How to Calculate Fatigue Strength: Formula, Equations, and Worked Example

What You Need Before You Calculate Fatigue Strength

Before using any formula, define the calculation case clearly.

You need:

• Material data: ultimate tensile strength (UTS), uncorrected endurance limit (Se′) for steels or fatigue strength at a specified cycle count for materials without a true endurance limit
• Loading data: maximum stress, minimum stress, stress amplitude, mean stress, and stress ratio
• Target life: the number of cycles you want the component to survive
• Correction factors: surface finish, size, load type, temperature, reliability, and any other relevant reduction factors

If those inputs are vague, the result will be vague too.

Common Fatigue Strength Symbols

The same terms appear repeatedly in fatigue calculations, so the notation should be clear from the start.

• σ_a — stress amplitude
• σ_m — mean stress
• σ_max — maximum cyclic stress
• σ_min — minimum cyclic stress
• R — stress ratio, defined as σmin / σmax
• UTS — ultimate tensile strength
• S_e′ — uncorrected endurance limit for the test specimen
• S_e — corrected endurance limit for the real component
• σ^′_f — fatigue strength coefficient
• b — fatigue strength exponent
• N_f — number of cycles to failure
• S_N — fatigue strength at N cycles

These are the symbols most engineers encounter when working with stress-life fatigue calculations.

Step 1: Calculate the Stress Amplitude, Mean Stress, and Stress Ratio

Hand calculations start with the loading cycle.

Stress amplitude

\(\sigma_a = \frac{\sigma_{\max} – \sigma_{\min}}{2}\)

This is the alternating part of the stress cycle. In most fatigue calculations, this is the stress you are ultimately trying to compare with an allowable fatigue value.

Mean stress

\(\sigma_m = \frac{\sigma_{\max} + \sigma_{\min}}{2}\)

Mean stress matters because tensile mean stress reduces fatigue resistance, while compressive mean stress can improve it.

Stress ratio

\(R = \frac{\sigma_{\min}}{\sigma_{\max}}\)

The stress ratio defines the loading cycle shape. A fully reversed cycle has R = -1. A pulsating tensile load has R = 0. Different fatigue data sets are tied to different stress ratios, so this value cannot be ignored.

Step 2: Estimate the Base Fatigue Strength

The next step depends on the material type.

For steels with an endurance limit

A common starting estimate for polished laboratory specimens is:

\(S_e’ \approx 0.5 \, UTS\)

This is only a first estimate, not a design-ready number. It applies to smooth test specimens, not real components with holes, welds, rough surfaces, or size effects.

For materials without a true endurance limit

For aluminium and other non-ferrous materials, use the fatigue strength at a specified cycle count from an S–N curve instead of assuming an endurance limit.

For aluminium, “what is the fatigue limit?” is usually the wrong question. The practical question is: what stress amplitude is acceptable for the required life?

Step 3: Correct the Endurance Limit for Real-World Conditions

For steel components, the test-specimen endurance limit must be reduced before it can be used in design.

The standard Marin-style correction form is:

\(S_e = k_a \cdot k_b \cdot k_c \cdot k_d \cdot k_e \cdot k_f \cdot S_e’\)

Where:

• k_a — surface finish factor
• k_b — size factor
• k_c — load factor
• k_d — temperature factor
• k_e — reliability factor
• k_f — miscellaneous-effects factor or fatigue stress-concentration treatment, depending on method used

This is the step many simplified explanations skip, even though it is what makes the estimate usable for a real component. They jump from 0.5 × UTS straight to a design conclusion, skipping the part where the lab value gets cut down to something realistic.

Step 4: Apply a Mean-Stress Correction

If the mean stress is not zero, adjust the allowable stress amplitude.

One of the most widely used practical equations is the modified Goodman equation:

\(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{UTS} = 1\)

Rearranged to solve for allowable stress amplitude:

\(\sigma_a = S_e \left(1 – \frac{\sigma_m}{UTS}\right)\)

This gives a quick hand-calculation estimate of fatigue strength under non-zero mean stress.

What the Goodman equation means in practice

• If σ_m = 0, the allowable alternating stress is about S_e
• If σ_m increases in tension, the allowable σ_a drops
• If σ_m approaches UTS, the allowable alternating stress collapses toward zero

That is why a component under steady tensile preload usually has worse fatigue performance than one under a fully reversed cycle with the same stress amplitude.

Step 5: Use the Basquin Equation for Finite-Life Calculations

If you need the fatigue strength at a specified number of cycles, use the stress-life relationship.

Basquin equation

\(\sigma_a = \sigma_f’ \left(2N_f\right)^b\)

Where:

• σ_a is the stress amplitude
• σ^′_f is the fatigue strength coefficient
• N_f is the number of cycles to failure
• b is the fatigue strength exponent

This is the standard equation for high-cycle fatigue in the elastic regime.

Worked Example: Hand Calculation of Fatigue Strength

Take a steel component with:

• UTS = 600 MPa
• Corrected endurance limit Se = 300 MPa
• Mean stress σm = 100 MPa

Use the modified Goodman equation to estimate the allowable fatigue stress amplitude.

