CHAPTER 10
THICK CYLINDERS
Summary
The hoop and radial stresses at any point in the wall cross-section of a thick cylinder at
radius r are given by the Lam6 equations:
B
hoop stress OH = A + -
r2
B
radial stress cr, = A - -
r2
With internal and external pressures P , and P, and internal and external radii R, and R,
respectively, the longitudinal stress in a cylinder with closed ends is
P1R: - P2R:
aL =
= Lame constant A
(R: - R:)
Changes in dimensions of the cylinder may then be determined from the following strain
formulae:
circumferential or hoop strain = diametral strain
'JH
c r
O L
v- - v-
=--
E
E
E
O L
or
OH
longitudinal strain = - - v- - v-
E
E
E
For compound tubes the resultant hoop stress is the algebraic sum of the hoop stresses
resulting from shrinkage and the hoop stresses resulting from internal and external pressures.
For force and shrink fits of cylinders made of diferent materials, the total interference or
shrinkage allowance (on radius) is
CEH, - 'Hi 1
where E", and cH, are the hoop strains existing in the outer and inner cylinders respectively
at the common radius r. For cylinders of the same material this equation reduces to
For a hub or sleeve shrunk on a solid shaft the shaft is subjected to constant hoop and radial
stresses, each equal to the pressure set up at the junction. The hub or sleeve is then treated as a
thick cylinder subjected to this internal pressure.
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Wire-wound thick cylinders
If the internal and external radii of the cylinder are R, and R, respectively and it is wound
with wire until its external radius becomes R,, the radial and hoop stresses in the wire at any
radius r between the radii R, and R3 are found from:
radial stress = ( -27i-)
r - R: Tlog, (-) Ri - R:
r2 - Rt
r2 + R:
R; - R:
hoop stress = T { 1 - ( -
2r2 )'Oge(r2-Rf)}
where T is the constant tension stress in the wire.
The hoop and radial stresses in the cylinder can then be determined by considering the
cylinder t