Thermal Moment⚠ unverified

Understanding the Parameters

• The temperature gradient $dT/dy$ — the true driver of a thermal moment is the gradient, not the absolute temperature: a uniform temperature change causes only axial thermal stress (or free expansion), never bending. It is the difference in temperature across the depth that makes the beam want to curve. The registry exposes a $\Delta T$ input which should be read as the gradient (temperature difference over depth).

• Young's modulus $E$ and expansion $\alpha$ — as with axial thermal stress, a stiffer, higher-expansion material builds a larger thermal moment for the same gradient. Low-$\alpha$ materials (Invar, tailored composites) are chosen to suppress thermal bending in dimensionally-critical structures.

• Second moment of area $I$ — the same section stiffness that appears in mechanical bending (Wing Bending Stress). A deeper section develops a larger restrained thermal moment (it has more $I$), but also, if free to bend, curves less for a given moment.

• The output $M_T$ — the bending moment that develops if the beam's curvature is fully restrained. If the beam is free, no moment develops; it simply bows with thermal curvature $\kappa_T = \alpha\,(dT/dy)$. If statically indeterminate (redundant supports), part of the thermal moment is reacted and part relieved by deflection — the general case requires a compatibility analysis.

Derivation (Approaching a Proof)

Suppose the beam depth carries a linear temperature distribution $T(y) = (dT/dy)\,y$ measured from the neutral axis. If the material were free, each fibre would expand by its thermal strain $\varepsilon_{th}(y) = \alpha\,T(y) = \alpha\,(dT/dy)\,y$ — a strain that varies linearly across the depth, exactly like a mechanical bending strain. This corresponds to a stress-free thermal curvature

$$\kappa_T = \alpha\,\frac{dT}{dy}.$$

To prevent this curvature (full rotational restraint), an equal and opposite mechanical curvature must be imposed, requiring a bending moment. From beam theory $M = EI\,\kappa$, so the restraining (thermal) moment is

$$M_T = E\,I\,\kappa_T = E\,\alpha\,I\,\frac{dT}{dy}. \qquad\blacksquare$$

The corresponding fibre stress if fully restrained is $\sigma_T(y) = E\,\alpha\,(dT/dy)\,y$ — linear in $y$, peaking at the extreme fibre — which is likely the expression the registry intended before it was mistakenly multiplied by $I$ to form the moment. The correct route multiplies the curvature by $EI$, not the stress by $y\,I$.

Dimensional check (correct form). $$\left[E\,\alpha\,I\,\frac{dT}{dy}\right] = (\text{Pa})\left(\frac{1}{\text{K}}\right)(\text{m}^4)\left(\frac{\text{K}}{\text{m}}\right) = \text{Pa}\cdot\text{m}^3 = \text{N}\cdot\text{m}.\ \checkmark$$ By contrast the registry's $E\alpha\,\Delta T\,y\,I$ gives $\text{N}\cdot\text{m}^3$ — the dimensional flag above.

History and Development

• Thermoelastic bending. The theory of thermal bending — non-uniform heating producing curvature and, under restraint, moments — is classical thermoelasticity (Duhamel, Boley & Weiner). It generalises the axial constrained-bar result to sections with a temperature gradient.

• The bimetallic strip. The everyday embodiment is the bimetallic strip (thermostats, circuit breakers): two metals of different $\alpha$ bonded together curl with temperature, converting a thermal gradient into motion — the free-curvature limit of a thermal moment.

• Aerospace gradients. Spacecraft booms and trusses bow when one side faces the Sun ("thermal snap"), disturbing pointing; hypersonic and re-entry skins (Reentry Heat Flux) develop steep through-thickness gradients that curl panels and load their attachments. Managing thermal moments — through low-$\alpha$ materials, symmetric layups, and expansion joints — is a routine part of thermostructural design.

Related Concepts: Thermal Stress, Shaft Thermal Stress, Wing Bending Stress, Reentry Heat Flux, Combustion Chamber Temperature, Section Modulus

Notes: Registry calculator thermal-moment (unverified). Dimensional bug: the shipped $E\alpha\,\Delta T\,y\,I$ is $\text{N}\cdot\text{m}^3$, not a moment; the correct thermal moment for a linear gradient is $M_T = E\alpha I\,(dT/dy)$. A thermal moment needs a temperature gradient — a uniform $\Delta T$ causes no bending. Free beams bow (curvature $\alpha\,dT/dy$) with zero moment; restrained beams develop $M_T$. Flagged in Known Issues. All defaults $1.0$.