The reader may find the following review material helpful before proceeding on to the discussion of the rigid body motion for a one-dimensional beam element below:
Rigid body motion occurs when forces and/or moments are
applied to an unrestrained mesh (body), resulting in motion that
occurs without any deformations in the entire mesh (body). Since
no strains (deformations) occur during rigid body motion, there
can be no stresses developed in the mesh. In order to obtain a
unique FEM solution, rigid body motion must be constrained. If
rigid body motion is not constrained, then a singular system of
equations will result, since the determinate of the mesh
stiffness matrix is equal to zero (i.e., 
).
There are two rigid body modes for the one-dimensional beam element, a translation (displacement) only and a rotation only. These two rigid body modes can occur at the same time resulting in a displacement and a rotation simultaneously. In order to eliminate rigid body motion in a 1-D beam element (body), one must prescribe at least two nodal degrees of freedom (DOF), either two displacements or a displacement and a rotation. A DOF can be equal to zero or a non-zero known value, as long as the element is restrained from rigid body motion (deformation can take place when forces and moments are applied) .
For simplicity we will introduce the rigid body modes using a mesh composed of a single element. If only translational rigid body motion occurs, then the displacement at local node I will be equal to the displacement at local node J. Since the displacements are equal there is no strain developed in the element and the applied nodal forces cause the element to move in a rigid (non-deflected) vertical motion (which can be either up as shown below or it can be in the downward direction depending on the direction of the applied forces).
This rigid body mode can be suppressed by prescribing a vertical nodal displacement.
If rotational rigid body motion occurs, then the rotation at local node I will be equal to the rotation at local node J (i.e., in magnitude and direction). In this situation the nodal forces and/or moments applied to the element, cause the element to rotate as a rigid body (either clockwise as shown below or counterclockwise depending on the direction of the applied forces and/or moments).
This rigid body mode can be suppressed by prescribing a nodal translation or rotation.
If translational and rotational rigid body motion occurs simultaneously then:
| Simple Examples of
Beam Problems with and without Rigid Body Motion |
||||
|---|---|---|---|---|
Case |
Stable/Unstable |
Rigid Body Mode(s) |
Determinant of Mesh Stiffness Matrix |
Equations |
 |
Unstable |  and  |
|
Dependent Equations |
 |
Unstable | |
|
Dependent Equations |
 |
Unstable | |
|
Dependent Equations |
 |
Stable | None |  |
Independent Equations |
 |
Stable | None | |
Independent Equations |