The Finite Element Method - Mechanics of Structures: Variational and Computational Methods - page 335

In fact, the displacement method is only one of many solution methods in which “finite
elements” can be utilized. In terms of commercially available general purpose analysis
software, the displacement method is the most popular approach.

6.2 Summary of the Displacement Method

The displacement method as developed in Chapter 5 for bar structures is appropriate for
the analysis of any structure modeled using any type of element such as those displayed
in Fig. 6.2. A brief summary of the displacement method was given in Chapter 5, Section
5.3.9. We choose to expand on this summary here. An outline of the method follows:

1. From the system and loading, develop a model that is discretized into elements.
2. Determine for each element i the element stiffness matrix k i and the loading vector
p i 0 for distributed loads.
3. Transform the element properties from local to global coordinates

p i 0 k i T i ,

p i T iT p i , k i T iT p i 0 T iT = = = (6.1)

where T i is the coordinate transformation matrix. In most of this chapter it is as-
sumed that all local element forces, displacements, and stiffness matrices have been
transformed to the global coordinate system.
4. Develop the system equations:
— Establish the nodal displacements V .
— Develop the system equations, without consideration of the boundary conditions,
by assembling the element stiffness matrices k i , leading to the global stiffness
matrix K . These equations represent the equations of equilibrium in the global
coordinate system. If a consistent global node numbering system has been estab-
lished, then the assembly is accomplished by summing element stiffness coeffi-
cients with identical subscripts. Thus, for example, for elements 1 and 2,

= k jk + k jk

K jk (6.2)

— Develop the loading vector P from the applied nodal forces P ∗ and an assembled
distributed load vector P 0 .
— Introduce the boundary conditions by eliminating appropriate columns and rows.
Alternatively, introduce the boundary conditions during the assembly of the
system matrices.
5. Solve the system of algebraic equations,

P ∗ P 0

KV = + = P (6.3)

yielding the unknown displacements V of the nodes. These are in the global coordinate
system.
6. Postprocess these displacements by computing the forces and displacements for the
elements.