Bimetallic strip · CTE differential bending · Thermal stress σ = EαΔT
σ = E · α · ΔT (bar fixed at both ends)
Interactive bimetallic strip simulation exploring how different thermal expansion coefficients cause bending when heated or cooled. Also calculates thermal stress in a constrained bar.
A bimetallic strip is made of two metals bonded together with different thermal expansion coefficients (α). When heated, the metal with higher α expands more, causing the strip to bend. Curvature depends on ΔT, strip thickness, and the ratio of expansion coefficients.
Adjust the temperature with the slider. Watch the strip bend toward the metal with lower expansion coefficient. Toggle the constrained-bar mode to see thermal stress build up when expansion is prevented.
Bimetallic strips are used in thermostats, circuit breakers and old car turn signals. The original thermostat design by Warren Johnson (1883) used exactly this principle — and similar designs are still used in billions of devices.
This simulation shows how a bimetallic strip — two bonded metal layers of equal thickness but different coefficients of thermal expansion (CTE) — bends when its temperature changes. The curvature κ is computed from the classic Timoshenko equal-thickness formula, κ = 6(α₂ − α₁)ΔT(1+m)² / (h[3(1+m)² + (1+mn)(m² + 1/mn)]), with m = 1 and n = E₁/E₂.
You pick a material pair (Brass/Steel, Aluminium/Steel or Invar/Brass), set the temperature change ΔT from a 20°C baseline using a slider (−60 to +200°C), and adjust strip length (20–250 mm). The panel reports tip deflection δ = L²κ/2, radius of curvature R = 1/κ, curvature κ, and the constrained-bar thermal stress σ = EαΔT for each layer. This principle drives thermostats, circuit breakers and temperature gauges.
What is a bimetallic strip?
It is two thin metal layers bonded face-to-face, each made of a metal with a different coefficient of thermal expansion. Because one metal expands more than the other for the same temperature change, the bond forces the strip to curl. The metal with the higher CTE ends up on the outside of the curve.
How does the simulation calculate the bending?
It uses the Timoshenko bimetallic-strip formula for two layers of equal thickness, which gives the curvature directly from the CTE difference, the temperature change, the moduli ratio and the total thickness. The strip is drawn as a circular arc of radius R = 1/κ, and the free tip deflection is approximated as δ = L²κ/2.
What do the controls change?
The material-pair buttons swap in different values of α and Young's modulus E for the two layers. The ΔT slider sets the temperature rise or fall from 20°C, and the length slider sets the strip length, which scales the tip deflection. The results panel and the on-canvas drawing update instantly.
Invar is a nickel-iron alloy with an unusually low CTE of roughly 1.2 × 10⁻⁶ per kelvin, far below ordinary steel. Pairing it with brass still produces some curvature because brass moves, but the small difference and Invar's stiffness keep the overall response modest, which is exactly why Invar is used where dimensional stability matters.
If a bar is rigidly fixed at both ends and cannot lengthen, heating it generates an internal compressive stress σ = EαΔT. The panel computes this separately for each layer's material using its own E and α. For steel at ΔT = 80°C this is on the order of 190 MPa, which is why thermal expansion joints are vital in bridges and rails.
On heating, the higher-CTE layer wants to grow longer than its partner. Since they are bonded, the longer layer is forced onto the outside of the bend while the shorter, lower-CTE layer sits on the inside. The result is that the strip curls towards the metal that expands the least.
The Timoshenko equal-thickness model is a standard, well-validated analytical result for thin bonded strips, accurate for small deflections and uniform temperature. The simulation simplifies by fixing total thickness at 2 mm and using a small-angle tip-deflection approximation, so very large curvatures are illustrative rather than exact.
A negative ΔT represents cooling below the 20°C baseline. The curvature reverses sign, so the strip bends the opposite way as the higher-CTE metal now contracts more than its partner. This is exactly how a bimetallic thermostat closes a contact when a room cools down.
No — curvature κ depends only on the materials, the temperature change and the strip thickness, not on its length. Length does matter for the visible tip deflection, which grows with the square of length through δ = L²κ/2. That is why sliding the length control changes the deflection reading while the radius of curvature stays the same.
They appear in mechanical thermostats, thermal circuit breakers, over-temperature cut-outs, dial thermometers and older car turn-signal flashers. Warren Johnson's 1883 thermostat used this principle, and billions of devices still rely on the strip's predictable, temperature-driven movement to open or close electrical contacts.