Mechanics of Materials: Torsional deformation at free end with torque at middle

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SUMMARY

The discussion focuses on determining the twist angle of a circular shaft AB subjected to a torque T at its midpoint (plane C), with end A fixed and end B free. The relevant equation for calculating the twist angle is φ = (T*L)/(G*I_p), where G is the shear modulus and I_p is the polar moment of inertia. Participants express confusion regarding the behavior of the twist angle from point C to B, particularly in the absence of reaction torques at the free end. The consensus is that the twist angle remains constant from C to B due to the lack of torque in that section.

PREREQUISITES
  • Understanding of torsional deformation in materials
  • Familiarity with the mechanics of materials, specifically shear modulus and polar moment of inertia
  • Knowledge of torque and its effects on structural elements
  • Ability to apply the twist angle formula φ = (T*L)/(G*I_p)
NEXT STEPS
  • Study the effects of fixed and free ends on torsional deformation in shafts
  • Learn about the shear modulus (G) and its role in material deformation
  • Investigate the concept of polar moment of inertia (I_p) and its calculation for different shaft geometries
  • Explore examples of torsional loading in engineering applications
USEFUL FOR

Mechanical engineers, students of mechanics of materials, and professionals involved in structural analysis and design will benefit from this discussion.

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Homework Statement


A circular shaft AB has a torque T acting at the middle of the shaft, defined as plane C. Shaft end A is fixed while shaft end B is free to rotate (mounted in a thrust bearing). Finding the twist angle from A to C is not difficult, but the question requires that the twist angle from A to B be determined.


Homework Equations


phi=(T*L)/(G*I_p)


The Attempt at a Solution


I have looked back through all my class notes, homework, and mechanics of materials textbooks and can't seem to find any examples where a shaft has a free end with no reaction torques and where the twist angle is determined there. I have been able to find several sources which say that the twist angle rate should remain constant for the length of the rod, but I am not sure if this applies for a rod where there is no reaction torque on the free end. From a materials perspective, I would also not expect the twist angle rate to instantly change after plane C where the torque is applied. Can anyone help me out with understanding what would happen from C to B on the shaft?
 
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Hint: Is there any torque in the section from C to B? What does that tell you about the twist at B relative to C? Or the twist at B relative to A?
 

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