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

From this equilibrium equation, an expression for the tangential normal stress in the sphere
wall can be derived in terms of the inside radius r or the inside diameter d:
587
THIN-wALLEd SPHERICAL
PRESSuRE VESSELS
pr pd
σ t = =
(14.1)
2t
4t
Here, t is the wall thickness of the vessel.
By symmetry, a pressurized sphere is subjected to uniform tangential normal stresses σ t
in all directions.
Stresses on the outer Surface
Commonly, pressures specified for a vessel are gage pressures, meaning that the pressure
is measured with respect to atmospheric pressure. If a vessel at atmospheric pressure is
subjected to a specified internal gage pressure, then the external pressure on the vessel is
taken as zero while the internal pressure is equal to the gage pressure. Internal pressure in
a spherical pressure vessel creates normal stress σ t that acts in the circumferential direction
of the shell. Since atmospheric pressure (i.e., zero gage pressure) exists on the outside of
the sphere, the normal stress in the radial direction will be zero.
Pressure in the sphere creates no shear stress; therefore, the principal stresses are
σ p1 = σ p2 = σ t . Furthermore, no shear stress exists on free surfaces of the sphere, which
means that any normal stress in the radial direction (perpendicular to the sphere wall) is
also a principal stress. Since pressure outside the sphere is zero (assuming that the sphere
is surrounded by 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. Consequently,
the outer surface of the sphere (Figure 14.2) is in a condition of plane stress, which
is also termed biaxial stress here.
Mohr’s circle for the outer surface of a spherical pressure vessel (subjected to an internal
gage pressure) is shown in Figure 14.3. Mohr’s circle describing stresses in the plane
of the sphere wall is a single point. Therefore, the maximum shear stress in the plane of the
sphere wall is zero. The maximum out-of-plane shear stress is
1
τ = σ − σ =
⎛ pr
−
⎞
⎝ t ⎠ = pr
2 ( ) 1
t
0
2 2 4t
absmax radial (14.2)
Stresses on the Inner Surface
The stress σ t on the inner surface of the spherical pressure vessel is the same as σ t on the
outer surface because the vessel is thin walled (Figure 14.2). Pressure exists inside the
σ t = pr
2t
σ t = pr
2t
σ radial = 0 σ radial = p
Outer surface
Inner surface
stress state
stress state
FIGURE 14.2 Stress elements on the outer and inner surfaces of a spherical pressure vessel.
σ t = pr
2t
σ t = pr
2t