MEC-36

Gerber fatigue parabola

(σa/Se)² + σm/Sut = 1/n. A parabola through Se and Sut.

Reading speed

FatigueGerber 1874

Governing equation

\left(\dfrac{\sigma_a}{S_e}\right)^2+\dfrac{\sigma_m}{S_{ut}}=\dfrac{1}{n}

$\sigma_a \in \left[1, 400\right]$

where

\sigma_m: Mean stress (MPa)
\sigma_a: Alternating stress (MPa)
S_{ut}: Ultimate tensile (MPa)
S_e: Endurance limit (MPa)
n: Factor of safety (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

\left(\dfrac{\sigma_a}{S_e}\right)^2+\dfrac{\sigma_m}{S_{ut}}=\dfrac{1}{n}

$\sigma_a \in \left[1, 400\right]$

$\sigma_a = 1$ $n = 5.999$

Along the curve

②

Domain of the variable — blue

\sigma_a_{\min} = 1,\quad \sigma_a_{\max} = 400

$\sigma_a \in \left[1, 400\right]$

$\sigma_a = 1$ $\sigma_a = 1$

Along the curve

③

Response curve — green

\left(\dfrac{\sigma_a}{S_e}\right)^2+\dfrac{\sigma_m}{S_{ut}}=\dfrac{1}{n}

$\sigma_a \in \left[1, 400\right]$

$\sigma_a = 1$ $n = 5.999$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}n}{\mathrm{d}\sigma_a}

$\sigma_a \in \left[1, 400\right]$

$\sigma_a = 3.509$ $n^{\prime} = -0.006$

Along the curve

Lecture brief

Historical brief

Machine design grew from Coulomb torsion and Hertz contact (1881) through Soderberg fatigue and the heat-engine cycle. The lab writes shaft, bearing, contact and thermodynamic limits in SI. This sheet (MEC-36 — Gerber fatigue parabola) is the form associated with Gerber 1874. Working symbols: $\sigma_m$ , $\sigma_a$ , $S_{ut}$ , $S_e$ $\rightarrow$ $n$ . Gerber fitted 19th-century fatigue data with a parabola, sitting between Goodman and Soderberg.

Purpose

Purpose: compute

n

from

\sigma_m

\sigma_a

S_{ut}

S_e

in Mechanical via

\left(\dfrac{\sigma_a}{S_e}\right)^2+\dfrac{\sigma_m}{S_{ut}}=\dfrac{1}{n}

(σa/Se)² + σm/Sut = 1/n. A parabola through Se and Sut. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\sigma_m = 100.000\,\mathrm{MPa}

\sigma_a = 80.000\,\mathrm{MPa}

S_{ut} = 600.000\,\mathrm{MPa}

S_e = 200.000\,\mathrm{MPa}

, the governing relation

\left(\dfrac{\sigma_a}{S_e}\right)^2+\dfrac{\sigma_m}{S_{ut}}=\dfrac{1}{n}

yields

n = 3.061\,\mathrm{—}

. A Haigh plot, a parabola, a safety tick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mean stress \sigma_m100 MPaAlternating stress \sigma_a80 MPaUltimate tensile S_{ut}600 MPaEndurance limit S_e200 MPa

Outputs

Factor of safety n3.061 —

Reading speed

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Narration of this film

A Haigh plot, a parabola, a safety tick.

Gerber fitted 19th-century fatigue data with a parabola, sitting between Goodman and Soderberg.

Reading speed

Watch on YouTube