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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

Diameter of sphere #1, d1		m
Diameter of sphere #2, d2		m
Modulus of elasticity of sphere #1, E1		109N/m2
Poisson's ratio of sphere #1, υ1
Modulus of elasticity of sphere #2, E2		109N/m2
Poisson's ratio of sphere #2, υ2
Force applied, F		N

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OUTPUT   VARIABLES   &   GRAPHS

VARIABLES	Values	Units
♦ Radius of area of contact, a		10-6m	Graphs:  Stresses Vs Distance in z direction, Sphere 1    Stresses Vs Distance in z direction, Sphere 2
♦ Max contact Pressure, pmax		106N/m2
♦ Relative displacement, δ		10-6m
♦ Normalized shear stress in sphere#1, τmax,1		pmax
♦ τ max,1 acting at normalized distance		a
♦ Normalized shear stress in sphere#2, τmax,2		pmax
♦ τmax,2 acting at normalized distance		a

Stresses And Strains Due to Pressure Between Elastic Bodies

Contact loading occurs between machine elements such as rolling metal wheels, meshing gear teeth and bearings. Consider two spheres of different diameters being compressed by force F. The initial point of contact develops into an area of contact over which the load is distributed. The resulting stresses and deformation are described by the following equations:

where
     F = Compressive force
     z = direction of application of the force
     a = radius of contact area
     p_max = maximum contact pressure
     δ = displacement along axis of loading of the two spheres
     d₁ = diameter of sphere 1
     d₂ = diameter of sphere 2
     E₁ = modulus of elasticity of sphere 1
     υ₁ = Poisson's ratio of sphere 1
     E₂ = modulus of elasticity of sphere 2
     υ₂ = Poisson's ratio of sphere 2
     υ = Poisson's ratio of sphere of interest
     σ_x = principal stress in x direction, for sphere of interest
     σ_y = principal stress in y direction
     σ_z = principal stress in z direction
     τ_max = maximum shear stress

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point

BIBLIOGRAPHY

• Brown TH; Marks' Calculations for Machine Design; McGraw Hill, New York, 2005, Chapter 3.
• Young WC & Budynas RG; Roark's Formulas for Stress and Strain; 7th Edition, McGraw Hill, New York, 2002, Chapter 14.
• Moulton A, Grosjean J & Owen G; The Moulton Formulae and Methods - directly usable for calculations in mechanical engineering; Professional Engineering Publishing Ltd, London, UK, 2005.