Degrees Of Freedom With No Local Stiffness

It is possible to construct a model and find that there is no stiffness associated with particular degrees of freedom, either for translation or rotation. For example a model made up of shell elements in a general plane 6 degrees of freedom will be assigned per node, but there is only stiffness in 5 of these. There are a number of approaches to avoid this problem.

Geometry based automatic constraints

At each node in the structure the attached elements are identified. A pseudo stiffness matrix associated with rotations is set up with a value of one on the diagonal if the element is stiff in that direction or zero if there is no stiffness. All off-diagonal terms are set to zero. The pseudo stiffness is transformed into the nodal axis system (so the off-diagonal terms are, in general, no longer zero) and added to a nodal pseudo stiffness matrix.

Once this has been done for all the attached elements an eigenvalue analysis of the resulting pseudo stiffness is carried out to reveal the principal pseudo stiffnesses and their directions. If any of the principal pseudo stiffnesses are less that the pre-set “flatness tolerance” then those degrees of freedom are removed from the solution and an appropriate rotation to apply to the stiffness matrix at the node is stored.

Stiffness based automatic constraints

This is similar to the geometry based automatic constraints but instead of a value of one or zero assigned to degrees of freedom the actual stiffness matrix is used. The resulting stiffness matrix is the same as would result from restraining the whole model except from the rotations at the node of interest.

Again an eigenvalue analysis is carried out to reveal the principal stiffnesses and their directions. If any of the principal stiffnesses are less that the pre-set “stiffness tolerance” then those degrees of freedom are removed from the solution and an appropriate rotation to apply to the stiffness matrix at the node is stored.

Artificial stiffness in shells

An alternative and cruder approach is to make sure that there is some stiffness in all directions by applying an artificial stiffness in the directions that are not stiff. This is done by constructing the element stiffness matrix for shell elements and then replacing the zeros on the leading diagonal with a value of 1/1000th of the minimum non-zero stiffness on the diagonal.

Since this approach introduces an artificial stiffness term that has not physical basis it should be used with care.