Hertzian Contact Stress | Sphere in a Cone of Spheres

	Hertzian Contact: Sphere in a Cone of Spheres
	Date:


Assumptions

1. Surfaces in contact are perfectly smooth.
2. Material limits are not exceeded.
3. Materials are homogeneous.
4. No frictional forces within the contact area.

Geometry

D₁		mm	Diameter of the Top Sphere (Body 1)
D₂		mm	Diameter of the Bottom Spheres (Body 2)
θ_C		°	Angle of Contact

Applied Load

F_a		N	Applied Force

Material Properties
Body 1:	Sphere
Mat:			Material
Type:		-	Material Type (Ductile or Brittle)
E		GPa	Modulus of Elasticity
ν		-	Poisson's ratio
σ_a		MPa	Allowable stress (tensile or flexural)


Body 2:	Cone of Spheres
Mat:			Material
Type:		-	Material Type (Ductile or Brittle)
E		GPa	Modulus of Elasticity
ν		-	Poisson's ratio
σ_a		MPa	Allowable stress (tensile or flexural)
Equivalent and Allowable Properties

E_e		GPa	Effective Young's Modulus
D_e		mm	Equivalent diameter for the two bodies
P_f		MPa	Hertzian contact pressure at failure

Results

a		μm	Contact zone radius
P_max		MPa	Maximum Hertzian contact pressure
δ_c		μm	Deflection of the contact zone
δ_z		μm	Displacement of the sphere
K_c		10⁶N/m	Contact zone stiffness


Body 1 Stress
σ_tmax,1		MPa	Max tensile stress in Body 1
τ_max,1		MPa	Max shear stress in Body 1
z₁		μm	Depth of max shear stress in Body 1


Body 2 Stress
σ_tmax,2		MPa	Max tensile stress in Body 2
τ_max,2		MPa	Max shear stress in Body 2
z₂		μm	Depth of max shear stress in Body 2

Limits: Based on Max Allowable Hertzian Stress / Pressure

F_c, max		N	Max allowable contact force
D_min		mm	Minimum sphere diameter

Directions

These calculators are designed to resemble spreadsheet calculators that can be printed as reports for project recordkeeping. The term project, with regards to these calculators, may refer to an engineered design, homework problem, tutorial problem, hobby design or self-study. The workflow for each calculator is from top to bottom.

Enter the diameter of the top sphere or ball.
Enter the diameter of the bottom three spheres.
Enter the applied load or preload.
Select or enter the material properties for Body I, i.e., the sphere.
Select or enter the material properties for Body II, i.e., the flat plate.
Press the calculate button.
Print a report for your project recordkeeping by pressing the print button.

Assumptions

Surfaces in contact are perfectly smooth
Material limits are not exceeced
Materials are homogeneous
No frictional forces within the contact area

Equations

Equivalent Modulus of the Two Bodies
$$E_e = {1\over {{{1-\nu_1^2}\over E_1} + {{1-\nu_2^2}\over E_2}}}$$
Equivalent Diameter of the Two Bodies
$$D_e = \frac{D_1D_2}{D_1+D_2}$$
Force Normal to the Contact Zone
$$F_c = \frac{F_a}{3sin\left(\theta_C\right)}$$
Radius of Circular Contact
$$a = \left(\frac{3FD}{8E_e}\right)^{1\over 3}$$
Maximum Hertzian Pressure
$$P_{max} = \left(\frac{24FE_e^2}{\pi^3D^2}\right)^{1\over 3}$$
Deflection
$$\delta = \left(\frac{9F^2}{8DE_e^2}\right)^{1\over 3}$$
Stiffness
$$k_z = \left({3FDE_e^2}\right)^{1\over 3}$$
Max Tensile Stress, Body 1
$$\sigma_{tmax, 1} = \frac{P_{max}}{3}\left(1-2\nu_1\right)$$
Max Tensile Stress, Body 2
$$\sigma_{tmax, 2} = \frac{P_{max}}{3}\left(1-2\nu_2\right)$$
Max Shear Stress in both Bodies
$$\tau_{max} = 0.313P_{max}$$
Depth of Max Shear Stress in both Bodies
$$z = 0.48a$$
Max Allowable Contact Force
$$F_{c,~max} = \frac{P_f^3\pi^3D_e^2}{24E_e^2}$$

Background

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Hertzian Contact: Sphere in a Cone of Spheres

Directions

Assumptions

Equations

Background