• Brief Report •

### Geometrical nonlinear analysis of truss structures with random parameters utilizing recursive stochastic finite element method

Bin Huang Jianchen Suo Wenjun Huang

• Received:2007-04-03 Revised:2007-04-28 Online:2007-11-18 Published:2007-11-18

Abstract: Geometrical nonlinear analysis of truss structures with random parameters is carried out using a new stochastic finite element method that is called as recursive stochastic finite element method in this paper. Combining nonorthogonal polynomial expansion and perturbation technique, RSFEM has been successfully used to solve static linear elastic problems, eigenvalue problems and elastic buckling problems. Although such method is similar in form to traditional the second order perturbation stochastic finite element method, it can deal with mechanical problems involving random variables of relatively large fluctuation levels. Different from spectral stochastic finite element method utilized widely that transforms the random different equation into a large deterministic equation through projecting the unkown random variables into a set of orthogonal polynomial bases, the new method is more suitable for solving large dimensional random mechanical problem because of recursive solution method. The structural response can be explicitly expressed by using some mathematical operators defined to transform the random different equation into a series of same dimensional deterministic equations. And more important point is that the above advantages of this presented method make it more helpful for solving static nonlinear problem than SSFEM. In the present paper, the stochastic equilibrium equation of geometrical nonlinear analysis of random truss structures under static load is firstly set up. Apart from that the random loads and the random area parameters are expanded using the first order Taylor series, both of the modulus and structural responses are expressed using nonorthogonal polynomial expansions. Then a set of deterministic recursive equations is obtained utilizing perturbation method. Transposition technique is given for solving the equations containing unknown coefficients according to operation rule of matrix and characteristics of truss structures. After the unknown coefficients are gotten, the second statistic moment can be easily obtained according to relationship matrix between orthogonal and nonorthogonal polynomial expansions. In 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 that of Monte-Carlo simulation when fluctuation of random variables becomes 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 significant conclusions are obtained.