Lecture 5 Contact Stress

Contact Stress
(3.19)
MAE 316 – Strength of Mechanical Components
Y. Zhu
1
Contact Stress

Introduction
Where does contact stress occur
Ball bearings
Railroad wheel on a track
Bowling ball on an alley
Want to find the local stress at the point (region) of
contact.
This will depend on elasticity of contacting materials (E &
ν), loading, and geometry.
2
Contact Stress

Spherical Contact Surfaces (3.19)
Where a = radius of circular contact area and p o = p max =
maximum pressure.
3
Contact Stress

Spherical Contact Surfaces (3.19)
For spheres in contact, the contact patch is circular
(radius a).
a
3
3F
(1
) / E (1
) / E
8 1/ d 1/
d
2 2
1 1 2 2
1 2
Where:
F = force pressing the two spheres together
d 1 and d 2 = diameters of the two solid spheres in contact
E 1 , ν 1 , E 2 , ν 2 = respective elastic constants of the two spheres
4
Contact Stress

Spherical Contact Surfaces (3.19)
The maximum contact pressure is
The stress distribution is
p
3F
2 a
max 2
z 1 1
1
1 2 x
y
pmax
1 tan (1 )
2
a z a 21 ( za)
pmax
3 z
2
1 ( za)
12
0
1 1
max 13 23 yz
xz
( 1 3) ( 2 3)
2 2
xy
5
Contact Stress

Spherical Contact Surfaces (3.19)
Figure 3-37 in the
textbook shows the
magnitude of the stress
components below the
surface as a function of
p max of contacting
spheres with ν = 0.3.
6
Contact Stress

Cylindrical Contact Surfaces (3.19)
2b
l
Where b = half-width of rectangular contact area and p o = p max
= maximum pressure.
7
Contact Stress

Cylindrical Contact Surfaces (3.19)
For cylinders in contact, the contact patch is rectangular
(half-width b).
b
2F
(1
) / E (1
) / E
l 1/ d 1/
d
2 2
1 1 2 2
1 2
Where:
l = length of contact area
F = force pressing the two spheres together
d 1 and d 2 = diameters of the two solid spheres in contact
E 1 , ν 1 , E 2 , ν 2 = respective elastic constants of the two spheres
8
Contact Stress

Cylindrical Contact Surfaces (3.19)
The maximum contact pressure is
The stress distribution is
p
max
2F
bl
2
1
x
2 pmax
1 ( zb)
zb
2
1
2( zb)
p
2
2
1 ( zb)
pmax
3
z
2
1 ( zb)
2 y max
zb
9
Contact Stress

Cylindrical Contact Surfaces (3.19)
Figure 3-39 in the
textbook shows the
magnitude of the stress
components below the
surface as a function of
p max of contacting
cylinders with ν = 0.3.
10
Contact Stress

Example
Two carbon steel balls, each 30 mm in diameter, are
pressed together by a force F. Find the maximum values
of the principal stress and the maximum shear stress if F
= 50 N, ν = 0.3, and E = 207 GPa.
11
Contact Stress