Lecture 3 – Contact Stresses
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Lecture Video
• Part 1
• https://web.microsoftstream.com/video/36e94e5b-fb0e-4efd-97ff77c26cf5eb86
• Part 2
• https://web.microsoftstream.com/video/5c99524e-6f97-4fed-b0659671aee14fda
• Visual Aid
• https://web.microsoftstream.com/video/5c408d38-92bf-4b84-8eb583413adb955f
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Lecture Plan
•
•
•
•
L1 - Introduction
L2 - Bearings
L3 - Contact stresses
L4* – Contact stresses examples
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Introduction
• Background
• Contact Stresses
– Spherical bodies
– Cylinders in contact
– General Case
• Radii of curvature at point of contact
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Background
• The chain on a bicycle or a motor-bike consists of a row of cylindrical
rollers, joined together by links and pins.
• The sprocket teeth are essentially flat at the point where the rollers
press on them.
• The contact between the roller and the sprocket tooth is a line of
theoretically zero width.
• With the chain in tension the stress at the contact is theoretically
infinite.
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Similarly with Bearings
• A cylindrical roller bearing has line contact between the rollers and the
tracks. The length of the line is the length of the roller, but the width of
it is very small.
• In a deep-groove ball bearing the contact between a ball and the tracks
is theoretically a point. This is also extremely small, but we might
expect it to be even smaller than the line contact in a roller bearing.
• A point is infinitely small in both directions. So the stresses in a deepgroove ball bearing are larger than those in roller bearings, for similar
radial loads.
• How do these components support realistic loads?
• Deformation of the surfaces around the contact point, so that
the area of contact is not zero, but very small.
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Two equal spherical bodies
• Consider two equal spheres, each of radius r, pressed together with force F.
r
b
Unloaded
Load applied
The square is at the edge of the circular contact patch. One of its sides is up
against the radius and remains at the same angle, but the other side rotates
through an angle. Strictly, this is tan-1 (b/r), but provided angles are small
and measured in radians, then it can be approximated to (b/r)
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Shear Strains
• These rotations are in effect shear strains. So shear strains are proportional to
(b/r), and so are all other strains.
• Stresses σ are proportional to
(E is the modulus of elasticity for that material)
E(b/r)
(1)
• F is proportional to
E (b3/r)
(2)
• Rearranging this, radius b of contact patch is proportional to
(rF/E)⅓
(3)
• Substituting for b with (3) into (1) above, we find
σ is proportional to (E2F/r2)⅓.
(4)i
• This shows that the contact stress at a point contact is proportional to the cube
root of the load, and inversely proportional to the ⅔ power of the radius.
• (The stress also depends on Poisson’s ratio ν , but as this is equal to 0.3 for most
engineering metals its effect is the same for most metals.)
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Spheres of different radii in contact
• Now we consider two different spherical bodies in contact, one of
radius r1 and the other of radius r2. They are pressed together with
force F as before.
• When the two little squares come together as shown above, the total
of the shear strains in the two of them is
(5)
• In the previous case of equal spheres each of radius r, the total was
2b/r, so we can replace r2 in the previous working (Eqn 4) by a radius
(6)
• This is called the relative radius of curvature of the two bodies.
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Cylinders in contact with axes parallel
• The length of the two cylinders is l
• The end view of the pair of cylinders is two circles
• If the cylinders have equal diameters, then in the same way as
before, the stress σ is proportional to E(b/r).
• The contact area is a rectangle, length l and width 2b.
• Therefore, the load F is proportional to (Eb2l/r).
• The half-width of the rectangle, b, is proportional to
• Therefore, the stress σ is proportional to
(7)
• For unequal radii, r1 and r2, we replace r1/2 in the working above
with the relative radius of curvature
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Formulae for Contact Stresses
Spheres in contact:
Where the contact area is a circle, and for material with Poisson’s ratio ν = 0.3, e.g
steel, aluminium etc:
1/ 3
 max
 E2F 
= 0.39 2 
 R 
E is modulus of elasticity, F is the normal load applied at the contact, and R is the
relative radius of curvature of the two bodies at their point of contact.
Cylinders with axes parallel:
Where the contact area is a rectangle, and for material with Poisson’s ratio ν = 0.3.
 max
where ℓ is the face width.
 EF 
= 0.42

 R 
1/ 2
The non-dimensional curves for the semi-axes of the elliptical contact area between
two bodies (in the Data Book) can be used to provide the same information, as two
special cases of the more general results.
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Contact Stresses: General Case
• In general, the contact patch between two solid objects is an ellipse, with
semi-axes a and b.
• The maximum stress σ0 occurs at the centre of the ellipse, and the stress is
zero along the elliptical boundary. If plotted on the vertical axis, it traces out
an ellipsoidal surface, or, to be more precise, a half-ellipsoid:
σ0
b
a
• The area of the elliptical contact patch is πab. (πr2 Circle)
• The total force on the area is given by summing all the products σ δA. Each of
these products is a little volume, area δA and height σ; so the sum, or
integral, of them is simply the volume of the half-ellipsoid.
• The volume of a sphere is 4/3 πr3, so by analogy, the volume of an ellipsoid is
4/3 πabσ0, and for half-ellipsoid is 2/3 πabσ0.
• The average stress on the contact area is thus (load/area) = (2/3 πabσ0)/πab ,
which is equal to 2/3 σ0. In words, the mean stress on the contact is 2/3 of
the maximum stress.
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Radii of Curvature at the Point of Contact
• The two bodies will have a common normal at their point of contact - a
line perpendicular to the contact area.
• Any plane containing this common normal can be thought of as slicing
through the bodies. In the region of the point of contact, this will reveal
the curved surfaces of the two bodies, with in general different radii.
The relative radius of curvature in this plane can be calculated in the
usual way.
• There will always be two principal planes of relative curvature, one
containing the maximum relative curvature and the other the minimum.
These correspond to the major and minor axes of the elliptical contact
area.
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Radii of Curvature at the Point of Contact
• Let the relative radii of curvature in these planes be RA, RB.
• The major semi-axis a of the contact patch depends chiefly on the
corresponding relative radius of curvature, RA, and not very much on RB.
Similarly, semi-axis b depends chiefly on RB.
• If we calculate the dimensionless ratios a/(ε0 RA) and b/(ε0 RB), we find that
each of them varies only slowly with ‘other’ relative radius of curvature.
• [Note: ε0 = σ0/E’, where E’ = E/(1 – ν2)]
• These two ratios have been plotted against the ratio RA/RB, and the resulting
curves are given in the module Formula Sheet.
• It can be seen that, if RA/RB is varied by a factor of 105, the two ratios vary
only by a factor of around 13 - they are relatively invariant, and it is possible
to make a reasonable guess of their value to enable a start to be made on a
design.
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Relative Radius of Curvature
•
•
•
•
(relative diameter same as relative radius)
For flat surfaces the radius is infinite
For internal surfaces the radius is negative
These diagrams show single view plane only (end view) there is also a RRC for the side view!
http://www.esierra.me/ContactStress_calc2.html Accessed 03/10/18
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Summary
• Contact Stresses
– Spherical bodies
– Cylinders in contact
– General Case
• Relative Radii
• Radii of curvature at point of contact
• Next session contact stresses examples.
• Please read through this lecture again to ensure you understand.
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