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Concept explainers
The 20-mm-diameter steel rod CD is welded to the 20-mm- diameter steel shaft AB as shown. End C of rod CD is touching the rigid surface shown when a couple TB is applied to a disk attached to shaft AB. Knowing that the bearings are self aligning and exert no couples on the shaft, determine the angle of rotation of the disk when TB = 400 N ∙ m. Use E = 200 GPa and G = 77.2 GPa. (Consider the strain energy due to both bending and twisting in shaft AB and to bending in arm CD.)
Fig. P11.69
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The angle of rotation of the disk when
Answer to Problem 69P
The angle of rotation of the disk at B is
Explanation of Solution
Given information:
The diameter of the shaft AB and the steel rod CD is
The modulus of rigidity
The torque applied at B is
The modulus of elasticity
The length of steel rod CD is
The length of shaft AB is
Calculation:
Calculate the moment of inertia
Substitute
Consider the bending of rod CD.
Sketch the Free Body Diagram as shown in Figure 1.
Refer to Figure 1.
Take moment about rod D is Equal to zero.
Substitute
Summation of forces along y direction is Equal to zero.
Calculate the bending moment at a distance x from C as shown below.
Calculate the strain energy as shown below.
For the steel rod CD.
Substitute
Consider the bending of shaft ADB.
Sketch the Free Body Diagram of the shaft as shown in Figure 2.
Refer to Figure 2.
Take moment about A is Equal to zero.
Take moment about B is Equal to zero.
Bending moment at a distance x from A
Bending moment at a distance x from B
Calculate the strain energy for shaft AB using Equation (1) as shown below.
Substitute
Substitute
Consider the portion DB of shaft ADB carries the torque.
Calculate the polar moment of inertia
Substitute
Calculate the strain energy
Substitute
Calculate the total strain energy
Substitute
Calculate the angle
Provide the work energy equation at disk B as shown below.
Substitute
Therefore, the angle of rotation of the disk at B is
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Chapter 11 Solutions
Mechanics of Materials, 7th Edition
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