Calculation
pH = √[(FN/L)·E* / (π·D/2)]
The special case of line contact with a flat surface as the second partner (infinite diameter); typical for a roller running on a flat rail or track.
pH = √((FN/L)·E*/(π·D/2))
Hertz contact pressure for cylinder-on-flat contact
The special case of line contact with a flat surface as the second partner (infinite diameter); typical for a roller running on a flat rail or track.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Maximum Hertz pressure pH — Peak pressure at the center of the contact band.
- Normal force FN — Contact force acting normal to the contact surface.
- Contact length L — Length of the line of contact between cylinder and flat.
- Cylinder diameter D — Diameter of the cylinder resting or rolling on the flat surface.
- Reduced elastic modulus E* — Effective elastic modulus of the material pair, see the linked calculator.
Example
5,000 N over a 50 mm contact length on a 40 mm steel cylinder (E* ≈ 115.4 GPa) give pH ≈ 429 MPa.
Assumptions and limits
Classical Hertz theory with an ideally flat counter-body (infinite diameter); real tracks have finite curvature and roughness that additionally affect pressure.
Understand Hertz contact pressure for cylinder-on-flat contact
This calculator finds the maximum Hertz contact pressure between a cylinder and a flat surface from normal force, contact length, cylinder diameter and the reduced elastic modulus.
What does this quantity describe?
Cylinder-on-flat contact is the special case of line contact where the second contact partner has an infinitely large diameter (a flat surface). The reduced radius then simplifies to R = D/2, and maximum pressure follows from pH = √[(FN/L)·E*/(π·D/2)].
This is like a roller running on a flat track or rail: the track itself effectively has no curvature, so only the roller's curvature enters the calculation — a simpler special case of the general cylinder-on-cylinder contact.
Formula and variables
pH = √[(FN/L)·E* / (π·D/2)]
pH = √[(FN/L)·E* / (π·D/2)]R = D/2
| Symbol / input | Meaning |
|---|---|
| Maximum Hertz pressure pH | Peak pressure at the center of the contact band. |
| Normal force FN | Contact force acting normal to the contact surface. |
| Contact length L | Length of the line of contact between cylinder and flat. |
| Cylinder diameter D | Diameter of the cylinder resting or rolling on the flat surface. |
| Reduced elastic modulus E* | Effective elastic modulus of the material pair, see the linked calculator. |
Choose the inputs correctly
Normal force FN, contact length L, cylinder diameter D and the reduced elastic modulus E* set the maximum pressure.
How to use the calculator
First compute E* with the reduced elastic modulus calculator if not already known. Then enter force, contact length and cylinder diameter to find maximum pressure.
Worked example
5,000 N over a 50 mm contact length on a 40 mm steel cylinder against a flat steel surface (E* ≈ 115.4 GPa) give pH ≈ 429 MPa.
Understand the result and units
Comparison with the general cylinder-on-cylinder calculator shows: at the same force, length and a 40 mm first diameter, pressure against a flat (429 MPa) is lower than against a second, finite 80 mm cylinder (525 MPa) — a larger counter-body curvature radius always spreads load over a larger area.
Force is given in N, length and diameter in mm, the reduced elastic modulus in GPa, and the resulting pressure in MPa.
A special case with one simplification
Mathematically, cylinder-on-flat contact follows from general cylinder-on-cylinder contact as the second diameter D2 approaches infinity: the term 1/D2 in the reduced-radius formula 1/R = 2/D1 + 2/D2 vanishes, leaving only R = D1/2. This simplification makes the calculator more compact, needing only one diameter instead of two, but is physically valid only when the second body is actually approximately flat.
Typical applications
The calculation is used to design rollers on tracks, cam followers against flat plungers, linear guides with cylindrical rolling elements on a guide rail, and similar cylinder-on-flat contacts in machine design.
Assumptions, limits and common mistakes
Classical Hertz theory with an ideally flat, infinitely extended counter-body. Real tracks or rails have finite width and possibly slight transverse curvature, which can alter actual pressure distribution compared with this idealized model.
Common mistake: A common mistake is applying this special case where the "flat" surface actually has noticeable transverse curvature of its own (for example a slightly crowned running surface) — the general cylinder-on-cylinder calculator with the actual second diameter should be used instead.
Frequently asked questions
When is the cylinder-on-flat special case valid?
When the second contact partner is actually approximately flat, meaning a curvature radius much larger than the cylinder's, or infinite.
How does the result differ from the general cylinder-on-cylinder case?
At the same force and first diameter, pressure against a flat is always lower than against a second, finite cylinder, since a flat surface accommodates deformation less than a curved one.
What if the track has slight curvature of its own?
Then the general cylinder-on-cylinder calculator should be used instead, with the actual (possibly very large) second diameter.
What role does contact length L play?
A larger contact length spreads the same force over a larger area, lowering pressure; L enters the formula directly under the square root.
Is this calculation directly usable for linear guides?
As pre-sizing, yes; a complete design additionally needs rolling-element count, load distribution and service life per manufacturer data.