### 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).

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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