Step 1: Write the equation

\(\sigma_a = S_e \left(1 – \frac{\sigma_m}{UTS}\right)\)

Step 2: Substitute the values

\(\sigma_a = 300 \left(1 – \frac{100}{600}\right)\)

Step 3: Simplify

\(\sigma_a = 300 \left(1 – 0.1667\right)\)

\(\sigma_a = 300 \cdot 0.8333\)

Step 4: Final answer

\(\sigma_a \approx 250 \, \text{MPa}\)

So the estimated allowable fatigue stress amplitude for this loading condition is 250 MPa.

That does not mean the part is automatically safe. It only means the hand-calculated fatigue strength estimate under the stated assumptions is 250 MPa.

Example Using the Basquin Equation

Suppose a material has:

• σ^′_f = 900 MPa
• b = -0.09
• target life N_f = 10^6 cycles

Then:

\(\sigma_a = 900 \left(2 \times 10^6\right)^{-0.09}\)

This gives the estimated fatigue strength at one million cycles based on the stress-life relation.

In practice, engineers usually take σ^′_f and b from material data, standards, or validated fatigue test results, not guess them.

When Hand Calculations Are Good Enough

Hand calculations are useful when:

• you need a quick concept-stage estimate
• the geometry is simple
• the loading is clear and approximately constant-amplitude
• the goal is to sense-check a result or compare options

They are especially useful for understanding how mean stress, stress ratio, and correction factors change the answer.

When Hand Calculations Stop Being Enough

Hand calculations are not enough when:

• welds dominate fatigue behavior
• stress concentration is difficult to idealize
• the geometry is complex or three-dimensional
• multiple load cases interact
• variable-amplitude loading must be counted
• code compliance has to be demonstrated formally
• reporting has to be traceable and repeatable

That is where fatigue software, detailed FEA results, S–N class selection, rainflow counting, and code-based verification become necessary.

Common Mistakes in Fatigue Strength Calculations

Using UTS as if it were fatigue strength

Ultimate tensile strength is not fatigue strength. Static capacity and cyclic resistance are different problems.

Using 0.5 × UTS as a final design answer

That value is only a smooth-specimen starting estimate for some steels. It is not the usable fatigue strength of a real component.

Ignoring mean stress

A tensile mean stress reduces allowable fatigue stress amplitude. If you skip that correction, the estimate is too optimistic.

Applying steel logic to aluminium

Aluminium does not usually have a true endurance limit. Use finite-life S–N data instead of pretending there is a safe infinite-life plateau.

Treating hand calculations as code compliance

They are not. Hand calculations are screening tools, not a replacement for a proper fatigue assessment when standards, weld classes, and real stress fields matter.

Fatigue Calculators and Software

Fatigue calculators are useful when you want faster iteration or when the number of interacting variables starts getting large. They help when:

• several load cases must be checked
• correction factors have to be applied repeatedly
• local stresses come from FEA
• welded details must be recognized and classified
• a formal report is required

The SDC Verifier supports fatigue checks according to standards such as DNV-RP-C203, Eurocode 3, and DIN 15018. It also includes automated tools for weld recognition and fatigue-detail setup, which is where manual workflows usually become slow and error-prone.

Final Takeaway

If you want to calculate fatigue strength by hand, the workflow is straightforward:

1. define the loading cycle
2. calculate σ_a, σ_m, and R
3. estimate the base fatigue value from material data
4. apply correction factors
5. correct for mean stress
6. use Basquin when you need fatigue strength at a target cycle count

That is enough for a serious first estimate.

It is not enough for every real structure.

When welds, variable-amplitude loading, code compliance, and complex geometry start driving the result, hand calculations should stop being the main method and become the sense-check.

Fundamentals of Fatigue — Article Series

1. What Is Fatigue? (Definitions, Types, Causes)
2. Fatigue Strength vs Fatigue Limit: Formula, Symbols & Material Data
3. Fatigue Life: Key Influencing Factors and Advanced Prediction Methods
4. Fatigue Stress and Its Role in Structural Failure
5. How to Calculate Fatigue Strength: Formula, Equations, and Worked Example

FAQ

What is the formula for fatigue strength?

There is no single universal fatigue-strength formula for every case. The most common equations are the Basquin equation for finite-life stress-life calculations and the modified Goodman equation for mean-stress correction.

How do you calculate fatigue strength by hand?

At minimum, calculate the stress amplitude, mean stress, and stress ratio; estimate or obtain the base fatigue value from material data; apply correction factors; then use a mean-stress equation such as Goodman or a stress-life equation such as Basquin.

What is the fatigue strength coefficient?

The fatigue strength coefficient σ^′_f is a material parameter used in the Basquin equation. It helps define the stress-life curve in the high-cycle regime.

What is the Goodman equation for fatigue?

The modified Goodman equation is:

\(\frac{\sigma_a}{S_e} + \frac{\sigma_m}{UTS} = 1\)

It is used to reduce the allowable stress amplitude when tensile mean stress is present.

What is the difference between fatigue strength and fatigue limit?

Fatigue strength is the stress causing failure at a specified number of cycles. Fatigue limit is a threshold below which some materials can survive very large numbers of cycles without fatigue failure. For the full comparison, see Fatigue Strength vs Fatigue Limit: Formula, Symbols, and Material Data.

Can you calculate fatigue strength for aluminium the same way as steel?

Not exactly. Steel calculations often start from an endurance-limit concept. Aluminium calculations usually rely on finite-life S–N data because aluminium generally does not show a true endurance limit.