› Miner Rule Simulator Back
Fatigue Analysis Simulator

Miner Rule Simulator — Linear Cumulative Damage and Fatigue Life

From the Palmgren-Miner linear cumulative damage rule D = Sum(n_i/N_i) and the Basquin S-N curve N = 0.5*(sigma_a/sigma_f')^(1/b), this tool takes the stress amplitudes and actual cycle counts of two load levels and computes the allowable cycles N_i, the cumulative damage D, and the safety factor S = 1/D in real time. Two operating points are plotted on a log-log Basquin S-N curve, and a stacked bar chart visualises D_1, D_2, D_total and the failure criterion D = 1.

Parameters
Stress level 1 sigma_a1
MPa
Applied cycles n_1
k cyc
Stress level 2 sigma_a2
MPa
Applied cycles n_2
k cyc

Defaults: sigma_a1 = 250 MPa x 5,000 cycles, sigma_a2 = 200 MPa x 10,000 cycles. Material constants used internally: Basquin exponent b = -0.10, fatigue strength coefficient sigma_f' = 1,000 MPa (typical steel).

Results
—
Cumulative damage D = Sum(n_i/N_i)
—
Allowable N_1 at level 1
—
Allowable N_2 at level 2
—
Safety factor S = 1/D
Basquin S-N curve (log-log) with operating points

x-axis = cycles N (log, 10^2 to 10^8) / y-axis = stress amplitude sigma_a (MPa, 50 to 500) / blue curve = Basquin S-N / red dot = level 1 (sigma_a1, N_1) / orange dot = level 2 (sigma_a2, N_2). The Basquin line is straight on log-log axes.

Damage decomposition (D_1, D_2, D_total)

Bars = damage fractions D_i = n_i/N_i and their total D / red dashed line = failure criterion D = 1 (Miner limit). The closer D is to 1 the closer the structure is to end of life; S = 1/D tells how many more repetitions of the same spectrum are allowed.

Theory & Key Formulas

The Palmgren-Miner linear cumulative damage rule is a classical method for predicting the fatigue life of structures under variable-amplitude loading. At each stress level $i$ it defines the damage fraction as the ratio between the applied cycles $n_i$ and the allowable cycles $N_i$ to failure, and adds the fractions linearly:

$$D = \sum_i \frac{n_i}{N_i}$$

Fatigue failure is predicted at $D=1$ (Miner criterion). The allowable cycles $N_i$ come from the Basquin S-N curve:

$$\sigma_a = \sigma_f' \cdot (2N_f)^{b} \;\;\Longrightarrow\;\; N = \tfrac{1}{2}\!\left(\frac{\sigma_a}{\sigma_f'}\right)^{1/b}$$

This tool uses typical steel values $\sigma_f' = 1000$ MPa and $b = -0.10$ (so $1/b = -10$), and defines the safety factor as $S = 1/D$:

$$S = \frac{1}{D} = \frac{1}{\sum_i n_i / N_i}$$

$\sigma_a$ is the stress amplitude, $\sigma_f'$ the fatigue strength coefficient (about $1.5 S_u$ for steels), $b$ the Basquin exponent (typically $-0.05$ to $-0.15$), $n_i$ the applied cycles, and $N_i$ the allowable cycles at the same stress. With $b=-0.10$, doubling the stress amplitude makes $N$ about $2^{10} \approx 1000$ times smaller, which is why the Miner rule is so sensitive to high-stress events.

What is the Miner Rule Simulator?

