9 Contact Stresses

9.2 HERTZIAN CONTACT STRESSES 429
5. Suppose that the operating load is fixed at 20 367 N and that the rail and wheel
radii are fixed at 35 and 45 cm, respectively. Find by what factor the tensile strength
of the steel must be increased to make the maximum octahedral shear stress one-half
the yield point value.
Since E and ν of steel are essentially constant for steels of all strengths and A+B
is determined by the fixed radii of rail and wheel, the quantity in Eq. (9.10b) is a
fixed value. Therefore, from (5), the maximum octahedral shear stress would remain
at 195.07 MPa for all steels. Because tensile yield strength and octahedral shear
stress at yield are directly proportional, doubling the tensile strength would result
in the maximum octahedral shear stress being one-half the value that causes yield.
From (1), the strength of the steel would be increased to
σys = (3/ √ 2)(τoct)ys = (3/ √ 2)(2 × 195.07) = 827.6 MPa (8)
6. Determine for which of the three quantities (load, radii of curvature, or steel
strength) would a change be most effective in producing a system with an acceptable
value of maximum octahedral shear stress.
Reducing the load is most ineffective in reducing the maximum octahedral shear
stress because, from (2), the stress varies directly as the cube root of the load. When
the radii of curvature are increased in constant proportion, the maximum octahedral
shear stress varies inversely as the two-thirds power of the radii factor λ [see (6)];
hence changing the radii is more effective than changing the load. However, if large
reductions in stress are required, it is doubtful that the necessarily large changes in
radii (λ = √ 8 = 2.83-fold increase for a halving of the shear stress) would be
feasible. It appears from the previous question that increasing the tensile strength
of the material of construction is the most effective alternative when the stress is
significantly higher than an acceptable level.
Two Bodies in Line Contact
Two bodies in contact along a straight line before loading are said to be in line contact.
For instance, a line contact occurs when a circular cylinder rests on a plane
or when a small circular cylinder rests inside a larger hollow cylinder. In these line
contact cases, Eqs. (9.5) and (9.6) become
and
A = 0, B = 1 2 (1/R1 + 1/R2)
B/A =∞ (9.16)
It can be shown that in this case, the quantity k in Eq. (9.10c) approaches zero. When
a distributed load q (force/length) is applied, the area of contact is a long narrow
rectangle of width 2b in the x direction and a length 2a in the y direction.