Strain measures
The internal virtual work rate associated with axial stress is
where and are any material stress and strain measures associated with axial deformation at the point of the beam, since strains are assumed to be small. For this purpose we will use Green's strain so that
where , the square of the ratio of current configuration length to reference configuration length in the axial direction on the fiber. From Equation 1 and its equivalent in the reference configuration, we have
Again, neglecting all but first-order terms in and because of the slenderness assumption, this becomes
where
This simplification allows us to write the internal virtual work rate associated with axial stress as
where
with
and
Now the cross-sectional base vectors and are assumed to remain normal to the beam axis, so
Hence,
So we have
and
This defines the generalized strains associated with axial stretch. For torsional strain the internal virtual work rate is
where
and is the proportionality constant between shear strains and torsional strains (see Beam element formulation for details). For computational simplicity the form of the torsional strains is taken to be
where
and
This assumes a linear interpolation of the finite rotation vector, along the beam. Thus, the generalized strain measures for these beams are
-
axial strain,
- and
-
the beam curvature change measures, and
-
the torsional strain.
With these measures, the internal virtual work rate can be written