🙋
I learned about S-N curves at university, but real machines never see a constant-amplitude load. A car suspension takes huge hits over potholes and almost nothing on smooth tarmac — it's all over the place. How do we even handle that?
🎓
Great question. The classic tool for that is the Miner rule (Palmgren-Miner linear cumulative damage rule). The formula is just D = Sum(n_i/N_i). At each stress level you apply n_i cycles, the S-N curve tells you the allowable life N_i for that level, so n_i/N_i is the fraction of life consumed. You sum the fractions and the structure fails when D hits 1. With this tool's defaults (sigma_a1 = 250 MPa x 5,000 cycles, sigma_a2 = 200 MPa x 10,000 cycles) you get N_1 ≈ 524k, N_2 ≈ 4.88M, D ≈ 0.0116, so the safety factor S = 1/D ≈ 86 means you can repeat that spectrum 86 more times.
🙋
OK but the allowable cycles change by a factor of 10 just going from 250 to 200 MPa. That feels almost too sensitive, doesn't it?
🎓
That is exactly the point of an S-N curve. Basquin's relation is straight in log-log axes with slope b, usually -0.05 to -0.15. This tool uses a typical-steel value of b = -0.10, so 1/b = -10 and a 25% stress increase (250/200) divides N by (1.25)^10 ≈ 9.3. In other words, a 25% stress rise shortens life by roughly a factor of 10. That is why aerospace and spring designers fight to shave another 20 MPa off the operating stress. Try moving the sigma_a1 slider from 250 to 300 in the tool — N_1 collapses to about one sixth (around 87k) and D shoots up.
🙋
So in practice do we just check "D less than 1 means safe"?
🎓
In theory yes, but experiments show actual failure scatters from D = 0.5 to 2.0, not exactly at 1. Load sequence (high-then-low vs the reverse), mean stress, and how you count cycles below the fatigue limit all violate the linear assumption. So safe designs tighten the criterion to D ≤ 0.3 or 0.5, and aerospace structures sometimes demand D ≤ 0.1. This tool is meant to teach the linear-accumulation idea; production design combines it with rainflow counting, Goodman/Gerber mean-stress correction and modified Miner rules such as Manson-Halford or Haibach.
🙋
The bar chart at the bottom right that splits D into D_1 and D_2 is really helpful. How is that decomposition used in real design?
🎓
Good catch. Identifying which stress level eats most of the life is arguably the most useful output of the Miner rule. With the defaults you get D_1 ≈ 0.00954 and D_2 ≈ 0.00205, so level 1 (250 MPa) accounts for 82% of the damage even though it has fewer cycles. The message is clear: cutting level 2 by half hardly helps, but shaving 20 MPa off level 1 is a huge win. Real designs split the spectrum into 5 to 10 levels, draw the same bar chart, and concentrate countermeasures on the dominant level. Slide sigma_a1 from 250 to 240 in the tool and watch D drop by about 35%.

Frequently Asked Questions

The Miner rule is the most basic method for predicting fatigue life of structures under variable-amplitude loading, also known as the linear cumulative damage rule. The formula is D = Sum(n_i/N_i), where n_i is the actual applied cycle count and N_i is the allowable cycle count to failure from the S-N curve at the same stress amplitude. The damage fractions n_i/N_i are added linearly across all stress levels, and the structure is predicted to fail when D reaches 1. With this tool's defaults (sigma_a1=250 MPa x 5,000 cycles, sigma_a2=200 MPa x 10,000 cycles) we obtain D ≈ 0.0116 and safety factor S = 1/D ≈ 86, meaning the spectrum can be repeated 86 more times before failure. Palmgren proposed the concept for rolling bearings in 1924 and Miner generalised it in 1945.
Because the Miner rule is purely linear, it (1) ignores the load-sequence effect (high-then-low vs low-then-high give different lives), (2) does not include mean-stress effects directly, and (3) usually ignores cycles below the fatigue limit. Experimental data show that failure happens not at D=1 but anywhere between D=0.5 and D=2.0. In safe design it is common to tighten the criterion to D ≤ 0.3 to 0.5, with aerospace structures often demanding D ≤ 0.1. This tool is meant as a teaching aid for the linear-accumulation concept; real design must combine it with rainflow counting, Goodman/Gerber correction and modified Miner rules (Manson-Halford, Haibach, etc.).
The Basquin rule assumes sigma_a = sigma_f' * (2 N_f)^b; solving for N gives N = 0.5 * (sigma_a/sigma_f')^(1/b). This tool uses typical steel constants: fatigue strength coefficient sigma_f' = 1000 MPa and Basquin exponent b = -0.10 (so 1/b = -10). For example sigma_a = 250 MPa gives N = 0.5 * (0.25)^(-10) = 0.5 * 4^10 = 524,288 cycles, while sigma_a = 200 MPa gives N = 0.5 * (0.2)^(-10) = 0.5 * 5^10 ≈ 4.88 x 10^6 cycles. The S-N curve is a straight line in log-log axes, and the allowable cycles grow exponentially as the stress amplitude is lowered.
A smaller accumulated damage D means more remaining fatigue life, and its reciprocal S = 1/D represents how many more times the same load spectrum can be repeated before failure. For example D = 0.0116 gives S ≈ 86, so 86 repetitions of the current load block bring you to the failure prediction D=1. In practice the required S follows the design life: S ≥ 2 for automotive and industrial machinery, S ≥ 4 to 10 for aircraft structures, and S ≥ 10 to 20 for nuclear pipework. Raising sigma_a1 from 250 to 350 MPa in this tool collapses N_1 by more than 10x and shows how dramatically D and S change.

