据不确定性识别方法识别薄层单元参数。以四螺栓搭接结构试验模型为研究对象,开展接触面不确定性参数识别方法仿真研究。采用Monte-Carlo方法构造待识别参数真实值样本,代入基准有限元模型中计算获得具有统计意义的仿真试验数据;采用不确定性参数识别方法预测薄层单元参数均值与标准差,仿真结果表明：该方法能够较为准确的模拟接触面法向和切向接触刚度,并显著提高连接结构的建模效率,建立反映真实结构动态性能统计特征的有限元模型。     

Probabilistic modeling of fatigue crack growth in Ti–6Al–4V

This paper proposes an approximate approach to efficient estimation of some variabilities caused by the material microstructural inhomogeneities. The approach is based on the results of a combined experimental and analytical study of the probabilistic nature of fatigue crack growth in Ti–6Al–4V. A simplified experimental fracture mechanics framework is presented for the determination of statistical fatigue crack growth parameters from two fatigue tests. The experimental studies suggest that the variabilities in long fatigue crack growth rate data and the Paris coefficient are well described by the log-normal distributions. The variabilities in the Paris exponent are also shown to be well characterized by a normal distribution. The measured statistical distributions are incorporated into a probabilistic fracture mechanics framework for the estimation of material reliability. The implications of the results are discussed for the probabilistic analysis of fatigue crack growth.     

基于LHS-MC方法的矮塔斜拉桥地震风险概率分析

提出一种既能避免繁琐积分,又能综合考虑结构材料和地震波动随机性问题以及地震危险性的地震风险概率计算方法。基于结构材料参数的概率分布,采用拉丁超立方抽样(LHS)方法考虑结构构件材料参数的随机性,并结合选取的地震波,形成地震动-桥梁组合样本集。针对典型矮塔斜拉桥结构体系建立非线性有限元纤维模型,确定各主要构件的损伤指标,与增量动力分析方法相结合进行了地震易损性分析,选取合适的分布函数,拟合加速度峰值-结构损伤概率曲线。采用蒙特卡罗(MC)抽样方法离散地震危险性概率模型,避免了繁琐的积分,针对典型矮塔斜拉桥的地震风险概率进行了评估。     

Weight function based Dugdale model for mixed-mode crack problems with arbitrary crack surface tractions

Weight function theory states crack surface displacements can be found for any arbitrary distribution of mode I, or mixed-mode crack face tractions via that geometry’s weight functions. This statement is validated via finite element analysis of the infinite center-cracked plate for various mixed mode loadings. An elastic-perfectly plastic material is considered using a Dugdale approach and compared to elastic-plastic finite element simulations. The weight function method in all cases agrees well with the finite element simulations for small scale yielding at the crack tip. As the maximum traction value approaches one-half the yield strength discrepancies become larger due to violation of small scale yielding.     

A probabilistic model for spatial distribution of material properties

This paper deals with probabilistic characteristics of nonhomogeneous properties of structural materials such as structural ceramics and proposes a probabilistic model that can be used for digital-analytical simulation of such material properties. The model will be compatible with the finite element method and therefore extremely useful for the analysis and design of nonhomogeneous structural systems. A technique is also presented to consider simultaneously random variations of two material properties. A particular emphasis is placed, however, on the probabilistic model for spatial variation of material strength which results in a corresponding statistical size effect. The principal idea lies in the interpretation that the material strength is a random function of the space variables. This interpretation together with a digital simulation technique of random function makes it possible to demonstrate the statistical size effect in terms of numerical examples. Specific examples considered here correspond to hot pressed silicone nitride for whch some experimental results are available.