Page 285 - Introduction to Continuum Mechanics

The Elastic Solid 269

Example 5.14.1
For an elliptic cylindrical bar in torsion, (a) find the magnitude of the maximum normal and
shearing stress at any point of the bar, and (b) find the ratio of the maximum shearing stresses
at the extremities of the elliptic minor and major axes.
Solution. As in Example 5.13.1, we first solve the characteristic equation

The principal values are

which determines the maximum normal and shearing stresses:

(b) Supposing that b >a, we have at the end of the minor axis (x% - a, *3 = 0),

and at the end of major axis fa = 0, x$ = b )

The ratio of the maximum stresses is therefore b/a and the greater stress occurs at the end of
the minor axis.

5.15 Pure Bending of a Beam

A beam is a bar acted on by forces or couples in an axial plane, which chiefly cause bending
of the bar. When a beam or portion of a beam is acted on by end couples only, it is said to be
in pure bending or simple bending. We shall consider the case of cylindrical bar of arbitrary
cross-section that is in pure bending.
Figure 5.13 shows a bar of uniform cross-section. We choose the*i axis to pass through the
cross-sectional centroids and let jcj = 0 andjcj = / correspond to the left- and right-hand faces
of the bar.