Page 282 - Introduction to Continuum Mechanics

266 Torsion of a Noncircular Cylinder

Fig. 5.12

5.14 Torsion of a Noncircular Cylinder

For cross-sections other than circular, the simple displacement field of Section 5.13 will not
satisfy the tractionless lateral surface boundary condition (see Example 5.13.4). We will show
that in order to satisfy this boundary condition, the cross-sections will not remain plane.
We begin by assuming a displacement field that still rotates each cross-section by a small
angle 0, but in addition there may be a displacement in the axial direction. This warping of
the cross-sectional plane will be defined by u\ - (pfa, x$). Our displacement field now has
the form

The associated nonzero strains and stresses are given by

The second and third equilibrium equations are still satisfied if 0' = constant. However,
the first equilibrium equation requires that

Therefore, the displacement field of Eq. (5.14.1) will generate a possible state of stress if <p
satisfies Eq. (5.14.3). Now, we compute the traction on the lateral surface. Since the bar is