Exact elasticity-based finite element for circular plates

Abstract

In this paper, a general elasticity solution for the axisymmetric bending of a linearly elastic annular plate is used to derive an exact finite element for such a plate. We start by formulating an interior plate problem by employing Saint Venant's principle so that edge effects do not appear in the plate. Then the elasticity solution to the formulated interior problem is presented in terms of mid-surface variables so that it takes a form similar to conventional engineering plate theories. By using the mid-surface variables, the exact finite element is developed both by force-and energybased approaches. The central, non-standard feature of the interior solution, and the finite element based on it, is that the interior stresses of the plate act as surface tractions on the plate edges and contribute to the total potential energy of the plate. Finally, analytical and numerical examples are presented using the elasticity solution and the derived finite element.

Shear deformable plate elements based on exact elasticity solution

“…where the strain energy stored in the interior plate is (28) and the work W s due to the interior stresses on the lateral edges of the plate consists of…”

“…However, this statement is generally based on analyses in which work W s , in the form given above, is absent. For example, the stiffness matrix of a Full Interior circular plate element is in fact not symmetric, nevertheless, the general interior solution (2)-(4) from which the element is derived without making any assumptions satisfies the reciprocal theorem when W s is used [28]. The stiffness matrix of an anisotropic interior beam element is not symmetric either [40].…”

“…In the remaining sections, we first present a brief overview on the interior framework for plates on the basis of our earlier works on the topic [27,28]. This is followed by the formulation of the rectangular finite elements for Mindlin, Levinson, and Full Interior plates by Galerkin's method of weighted residuals.…”

About Solution of Multipoint Boundary Problems of Three-Dimensional Structural Analysis With the Use of Combined Application of Finite Element Method and Discrete-Continual Finite Element Method. Part 1: Formulation and Basic Principles of Approximation

“…Расчетная модель -трехмерная (пространственная) задача теории упругости. Математическая постановка этой задачи приведена, например, в [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Пусть : -область, занимаемая рассматрива-емой конструкцией, …”

Uncovering the finite difference model equivalent to Hencky bar-net model for axisymmetric bending of circular and annular plates

“…Circular and annular plates are very important structural elements in aerospace, ocean, mechanical and civil engineering applications [1]. There have been many studies conducted on bending, buckling and vibration of circular and annular plates using the finite element method [2,3], differential quadrature method [4], Ritz method [5,6], perturbation method [7], the differential transformation method [8]. However, the finite difference method is rarely adopted for analyses of circular and annular plates.…”