Higher Order Strain-Based Quadrilateral Membrane Finite Element for Static and Free vibration Analysis

This paper presents a new membrane finite element with in-plane rotational degrees of freedom and an additional internal node formulated using the assumed strain formulation. Designed for static and free vibration analyses, the proposed element is a quadrilateral with five nodes, which contains two translational and one in-plane rotational degrees of freedom at each corner node and only two in-plane translations for the central node. To improve computational efficiency, the static condensation concept is applied to remove the degrees of freedom of the central node, reducing system complexity without affecting accuracy or stability. A series of extensive static tests, including well-known reference problems such as cantilever beams with different shapes, and thick cylinders subjected to internal pressure, demonstrated the element’s superior performance on both regular and distorted meshes. In addition, the present element showed good agreement in free vibration studies of a cantilever wall, a cantilever beam, a variable-section cantilever beam, and a wall with openings. Results indicate that the proposed element achieves an accuracy comparable to higher-order elements, making it an effective and reliable choice for modeling thin-walled structures with significant in-plane behavior.