Fatigue Life Estimator

1. Overview

Fatigue life estimation links cyclic stress to expected cycles to failure. This tool implements the stress-life (S-N) approach, which is one of three primary fatigue analysis methods:

• Stress-Life (S-N): Used here. Best for high-cycle fatigue (N > 10⁴ cycles) where stresses remain nominally elastic. Simple to apply with readily available material data.
• Strain-Life (ε-N): Accounts for local plasticity at notches. Better for low-cycle fatigue (N < 10⁴ cycles) and crack initiation predictions.
• Fracture Mechanics (da/dN): Models crack propagation from an initial flaw size. Used when inspection intervals or damage tolerance are critical.

This tool combines a mean stress correction (Goodman, Gerber, or Soderberg) with a Basquin S-N relationship and a modified endurance limit. It is best suited for early design checks, trade studies, and comparison of alternatives rather than final certification analysis.

2. Historical Context

Fatigue as an engineering discipline emerged from catastrophic failures in the 19th century. The term "fatigue" was coined by Jean-Victor Poncelet around 1839 to describe the gradual weakening of materials under repeated loading.

Wöhler's Foundational Work (1858-1870)

August Wöhler, chief of rolling stock for the Royal Lower Silesian Railways in Prussia, conducted the first systematic fatigue experiments following a series of railway axle failures. His work, published between 1858 and 1870, established several fundamental principles:

• Fatigue life depends on the range of stress, not just the maximum value
• There exists a limiting stress amplitude below which failure does not occur (the endurance limit)
• The relationship between stress amplitude and cycles to failure can be plotted as a curve (the S-N or Wöhler curve)

Wöhler's original tests used rotating bending specimens, which is why the rotating-beam endurance limit (S_e') remains the standard reference for many correction factor methods.

Basquin's Power Law (1910)

O.H. Basquin observed that when Wöhler's S-N data was plotted on logarithmic axes, the high-cycle portion approximated a straight line. He proposed the empirical relationship:

This power-law form remains the basis for most high-cycle fatigue predictions today.

Mean Stress Corrections (1899-1930)

Early researchers recognized that tensile mean stress reduced fatigue strength. Several empirical corrections were developed:

• Gerber (1874): Heinrich Gerber proposed a parabolic relationship based on tests of ductile metals
• Goodman (1899): John Goodman proposed a simpler linear relationship in his book Mechanics Applied to Engineering
• Soderberg (1930): C.R. Soderberg proposed using yield strength instead of ultimate strength for additional conservatism

Modern Developments

The mid-20th century saw significant advances: Miner's rule for cumulative damage (1945), Coffin-Manson for low-cycle fatigue (1954), and Paris' law for crack growth (1961). These methods address limitations of the basic S-N approach but require additional material data and computational effort.

3. Model Assumptions

The stress-life approach implemented here makes several simplifying assumptions. Understanding these is essential for interpreting results correctly.

Loading Assumptions

• Uniaxial stress state: The model assumes a single principal stress direction. For multiaxial loading, equivalent stress criteria (von Mises, Tresca, or critical plane methods) should be used to reduce the stress state to an equivalent uniaxial value.
• Constant amplitude: Each cycle has identical max/min stress. Real service loading is rarely constant-amplitude; variable loading requires cumulative damage methods like Palmgren-Miner's rule.
• No sequence effects: The model ignores load history effects such as overload retardation or underload acceleration. These can significantly affect crack initiation and propagation.
• Proportional loading: Principal stress directions remain fixed. Non-proportional multiaxial loading requires specialized models.

Material Assumptions

• Linear elastic behavior: Stresses remain below yield throughout the cycle. If local plasticity occurs (e.g., at notch roots), strain-life methods are more appropriate.
• Homogeneous, isotropic material: Properties are uniform throughout. Welds, castings, and composites may require special treatment.
• No initial defects: The model predicts crack initiation life, assuming no pre-existing flaws. For damage-tolerant design, fracture mechanics methods are required.
• Room temperature, ambient environment: Elevated temperatures and corrosive environments can dramatically reduce fatigue life through creep-fatigue interaction or corrosion fatigue.

Stress Interpretation

• Hot-spot stress: Input stresses should represent the local stress at the critical location, including any stress concentration effects. If using nominal stresses, apply an appropriate fatigue notch factor (K_f).
• Elastic stress: Even if local yielding occurs, the input should be the elastically-calculated stress. The S-N curve inherently accounts for local plasticity through the test data.

