Fatigue Analysis — Explorable Explanations

Introduction to Fatigue Analysis

Fatigue is a failure mechanism that occurs in materials subjected to cyclic loading, even when the stresses are well below the material's ultimate tensile strength. By common estimates it drives roughly 80–90% of all in-service mechanical failures.

Key aspects that differentiate fatigue from static failure:

Failure occurs after many loading cycles, not immediately
Failure can occur at stress levels well below the yield strength
Failure typically begins at a localised defect or stress concentration
The process involves three stages: crack initiation, crack propagation, and final fracture

The tools below work through fatigue in four stages: characterising a stress cycle, reading life off an S-N curve, correcting for mean stress, and summing damage under variable-amplitude loading.

Fundamentals of Fatigue Loading

Stress Cycle Parameters

In fatigue analysis, we characterise cyclic loading using several key parameters:

\[\sigma_{max} = \text{Maximum stress in cycle}\]

\[\sigma_{min} = \text{Minimum stress in cycle}\]

\[\sigma_a = \frac{\sigma_{max} - \sigma_{min}}{2} = \text{Stress amplitude}\]

\[\sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2} = \text{Mean stress}\]

\[R = \frac{\sigma_{min}}{\sigma_{max}} = \text{Stress ratio}\]

Loading Parameters

Max Stress (MPa): 200

Min Stress (MPa): -100

Frequency (Hz): 0.5

Common Loading Types:

Stress-Time History

S-N Curves and Fatigue Life Prediction

The Wöhler Experiment

August Wöhler conducted the first systematic investigation of fatigue failure in the 1850s whilst studying railway axle failures. His experiments involved applying cyclic loads to test specimens until failure and plotting the results.

Basquin's Equation

The relationship between stress amplitude and fatigue life:

\[\sigma_a = \sigma'_f (2N_f)^b\]

Where:

\(\sigma_a\) = Stress amplitude
\(\sigma'_f\) = Fatigue strength coefficient
\(N_f\) = Number of cycles to failure
\(b\) = Fatigue strength exponent (typically −0.05 to −0.12; some alloys such as magnesium reach about −0.13)

On log-log scale: Appears as a straight line with slope = b

On semi-log scale: Appears as a curved (power law) line

Material Parameters

Fatigue Strength Coeff. (MPa): 1000

Fatigue Strength Exponent: -0.09

S-N Curve (Wöhler Diagram)

Use Log-Log Scale (straight line)

Log-log: Both axes logarithmic - shows power law as straight line
Semi-log: Only N-axis logarithmic - shows power law as curve

Understanding Scale Types:

Log-log scale: Makes power laws appear linear, useful for determining material constants (slope = b)
Semi-log scale: Better shows absolute stress values, commonly used in design handbooks
Toggle between scales to understand how the same data can be visualised differently

Effect of Mean Stress on Fatigue Life

In practical applications, many components experience cyclic loading with a non-zero mean stress. Tensile mean stress generally reduces fatigue life, whilst compressive mean stress tends to improve it.

Mean Stress Correction Models

Goodman: \(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = 1\)

Gerber: \(\frac{\sigma_a}{S_e} + \left(\frac{\sigma_m}{S_{ut}}\right)^2 = 1\)

Soderberg: \(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_y} = 1\)

Material Properties

\(S_{ut}\) (MPa): 800

\(S_y\) (MPa): 600

\(S_e\) (MPa): 300

Assessment Point:

\(\sigma_m\) (MPa): 200

\(\sigma_a\) (MPa): 150

Haigh Diagram

Cumulative Damage Theory

In real applications, components often experience variable amplitude loading. The Palmgren-Miner rule provides a way to assess cumulative damage.

Palmgren-Miner Linear Damage Rule

\[D = \sum_{i=1}^{k} \frac{n_i}{N_i}\]

Where:

\(D\) = Cumulative damage (failure when D ≥ 1)
\(n_i\) = Cycles applied at stress level i
\(N_i\) = Cycles to failure at stress level i

Stress Block 1:

\(\sigma_a\) (MPa): 300

\(n_1\): 10000

Stress Block 2:

\(\sigma_a\) (MPa): 275

\(n_2\): 50000

Stress Block 3:

\(\sigma_a\) (MPa): 325

\(n_3\): 100000

Palmgren-Miner Damage Accumulation

Cycles to failure \(N_i\) for each block come from Basquin's equation for Steel 1045 (\(\sigma'_f = 1120\) MPa, \(b = -0.092\)), matching the Steel 1045 curve in the S-N section above.

