7 - FEM for two-dimensional solids

Abstract

This chapter focuses on the development of finite element (FE) equations for the stress analysis of two-dimensional (2D) solids subjected to external loads. The element developed is called a 2D solid element that is used for structural problems where the loading-and hence the deformation-occur within a plane. Though no real life structure can be truly 2D, experienced analysts can often idealize many practical problems to 2D problems to obtain satisfactory results by carrying out analyses using 2D models, which can be very much more efficient and cost-effective compared to conducting full 3D analyses. A 2D solid element, be it plane strain or plane stress, can be triangular, rectangular or quadrilateral in shape with straight or curved edges. The order of the 2D element is determined by the order of the shape functions used. A linear element uses linear shape functions, and therefore the edges of the element are straight. A quadratic element uses quadratic shape functions, and their edges can be curved. The same can be said for elements of the third order or higher. In a 2D model, the elements can only deform in the plane where the model is defined, and in most situations, this is the x-y plane. At any point, the variable, that is the displacement, has two components in the x and y directions, and so do the external forces. For plane strain problems, the thickness of the true structure is usually not important, and is normally treated as a unit quantity uniformly throughout the 2D model.