Page 146 - Marks Calculation for Machine Design

P1: Rakesh
January 4, 2005
14:16
Brown.cls
Brown˙C03
STRENGTH OF MACHINES
128
3.2.1 Thin-Walled Vessels
Thin-walled vessels are typically either spherical or cylindrical. Other geometries are pos-
sible, but their complexity precludes their inclusion in this book. Pressure vessels can be
considered thin if the diameter is greater than ten times the thickness of the wall.
Spheres. For the thin-walled spherical pressure vessel shown in Fig. 3.1, the normal stress
(σ sph ) in the wall of the sphere is given by Eq. (3.1),
p i r m
σ sph = (3.1)
2t
where p i = internal gage pressure (meaning above atmospheric pressure)
r m = mean radius (can be assumed to be the inside radius of the sphere)
t = wall thickness

r m s
sph
p
i
s sph s sph
s sph
Geometry t Stress
FIGURE 3.1 Spherical pressure vessel.

Caution. External pressure on any thin-walled vessel causes buckling of the vessel wall
long before excessive stress is reached. The study of the buckling of thin-walled vessels is
very complex, and is beyond the scope of this book.

U.S. Customary SI/Metric
Example 1. Determine the normal stress Example 1. Determine the normal stress
(σ sph ) in a thin-walled spherical vessel, where (σ sph ) in a thin-walled spherical vessel, where
p i = 200 psi p i = 1.4 MPa = 1,400,000 N/m 2
r m = 3ft = 36 in r m = 1m
t = 0.25 in t = 0.6 cm = 0.006 m
solution solution
Step 1. Using Eq. (3.1), calculate the normal Step 1. Using Eq. (3.1), calculate the normal
stress (σ sph ) as stress (σ sph ) as
2
2
p i r m (200 lb/in )(36 in) p i r m (1,400,000 N/m )(1m)
σ sph = = σ sph = =
2 t 2 (0.25 in) 2 t 2 (0.006 m)
7,200 lb/in 1,400,000 N/m
= =
0.5in 0.012 m
2
8
2
= 14,400 lb/in = 14.4 kpsi = 1.167 × 10 N/m = 116.7MPa