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Mechanical

Stress amplitude of a fatigue cycle

Half the range of a repeating stress cycle.

Ref: MEC-87
Rev: v1
Field: Mechanical
Result: σ_a [MPa]

σ_{a} = \frac{σ_{\max} - σ_{\min}}{2}

A fatigue cycle rises and falls between a maximum and minimum stress. The stress amplitude is half the range: σ_a = (σ_max − σ_min)/2. Fatigue life is governed mainly by the amplitude, with the mean stress having a secondary effect. For fully reversed loading (σ_min = −σ_max) the amplitude equals σ_max.

Half the range of a repeating stress cycle. It is the alternating part of the stress and is never negative, and it controls the damage done by each cycle.

Calculator

Solve for

σ_aStress amplitudeComputed

MPa

σ_maxMaximum stress in the cycle

MPa

−400600

σ_minMinimum stress in the cycle

MPa

−400600

Stress range, σ_max − σ_min160 MPa
Stress amplitude80 MPa
Mean stress40 MPa
Amplitude ratio, σ_a/σ_m2

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Stress amplitude (MPa) current value 80 MPa

σ_maxMaximum stress in the cycleKey driver

3.33320

σ_minMinimum stress in the cycle

6.67260

A cycle from −40 to 120 MPa has an amplitude of 80 MPa about a mean of 40 MPa.
The same amplitude can come with different means: a cycle of 0 to 160 MPa and one of −80 to 80 MPa both have 80 MPa amplitude, but the first (tensile mean) is the more damaging.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (MPa)	Local step	Change in result
σ_max Maximum stress in the cycle	−400 to 600 MPa	3.333	320	316.7	+20 MPa	+12.5 %
σ_min Minimum stress in the cycle	−400 to 600 MPa	6.667	260	253.3	+20 MPa	−12.5 %

Sweep of Maximum stress in the cycle

Maximum stress in the cycle (MPa)	Stress amplitude (MPa)
−400	180
−275	117.5
−150	55
−25	7.5
100	70
225	132.5
350	195
475	257.5
600	320

Sweep one variable

Vary

Stress amplitude (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_a

Stress amplitude

Result · MPa

What the equation gives you. Everything else feeds it.

σ_max

Maximum stress in the cycle

MPa · typical −400 to 600

The maximum stress of the cycle, in MPa (tension positive). Raises the amplitude by half the change, because the range grows. Raising it by 10 MPa (to 130 MPa) moves the result by +6.25 %, from 80 to 85 MPa.

σ_min

Minimum stress in the cycle

MPa · typical −400 to 600

The minimum stress of the cycle, in MPa (compression negative). Lowers the amplitude by half the change when raised, because the range narrows. Raising it by 10 MPa (to −30 MPa) moves the result by −6.25 %, from 80 to 75 MPa.

Worked example with the values above

Range: σ_max − σ_min = 120 − (−40) = 160 MPa.
Halve it: 160 / 2.
σ_a = 80 MPa.

Assumptions and limits

A constant-amplitude cycle with nominally constant mean stress; stresses are the nominal local stresses.
Tension is positive and compression negative, with σ_max the algebraically larger value.

Common mistakes

Using the full range instead of half of it.
Entering compression as a positive number.

Statistics and uncertainty

The amplitude is the difference of two stresses, so errors in each add: ±5 MPa in both extremes gives about ±5 MPa in the amplitude. Fatigue life depends on the amplitude to a high power (often 3 to 10), so a small amplitude error becomes a large life error.

Where it is used

The first step in a fatigue check of any part under cyclic load.
Reading the amplitude from a strain-gauge or load record.

Maths you need

Stress cycle: the repeated rise and fall of stress
Tension positive: compression is entered as a negative stress

Related formulas

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