Real-World Applications

Automotive coil springs: a suspension spring takes a continuous mix of mild road vibration and large pothole hits. Vehicle test data are processed by accelerometer-to-stress conversion and rainflow counting into a histogram of 5 to 10 stress levels; each sigma_a_i x n_i is fed into a Basquin S-N curve to obtain N_i and the Miner rule then gives D. A 100,000 km design typically requires D ≤ 0.5. Slide sigma_a1 from 250 to 300 in this tool and the damage roughly multiplies by six — exactly the kind of trade-off engineers face when choosing between SUP9 and SUP12 spring steels or thicker wire.

Wind-turbine tower and blade 20-year life: wind loads are modelled by a Weibull distribution, and stress cycles arrive at the blade root every second to minute, totaling 10^8 to 10^9 over 20 years. IEC 61400-1 requires that the annual damage D_year evaluated from the wind-speed histogram satisfies 20*D_year ≤ 1. This tool handles only two levels, but the workflow scales to many levels and is usually combined with Goodman correction for the dead-weight mean stress.

Aircraft structure durability testing: every take-off and landing produces a pressurisation cycle (ground-air-ground, GAG cycle) plus in-flight gust cycles. A typical flight spectrum mixes amplitudes from 1 to 100 MPa, and design lives of 60,000 to 100,000 flights must satisfy D ≤ 0.5, often D ≤ 0.1 to 0.3 for primary aircraft structure. Since the Comet accidents of 1954, airframes are double-checked with Miner-based damage accumulation and a damage-tolerance crack-growth analysis.

Pressure-vessel and piping operating history: nuclear and chemical-plant piping experiences hundreds of MPa thermal stress cycles per startup, shutdown or trip. ASME Boiler & Pressure Vessel Code Section III classifies the operating history into 5 to 8 levels and requires the cumulative usage factor U = Sum(n_actual/N_design) ≤ 1.0 for the entire design life — the American formulation of the Miner rule. D in this tool is the same quantity as U.

Common Misconceptions and Pitfalls

The most common mistake is assuming that failure happens exactly at D = 1. In experiments only about 50% of specimens fail at D = 1; the rest spread between roughly D = 0.5 and D = 2.0 (lognormal). In particular, a high-then-low load sequence often fails earlier than the linear rule predicts because an early overload nucleates micro-cracks that then propagate faster under the smaller cycles. This effect is called the "sequence effect" and is not captured by the pure linear assumption used here.

The second pitfall is oversimplifying "no damage below the fatigue limit". Classical Miner does assume so, but modern very-high-cycle-fatigue tests show that failure does occur beyond N > 10^9 — the so-called gigacycle regime. Rail wheels, train axles and turbine blades cannot ignore this effect. This tool uses the Basquin straight line everywhere and does not include a horizontal fatigue limit, but real design often applies the modified Miner rule (CITA) or the Haibach approach to gently bend the curve below the knee.

The third misunderstanding is that the Miner rule is "old-fashioned" and that modern CFD/FEA computes fatigue life directly. In reality commercial fatigue post-processors such as Ansys Workbench, Abaqus FE-SAFE and MSC Fatigue all use the Miner rule internally. They take the FE stress history, rainflow-count it, apply an S-N curve at each level and sum n_i/N_i — exactly what this tool does. What they add is multi-axial stress combination, mean-stress correction and surface/size correction factors. The simulator focuses on this "core" of the workflow so you can build intuition before reaching for a commercial code.

Related Tools

Variable Amplitude Fatigue Calculate fatigue life for variable loads using Rainflow counting and Miner's Rule. Visuali... Fatigue Notch Calculate fatigue notch factor Kf instantly. Compare Neuber and Peterson methods using Kt, ... Fatigue Spectrum Calculate cumulative fatigue damage, safety factors, and remaining life in real-time using ... Creep Analysis Calculate creep strain rate using the Norton power law and visualize primary-to-tertiary cr... Creep Deformation Calculate creep strain rate and rupture life for alloys like 316L stainless. Use Norton's L... Creep Relaxation Simulate creep strain and stress relaxation in real time for 316SS or IN718 alloys using th...