对称几何非线性三角形平面单元研究

Study on geometrically nonlinear triangle plane element with symmetry

为克服现有共旋法的几何非线性平面单元切线刚度矩阵的不对称给非线性方程组求解带来的困难,基于共旋坐标法与几何一致性相结合的原则,对三角形平面单元进行了研究。将单元共旋坐标系的原点坐标分量标始终设为三角形的3个顶点坐标分量的平均值,合理选择共旋坐标系初始时刻及任意计算时刻的坐标轴方向,基于虚功原理,利用微分方程,建立三角形平面单元在大转动、小应变条件下的几何非线性单元切线刚度矩阵和相应的节点抗力算法。研究结果表明:该切线刚度矩阵是对称的;且将该切线刚度矩阵与一种能将位移增量法与荷载增量法形成统一迭代格式的非线性方程组求解方法进行结合,能大幅简化求解过程,依据推导公式与算法开发了相应的计算程序,并利用该计算程序对柔性立柱和大变形悬臂梁的经典算例进行了计算和比较,验证了该研究理论推导结果的正确性与计算程序的可靠性,且算例2显示该算法的计算精度高于ANSYS商业数值软件的。

In order to overcome the difficulty of solving nonlinear equations caused by the asymmetry of the past element in present co-rotational approach,the triangular plane elements are studied based on the principle of the combination of co-rotational method and geometric consistency.The origin of the element co-rotation coordinate system is located at the average value point of the three node coordinate values,and coordinate axis directions at the initial time and random calculation time of co-rotational coordinate system are reasonably selected.Based on the principle of virtual work and differentiation,the tangent stiffness matrix of geometric nonlinear element and the corresponding node resistance algorithm of triangular plane element under the condition of large rotation and small strain are established.It shows that the tangent stiffness matrix is symmetric from the calculation formula,and a corresponding program is developed combined with a unified incremental iteration scheme of incremental displacement method and incremental loading method for solving nonlinear equations.The classical examples of flexible column and large deformation cantilever beam are analyzed to verify the correctness of theoretical deduction and the program reliability.