Civil Engineering Reference
In-Depth Information
2.9 Two-dimensional elements: plane strain
and plane stress
The elements so far described have not been true finite elements because they have been
used to solve differential equations in one space variable only. Thus the real problem
involving two or three space variables has been replaced by a hypothetical, equivalent
one-dimensional problem before solution. The elements we have considered can be joined
together at points (the nodes) and complete continuity (compatibility) and equilibrium
achieved. In this way we can sometimes obtain exact solutions to our hypothetical problems
(especially at the nodes) in which solutions will be unaffected by the number of elements
chosen to represent uniform line segments.
This situation changes radically when problems in two or three space dimensions are
analysed. For example, consider the plane shear wall with openings shown in Figure 2.5(a).
The wall has been sub-divided into rectangular elements of side lengths
a
and
b
of which
Figure 2.5(b) is typical. These elements have 4 corner nodes so that when the idealised
wall is assembled, the elements will only be attached at these points.
If the wall can be considered to be of unit thickness and in a state of plane stress,
(Timoshenko and Goodier, 1982), the equations to be solved are the following:
1. Equilibrium
∂σ
x
∂x
∂τ
xy
∂y
+
+
F
x
=
0
∂τ
xy
∂x
∂σ
y
∂y
+
+
F
y
=
0
(2.50)
y,v
y
2
3
b
1
4
x
a
x,u
(a)
(b)
Figure 2.5
(a) Shear wall with openings. (b) Typical rectangular 4-node element
