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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!