Concept explainers
The cantilevered handle in Prob. 3–84 is made from mild steel. Let Fy = 250 lbf and Fx = Fz = 0. Using Castigliano’s theorem, determine the vertical deflection (along the y axis) at the tip. Repeat the problem with shaft OC simplified to a uniform diameter of 1 in for its entire length. What is the percent error from this simplification?
The vertical deflection at the tip along the y-axis.
The percentage error of the deflection in case of simplified shaft.
Answer to Problem 73P
The vertical deflection at the tip along the y-axis is
The percentage error of the deflection in case of simplified shaft is
Explanation of Solution
The given force is the y-direction is
Write the moment of inertia of the rod
Here, the diameter of the rod
The moment of inertia of the rod
Write the polar moment of inertia of the rod
The polar moment of inertia of the rod
Write the moment of inertia of the rod
Here, the diameter of the rod
Write the polar moment of inertia of the rod
Write the moment of inertia of the rod
Here, the diameter of the rod
Write the deflection at the tip by Castigliano’s theorem.
Here, the strain energy is
The dummy variable
Write the moment function for
Differentiate the Equation (VII) with respect to
Write the torque on rod
Differentiate the Equation (VIII) with respect to
Write the moment function of the link
Differentiate the Equation (IX) with respect to
Substitute
For the case of simplified shaft
Substitute
Calculate the percentage error of the simplified shaft deflection.
Conclusion:
Refer to the Table A-5 “Physical Constants of Materials” and obtain the value of
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Substitute
Thus, the vertical deflection at the tip along the y-axis is
Substitute
Thus, the percentage error of the deflection in case of simplified shaft is
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Chapter 4 Solutions
Shigley's Mechanical Engineering Design (McGraw-Hill Series in Mechanical Engineering)
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