摘要
建立了随机静力作用下考虑几何非线性的随机杆系结构的随机非线性平衡方程.将和位移耦合的随机割线弹性模量以及随机响应量表示为非正交多项式展开式,运用传统的摄动方法获得了关于非正交多项式展式的待定系数的确定性的递推方程.在求解了待定系数后,利用非正交多项式展开式和正交多项式展开式的关系矩阵,可以很方便地得到未知响应量的二阶统计矩.两杆结构和平面桁架拱的算例结果表明,当随机量涨落较大时,递推随机有限元方法比基于二阶泰勒展开的摄动随机有限元方法更逼近蒙特卡洛模拟结果,显示了该方法对几何非线性随机问题求解的有效性.
Geometrical nonlinear analysis of truss structures with random parameters is carried out using a new stochastic finite element method called the recursive stochastic finite element method （RSFEM） in this paper. Combining nonorthogonal polynomial expansion and perturbation technique, RSFEM is successfully used to solve static linear elastic problems, eigenvalue problems and elastic buckling problems. Although such method is similar in form to the traditional second order perturbation stochastic finite element method, it can deal with problems involving random variables of relatively large fluctuation levels. Different from the spectral stochastic finite element method （SSFEM） used widely that transforms the random differential equation into a large set of deterministic equations through projecting the unknown random variables into a set of orthogonal polynomial bases, the new method is more suitable for solving large dimensional random mechanical problems because of its recursive solution method. The structural response can be explicitly expressed by using some mathematical operators defined to transform the random differential equation into a series of deterministic equations. And it is more important that the above advantages of the method make it more useful for solving static nonlinear problems than SSFEM. In the present paper, the stochastic equilibrium equation for geometrical nonlinear analysis of random truss structures under static load is first set up. The random loads and the random area parameters are expanded using the first order Taylor series, and both modulus and structural responses are expressed using nonorthogonal polynomial expansions. Then a set of deterministic recursive equations is obtained utilizing the perturbation method. Transposition technique is used for solving the equations containing unknown coefficients according to the rules of matrix operations and characteristics of truss structures. After the unknown coefficients are obtained, the second statistic moment can be easily obtained according to relationship matrix between orthogonal and nonorthogonal polynomial expansions. As examples, the geometrical nonlinear analysis of a two bar structure and a plane truss arch are investigated. The numerical results show that compared with traditional perturbation stochastic FEM based on the second Taylor series, the results obtained using the new method are more close to those of Monte-Carlo simulation when the fluctuation of random variables is large. The interesting thing is that in the static geometrical nonlinear problem, when the second order perturbation stochastic finite element method is utilized, the divergence trend of the structural response also appears along with the increase of standard deviation of random cross sectional areas. However, this phenomenon disappears when the fourth order RSFEM is used to solve this problem. In the end, some conclusions are drawn.
出处
《力学学报》
EI
CSCD
北大核心
2007年第6期835-842,共8页
Chinese Journal of Theoretical and Applied Mechanics
基金
国家自然科学基金(50208016).~~
关键词
随机杆系结构
几何非线性
递推随机有限元方法
非正交多项式展开式
二阶泰勒展开
truss structures with random parameters, geometrical nonlinear, recursive stochastic finite element method, perturbation stochastic finite element method, the second Taylor series
