9 Contact Stresses

9.2 HERTZIAN CONTACT STRESSES 425
The radii of concern are given in Fig. 9-6 as
From Eqs. (9.5) and (9.6),
R1 = 1 2 d0 = 19 mm, R ′ 1 = 1 2 d0 = 19 mm
R2 =−r =−20 mm, R ′ 2 = 1 2 D = 100 mm
A = 1
1 1 1 1
− + +
4 0.019 0.020 0.019 0.100
− 1
1 1
− + −
4 0.019 0.019
1
2 1
−
0.020 0.100
1 1
− 4 − −
0.019 0.019
1
1
− sin
0.020 0.100
2 1/2
(0) = 1.316 m −1
B = 31.32 m −1
Then
B/A = 23.78
=
(3)
2 1 − ν
A + B
2
E =
2(1 − 0.32 )
(31.32 + 1.316)(200 × 109 )
= 2.79 × 10 −13 m 3 /N (4)
From Fig. 9-3, the coefficients are found. The variables of interest are then computed
using the appropriate formulas:
k = 0.13, Coct = 0.27, Cτ = 0.3
Cb = 0.394, Cσ = 1.0, Cz = 0.8
b = Cb(F) 1/3 = 0.394[(4500)(2.79 × 10 −13 )] 1/3 = 4.250 × 10 −4 m
= 0.425 mm
a = 4.250 × 10 −4 /0.13 = 3.269 × 10 −3 m = 3.269 mm
b/ = 4.250 × 10
(5)
−4 /(2.79 × 10 −13 ) = 1523 MPa
σc =−Cσ (b/) = (−1.0)(1523) =−1523 MPa
τmax = Cτ (b/) = 0.3(1523) = 456.9 MPa
τoct = Coct(b/) = (0.27)(1523) = 411.2 MPa
Zs = Czb = (0.8)(4.250 × 10 −4 ) = 3.40 × 10 −4 m = 0.34 mm
(1)
(2)