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

14.6 Deformations in Thick-Walled cylinders
Radial and circumferential deformations δ r and δ θ are important considerations in
interference-fit connections such as the attachment of a gear to a shaft. When a thick-walled
cylinder is subjected to internal pressure p i or external pressure p o , the circumference of the
thin ring shown in Figure 14.9a will either elongate or contract. This deformation may be
expressed in terms of the radial displacement δ r of a point on the ring as
δ
θ
=
2πδ
r
The circumferential deformation δ θ may also be expressed in terms of the circumferential
strain ε θ as
δ = ε c
where c = 2πr is the circumference of the thin ring. Thus,
θ
δr
θ
= θ
For most interference-fit connections, the cylinder is open, which means that the longitudinal
stress in the cylinder is σ long = 0. As a result, a state of plane stress exists in the cylinder
wall. Using the generalized Hooke’s law for plane stress, we can express the circumferential
strain in terms of the radial stress σ r and the circumferential stress σ θ :
ε r
1
εθ
= ( σθ
−νσ r )
E
The radial displacement of a point in the cylinder wall is then obtained in terms of the radial
and tangential stresses at that same point:
Radial Displacement for Internal pressure only
r
δr
=
E ( σθ −νσ
r )
(14.28)
If the thick-walled cylinder is loaded only by an internal pressure, Equations (14.23) and
(14.24) can be substituted into Equation (14.28) to obtain the following expression for the
radial deformation:
2
a pi
2 2
δr
=
[(1 − ν) r + (1 + ν) b ]
(14.29)
2 2
( b − a ) rE
Radial Displacement for External pressure only
If only an external pressure acts on a thick-walled cylinder, Equations (14.25) and (14.26) can be
substituted into Equation (14.28) to obtain the following expression for the radial deformation:
2
b po
2 2
δr
=−
[(1 − ν) r + (1 + ν) a ]
2 2
(14.30)
( b − a ) rE
Radial Displacement for External pressure on Solid circular cylinder
For a solid circular cylinder loaded by an external pressure, the radial and circumferential
stresses equal the external pressure and both stresses are compressive. Thus, Equation
(14.28) yields the following expression for the radial deformation:
(1 − ν)
pr o
δ r =−
E
(14.31)
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