Visual walkthrough — Thermal analysis — conduction in structures, thermal stress

This page rebuilds the parent result Thermal Stress one picture at a time. We start with a bar of metal and a warm Sun, and we do not use a single symbol until we have drawn it. By the end you will see why a heated, bolted-down beam pushes back — and how hard.

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Step 1 — What "expansion" even means

WHAT. Take a straight metal bar. Warm it uniformly. It gets a little longer. We call the extra length $\Delta L$ (read: "the change in length", the triangle $\Delta$ just means change in).

WHY. Before we talk about forces we need a clean number for how much longer the bar gets. Everything downstream is built on this one measurement.

PICTURE. The top bar is cold, length $L$ (the original length). The bottom bar is warm — its right end has crept to the right by $\Delta L$.

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Step 2 — How temperature turns into strain

WHAT. Experiment says: heat a bar by $\Delta T$ degrees and its strain grows in direct proportion. Double the heating, double the stretch. The proportionality constant is the material's coefficient of thermal expansion, written $\alpha$ (Greek "alpha").

WHY. We need a bridge from temperature (what the orbit gives us) to strain (what mechanics cares about). $\alpha$ is that bridge, and it is a fixed number for each material — see Material Selection for Spacecraft.

PICTURE. A straight line: horizontal axis is temperature change $\Delta T$, vertical axis is thermal strain ${\epsilon }_{\text{th}}$. The steeper the line, the "twitchier" the material. Aluminium's line is steep; carbon-fibre's is nearly flat.

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Step 3 — How force turns into strain (Hooke's Law)

WHAT. Forget heat for a moment. Grab a cold bar and pull it with a force. It stretches. Push it and it squashes. Over the elastic range, strain is proportional to the stress you apply.

WHY. In Step 4 the bar will be forbidden to stretch thermally — so something else must cancel that stretch. That "something else" is mechanical squashing. To use it, we need the force↔strain rule by itself, clean.

PICTURE. Left: a bar pulled by an arrow (tension) grows and turns teal. Right: a bar pushed by inward arrows (compression) shrinks and turns orange. The graph beside them: stress $\sigma$ up the axis, strain $\epsilon$ across — a straight line through the origin whose slope is $E$.

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Step 4 — The trap: forbid the bar to move

WHAT. Now bolt both ends of the bar to a rigid wall, then heat it by $\Delta T$. The bar wants to grow by ${\epsilon }_{\text{th}}=\alpha \Delta T$ — but the walls refuse. Final length change: zero.

WHY. This is the whole point of the parent note. A free bar builds no stress (see the mistake box below). Stress appears only when the wanted motion is blocked. The blocking is what stores force in the material.

PICTURE. Two walls (hatched) clamp the bar. A faint "ghost" bar shows where the warm bar wanted to reach (dashed, past the wall). The real bar is stuck at length $L$. Inward orange arrows at each wall show the material shoving outward and the wall shoving back.

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Step 5 — Add the two strains and set the total to zero

WHAT. The bar experiences two strains at once: the thermal stretch it wants (${\epsilon }_{\text{th}}$) and an elastic strain (${\epsilon }_{\text{el}}$) forced on it by the wall's push. The wall guarantees the total length change is zero: ${\epsilon }_{\text{total}}={\epsilon }_{\text{th}}+{\epsilon }_{\text{el}}=0.$

WHY. This single equation is the hinge of the whole derivation. "Total motion = 0" is the mathematical statement of "the walls don't let it move." Rearranged, it says the elastic strain must exactly undo the thermal strain.

PICTURE. A number line of strain. A right-pointing teal arrow of length ${\epsilon }_{\text{th}}=\alpha \Delta T$ (the wanted growth). A left-pointing orange arrow of length ${\epsilon }_{\text{el}}$ (the forced squash). They are the same length, tip-to-tail landing back at $0$.

