Civil Engineering Reference
In-Depth Information
There are 64 equations and the skyline storage is 1050
Node x-disp y-disp
1 0.0000E+00 -0.1591E-04
2 0.0000E+00 -0.1158E-04
3 0.0000E+00 -0.7226E-05
4 0.0000E+00 -0.3354E-05
5 0.0000E+00 0.0000E+00
6 -0.9321E-06 -0.1559E-04
7 0.1493E-06 -0.1128E-04
8 0.4540E-06 -0.7019E-05
9 0.3347E-06 -0.3255E-05
10 0.0000E+00 0.0000E+00
.
.
.
44 0.0000E+00 -0.1906E-07
45 0.0000E+00 0.0000E+00
The integration point (nip= 1) stresses are:
Element x-coord y-coord sig_x sig_y tau_xy
1 0.3333E+00 -0.6667E+00 -0.8302E-01 -0.9098E+00 0.7671E-01
2 0.6667E+00 -0.1333E+01 -0.4434E-01 -0.6555E+00 0.1123E+00
3 0.2667E+01 -0.6667E+00 -0.2042E-01 0.3240E-01 -0.1323E-01
4 0.4333E+01 -0.1333E+01 -0.7382E-02 0.1345E-01 -0.3256E-02
Figure 5.8 Results from second Program 5.1 example
number of integrating points for this element in plane strain is
nip=12
. The data follow
a similar pattern to the previous example. In this case, the equivalent nodal loads for a
15-noded triangle are not intuitive, and Appendix A gives the required values to reproduce
a unit pressure under the “footing”.
The computed results for this example, given in Figure 5.8, indicate a centreline dis-
placement of
10
−
4
m. This is in good agreement with the solution of
×
10
−
4
m given by Poulos and Davis (1974). In order to minimise the volume of output,
Program 5.1 always computes and prints stresses at element centroids. This is easily
achieved in the main program by redefining
nip=1
, followed by a reallocation of the
points
and
weights
arrays (having first been “deallocated”). Users are of course free
to remove these lines, and print the stresses at other locations if required.
The third example demonstrates the 4-node “linear strain” quadrilateral. Figure 5.9
shows a typical mesh of elements, together with the node and element numbering in the
case of numbering in the
y
-direction (
dir='y'
). Figure 5.10 gives the node numbering
system adopted for the 4-node quadrilateral and also the order in which the recommended
number of integrating points
nip=4
are visited. Consistent with triangular elements, nodal
numbering always starts at a corner and proceeds clockwise.
Figure 5.11 shows the mesh and data for a rigid strip footing resting on a uniform elastic
layer. In this case the footing is given a fixed displacement in the
y
-direction of
−
0
.
1591
×
−
0
.
153
×
10
−
5
m at nodes 1 and 4 into the layer. Since there are no applied loads,
loaded nodes
is read as zero. The two fixed displacements are entered by reading
fixed freedoms
as 2, followed by, for each fixed freedom, the node to be fixed (1 and 4), the sense of the
fixity (2), and the magnitude of the fixed displacement (
−
1
10
−
5
m).
The computed results in Figure 5.12 confirm the fixed
y
-displacements at nodes 1 and
4 have the expected value of
−
1
×
10
−
5
m. Node 7 has moved up by 0
.
1258
10
−
5
m,
−
1
×
×
