Calculation
b = pH·D/E*; full width = 2b
Rather than starting from force and length, contact width can also be derived directly from an already known Hertz pressure, for example from a manufacturer rating or the linked pressure calculator.
b = pH·D/E*
Hertz line-contact width
Rather than starting from force and length, contact width can also be derived directly from an already known Hertz pressure, for example from a manufacturer rating or the linked pressure calculator.
MINTSI
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Result
Select a target and calculate.
Understand the inputs
- Half contact width b — Half-width of the elastic pressure band.
- Full contact width — Total width of the elastic pressure band, 2·b.
- Hertz pressure pH — Known maximum pressure, e.g. from the line-contact calculator or a manufacturer rating.
- Reduced diameter D — Reduced diameter combining both curvature diameters, 1/D = 1/D1 + 1/D2.
- Reduced elastic modulus E* — Effective elastic modulus of the material pair.
Example
A Hertz pressure of 524.8 MPa at a reduced diameter of 26.67 mm and E* ≈ 115.4 GPa give a half contact width of b ≈ 0.121 mm, a full width of about 0.243 mm.
Assumptions and limits
Applies to line contact (cylinder-cylinder or cylinder-flat) under classical Hertz theory; for point contact (sphere contact), the contact radius from the sphere-on-flat calculator applies instead.
Understand Hertz line-contact width
This calculator finds the half and full width of the elastic contact band for Hertz line contact from a known Hertz pressure, the reduced diameter and the reduced elastic modulus.
What does this quantity describe?
Rather than deriving contact width from force and contact length, it can be derived directly from an already known Hertz pressure: b = pH·D/E*, with reduced diameter D from 1/D = 1/D1 + 1/D2. The full width of the contact band is 2b.
This is like inverting an already known relationship: knowing a calculation's result (pressure, for example from a bearing datasheet) lets you infer another quantity of interest (contact width) directly, without repeating the original derivation via force and length.
Formula and variables
b = pH·D/E*; full width = 2b
b = pH·D/E*full width = 2bpH = b·E*/D
| Symbol / input | Meaning |
|---|---|
| Half contact width b | Half-width of the elastic pressure band. |
| Full contact width | Total width of the elastic pressure band, 2·b. |
| Hertz pressure pH | Known maximum pressure, e.g. from the line-contact calculator or a manufacturer rating. |
| Reduced diameter D | Reduced diameter combining both curvature diameters, 1/D = 1/D1 + 1/D2. |
| Reduced elastic modulus E* | Effective elastic modulus of the material pair. |
Choose the inputs correctly
A known Hertz pressure pH, the reduced diameter D and the reduced elastic modulus E* set the half contact width b.
How to use the calculator
First find the Hertz pressure with the line-contact calculator, or take it from a manufacturer rating. Enter it along with the reduced diameter and E* to find contact width.
Worked example
A Hertz pressure of 524.8 MPa at a reduced diameter of 26.67 mm and E* ≈ 115.4 GPa give a half contact width of b ≈ 0.121 mm, a full width of about 0.243 mm.
Understand the result and units
A full contact width of only 0.243 mm at a 26.67 mm reduced diameter shows how narrow the actual pressure band is compared with component size — a vivid measure of why pressure within this narrow band is so much higher than a rough force-over-area estimate would suggest.
Hertz pressure is given in MPa, reduced diameter in mm, the reduced elastic modulus in GPa, and the resulting contact width in mm or µm for small values.
How contact width follows directly from pressure
Substituting the Hertz pressure formula pH = √[(FN/L)·E*/(π·R)] into the half-width formula b = 2·√[(FN/L)·R/(π·E*)] and solving for FN/L, force and length cancel out completely: what remains is the simple relationship b = pH·D/E* (with D = 2R). This is practical when pressure is already known from another source — for example an allowable limit pressure from a bearing datasheet — without having to re-derive it via force and contact length.
Typical applications
The calculation is used to estimate the contact width corresponding to a known or allowable Hertz pressure, for example to assess lubricant film extent, to estimate the rolling contact area for wear considerations, or as a cross-check against a force-based pressure calculation.
Assumptions, limits and common mistakes
The formula applies to line contact (cylinder-cylinder or cylinder-flat) under classical Hertz theory. For point contact (sphere contact), the contact radius from the sphere-on-flat calculator should be used instead, since a circular rather than strip-shaped contact area forms there.
Common mistake: A common mistake is applying this formula to point contact (sphere) instead of line contact (cylinder) — the underlying geometry differs fundamentally. It is also easy to confuse half contact width with full contact width.
Frequently asked questions
Why does pressure alone suffice to find width?
Because force and contact length cancel out exactly when substituting the pressure formula into the width formula; see the section above.
What's the difference between half and full contact width?
Half-width b is the distance from the pressure band's center to its edge; full width 2b is the band's total extent transverse to the direction of motion.
Does this formula also apply to point contact?
No, for point contact (sphere-on-flat or sphere-sphere), the contact radius from the corresponding calculator applies instead, since a circular rather than strip-shaped area forms there.
Where do I get the reduced diameter D?
From 1/D = 1/D1 + 1/D2 for two cylinders, or directly D = D1 for the cylinder-on-flat special case.
What is contact width used for in practice?
To estimate lubricant film extent, the rolling contact area for wear estimates, or as an independent cross-check against a force-based pressure calculation.