From ${\epsilon }_{\text{th}}+{\epsilon }_{\text{el}}=0$: ${\epsilon }_{\text{el}}=-{\epsilon }_{\text{th}}=-\alpha \Delta T.$

• The minus sign: the elastic strain points opposite to the thermal one. Heating (wants to grow) is answered by squashing.

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Step 6 — Convert that forced strain into stress

WHAT. Feed ${\epsilon }_{\text{el}}=-\alpha \Delta T$ into Hooke's Law $\sigma =E{\epsilon }_{\text{el}}$ from Step 3.

WHY. Strain is invisible; stress is the thing that cracks metal, warps mirrors, and fatigues joints. Hooke's Law is the only tool that turns a known strain into the stress we actually design against — that is exactly why we isolated it back in Step 3.

PICTURE. A flow strip: box "$\Delta T$" → (×$\alpha$) → box "${\epsilon }_{\text{th}}$" → (flip sign, the wall) → box "${\epsilon }_{\text{el}}$" → (×$E$) → box "$\sigma$". Each arrow labelled with the operation.

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Step 7 — Cover every sign (all the cases)

WHAT. The formula has one term whose sign flips: $\Delta T$. That single sign controls the entire physical story. Walk all three cases.

WHY. Contract rule: never leave the reader in a scenario you didn't draw. Orbit takes a strut from sunlight to shadow — it will see both signs, and the crossing point too.

PICTURE. Three clamped bars stacked.

• Top, $\Delta T>0$ (heated): wants to grow → wall squashes it → compression (orange, inward arrows). $\sigma <0$.
• Middle, $\Delta T=0$: wants nothing → wall does nothing → zero stress. The degenerate case.
• Bottom, $\Delta T<0$ (cooled): wants to shrink → wall pulls it back out → tension (teal, outward arrows). $\sigma >0$.

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Step 8 — When the bar is free on one end (degenerate limit)

WHAT. Loosen one bolt. Now the bar can grow. What happens to the stress?

WHY. Real spacecraft joints range from rock-solid to floppy. The two extremes — perfectly clamped and perfectly free — bracket everything in between. We must show both ends of that spectrum.

PICTURE. A dial from "free" to "fixed." At free: the ghost and the real bar coincide, arrows vanish, $\sigma =0$. At fixed: the ghost overshoots, full arrows, $\sigma =-E\alpha \Delta T$. A curve between shows stress rising with constraint stiffness.

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The one-picture summary

Everything above, on one canvas: temperature on the far left, driving strain, blocked by a wall, becoming stress on the far right — with the sign map underneath.

Recall Feynman retelling — say it like a story

A metal bar is basically a spring made of the same stuff all the way through. Heat it and every atom jiggles a bit wider apart, so the whole bar tries to get longer — by a fraction $\alpha$ for each degree, times how many degrees you added. That's the wanting-to-grow. Now bolt both ends to a wall. The bar tries to grow, the wall says no. So the bar ends up squashed by exactly the amount it wanted to grow — the two cancel to zero motion. A squashed spring pushes back, and how hard it pushes depends on how stiff it is, which we call $E$. Multiply stiffness by the amount of squash and you get the push per unit area: that's the stress, $\sigma =-E\alpha \Delta T$. The minus sign is just bookkeeping — heat means push (compression), cold means pull (tension). Free the bar and the squash disappears, so the stress disappears too. That last fact is the designer's escape hatch: don't fight the heat, let the bar move.

Recall Quick self-test

What single quantity's sign decides tension vs compression? ::: $\Delta T$ — positive (heating) → compression; negative (cooling) → tension. A bar heats up but is free on both ends. What is its thermal stress? ::: Zero — nothing blocks the expansion. Which two material numbers multiply $\Delta T$ in the stress formula? ::: Young's modulus $E$ and the expansion coefficient $\alpha$. Why do engineers add flexible mounts? ::: To lower the constraint fraction $c$, which lowers stress in direct proportion.

Prerequisite links: Structural Dynamics, Finite Element Analysis, Composite Materials in Spacecraft, Thermal Environment in Orbit.