Civil Engineering Reference
In-Depth Information
g(5)
g(9)
y
g(11)
2
3
g(7)
g(6)
g(12)
g(8)
g(10)
bb
g(13)
g(1)
g(2)
g(16)
g(4)
g(14)
x
1
4
g(15)
g(3)
aa
Figure 4.40
Node and freedom numbering for plate element
generated automatically by a geometry subroutine called
geom rect
. This subroutine will
be used frequently in later chapters of the topic, having the ability to generate rectangular
meshes for a variety of 2D elements.
In the “element stiffness integration and assembly” loop, all the derivative arrays men-
tioned above are delivered for each Gauss point by the library subroutine
fmplat
. Once
the Gauss point loop is completed, the element stiffness matrix is held in
km
. This is fol-
lowed by assembly into a global stiffness matrix, stored as usual as a skyline vector
kv
.
Nodal loads (forces, moments) and/or fixed displacements (translations, rotations) are then
read and the equilibrium equations solved.
Following calculation of the global displacements, a post-processing phase begins in
which the elements are scanned once more. Element nodal displacements are retrieved and
the three moments (
M
x
) are computed at the centroid of each element. It
should be noted that since
nip=16
was needed for the integration phase (four Gauss points
in each of the coordinate directions for exact integration), and
nip=1
is required to find
the centroid of each element, it was necessary to reset
nip
to unity and “reallocate” the
points
array.
The example shown in Figure 4.41 illustrates a symmetrical quadrant of a square plate
simply supported at its edges and modelled by four square elements. The plate supports a
central unit load so one quarter of this value is applied to node 9.
The results in Figure 4.42 show the central deflection of the plate (node 9) to be
0.01147 which can be compared with the “exact” solution of 0.01160 (Timoshenko and
Woinowsky-Krieger, 1959). By increasing the number of elements, better approximations
to the exact solution are obtained.
In addition to the results file
fe95.res
, Program 4.7 is the first program in the topic
to output a graphics file, generically called
fe95.msh
, which holds a PostScript image of
the mesh. This file is generated by subroutine
mesh
, which is one of a suite of graphics
subroutines held in the library
main
. Some of the other subroutines will be described in
the next chapter, and are useful for visualising results and debugging data.
,
M
y
and
M
xy
