2026-07-30 8 min

Hertz Contact Stress Calculator

Calculate Hertzian contact stress, contact area, and subsurface stress for curved surfaces in contact. Critical for bearing, gear, and cam design.

Calculator

Unit System

Contact Geometry

Load F (N)

Radius R₁ (mm)

Radius R₂ (mm)

Young\'s Modulus E₁ (GPa)

Young\'s Modulus E₂ (GPa)

Poisson Ratio ν₁

Poisson Ratio ν₂

Results

Normal

Max Contact Pressure (p_max)

1552.0759GPa

Maximum Hertzian contact pressure at center of contact

Contact Radius (a)

0.5546mm

Radius of circular (or half-width of rectangular) contact patch

Contact Area

0.9664mm²

Total area of contact between curved surfaces

Max Subsurface Shear Stress (τ_max)

481143.5157MPa

Maximum shear stress below surface — governs yielding

Depth of τ_max

0.2662mm

Depth below surface where τ_max occurs

Approach Distance (δ)

12.3052μm

Elastic approach of the two bodies

Visualizations

Results Overview

1,552GPaMax Contact Pressure

0.6GPaContact Radius

1GPaContact Area

481,144GPaMax Subsurface Shear Stress

0.3GPaDepth of τ_max

12.3GPaApproach Distance

Formula

Hertz theory describes the local stress and deformation when two curved elastic bodies are pressed together.

a = (3FR_red / (4E_red))^(1/3), p_max = 3F / (2πa²)

Alternative Form

τ_max = 0.31 × p_max (spherical), τ_max = 0.30 × p_max (cylindrical)

Variables

Symbol	Name	Unit	Description
p_max	Max Contact Pressure	GPa	Peak compressive stress at center of contact
a	Contact Radius	mm	Radius of circular contact patch (or half-width for cylinders)
τ_max	Max Shear Stress	MPa	Critical subsurface stress — governs plastic yielding
δ	Approach Distance	μm	Relative displacement of the two bodies

Explanation

Hertzian contact pressure follows a semi-elliptical distribution: p(r) = p_max × √(1 - (r/a)²). The maximum shear stress occurs below the surface (not at the surface), which explains why subsurface-initiated fatigue (pitting) is common in rolling contacts like bearings and gears.

Derivation

From Hertz (1882): For two spheres of radii R₁, R₂ under load F, the reduced radius R_red = (1/R₁ + 1/R₂)⁻¹ and reduced modulus E_red = ((1-ν₁²)/E₁ + (1-ν₂²)/E₂)⁻¹. The contact radius a = (3FR_red/(4E_red))^(1/3) and p_max = 3F/(2πa²). The maximum shear stress τ_max = 0.31 × p_max at depth z = 0.48a below the surface.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Calculate the reduced radius and reduced modulus from geometry and material properties.

R₁:50.0000 mm

R₂:50.0000 mm

E₁:200.0000 GPa

E₂:200.0000 GPa

Result: Reduced parameters computed

Step 2

Determine the contact patch dimensions from the load and reduced properties.

a = (3FR_red / (4E_red))^(1/3)

Load:1000.0000 N

Intermediate: a = (3 × 1000.0000 × R_red / (4 × E_red))^(1/3)

Result: a = 0.5546 mm

Step 3

Compute the maximum contact pressure and subsurface stress distribution.

p_max = 3F / (2πa²)

Contact Radius a:0.5546 mm

Contact Area:0.9664 mm²

Intermediate: p_max = 3 × 1000.0000 / (2π × 0.5546²)

Result: p_max = 1552.0759 GPa

Final Answer

1552.0759

Worked Examples

See how the calculator works with different scenarios.

Scenario

A 6204 ball bearing with 10 mm diameter balls in contact with a 40 mm inner race. Load 500 N per ball. Steel (E = 200 GPa, ν = 0.3).

