Bimetallic Strip Radius of Curvature Calculator

Calculate the radius of curvature $R$ due to thermal expansion using: $$ R = \frac{t}{\Delta\alpha\, \Delta T} $$

* Ensure all units are consistent: thickness $t$ in meters, Δα in 1/K, and ΔT in Kelvin.

Step 1: Enter Parameters

Total Thickness, $t$ (m):

e.g., 0.005 m (5 mm)

Difference in Expansion Coefficients, $\Delta\alpha$ (1/K):

e.g., 5×10⁻⁶ 1/K

Temperature Change, $\Delta T$ (K):

e.g., 50 K

Bimetallic Strip Radius of Curvature Calculator (In-Depth Explanation)

When a bimetallic strip is exposed to a change in temperature, the different thermal expansion coefficients of the two metals cause the strip to bend. The resulting curvature is characterized by its radius of curvature, which is a key parameter in designing thermostatic devices and sensors. This guide explains how to calculate the radius of curvature using a specific formula based on Timoshenko’s analysis.

Overview of Bimetallic Strips
Fundamental Principles and Key Parameters
The Radius of Curvature Formula
Step-by-Step Calculation Process
Practical Examples
Common Applications
Conclusion

1. Overview of Bimetallic Strips

A bimetallic strip is composed of two metals bonded together that have different coefficients of thermal expansion. When the temperature changes, the metal with the higher expansion coefficient expands or contracts more than the other, causing the strip to bend. The degree of bending is quantified by its radius of curvature (R).

2. Fundamental Principles and Key Parameters

The bending of a bimetallic strip is governed by the differences in thermal expansion and the mechanical properties of the metals. Key parameters include:

$ \alpha_1 $ and $ \alpha_2 $: Thermal expansion coefficients of metal 1 and metal 2, respectively.
$\Delta T$: The change in temperature (in °C or K).
$ t $: The total thickness of the bimetallic strip.
$ m $: The thickness ratio, defined as $ m = \frac{t_2}{t_1} $ (with $t_1$ and $t_2$ being the thicknesses of the two metals).
$\eta$: The ratio of the Young's moduli, $ \eta = \frac{E_2}{E_1} $, which reflects the relative stiffness of the metals.

3. The Radius of Curvature Formula

Timoshenko’s analysis provides a formula for the radius of curvature $ R $ of a bimetallic strip due to a temperature change:

$ R = \frac{t}{6(\alpha_2 - \alpha_1)\Delta T} \cdot \frac{(1 + m)^2}{m(1 + 4\eta + m\eta^2)} $

Where:

$t$ is the total thickness of the strip.
$\alpha_1$ and $\alpha_2$ are the thermal expansion coefficients of the two metals (with $\alpha_2 > \alpha_1$).
$\Delta T$ is the temperature change.
$m = \frac{t_2}{t_1}$ is the ratio of the thicknesses of the two metals.
$\eta = \frac{E_2}{E_1}$ is the ratio of their Young's moduli.

This formula combines the effects of differential expansion and the mechanical resistance of the metals to bending, yielding the radius of curvature.

4. Step-by-Step Calculation Process

Gather the Parameters:
- Measure or obtain the total thickness $t$ of the bimetallic strip.
- Determine the thermal expansion coefficients $\alpha_1$ and $\alpha_2$ for each metal.
- Note the temperature change $\Delta T$.
- Determine the thickness ratio $ m = \frac{t_2}{t_1} $ and the modulus ratio $ \eta = \frac{E_2}{E_1} $ if available.
Plug the Values into the Formula:
Use the formula:

$ R = \frac{t}{6(\alpha_2 - \alpha_1)\Delta T} \cdot \frac{(1 + m)^2}{m(1 + 4\eta + m\eta^2)} $
Calculate $R$:
Perform the arithmetic to obtain the radius of curvature in meters (or the desired unit).

