Civil Engineering Reference
In-Depth Information
2
Spatial Discretisation
by Finite Elements
2.1
Introduction
The finite element method is a technique for solving partial differential equations by first
discretising these equations in their space dimensions. The discretisation is carried out
locally over small regions of simple but arbitrary shape (the finite elements). This results
in matrix equations relating the input at specified points in the elements (the nodes) to
the output at these same points. In order to solve equations over large regions, the matrix
equations for the smaller sub-regions can be summed node by node, resulting in global
matrix equations, or “element-by-element” techniques can be employed to avoid creat-
ing (large) global matrices. The method is already described in many texts, for example,
Zienkiewicz and Taylor (1989), Strang and Fix (1973), Cook
et al
. (1989), and Rao (1989),
but the principles will briefly be described in this chapter in order to establish a notation
and to set the scene for the later descriptions of programming techniques.
2.2 Rod element
2.2.1 Rod stiffness matrix
Figure 2.1(a) shows the simplest solid element, namely an elastic rod, with end nodes 1
and 2. The element has length
L
while
u
denotes the longitudinal displacements of points
on the rod which is subjected to axial loading only.
If
P
is the axial force in the rod at a particular section and
F
is an applied body force
(units of force/length) then,
EA
d
u
d
x
P
=
σA
=
EAε
=
(2.1)
