Spur Gear Contact & Bending Stress Calculator — Safety Factors & Torque Capacity
Screen a 20° external spur-gear pair using geometry, torque, load factors and material reference stresses to estimate contact stress, Lewis bending stress, safety margins and governing torque capacity.
Enter the known values and review the calculated result
Input parameters
Use consistent values and select the intended engineering units.
Gear geometry — standard 20° spur teeth
Applied load
Load factors
Stress reference values and elasticity
Project acceptance criteria
Results
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Method, application and limitations
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Formula and calculation method
Spur gear contact and tooth bending stress calculation
This calculator performs a preliminary strength screening of an external, unshifted, 20° full-depth spur gear pair. It calculates nominal pitch-point contact stress, Lewis tooth bending stress for both the pinion and gear, stress-ratio safety margins, and the torque capacity permitted by the entered acceptance criteria.
1. Reference geometry
The pressure angle is fixed at:
α = 20°
Reference diameters:
d1 = m · z1
d2 = m · z2
Tooth-count ratio and nominal centre distance:
u = z2 / z1
a = (d1 + d2) / 2
where:
- m — transverse module [m internally]
- z1 — pinion tooth count [-]
- z2 — gear tooth count [-]
- d1, d2 — reference diameters [m]
- u — tooth-count ratio [-]
- a — nominal centre distance [m]
2. Tangential and radial tooth forces
The pinion torque is converted into tangential tooth force:
Ft = 2T / d1
The corresponding radial separation force is:
Fr = Ft · tan(α)
where T is the torque applied to the pinion [N·m].
3. Transverse contact ratio
Base-circle and addendum radii are calculated as:
rb1 = d1 · cos(α) / 2
rb2 = d2 · cos(α) / 2
ra1 = d1 / 2 + m
ra2 = d2 / 2 + m
The transverse contact ratio is:
εα = [√(ra12 − rb12) + √(ra22 − rb22) − a · sin(α)] / [π · m · cos(α)]
The implemented screening model accepts only:
1 ≤ εα < 2
4. Combined load factor
The entered load factors are multiplied:
Ktotal = KA · KV · Kβ
- KA — application factor [-]
- KV — dynamic factor [-]
- Kβ — simplified common face-load factor [-]
These factors are direct calculator inputs or may be populated from the available operating-condition presets. The calculator does not derive KV from rotational speed or Kβ from detailed alignment, lead modification, housing stiffness, or tooth-contact analysis.
5. Nominal pitch-point contact stress
The implemented zone factor is:
ZH = √[2 / (cos(α) · sin(α))]
The contact-ratio factor is:
Zε = √[(4 − εα) / 3]
Nominal contact stress is calculated from:
σH0 = ZE · 1000 · ZH · Zε · √{[Ft · Ktotal / (b · d1)] · [(u + 1) / u]}
where:
- σH0 — nominal pitch-point contact stress [Pa internally]
- ZE — elasticity coefficient entered numerically in √MPa
- b — effective face width [m internally]
- Ft — tangential tooth force [N]
The factor 1000 converts the numerical ZE value from √MPa to √Pa for the metre-based internal equation.
6. Lewis tooth bending stress
For standard 20° full-depth teeth, the calculator uses the following Lewis form-factor approximation:
Y1 = 0.484 − 2.87 / z1
Y2 = 0.484 − 2.87 / z2
Pinion bending stress:
σF1 = Ft · Ktotal / (b · m · Y1)
Gear bending stress:
σF2 = Ft · Ktotal / (b · m · Y2)
This is a Lewis bending approximation. It is not an ISO 6336-3 Method B tooth-root stress calculation.
7. Contact and bending safety margins
The calculated stresses are compared with the entered reference stress values:
SH1 = σH,ref1 / σH0
SH2 = σH,ref2 / σH0
SF1 = σF,ref1 / σF1
SF2 = σF,ref2 / σF2
If the optional gear contact or bending reference stress is omitted, the corresponding pinion reference value is used for both members.
If SH,min or SF,min is omitted, the calculator uses a required value of 1.0.
