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Fatigue Mean-Stress Explorer

Compare Goodman, Gerber, and Soderberg mean-stress corrections side by side. See how mean stress shifts the Haigh diagram, changes predicted life, and separates the three classical methods.

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Haigh Diagram

What this shows: The allowable alternating stress as a function of mean stress for each method. The operating point (red marker) should lie below all active envelope lines for the design to pass at the selected reference stress. Yield line (gray dashed) bounds the envelope regardless of method.

S-N Curve

What this shows: The base material S-N curve with each method's predicted operating point plotted. Methods that predict infinite life are not shown as markers. The vertical dashed line is the target life; the horizontal dashed line is the endurance limit.

Life vs Mean Stress Sweep

What this shows: How predicted fatigue life varies as mean stress is swept across the range, holding alternating stress constant. Null/invalid regions appear as line breaks. The vertical dashed line marks the current operating mean stress; the horizontal dashed line marks target life.
Contents
  1. Stress State Definitions
  2. The Haigh Diagram
  3. Mean-Stress Correction Methods
  4. Why Tensile Mean Stress Hurts Fatigue
  5. Compressive Mean Stress: Not a Free Credit
  6. When Not to Use This Tool
  7. Worked Example
  8. References

1. Stress State Definitions

A fluctuating stress cycle is characterized by four quantities that are all interrelated:

Mean and Alternating Stress $$\sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2}, \quad \sigma_a = \frac{\sigma_{max} - \sigma_{min}}{2}$$
  • σm: Mean (steady) stress component
  • σa: Alternating (amplitude) stress component
  • σmax: Maximum stress in the cycle
  • σmin: Minimum stress in the cycle
Stress Ratio R $$R = \frac{\sigma_{min}}{\sigma_{max}} = \frac{\sigma_m - \sigma_a}{\sigma_m + \sigma_a}$$
  • R = -1: Fully reversed (zero mean stress, pure bending)
  • R = 0: Pulsating (minimum stress is zero)
  • R = +1: Static load (no alternating component)
Convention: Standard S-N curves are generated at R = -1 (fully reversed loading). Any other R ratio introduces a mean stress that must be accounted for — that is exactly what the methods below do.

2. The Haigh Diagram

The Haigh diagram (also called the Modified Goodman diagram) plots allowable alternating stress on the y-axis against mean stress on the x-axis. Any design point lying below and to the left of the envelope line is considered acceptable under the chosen criterion.

The envelope always starts at the endurance limit S_e (or a finite-life equivalent) on the y-axis (zero mean stress) and terminates at the material's characteristic strength on the x-axis (S_ut for Goodman, S_y for Soderberg, or S_ut with a parabolic path for Gerber). The yield line σm + σa = S_y is also plotted as a practical upper bound.

3. Mean-Stress Correction Methods

No Correction (R = -1 baseline)

The alternating stress is compared directly to the endurance limit or finite-life S-N curve without any mean-stress correction. This is appropriate only when the mean stress is negligible or for initial screening. The envelope is a horizontal line at S_e.

No Correction $$\sigma_{a,eq} = \sigma_a$$
  • Ignores mean stress entirely
  • May be non-conservative for large tensile mean stresses

Goodman (Modified Goodman)

The most widely used method in industry. It assumes a linear relationship between the endurance limit and the ultimate tensile strength on the Haigh diagram. Typically regarded as slightly conservative (safe) for steels.

Modified Goodman $$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = 1 \quad \Longrightarrow \quad \sigma_{a,eq} = \frac{\sigma_a}{1 - \sigma_m / S_{ut}}$$
  • σa: Applied alternating stress (MPa)
  • σm: Applied mean stress (MPa)
  • S_e: Modified endurance limit (MPa)
  • S_ut: Ultimate tensile strength (MPa)

Gerber Parabola

Uses a parabolic (quadratic) envelope that passes through the same endpoints as Goodman but curves upward. It tends to be less conservative than Goodman for steels with moderate-to-high mean stress ratios, and fits experimental data for ductile metals more closely on average.

Gerber Criterion $$\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_{ut}}\right)^2 = 1 \quad \Longrightarrow \quad \sigma_{a,eq} = \frac{\sigma_a}{1 - (\sigma_m / S_{ut})^2}$$
  • Parabolic envelope — less conservative than Goodman
  • Better statistical fit to ductile metal test data
  • Does not distinguish tensile vs compressive mean stress as clearly

Soderberg

Replaces the ultimate strength with the yield strength as the x-intercept, producing the most conservative of the three methods. It simultaneously prevents yielding under the peak cycle stress. Rarely used in modern practice because it is overly penalizing for high-strength steels.