4. Mean Stress Corrections

Experimental evidence shows that tensile mean stress reduces fatigue strength, while compressive mean stress can be beneficial. Mean stress corrections transform a stress cycle with non-zero mean into an equivalent fully-reversed (R = -1) stress amplitude.

Stress Cycle Parameters

Common stress ratios and their meanings:

• R = -1: Fully reversed loading (σ_m = 0). The baseline condition for most S-N data.
• R = 0: Pulsating tension (zero-to-max). Common in rotating machinery and pressure vessels.
• R = 0.1: Tension-dominated with small unloading. Standard for many aerospace specifications.
• R < -1 or R > 1: Compression-dominated cycles. Generally beneficial for fatigue but may cause buckling.

Goodman Diagram (Modified Goodman Line)

Proposed by John Goodman in 1899, this linear relationship connects the endurance limit at R = -1 to the ultimate tensile strength at R = 1. It is widely used in design codes (e.g., ASME, DIN) because of its simplicity and moderate conservatism.

• Conservative for most ductile steels and aluminum alloys
• May be unconservative for very high-strength steels (S_ut > 1400 MPa)
• Assumes failure at S_ut for static mean stress, which is appropriate for ductile materials

Gerber Parabola

Heinrich Gerber's parabolic relationship (1874) often provides a better fit to experimental data for ductile materials, particularly at moderate mean stresses.

• Better correlation with test data for ductile metals
• Less conservative than Goodman, often used for analysis rather than design
• May be unconservative for brittle materials

Soderberg Line

C.R. Soderberg's criterion (1930) uses yield strength instead of ultimate strength, providing an additional margin against both fatigue failure and yielding.

• Most conservative of the three classical methods
• Ensures no yielding at any point in the cycle
• Often overly conservative for many applications

Choosing a Correction Method

Compressive Mean Stress

For compressive mean stress (σ_m < 0), the correction methods above predict an increase in allowable amplitude. This is physically reasonable—compressive mean stress is generally beneficial for fatigue. However, this tool applies the correction as written, which may overestimate the benefit. Many codes conservatively ignore the benefit and use σ_a,eq = σ_a for compressive mean stress.

5. Basquin S-N Model

The Basquin equation describes the relationship between stress amplitude and cycles to failure in the high-cycle fatigue regime. When plotted on log-log axes, the S-N curve approximates a straight line.

The Basquin Equation

Parameters

• σ_f' (Fatigue Strength Coefficient): The stress intercept at 2N_f = 1 reversal (0.5 cycles). For steels, typically ranges from 1.0 to 1.75 times S_ut. Often approximated as the true fracture strength.
• b (Fatigue Strength Exponent): The slope of the S-N curve on a log-log plot. Always negative. For most metals, b ranges from -0.05 to -0.15, with -0.09 being a common approximation for steels.
• 2N_f (Reversals to Failure): The factor of 2 accounts for counting reversals rather than cycles. One cycle = two reversals.

Typical Values for Steels

Relationship to S-N Curve Regions

The complete S-N curve has three distinct regions:

• Low-Cycle Fatigue (LCF): N < 10³ to 10⁴ cycles. Plastic strain dominates. The Coffin-Manson equation is more appropriate here.
• High-Cycle Fatigue (HCF): 10⁴ < N < 10⁶ to 10⁷ cycles. The Basquin equation applies well in this region. Stresses are nominally elastic.
• Endurance Region: N > 10⁶ to 10⁷ cycles. For ferrous materials, the curve may flatten to a horizontal asymptote (endurance limit). Non-ferrous materials often continue to decline.

Combined Strain-Life Equation

For a complete picture spanning both LCF and HCF, the Coffin-Manson-Basquin equation combines elastic (Basquin) and plastic (Coffin-Manson) components:

This tool uses only the stress-based Basquin portion, which is sufficient for high-cycle fatigue where elastic strains dominate.

6. Endurance Limit and Modifiers

The endurance limit (S_e) represents a stress amplitude below which fatigue failure will theoretically not occur, regardless of the number of cycles. However, the laboratory endurance limit must be modified to account for real-world conditions.

Does an Endurance Limit Exist?

