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Contact Stress (3.19)
MAE 316 – Strength of Mechanical ComponentsNC State University Department of Mechanical and Aerospace Engineering
Contact Stress1
Introduction
Contact Stress2
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.
Spherical Contact Surfaces (3.19)
Contact Stress3
Where a = radius of circular contact area and po = pmax = maximum pressure.
Spherical Contact Surfaces (3.19)
Contact Stress4
For spheres in contact, the contact patch is circular (radius a).
2 21 1 2 23
1 2
(1 ) / (1 ) /3
8 1/ 1/
E EFa
d d
Where:F = force pressing the two spheres togetherd1 and d2 = diameters of the two solid spheres in contactE1, ν1, E2, ν2 = respective elastic constants of the two spheres
Spherical Contact Surfaces (3.19)
Contact Stress5
The maximum contact pressure is
The stress distribution is
max 2
3
2
Fp
a
11 2 max 2
max3 2
12
max 13 23 1 3 2 3
1 11 tan (1 )
2 1 ( )
1 ( )
0
1 1( ) ( )
2 2
x y
z
xy
yz xz
zp
a z a z a
p
z a
Spherical Contact Surfaces (3.19)
Contact Stress6
Figure 3-37 in the textbook shows the magnitude of the stress components below the surface as a function of pmax of contacting spheres with ν = 0.3.
Cylindrical Contact Surfaces (3.19)
Contact Stress7
Where b = half-width of rectangular contact area and po = pmax = maximum pressure.
2b
l
Cylindrical Contact Surfaces (3.19)
Contact Stress8
For cylinders in contact, the contact patch is rectangular (half-width b).
2 21 1 2 2
1 2
(1 ) / (1 ) /2
1/ 1/
E EFb
l d d
Where:l = length of contact areaF = force pressing the two spheres togetherd1 and d2 = diameters of the two solid spheres in contactE1, ν1, E2, ν2 = respective elastic constants of the two spheres
Cylindrical Contact Surfaces (3.19)
Contact Stress9
The maximum contact pressure is
The stress distribution is
max
2Fp
bl
21 max
2
2 max 2
max3 2
2 1 ( )
1 2( )2
1 ( )
1 ( )
x
y
z
p z b z b
z bp z b
z b
p
z b
Cylindrical Contact Surfaces (3.19)
Contact Stress10
Figure 3-39 in the textbook shows the magnitude of the stress components below the surface as a function of pmax of contacting cylinders with ν = 0.3.
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.
Contact Stress11
