Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

τ
Stress
xy
τ
nt
σ
n
141
STRESSES ON ObLIQuE PLANES
Angle
0° 45° 90° 135° 180°
τ
– xy
FIGURE 6.9 Variation of normal and shear stresses with angle θ on the surface of a shaft.
Any of the stresses discussed in the preceding paragraph may be significant in a particular
problem. Compare, for example, the failures shown in Figure 6.10. In Figure 6.10a,
the steel axle of a truck split longitudinally. One would expect this type of failure to occur
also in a shaft of wood with the grain running longitudinally. In Figure 6.10b, compressive
stress caused the thin-walled aluminum alloy tube to buckle along one 45° plane while tensile
stress caused tearing on the other 45° plane. Buckling of thin-walled tubes (and other
shapes) subjected to torsional loading is a matter of great importance to the designer. In
Figure 6.10c, tensile normal stresses caused the gray cast iron shaft to fail in tension—a
failure typical of any brittle material subjected to torsion. In Figure 6.10d, the low-carbon
steel failed in shear on a plane that is almost transverse—a typical failure for ductile material.
The reason the fracture in Figure 6.10d did not occur on a transverse plane is that, under
the large plastic twisting deformation before rupture (note the spiral lines indicating elements
originally parallel to the axis of the bar), longitudinal elements were subjected to
axial tensile loading. This axial loading was induced because the testing machine grips
would not permit the torsion specimen to shorten as the elements were twisted into spirals.
This axial tensile stress (not shown in Figure 6.8) changes the plane of maximum shear
stress from a transverse to an oblique plane (resulting in a warped surface of rupture). 3
Buckling is a stability failure.
The phenomenon of stability
failure is discussed in Chapter 16.
(a)
(b)
(c)
(d)
FIGURE 6.10 Photos of actual shaft failures.
3
The tensile stress is not entirely due to the grips, because the plastic deformation of the outer elements of the bar
is considerably greater than that of the inner elements. This difference in deformation results in a spiral tensile
stress in the outer elements and a similar compressive stress in the inner elements.