Shear center, tension axis and cross-sectional stiffness/flexibility

Most often a genreral beam theory requires a 6×6 cross-sectional stiffness/flexibility (s/f). Though it is great to be able to give this completely-general, coupled s/f matrix, it is sometimes difficult to figure out how to specify the s/f matrix if one has a simple (isotropic/quasi-isotropic) cross-section. This post tries to address this issue. Consider a cross-section as shown below.
Figure A: Beam cross-section showing the shear center (s) and the tension center (t)

For a simple cross-section, one can always find two points on the cross-section, viz., shear center (s) and the tension center (t). By defining the shear forces and torque at the shear center while defining the axial force and the bending moments at the tension center, we have the relation between the beam generalized forces and the corresponding beam generalized strains as:



The equations can be written as:

and, the corresponding inverse relation:

Figure B: Forces and moments at the beam reference line

Now to solve the beam equations one needs to transform all the variables to the beam reference axis. The transformations are:

where:


The above constitutive law can now be written for the beam reference axis as: