Page 166 - Marks Calculation for Machine Design

P1: Rakesh
January 4, 2005
14:16
Brown.cls
Brown˙C03
148
STRENGTH OF MACHINES
Stress element
r i
r o
t
FIGURE 3.13 Thin rotating disk.
where r o = outside radius
r i = inside radius
t = thickness of disk
For the disk to be treated as thin, the outside radius (r o ) should be at least 25 times greater
than the thickness (t). Also, it is assumed that the disk is a constant thickness (t) and the
inside radius (r i ) is very small compared to the outside radius (r o ).
The rotational loading develops both a tangential stress (σ t ) and a radial stress (σ r ).
These two stresses form a biaxial stress element, shown in Fig. 3.14.
As this is a biaxial stress element, the shear stress (τ xy ) is zero; however, there will be
a maximum shear stress (τ max ) that is determined either mathematically or using Mohr’s
circle.
The tangential stress (σ t ) is given in Eq. (3.41),
2 2
3 + ν r i r i 1 + 3ν r 2
σ t = σ o 1 + 2 + 2 − 2 (3.41)
8 r o r 3 + ν r o
and the radial stress (σ r ) is given in Eq. (3.42).

3 + ν r i 2 r i 2 r 2
σ r = σ o 1 + − − (3.42)
8 r 2 r 2 r 2
o o
where (ν) is Poisson’s ratio and using the quantity labeled (σ o ), which has units of stress,
makes these two equations more compact and mathematically manageable, where
2 2
σ o = ρω r (3.43)
o
s yy s r
t xy
0
t xy
s xx
¨
s xx s t s t
t xy
0
t xy
s yy s r

FIGURE 3.14 Biaxial stress element.