Developing Finite Element Mathematical Model (Formal Approach) - Part 1

 In this article, we will be covering how to drive the element mathematical model using formal approach that involves the concepts of shape functions, natural coordinates, minimization of potential energy and numerical integration using Gauss-Legendre quadrature method. The article has 2 parts. Part-1 covers the how to drive element stiffness matrix. Part-2, coming next, will be focused on how to make use of numerical integration to evaluate the entries of element stiffness matrix.

Let’s dig out the formal approach by considering a simple 1D bar element with 2 displacement dofs, as show in Figure 1. There are 03 types of coordinate systems (CS) to define the element mathematical model: global CS, local/element CS and natural CS. The global CS is used for element assembly process. The local CS is used to define the element stiffness matrix, and the natural CS is used to conveniently solve for the entries of element stiffness matrix etc. The concept of using natural CS will be further explained in the following text.

In case of formal approach, we begin with choosing with a displacement function that is expressed in terms of nodal values. Such functions are named as shape functions as they define the distribution of displacement field within the element domain by interpolating nodal values. The number of shape functions depends on the number of element nodes and associated dofs. For example, 2 shape functions are needed for 1D bar element with 2 nodes, each having 1 displacement dof.

Pay attention to the fact that:

·      Shape function are expressed in terms of natural coordinate system (CS), which are normalized in nature as they cover the domain space within range [-1, 1]. Why is there need of natural CS? It is needed to compute the entries of element stiffness matrix conveniently using numerical integration method.

·    Shape functions are not arbitrary selected. They must fulfill the 02 criteria which are: (1) the value of shape function at a given node is 1 and 0 at other node(s), and (2) the sum of shape functions at any point with the element domain must be equal to 1.

Here is demonstration of how to choose shape functions and recheck them.

One the choice of shape functions is made, then, we express the displacement and strain fields within elements in terms of shape functions and natural coordinates, present them in matrix form as following.

Now, to drive the element stiffness matrix, we can deploy various methods i.e. weighted residual, minimization of potential energy etc. The minimization of potential energy principle is very commonly used in the field of engineering structures, and the same we will be using here. Using this principle, we equate the work done on the structure with the corresponding strain energy stored in the structure. After doing tedious mathematics, we can easily derive the element stiffness matrix as following.

For 1D bar element with 2 nodes, we solve this integral to get the element stiffness matrix. Also, we can nodal force and displacement vectors, as following.