Civil Engineering Reference
In-Depth Information
New scalar reals:
aa
element dimension in
x
-direction
bb
element dimension in
y
-direction
plate flexural stiffness
d
set to 4.0
d4
set to 12.0
d12
Young's modulus
e
set to 1.0
one
plate thickness
th
set to 2.0
two
Poisson's ratio
v
New dynamic real arrays:
bm
bending moments
integration point contribution to
km
dtd
second derivatives of shape functions with respect to
ξ
d2x
“mixed” second derivatives of shape functions with respect to
ξ
and
η
d2xy
second derivatives of shape functions with respect to
η
d2y
holds sampling points in local coordinates
points
x
coords
x
-coordinates of mesh layout
y
coords
y
-coordinates of mesh layout
holds weighting coefficients for numerical integration
weights
The previous examples have illustrated the principles of finite elements applied to
“structures” made up of one-dimensional elements. Solutions to these idealised problems
were not usually dependent upon the number of elements, which were chosen conveniently
to reflect the positions of the load applications and changes of geometry. This example
models a two-dimensional thin plate structure using a genuine finite element approximation.
The number of elements used to model the plate is decided by the user, but as the number
increases, so the solution should improve. The success of a finite element analysis rests on
“close enough” solutions being found using a reasonable number of elements. Section 2.14
describes the governing differential equation and the finite element discretisation.
The formulation described here enforces complete compatibility of displacements bet-
ween elements and equilibrium at the nodes, but there will in general be some loss of
equilibrium between the nodes. Figure 4.40 illustrates a typical element and gives the
freedom numbering of the
g
vector. It can be seen that there are 16 degrees of freedom per
element comprising a vertical translation (
w
), two ordinary rotations (
θ
x
,
θ
y
) and a “twist”
rotation (
), at each node.
The structure of the program is similar to several of the earlier programs in this chapter,
except that the element stiffness matrix is calculated using (Gaussian) numerical integration
in the
θ
xy
-directions. Property data involves Young's modulus and Poisson's ratio, read
into the array
prop
, the plate thickness
th
and the rectangular element dimensions
aa
and
bb
(the dimensions of the elements in the
x
-and
y
-directions respectively). The flexural
stiffness of the plate
d
is calculated from the plate thickness and elastic properties. All
elements are assumed to be the same size in this program and arranged into a rectangular
mesh. This enables the nodal coordinates and “steering” vector for each element to be
x
-and
y
