转换矩阵在非线性材料杆系分析中的应用

Application of transformation matrix in the analysis of nonlinear material bar system

证明了在杆系中,力的转换矩阵与位移的转换矩阵互为转置矩阵,当静不定非线性杆系静力平衡方程确定,而变形协调条件难以确定时,利用转置矩阵可以方便求得静不定非线性杆系的内力及有关节点位移。非线性材料杆系应力-应变关系σ=Bε^(1/n)中的幂n=2时,非线性材料静不定桁架有可能存在两个解;而采用常规方法求解静不定非线性杆系内力时有可能存在漏解现象,即出现仅能得到一个解的现象。非线性材料杆系应力-应变关系σ=Bε^(1/n)中的幂n=1时,假设非线性材料杆系各杆内力全部受拉力,或按各杆内力真实受力情况去求各杆内力及节点位移,求得结果的绝对值都是相同的,仅存在符号的差异;与按非线性材料杆系应力-应变关系σ=Bε^(1/n)中幂n=2时,求得的各杆内力及节点位移的其中一个解的绝对值是一致的。

It is proved that the force transformation matrix and the displacement transformation matrix in the bar system are transposed matrices of each other.When it is easy to determine the static equilibrium equation but difficult to obtain the deformation coordination condition in the statically indeterminate nonlinear bar system,the internal forces of statically indeterminate nonlinear bar system and the displacement of related nodes can be easily obtained by using the transposed matrix.In solving the internal forces of statically indeterminate nonlinear bar system,there may be two solutions when n=2 in the stress-strain relationship σ=Bε^(1/n).However,the conventional methods may lose some results,that is,only one solution can be obtained.Assuming that when n=1,all the internal forces of each rod in the nonlinear material rod system are subjected to tensile force,or the internal forces and node displacements of each rod are calculated according to the actual stress,the absolute values of the results are the same except for the sign difference.The result is consistent with the absolute value of one solution n=2.