The existence of a true endurance limit is material-dependent:

• Ferrous metals (steels, cast irons): Generally exhibit an endurance limit between 10⁶ and 10⁷ cycles, attributed to strain aging of interstitial atoms (carbon, nitrogen) that pin dislocations.
• Non-ferrous metals (aluminum, copper, titanium): Generally do not exhibit a true endurance limit. The S-N curve continues to decline, though the slope may decrease. For these materials, a "fatigue strength at N cycles" (e.g., 10⁸ cycles) is specified instead.
• Gigacycle fatigue: Recent research (N > 10⁹ cycles) has shown that even steels can fail below the traditional endurance limit due to subsurface crack initiation. This is relevant for high-frequency applications (ultrasonic, turbine blades).

Modified Endurance Limit

The Marin factors account for differences between the laboratory test specimen and the actual part:

Surface Factor (k_s or k_a)

Surface finish significantly affects fatigue life because fatigue cracks typically initiate at surface irregularities. The surface factor relates the part's surface to a polished specimen.

Size Factor (k_d or k_b)

Larger parts have lower fatigue strength due to:

• Greater probability of critical flaws in a larger volume
• Smaller stress gradients (less constraint on crack growth)
• Surface area effects

Reliability Factor (k_r or k_c)

Published endurance limits represent mean values (50% survival probability). The reliability factor adjusts for higher survival probability requirements.

Other Marin Factors (Not Included in This Tool)

• k_c (Load Factor): Accounts for loading type. Axial loading ≈ 0.85 relative to rotating bending; torsion ≈ 0.58.
• k_d (Temperature Factor): For elevated temperatures, fatigue strength generally decreases. Below room temperature, strength may increase but ductility decreases.
• k_e (Miscellaneous Effects): Corrosion, fretting, residual stresses, plating, etc.

7. Interpreting Results

This tool provides three key metrics for assessing fatigue durability. Understanding their meaning and appropriate target values is essential for sound engineering judgment.

Estimated Life (N_f)

The predicted number of cycles to failure at the specified stress level. This is a median estimate—50% of specimens would be expected to fail before this life, 50% after.

• Infinite life: Displayed when equivalent stress is at or below the modified endurance limit. Indicates stress is low enough that fatigue failure is not expected.
• Finite life: Calculated using the Basquin equation. Compare to your target service life.

Life Safety Factor (SF_L)

The ratio of predicted life to required life. Because fatigue data has significant scatter, substantial safety factors are typically required:

Fatigue Safety Factor (SF_D)

The ratio of endurance limit to equivalent alternating stress. Also called the "factor of safety for infinite life" or "endurance safety factor."

• SF_D > 1.0: Operating stress is below endurance limit—infinite life expected.
• SF_D < 1.0: Operating stress exceeds endurance limit—finite life predicted.

Yield Safety Factor (SF_y)

The ratio of yield strength to peak stress (maximum absolute stress in the cycle). This ensures gross yielding does not occur.

What Safety Factor Should I Use?

Appropriate safety factors depend on many factors:

• Consequence of failure: Critical/safety-related → higher SF
• Loading certainty: Well-defined loads → lower SF; unknown/variable → higher SF
• Material data quality: Tested specimens → lower SF; handbook estimates → higher SF
• Analysis confidence: Detailed FEA with validation → lower SF; simple estimates → higher SF
• Inspection/maintenance: Regular inspection possible → lower SF; inaccessible → higher SF

8. Limitations

This tool provides useful estimates for preliminary design but has important limitations. Understanding these will help you decide when more sophisticated analysis is needed.

Loading Limitations

• Constant amplitude only: Real loading is rarely constant. For variable amplitude loading, use Palmgren-Miner's linear damage rule or more advanced spectral methods.
• No sequence effects: Load order can significantly affect life. High overloads can retard crack growth; underloads can accelerate it.
• Uniaxial stress only: Multiaxial stress states require equivalent stress criteria or critical plane approaches.
• No frequency effects: At very high frequencies or elevated temperatures, time-dependent effects (creep-fatigue) become important.

Material Limitations

• Approximate parameters: Default Basquin coefficients are estimates. Actual material properties can vary significantly with heat treatment, composition, and processing.
• Isotropic assumption: Rolled, forged, or composite materials may have directional properties.
• No environmental effects: Corrosion fatigue, hydrogen embrittlement, and elevated temperature effects are not modeled.
• Mean stress simplification: The classical corrections may not capture all mean stress effects, especially at high R-ratios or compressive mean stress.

Geometric Limitations

• Stress concentration: If using nominal stress, you must separately apply a fatigue notch factor (K_f). The tool assumes input stress is the local hot-spot value.
• No crack growth: This is a crack initiation model. If a pre-existing flaw is present, fracture mechanics methods are required.
• Size effects simplified: The size factor is a simplified correction. Large structures may require more detailed analysis.

