## 210 The Finite Element Method

For problems involving complex material properties and boundary conditions, numerical methods are employed to provide approximate but acceptable solutions. Of the many numerical methods developed before and after the advent of computers, the finite element method has proven to be a powerful tool. This method can be regarded as a natural extension of the matrix methods of structural analysis. It can accommodate complex and difficult problems such as nonhomogeneity, nonlinear stress-strain behavior, and complicated boundary conditions. The finite element method is applicable to a wide range of boundary value problems in engineering and it dates back to the mid-1950s with the pioneering work of Argyris (1960), Clough (1993), and others. The method was first applied to the solution of plane stress problems and extended subsequently to the solution of plates, shells, and axisymmetric solids.

### 2.10.1 Basic Principle

The finite element method is based on the representation of a body or a structure by an assemblage of subdivisions called finite elements, as shown in Figure 2.81. These elements are considered to be connected at nodes. Displacement functions are chosen to approximate the variation of displacements over each finite element. Polynomials functions are commonly employed to approximate these displacements. Equilibrium equations for each element are obtained by means of the principle of minimum potential energy. These equations are formulated for the entire body by combining the equations for the individual elements so that the continuity of displacements is preserved at the nodes. The resulting equations are solved by satisfying the boundary conditions to obtain the unknown displacements.

Typical element

Boundary of a region

Typical element

Boundary of a region

Typical node

FIGURE 2.81 Assemblage of subdivisions.

The entire procedure of the finite element method involves the following steps:

1. The given body is subdivided into an equivalent system of finite elements.

2. A suitable displacement function is chosen.

3. The element stiffness matrix is derived using a variational principle of mechanics such as the principle of minimum potential energy.

4. The global stiffness matrix for the entire body is formulated.

5. The algebraic equations thus obtained are solved to determine unknown displacements.

6. Element strains and stresses are computed from the nodal displacements.

### 2.10.2 Elastic Formulation

Figure 2.82 shows the state of stress in an elemental volume of a body under a load. It is defined in terms of three normal stress components sx, oy, and sz and three shear stress components ixy, tyz, and tzx. The corresponding strain components are three normal strains ex, Ey and ez and three shear strains gxy, gyz, and gzx. These strain components are related to the displacement components u, v, and w at a point as follows:

0 0