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

Similarly, the change in radial stresses from the inner surface to the outer surface
always equals the difference between the external and internal pressures:
603
STRESSES IN THICk-wALLEd
CyLINdERS
σ
ri
2 2 2 2
( b a ) pi
( b a ) p
− σro
= − − + −
2 2
b − a
o
= po
− pi
(14.22)
maximum Stresses for Internal pressure only
If the internal pressure is p i and the external pressure is zero (p o = 0), as in most hydraulic
pipes, tanks, and other machinery, then Equations (14.14) and (14.16) respectively reduce to
and
2
a pi
σ r =
2 2
b − a
⎛ b
1 −
⎝
⎜
r
2
2
⎞
⎠
⎟ (14.23)
2
2
a pi
⎛ b ⎞
σ θ = 1 +
2 2 2
b − a ⎝
⎜
r ⎠
⎟
(14.24)
The radial and circumferential normal stresses for this case are plotted in Figure 14.11a.
The maximum radial stress occurs at r = a. Since b 2 /r 2 ≥ 1, the radial stress σ r is negative
(i.e., compressive) for all values of r except r = b, in which case the radial stress is zero. The
circumferential stress σ θ is positive (i.e., tensile) for all radii. Its maximum value also
occurs at r = a.
Comparison of circumferential stresses in thick- and thin-walled cylinders subjected
to internal pressure only: The average circumferential stress (i.e., hoop stress) in a thinwalled
cylinder is given by Equation (14.5):
σ hoop =
pd
2t
p o
p o
0
–25
–50
–75
–100
Radial
normal stress
–
(b 2 + a 2 ) p 2a 2 p i
i
b 2 – a 2 b 2 – a 2
b
125
100
75
50
a
+
25
p 0
p i
i Circumferential
normal stress
–p i
b
Radial
a
Circumferential
normal stress
normal stress
0
0
–25
–50 –
–75
–
–100
–p o
p o
2b 2 p
– o
b 2 – a 2
–
(b 2 + a 2 ) p o
b 2 – a 2
–25
–50
–75
–100
–125
–150
–175
–200
–225
(a) Internal pressure only.
(b) External pressure only.
FIGURE 14.11 Radial and circumferential normal stresses in thick-walled cylinders subjected to internal and external
pressure.