Page 463 - Mechanical Behavior of Materials

Section 9.8 Multiaxial Stresses 463

9.8 MULTIAXIAL STRESSES

In engineering components, cyclic loadings that cause complex states of stress are common. Some
examples are biaxial stresses due to cyclic pressure in tubes or pipes, combined bending and torsion
of shafts, and bending of sheets or plates about more than one axis. Steady applied loads that
cause mean stresses may also be combined with such cyclic loads. An additional complexity is
that different sources of cyclic loading may differ in phase or frequency or both. For example, if a
steady bending stress is applied to a thin-walled tube under cyclic pressure, there are different stress
amplitudes and mean stresses in two directions, as shown in Fig. 9.40. The axial and hoop directions
are the directions of principal stress and remain so as the pressure fluctuates.
If a steady torsion is instead applied, a more complex situation exists, as illustrated in Fig. 9.41.
At times when the pressure is momentarily at zero, the principal stress directions are controlled
◦
by the shear stress and are oriented 45 to the tube axis. However, for nonzero values of pressure,
these directions rotate to become more closely aligned with the axial and hoop directions, but never
reaching them, except for the limiting case where the stresses σ x and σ y due to pressure are large

M

σ p
1 r
σ
2
t

σ
p 1
pr/t
p
0
0 time time

σ
M 2 pr/(2t)

Mr/I
0 time 0 time

Figure 9.40 Combined cyclic pressure and steady bending of a thin-walled tube with
closed ends. The principal directions are constant.