Fatigue damage accumulates over time as components experience varying stress levels. Miner's rule helps engineers estimate this cumulative damage by summing up the damage fractions at each stress level. It's a simple yet powerful tool for predicting component failure.

Cycle counting methods like rainflow counting simplify complex loading histories into manageable sets of stress cycles. This allows engineers to apply Miner's rule and predict fatigue life in real-world scenarios with irregular loading, like wind turbines or aircraft components.

Cumulative Damage and Miner's Rule

Assessing Cumulative Damage

• Cumulative damage assesses the total damage a component experiences when subjected to varying stress levels over its lifetime
• Fatigue damage accumulates over time due to repeated stress cycles at different stress amplitudes
• Each stress cycle contributes a small amount of damage, which adds up over the component's life
• Cumulative damage models predict the total damage and estimate the remaining life of a component (gears, shafts, bearings)

Miner's Rule and Linear Damage Hypothesis

• Miner's rule, also known as the Palmgren-Miner rule, is a widely used method for estimating cumulative fatigue damage
• Based on the linear damage hypothesis, which assumes that fatigue damage accumulates linearly with the number of stress cycles
• According to Miner's rule, the total damage $D$ is the sum of the damage fractions at each stress level: $D = \sum_{i=1}^{k} \frac{n_i}{N_i}$
  • $n_i$ is the number of cycles at stress level $i$
  • $N_i$ is the number of cycles to failure at stress level $i$
• Failure is predicted to occur when the total damage $D$ reaches unity (1)
• Miner's rule provides a simple and practical approach to estimate cumulative damage, despite its limitations (ignores load sequence effects and assumes linear damage accumulation)

Cycle Counting and Variable Amplitude Loading

Cycle Counting Methods

• Cycle counting methods are used to simplify variable amplitude loading histories into a set of constant amplitude cycles
• Rainflow counting is the most widely used cycle counting method
  • Identifies closed stress-strain hysteresis loops in the loading history
  • Counts the number of cycles at each stress level and amplitude
• Other cycle counting methods include range counting, peak counting, and level crossing counting
• Cycle counting reduces complex loading histories into a manageable set of stress cycles for fatigue analysis (irregular loading in wind turbines, aircraft, and vehicles)

Damage Fraction and Fatigue Life Prediction

• The damage fraction is the ratio of the number of cycles applied at a specific stress level to the total number of cycles to failure at that stress level
• Damage fraction $d_i$ at stress level $i$ is calculated as: $d_i = \frac{n_i}{N_i}$
• The total damage $D$ is the sum of the damage fractions across all stress levels: $D = \sum_{i=1}^{k} d_i$
• Fatigue life can be predicted using the total damage and Miner's rule
  • Estimate the number of cycles to failure $N_i$ at each stress level using S-N curves or other fatigue data
  • Calculate the damage fraction $d_i$ for each stress level based on the applied number of cycles $n_i$
  • Sum the damage fractions to obtain the total damage $D$
  • Failure is expected when $D$ reaches unity (1)
• Variable amplitude loading and cycle counting are essential for accurate fatigue life prediction in real-world applications (suspension components, offshore structures)