Two steel lubes are shrink-filled together where the nominal diameters are 40, 45, and 50 mm. Careful measurement before fitting determined the diametral interference between the tubes to be 0.062 mm. After the fit. the assembly is subjected to a torque of 900 N · m and a bending-moment of 675 N · m. Assuming no slipping between the cylinders, analyze the outer cylinder at the inner and outer radius. Determine the factor of safety using distortion energy with Sv = 415 MPa.
The factor of safety using distortion energy theory.
Answer to Problem 74P
The factor of safety using distortion energy theory for inner radius is
The factor of safety using distortion energy theory for outer radius is
Explanation of Solution
Write the expression for contact pressure.
Here, the contact pressure is
Write the expression for inner radius.
Here, the inner radius is
Write the expression for outer radius.
Here, the outer radius is
Write the expression for tangential stress at outer radius for outer member.
Here, the tangential stress at outer radius for outer member is
Write the expression for tangential stress for inner member.
Here, the tangential stress for inner member is
Write the expression for radial stress for inner member.
Here, the radial stress for inner member is
Write the expression for second moment of area.
Here, the second moment of area is
Write the expression for stress.
Here, the stress in
Write the expression for second polar moment of area.
Here, the second polar moment of area is
Write the expression for shear stress.
Here, the shear stress is
Write the expression for von Mises stress for outer radius.
Here, the von Mises stress is
Calculate factor of safety for outer radius.
Here, the factor of safety for outer radius is
Write the expression for von Mises stress for inner radius.
Here, the von Mises stress for inner radius is
Calculate the factor of safety for inner radius.
Here, the factor of safety for inner radius is
Write the expression for nominal radius.
Here, the nominal radius is
Conclusion:
Substitute
Substitute
Substitute
Substitute
Substitute
The radial stress for outer radius is zero. Thus,
Substitute
Substitute
Substitute
Substitute
Here, the stress in outer member is
Substitute
Substitute
Here, the stress for inner member is
Substitute
Substitute
Substitute
Here, the shear stress for outer member is
Substitute
Substitute
Here, the shear stress for inner member is
Substitute
Substitute
Substitute
Thus, the factor of safety for outer radius is
The following diagram shows the 3D stress for outer radius.
Figure (1)
Substitute
Substitute
The value of
Substitute
Thus, the factor of safety for inner radius is
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Chapter 5 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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