Inputs

Geometry:Sphere on Sphere

Ball Radius:5 mm

Race Radius:20 mm

Load:500 N

Material:Steel

Calculation

R_red = (1/5 + 1/20)⁻¹ = 4 mm E_red = 200/(2×(1-0.3²)) = 109.9 GPa a = ((3×500×4)/(4×109900))^(1/3) = 0.258 mm p_max = 3×500/(2π×0.258²) = 3585 MPa = 3.59 GPa τ_max = 0.31×3.59 = 1.11 GPa at depth 0.124 mm

Result

p_max = 3.59 GPa, a = 0.258 mm, τ_max = 1.11 GPa

Interpretation

The high contact pressure (3.59 GPa) is typical for ball bearings. The subsurface shear stress (1.11 GPa) determines the fatigue life — bearings are rated for millions of cycles at this stress level with proper lubrication.

Understanding Your Result

Your Result: Max Contact Pressure (p_max): 1552.0759 GPa

Typical range: Ball bearings: p_max 1-4 GPa. Roller bearings: p_max 1-3 GPa. Gear teeth: p_max 0.5-2 GPa. Cam followers: p_max 0.5-1.5 GPa. Human joints: p_max 3-8 MPa.

What This Means

Hertz contact stress quantifies the extreme local pressures that occur when curved surfaces contact. Even at modest loads, the pressure at the contact patch can reach several GPa.

Why It Matters

Contact stress determines bearing life (pitting fatigue), gear durability, cam wear, and tribological performance. Understanding Hertz stress is fundamental to designing reliable rolling contacts.

Professional Interpretation

The maximum subsurface shear stress (τ_max ≈ 0.31 × p_max) is the critical value for yielding per von Mises or Tresca criteria. For bearing steels, p_max should not exceed 4 GPa for infinite life with proper lubrication. Surface-initiated fatigue can occur if the surface is damaged or poorly lubricated.

Recommendations

•For bearing design: use dynamic load ratings and L₁₀ life calculations, not just static stress
•Consider EHL film thickness: λ ratio (film thickness/roughness) > 3 for full film separation
•Surface finish matters: smoother surfaces allow lower λ ratios before asperity contact
•Apply safety factors: 1.2-1.5 for steady loads, 2-3 for impact loads

Educational Content

Introduction

Hertz contact theory, developed by Heinrich Hertz in 1882, describes the stresses and deformation when two curved elastic bodies are pressed together. It is one of the most important theories in contact mechanics and forms the basis for bearing, gear, and cam design.

Concept Explanation

The theory assumes small strains, frictionless surfaces, and elliptical pressure distribution. For spherical contact, the pressure distribution is p(r) = p_max × √(1 - (r/a)²). The maximum tensile stress occurs at the edge of the contact circle (radial tension), while the maximum shear stress occurs at depth below the surface, explaining why pitting fatigue initiates subsurface.

Applications

▸Rolling element bearing design (ball, roller, tapered)

▸Spur, helical, and bevel gear contact analysis

▸Cam and follower design

▸Rail-wheel contact in railway engineering

▸Indentation hardness testing (Brinell, Rockwell)

▸Prosthetic joint design (hip, knee implants)

▸Mechanical seal face contact analysis

Benefits

•Provides closed-form analytical solution for contact stress
•Enables bearing life prediction (ISO 281, AFBMA)
•Guides material selection for high-contact-stress applications
•Predicts subsurface fatigue initiation depth
•Establishes allowable load limits for rolling contacts

Limitations

•Assumes perfectly smooth surfaces (no roughness effects)
•Frictionless contact assumption (no tangential tractions)
•Small strain assumption limits applicability at high loads
•Does not account for plastic deformation or residual stresses

Best Practices

•Verify that p_max < material hardness/3 for elastic contact
•Consider the λ ratio (EHL film thickness / composite roughness)
•Account for residual stresses from heat treatment
•Use dynamic load ratings (C) for bearing selection, not static analysis alone
•Include lubrication effects for realistic film thickness predictions

Common Misconceptions

Myth: Maximum stress is at the contact surface.

Truth: The maximum shear stress occurs beneath the surface (at depth z ≈ 0.48a for spherical contact). This is why fatigue cracks often initiate below the surface in rolling contacts.

Myth: Hertz theory applies to all contact problems.

Truth: Hertz theory assumes frictionless, elastic contact of smooth surfaces. It does not apply to conformal contacts (e.g., plain bearings), frictional sliding, or contacts with plastic deformation.

