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TORSION
Consider a bar to be rigidly attached at one end and twisted at the other end by a torque or twisting moment T equivalent to F × d, which is applied perpendicular to the axis of the bar, as shown in the figure. Such a bar is said to be in torsion.

TORSIONAL SHEARING STRESS, τ
For a solid or hollow circular shaft subject to a twisting moment T, the torsional shearing stress τ at a distance ρ from the center of the shaft is and where J is the polar moment of inertia of the section and r is the outer radius. For solid cylindrical shaft:

For hollow cylindrical shaft:

ANGLE OF TWIST
The angle θ through which the bar length L will twist is

where T is the torque in N·mm, L is the length of shaft in mm, G is shear modulus in MPa, J is the polar moment of inertia in mm4, D and d are diameter in mm, and r is the radius in mm. POWER TRANSMITTED BY THE SHAFT
A shaft rotating with a constant angular velocity ω (in radians per second) is being acted by a twisting moment T. The power transmitted by the shaft is

where T is the torque in N·m, f is the number of revolutions per second, and P is the power in watts.

Problem 307
A solid steel shaft 5 m long is stressed at 80 MPa when twisted through 4°. Using G = 83 GPa, compute the shaft diameter. What power can be transmitted by the shaft at 20 Hz? Solution 307

answer

answer

Problem 308
A 2-in-diameter steel shaft rotates at 240 rpm. If the shearing stress is limited to 12 ksi, determine the maximum horsepower that can be transmitted. Solution 308

answer

Problem 310
Show that the hollow circular shaft whose inner diameter is half the outer diameter has a torsional strength equal to 15/16 of that of a solid shaft of the same outside diameter. Solution 310
Hollow circular shaft:

Solid circular shaft:

(okay!)

Problem 311
An aluminum shaft with a constant diameter of 50 mm is loaded by torques applied to gears attached to it as shown in Fig. P-311. Using G = 28 GPa, determine the relative angle of twist of gear D relative to gear A.

Solution 311

Rotation of D relative to A:

answer

Problem 316
A compound shaft consisting of a steel segment and an aluminum segment is acted upon by two torques as shown in Fig. P-316. Determine the maximum permissible value of T subject to the following conditions: τst ≤ 83 MPa, τal ≤ 55 MPa, and the angle of rotation of the free end is limited to 6°. For steel, G = 83 GPa and for aluminum, G = 28 GPa.

Solution 316
Based on maximum shearing stress, τmax = 16T / πd3:
Steel

Aluminum

Based on maximum angle of twist, θmax = 6°:

Use the least value of T. Thus, T = 679.04 N·m answer

Problem 322
A solid steel shaft is loaded as shown in Fig. P-322. Using G = 83 GPa, determine the required diameter of the shaft if the shearing stress is limited to 60 MPa and the angle of rotation at the free end is not to exceed 4 deg.

Solution 322
Based on maximum allowable shear:

For the 1st segment:

For the 2nd segment:

Based on maximum angle of twist:

Use answer

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Verse 3: A B C#m7 You are the voice that called the universe to be A B C#m7 You are the whisper in my heart that speaks to me A B And 'til I see You face to face C#m7 And grace amazing takes me home A I'll trust in You Bridge: A E B C#m7 You alone are God of all A E B C#m7 You alone are worthy Lord A E B C#m7 And with all I am my soul will bless A Your Name (repeat)

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