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在导护型牛顿法求得分叉点和分叉点上失稳模态的基础上,该文提出一个分叉路径的求解算法。将解路径上的解视为解路径弧长的连续光滑函数,由结构平衡方程对解路径弧长的一阶导数建立起分叉方向满足的控制方程。由该控制方程知,分叉点上结构结点位移向量对解路径弧长的导数可分解为分叉点上失稳模态和控制方程特解的线性组合,从而将分叉方向的求解转化为线性组合系数的求解。通过考虑结构平衡方程对解路径弧长的二阶导数与各失稳模态的向量点积,建立起线性组合系数满足的二次方程组,用牛顿法求得组合系数的解答,从而获得各分叉方向。沿各分叉方向作弧长延拓,即可从基本路径转入各分叉路径。通过跟踪各分叉路径,可对结构屈曲后的受力性能获得较全面的了解。数值算例表明该文方法准确、可靠、高效,能很好地处理大型杆系结构的分叉失稳问题。
Based on the derivation of the bifurcation point and the instability mode at the bifurcation point, the paper proposes a solution to the bifurcation path. The solution on the solution path is considered as a continuous smooth function of the arc length of the solution path. The governing equation satisfying the bifurcation direction is established by the structural balance equation for the first derivative of the arc length of the solution path. From this control equation, the derivative of the displacement vector of the structure node at the bifurcation point to the arc length of the solution path can be decomposed into a linear combination of the solutions of the instability modes and the governing equations at the bifurcation point, so that the solution of the bifurcation direction Solving Linear Combination Coefficients. By considering the second derivative of the arc-length of the solution path and the vector-dot product of each instability mode, the quadratic system satisfying the linear combination coefficient is established and the solution of the combination coefficient is obtained by Newton’s method so as to obtain the Bifurcation direction. The arc length extension along each bifurcation direction can be transferred from the basic path to each bifurcation path. By tracking the bifurcation paths, a more complete understanding of the mechanical behavior of the structure after buckling can be obtained. Numerical examples show that the proposed method is accurate, reliable and efficient, and can well handle the bifurcation instability of large bar structures.