To characterize the dependency between basic random variables is of paramount importance in engineering practice. However, it is usually difficult to deal with joint probability density function of dependent variates directly. In the present paper, a random function model is proposed for the probabilistic description of 2-dimensional dependent random variables. This random function converts a 2-dimensional dependent random vector into an independent random vector. The undetermined functions in the random function model are found to be the conditional mean and conditional standard deviation function, which could be further specified by observed data. Specifically, in the present paper it is suggested that the shape of the undetermined function, i.e., the conditional mean and standard deviation function in this case, be extracted based on the insight into the embedded physical mechanism, and then the undetermined parameters be identified from observed data. The procedure is illustrated in detail by adopting the relationship between the modulus of elasticity and compressive strength of concrete as an example. The one-dimensional damage evolution mechanism is firstly introduced, yielding the lower limit of the data. Then the viscoelasticity mechanism is advocated to determine the shapes of conditional mean and conditional standard deviation function. The parameters of the model are consequently identified from experimental data. Thereby, a pragmatic model, which could character the dependency between the modulus of elasticity and compressive strength, is proposed for engineering purposes. The comparison to the Copula model shows that the proposed model could capture the probabilistic characteristics of observed data. It is noted that in the proposed model the random function is weakly nonlinear and thus will not worsen the well-posedness of the problem. Besides, the direct dealing with joint probability density function is avoided. The proposed approach could be extended to the probabilistic description of more non-independent random variables.
国家自然科学基金(11672209)
科技部国家重点实验室基金(SLDRCE14-B-17资助项目)
[1] Ghanem R, Spanos P D. Stochastic Finite Elements: A Spectral Approach. Berlin: Springer-Verlag, 1991. Google Scholar
[2] Ang A H-S, Tang W. Probability Concepts in Engineering. Hoboken: John Wiley & Sons, 2006 [Ang A H-S, Tang W. 工程中的概率概念. 陈建兵, 彭勇波, 刘威, 艾晓秋. 北京: 中国建筑工业出版社, 2017]. Google Scholar
[3] Li G Q, Li J H. Method of second moment matrix—on the reliability calculation of dependent stochastic vector (in Chinese). J Chongqing Inst Architech Eng, 1987, 1: 55–67 [李国强, 李继华. 二阶矩矩阵法——关于相关随机向量的结构可靠度计算. 重庆建筑工程学院学报, 1987, 1: 55–67]. Google Scholar
[4] Rosenblatt M. Remarks on a multivariate transformation. Ann Math Statist, 1952, 23: 470-472 CrossRef Google Scholar
[5] Nelsen R B. An Introduction to Copulas. 2nd ed. New York: Springer, 2006. Google Scholar
[6] Das S, Ghanem R, Finette S. Polynomial chaos representation of spatio-temporal random fields from experimental measurements. J Comp Phys, 2009, 228: 8726-8751 CrossRef ADS Google Scholar
[7] Xu L, Cheng G D. Discussion on: Moment methods for structural reliability. Struct Safety, 2003, 25: 193-199 CrossRef Google Scholar
[8] Li D Q, Zhang L, Tang X S, et al. Bivariate distribution of shear strength parameters using copulas and its impact on geotechnical system reliability. Comput Geotech, 2015, 68: 184–195. Google Scholar
[9] Papoulis A, Pillai S U. Probability, Random Variables, and Stochastic Process. New York: McGraw-Hill, 2001 [Papoulis A, Pillai S U. 概率、随机变量与随机过程. 保铮. 冯大政, 水鹏朗. 西安: 西安交通大学出版社, 2015]. Google Scholar
[10] China Academy of Building Research-Research Group on Basic Mechanical Properties of Concrete. Research report on reinforced concrete-some basic mechanical properties of concrete (in Chinese). Beijing: China Architecture & Building Press, 1977 [国家建委建筑科学研究院 混凝土基本力学性能研究组. 钢筋混凝土研究报告选集 混凝土的几个基本力学指标. 北京: 中国建筑工业出版社, 1977]. Google Scholar
[11] Ministry of Housing and Urban-Rural Development of the People’s Republic of China. GB 50010-2010 Code for design of concrete structures (in Chinese). Beijing: China Architecture & Building Press, 2010 [中华人民共和国住房和城乡建设部组织. GB 50010-2010 混凝土结构设计规范. 北京: 中国建筑工业出版社, 2010]. Google Scholar
[12] Wan Z Y, Ren X D, Li J. Uniaxial elasto-plastic damage model for confined concrete (in Chinese). J Build Struct, 2014, 35: 178–184 [万增勇, 任晓丹, 李杰. 箍筋约束混凝土单轴拉压弹塑性损伤本构模型. 建筑结构学报, 2014, 35: 178–184]. Google Scholar
[13] Li J, Wu J Y, Chen J B. Stochastic Damage Mechanics of Concrete Structures (in Chinese). Beijing: Science Press, 2014 [李杰, 吴建营, 陈建兵. 混凝土随机损伤力学. 北京: 科学出版社, 2014]. Google Scholar
[14] Li J. Physical stochastic models for the dynamic excitations of engineering structures. In: Li J, Chen J B, eds. Advances in Theory and Application of Random Vibration (in Chinese). Shanghai: Tongji University Press, 2009: 119–132 [李杰. 工程结构随机动力激励的物理模型. 见: 李杰, 陈建兵, 编. 随机振动理论与应用新进展. 上海: 同济大学出版社, 2009. 119–132]. Google Scholar
[15] Li J, Yan Q, Chen J B. Stochastic modeling of engineering dynamic excitations for stochastic dynamics of structures. Probab Eng Mech, 2012, 27: 19-28 CrossRef Google Scholar
[16] Wei J. An approach on the relation between elastic modulus and strength of concrete (in Chinese). J Zhengzhou Univ (Eng Sci), 1987, 3: 13–16 [卫军. 关于砼的弹性模量与强度的关系之探讨. 郑州大学学报(工学版), 1987, 3: 13–16]. Google Scholar
[17] Wu L X, Huo J H. A theoretical basis of equation of correlation between strength and elastic modulus of concrete (in Chinese). J Chongqing Instit Architect Eng, 1993, 15: 31–35 [吴礼贤, 霍津海. 混凝土抗压强度与弹模关系式的理论基础. 重庆建筑工程学院学报, 1993, 15: 31–35]. Google Scholar
[18] Li J, Chen J B. Stochastic Dynamics of Structures. Hoboken: John Wiley & Sons, 2009. Google Scholar
Figure 1
(Color online) Experimental data
Figure 2
(Color online) Uniaxial constitutive relation of concrete in compression.
Figure 3
(Color online) Lower limit constraint.
Figure 4
(Color online) The absolute value of residual values.
Figure 5
(Color online) Fitted result of standard deviation.
Figure 6
(Color online) Histogram of the random variable
Figure 7
(Color online) Histogram of the compressive strength of concrete and fitted curve of PDF.
Figure 8
(Color online) Contour of joint probability density function of compressive strength and modulus of elasticity based on random function model.
Figure 9
(Color online) Histogram of modulus of elasticity concrete and fitted PDF curve.
Figure 10
(Color online) Contour of joint probability density function of compressive strength and modulus of elasticity of concrete based on Copula function.
Figure 11
(Color online) Four random samples based on random function model (in the figure, “×” data is the simulation result by random function model, “·” data is the experimental data ).
Figure 12
(Color online) 10000 Monte Carlo simulation results.
参数 |
数值 |
0.2234 |
|
3.0305 |
|
1.057 |
|
24.09 |
|
0.00152 |
|
1.38 |
|
0.118 |
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