5. Practical Examples

Example 1: Symmetrical Bimetallic Strip

Assumptions: For a simple case, assume that the two metals have equal thicknesses (i.e., $ m = 1 $) and similar stiffness such that $ \eta \approx 1 $. Then the formula simplifies to:

$ R \approx \frac{t}{6(\alpha_2 - \alpha_1)\Delta T} $

Given: A bimetallic strip of total thickness $t = 0.002\,m$ (2 mm), with $\alpha_1 = 12 \times 10^{-6}/^\circ C$ and $\alpha_2 = 24 \times 10^{-6}/^\circ C$. If the temperature increases by $\Delta T = 50^\circ C$, then:

$ R \approx \frac{0.002}{6(24 \times 10^{-6} - 12 \times 10^{-6}) \times 50} $

Simplify the denominator:

$ 6(12 \times 10^{-6}) \times 50 = 6 \times 12 \times 50 \times 10^{-6} = 3600 \times 10^{-6} = 0.0036 $

Now, calculate $R$:

$ R \approx \frac{0.002}{0.0036} \approx 0.556\,m $

Thus, the radius of curvature is approximately 0.556 meters.

Example 2: Asymmetrical Bimetallic Strip

Given: A bimetallic strip with total thickness $t = 0.003\,m$ (3 mm), where the two metals are not of equal thickness. Suppose the thickness ratio is $ m = \frac{t_2}{t_1} = 1.5 $ and the Young's moduli ratio is $\eta = 1.2$. The thermal expansion coefficients are $\alpha_1 = 10 \times 10^{-6}/^\circ C$ and $\alpha_2 = 20 \times 10^{-6}/^\circ C$, and the temperature change is $\Delta T = 40^\circ C$.

Calculation: First, compute the difference in thermal expansion:

$\alpha_2 - \alpha_1 = (20 - 10) \times 10^{-6} = 10 \times 10^{-6}/^\circ C$

Next, compute the effective modifier from geometry and material properties:

$ \frac{(1 + m)^2}{m(1 + 4\eta + m\eta^2)} $

Substitute $ m = 1.5 $ and $\eta = 1.2$:

$ (1 + 1.5)^2 = (2.5)^2 = 6.25 $

Denom.: $ m(1 + 4\eta + m\eta^2) = 1.5 \times (1 + 4(1.2) + 1.5 \times (1.2)^2) $

Calculate inside the parentheses:

$ 1 + 4(1.2) = 1 + 4.8 = 5.8 $

$ 1.2^2 = 1.44,\; 1.5 \times 1.44 = 2.16 $

So, $ 1 + 4\eta + m\eta^2 = 5.8 + 2.16 = 7.96 $

Then, $ m \times 7.96 = 1.5 \times 7.96 \approx 11.94 $

Now, compute the modifier:

$ \frac{6.25}{11.94} \approx 0.5238 $

Next, plug all values into the full formula:

$ R = \frac{t}{6(\alpha_2-\alpha_1)\Delta T} \times 0.5238 $

Substitute $ t = 0.003\,m $, $\alpha_2-\alpha_1 = 10 \times 10^{-6}/^\circ C$, and $\Delta T = 40^\circ C$:

$ R = \frac{0.003}{6 \times 10 \times 10^{-6} \times 40} \times 0.5238 $

Compute the denominator first:

$ 6 \times 10 \times 40 \times 10^{-6} = 2400 \times 10^{-6} = 0.0024 $

$ \frac{0.003}{0.0024} = 1.25 $

Now, $ R \approx 1.25 \times 0.5238 \approx 0.655\,m $

Therefore, the estimated radius of curvature is approximately 0.66 meters.

6. Common Applications

Thermostats: Bimetallic strips are used in temperature control devices, where precise bending triggers switching mechanisms.
Temperature Sensors: Converting thermal expansion into an electrical signal for monitoring.
Mechanical Relays: Using the curvature of bimetallic strips to open or close electrical circuits.
Structural Engineering: Designing components that adjust shape with temperature changes.

7. Conclusion

The Bimetallic Strip Radius of Curvature Calculator provides a systematic approach for estimating how a bimetallic strip will bend due to thermal expansion. Using Timoshenko’s formula, the calculator takes into account the total thickness, the difference in thermal expansion coefficients, the temperature change, and geometric and material property ratios. Whether for designing thermostats, sensors, or other temperature-responsive devices, understanding the radius of curvature is crucial for ensuring reliable and predictable performance.