Page 163 - Marks Calculation for Machine Design
P. 163
P1: Rakesh
14:16
January 4, 2005
Brown˙C03
Brown.cls
ADVANCED LOADINGS
To determine the maximum shear stress (τ max ) at the plane of contact between the two
cylinders, subsitute (z = 0) in Eqs. (3.32), (3.33), and (3.34) to give 145
1 0 2 0
σ x =−p max 2 − 1 + − 2
0 2 b 2 b
1 +
b 2
(3.36)
√
=−p max [(2 − 1) 1 − 0] =−p max
0 2 0
σ y =−p max (2ν) 1 + −
b 2 b
√
=−p max (2ν)( 1 − 0) =−p max (2ν) (3.37)
−p max −p max
= √ =−p max (3.38)
σ z = $
0 2 1
1 +
b 2
As it is the largest, substitute (σ x ) from Eq. (3.36) and (σ z ) from Eq. (3.38) in Eq. (3.35)
to give the maximum shear stress (τ max ) at the plane of contact as
σ x − σ z [−p max ] − [−p max ]
τ max = =
2 2
(3.39)
− p max + p max 0
= = = 0
2 2
Even though the principal stresses are a maximum at the plane of contact (z = 0), it turns
out that the maximum value of the maximum shear stress (τ max ) does not occur at (z = 0)
but at some small distance into the cylinder. Typically this small distance is between one-half
and one times the distance (b). This explains what is seen in practice where a cylindrical
roller bearing develops a crack internally, then as the crack propagates to the surface of the
roller it eventually allows lubricant in the bearing to enter the crack and fracture the roller
bearing catastrophically by hydrostatic pressure.
The relative distributions of the principal stresses, normalized to the maximum pressure
(p max ), are shown in Fig. 3.12, where a Poisson ratio (ν = 0.3), which is close to that for
steel, has been used.
Again, notice that the maximum value of the maximum shear stress (τ max ) does not occur
at the surface (z = 0), but is at a distance of about 0.75 times the distance (b), and has a
value close to 0.3 times the maximum pressure (p max ). Also, observe that the values of the
principal stresses at the plane of contact (z = 0) agree with the calculations in Eqs. (3.36),
(3.37), and (3.38) for a Poisson ratio (ν = 0.3).
Remember that the curves for the three principal stresses shown in Fig. 3.12 are for a
Poisson ratio (ν = 0.3). For other Poisson ratios, a different set of curves would need to be
drawn. Also, as the stress elements along the (z) axis are triaxial, the design is safe if the
maximum value of the maximum shear stress (τ max ) is less than the shear yield strength
in compression, that was found in the previous section to be half the yield stress (S y ).
Converting this statement into a factor-of-safety expression is given in Eq. (3.40) as
S y τ max 1
τ max < S sy = → = (3.40)
2 S y n
2
