### Minimum Constraints in FEA – 1. Overview by Kotur Raghavan

**Description**

1.
In any finite element structural analysis
model, minimum number of constraints refer to the constraints needed just to
prevent rigid body motion. In a mathematical sense, it is the minimum number of
degrees of freedom to be arrested so that the stiffness matrix becomes
non-singular so that a solution is possible.

2.
Any object in space has six degrees of
freedom. They are three translations and three rotations. Any object in
two-dimensional space has three degrees of freedom. They are the two
translation and one in-plane rotation.

3.
Therefore it follows that for structures
in three-dimensional space, six constraints are needed. Likewise for structures
in two-dimensional space, three constraints are needed.

4.
In structural models involving flexural
elements, that is beams and shells, all the required constraints can be applied
at a single node and a solution will be possible.

Fig. 1

6.
In the case of continuum elements, which do
not have rotational DOF’s, one has to choose translational DOF’s in suitable
combinations in order to effect arresting of rotations.

Fig. 2

7.
In the case of structures in
two-dimensional plane, the approach is to choose two arbitrary nodes. In one of
the nodes (say N1) translations in X- and Y- directions are to be constrained.
At the other node (say N2) it is necessary to constrain the translation in the
direction normal to the line joining N1 to N2. Two of the many possible methods
are shown in Fig. 2 above. It is easy to visualize that constraints at N1
arrest rigid body translations. The constraint at N2, in combination with the
ones at N1, arrest the in-plane rotation.

Fig. 3

9.
Just as the points (nodes) to be chosen
are arbitrary, the lines joining them need not necessarily be along the global
directions. All that is necessary is the choice of three nodes, N1, N2 and N3,
which are non-collinear. It is then necessary to define a local coordinate
system with origin at N1 and the local X- axis along the line joining N1 to
N2. The third node N3 will define the
local XY plane. It will be necessary also to “orient” the three nodes to this
local coordinate system. The
constraints, using the same scheme as above, will get applied in the local
directions with the same effect. This procedure can be easily coded and will be
handy while dealing with large size problems.

**Applications**

The principle of
minimum constraints find the following important uses.

1.
Sanity checks.

2.
Analysis of structures subject to
self-equilibrated forces.

3.
Submodeling with cut-boundary forces.

4.
Inertia relief applications.

The foregoing
applications will be discussed in future publications.

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