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EMA 405, Fall 2003
Solutions for HW #1
1. a)
The different arrangements of constraints on vertical displacement are outside the scope of
the preliminary analysis, which has been performed in class with the equations for a long thin
beam.
What we do know is that the sum of the reaction forces in the xdirection is zero and the
sum of the reaction forces in the ydirection is +1x10
6
N.
To some extent, intuition on the
distribution of forces can be guided by our preliminary computations for the distribution of
σ
x
and
τ
xy
from the long beam model.
The
σ
x
distribution is linear in y, with a maximum value of
36 MPa at y=h/2 and a minimum value of –36 MPa at y=h/2 at the wall end of the beam (x=0).
The
τ
xy
distribution is parabolic in y with a peak amplitude of 3 MPa at y=0.
If we divide the
wall end of the beam into a set of horizontal strips, and integrate the stresses from the
preliminary analysis over the strips, we can determine the average force over each strip to
compare with the nodal reactions from the FEA.
The top and bottom strips will have half the
area of the rest of the strips, as shown below for the five nodes along the wall for the 2x6 mesh.
Wall end of beamcircles represent nodes,
horizontal lines indicate areas for computing
reaction forces from stresses.
The expected reaction forces based on stresses from the preliminary analysis are given in Table
1.1.
Since these results are computed from the long beam approximation, they DO NOT include
effects associated with a nonzero Poisson ratio.
Table 1.1.
Predicted Reaction Forces for a 2x6 Mesh—Integrated Stresses from the P.A.
vertical position (m)
Fx (N)
Fy
1
2.0x10
6
+0.04x10
6
0.75
2.3x10
6
+0.27x10
6
0.5
0
+0.37x10
6
0.25
+2.3x10
6
+0.27x10
6
0
+2.0x10
6
+0.04x10
6
1/7
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View Full DocumentThe results for the 1x3, 2x6, and 4x12 FEA computations are provided in Tables 1.21.4.
In all
cases the sum of the x and ycomponents meet our expectations for the net reaction forces.
In
addition, the distribution of Fx is close to our predicted distribution for the 1x3 and 2x6 meshes.
However, the more refined mesh produces an Fx distribution that does not appear to result from
a purely lineariny distribution of
σ
x
.
The computed Fy distributions are further from the
predictions than the computed Fx distributions.
For the 1x3 and 2x6 meshes, the sign of the
computed Fy changes from node to node.
This does not meet expectation, and is an indication of
poor resolution.
The Fy reactions from the 4x12 mesh also shows significant changes from one
node to the next, but the largest forces remain at the extreme ylocations, where the largest
strains associated with the nonzero Poisson ratio occur.
[The Fy reaction forces in the
computation with Poisson’s ratio set to zero, given in Table 1.9, are closer to the predicted
distribution, but the largest forces do not occur at the midplane. The reason for this will become
apparent when we discuss distributed loading.]
Table 1.2.
Computed Reaction Forces with the 1x3 mesh
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 Spring '04
 Witt

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