Element Stiffness Matrix Analysis for Variable Curvature Curved Beam

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 the correctness of the element stiffness matrix.