Fatigue Life S-N Curve Analyzer: Stress-Life Prediction & Damage Accumulation

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Introduction

On April 28, 1988, the roof of Aloha Airlines Flight 243 tore off at 24,000 feet, exposing passengers to the open sky. The aircraft had accumulated nearly 90,000 pressurization cycles over 19 years of island-hopping service. The fatigue cracks that caused this catastrophic failure had been growing invisibly for thousands of flights, each cycle of takeoff and landing adding microscopic damage that eventually became catastrophic.

This is where structures fail: not from single overloads, but from the relentless accumulation of small damages that add up over millions of cycles. Any structural engineer will tell you: fatigue is the silent killer. Components fail at stress levels far below the material's yield strength (as determined from the stress-strain curve), without warning, after years of apparently normal service. The load has to go somewhere, and when that load cycles repeatedly, even small stress amplitudes eventually nucleate cracks that grow with each repetition.

The S-N curve (Stress vs. Number of cycles) maps this treacherous territory, showing exactly how many cycles your component can survive at a given stress level. Experienced designers know that a 10% increase in stress amplitude can reduce fatigue life by 50% or more, making this curve one of the most important tools for anyone designing rotating machinery, aircraft structures, or any component that sees repeated loading. This simulation lets you explore these relationships and develop the intuition that separates components that last from components that fail.

How to Use This Simulation

This interactive fatigue analyzer computes fatigue life predictions, mean stress corrections, and cumulative damage using industry-standard methods. The load has to go somewhere, and when that load cycles repeatedly, this tool shows you exactly how many repetitions your component can survive before failure initiates.

Main Controls

Input Parameters

Results Display

The simulation provides comprehensive fatigue analysis:

• S-N Curve Plot: Log-log plot of stress amplitude versus cycles to failure. The curve's slope reveals how sensitive the material is to stress changes. This is where structures fail: on the steep portion where small stress increases dramatically reduce life.
• Fatigue Life Nf: Predicted cycles to failure at the current stress amplitude, displayed prominently.
• Modified Endurance Limit Se': Corrected endurance limit after applying Marin factors.
• Goodman Diagram: Mean stress correction showing the safe operating envelope. Any structural engineer will tell you that tensile mean stress is the enemy of fatigue life.
• Damage Fraction D: For Miner's Rule analysis, the cumulative damage from multiple stress levels (failure when D ≥ 1).
• Safety Factor: Ratio of allowable stress to applied stress at the specified life.

Tips for Exploration

1. Watch the log-log slope. On the S-N curve, a 10% stress increase typically reduces life by 50% or more because of the logarithmic relationship. Move the stress slider by small amounts and observe the dramatic life changes. This sensitivity is why fatigue design uses conservative factors of safety.

2. Explore mean stress effects. Start with fully reversed loading (σm = 0) and note the fatigue life. Then add tensile mean stress while keeping σa constant. Watch the life drop. Any structural engineer will tell you that residual stresses from welding create exactly this problem, hidden mean stresses that slash fatigue performance.

3. Apply Marin factors sequentially. Start with the baseline Se' (polished lab specimen), then apply surface finish correction, size correction, and reliability factor one at a time. Watch the endurance limit drop from perhaps 400 MPa to 200 MPa. The load has to go somewhere, and real-world conditions are far harsher than laboratory conditions.

4. Test stress concentration sensitivity. Increase Kf from 1.0 (smooth bar) to 2.5 (sharp notch) and observe the life reduction. This is where structures fail: at fillet radii, keyways, holes, and other geometric discontinuities where local stress is amplified far above nominal values.

5. Use Miner's Rule for variable loading. Switch to Miner's mode and input multiple stress levels with their cycle counts. Watch the damage fraction accumulate. Experienced designers know that Miner's Rule is approximate (damage accumulation is rarely perfectly linear), but it remains the foundation of most fatigue life predictions.

Types of Fatigue Loading

High-Cycle Fatigue (HCF)

High-cycle fatigue occurs when components experience a large number of load cycles (typically N > 10⁴) at relatively low stress levels within the elastic range. This is the regime where the S-N curve applies most directly. Examples include rotating shafts, turbine blades, and springs operating under normal conditions. In HCF, crack initiation dominates the total life, and the endurance limit concept is most relevant.

