Nonlinear Shallow Curved‐Beam Finite Element

Abstract

A geometrically nonlinear curved‐beam finite element is derived and is applied to the load‐deflection and buckling analysis of shallow and deep arches. The element stiffness properties are derived relative to a moving reference frame having one axis along the element chord, and they are then transformed to a global reference frame. The initial shape of the element is a shallow cubic polynomial, and its deformed shape is the exact shape caused by end actions within the context of shallow beam theory with shear deformation included. The element stiffness properties are exact within the context of that theory, and are described by explicit formulas similar to those of an initially straight element. In a global reference frame the element applies to deep arches with large rotations. Numerical applications are implemented for the second‐order theory. Good results are obtained for the snap‐through analysis of a shallow arch and for buckling of deep and shallow arches.