Page 150 - Marks Calculation for Machine Design

P1: Rakesh
14:16
January 4, 2005
Brown.cls
Brown˙C03
132
STRENGTH OF MACHINES
If the external pressure (p o ) is zero gage, meaning atmospheric, then the radial stress σ r
becomes that given in Eq. (3.6), and is always compressive, that is, negative.
p i r 2 r o 2
σ r = i 2 1 − (3.7)
2
r − r r
o i
The radial stress σ r distribution using Eq. (3.7) is shown in Fig. 3.5.
s r
r

FIGURE 3.5 Radial stress with p o = 0.

Note that the radial stress (σ r ) is a maximum at the inside radius and zero at the outside
radius. Also notice that there are no arrows on the lines displaying the distribution like there
were for the tangential stress distribution. This is because the radial stress is in the radial
direction, so the length of the plain lines on the distribution curve represent the magnitude
of the radial stress. (The arrows on the tangential stress distribution curve represented both
magnitude and direction.)

U.S. Customary SI/Metric
Example 4. Calculate the maximum tangen- Example 4. Calculate the maximum tangen-
tial stress (σ t ) and the maximum radial stress tial stress (σ t ) and the maximum radial stress
(σ r ) for a thick-walled cylinder, where (σ r ) for a thick-walled cylinder, where
p i = 450 psi p i = 3.15 MPa = 3,150,000 N/m 2
p o = 0 psi (atmospheric) p o = 0 psi (atmospheric)
r i = 2in r i = 5cm = 0.05 m
r o = 4in r o = 10 cm = 0.1 m
solution solution
Step 1. Note that both the tangential stress (σ t ) Step 1. Note that both the tangential stress (σ t )
and the radial stress (σ r ) are a maximum at the and the radial stress (σ r ) are a maximum at the
inside radius (r i ). So substitute the inside radius inside radius (r i ). So substitute the inside radius
in Eq. (3.5) to obtain the tangential stress as in Eq. (3.5) to obtain the tangential stress as
2 2
p i r 2 i r o p i r i 2 r o
σ t = 1 + σ t = 1 +
2
2
r − r i 2 r i r − r i 2 r i
o
o
and in Eq. (3.7) to obtain the radial stress as and in Eq. (3.7) to obtain the radial stress as
2 2
p i r i 2 r o p i r i 2 r o
σ r = 1 − σ r = 1 −
2
2
r − r i 2 r i r − r i 2 r i
o
o