Myth: Higher contact pressure always means faster wear.

Truth: While higher pressure accelerates adhesive and abrasive wear, in EHL contacts, adequate film thickness can prevent wear even at high contact pressures. Bearings operate successfully at 3-4 GPa with proper lubrication.

Practical Tips

•For a sphere on a flat, use R₂ = a very large value (e.g., 9999 mm) to approximate infinite radius
•The reduced modulus combines both materials: softer material dominates the compliance
•Contact area increases as F^(2/3) for spheres and F^(1/2) for cylinders — not linearly with load
•The maximum tensile stress occurs at the contact edge (radial direction) and can cause cone cracks in brittle materials
•For conformal contacts (bearing race with ball), use the exact radii (not effective curvature)
•The approach distance δ is typically a few micrometers — measurable with precision sensors
•Lubricant viscosity increases exponentially with pressure in EHL contacts, enabling film formation at high loads

Common Mistakes

Using nominal pressure instead of Hertz pressure for curved contacts

Always use Hertz theory for curved surface contacts. The peak pressure can be 3-5× the average pressure, and using average values significantly underestimates contact stress.

Neglecting the effect of Poisson ratio on reduced modulus

Include the (1-ν²) terms in the reduced modulus calculation. Neglecting this overestimates the stiffness by 8-10% for typical metals.

Applying Hertz theory to conformal contacts

Hertz theory is valid for non-conforming (curved-on-curved) contacts only. For conformal contacts like journal bearings, use different analysis methods (e.g., Petrow's equation, Sommerfeld number).

Ignoring the effect of surface roughness on contact stress

Real surfaces have asperities that locally increase contact pressure. The λ ratio (EHL film / roughness) determines whether the Hertz pressure is accurate or asperity contacts dominate.

Assumptions

This calculator makes the following assumptions to provide estimates:

Frictionless Surfaces: Hertz theory assumes no friction between contacting surfaces. Tangential tractions from friction alter the stress distribution, particularly the location and magnitude of maximum shear stress.

Small Strains: The deformation is assumed small compared to body dimensions. This limits applicability for soft materials or very high loads where finite deformation occurs.

Smooth Surfaces: Surfaces are assumed perfectly smooth. Surface roughness creates local stress concentrations at asperities that can exceed the theoretical Hertz pressure.

Elastic Material Behavior: Both materials are assumed linear elastic. Plastic deformation at the contact changes the pressure distribution and invalidates the Hertz solution.

Limitations

Be aware of these limitations when using this calculator:

No Lubrication Effects: Hertz theory does not account for lubricant pressure or film thickness. In EHL contacts, the lubricant pressure follows the Hertz distribution but with a secondary pressure spike at the outlet.

Static Loading Only: The theory addresses static contact. Dynamic loading, vibration, and impact require different analytical or numerical approaches.

Geometry Restrictions: Hertz theory applies only to non-conforming surfaces. Conformal contacts (e.g., pin in a bushing, O-ring in a groove) require different analysis methods.

No Surface Tractions: Friction and surface shear stresses alter the subsurface stress field, moving the location of maximum shear stress toward the surface and potentially causing surface-initiated fatigue.

Frequently Asked Questions

Answers to common questions about this calculator.

What is Hertz contact stress?

Hertz contact stress is the localized compressive stress that occurs when two curved elastic surfaces are pressed together. It follows an elliptical pressure distribution with the maximum at the center of the contact patch.

How is Hertz contact stress calculated?

For spheres: p_max = 3F/(2πa²) where a = (3FR_red/(4E_red))^(1/3). For cylinders: p_max = 2F/(πaL) where a = √(4FR_red/(πE_redL)). R_red and E_red combine geometry and material properties.

What is the reduced modulus E_red?

E_red = ((1-ν₁²)/E₁ + (1-ν₂²)/E₂)⁻¹. It combines the elastic properties of both materials. The softer material dominates the contact compliance.

What is the contact radius a?

The contact radius a is the radius of the circular contact area between two spheres (or half the contact width for cylinders). It varies with the cube root of load for spherical contacts.

Where does maximum shear stress occur in Hertz contact?

