Download - contact stress theories.ppt
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Bahir Dar Institute of Technology
Faculty of Mechanical and Industrial Engineering
Program (Msc ):Mechanical Design Engineering
Engineering Design Laboratory
Tittle:-contact Stress Theories
Prepared By: Tibebu Meride
ID NO. BDU 0702325 PR
outline
Discussion contact stress theoriesContact b/n two elastic bodiesContact b/n spheresContact b/n two cylindersContact b/n two parallel cylinderContact b/n crossed cylinder
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 & v),loading, and geometry.
Curved Surfaces in Contact
The theoretical contact area of two spheres is a point (=0-
dimensional)
The theoretical contact area of two parallel cylinders is a line (=
1-dimensional)
As a result, the pressure between two curved surfaces should
be infinite
The infinite pressure at the contact should cause immediate
yielding of both surfaces
Curved Surfaces in Contact – Examples
Spherical Contact Surfaces
Where: F = force pressing the two spheres together
E1, V1, E2, V2 = respective elastic constants of the two spheres
d1 and d2=diameters of the two solid spheres in contact
Spherical Contact Surfaces
Curved Surfaces in Contact – Examples (contd.)
Spheres in Contact
The radius of the contact area is given by:
Cont..
Cylindrical Contact Surfaces
Cont..
Spheres in Contact – Principal Stresses
• The principal stresses σ1, σ2, and σ3 are generated on the z-axis:
Spheres in Contact – Vertical Stress Distribution atCenter of Contact Area
Depth below contact area
Cont..
• Two Spheres in Contact
The suffixes 1 and 2 relate to spheres 1 and 2 respectively
• General Case
Spheres of Same Material
cont..• Sphere in Contact with a Plane
General Case
Cont..
• Sphere Between Two Parallel Planes
Cont..
• Equal Diameter Cylinders Crossed with Their Axes at Right Angles
Cont..
Cont..
• Unequal Diameter Cylinders Crossed with Their Axes at Right Angles
Cont..
Cont..
• Unequal Diameter Cylinders Crossed with Their Axes at Any Angle.
Cont..
• The suffix 1 refers to the larger diameter cylinder, the suffix 2 to the smaller.
• Let the axes be inclined at an acute angle e to one another.
• It is first necessary to obtain the ratio A/B by solving the following equations, for A and B.
Cont.
From the calculated value of A/B the values of K and – 1/e dE/de
Cont..
• The maximum shear and Von Mises stress are reached below the contact area.
This causes pitting where little pieces of material break out of the surface.
Cylinders in Contact
• The half-width b of the rectangular contact area of two parallel cylinders is found as:
Cont..
• Where E1 and E2 are the moduli of elasticity for cylinders 1 and 2 and ν1and ν2 are the Poisson’s ratios, respectively. L is the length of contact.
• The maximum contact pressure along the center line of the rectangular contact area is:
Cylinders in Contact (contd.)
The equations for two cylinders in contact are also valid for:Cylinder on a flat plate (a flat plate is a cylinder
with an infinitely large radius) Cylinder in a cylindrical groove (a cylindrical
groove is a cylinder with a negative radius)
Cont.
Effects of Material Combinations
The maximum contact pressure between two curved surfaces depends on:Type of curvature (sphere vs. cylinder) Radius of curvatureMagnitude of contact force Elastic modulus and Poisson’s ratio of contact
surfaces
Thanks!