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

590
PRESSuRE VESSELS
Stresses on the outer Surface
Pressure in a cylindrical pressure vessel creates stresses in the longitudinal direction
and in the circumferential direction. If atmospheric pressure (i.e., zero gage pressure)
exists outside the cylinder, then no stress will act on the cylinder wall in the radial
direction.
Since pressure in the vessel creates no shear stress on longitudinal or circumferential
planes, the longitudinal and hoop stresses are principal stresses: σ p1 = σ hoop and σ p2 = σ long .
Furthermore, since no shear stress exists on free surfaces of the cylinder, any normal stress
in the radial direction (perpendicular to the cylinder wall) is also a principal stress. Because
pressure outside the cylinder is zero (assuming atmospheric pressure), the normal stress in
the radial direction due to external pressure is zero. Therefore, the third principal stress is
σ p3 = σ radial = 0. The outer surface of the cylinder (Figure 14.6) is thus said to be in a state
of plane stress, which can be termed biaxial stress.
Mohr’s circle for the outer surface of a cylindrical pressure vessel (with internal
pressure) is shown in Figure 14.7. The maximum in-plane shear stresses (i.e., stresses in
the plane of the cylinder wall) occur on planes that are rotated at 45° with respect to the
circumferential and longitudinal directions. From Mohr’s circle, the magnitude of these
shear stresses is
1
pr pr pr
2 ( ) 1
τmax = σhoop − σ long =
⎛
−
⎞
2⎝
t 2t
⎠ = (14.6)
4t
σ radial = 0
σ hoop =
pr
t
σ long =
pr
2t
σ radial = p
σ hoop =
pr
t
pr
σ long =
2t
Outer surface
stress state
Inner surface
stress state
FIGURE 14.6 Stress elements on the outer and inner surfaces of a cylindrical pressure vessel.
τ
pr
τ abs max =
2t
Out-of-plane
Mohr’s circles
pr
R =
2t
In-plane
Mohr’s circle
σ radial = 0
pr
σ long =
2t
σ hoop =
σ
pr
t
FIGURE 14.7 Mohr’s circle for outer surface of a cylinder.