Concept explainers
A hollow cylindrical shaft of length L, mean radius cm, and uniform thickness t is subjected to a torque of magnitude T. Consider, on the one hand, the values of the average shearing stress τave and the angle of twist ϕ obtained from the elastic torsion formulas developed in Secs. 3.1C and 3.2 and, on the other hand, the corresponding values obtained from the formulas developed in Sec. 3.10 for thin-walled shafts, (a) Show that the relative error introduced by using the thin-walled-shaft formulas rather than the elastic torsion formulas is the same for τave and ϕ and that the relative error is positive and proportional to the ratio t/cm·(b) Compare the percent error corresponding to values of the ratio t/cm of 0.1, 0.2, and 0.4.
Fig. P3.150
(a)
Show that the relative error introduced by using the thin walled shaft formulas rather than the elastic torsion formulas is the same for
Answer to Problem 150P
The ratio
The ratio
Explanation of Solution
Given information:
The hollow cylindrical shaft of length is (L).
The uniform thickness of the hollow cylinder is (t).
The mean radius of the hollow cylinder is
The magnitude of torque is (T).
Calculation:
Writ the equation for outer radius
Writ the equation for inner radius
Calculate the polar moment of inertia (J) using the relation:
Substitute
Calculate the maximum shearing stress
Substitute
Calculate the angle twist
Here, G is rigidity modulus.
Substitute
Write the expression to calculate the area bounded by centerline
Calculate the shearing stress at tube ‘a’
Substitute
Calculate the angle twist
Substitute
Calculate the ratio
Substitute
Calculate the ratio
Substitute
Thus, the ratio
Thus, the ratio
(b)
Compare the percent error corresponding to value of the ratio
Answer to Problem 150P
The ratio
Explanation of Solution
Calculation:
Calculate the percent error corresponding to value of the ratio
Substitute
Calculate the ratio for value 0.1 using the Equation (1).
Substitute 0.1 for
Calculate the ratio for value 0.2 using the Equation (1).
Substitute 0.2 for
Calculate the ratio for value 0.3 using the Equation (1).
Substitute 0.3 for
Thus, the ratio
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Chapter 3 Solutions
Mechanics of Materials, 7th Edition
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