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Mechanical

Mean stress of a fatigue cycle

Average of the maximum and minimum stress in a cycle.

Ref: MEC-88
Rev: v1
Field: Mechanical
Result: σ_m [MPa]

σ_{m} = \frac{σ_{\max} + σ_{\min}}{2}

The mean stress of a cycle is the average of its extremes: σ_m = (σ_max + σ_min)/2. A tensile mean stress shortens fatigue life and a compressive one lengthens it, for the same amplitude. For fully reversed loading the mean is zero.

Average of the maximum and minimum stress in a cycle. A tensile mean stress shortens fatigue life and a compressive mean stress usually lengthens it.

Calculator

Solve for

σ_mMean stressComputed

MPa

σ_maxMaximum stress in the cycle

MPa

−400600

σ_minMinimum stress in the cycle

MPa

−400600

Stress amplitude80 MPa
Stress range160 MPa
Stress ratio R−0.3333

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.

Mean stress (MPa) current value 40 MPa

σ_maxMaximum stress in the cycleKey driver

−220280

σ_minMinimum stress in the cycle

−140360

A cycle from −40 to 120 MPa has a mean stress of 40 MPa. The tensile mean makes it worse than a fully reversed ±80 MPa cycle with the same amplitude.
Raising both extremes by the same amount shifts the mean by that amount and leaves the amplitude unchanged.

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	−220	280	500	+20 MPa	+25 %
σ_min Minimum stress in the cycle	−400 to 600 MPa	−140	360	500	+20 MPa	+25 %

Sweep of Maximum stress in the cycle

Maximum stress in the cycle (MPa)	Mean stress (MPa)
−400	−220
−275	−157.5
−150	−95
−25	−32.5
100	30
225	92.5
350	155
475	217.5
600	280

Sweep one variable

Vary

Mean stress (MPa)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

σ_m

Mean stress

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 mean stress by half the change. Raising it by 10 MPa (to 130 MPa) moves the result by +12.5 %, from 40 to 45 MPa.

σ_min

Minimum stress in the cycle

MPa · typical −400 to 600

The minimum stress of the cycle, in MPa (compression negative). Raises the mean stress by half the change. Raising it by 10 MPa (to −30 MPa) moves the result by +12.5 %, from 40 to 45 MPa.

Worked example with the values above

Sum: σ_max + σ_min = 120 + (−40) = 80 MPa.
Halve it: 80 / 2.
σ_m = 40 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

Forgetting that residual stresses change the mean stress.
Adding the signs wrongly for a compressive minimum.

Statistics and uncertainty

The mean stress is the average of the two extremes, so equal and independent errors of ±u in each give an error of about ±0.7u. Residual stress (for example from shot peening, autofrettage or welding) adds directly to the mean stress.

Where it is used

Fitting mean-stress corrections such as Goodman and Gerber.
Understanding why preloaded bolts and shot-peened parts last longer.

Maths you need

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

Related formulas

Practise this formula