8. Screened torque capacity
The governing contact ratio relative to the project criterion is:
RH = min(SH1, SH2) / SH,min
Because the implemented contact stress varies with the square root of torque, the contact-limited torque is:
TH,allow = T · RH2
For bending:
RF = min(SF1, SF2) / SF,min
Since the Lewis bending stress varies directly with torque:
TF,allow = T · RF
The reported screened torque capacity is:
Tallow = min(TH,allow, TF,allow)
When to use this calculator
When to use this calculator
Use this calculator for preliminary strength screening of a conventional external spur gear pair with standard unshifted 20° full-depth involute teeth. It is intended for early design, comparison of candidate gear geometries, and checking whether a selected torque is compatible with user-defined contact and bending reference stresses.
Typical applications
- Estimate nominal gear contact stress at the pitch point.
- Estimate Lewis bending stress for both pinion and gear teeth.
- Calculate tangential tooth force from the applied pinion torque.
- Calculate the radial separating force generated by the 20° pressure angle.
- Evaluate the influence of module, tooth count, face width, torque, application factor, dynamic factor, and face-load factor.
- Compare calculated stresses with separate pinion and gear reference stress values.
- Perform a safety factor calculation using project-selected minimum contact and bending ratios.
- Estimate whether contact stress or tooth bending controls the screened torque capacity.
- Compare alternative spur gear sizes during concept or preliminary gearbox design.
Required design information
The calculation requires module, pinion and gear tooth counts, effective face width, pinion torque, KA, KV, Kβ, contact and bending reference stresses, and the elasticity coefficient ZE. Material and operating-condition presets can populate some of these values, while custom values can also be entered.
When not to use this calculator
Do not use this calculator as a complete gear-rating procedure for final release of a transmission. It does not implement the full ISO 6336 contact or tooth-root bending procedures and does not determine gear life, reliability, lubrication suitability, scuffing resistance, micropitting resistance, wear, tooth-flank fracture, or manufacturing-quality effects.
It is also not intended for helical gears, bevel gears, worm gears, internal gears, profile-shifted tooth systems, non-standard pressure angles, or detailed tooth-root geometry.
A final gear design may require a complete calculation according to the governing gear-rating standard, verified material data, operating spectrum, manufacturing accuracy, lubrication conditions, shaft and bearing deflections, housing stiffness, alignment, tooth modifications, and appropriate fatigue-life criteria.
How to interpret the result
How to interpret the results
Nominal contact stress, σH0
σH0 is the calculated nominal pitch-point contact stress for the gear pair. Compare it separately with the entered pinion and gear contact reference stresses. A lower σH0 increases both SH1 and SH2.
For unchanged geometry and load factors, contact stress increases approximately with the square root of torque and decreases approximately with the square root of effective face width.
Pinion and gear bending stresses, σF1 and σF2
These are simplified Lewis bending stresses. They use the same transmitted tangential force but separate Lewis form factors based on each tooth count.
For unchanged geometry factors, bending stress varies directly with torque and total load factor, and inversely with face width and module. Tooth count also affects the Lewis form factor and reference diameter, so its overall influence should be evaluated through the complete calculation rather than from one factor alone.
Contact safety margins, SH1 and SH2
These ratios compare the entered contact reference stress for each gear member with the common calculated nominal contact stress:
SH = reference contact stress / calculated contact stress
The lower of SH1 and SH2 governs the contact screening decision.
Bending safety margins, SF1 and SF2
Each bending margin compares the applicable bending reference stress with the corresponding pinion or gear Lewis stress:
SF = reference bending stress / calculated bending stress
The lower of SF1 and SF2 governs the bending screening decision.
Required safety criteria
SH,min and SF,min are project-selected acceptance criteria. They are not universal values automatically prescribed by the calculator. If either field is left blank, the implemented default is 1.0.
The entered reference stresses and required safety criteria must be selected consistently. Do not apply the same safety reduction twice through both a reduced reference stress and an additional required margin unless that is intentional in the project design basis.
