Equivalent torque of hollow shaft

AI Thread Summary
To determine the dimensions of a hollow shaft that can replace a solid circular shaft with a diameter of 200mm while transmitting the same torque and maximum shear stress, the internal and external diameters of the hollow shaft must maintain a ratio of 4:5. The calculation involves equating the polar moment of inertia (J) for both shafts, leading to the conclusion that the hollow shaft's internal diameter should be approximately 19.07 cm and the external diameter about 23.84 cm. This configuration results in a significant material reduction of approximately 48.9%. The calculations confirm that the hollow shaft can effectively replace the solid shaft while optimizing material usage. Understanding these principles is crucial for efficient mechanical design.
Deathfish
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Homework Statement



A solid circular shaft has a diameter of 200mm. What would be the dimension of a hollow
shaft which could replace it (i.e. transmit the same torque for the same maximum shear
stress), and what would be the percentage reduction in the material required? The internal
and external diameters of the hollow shaft are to be in the ratio of 4:5. (Ans: 23.84 cm,
19.07 cm; 48.9%)

Homework Equations



T/J = G(theta)/L = tao/R
J = pi(d)^4/32


The Attempt at a Solution



J = pi(0.2)^4/32 = 0.000157

pi(5d)^4/32 - pi(4d)^4/32 = 0.000157

d = 0.2m

5d = 1m (wrong)
 
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Deathfish said:
The internal and external diameters of the hollow shaft are to be in the ratio of 4:5. (Ans: 23.84 cm,
19.07 cm; 48.9%)

You are right in that you want Jsolid = Jhollow

But this statement means that d/D = 4/5 or d = 0.8D (d = internal, D= external)

And Jhollow = π/32 (D4-d4)
 

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