分析梁大变形问题的一种新算法

梁发生大变形时,在变形前的平衡位置建立平衡方程必然引起很大误差,因此除了考虑位移与变形之间的非线性关系外,几何非线性问题还需要考虑大变形所引起的平衡方程的变化。文中提出了一种在变形后的平衡位置分析几何非线性梁的新算法,建立梁发生大变形一阶微分控制方程的矩阵形式,结合迭代修正齐次扩容精细积分法和迭代优化算法进行求解。由于考虑大变形（挠度和转角）对平衡方程的影响,文中方法的模型较为精确,采用精细积分法可以实现高精度求解。以悬臂梁为例,验证方法的正确性和精确性。其方法求解简单,对于不同的边界条件,只需修改优化目标函数。

New Algorithm for Large Deformation Beam

When large deformation of the beams occurrs,the establishment of equilibrium equations on the equilibrium position before deformation inevitably lead to large errors,in addition to considering the nonlinear relationship between displacement and deformation,the changes of balance equations caused by large deformation need to be considered in the nonlinear problems. This paper proposes a new algorithm in the equilibrium position after deformation analysis of nonlinear beam geometry,establishes the first order matrix differential equations of the large-deformation beams,which is solved combined with an iterative correction homogeneous capacity precision integration and iterative optimization algorithm. Due to the consideration of large deformation（deflection and rotation）on the equilibrium equations, the model of the proposed method becomes more accurate,can achieve high accuracy solution using precise integration method. Taking cantilever beam as an example, the correctness and accuracy of this method are proven. The solution is simple,for different boundary conditions,just change the optimization objective function.