MEC-24

Goodman fatigue line

1/n = σm/Sut + σa/Se. Mean stress referred to ultimate, not yield.

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FatigueGoodman 1899

Governing equation

\dfrac{1}{n}=\dfrac{\sigma_m}{S_{ut}}+\dfrac{\sigma_a}{S_e}

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

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

\dfrac{1}{n}=\dfrac{\sigma_m}{S_{ut}}+\dfrac{\sigma_a}{S_e}

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

$\sigma_a = 1$ $n = 7.229$

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

\dfrac{1}{n}=\dfrac{\sigma_m}{S_{ut}}+\dfrac{\sigma_a}{S_e}

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

$\sigma_a = 1$ $n = 7.229$

Along the curve

④

Rate of change — amber

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

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

$\sigma_a = 3.509$ $n^{\prime} = -0.221$

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-24 — Goodman fatigue line) is the form associated with Goodman 1899. Working symbols: $\sigma_m$ , $\sigma_a$ , $S_{ut}$ , $S_e$ $\rightarrow$ $n$ . Goodman is less conservative than Soderberg and the usual first check in machine design.

Purpose

Purpose: compute

n

from

\sigma_m

\sigma_a

S_{ut}

S_e

in Mechanical via

\dfrac{1}{n}=\dfrac{\sigma_m}{S_{ut}}+\dfrac{\sigma_a}{S_e}

1/n = σm/Sut + σa/Se. Mean stress referred to ultimate, not yield. 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 = 80.000\,\mathrm{MPa}

\sigma_a = 70.000\,\mathrm{MPa}

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

S_e = 200.000\,\mathrm{MPa}

, the governing relation

\dfrac{1}{n}=\dfrac{\sigma_m}{S_{ut}}+\dfrac{\sigma_a}{S_e}

yields

n = 2.069\,\mathrm{—}

. A Haigh plot, a Goodman line, an intercept at Sut. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Mean stress \sigma_m80 MPaAlternating stress \sigma_a70 MPaUltimate tensile S_{ut}600 MPaEndurance limit S_e200 MPa

Outputs

Factor of safety n2.069 —

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Narration of this film

A Haigh plot, a Goodman line, an intercept at Sut.

Goodman is less conservative than Soderberg and the usual first check in machine design.

Reading speed

Watch on YouTube