Civil Engineering Reference
In-Depth Information
3.7.3 Plane elastic analysis using quadrilateral elements
As an example of element matrix calculation, consider the computation of the element
stiffness matrix for plane elasticity given by (2.69)
[
B
]
T
[
D
][
B
] d
x
d
y
[
k
m
]
=
(3.40)
This formulation in the programs is described by the inner loop of the structure chart
in Figure 3.8.
It is assumed for the moment that the element nodal coordinates
(x, y)
have been
calculated and stored in the array
coord
. For example, for a 4-node quadrilateral (
nod=4
),
x
1
y
1
x
2
y
2
x
3
y
3
x
4
y
4
coord
=
(3.41)
The shape functions [
N
] are held in array
fun
, as specified in (3.1) by,
1
4
(
1
−
ξ)(
1
−
η)
1
4
(
1
−
ξ)(
1
+
η)
fun
=
(3.42)
1
4
(
1
+
ξ)(
1
+
η)
1
4
(
1
+
ξ)(
1
−
η)
For all elements
Find the element nodal coordinates.
Form the stress
−
strain matrix [D]
Null the stiffness matrix [k
m
]
For all integrating points
Find the integrating point coordinates and
weighting coefficients
Form the element shape functions and
derivatives with respect to local coordinates
Transform to derivatives with respect
to global coordinates
Form the strain-displacment matrix [B]
Form product [B]
T
[D][B]
Weight this contribution and add it to the
element stiffness matrix [k
m
]
Figure 3.8 Structure chart for element matrix assembly assuming numerical integration
