基于变分原理的直接解法求压杆的临界载荷

The calculation of critical load of a compressive bar by direct method based on variation principle

稳定性是结构工程的重要课题 ,因此压杆临界载荷的计算也显得非常重要 .求压杆临界载荷的方法很多 :有静力法、矩量法、子域法、最小二乘法 .利用变分原理的直接解法 ,针对不同支承的压杆假设弯曲函数～试验函数 ,求出了压杆临界载荷的计算公式 .计算结果非常接近精确解 .直接解法不依赖于微分方程的积分而是直接利用变分原理 ,通过假设弯曲函数～试验函数求得近似解 .它的优点是把微分方程的积分过程转化为代数方程组的求解过程 ,避免了求解微分方程的麻烦 .如果试验函数满足压杆的自然边界条件 ,在试验函数中取前一项或前两项就能有比较高的精度 ;如果试验函数能满足位移边界条件 ,但不能完全满足力的边界条件 ,在试验函数中可以多取几项 ,也可以达到比较高的精度 .最后结果表明 ,基于变分原理的直接解法原理简单 。

Stability is an important problem in structu ral engineering, therefore calculating critical load is very important. Many methods, e.g. static force method, moment method, sub-region method and least square method, have been us ed to g et the critical load. A direct method based on the variation principle is used to obtain the formulae of critical loads by ass uming trial functions of variation supported compressive bars. The results are c lose to analytic ones. The direct method, u sing variation principle directly to get an approximate solution by assuming tri al function, is independent of the integrating of differential equation. Its advant age is converting differential equations to algebraic equations avoiding dif f iculty in solving differential equations. If the trial function satisfies the na tural boundary conditions, the result is exact enough with one or two terms. If the d isplacement boundary condition is satisfied but the force boundary condition is not satisfied, the result is exact enough with more terms. The fi nal resu lts show that the direct method based on the variation principle is simple with high preasion.