Introduction

To a close approximation, when a beam bends and extends, each planar cross-section translates in each of the three Cartesian coordinate directions, and rotates about the y and z axes. The one motion that is excluded from those approximations is the twisting of the beam cross-section about the x axis, called φx. The primary reason for separating this one motion from the other five is that its inclusion substantially complicates the treatment of finite beam bending deflections. However, for small deflections, there is no interaction between the twisting motion and the other five motions. Thus by limiting the discussion of this chapter and Chapter 13 to the situation where the beam bending deflections, if present, are small, the following discussion of twisting deflections and torsional loadings can proceed without taking any notice at all of the extensional and bending deflections caused by axial and shearing forces and bending moments.

Recall that a bar is a beam that is loaded only in extension or torsion. Since only twisting deflections are to be discussed in this chapter and Chapter 13, here the terms beam and bar can be, and are, used interchangeably. Consider a bar with a noncircular cross-section. When the bar is twisted, the bar cross-section of arbitrary shape does not remain plane after twisting. On the contrary, the cross-section warps out of its original plane in apparently complicated ways.