Bimetal Thermostat Simulator — Thermal Bending of a Two-Layer Strip
Visualize the bending of a two-layer strip with the Timoshenko bimetal formula. Adjust the temperature change, length, thickness and expansion mismatch to learn how a temperature switch works.
Parameters
Temperature change ΔT
°C
Strip length L
mm
Total thickness h
mm
Expansion mismatch Δα
×10⁻⁶/°C
The layers are assumed to have equal thickness and equal elastic modulus (symmetric bimetal). The bending in the figure is exaggerated for clarity.
Results
—
Tip deflection δ
—
Curvature κ
—
Radius of curvature ρ
—
Tip slope angle θ
Deformation of the Bimetal Strip
Left end clamped. Orange = high-expansion side, blue = low-expansion side / dashed = straight position before heating, yellow dot = tip
Tip Deflection vs Temperature Change δ(ΔT)
X axis = temperature change ΔT / Y axis = tip deflection δ (yellow dot = current state, straight line = perfect proportionality)
Theory & Key Formulas
If the temperature is uniform, a bimetal deforms into an arc of constant curvature. For a symmetric bimetal whose layers have equal thickness and equal elastic modulus, the curvature reduces to a simple expression.
Curvature κ. Δα is the expansion mismatch, ΔT is the temperature change, h is the total thickness:
The curvature is proportional to the temperature change and the expansion mismatch, and inversely proportional to the thickness. The tip deflection scales with the square of the length, so a longer strip moves more.
What is the Bimetal Thermostat Simulator
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An electric kettle or an old iron seems to have no battery and no circuit board. How does it measure temperature and switch off?
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The star of that show is the bimetal. Roughly speaking, it is a strip made of two metals with different thermal expansion rates bonded together. When the temperature rises, the side that expands more wants to elongate, but being bonded to its partner it cannot do so freely. As a result, the whole strip curls toward the side that expands less. Move the "temperature change ΔT" slider in the simulator above and you will see the strip bend and the tip move.
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I see. So how does that bending change with temperature?
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It is beautifully proportional. The curvature is $\kappa = \tfrac{3}{2}\,\Delta\alpha\,\Delta T / h$, directly proportional to the temperature change ΔT. That is why the graph on the lower right is a perfectly straight line. This "proportional to temperature" property is exactly why a bimetal can serve as a thermometer or a temperature switch — with no power supply and no electronics.
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Looking at the tip-deflection card, the deflection changes a lot when I change the length.
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Good catch. The tip deflection is $\delta = \kappa L^2/2$, so it scales with the square of the length. Doubling the strip length increases the deflection roughly fourfold. But you cannot fit a strip tens of centimetres long inside a thermometer. So real products coil the bimetal into a spiral to gain effective length while staying compact. That coil is exactly what is inside a dial-type thermometer.
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When I move the thickness slider, a thinner strip bends more. So is thinner always better?
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That is the trade-off. The curvature is inversely proportional to thickness, so a thinner strip bends more — but a thin strip generates less force as it bends. A thermostat has to bend and push a contact, so it needs not just deflection but also force. If you only need to show motion, like a thermometer, go thin; if you need to switch a contact firmly, like a circuit breaker, go thicker. Design is always a balance between deflection and force.
Frequently Asked Questions
Practical bimetals use Invar (an iron-nickel alloy, about 1.2×10⁻⁶/°C) for the side that barely expands, and brass or a manganese-copper alloy (about 18 to 27×10⁻⁶/°C) for the expanding side. Depending on the combination, the effective Δα lies roughly in the 10 to 26×10⁻⁶/°C range, and standards specify it through a "specific curvature" figure. The simulator's default of 14×10⁻⁶/°C corresponds to a typical general-purpose bimetal.
In general you use Timoshenko's formula (1925). It is a somewhat complex expression containing the thickness ratio m and the modulus ratio n that determines the curvature. Substituting a symmetric bimetal with equal layer thickness and modulus (m=1, n=1) reduces it to the κ = (3/2)Δα ΔT / h used in this simulator. Practical bimetals are usually made with nearly equal layer thickness to maximize curvature, so this reduced formula is a good approximation.
If a bimetal is used as a plain strip, the contact moves gradually with temperature and the contact becomes unstable. So real thermostats form the bimetal into a shallow dome (like a Belleville spring). When a certain temperature is reached, the stored elastic energy is released all at once and the dome snaps through (buckles), switching abruptly. This prevents contact chatter and gives well-defined on and off temperatures.
This simulator assumes the whole strip reaches a uniform temperature. In reality heat takes time to spread, so the response lags behind rapid temperature changes. And if there is a temperature gradient along the strip, the curvature varies from place to place and it no longer forms a simple arc. Where response speed matters, designers use a thin bimetal with low heat capacity and ensure good thermal contact with the object being measured.
Real-World Applications
Temperature switches in appliances: Electric kettles, irons, toasters, hair dryers, electric blankets — almost every appliance involving heating contains a bimetal thermostat. When the set temperature is reached, the bimetal bends and opens a contact, cutting power to the heater; when it cools, it returns and reconnects. This simple, power-free mechanism delivers cheap and robust temperature control.
Thermal circuit breakers for circuit protection: When excessive current flows, the bimetal heats up (by self-heating or an adjacent heater element) and bends, interrupting the circuit. It is used for motor overload protection, in miniature circuit breakers, and in the protection circuits of lithium battery packs. It stays still while current is within rating and trips reliably only on dangerous overcurrent — a fail-safe protective element.
Dial thermometers and thermostats: In oven thermometers, refrigerator thermometers and HVAC thermostats, the bimetal is coiled into a spiral to gain effective length and convert temperature change into rotation of a pointer. Even now that electronic types dominate, bimetals remain in service for cooking and some HVAC uses because they work during power outages and hold their calibration well.
Automotive and industrial equipment: Bimetals have long been used in the classic engine automatic choke, the flasher mechanism of turn signals, and warm-up control. In industry they continue to work quietly inside many machines as overheat protection, temperature alarms and simple temperature-compensation mechanisms.
Common Misconceptions and Cautions
The most common misconception is to think that "a thicker bimetal is stronger, so it bends more". The curvature is inversely proportional to thickness — a thicker strip bends less. What a thick bimetal has is a larger "force (moment) that tries to bend it", not a larger "amount of bending". Move the thickness slider in the simulator and you can confirm that the tip deflection shrinks as the strip gets thicker. If you want deflection go thin; if you want force go thick — that distinction is the starting point of design.
The next most common error is to assume the tip deflection is "proportional" to the length. In reality it scales with the square of the length. For the same temperature change, making the strip 1.5 times longer increases the deflection about 2.25 times, and doubling it increases it about fourfold. Conversely, if you estimate "just a slightly longer strip will do" when you need a given deflection in a small space, you will be far off. Moving the length slider in equal steps while watching the tip-deflection card lets you feel how it shoots up faster toward the higher end.
Finally, take care not to use this calculation without realizing it is the value for the ideal state of a uniform, slowly changing temperature. A real bimetal has heat capacity, so even if the ambient temperature changes suddenly it takes time for the strip's temperature to catch up, and the response lags. And if there is a temperature difference along the strip, the curvature varies from place to place and departs from a simple arc. The simulator's value gives the "final deformation once the set temperature is reached"; the speed of response and the transient behavior must be examined separately from the standpoint of heat transfer.