影响函数与有限元应力计算

Influence function and finite element stress calculation

用有限元法得到位移场后,总要计算应力场。通常的做法是对位移进行微商计算应变,再根据应力-应变关系计算应力。有限元位移计算的精度比较高,但通过用位移微商来计算应力,精度会大大降低。本文利用Hamilton对偶体系的已有成果,解析求解位移和应力的影响函数,利用有限元法计算得到的位移和节点力,通过功的互等定理,可以求得一点的应力值。因影响函数是分析解,而且计算应力时不必进行微商,应力精度大幅提高。数值结果表明该方法是可行的和有效的。由该方法编制成的计算程序,可作为有限元通用程序应力计算的一个模块,将较大地提高有限元应力计算的精度和稳定性。

The stress filed will be calculated accordingly after the displacement field is obtained with the finite element method. As usual, the strain is firstly obtained through differentiating displacements, then the stress field is calculated by using the relationship of the stress-strain. The accuracy of displace- ments by FEM is relatively higher. However through the displacement of derivation to calculate the stress, the accuracy will be much lower. Based on the achievements of Hamiltonian dual system, the stresses of one point are obtained through 3 steps of solving the influence functions of one point＇s dis- placements and stresses, getting displacements and nodal forces by FEM, and applying the reciprocal theorem of works. The accuracy increase of stress is obvious because the influence functions are analytic solutions and they are needless to differentiate displacements. Numerical results show that the method is feasible and effective. The program is compiled by the influence function method. It can be used as a module of finite element general program for stress calculation.