ABSTRACT
The general problem of a shaft under torque is obviously of great practical importance. The linear solution, as presented in a first course in solid mechanics, assumes that for a circular cross-section, radical lines remain straight when the torque is applied. Thus, as shown in Figure 9.1, two neighboring cross-sections,
dz
apart rotate relative to each other by a small angle of twist
d
. Thus, longitudinal straight lines become helixes at the angle of shear strain,
, related to the angle of twist by the circumferential movement:
(9.1)
where the rate of twist or “unit angle of twist,” , is a constant. Thus, for an elastic material
ds dz rd and r d dz ------ r
(9.2)
and from equilibrium
(9.3)
where
I
r
dA
is the polar moment of inertia (the first invariant of the
moment-of-inertia tensor). Thus
(9.4)
with all other stress and strain components zero.*
The approach taken by St. Venant to the torsion problem provides a neat demonstration of the semi-inverse method of solving the elasticity equations for the displacement vector field. Although results in closed form can only be found for a few simple shapes, the St. Venant formulation, because it deals with displacements, is intuitive and gives physical insight into the complications introduced by nonconstant axial displacements required to satisfy the boundary conditions for noncircular cross-sections.**
That longitudinal cross-sections, if not circular, deform or “warp” is shown in Figure 9.2. Assuming the rod is cylindical in the
z
direction (and not tapered) and that projections of the cross-section on the
xy
plane rotate rigidly at a constant rate of twist,
, St. Venant, pioneering the inverse method, postulated the displacement field (shown in Figure 9.3):
(9.5)
z G Gr
Mt r Ad r 2G A Gd r2 A GIpd
Mt GIp -------- and z z
Mtr Ip
---------
u zy, v zx, w w x,y( )
as the solution (the answer). Thus the warp of the cross-section,
w
, is left as an unknown function (i.e., the warping function) to be adjusted to satisfy the boundary condition dictated by the shape. In terms of strains, he therefore postulated that:
(9.6)
with the elastic rotation
(9.7)
being the total angle of twist of the cross-section. Geometric compatibility is automatically satisfied because this formulation assumes continuous and smooth displacements.
