Page 149 - Marks Calculation for Machine Design

P1: Rakesh
14:16
January 4, 2005
Brown.cls
Brown˙C03
ADVANCED LOADINGS
p 131
o
p
i r i
r o
FIGURE 3.3 Geometry of a thick-walled cylinder.
The major difference between the stresses in a thin-walled cylinder and a thick-walled
cylinder is that the hoop stress, also called the tangential stress, for the thick-walled cylinder
varies in the radial direction, and there is a radial stress across the thickness of the cylinder
that also varies radially.

Tangential Stress. For the geometry and pressures shown in Fig. 3.3, the tangential (hoop)
stress (σ t ) is given by Eq. (3.4).
2 2
2
2

p i r − p o r + (p i − p o ) r r /r 2
i o
o
i
σ t = 2 (3.4)
2
r − r i
o
If the external pressure (p o ) is zero gage, meaning atmospheric, then the tangential stress
(σ t ) becomes that given in Eq. (3.5)
p i r i 2 r o 2
σ t = 2 1 + (3.5)
2
r − r r
o i
The tangential stress (σ t ) distribution using Eq. (3.5) is shown in Fig. 3.4.
s t
r

FIGURE 3.4 Tangential stress with p o = 0.

Note that the tangential stress (σ t ) is maximum at the inside radius and a lower value at
the outside radius. Also, if the outside radius is twice the inside radius, by a few algebra
steps it can be shown that the tangential stress at the inside radius is two and a half times
greater than the tangential stress at the outside radius.

Radial Stress. For the geometry and pressures shown in Fig. 3.3, the radial stress (σ r ) is
given by Eq. (3.6).
2
2 2
2

p i r − p o r − (p i − p o ) r r /r 2
o
o
σ r = i 2 i (3.6)
2
r − r
o i