BIZOMAT MAKEOS · Strength
Hertz Contact Stress
What contact pressure develops between these curved bodies under load?
Contact width, maximum contact pressure and maximum subsurface shear stress for sphere (point) or parallel cylinder (line) contact by Hertz theory.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Contact type | choice | Point contact for balls; line contact for rollers and wheels on flat tracks. |
| Radius of body 1 | mm | Radius of the convex body (ball or roller radius). |
| Radius of body 2 (0 = flat, negative = concave) | mm | Radius of the mating body: 0 for flat, negative for a concave seat or groove. |
| Normal load | N | Normal load across the contact. |
| Contact length (line contact only) | mm | Length of line contact (roller or wheel width). |
| Materials | ||
| Elastic modulus, body 1 | GPa | Modulus of body 1. |
| Poisson’s ratio, body 1 | Poisson’s ratio of body 1. | |
| Elastic modulus, body 2 | GPa | Modulus of body 2. |
| Poisson’s ratio, body 2 | Poisson’s ratio of body 2. | |
| Allowable contact pressure | MPa | Allowable contact pressure for the material, hardness and number of cycles, from your design data. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Contact type
- What it is
- Whether the bodies touch at a point (balls or spheres) or along a line (rollers, cylinders or wheels on a flat track).
- Units and format
- Choose one of 2 options from the list: Sphere on sphere or flat (point contact); Parallel cylinders or cylinder on flat (line contact).
- Example
- The form opens with “Parallel cylinders or cylinder on flat (line contact)” selected as a worked example. Choose the option that matches your case.
- Get it right
- The contact length input is used only for line contact.
What this means: Radius of body 1
- What it is
- The radius of the convex body at the contact, for example the radius of a ball or roller.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter the radius, not the diameter.
What this means: Radius of body 2 (0 = flat, negative = concave)
- What it is
- The radius of the mating body at the contact. Zero means a flat surface; a negative value means a concave seat or groove that wraps round body 1.
- Units and format
- Enter a number in mm: millimetres. The form accepts -1000000 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A concave radius must be larger than the convex radius of body 1, or the tool rejects it.
What this means: Normal load
- What it is
- The total load pressing the two bodies together, at right angles to the contact surface.
- Units and format
- Enter a number in N: newtons. The form accepts 0.000001 N or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the load on this one contact, not the total load shared by several balls or rollers.
What this means: Contact length (line contact only)
- What it is
- The length of the line of contact, normally the width of the roller or wheel that touches the track. Used only for line contact.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the effective contact length excluding end chamfers; edge loading raises the real pressure.
Materials
What this means: Elastic modulus, body 1
- What it is
- The elastic modulus (Young’s modulus) of body 1, from the material data sheet.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1500 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 210 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Poisson’s ratio, body 1
- What it is
- Poisson’s ratio of body 1, from the material data sheet.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It is a dimensionless ratio, not a percentage.
What this means: Elastic modulus, body 2
- What it is
- The elastic modulus (Young’s modulus) of body 2, from the material data sheet.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 1500 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 210 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter GPa, not MPa: 1 GPa is 1000 MPa.
What this means: Poisson’s ratio, body 2
- What it is
- Poisson’s ratio of body 2, from the material data sheet.
- Units and format
- Enter a plain number with no unit. The form accepts 0 to 0.5. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.3 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It is a dimensionless ratio, not a percentage.
What this means: Allowable contact pressure
- What it is
- The highest contact pressure allowed for the material, its hardness and the number of load cycles, from your design data. The peak contact pressure is compared with it.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 100000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1500 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Contact pressure limits are much higher than ordinary stress limits; take them from contact-fatigue data, not from yield strength.
Method and formulas
- E* = 1 ÷ ((1 − ν₁²) ÷ E₁ + (1 − ν₂²) ÷ E₂); 1/R = 1/R₁ + 1/R₂.
- Point: a = (3FR ÷ 4E*)^(1/3); p₀ = 3F ÷ 2πa²; τmax ≈ 0.31 p₀ at 0.48a.
- Line: b = √(4FR ÷ πLE*); p₀ = 2F ÷ πbL; τmax ≈ 0.30 p₀ at 0.786b.
- Example: a 50 mm radius roller, 20 mm long, on a flat with 10000 N gives p₀ = 606.04 MPa.
Limitations
- Elastic, frictionless, smooth contacts with dimensions much larger than the contact area.
- Edge loading at roller ends raises pressure well above the line-contact value.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check wheels on rails, cam followers and rollers
- Size ball transfer units and pivots
- Compare materials and radii
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Contact radius or half-width
- Maximum contact pressure
- Maximum subsurface shear and its depth
- Approach (point contact)
Interpreting the result
- Compare with allowable values for the actual hardness and cycle count; surface fatigue governs repeated contacts.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2301–2309. Topic reference; no source tables or recommended settings are reproduced.
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