变曲率曲线梁的单元刚度矩阵分析

Element Stiffness Matrix Analysis for Variable Curvature Curved Beam

针对单元刚度矩阵多为隐函数,不便于直接应用的问题,在极坐标系下,假定变曲率曲线梁剪心和形心重合,根据卡氏第二定理,推导了一种显式变曲率曲线梁单元悬臂端柔度矩阵的解析解公式.该公式先将其柔度矩阵退化到经典的形式;再通过求逆运算得到变曲率曲线梁单元悬臂端刚度矩阵;最后根据静力平衡条件与结点位移的任意性获得曲线梁的单元刚度矩阵.以两端固定曲线梁为例,利用MATLAB编程与ANSYS有限元计算结果进行了对比,结果表明:竖向位移和扭转角相差都在5%以内,两者的误差较小,验证了单元刚度矩阵对变曲率曲线梁计算的有效性;矩阵中的元素可用带参数的显函数表达,且所有参数都可直接引用,说明了它的正确性.

Most element stiffness matrixes are implicit functions, and hence are inconvenient to apply directly. To overcome this deficiency, assuming that the shear center of a variable-curvature curved beam coincides with its centroid, an explicit analytical solution formula for flexibility matrix of a kind of variable-curvature curved beam element with cantilever end condition is derived in polar coordinates by Castigliano’s displacement theorem. First, the flexibility matrix is degraded to classical forms. Then, the stiffness matrix of the variable-curvature curved beam element with cantilever end condition is obtained by inversion of the flexibility matrix. Finally, according to conditions of static balance and arbitrariness of node displacement, the element stiffness matrix is obtained. In addition, taking a curved girder with two clamped ends as an example, comparisons are conducted between the calculation results by MATLAB program and those by ANSYS. The results show that the values of vertical displacement and torsion angle generated by MATLAB deviate from those by ANSYS within 5% , the small error between them verifying the effectiveness of the stiffness matrix in calculations of variable-curvature curved beam. What’s more,elements in the matrix can be expressed as explicit functions of parameters and all the parameters can be directly referenced, which also proves thecorrectness of the element stiffness matrix.