BOUNDARY CONDITIONS FOR TORSION

Torsion is one of the most sophisticated problems in structures. Here, the focus is on uniform torsion. Note that this assumption is only accurate for circular cross sections and in this section boundary conditions for uniform torsion are of interest. The following equation shows the free vibration of uniform torsion:

In this equation GJ is the torsional stiffness, θ is the twist angle, and ρI_p is polar mass moment of inertia about axis of beam (x axis).

If GJ is constant along the beam, this equation can be simplified as:

which is in the form of 1-D wave equation with c² = (GJ)/(ρI_p).

This equation needs two boundary conditions and two initial conditions.

One can write twisting moment (torsion) as

Boundary conditions on the twist angle is called geometric boundary condition and boundary conditions on the twisting moment is called natural boundary condition. This classification will become useful in the derivation of approximate solutions.

Notice that at each boundary either twisting angle or twisting moment can be prescribed, not both at the same end.

For further illustration of the boundary conditions, please refer to the Examples section.