Hertzian Contact Stress Calculator: Cylinder, Roller & Sphere

Line Contact (Cylinders & Cam Rollers) vs. Point Contact (Spheres & Ball Bearings)

When designing cam-and-roller followers, linear guideways, or dowel pins in bores from the gear pitch diameter and module calculator, simple projected bearing area (F / (D · L)) grossly underestimates the real local stress. Because two convex surfaces initially touch along a theoretical line of zero width, elastic compression governed by Young's modulus E and Poisson's ratio ν flattens a microscopic rectangular strip of width 2b across which the pressure follows a semi-elliptical cylinder distribution:

• Reduced Contact Modulus E*: 1 / E* = (1 − ν1²) / E1 + (1 − ν2²) / E2 (for steel on steel with E = 210 GPa, ν = 0.30, E* = 115.38 GPa).
• Equivalent Curvature Radius Req: For two external convex cylinders, Req = (R1 · R2) / (R1 + R2); for a cylinder on a flat way (R2 → ∞), Req = R1; and for a pin inside a conformal bore socket (R2 > R1), Req = (R1 · R2) / (R2 − R1).

Wall edge and short line. The line formulas assume that the contact band lies clear of the part's edge and that the band is short compared with the line length. If the load acts within the half-width b of a shoulder, a bore edge or a corner, the local pressure is higher than the formula gives, so check that edge separately. When 2b exceeds L, the calculator withholds p_max and τ_max; when L/b is low, end effects make the formula optimistic, so read the L/b value shown with each result before you treat the contact as a pass.

Why Spalling Fatigue Starts Below the Surface at Depth z_τ = 0.786b

At the center of the contact patch on the surface (z = 0), all three principal stresses are compressive (σz = −pmax, σx = −pmax, σy = −2ν·pmax), so the surface shear stress is nearly zero. As you move below the surface along the load axis, horizontal compression decays faster than vertical compression, creating a peak Tresca shear stress of τmax = 0.300 · pmax at depth zτ = 0.786 · b for line contact (or τmax = 0.310 · pmax at zτ = 0.480 · a for spherical point contact).

As detailed in Machinery's Handbook (31st Ed., Industrial Press), this subsurface shear peak is why rolling element bearings and cams fail by subsurface spalling (pitting) rather than surface crushing—and why the depth z_τ matters for case design. This page reports z_τ only: the case depth comes from your heat-treat specification, CHD measured at 550 HV1 as defined in ISO 2639, and the HRC / HV hardness converter relates the two scales.

Worked Examples: External Cam Follower, Conformal Pin Socket & Steel Ball on Flat Race

Worked Example 1 (External Cam Roller vs Conformal Pin in Bore at 5 kN): Pressing a R1 = 20 mm steel cam roller against a R2 = 40 mm convex cam over L = 25 mm at F = 5,000 N yields Req = 13.33 mm, half-width b = 0.172 mm, peak pressure p_max = 742.2 MPa, and subsurface shear τ_max = 222.7 MPa at z = 0.135 mm. If that same R1 = 20 mm pin seats inside a conformal R2 = 20.5 mm bore socket, Req jumps to 820.0 mm, spreading the contact half-width to 1.345 mm and dropping peak pressure to 94.6 MPa!

Edge and length check for Example 1: the band is 2b = 0.343 mm against L = 25 mm, so the line formula applies (L/b = 145.739, shown in the output). The band must still stay clear of the shoulder: if the roller runs near a wall edge, check that edge pressure separately.

Worked Example 2 (Sphere on Flat Race at 500 N): A 1" diameter steel ball (R1 = 12.7 mm) loaded to 500 N against a flat hardened plate forms a circular contact patch of radius a = 0.346 mm with peak pressure 1998.9 MPa (289.9 ksi) and maximum shear 619.7 MPa at depth 0.166 mm. At this load the peak pressure of 1998.9 MPa exceeds the 1500 MPa steel-on-steel allowable, so the calculator flags it (p_max above the material limit; elastic Hertz theory no longer valid): reduce the load or choose a pair with a higher allowable. For precision spherical gaging, see the internal taper bore two-ball calculator and dowel pin press-fit and slip-fit chart.

The metric tolerances referenced for the contact geometry follow International Organization for Standardization (ISO) ISO 286, and unit conversions follow NIST Special Publication 811. Before you rely on p_max, confirm that the contact is a line (2b ≤ L) and that p_max stays below the allowable of your material. Above that limit the elastic Hertz result no longer describes the contact, and the calculator flags it instead of publishing a safe number.

For the bronze pair, the allowable is the minimum yield strength of C93200 (138 MPa), so the factor shown is a yield-based site criterion, not a contact-fatigue life. Use it to screen a design, and check the wear and fatigue life of the bronze bearing separately.

Next step: Convert required 58–62 HRC case hardness in the ASTM E140 hardness converter, and size gear tooth geometry in the spur gear module calculator.