In spherical contact, τ_max occurs at depth z ≈ 0.48a below the surface. In cylindrical contact, it occurs at z ≈ 0.78a. This subsurface location explains why pitting fatigue initiates below the surface.

What is the relationship between p_max and average pressure?

For spherical contact, p_avg = 2/3 × p_max. The average contact pressure is 2/3 of the peak pressure due to the elliptical pressure distribution.

How does load affect contact area?

Contact area increases with load as Area ∝ F^(2/3) for spheres and Area ∝ F^(1/2) for cylinders. The nonlinear relationship means contact pressure increases less rapidly than load.

What is a typical Hertz stress for ball bearings?

Ball bearings typically operate at p_max of 2-4 GPa. High-performance bearings may reach 4.5 GPa. The exact value depends on bearing geometry, load, and material (bearing steel hardness ~60 HRC).

How does Hertz stress differ for cylinders vs spheres?

Cylindrical contact produces a rectangular contact area with elliptical pressure distribution in the width direction. Spherical contact produces a circular contact area. Cylindrical contacts have slightly lower τ_max/p_max ratio (0.30 vs 0.31).

What is the significance of subsurface shear stress?

The subsurface shear stress drives plastic yielding (per von Mises/Tresca) and causes subsurface fatigue crack initiation. This leads to pitting — the primary failure mode in rolling contacts.

How does Poisson ratio affect contact stress?

Poisson ratio affects the reduced modulus and therefore the contact stiffness. Higher ν increases the (1-ν²) term, reducing E_red and increasing contact area. Typical values: steel 0.3, aluminum 0.33, rubber 0.5.

What is the approach distance δ?

The approach distance is the relative displacement of the centers of the two bodies due to elastic deformation at the contact. It is typically small (micrometers to millimeters) even under substantial loads.

How does material hardness relate to allowable contact stress?

For elastic contact, p_max should not exceed approximately 0.4× material hardness. Above this, significant plastic deformation occurs. Bearing steels with hardness 60 HRC (≈ 700 HV) can support up to about 280 MPa nominal stress.

What is the role of lubrication in Hertz contact?

Lubrication in EHL contacts generates a thin film (0.1-1 μm) that separates surfaces. The lubricant pressure follows the Hertz distribution but with a characteristic pressure spike at the outlet. Adequate film thickness (λ > 3) prevents asperity contact.

How does surface roughness affect Hertz contact?

Surface roughness causes local stress concentrations at asperities that exceed the theoretical Hertz pressure. The composite roughness and λ ratio determine whether the Hertzian solution is valid or asperity contact dominates.

What is the difference between static and dynamic contact stress?

Hertz theory describes static contact. Dynamic effects (vibration, impact) can produce instantaneous pressures much higher than static values. Bearing dynamic capacity (C) accounts for rotating loads.

How do I calculate contact stress for crossed cylinders?

Crossed cylinders (e.g., roller in a concave race) can be approximated using equivalent radii. The contact is elliptical, requiring elliptical integral solutions. For preliminary design, the sphere-sphere approximation with equivalent radius often suffices.

What is the maximum contact stress before plastic yielding?

Yielding begins when τ_max exceeds the shear yield strength (≈ 0.5-0.6 × tensile yield strength). For steel (yield = 500 MPa), τ_max should not exceed about 300 MPa, limiting p_max to about 1000 MPa for the first yield.

How does contact stress affect bearing fatigue life?

Bearing fatigue life is inversely proportional to p_max^3 to p_max^5 depending on bearing type. Reducing contact stress by 10% can increase bearing life by 30-60%. This is why bearing selection is critical.

What are the assumptions of Hertz contact theory?

The key assumptions are: isotropic linear elastic materials, frictionless surfaces, small strains, continuous (smooth) surfaces, and radii much larger than the contact area. Deviations from these require more advanced analysis.

How is Hertz stress measured experimentally?

Contact pressure cannot be measured directly. Methods include: pressure-sensitive film (Fujifilm Prescale), ultrasonic measurement, photoelasticity, and FEM validation. Strain gauges on the back surface can infer contact loads.

What is the effect of temperature on contact stress?

Temperature affects material properties (E, ν, yield strength) and thermal expansion. At elevated temperature, reduced modulus and yield strength increase contact area and decrease allowable stress.