Soderberg Criterion $$\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = 1 \quad \Longrightarrow \quad \sigma_{a,eq} = \frac{\sigma_a}{1 - \sigma_m / S_y}$$
  • σa: Applied alternating stress (MPa)
  • S_y: Yield strength (MPa)
  • Most conservative — effectively guarantees no yielding on peak half-cycle

4. Why Tensile Mean Stress Hurts Fatigue Performance

A tensile mean stress holds a crack-tip open during the compressive half of a cycle, allowing fatigue crack growth to occur over a larger portion of each cycle. Empirically, all three classical methods predict a monotonically decreasing allowable alternating stress as mean stress increases toward S_ut (or S_y for Soderberg).

In fracture-mechanics terms, the effective stress-intensity factor range ΔK depends on the applied stress ratio R — higher R (more tensile mean) increases ΔK for the same amplitude, accelerating fatigue crack growth per cycle.

5. Compressive Mean Stress: Not a Free Design Credit

Caution with compressive mean stress: Classical methods (Goodman, Gerber, Soderberg) may predict beneficially infinite allowable alternating stress for compressive mean stresses. This is not reliable in practice.

Compressive residual stresses (from shot-peening, cold-rolling, etc.) do improve fatigue life in practice, but classical Haigh-diagram methods were not designed to quantify this reliably. Additional considerations include:

  • Relaxation of compressive residual stress under cyclic loading
  • Surface contact and fretting fatigue interactions
  • Multiaxial stress states near stress concentrations

Treat predictions for σm < 0 as upper-bound estimates only.

6. When Not to Use This Tool

Not for design sign-off: This tool is for education and exploratory comparison only. Do not use these results as the sole basis for a fatigue design decision.

The classical mean-stress methods implemented here assume:

  • Uniaxial, fully sinusoidal, constant-amplitude loading
  • Smooth, unnotched specimens (or that you have already applied a stress concentration factor to σa)
  • Wrought, ductile metallic material with a well-defined endurance limit
  • Room temperature, non-corrosive environment
  • No residual stresses beyond what the modifying factors capture

For variable-amplitude loading, use a damage-accumulation rule (Palmgren-Miner). For notched components, apply a fatigue stress concentration factor K_f. For multiaxial loading, use von Mises equivalent stress or a critical-plane approach.

7. Worked Example

Material: SAE 1045 steel (S_ut = 625 MPa, S_y = 530 MPa)
Loading: σa = 180 MPa, σm = 120 MPa
Endurance limit ratio: 0.50 → S_e' = 312.5 MPa (before modifying factors)
Modifying factors: k_s = k_d = k_r = 1.0 (ideal specimen, S_e = 312.5 MPa)

Goodman equivalent stress

Calculation $$\sigma_{a,eq} = \frac{180}{1 - 120/625} = \frac{180}{0.808} \approx 222.8 \text{ MPa}$$
  • Compare 222.8 MPa against S_e = 312.5 MPa
  • Fatigue SF = 312.5 / 222.8 ≈ 1.40
  • The operating point lies inside the Goodman envelope

Soderberg equivalent stress (most conservative)

Calculation $$\sigma_{a,eq} = \frac{180}{1 - 120/530} = \frac{180}{0.774} \approx 232.6 \text{ MPa}$$
  • Higher equivalent stress because S_y < S_ut
  • Fatigue SF = 312.5 / 232.6 ≈ 1.34

Gerber equivalent stress (least conservative)

Calculation $$\sigma_{a,eq} = \frac{180}{1 - (120/625)^2} = \frac{180}{0.963} \approx 186.9 \text{ MPa}$$
  • Parabolic denominator gives a much smaller penalty
  • Fatigue SF = 312.5 / 186.9 ≈ 1.67

The spread between Soderberg (most conservative) and Gerber (least conservative) is about 45 MPa equivalent stress and reflects genuine model uncertainty about how mean stress interacts with fatigue damage in this steel.

8. References

  • Shigley, J.E., Mischke, C.R., Budynas, R.G. Mechanical Engineering Design, 8th ed. McGraw-Hill, 2006. — Chapter 6.
  • Juvinall, R.C. & Marshek, K.M. Fundamentals of Machine Component Design, 5th ed. Wiley, 2012. — Chapter 8.
  • Dowling, N.E. Mechanical Behavior of Materials, 4th ed. Pearson, 2012. — Chapters 9 & 14.
  • Morrow, J.D. "Fatigue properties of metals," in Fatigue Design Handbook, SAE, 1968.
  • Goodman, J. Mechanics Applied to Engineering, Longmans, Green & Co., 1899.