MEC-23

Basquin S–N law

σa = σf' (2N)^b in high-cycle fatigue.

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FatigueBasquin 1910

Governing equation

\sigma_a=\sigma_f'(2N)^b

$N \in \left[1000, 1\times 10^{8}\right]$

where

\sigma_f': Fatigue coefficient (MPa)
b: Basquin exponent (—)
N: Reversals/2 (cycles)
\sigma_a: Allowable amplitude (MPa)

The figure, part by part

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①

Governing law — red

\sigma_a=\sigma_f'(2N)^b

$N \in \left[1000, 1\times 10^{8}\right]$

$N = 1000$ $\sigma_a = 454.1$

Along the curve

②

Domain of the variable — blue

N_{\min} = 1000,\quad N_{\max} = 1\times 10^{8}

$N \in \left[1000, 1\times 10^{8}\right]$

$N = 1000$ $N = 1000$

Along the curve

③

Response curve — green

\sigma_a=\sigma_f'(2N)^b

$N \in \left[1000, 1\times 10^{8}\right]$

$N = 1000$ $\sigma_a = 454.1$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma_a}{\mathrm{d}N}

$N \in \left[1000, 1\times 10^{8}\right]$

$N = 6.3\times 10^{5}$ $\sigma_a^{\prime} = -1.71\times 10^{-4}$

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-23 — Basquin S–N law) is the form associated with Basquin 1910. Working symbols: $\sigma_f'$ , $b$ , $N$ $\rightarrow$ $\sigma_a$ . Log stress versus log reversals is linear in the elastic-life regime.

Purpose

Purpose: compute

\sigma_a

from

\sigma_f'

b

N

in Mechanical via

\sigma_a=\sigma_f'(2N)^b

σa = σf' (2N)^b in high-cycle fatigue. 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_f' = 900.000\,\mathrm{MPa}

b = -0.090\,\mathrm{—}

N = 1.000e+6\,\mathrm{cycles}

, the governing relation

\sigma_a=\sigma_f'(2N)^b

yields

\sigma_a = 243.87\,\mathrm{MPa}

. An S–N curve, a power slope, a life tick. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Fatigue coefficient \sigma_f'900 MPaBasquin exponent b-0.09 —Reversals/2 N1000000 cycles

Outputs

Allowable amplitude \sigma_a243.87 MPa

Reading speed

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Narration of this film

An S–N curve, a power slope, a life tick.

Log stress versus log reversals is linear in the elastic-life regime.

Reading speed

Watch on YouTube