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

The radial stress in the thick-walled cylinder is obtained by substituting the constants
in Equation (e) into Equation (d):
601
STRESSES IN THICk-wALLEd
CyLINdERS
σ =
r
a p
b
− b p
− a
2 2
ab( p − p )
−
2 2 2
( b − a ) r
2
i
2
o i o
2 2
(14.13)
Separating Equation (14.13) into portions pertaining to the internal pressure p i and the
external pressure p o yields
σ =
r
2
a pi
b − a
2 2
2 2
⎛ b ⎞ b po
a
1−
1
⎝
⎜ 2 2 2
r ⎠
⎟ − ⎛
−
b − a ⎝
⎜
r
2
2
⎞
⎠
⎟ (14.14)
The circumferential stress in the thick-walled cylinder is obtained by substituting the
constants in Equation (e) into Equation (a):
2 2 2 2
a pi − b po ab( pi − po)
σ θ =
+
2 2
2 2 2
(14.15)
b − a ( b − a ) r
Separating Equation (14.15) as we did Equation (14.13) gives
2
2 2
2
a pi
⎛ b ⎞ b po
a
σ = 1+
1
2 2 2 2 2 2
b − a ⎝
⎜
r ⎠
⎟ − ⎛ ⎞
θ +
b − a ⎝
⎜
r ⎠
⎟
(14.16)
The stress equations (14.13) through (14.16) are together known as the Lamé solution.
Equations (14.13) through (14.16) can also be stated in terms of (b/a) 2 by dividing
numerators and denominators by a 2 . From the resulting equation, we observe that the radial
and circumferential stresses depend on the ratio b/a of the outside radius b to the inside
radius a of the cylinder, rather than on its absolute size.
maximum Stresses when Internal and External pressures Act Together
The maximum stresses occur either on the inner surface or the outer surface of a thickwalled
cylinder.
If the internal pressure is greater than the external pressure, then the maximum radial
stress occurs on the inner surface at r = a:
( σ ) =− p
(14.17)
r
max
So does the maximum circumferential stress:
i
2 2 2
( b + a ) pi
− 2b po
( σ θ ) max =
(14.18)
2 2
b − a
The maximum circumferential stress will be tensile.
If the external pressure is greater than the internal pressure, then the maximum radial
stress occurs on the outer surface at r = b:
( σ ) =− p
(14.19)
r
max
The maximum circumferential stress still occurs on the inner surface at r = a; however, the
circumferential stress will be compressive rather than tensile.
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