Civil Engineering Reference
In-Depth Information
As soon as the element matrix has been formed from (3.51) it can be assembled into
the global system matrix (or matrices) by special subroutines described later in this chapter.
Following equation solution, once the global nodal displacements are known, the ele-
ment displacements
eld
are retrieved, and the strains
eps
given by the strain-displacement
relations,
(3.52)
eps = MATMUL(bee,eld)
where, in the case of a 4-node quadrilateral
eld =
[
u
1
v
1
u
2
v
2
u
3
v
3
u
4
v
4
]
T
(3.53)
and stresses
sigma
from the stress-strain relations,
(3.54)
sigma = MATMUL(dee,eps)
The variables
u
and
v
are simply the nodal displacements in the
x
and
y
directions respec-
tively assuming the nodal ordering of Figure 3.9(a).
In cases where the stiffness matrix
km
of a 4-node quadrilateral is required “ana-
lytically”, the integration loop in Figure 3.8 is replaced by a single call to the subroutine
stiff4
(see Program 11.5). Similarly, when strains and stresses are back-calculated using
8-node quadrilateral elements (usually at the sampling points) the “analytical”, subroutine
bee8
can be used to replace the lines of program given by (3.47) to (3.49) (see e.g.
Programs 6.3, 6.8, and 6.9).
The shape functions and derivatives provided by subroutines
shape_fun
and
shape_
der
allow analyses to be performed using quadrilateral elements with 4, 8 or 9 nodes (e.g.
Program 5.1). A summary of the shape functions for all the elements used in this topic is
given in Appendix B.
Before describing the assembly process, which is common to all elements, modifications
to the element matrix calculation for different situations will first be described.
3.7.4 Plane elastic analysis using triangular elements
The previous section showed how the stiffness matrix of a typical 4-node quadrilateral
could be built up. In order to use triangular elements, very few alterations are required. For
example, for a 3-node triangular element
(nod=3)
,
x
1
y
1
x
2
y
2
x
3
y
3
coord
=
(3.55)
The shape functions [
N
] and their derivatives with respect to local coordinates at a
particular location
(L
1
,L
2
,L
3
)
(where
L
3
=
−
L
1
−
L
2
) are held in the arrays
fun
and
der
. Subroutine
shape_fun
delivers the shape functions,
1
L
1
L
3
L
2
fun
=
(3.56)