Low-Cycle Fatigue (LCF)

Low-cycle fatigue involves high stress levels that cause significant plastic deformation, resulting in failure at fewer than 10⁴ cycles. LCF is characterized using strain-life relationships (ε-N curves) rather than stress-life. Common in pressure vessels, aircraft landing gear, and components experiencing thermal cycling, LCF analysis requires different approaches like the Coffin-Manson equation.

Fully Reversed Loading (R = -1)

In fully reversed loading, the stress alternates between equal tension and compression with zero mean stress. This is the baseline condition for most S-N data and represents the most severe fatigue condition for a given stress amplitude. Examples include rotating bending specimens and oscillating beams.

Zero-to-Maximum Loading (R = 0)

When stress cycles from zero to a maximum value and back, R = 0. This is common in machinery where loads are applied and released, such as crane hooks and press frames. The mean stress equals half the maximum stress, which generally improves fatigue life compared to fully reversed loading.

Variable Amplitude Loading

Real components rarely experience constant amplitude loading. Instead, loads vary randomly or in complex patterns. Miner's rule and more sophisticated approaches like rainflow counting are used to predict life under variable amplitude conditions by accumulating damage from individual cycles.

Key Parameters

Key Formulas

Basquin's Equation (S-N Relationship)

Formula: σₐ = σ'f (2Nf)^b

Where:

• σₐ = stress amplitude (MPa)
• σ'f = fatigue strength coefficient (≈ Sᵤ for steels)
• Nf = cycles to failure
• b = fatigue strength exponent (typically -0.05 to -0.12)

Used when: Determining fatigue life from known stress amplitude, or finding allowable stress for target life.

Modified Endurance Limit (Marin Equation)

Formula: Sₑ = kₐ × kb × kc × kd × kₑ × S'ₑ

Where:

• S'ₑ = uncorrected endurance limit (≈ 0.5Sᵤ for steels ≤ 1400 MPa)
• kₐ = surface finish factor (0.35-1.0)
• kb = size factor (0.6-1.0)
• kc = reliability factor (0.753-1.0)
• kd = temperature factor
• kₑ = miscellaneous effects factor

Used when: Converting laboratory specimen data to real component conditions.

Goodman Mean Stress Correction

Formula: σₐ/Sₑ + σₘ/Sᵤ = 1/nf

Where:

• nf = factor of safety against fatigue failure

Used when: Accounting for non-zero mean stress; linear relationship provides conservative estimate.

Gerber Mean Stress Correction

Formula: σₐ/Sₑ + (σₘ/Sᵤ)² = 1/nf

Where:

• The parabolic relationship better fits experimental data for ductile materials

Used when: More accurate prediction for ductile metals; less conservative than Goodman.

Soderberg Mean Stress Correction

Formula: σₐ/Sₑ + σₘ/Sᵧ = 1/nf

Used when: Most conservative approach; guards against both fatigue and yielding; uses yield strength.

Miner's Rule (Cumulative Damage)

Formula: D = Σ(nᵢ/Nfᵢ) ≤ 1

Where:

• nᵢ = number of cycles at stress level i
• Nfᵢ = fatigue life at stress level i
• D = cumulative damage (failure predicted at D = 1)

Used when: Predicting life under variable amplitude loading; assumes linear damage accumulation.

Learning Objectives

After completing this simulation, you will be able to:

1. Interpret S-N curves on log-log scales, identifying the finite life region, transition zone, and endurance limit for different materials.

2. Apply Marin factors to convert laboratory endurance limits to real-world component conditions, accounting for surface finish, size, and reliability requirements.

3. Calculate fatigue life using Basquin's equation and determine safe operating stress levels for target design lives.

4. Evaluate mean stress effects using Goodman, Gerber, and Soderberg criteria, selecting the appropriate method for different applications.

5. Predict cumulative damage under variable amplitude loading using Miner's rule, assessing remaining life and failure risk.

6. Design for fatigue resistance by understanding how material selection, stress concentrations, and surface treatments affect fatigue performance.

Exploration Activities

Activity 1: S-N Curve Fundamentals

Objective: Understand the relationship between stress amplitude and fatigue life.

Steps:

1. Select "Steel (Low Strength)" material preset
2. Start with alternating stress σₐ = 250 MPa (near Se')
3. Gradually increase stress and observe life reduction
4. Find the stress that gives N = 10⁵ cycles
5. Reduce stress until you reach infinite life region

Observe: The steep slope of the S-N curve in log-log space; the transition at the endurance limit.

