Page 148 - Marks Calculation for Machine Design

P1: Rakesh
January 4, 2005
14:16
Brown.cls
Brown˙C03
STRENGTH OF MACHINES
130
where p i = internal gage pressure (meaning above atmospheric pressure)
r m = mean radius (can be assumed to be the inside radius of cylinder)
t = wall thickness
Notice that the hoop stress (σ hoop ) is twice the axial stress (σ axial ). This is why metal stress
rings, or hoops, are seen in cylindrical pressure vessels constructed of low-strength materials
such as fiberglass. Fiberglass is chosen because of the corrosive effects of certain liquids
and gases, and the metal hoops provide the strength not present in the fiberglass. Also notice
that the axial stress (σ axial ) in a thin-walled cylinder is the same as the stress (σ sph ) in a
thin-walled sphere. This means that there will be no discontinuity at the welded seams of a
cylindrical pressure vessel with spherical end caps.
U.S. Customary SI/Metric
Example 3. Calculate the axial stress (σ axial ) Example 3. Calculate the axial stress (σ axial )
and hoop stress (σ hoop ) for a thin-walled cylin- and hoop stress (σ hoop ) for a thin-walled cylin-
drical pressure vessel, where drical pressure vessel, where
p i = 300 psi p i = 2.1 MPa = 2,100,000 N/m 2
r m = 2.5 ft = 30 in r m = 0.8 m
t = 0.375 in t = 1cm = 0.01 m
solution solution
Step 1. Using Eq. (3.2), calculate the axial Step 1. Using Eq. (3.2), calculate the stress
stress (σ axial ) as (σ axial ) as
2
2
p i r m (300 lb/in )(30 in) p i r m (2,100,000 N/m )(.8m)
σ axial = = σ axial = =
2 t 2 (0.375 in) 2 t 2 (0.01 m)
9,000 lb/in 1,680,000 N/m
= =
0.75 in 0.02 m
2
7
2
= 12,000 lb/in = 12 kpsi = 8.4 × 10 N/m = 84 MPa
Step 2. Using Eq. (3.3), calculate the hoop Step 2. Using Eq. (3.3), calculate the hoop
stress (σ hoop ) as stress (σ hoop ) as
2
2
p i r m (300 lb/in )(30 in) p i r m (2,100,000 N/m )(.8m)
σ hoop = = σ hoop = =
t (0.375 in) t (0.01 m)
9,000 lb/in 1,680,000 N/m
= =
0.375 0.01 m
8
2
2
= 24,000 lb/in = 24 kpsi = 1.68 × 10 N/m = 168 MPa
3.2.2 Thick-Walled Cylinders
Thick-walled cylinders have application in all sorts of machine elements and will be the basis
for the presentation in Sec. 3.1.3 on shrink or press fits. Typically a cylinder is considered
thick if the diameter is less than ten times the wall thickness.
Geometry. The geometry of a thick-walled cylinder is shown in Fig. 3.3.
There is an internal pressure (p i ) associated with the inside radius (r i ), and an external
pressure (p o ) associated with the outside radius (r o ). Unlike thin-walled vessels, thick-
walled cylinders do not tend to buckle under excessive external pressure, but merely crush.