How does a bimetal bend?

Two metals with different expansion coefficients are bonded; when heated, one tries to expand more and the other holds it back, so the strip curls with the high-expansion side outward. Timoshenko's formula gives the curvature, proportional to the expansion difference and temperature change and inversely proportional to total thickness.

How should the two thicknesses be chosen?

Curvature is maximised when the thickness ratio equals the square root of the modulus ratio. For a high-expansion alloy (170 GPa) on Invar (140 GPa), the Invar layer should be 1.10 times thicker. But curvature differs from equal thickness by under 1%, so in practice ease of manufacture can decide.

Where does delamination tend to start?

At the free end. A 2D FE model shows interface shear stress near zero along the middle and concentrated within 0.5 mm of the free end (about the total thickness). The end corner is theoretically singular, so values rise as elements shrink. Delamination usually starts at the ends, so end geometry and bond quality govern life.

Bimetal Thermal Deformation Analysis

Category: Coupled Analysis – Thermal-Structural | Revised 2026-10-01

CAE visualization for bimetallic strip theory - technical simulation diagram

Bimetallic Thermal Deformation Analysis

Theory: turning expansion difference into curvature

Overview

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Bimetals are used in thermostats. What do I need to calculate to design one?

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Four things: how much it curls with temperature (curvature), how far the tip moves (deflection), how hard it pushes when blocked (force), and how much stress the bond interface sees (delamination risk). Beam formulas handle the first three, but interface stress concentrates at the ends and needs FEM. Let's walk through a thermostat high-expansion alloy on Invar.

Timoshenko's formula

$$ \kappa = \frac{6\,\Delta\alpha\,\Delta T\,(1+m)^2}{h\left[3(1+m)^2 + (1+mn)\left(m^2 + \dfrac{1}{mn}\right)\right]},\quad m = \frac{t_1}{t_2},\ n = \frac{E_1}{E_2} $$

$h$ is total thickness, $t_1, t_2$ the layer thicknesses, $\Delta\alpha$ the expansion difference. With equal moduli and thicknesses, $\kappa = 1.5\,\Delta\alpha\,\Delta T / h$.

Tip deflection and force

$$ \delta = \frac{\kappa L^2}{2},\qquad F_{block} = \frac{3\,(EI)\,\delta}{L^3} $$

A cantilevered bimetal bends at uniform curvature, so tip deflection scales with length squared. The blocked force is the force needed to push the tip back by that deflection; $EI$ is the composite bending stiffness.

Coffee Break Trivia

From Harrison's clocks to thermostats

The bimetal was first put to practical use by the 18th-century English clockmaker John Harrison. In his marine timekeeper H3 he used a bonded brass-and-steel strip to compensate for the balance spring's temperature sensitivity. The curvature formula came much later: in 1925 Timoshenko published it including both layers' thicknesses and moduli. It is still widely used, from bimetals to thin-film warpage in semiconductors and substrate deformation.

Worked examples

Example 1: thickness ratio and curvature

High-expansion alloy (α = 20×10⁻⁶ /K, E = 170 GPa) on Invar (α = 1.5×10⁻⁶ /K, E = 140 GPa), total 1 mm, ΔT = 50 K. m is Invar thickness over high-expansion thickness.

Ratio m	Curvature [1/m]
0.5	1.19
0.8	1.35
1.0	1.38
1.10 (optimum)	1.39
1.5	1.36
2.0	1.27

The optimum matched the square root of the modulus ratio, √(170/140) = 1.10. The peak curvature, 1.39 1/m, equals the equal-modulus value 1.5ΔαΔT/h. Anywhere from m = 0.8 to 1.5 stays within 3% of the peak, so curvature is insensitive to the ratio.

Example 2: cantilever deflection and force

Item	Value
Radius of curvature (equal thickness, 50 K)	722 mm
Tip deflection (length 50 mm)	1.73 mm
Deflection per kelvin	0.035 mm/K
Blocked tip force (width 5 mm)	2.7 N

Example 3: checking with 2D FEM

The same cantilever solved with plane-stress 4-node elements (400 along the length × 20 through the thickness):

Item	Result
Tip deflection	1.721 mm (formula 1.730 mm, 0.5% apart)
Axial stress at interface, mid-length	Invar side +62 MPa / high-expansion side −71 MPa
Interface shear (mid-length to 2 mm from end)	About zero
Interface shear (0.44 mm from end)	4.3 MPa
Interface shear (0.06 mm from end)	24 MPa

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The curvature matches the beam formula exactly, yet interface shear doesn't appear in it at all.

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Timoshenko's formula describes the average deformation, treating the pair as bending together. Force is transferred between the layers only near the ends. Mid-length, the expansion mismatch is locked in equilibrium, so no shear is needed. At the free end the axial force must return to zero, so it's handed over within roughly the total thickness, concentrating shear. The corner is theoretically singular, so finer elements give larger values. What matters is not the number but that it starts at the end — and real delamination almost always grows from there.

Design and analysis tips

For more curvature, increase the expansion difference and reduce total thickness.
Small deviations from the optimum thickness ratio barely matter.
At high temperature, α and E change; Invar's expansion rises sharply near its Curie point.
When using tip force, account for contact reaction reducing deflection.
For interface stress, refine near the ends and judge by distribution rather than peak value.

Coffee Break Trivia

“The snapping bimetal”

Safety cut-outs in electric kettles and blankets often use a shallow dished bimetal disc. A flat strip bends gradually in proportion to temperature, but a dished disc snaps through suddenly at a set temperature. The contacts open instantly, reducing arcing and avoiding contact burn from slow opening. The design exploits snap-through buckling, so analysis must include geometric nonlinearity to reproduce it.

Common mistakes

Mistakes and fixes

Mistake	Effect	Fix
Beam formula for wide plates	Curvature error	Use E/(1−ν)
Room-temperature properties	High-temperature error	Use temperature-dependent data
Judging by interface peak	Mesh-dependent	Assess distribution or energy
Linear analysis of dished discs	Snap-through missed	Geometric nonlinearity
One element through thickness	Too stiff in bending	Several elements per layer

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I'd like to learn related topics.

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Try the bimetal thermostat simulator and thermal expansion tool. Related pages include thermal expansion mismatch, thermal stress analysis, solder reliability and CTE gradient design.

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