Screened torque capacity
TH,allow is the torque corresponding to the required contact criterion under the current geometry, material references, and load factors. TF,allow is the equivalent bending-limited torque. The lower value becomes Tallow.
If TH,allow is lower, the contact criterion governs the current screening case. If TF,allow is lower, tooth bending governs.
Engineering status
- Safe — both the governing contact margin and governing bending margin meet or exceed the entered required values.
- Unsafe — at least one of the two screening criteria does not meet its required value.
- Invalid — the input or calculated geometry lies outside the implemented numerical or geometric domain.
Invalid conditions include non-positive module, face width, torque, load factors, material reference stresses or elasticity coefficient; non-integer or non-positive tooth counts; tooth counts below the implemented no-undercut limit; an unsupported transverse contact ratio; non-positive Lewis factors; or non-finite calculated stresses.
For the fixed 20° unshifted tooth model, the implemented no-undercut condition requires at least 18 teeth.
A Safe result means only that the two implemented screening criteria satisfy the supplied reference values and required ratios. It does not confirm complete gear safety, required service life, compliance with a gear-rating standard, acceptable lubrication, adequate manufacturing accuracy, or absence of other gear failure modes.
Calculation example
Worked example — preliminary spur gear stress screening
Consider a 20° external spur gear pair transmitting 100 N·m at the pinion. The objective is to estimate contact stress, tooth bending stress, safety margins, and the screened torque capacity.
Input data
| Parameter | Value |
|---|---|
| Module, m | 3.00 mm = 0.00300 m |
| Pinion teeth, z1 | 20 |
| Gear teeth, z2 | 60 |
| Effective face width, b | 30.0 mm = 0.0300 m |
| Pinion torque, T | 100 N·m |
| Application factor, KA | 1.25 |
| Dynamic factor, KV | 1.10 |
| Face-load factor, Kβ | 1.10 |
| Pinion contact reference stress, σH,ref1 | 1200 MPa |
| Gear contact reference stress, σH,ref2 | 1100 MPa |
| Pinion bending reference stress, σF,ref1 | 350 MPa |
| Gear bending reference stress, σF,ref2 | 300 MPa |
| Elasticity coefficient, ZE | 189.8 √MPa |
| Required contact ratio, SH,min | 1.20 |
| Required bending ratio, SF,min | 1.50 |
Step 1 — gear geometry
d1 = 0.003 × 20 = 0.060 m = 60 mm
d2 = 0.003 × 60 = 0.180 m = 180 mm
u = 60 / 20 = 3.000
a = (0.060 + 0.180) / 2 = 0.120 m = 120 mm
Step 2 — tooth forces
Ft = 2 × 100 / 0.060 = 3333.3 N
Fr = 3333.3 × tan(20°) = 1213.2 N
Step 3 — contact ratio and load factor
Using the implemented base-circle and addendum geometry:
εα = 1.6708
The value is inside the supported 1 ≤ εα < 2 screening range.
Ktotal = 1.25 × 1.10 × 1.10 = 1.5125
ZH = 2.4946
Zε = √[(4 − 1.6708) / 3] = 0.8811
Step 4 — nominal contact stress
σH0 = 189.8 × 1000 × 2.4946 × 0.8811 × √{[(3333.3 × 1.5125) / (0.030 × 0.060)] × (4 / 3)}
σH0 ≈ 806.2 MPa
Step 5 — Lewis bending stresses
Y1 = 0.484 − 2.87 / 20 = 0.3405
Y2 = 0.484 − 2.87 / 60 = 0.4362
σF1 = (3333.3 × 1.5125) / (0.030 × 0.003 × 0.3405) ≈ 164.5 MPa
σF2 = (3333.3 × 1.5125) / (0.030 × 0.003 × 0.4362) ≈ 128.4 MPa
Step 6 — safety margins
SH1 = 1200 / 806.2 = 1.488
SH2 = 1100 / 806.2 = 1.364
SF1 = 350 / 164.5 = 2.127
SF2 = 300 / 128.4 = 2.336
The governing contact margin is 1.364, which exceeds the required contact ratio of 1.20. The governing bending margin is 2.127, which exceeds the required bending ratio of 1.50.