When to Use More Advanced Methods

• Variable amplitude loading: Use cumulative damage (Miner's rule) with rainflow cycle counting
• Low-cycle fatigue (N < 10⁴): Use strain-life methods (Coffin-Manson)
• Crack propagation: Use fracture mechanics (Paris law, NASGRO)
• Multiaxial loading: Use critical plane methods or equivalent stress criteria
• Welded joints: Use hot-spot stress methods, structural stress, or notch stress approaches per IIW recommendations
• Probabilistic requirements: Use statistical methods (P-S-N curves, reliability analysis)

9. References

Textbooks

• Budynas, R.G. and Nisbett, J.K. (2020). Shigley's Mechanical Engineering Design, 11th ed., McGraw-Hill. [Chapters 6-7 cover fatigue analysis in detail with practical examples and empirical correlations]
• Juvinall, R.C. and Marshek, K.M. (2019). Fundamentals of Machine Component Design, 7th ed., Wiley. [Good treatment of fatigue design with extensive problem sets]
• Dowling, N.E. (2012). Mechanical Behavior of Materials, 4th ed., Pearson. [Comprehensive coverage of fatigue from a materials science perspective]
• Bannantine, J.A., Comer, J.J., and Handrock, J.L. (1990). Fundamentals of Metal Fatigue Analysis, Prentice Hall. [Classic text covering both stress-life and strain-life methods]
• Suresh, S. (1998). Fatigue of Materials, 2nd ed., Cambridge University Press. [Advanced treatment including micromechanisms and crack growth]

Historical Papers

• Wöhler, A. (1870). "Über die Festigkeitsversuche mit Eisen und Stahl," Zeitschrift für Bauwesen, Vol. 20, pp. 73-106. [The foundational work establishing S-N curves]
• Basquin, O.H. (1910). "The Exponential Law of Endurance Tests," Proceedings of ASTM, Vol. 10, pp. 625-630. [Introduction of the power-law S-N relationship]
• Goodman, J. (1899). Mechanics Applied to Engineering, Longmans, Green and Co., London. [Source of the Goodman diagram]
• Gerber, H. (1874). "Bestimmung der zulässigen Spannungen in Eisenkonstruktionen," Zeitschrift des Bayerischen Architekten- und Ingenieur-Vereins, Vol. 6, pp. 101-110. [Origin of the Gerber parabola]
• Soderberg, C.R. (1930). "Factor of Safety and Working Stress," Transactions of ASME, Vol. 52, pp. 13-28. [Introduction of the Soderberg criterion]
• Miner, M.A. (1945). "Cumulative Damage in Fatigue," Journal of Applied Mechanics, Vol. 12, pp. A159-A164. [Linear damage accumulation rule]

Standards

• ASTM E466 - Standard Practice for Conducting Force Controlled Constant Amplitude Axial Fatigue Tests of Metallic Materials
• ASTM E606 - Standard Practice for Strain-Controlled Fatigue Testing
• ASTM E647 - Standard Test Method for Measurement of Fatigue Crack Growth Rates
• ASTM E739 - Standard Practice for Statistical Analysis of Linear or Linearized Stress-Life (S-N) and Strain-Life (ε-N) Fatigue Data
• ISO 12107 - Metallic materials - Fatigue testing - Statistical planning and analysis of data
• IIW Document XIII-2460-13 - Recommendations for Fatigue Design of Welded Joints and Components

Handbooks

• ASM Handbook, Volume 19: Fatigue and Fracture (1996). ASM International. [Comprehensive reference covering all aspects of fatigue]
• MMPDS (Metallic Materials Properties Development and Standardization). [Statistically-based material properties for aerospace applications]
• MIL-HDBK-5 (Historical). [Predecessor to MMPDS, still widely referenced]
• Peterson, R.E. (1974). Stress Concentration Factors, 2nd ed., Wiley. [Essential for determining K_t values]
• Pilkey, W.D. and Pilkey, D.F. (2008). Peterson's Stress Concentration Factors, 3rd ed., Wiley. [Updated edition of Peterson's classic]

Online Resources

• eFatigue (www.efatigue.com) - Fatigue analysis tools and material databases
• NASGRO - NASA/ESA fracture mechanics and fatigue crack growth software
• SAE Fatigue Design Handbook - Automotive fatigue applications
• Total Materia - Material property database including fatigue data