Expected Result: A 20% stress increase should reduce life by roughly 10×. At σₐ ≤ Se, life becomes infinite.

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Activity 2: Mean Stress Effects

Objective: Compare Goodman, Gerber, and Soderberg predictions.

Steps:

1. Switch to "Mean Stress Diagram" view
2. Set alternating stress σₐ = 100 MPa
3. Increase mean stress from 0 to 200 MPa
4. Observe how operating point moves toward failure lines
5. Note which criterion (Goodman/Gerber/Soderberg) is most conservative

Observe: The space between criterion lines represents different safety margins; operating point trajectory shows load line.

Expected Result: Soderberg (yellow dashed) is most conservative, Gerber (blue parabola) is least conservative.

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Activity 3: Marin Factors Impact

Objective: See how real-world conditions reduce fatigue strength.

Steps:

1. Note the Se value with Ground surface, kb = 0.85, 90% reliability
2. Change surface finish to "Hot-Rolled" - observe Se drop
3. Reduce size factor kb to 0.7 (larger component)
4. Change reliability to 99.9%
5. Calculate the combined reduction factor

Observe: Each factor multiplies the others; combined effect can reduce Se by 50% or more.

Expected Result: S'e × 0.5 × 0.7 × 0.753 = 0.264 × S'e (74% reduction from ideal conditions).

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Activity 4: Miner's Rule Variable Loading

Objective: Calculate cumulative damage under spectrum loading.

Steps:

1. Switch to "Miner's Rule" view
2. Keep default three loading blocks
3. Adjust Block 1: high stress (250 MPa), 10,000 cycles
4. Adjust Block 2: medium stress (180 MPa), 50,000 cycles
5. Adjust Block 3: low stress (120 MPa), 200,000 cycles
6. Add a fourth block if needed to approach D = 1

Observe: High stress cycles contribute disproportionately to damage; low stress cycles near Se contribute little.

Expected Result: Damage D < 1 indicates remaining life available; D ≥ 1 predicts failure.

Real-World Applications

1. Rotating Machinery: Shaft design for turbines, motors, and pumps requires fatigue analysis to ensure reliable operation over millions of cycles. Critical locations include keyways, fillets, and diameter changes where stress concentrations exist.

2. Aerospace Structures: Aircraft wings experience thousands of pressurization cycles during their service life. Fatigue-critical joints, fuselage skins, and landing gear components undergo rigorous S-N testing and damage tolerance analysis.

3. Automotive Components: Suspension components, engine connecting rods, and chassis parts experience variable amplitude loading from road conditions. Manufacturers use accelerated fatigue testing and simulation to validate designs.

4. Pressure Vessels: Chemical reactors and pressure equipment undergo cyclic pressure and temperature loading. ASME codes provide fatigue curves and design rules specific to vessel materials and loading conditions.

5. Offshore Structures: Wave loading creates continuous cyclic stresses on oil platforms, ship hulls, and offshore wind turbine foundations. Spectral fatigue analysis accounts for the random nature of wave-induced loading.

6. Biomedical Implants: Hip and knee replacements must survive millions of cycles over decades of use. Material selection (titanium, cobalt-chrome, UHMWPE) and surface finishing are critical for fatigue resistance.

Reference Data

Typical Endurance Limits and Fatigue Properties

Surface Finish Factors (kₐ)

Reliability Factors (kc)

Challenge Questions

1. Conceptual: Why do some materials (ferrous metals) have an endurance limit while others (aluminum) do not? What happens to aluminum at very long lives?

2. Calculation: A steel shaft (Sᵤ = 600 MPa, S'ₑ = 300 MPa) has a ground surface finish, 40mm diameter, and requires 99% reliability. Calculate the modified endurance limit Sₑ. If the shaft experiences σₐ = 150 MPa fully reversed, what is the predicted life?

3. Analysis: For a component with Kf = 2.0 operating at σₐ = 100 MPa and σₘ = 150 MPa, compare the safety factors from Goodman, Gerber, and Soderberg criteria. Which would you use for a safety-critical application and why?

4. Application: A pump shaft experiences the following loading spectrum each day: 1,000 cycles at 200 MPa, 5,000 cycles at 150 MPa, and 50,000 cycles at 100 MPa. If Sₑ = 180 MPa, estimate the shaft's service life in days using Miner's rule.

