Page 197 - Marks Calculation for Machine Design

P1: Shibu
January 4, 2005
14:25
Brown.cls
Brown˙C04
COMBINED LOADINGS
179
and the radial stress (σ r ) given in Eq. (4.25) becomes a minimum value (σ
r min ), and as stated
earlier is equal to zero, with the algebraic steps shown in Eq. (4.29).
pR 2
2
pR 2
min r o
σ r = 1 − = [1 − 1] = 0 (4.29)
2
2
r − R 2 r − R 2
o r o o

simplify bracket terms
Eq. (1.91) with r=r o
Stress Elements. The general stress element shown in Fig. 4.2 becomes the stress element
shown in Fig. 4.24, where the normal stress (σ xx ) is the tangential stress (σ t ) given by
Eq. (4.24) due to the interface pressure (P), the normal stress (σ yy ) is zero, and the shear
stress (τ xy ) is the shear stress due to the torque (T ) given by Eq. (4.20).
s yy 0
t xy t xy
t xy t xy
s xx s xx s = s t
xx
→
s xx
t xy t xy = Tr
t xy t xy J
s yy 0
FIGURE 4.24 Stress element for torsion and pressure.
The stress element in Fig. 4.25 is somewhat misleading in that it is not oriented according
to the arrangement of the elements in Fig. 4.23. Also, the radial stress (σ r ) cannot be shown
in this diagram. A better diagram is given in Fig. 4.26, where both the edge and plan views
are provided, aligned along the axis of the assembly.
2  2 
pR r  o 
s t =  1 +    s t
2
2
r − R    r   
o
t xy
t xy
2  2 
pR r  o 
s r s r =  1 −    0 0 Axis
r − R    r   
2
2
o
Tr Tr
t xy = t xy =
J J
t xy
s t s t
Edge view Plan view
FIGURE 4.25 Edge and plan views of stress element.
For the stress element at the outside radius (r o ), Fig. 4.25 becomes Fig. 4.26.
For the stress element at the radial interface (R), Fig. 4.25 becomes Fig. 4.27.