随机杆系结构几何非线性分析的递推求解方法

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 UTILIZING RECURSIVE STOCHASTIC FINITE ELEMENT METHOD

Geometrical nonlinear analysis of truss structures withrandom parameters is carried out using a new stochastic finite elementmethod that is called as recursive stochastic finite element method in thispaper. Combining nonorthogonal polynomial expansion and perturbationtechnique, RSFEM has been successfully used to solve static linear elasticproblems, eigenvalue problems and elastic buckling problems. Although suchmethod is similar in form to traditional the second order perturbationstochastic finite element method, it can deal with mechanical problemsinvolving random variables of relatively large fluctuation levels. Differentfrom spectral stochastic finite element method utilized widely thattransforms the random different equation into a large deterministic equationthrough projecting the unkown random variables into a set of orthogonalpolynomial bases, the new method is more suitable for solving largedimensional random mechanical problem because of recursive solution method.The structural response can be explicitly expressed by using somemathematical operators defined to transform the random different equationinto a series of same dimensional deterministic equations. And moreimportant point is that the above advantages of this presented method makeit more helpful for solving static nonlinear problem than SSFEM. In thepresent paper, the stochastic equilibrium equation of geometrical nonlinearanalysis of random truss structures under static load is firstly set up.Apart from that the random loads and the random area parameters are expandedusing the first order Taylor series, both of the modulus and structuralresponses are expressed using nonorthogonal polynomial expansions. Then aset of deterministic recursive equations is obtained utilizing perturbationmethod. Transposition technique is given for solving the equationscontaining unknown coefficients according to operation rule of matrix andcharacteristics of truss structures. After the unknown coefficients aregotten, the second statistic moment can be easily obtained according torelationship matrix between orthogonal and nonorthogonal polynomialexpansions. In examples, the geometrical nonlinear analysis of a two barstructure and a plane truss arch are investigated. The numerical resultsshow that compared with traditional perturbation stochastic FEM based on thesecond Taylor series, the results obtained using the new method are moreclose to that of Monte-Carlo simulation when fluctuation of random variablesbecomes large. The interesting thing is that in the static geometricalnonlinear problem, when the second order perturbation stochastic finiteelement method is utilized, the divergence trend of the structural responsealso appears along with the increase of standard deviation of random crosssectional areas. However, this phenomenon disappears when the fourth orderRSFEM is used to solve this problem. In the end, some significantconclusions are obtained.