5. Design: You need to increase the fatigue life of a notched component by a factor of 10 without changing the applied loads. List three design modifications that could achieve this goal, and estimate the effectiveness of each.

Common Mistakes to Avoid

This is where structures fail: not from obvious overloads, but from subtle errors in fatigue analysis that experienced designers learn to avoid:

1. Ignoring mean stress effects: Using S-N data (typically for R = -1) without correcting for non-zero mean stress overestimates fatigue life. Always apply Goodman, Gerber, or Soderberg corrections when σₘ ≠ 0. For σₐ = 100 MPa and σₘ = 100 MPa, Goodman predicts roughly 50% lower allowable alternating stress.

2. Forgetting to apply Marin factors: Laboratory S-N data uses small, polished, rotating beam specimens under controlled conditions. Real components have different surfaces, sizes, and environments. Failing to apply kₐ, kb, kc can overestimate Sₑ by 2-3× for large, rough-surfaced components.

3. Using stress concentration incorrectly: The fatigue stress concentration factor Kf (not the geometric Kt) should multiply the alternating stress component. Kf = 1 + q(Kt - 1), where q is the notch sensitivity. Using Kt directly is overly conservative for ductile materials.

4. Assuming Miner's rule is exact: Miner's rule (D = Σnᵢ/Nfᵢ = 1) is an approximation. Actual failure may occur at D = 0.7-2.0 depending on load sequence. High-low sequences (severe loads first) typically cause failure at D < 1, while low-high sequences may exceed D = 1.

5. Neglecting infinite life threshold: Below the endurance limit, damage does not accumulate in the classical sense (for ferrous metals). In Miner's rule calculations, cycles below Se should not be counted, or use modified approaches that assign very long Nf values to sub-endurance stresses.

Frequently Asked Questions

What is an S-N curve?

An S-N curve (also called Wöhler curve) plots stress amplitude (S) versus number of cycles to failure (N) on a log-log scale. It shows how fatigue life decreases as applied stress increases. The curve follows the Basquin relation: S = A × N^b, where A and b are material constants. S-N curves are obtained from standardized rotating beam tests and form the basis for fatigue life prediction [1].

What is the endurance limit?

The endurance limit (Se or σe) is the stress level below which a material can theoretically withstand an infinite number of cycles without fatigue failure. For ferrous metals (steel, iron), Se ≈ 0.4-0.5 × Su (ultimate tensile strength). Aluminum and copper alloys do NOT have a true endurance limit and will eventually fail even at low stresses—their "fatigue strength" is quoted at specific cycle counts like 10⁷ or 10⁸ [2].

How does Miner's rule work for variable amplitude loading?

Miner's rule (Palmgren-Miner rule) is a cumulative damage hypothesis: each load cycle consumes a fraction of fatigue life. Damage D = Σ(ni/Ni), where ni is cycles at stress level i and Ni is cycles to failure at that stress. Failure occurs when D = 1. In practice, failure often occurs at D = 0.7-2.0 depending on load sequence—Miner's rule doesn't account for sequence effects [3].

What are Marin modification factors?

Marin factors (ka, kb, kc, kd, ke, kf) correct laboratory endurance limit data for real-world conditions. ka = surface finish factor (machined < polished); kb = size factor (larger parts weaker); kc = reliability factor; kd = temperature factor; ke = stress concentration factor. Modified endurance limit: Se' = ka × kb × kc × kd × ke × Se. Real components often have Se' = 0.3-0.6 × laboratory Se [4].

How do I account for mean stress in fatigue?

Mean stress (σm) affects fatigue life—tensile mean stress is detrimental, compressive is beneficial. Common correction methods: Goodman line: σa/Se + σm/Su = 1; Gerber parabola: σa/Se + (σm/Su)² = 1; Soderberg (conservative): σa/Se + σm/Sy = 1. Goodman is most commonly used for ductile metals. These plot on σa vs σm diagrams to determine safe combinations [2].

What is the difference between high-cycle and low-cycle fatigue?

High-cycle fatigue (HCF) occurs at N > 10⁴ cycles with primarily elastic stresses—use S-N curves and stress-based analysis. Low-cycle fatigue (LCF) occurs at N < 10⁴ cycles with significant plastic strain—use strain-life (ε-N) curves and the Coffin-Manson relationship. Most rotating machinery experiences HCF; thermal cycling and pressure vessels often experience LCF [1].

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