9 Contact Stresses

422
CONTACT STRESSES
contact region, and the distance below the contact surface at which the maximum
shear stress and octahedral shear stress occur, the constants A and B must first be
evaluated. Denoting the wheel as body 1 and the railhead as body 2, the principal
radii of curvature are R1 = 45 cm, R ′ 1 =∞, R2 = 35 cm, and R ′ 2 =∞. The angle
between the principal planes of the two bodies is 90◦ . The physical constants of steel
are E = 200 GPa, ν = 0.29.
From Eqs. (9.5) and (9.6),
A = 1
1 1
+ −
4 0.45 0.35
1
2
1 1
1 1
+ − 4
4 0.45 0.35 0.45 0.35
1/2 = 1.2698 − 0.1587 = 1.111 m −1
(1)
B = 1.2698 + 0.1587 = 1.428 m −1
(2)
B/A = 1.428/1.111 = 1.285 (3)
When both bodies have the same physical properties, Eq. (9.10b) becomes
= 2(1 − ν2 )
E(A + B) =
2 1 − (0.29) 2
(2.0 × 1011 )(1.111 + 1.428) = 3.607 × 10−12 m 3 /N (4)
From knowledge of B/A, the constants for use in determining stresses and lengths
are read from Fig. 9-3, Cb = 0.84, k = 0.85, Cσ = 0.69, Cτ = 0.22, Coct =
0.21, Cz = 0.5, Cd = 2.2. The semiminor axis of the contact ellipse is given by
[Eq. (9.10a)]
b = Cb(F) 1/3
= 0.84 (40 000)(3.607 × 10 −12 1/3 )
= 0.00441 m = 4.41 mm (5)
The semimajor axis of the contact ellipse is [Eq. (9.10c)]
a = b/k = 0.00441/0.85 = 0.00519 m = 5.19 mm (6)
The compressive stress at the center of the contact ellipse (i.e., the maximum principal
stress) becomes [Eq. (9.12)]
σc =−Cσ (b/) =−0.69(0.00441/3.607 × 10 −12 ) =−843.6 MPa (7)
The maximum shear stress is [Eq. (9.13)]
τmax = Cτ (b/) = 269.0 MPa (8)
The maximum octahedral shear stress is given by [Eq. (9.14)]
τoct = Coct(b/) = 256.8 MPa (9)