The boundary conditions for coupled bending and torsion range from very simple to somewhat complex, depending on the type of restraint(s) imposed on the ends. For example, for a clamped end, we have v = dv/dx = в = 0, the same as for uncoupled bending and torsion. Similarly, a free end has zero bending moment, shear force, and twisting moment, respectively written as M = V = T = 0. Note the definitions of M and T in Eqs. (2.58) and that V = – BM/дx. Equations governing other restraints may be determined by appropriate kinematical or physical relationships. For example, a pinned connection may imply specification of an axis about which the moment vector (i. e., combination of bending and twisting moments) vanishes and perpendicular to which components of the rotation vector (i. e., combination of bending and twisting rotations) vanish. Relationships for both elastic and inertial restraints may be developed using Euler’s laws, as in the uncoupled cases herein.
The complexity of this class of problem provides excellent motivation for the introduction of approximate methods, which is undertaken in the next section.