A bimetallic strip of total thickness x is straight at...

A bimetallic strip of total thickness $x$ is straight at temperature $T$. What is the radius of curvature of the strip, $R$, when it is heated to temperature $T+\Delta T$ ? The coefficients of linear expansion of the two metals are $\alpha_1$ and $\alpha_2$, respectively, with $\alpha_2>\alpha_1$. You may assume that each metal has thickness $x / 2$, and you may assume that $x \ll R$.

Key Concepts

Thermal Expansion

Thermal expansion is the tendency of a material to change its dimensions when its temperature changes. The linear coefficient of thermal expansion quantifies how much a material expands per unit original length per degree of temperature change, and differences in these coefficients between materials are the driving force in many thermal deformation problems.

Bimetallic Composite Bending

A bimetallic strip is a composite structure made of two different metals bonded together. When subjected to a change in temperature, the two metals expand (or contract) by different amounts due to their distinct thermal expansion coefficients, resulting in internal stresses that force the strip to bend. This phenomenon is widely used in temperature-sensitive devices like thermostats.

Neutral Axis

In a bending beam, the neutral axis is the imaginary line within the cross?section where the material fibers experience no longitudinal strain during bending. In symmetric composite structures with equal thickness layers and similar mechanical properties, the neutral axis is typically located midway through the thickness, providing a reference point for analyzing the strain distribution caused by bending.

Geometric Relationship in Curved Beams

When a beam deforms into a circular arc, different layers of the beam travel along arcs of different radii. The outer layers have longer arc lengths than the inner layers. By relating these geometric differences to the differential expansion from heating, one can derive relationships that connect the curvature (1/R) of the composite beam to its thickness and the difference in thermal expansion coefficients.

Small Deflection Approximation

When the thickness of a beam is much smaller than the radius of curvature (x ? R), the geometry of bending can be simplified using linear approximations. This assumption allows the establishment of approximate linear relations between arc lengths, bending strains, and the radius of curvature, greatly simplifying the analysis of bending behavior in structures like bimetallic strips.

Transcript

00:01 We begin the first item of this question, noticing that the outer strip, which is formed by the red strip, measures r plus s times theta, while the inner strip composed by the blue strip measures r times theta.

00:21 Then we know that one r plus s times theta is equal to the expanded length.

00:31 Of the red strip, which is given by l plus l times alpha 2 times delta t.

00:38 And at the same time, you know that 2 r times theta is equal to l plus l times alpha 1 times delta t.

00:48 Now we have two equations.

00:51 We can eliminate theta by doing the following.

00:53 Pick equation 2 and solve it for theta.

00:56 So, theta is l plus l times alpha 1 times delta t.

01:01 Divided by r.

01:03 Then we can substitute theta in equation number one by this value of theta.

01:09 By doing that we get a new equation which tells us the following.

01:12 R plus s times theta which is l plus l times alpha 1 times delta t divided by r is equal to l plus l times alpha 2 times delta t.

01:31 Okay.

01:31 Now let me erase these two equations.

01:34 We will not need them anymore so that i can continue solving this problem.

01:39 Okay, now let me bring this equation here.

01:41 The first thing we should do is try to simplify this equation, and we can do that because we have a lot of factors of l.

01:48 So we can factor l here and we can factor l here.

01:52 By doing that, we get the following.

01:53 R plus s times l times 1 plus alpha 1 times delta t is equals to l times 1 plus alpha 2 times delta t and of course we have an r dividing the left hand side.

02:12 Now we can simplify l with l and this will make our life easier.

02:17 So now we have r plus s times 1 plus 1 plus alpha 1 times delta t equals to 1 plus alpha 2 times delta t and there is an r dividing the left -hand side.

02:35 Let me erase what i had just done so that i can have more space to continue solving this problem.

02:42 Okay now what can we do with this equation? we can solve it for r...

Please give Ace some feedback