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

586
PRESSuRE VESSELS
The wall comprising a pressure
vessel is sometimes termed
the shell.
Thin-walled pressure vessels are classified as shell structures. Shell structures derive
a large measure of their strength from the shape of the structure itself. They can be defined
as curved structures that support loads or pressures through stresses developed in two or
more directions in the plane of the shell.
Problems involving thin-walled vessels subject to fluid pressure p are readily solved
with free-body diagrams of vessel sections and the fluid contained therein. Thin-walled
spherical and cylindrical pressure vessels are considered in the sections that follow.
14.2 Thin-Walled Spherical pressure Vessels
A typical thin-walled spherical pressure vessel is shown in Figure 14.1a. If the weights of
the gas and vessel are negligible (a common situation), symmetry of loading and geometry
requires that stresses be equal on sections that pass through the center of the sphere. Thus,
on the small element shown in Figure 14.1a, σ x = σ y = σ t . Furthermore, there are no shear
stresses on any of these planes, since there are no loads to induce them. The normal stress
component in a sphere is referred to as tangential stress and is commonly denoted σ t .
The free-body diagram shown in Figure 14.1b can be used to evaluate the tangential
stress σ t in terms of the pressure p, the inside radius r, and the wall thickness t of the
spherical vessel. The sphere is cut on a plane that passes through the center of the sphere to
expose a hemisphere and the fluid contained within. The fluid pressure p acts horizontally
against the plane circular area of the fluid contained in the hemisphere. The resultant force P
from the internal pressure is the product of the fluid pressure p and the internal crosssectional
area of the sphere; that is,
P = pπ
r 2
where r is the inside radius of the sphere.
Because the fluid pressure and the sphere wall are symmetrical about the x axis, the
tangential normal stress σ t produced in the wall is uniform around the circumference. Since
the vessel is thin walled, σ t is assumed to be uniformly distributed across the wall thickness.
For a thin-walled vessel, the exposed area of the sphere wall can be approximated by
the product of the inner circumference (2πr) and the wall thickness t of the sphere. The
resultant force R from the internal stresses in the sphere wall can therefore be expressed as
R = σ (2 πrt)
From a summation of forces in the x direction,
t
2
∑ F = R − P = σ (2 πrt) − pπ
r = 0
x
y
t
σ t
z
x
σ t
P = pπr 2
σ x
σ y
(a) Typical sphere
FIGURE 14.1 Spherical pressure vessel.
(b) Free-body diagram exposing σ t