A JOINTC element consists of two nodes, referred to here as nodes 1 and 2. Each node has six degrees of freedom: displacements and rotations . A local orientation is defined for the element by the user. In a large-displacement analysis that local system rotates with the first node of the element.

We define the local system by its unit, orthogonal base vectors, , for . Then at any time in the analysis where is the rotation matrix defined by the rotation at the first node of the element.

The relative displacements in the element are then

with first variations where is a linearized rotation field (see “Rotation variables,” Section 1.3.1), and second variations

The relative rotation about the local -axis is defined as

with and defined by cyclic permutation of the local direction indices.

These rotation measures define only relative angular rotations for small relative rotations. They are simple to compute, increase monotonically for relative rotations up to 180°, and are taken as suitable for use in the elements for these reasons.

The first variation of is

and its second variation is

The relative translational velocities in the element are taken as

and the relative angular velocity about the local -axis is taken as

The virtual work contribution of the element is

We assume that the behavior of the joint is defined by

The contribution to the operator matrix for the Newton solution is

where is defined by the dynamic time integration operator.