The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 334

formed primarily of polynomials. Interpolation theory provides this relationship. The most
essential characteristic of the method of finite elements is that the trial functions need not
span the entire system; they apply only for an element. In the case of the displacement
method, the displacements u i in the i th element are related to the displacements v i at the

nodes by

u i N i v i =

where N i are the trial or shape functions. Thus, the finite element method is a technique for
solving boundary value problems in which the domain is subdivided into small elements
over which the solution is approximated by interpolation.
In the finite element method, information characterizing the whole system, e.g., applied
loading, is assembled in a form that is independent of the boundary conditions and other
properties. This leads to a versatile method that can be used to solve efficiently classes of
problems and that is well suited for solution using digital computers (Bathe, et al., 1977;
Wunderlich, et al., 1981).

6.1 The Finite Element Method Based on the Displacement Method

The displacement method was shown in the previous chapter to be a method for the calculation
of the forces and displacements at the nodes of a structure. In Chapter 5 this method
was used to analyze structures formed of bar elements. In this chapter, we describe the
finite element method, primarily the displacement formulation, beginning with structures
modeled with elements of the sort shown in Fig. 6.2. The fundamental step is to utilize
an approximate stiffness matrix to characterize an element. The structure is assumed to be
modeled with a fine grid of many small elements in order to achieve an acceptably accurate
solution in spite of the approximation involved in the stiffness matrix.
The most essential ingredient in a displacement analysis is the element stiffness matrix k i
which is a relationship between the forces and displacements at the nodes of the element.
The equilibrium conditions for the forces at common nodes for a structure modeled with
multiple elements leads to system (global) equations assembled by superposition of the
element stiffness matrices for connected elements.
In Chapter 4, exact stiffness matrices k i and loading vectors were derived for the exten-
sion, tension, and bending of simple bar elements. For most structural components it is not
possible to find exact expressions for k i . This is usually accomplished by assuming that
the displacements in the element can be replaced by a polynomial. In the case of bars, the
exact displacements are already polynomials so the assumed polynomials can lead to exact
stiffness matrices (Section 4.4.2). For most other elements the assumed polynomials gener-
ate approximate stiffness matrices. The solution procedure, when the approximate stiffness
matrices are used, is referred to as the finite element method . Thus, the finite element method
is a technique for solving boundary value problems in which the domain is subdivided into
small elements over which the solution is approximated by polynomials.
As mentioned in Chapter 4, the assumed solution (polynomial) for a response is referred
to as a trial function . The unknown quantities in the trial functions are calculated using
equations established at the nodes. For the displacement method these are nodal equilib-
rium conditions. In general, the smaller the domain for which the trial function applies, the
better the approximation.
In this chapter, we will consider the finite element method primarily as a displacement
method, beginning with a summary of the displacement method as developed in Chapter 5.