**Question: **A bimetallic beam used in a temperature-control switch consists of strips of aluminum and copper bonded together as shown in the figure, which is a cross-sectional view. The width of the beam is 1.0 in., and each strip has a thickness of 1/16 in.

Under the action of a bending moment *M* = 12 lb-in. acting about the *z *axis, what are the maximum stresses *σ*_{a}* *and *σ*_{c}* *in the aluminum and copper, respectively? (Assume *E*_{a} = 10.5 × 10^{6} psi and *E*_{c} = 16.8× 10^{6} psi .)

**Answer:**

# Step 1 of 13

Consider the beam as follows:

# Step 2 of 13

Calculate the distance from the center of the aluminum strip to the neutral axis as follows:

Here, is the thickness of the strip and is the distance from the neutral axis to the top of the beam.

Substitute for .

# Step 3 of 13

Calculate the area of the aluminum strip as follows:

Here, is the width of the beam.

Substitute for and for .

# Step 4 of 13

Calculate the distance from the center of the copper strip to the neutral axis as follows:

Substitute for .

# Step 5 of 13

Calculate the area of the copper strip as follows:

Here, is the width of the beam.

Substitute for and for .

# Step 6 of 13

Write the equation for locating the neutral axis.

Here, is the modulus of elasticity of aluminum and is the modulus of elasticity of copper.

Substitute for , for , for , for , for and for .

# Step 7 of 13

Calculate the value of as follows:

Here, is the distance from neutral axis to the bottom of the beam.

Substitute for .

# Step 8 of 13

Calculate the distance between the centroid of the aluminum strip and the neutral axis as follows:

Here, is the distance between the centroid of the aluminum strip and the neutral axis.

Substitute for and for .

# Step 9 of 13

Calculate the moment of inertia of the aluminum strip about the neutral axis using the parallel axis theorem as follows:

Substitute for , for , for and for .

# Step 10 of 13

Calculate the distance between the centroid of the copper strip and the neutral axis as follows:

Here, is the distance between the centroid of the copper strip and the neutral axis.

Substitute for and for .

# Step 11 of 13

Calculate the moment of inertia of the copper strip about the neutral axis using the parallel axis theorem as follows:

Substitute for , for , for and for .

# Step 12 of 13

The aluminum strip is above the neutral axis. So, maximum compressive stress occurs at the top of the aluminum strip.

Calculate the maximum compressive stress in the aluminum strip using the relation,

Here, *M* is the bending moment.

Substitute for , for , for , for , for and for .

Therefore, the maximum stress in aluminum trip is (Compressive).

# Step 13 of 13

Most of the copper strip is below the neutral axis. So, maximum tensile stress occurs at the bottom of the copper strip.

Calculate the maximum tensile stress in the copper strip using the relation,

Substitute for , for , for , for , for and for .

Therefore, the maximum stress in copper strip is (Tensile).