Rigid Element Approach for Deriving the Geometric Stiffness of Curved Beams for Use in Buckling Analysis

For a rigid element equilibrated by a set of initial forces, the geometric stiffness matrix derived is the same, whether it is straight or curved, as long as it has identical nodal degrees at the two ends. Thus, one can first derive the geometric stiffness matrix for a rigid straight beam that has the same ending points as those of the curved beam using a rigid displacement field, and then transform this matrix from the Cartesian coordinates to the cylindrical coordinates to obtain the one for the rigid curved beam. The geometric stiffness matrix so derived (for treating the rigid displacements) can be used along with the elastic stiffness matrix (for treating the natural deformations) in the buckling analysis of curved beams. The present approach is featured by the fact that no assumptions are made for the kinematic behavior of curved beams, while the procedure of derivation is simple, explicit, and physically meaningful. Good characteristics of convergence are achieved as the finite-element mesh is refined. The robustness of the proposed approach is demonstrated in the solution of some benchmark problems.