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

610
PRESSuRE VESSELS
could be cooled enough so that it fits inside of the jacket.) After the pieces are assembled
and the temperature of both parts returns to the ambient temperature, the pressure between
the two parts must be the same on the interface between the jacket and the tube. This pressure
is termed the contact pressure p c . The contact pressure creates stresses and deformations
in both the jacket and the tube.
The determination of stresses in a compound cylinder requires the solution of a statically
indeterminate problem that considers the radial interference δ and the radial deformations
of the jacket δ J (i.e., the inner cylinder) and the tube δ T (i.e., the outer cylinder). The
radial interference δ is related to radial deformations of the tube and jacket at the mating
surface by the equation
δ = δ + δ
(a)
T
After assembly, the inner cylinder is simply a thick-walled cylinder subjected to external
pressure only, where the external pressure is the contact pressure p c . For the tube, the
inner radius will be denoted a and the outer radius will be denoted b. The magnitude of the
radial deformation at the outer surface of the tube, d T , is expressed by Equation (14.30),
where we let r = b:
J
bpc
2 2
δT
=
[(1 − ν) b + (1 + ν) a ]
2 2
(b)
( b − a ) E
After assembly, the outer cylinder is a thick-walled cylinder subjected to internal pressure
only, where the internal pressure is the contact pressure p c . The magnitude of the radial
deformation at the inner surface of the jacket, d J , is expressed by Equation (14.29). We
will modify that equation to reflect the geometry shown in Figure 14.13, denoting the inner
radius of the jacket as b and the outer radius as c, and setting r = b. Accordingly,
bpc
2 2
δJ
= [(1 − ν) b + (1 + ν) c ]
2 2
(c)
( c − b ) E
Substituting Equations (b) and (c) into Equation (a) yields
δ =
3 2 2
2 b p ( c − a )
c
2 2 2 2
Ec ( − b )( b − a )
(14.32)
It is convenient to solve Equation (14.32) for the contact pressure p c :
p
c
=
2 2 2 2
Eδ
( c − b )( b − a )
3 2 2
2 b ( c − a )
(14.33)
In deriving Equations (14.32) and (14.33), we have assumed that both the tube and the
jacket are made of the same material, and hence, they have the same values for E and ν. An
expression for the contact pressure between cylinders of different materials will be presented
subsequently.
Equation (14.32) provides a means for establishing the radial interference that can be
tolerated without exceeding maximum allowable stress levels in the jacket or the tube.
Equation (14.33) offers the means to determine the contact pressure between the tube and
the jacket for a known interference. Once the contact pressure has been established, radial
and circumferential stresses in both the tube and jacket can be determined.