Limitations of the Palmgren-Miner Rule:

Does not account for sequence effects
Assumes linear damage accumulation
Observed failure often occurs at D values between 0.7 and 1.5
Conservative design often uses a target D value of 0.7 or lower

Engineering Applications

Fatigue analysis applies wherever components experience cyclic loading:

Aerospace: Aircraft fuselages, wing attachments, and turbine blades undergo millions of load cycles during service, making fatigue the primary design consideration for structural integrity.
Automotive: Suspension components, crankshafts, and connecting rods experience repeated loading cycles, requiring careful fatigue analysis to prevent catastrophic failures.
Civil Infrastructure: Bridges, railway tracks, and offshore platforms subjected to traffic and environmental loads must be designed against fatigue to ensure decades of safe operation.
Power Generation: Steam and gas turbine rotors, generator shafts, and pressure vessel components operate under cyclic thermal and mechanical stresses, so fatigue assessment sets their inspection intervals.
Medical Devices: Orthopaedic implants, stents, and prosthetic devices must withstand millions of loading cycles within the human body without fatigue failure.
Marine Engineering: Ship propeller shafts, mooring lines, and offshore drilling equipment face variable amplitude loading from waves and operational cycles, so their fatigue design must cover the full load spectrum.

Summary of Key Equations

Concept	Equation	Description
Stress Amplitude	\(\sigma_a = \frac{\sigma_{max} - \sigma_{min}}{2}\)	Half the stress range in cyclic loading
Mean Stress	\(\sigma_m = \frac{\sigma_{max} + \sigma_{min}}{2}\)	Average stress during cycle
Basquin's Equation	\(\sigma_a = \sigma'_f (2N_f)^b\)	S-N curve relationship for fatigue life
Goodman Criterion	\(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{SF}\)	Mean stress correction for fatigue design
Palmgren-Miner Rule	\(D = \sum_{i=1}^{k} \frac{n_i}{N_i}\)	Cumulative damage assessment (failure at D ≥ 1)

Practice Problems

Problem 1: Fatigue Life Prediction

A steel component (σ'f = 1000 MPa, b = -0.09) experiences fully reversed loading with a stress amplitude of 300 MPa. Calculate the expected fatigue life.

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Problem 2: Mean Stress Correction

A component experiences σₐ = 200 MPa and σₘ = 150 MPa. If Se = 400 MPa and Sut = 800 MPa, calculate the safety factor using the Goodman criterion.

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Problem 3: Cumulative Damage

A component experiences three stress blocks: Block 1: 400 MPa for 1000 cycles (Nf = 10,000), Block 2: 300 MPa for 20,000 cycles (Nf = 100,000), Block 3: 200 MPa for 500,000 cycles (Nf = 10^6). Calculate the cumulative damage.

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Stress-Based Fatigue Analysis

Introduction to Fatigue Analysis

Fundamentals of Fatigue Loading

Stress Cycle Parameters

Loading Parameters

Common Loading Types:

Stress-Time History

S-N Curves and Fatigue Life Prediction

The Wöhler Experiment

Basquin's Equation

Material Parameters

S-N Curve (Wöhler Diagram)

Effect of Mean Stress on Fatigue Life

Mean Stress Correction Models

Material Properties

Assessment Point:

Haigh Diagram

Cumulative Damage Theory

Palmgren-Miner Linear Damage Rule

Stress Block 1:

Stress Block 2:

Stress Block 3:

Palmgren-Miner Damage Accumulation

Engineering Applications

Summary of Key Equations

Practice Problems

Problem 1: Fatigue Life Prediction

Problem 2: Mean Stress Correction

Problem 3: Cumulative Damage

Knowledge Check Quiz

Question 1: Stress amplitude for a cycle between 50 MPa and 250 MPa is:

Question 2: For fully reversed loading, the stress ratio R equals:

Question 3: Which material typically shows a distinct endurance limit?

Question 4: According to Palmgren-Miner, failure occurs when D equals:

Question 5: Which mean stress correction is most conservative?