Step 7 — screened torque capacity
TH,allow = 100 × (1.364 / 1.20)2 ≈ 129.3 N·m
TF,allow = 100 × (2.127 / 1.50) ≈ 141.8 N·m
Therefore:
Tallow ≈ 129.3 N·m
The contact criterion governs this example because its screened torque capacity is lower than the bending-limited capacity. Both implemented acceptance criteria pass at the entered torque of 100 N·m.
This result is a preliminary screening result only. It does not establish ISO 6336 compliance, fatigue life, pitting life, scuffing resistance, lubrication suitability, or complete gearbox safety.
Assumptions and limitations
Assumptions and limitations
- The model is limited to external spur gears.
- The reference helix angle is zero: β = 0°.
- The pressure angle is fixed at 20°.
- Teeth are treated as standard full-depth and unshifted.
- The reference diameter is calculated directly from d = m·z.
- The addendum radius is modeled as ra = d/2 + m.
- The calculator rejects tooth counts below the implemented no-undercut limit for unshifted 20° teeth, corresponding to z ≥ 18.
- The transverse contact ratio must satisfy 1 ≤ εα < 2.
- Contact stress is a simplified nominal pitch-point calculation and does not include separate ZB and ZD factors.
- Tooth bending is calculated with the Lewis approximation Y = 0.484 − 2.87/z. It is not the ISO 6336-3 Method B root-stress calculation.
- The Lewis model does not explicitly resolve the generated root fillet, local root stress concentration, detailed load position, rim flexibility, or complete tooth-root geometry.
- KA, KV, and Kβ are supplied factors and are multiplied into one common Ktotal used by both implemented stress models.
- KV is not calculated from RPM, pitch-line velocity, gear quality, transmission error, or manufacturing accuracy.
- Kβ is not calculated from shaft deflection, bearing displacement, housing deformation, misalignment, crowning, lead correction, or a tooth-contact analysis.
- The model does not use separate transverse contact-load factors or separate contact and bending face-load factors.
- ZE is an input or preset value. The calculator does not calculate it from elastic modulus and Poisson's ratio inside the stress solver.
- Contact and bending reference stresses are inputs or preset values. The solver does not derive them from hardness, heat treatment, fatigue curves, load cycles, reliability, temperature, size, surface condition, lubrication, or material-quality factors.
- The optional gear reference stresses default to the corresponding pinion reference stresses when omitted.
- The optional required contact and bending ratios default to 1.0 when omitted.
- The screened torque-capacity calculation assumes that geometry, load factors, material reference values, and all other inputs remain constant as torque changes.
- The calculator does not calculate gear fatigue life or accumulated damage under a variable load spectrum.
- Pitting life, micropitting, scuffing, abrasive or adhesive wear, plastic deformation, tooth-flank fracture, case crushing, and other gear failure mechanisms are not evaluated.
- Lubricant type, viscosity, temperature, surface roughness, pitch-line velocity, and lubrication regime are not modeled.
- Backlash, tooth thickness tolerances, profile modifications, lead modifications, crowning, manufacturing deviations, and actual contact pattern are not modeled.
- Internal gears, helical gears, bevel gears, worm gears, racks, non-standard pressure angles, and profile-shifted gear pairs require separate verification.
- A favorable screening result must not be treated as certification or complete verification according to ISO 6336, AGMA, or another gear-design standard.
Applicable standards
Standards context
The implemented contact model is a simplified, ISO 6336-inspired nominal pitch-point screening calculation. The implemented bending model uses the classical Lewis form-factor approximation.
The calculator does not implement the complete procedures required for ISO 6336-1 general influence factors, ISO 6336-2 surface durability, or ISO 6336-3 tooth bending strength. It must therefore not be described as an ISO 6336-compliant gear rating.
For final design, select load factors, material limits, safety criteria, life factors, lubrication assumptions, manufacturing quality, and all other required parameters according to the governing design standard and the actual transmission application.