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Question: A bimetallic beam used in a temperature-control switch consists of strips of aluminum and copper bonded together as shown in the figure

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 axis, what are the maximum stresses σa and σc in the aluminum and copper, respectively? (Assume Ea = 10.5 × 106 psi and Ec = 16.8× 106 psi .)

Answer:

 

Step 1 of 13

Consider the beam as follows:

real pic

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).

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