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

604
PRESSuRE VESSELS
For convenience, let the ratio of the internal diameter to the wall thickness be denoted by
the parameter k:
k
d
= =
t
2a
b − a
(f)
Thus, the thin-walled circumferential stress can be expressed as
σ hoop =
and the thick-walled circumferential stress can be expressed as
pk i
2
σ =
θ
2
pi
( k + 2k
+ 2)
2( k + 1)
The percent error of the thin-walled average hoop stress relative to the thick-walled circumferential
stress is therefore
σhoop
− σθ
%error =
× 100% =
σ
θ
k
2
k + 2
× 100%
+ 2k
+ 2
(g)
Table 14.1 comparison of
Thin- and Thick-Walled
circumferential Stress
d
t
% error
1 60.0%
5 18.9%
10 9.8%
20 5.0%
50 2.0%
100 1.0%
Table 14.1 lists the percent error obtained from Equation (g) for a number of diameterto-thickness
ratios. This calculation shows that, when the internal diameter of a cylinder
is at least 20 times the wall thickness, the average circumferential stress calculated by
Equation (14.5) is approximately 5 percent less than the thick-walled stress value calculated
from Equation (14.24). Thus, Equations (14.4) and (14.5) are typically appropriate
for thin-walled pressure vessels when d/t ≥ 20. The thick-walled stress equations should
be used for cylinders when d/t < 20.
maximum Stresses for External pressure only
If the external pressure is p o and the internal pressure is zero (p i = 0), then Equations
(14.14) and (14.16) respectively reduce to
2
b po
σ r =−
b − a
2 2
⎛ a
1 −
⎝
⎜
r
2
2
⎞
⎠
⎟ (14.25)
and
2
2
b po
⎛ a ⎞
σ θ =− 1 +
2 2 2
b − a ⎝
⎜
r ⎠
⎟
(14.26)
The radial and circumferential normal stresses for this case are plotted in Figure 14.11b.
Note that, both σ r and σ θ are always compressive. The maximum radial stress occurs on
the outer surface of the cylinder. However, the maximum circumferential stress occurs
on the inner surface of the cylinder. The circumferential stress magnitude is always larger
than the radial stress magnitude.