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A thin-walled hollow tube AB of conical shape has constant thickness I and average diameters d A and d B at the ends (see figure). Determine the strain energy U oT the tube when it is subjected to pure torsion by torques T, Determine the angle of twist of the tube. Note: Use the approximate formula / ird^tlA for a thin circular ring; see Case 22 of Appendix E.

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Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347

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Section
BuyFindarrow_forward

Mechanics of Materials (MindTap Co...

9th Edition
Barry J. Goodno + 1 other
Publisher: Cengage Learning
ISBN: 9781337093347
Chapter 3, Problem 3.9.10P
Textbook Problem
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A thin-walled hollow tube AB of conical shape has constant thickness I and average diameters dAand dBat the ends (see figure).

  1. Determine the strain energy U oT the tube when it is subjected to pure torsion by torques T,
  2. Determine the angle of twist of the tube.

Note: Use the approximate formula / ird^tlA for a thin circular ring; see Case 22 of Appendix E.

  Chapter 3, Problem 3.9.10P, A thin-walled hollow tube AB of conical shape has constant thickness I and average diameters dAand

(a)

To determine

The strain energy of the tube.

Explanation of Solution

Write the expression for the variable diameter of the cone.

  dx=dA+(dBdAL)x

Here, the length of the tube is L , diameter of the tube at A is dA , and the diameter of the tube at B is dB .

Write the expression for the polar moment of inertia for hollow conical tube.

  Ipx=πt4dx3 ....... (I)

Here, thickness of the tube is t .

Substitute dA+(dBdAL)x for dx in Equation (I).

  Ipx=πt4[dA+( d B d AL)x]3

Write the expression for the amount of strain energy stored in the hollow conical tube.

  U=0LT2dx2GIpx....... (II)

Here, modulus of rigidity is G .

Substitute πt4[dA+( d B d AL)x]3 for Ipx in Equation (II)

(b)

To determine

The angle of twist of the tube.

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