Minimum Constraints in FEA – 2. Sanity Checks by Kotur Raghavan
General
An important property of “minimum
constraints” is that they permit free thermal expansion or contraction of the structure
when it is subjected to a uniform temperature field. This becomes evident by
visual examination of the applied constraints in a 2D problem (Fig. 1).
The node N1 is the fixed point. Free
thermal displacement is permitted in the X- direction as there are no other X-
direction constraints. The object will freely expand or contract in the Y-
direction relative to the line joining N1 to N2.
Fig. 2
Similar conditions can be
visualized in the case of three-dimensional objects (Fig. 2). The node N1 is the stationary point. The cube
has free thermal flow in the X- direction with respect to N1. Its free thermal
flow in the Y- direction is with respect to the line joining N1 to N2. The free
expansion in the Z- direction is relative to the XY- plane passing through the
origin.
The above property is exploited
for carrying out one of the recommended sanity checks in simulation models
(nodes and elements). The test is to confirm that the model with minimum
constraints experiences zero thermal stress when subjected to a uniform
temperature field. This will ensure that the model has no internal (unintended)
constraints. Internal constraints arise as a result of incorrectly coupling
degrees of freedom of two or more nodes.
Plane Stress Problem
A planar object of trapezoidal
shape, shown in Fig. 3, is analysed. Constraints and the applied temperature
field are also shown in the figure. The top right corner is located at (300,
300). The coefficient of thermal expansion is 12E-06/deg. C.
Fig. 3
The resulting displacement
patterns are shown in Fig. 4. The displacements are matching with the expected
value for free expansion.
Fig. 4
The computed stress fields are
shown in Fig. 5. The stresses are to be treated as having zero values.
Fig. 5
Any additional constraint will
hinder free thermal expansion and the object will experience thermal stresses.
Fig. 6
The same problem is solved with
an additional X- displacement constraint at the location shown as black dot in
Fig. 6 above. The figure also shows the resulting displacement field. High
local stresses are seen at two locations which have X- displacement
constraints.
Objects in 3D space
Qualitatively similar results
will be obtained in the case of objects in 3D space.
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