Angle of Twist and Peak Torsional Shearing Stress

In summary: When you solve for the reactions, you add the torques together and divide by the moment of inertia. In this case, the moment of inertia is J = (∏c^4)/2, so the total torque is Tc + Td = 2.5kN.m.
  • #1
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Homework Statement



A stepped solid shaft of circular cross section is rigidly clamped at ends A and D and loaded by twisting moments T1= 1kN.m and T2 = 1.5kN.m at points B and C. The material is steel for which G = 84x10^9 N/m^2, the length L of the shaft is 500mm, and the diameters of AB and CD are both 30 mm while that of BC is 50 mm. Determine the angle of twist at C and the peak torsional shearing stress.

http://images.4chan.org/sci/src/1362606506659.png [Broken]

Homework Equations


Angle of twist θ= TL/JG
max torsional shearing stress τmax = Tc/J
centroidal moment of inertia J = (∏c^4)/2

The Attempt at a Solution


I have not attempted a solution because I do not know how to take into account that the shaft is rigidly clamped at both ends and what side to start from. Also, I do not know how the Torque at B (T1) comes into play with the angle of twist at c, I would imagine it increases the angle because it is in the same direction of Torque at C (T2)
 
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  • #2
You have the start of a FBD. Use the equations of statics to find the reactions required at A and D to keep the shaft in equilibrium.
 
  • #3
Ok i understand by using the sum of the moment across the x plane Ta + Td = 2.5 kN.m. What's confusing me is if i break it into separate sections AB, BC and CD, or just AB and CD to find the reactions at A and D.

also, the total angle of the shaft is zero since both ends are restrained... so the angle of AB + angle of BC + angle of CD = 0 going from left to right angle of CD is negative.. I don't get where to add in the torque at B while finding the angle from b to c.
 
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  • #4
The shaft is statically indeterminate. Have you studied how to find the unknown reactions in such a case?
 
  • #5
I must correct myself, I have seen how to solve for statically indeterminate shafts but it was a single cylinder with a single torque applied to the center. The two torques is confusing me.

when i try to make my second static equation, I am not sure how to take into account both torques applied.
 
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  • #7
thanks steamking, I think I've figured it out now.
 

1. What is the angle of twist?

The angle of twist is the amount of rotation that occurs at one end of a structural member relative to the other end when a torsional force is applied.

2. How is the angle of twist calculated?

The angle of twist is calculated using the formula θ = TL/GJ, where θ is the angle of twist, T is the applied torque, L is the length of the structural member, G is the shear modulus, and J is the polar moment of inertia.

3. What is peak torsional shearing stress?

Peak torsional shearing stress is the maximum amount of shear stress that occurs in a structural member when a torsional force is applied. It is also known as maximum shear stress.

4. How is peak torsional shearing stress calculated?

Peak torsional shearing stress is calculated using the formula τ = Tr/J, where τ is the peak torsional shearing stress, T is the applied torque, r is the radial distance from the center of the structural member, and J is the polar moment of inertia.

5. What factors affect the angle of twist and peak torsional shearing stress?

The angle of twist and peak torsional shearing stress are affected by the material properties, cross-sectional shape and dimensions of the structural member, and the magnitude and direction of the applied torque. They are also affected by any changes in the structural member's geometry or loading conditions.

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