How do residual stresses affect Hertz contact?

Residual compressive stresses (from heat treating, shot peening) oppose tensile stresses at the contact edge, increasing fatigue resistance. Tensile residual stresses accelerate crack initiation and propagation.

What is the difference between Hertz contact and nominal contact?

Hertz contact applies to curved (non-conforming) surfaces where the contact area is small relative to body dimensions. Nominal contact applies to flat surfaces where the apparent area equals the geometric overlap.

How does contact stress distribution affect surface fatigue?

The cyclic subsurface shear stress range drives fatigue crack growth. The depth of τ_max determines where cracks initiate. If surface damage (scratches, dents) is present, surface-initiated fatigue can occur at lower loads.

What is the pressure spike in EHL contacts?

In elastohydrodynamic lubrication, a secondary pressure peak occurs at the outlet of the contact due to the sharp film thickness reduction. This spike can exceed the Hertz pressure by 10-30% and is a critical location for fatigue initiation.

How do I account for multiple contacts in bearings?

Bearings have multiple rolling elements sharing the load. The load distribution depends on internal clearance, preload, and applied forces. The maximum loaded rolling element carries approximately 4.37× the average load for radial ball bearings.

What is the role of the pressure-viscosity coefficient in EHL?

The pressure-viscosity coefficient (α) determines how much lubricant viscosity increases under pressure. Typical oils increase viscosity by 100-1000× at 1 GPa, enabling the formation of load-carrying EHL films at extreme pressures.

How does contact stress affect noise and vibration?

High contact stress with inadequate lubrication causes metal-to-metal contact, generating vibration and noise (bearing whine). Proper EHL film formation dampens vibration and reduces noise in rolling element bearings.

Related Calculators

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Related Guides

Deepen your understanding with these educational resources.

Hertz Contact Theory Overview

Educational resource on contact mechanics

Bearing Contact Stress Analysis Guide

SKF bearing stress analysis reference

Gear Contact Stress Calculation

AGMA gear rating standards

Glossary

Technical terms used in mortgage calculations explained in plain language.

Approach Distance (δ): Elastic displacement of the centers of two contacting bodies due to local deformation at the contact.
Contact Radius (a): Radius of circular contact area between two spheres, or half the contact width for cylindrical contact.
Elastohydrodynamic Lubrication (EHL): Lubrication regime where elastic deformation of surfaces and pressure-viscosity effects enable film formation under high contact pressure.
Hertzian Contact: Contact stress theory for curved elastic bodies with elliptical pressure distribution, developed by Heinrich Hertz in 1882.
Maximum Contact Pressure (p_max): Peak compressive stress at the center of the Hertzian contact patch.
Pitting: Subsurface fatigue failure in rolling contacts where material spalls away, leaving pits on the surface.
Reduced Modulus (E_red): Combined elastic modulus accounting for both materials: E_red = ((1-ν₁²)/E₁ + (1-ν₂²)/E₂)⁻¹.
Reduced Radius (R_red): Effective radius of curvature combining both surfaces: R_red = (1/R₁ + 1/R₂)⁻¹.
Subsurface Shear Stress (τ_max): Maximum shear stress occurring below the contact surface, responsible for plastic yielding and fatigue initiation.
Von Mises Criterion: Yield criterion predicting onset of plastic deformation based on distortional energy, commonly used for ductile materials.

Authoritative References

Trusted sources and standards referenced by this calculator.

Hertz, H. "On the Contact of Elastic Solids" (J. Reine Angew. Math., 1882) Johnson, K.L. "Contact Mechanics" (Cambridge University Press, 1985) ISO 281 - Rolling Bearings — Dynamic Load Ratings and Rating Life AGMA 2001-D04 - Fundamental Rating Factors for Involute Spur and Helical Gear Teeth Tribology Data Handbook (Bhushan) - Contact Mechanics Section SKF Bearing Design Guide - Hertz Contact Stress

Applicable Standards

ISO 281ISO 4287AGMA 2001-D04ASTM G99DIN 50324ASME B106.1M

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Last Updated:

2026-07-30

Version:

2.0.0

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