MEC-47

Thermal stress

Fully constrained thermal expansion.

Reading speed

MachinesThermal stress

Governing equation

\sigma=E\alpha\Delta T

$\Delta T \in \left[0, 400\right]$

where

E: Modulus (GPa)
\alpha: Alpha (1/K)
\Delta T: Rise (K)
\sigma: Stress (MPa)

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

\sigma=E\alpha\Delta T

$\Delta T \in \left[0, 400\right]$

$\Delta T = 0$ $\sigma = 0$

Along the curve

②

Domain of the variable — blue

\Delta T_{\min} = 0,\quad \Delta T_{\max} = 400

$\Delta T \in \left[0, 400\right]$

$\Delta T = 0$ $\Delta T = 0$

Along the curve

③

Response curve — green

\sigma=E\alpha\Delta T

$\Delta T \in \left[0, 400\right]$

$\Delta T = 0$ $\sigma = 0$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma}{\mathrm{d}\Delta T}

$\Delta T \in \left[0, 400\right]$

$\Delta T = 2.516$ $\sigma^{\prime} = 2.4$

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-47 — Thermal stress) is the form associated with Thermal stress. Working symbols: $E$ , $\alpha$ , $\Delta T$ $\rightarrow$ $\sigma$ . Fully constrained thermal expansion. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

\sigma

from

E

\alpha

\Delta T

in Mechanical via

\sigma=E\alpha\Delta T

Fully constrained thermal expansion. Use it when a real mechanical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

E = 200.000\,\mathrm{GPa}

\alpha = 1.200e-5\,\mathrm{1/K}

\Delta T = 80.000\,\mathrm{K}

, the governing relation

\sigma=E\alpha\Delta T

yields

\sigma = 192.00\,\mathrm{MPa}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Modulus E200 GPaAlpha \alpha0.000012 1/KRise \Delta T80 K

Outputs

Stress \sigma192.00 MPa

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Fully constrained thermal expansion. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube