A Finite Element Based on the Strain Approach Using Airy’s Function

Plane membrane elements of class C ∘ provide poor deflection and stress for problems where bending is dominant. They also encountered problems of continuity and compliance when connected to plate elements (elements of class C 1 ). The scope of this paper is to overcome these problems by developing a new triangular plane elastic element based on a strain formulation. The developed membrane element, denoted T43_Eq, has three nodes at the vertices of the triangle and the fourth one at its barycenter. Each node has three degrees of freedom, two translations and one rotation around the normal. The coefficients related to the degrees of freedom at the internal node are subsequently removed from the element stiffness matrix by using the static condensation technique. Interpolation functions of strain, displacements and stresses fields are developed from equilibrium conditions. These polynomial bi-harmonics functions are selected from the development of the Airy function solutions. The elementary stiffness matrix is evaluated by applying the variational principle and the analytical integration method. The results of the validation test show that the developed T43_Eq element is very efficient for treating bending problems. They are competitive compared to the triangular or rectangular elements available in the literature in terms of accuracy and convergence. The performance of T43_Eq element is observed in the presence of both regular and distorted meshes.