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4.55 and 4.56 Five metal strips, each 0.5 × 1.5-in. cross section, are bonded together to form the composite beam shown. The modulus of elasticity is 30 × 106 psi for the steel, 15 × 106 psi for the brass, and 10 × 106 psi for the aluminum. Knowing that the beam is bent about a horizontal axis by a couple of moment 12 kip∙in., determine (a) the maximum stress in each of the three metals, (b) the radius of curvature of the composite beam.
Fig. P4.55
(a)
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Find the maximum stress in Aluminum, Brass, and Steel.
Answer to Problem 55P
The maximum stress in Aluminum is
The maximum stress in Brass is
The maximum stress in Steel is
Explanation of Solution
Given information:
The dimension of each strip is
The modulus of elasticity of aluminum is
The modulus of elasticity of brass is
The modulus of elasticity of steel is
The beam is bent about a horizontal axis by a couple of moment
Calculation:
Consider aluminum as the reference material.
Calculate the modular ratio
For steel.
Substitute
For brass.
Substitute
For aluminum.
Substitute
Sketch the transformed cross section as shown in Figure 1.
Refer to Figure 1.
The moment of inertia of the cross sections
Calculate the moment of inertia for each section as shown below.
For section (1).
Substitute
Hence,
For section (2).
Substitute
Hence,
For section (3).
Substitute
Calculate the moment of inertia
Substitute
Calculate the maximum stress
For steel.
Substitute
Hence, maximum stress in steel is
For brass.
Substitute
Hence, maximum stress in brass is
For aluminum.
Substitute
Therefore, maximum stress in aluminum is
(b)
![Check Mark](/static/check-mark.png)
The radius of curvature of the composite beam.
Answer to Problem 55P
The radius of curvature of the composite beam is
Explanation of Solution
Given information:
The dimension of each strip is
The modulus of elasticity of aluminum is
The modulus of elasticity of brass is
The modulus of elasticity of steel is
The beam is bent about a horizontal axis by a couple of moment
Calculation:
Refer to part (a).
The moment of inertia of the beam is
Calculate the radius of curvature
Substitute
Therefore, the radius of curvature of the composite beam is
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