CONTACT STRESSES Between Two Surfaces Presses Together

Muhanned Mahdi

Instructor : Professor SatchiVenkataraman

Two Surfaces Pressed Together• The contact stresses created when surfaces of two bodies are pressed

together by external loads are the significant stresses; that is, the stresses onOr somewhat beneath the surface of contact are the major causes of failureof one or both of the bodies.

• The examples where contact stresses maybe significant at the area :1. between a locomotive wheel and the railroad rail.

2. between a roller or ball and its race in a bearing.

3. between the teeth of a pair of gears in mesh.

4. between the cam and valve tappets of a gasoline engine.

Two Surfaces Pressed Together• For example, a railroad rail

sometimes fails as a result of“contact stresses”; the failurestarts as a localized fracture inthe form of a minute transversecrack at a point in the head ofthe rail somewhat beneath thesurface of contact between therail and locomotive wheel. Thisfracture progresses outwardlyunder the influence of therepeated wheel loads until theentire rail cracks or fractures.This fracture is called atransverse fissure failure

Two Surfaces Pressed Together• In contrast, bearings and gear

teeth sometimes fail as a resultof formation of pits (pitting) atthe surface of contact. Thebottom of such a pit is oftenlocated approximately at thepoint of maximum shear stress.Steel tappets have beenobserved to fail by initiation ofmicroscopic cracks at thesurface that then spread andcause flaking. Chilled cast-irontappets have failed by cracksthat start beneath the surface,where the shear stress ishighest, and spread to thesurface, causing pitting failure

Assumptions1. Load

• It is assumed that there is no tendency for one body to slide withrespect to the other and, hence, no friction force is present.There is no effect of a friction force.

2. Properties of Materials• The material of each body is homogeneous, isotropic, and elastic

in accordance with Hooke’s law, but the two bodies are not necessarily made of the same material.

3. Shape of Surfaces near Point of Contact before Loading• If two bodies are in contact at a point, there is a common tangent

plane to the surfaces at The point of contact. In the solution forcontact stresses, an expression for the distance Betweencorresponding points on the surfaces near the point of contact isrequired.

Two Surfaces Pressed Together

Shape of Surfaces near point contact before Loading

similarly

Geometry of the contact

Geometry of the contact

should be equal to

Geometry of the contact

• The constants A and B depend on the principal radii of curvature of the two bodies atthe point of contact and on the angle a between the corresponding planes of theprincipal curvatures.

• A general equation for displacements accordingto geometrical considerations can be written as

• Since we know that , thenwe can write the equation as :

• The surface displacement for Hertzpressure can be written as :

• Then the equation for vertical displacements become :

Equations for Principal Stresses

• To satisfy the vertical displacement equilibrium, an elliptical stress distribution is required :

• Then the principal stresses become as follows :

Equations for Principal Stresses

• The major and minor axis of the area of contact can be defined as follows :

θ (degree) 30 35 40 45 50 55 60 65 70 75 80 85 90

m 2.731 2.397 2.136 1.928 1.754 1.611 1.486 1.378 1.284 1.202 1.128 1.061 1

n 0.493 0.53 0.567 0.604 0.641 0.678 0.717 0.759 0.802 0.846 0.893 0.944 1

Graph for Stresses