Contact Interaction Calculators

Sphere and Flat Plate Contact

Contact (Hertzian) stresses and deformations arise from the interaction of two contacting bodies. At that, significant lockal stresses arise in the contact zone due to the transfer of pressure over very small areas.

Contact stresses are determined when calculating critical parts, such us bearings, gear wheels, camshaft and jointed mechanisms. Calculations of contact stresses are performed on the assumption that the loads create only elastic strains in the contact zone.

Due to the fact that all three principal stresses in the center of the contact area are virtually identical, the material can withstand large pressures approximately equal to five times the yield strength without permanent strains. The most critical point in the contact zone is located deep in the material, at a depth of approximately 1/2 of the radius of the contact zone.

In this calculation, a contact of a sphere with a diameter D and a flat plate is considered. For the calculation, the elastic moduli E₁, E₂ and Poisson's ratios ν₁, ν₂ of the contacting bodies should be specified. The sphere is under load F, which is transmitted to a flat surface through the contact zone.

Following the calculations, the contact stresses σ in the contact zone, the diameter d of the contact zone, and the total deformations of the bodies Y in the direction of the load action are determined.

Sphere and flat plate contact

Sphere and flat plate contact

INITIAL DATA

Е₁ - Young's modulus of the flat plate;

ν₁ - Poisson's ratio of the flat plate;

D - Sphere diameter;

Е₂ - Young's modulus of the sphere;

ν₂ - Poisson's ratio of the sphere;

F - Load.

RESULTS DATA

σ - Stress at contact area ;

d - Diameter of contact area;

Y - Tolal deformation of two bodies.

Metric Imperial

Young's modulus (Е₁)

Poisson ratio (ν₁)

Sphere diameter (D)

Young's modulus (Е₂)

Poisson's ratio (ν₂)

Load (F)

Stress at contact area (σ)

Diameter of contact area (d)

Tolal deformalion (Y)

BASIC FORMULAS

Stress at contact area:

σ = 0.6[FE² / D²]^1/3;

Diameter of contact area:

d = 1.8[FD / E]^1/3;

Tolal deformalion:

Y = 1.6[F² / DE²]^1/3.

*The above formulas are valid for equal material with ν = 0.3.

INITIAL DATA

Е₁ - Young's modulus of the flat plate;

ν₁ - Poisson's ratio of the flat plate;

D - Sphere diameter;

Е₂ - Young's modulus of the sphere;

ν₂ - Poisson's ratio of the sphere;

F - Load.

RESULTS DATA

σ - Stress at contact area ;

d - Diameter of contact area;

Y - Tolal deformation of two bodies.

MATERIALS PROPERTIES

Material	Young’s modulus Pa (psi)	Poisson’s ratio
Steel	1.86÷2.1×10¹¹ (2.7÷3.05×10⁷)	0.25÷0.33
Cast iron	0.78÷1.47×10¹¹ (1.1÷2.1×10⁷)	0.23÷0.27
Copper	1.0÷1.3×10¹¹ (1.45÷1.9×10⁷)	0.34
Tin bronze	0.74÷1.22×10¹¹ (1.1÷1.8×10⁷)	0.32÷0.35
Brass	0.98÷1.08×10¹¹ (1.4÷1.6×10⁷)	0.32÷0.34
Aluminum alloy	0.7×10¹¹ (1.0×10⁷)	0.33
Magnesium alloy	0.4÷0.44×10¹¹ (5.8÷6.4×10⁶)	0.34
Nickel	2.5×10¹¹ (3.6×10⁷)	0.33
Titanium	1.16×10¹¹ (1.7×10⁷)	0.32
Lead	0.15÷0.2×10¹¹ (2.2÷2.9×10⁶)	0.42
Zinc	0.78×10¹¹ (1.1×10⁷)	0.27
Glass	4.9÷5.9×10¹⁰ (7.1÷8.5×10⁶)	0.24÷0.27
Concrete	1.48÷2.25×10¹⁰ (2.1÷3.3×10⁶)	0.16÷0.18
Wood (along the grain)	8.8÷15.7×10¹⁰ (12.8÷22.8×10⁶)	-
Wood (across the grain)	3.9÷9.8×10¹⁰ (5.7÷14.2×10⁶)	-
Nylon	1.03×10¹⁰ (1.5×10⁶)	-

OTHER CALCULATORS

AREA MOMENTS OF INERTIA

BEAM CALCULATORS

TORSION OF BARS

CIRCULAR FLAT PLATES

BUCKLING

ELASTIC CONTACT

IMPACT LOADS

NATURAL FREQUENCIES

PRESSURED SHELLS

FLUID DYNAMIC

COMPOSITES

SPRINGS

THREAD CONNECTIONS

SHAFT CONNECTIONS

BEARINGS

DRIVES

FATIGUE

HEAT TRANSFER