A new formulation based upon trigonometric function for finite circular arch elements

Abstract

In this paper, the formulation of the curved beam element by trigonometric functions for the curvature, which is an alternative to the displacement function is presented. Trigonometric function is chosen for the curvature to avoid the shear and membrane locking phenomena. In the developed formulation, the force—curvature relationships in polar coordinate system have been obtained first; then the curvature of the element has been assumed to have a trigonometric function form; and the radial and tangential displacements, and rotation of the cross-section have been found to be a function of the curvature. Moreover, the relationship between the nodal curvatures and the nodal deformations has been calculated and used for determining the deformations in terms of curvature at an arbitrary point. The total potential energy has been calculated accounting for bending, shear, and tangential deformations. Invoking the stationary condition of the system, the force—deformation relationship for the element has been obtained. Using this relationship, the stiffness matrix and the equivalent fixed loads applying at the nodes have been computed. In such an equilibrium equation, the locking phenomenon is eliminated. The formulation is applied to six examples to verify its capabilities. The results demonstrate that the presented element is capable of representing the behaviour of the curved beam with adequate accuracy and efficiency as compared with the previous methods.