January 2009

Torsion

Torsion

  1. Torsion
  2. Flanged Bolt Couplings
  3. Torsion of Thin-Walled Tubes
  4. Helical Springs

Torsion

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.
 

Bar in torsion

 

304 Maximum Shearing Stress and Angle of Twist of a Steel Shaft

Problem 304
A steel shaft 3 ft long that has a diameter of 4 in is subjected to a torque of 15 kip·ft. Determine the maximum shearing stress and the angle of twist. Use G = 12 × 106 psi.
 

305 Minimum Diameter of Steel Shaft With Allowable Angle of Twist

Problem 305
What is the minimum diameter of a solid steel shaft that will not twist through more than 3° in a 6-m length when subjected to a torque of 12 kN·m? What maximum shearing stress is developed? Use G = 83 GPa.
 

306 Maximum Shearing Stress of Marine Propeller Shaft

Problem 306
A steel marine propeller shaft 14 in. in diameter and 18 ft long is used to transmit 5000 hp at 189 rpm. If G = 12 × 106 psi, determine the maximum shearing stress.
 

307 Power Transmitted by Solid Steel Shaft of Unknown Diameter

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?
 

308 Maximum Horsepower That Can be Transmitted by Steel Shaft

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.
 

309 Finding the Diameter of a Propeller Shaft Transmitting a Power

Problem 309
A steel propeller shaft is to transmit 4.5 MW at 3 Hz without exceeding a shearing stress of 50 MPa or twisting through more than 1° in a length of 26 diameters. Compute the proper diameter if G = 83 GPa.
 

310 Strength Comparison of Hollow and Solid Steel Shafts of Equal Outside Diameters

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.
 

311 Aluminum Shaft Loaded by Multiple Torques Applied at Gears

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.
 

Three segments aluminum shaft

 

312 Deformation of Flexible Shaft Made From Steel Wire Encased in Stationary Tube

Problem 312
A flexible shaft consists of a 0.20-in-diameter steel wire encased in a stationary tube that fits closely enough to impose a frictional torque of 0.50 lb·in/in. Determine the maximum length of the shaft if the shearing stress is not to exceed 20 ksi. What will be the angular deformation of one end relative to the other end? G = 12 × 106 psi.
 

313 Maximum Torque That Can be Applied to a Hollow Steel Shaft

Problem 313
Determine the maximum torque that can be applied to a hollow circular steel shaft of 100-mm outside diameter and an 80-mm inside diameter without exceeding a shearing stress of 60 MPa or a twist of 0.5 deg/m. Use G = 83 GPa.
 

314 Maximum Shear and Relative Gear Rotation of a Compound Steel Shaft

Problem 314
The steel shaft shown in Fig. P-314 rotates at 4 Hz with 35 kW taken off at A, 20 kW removed at B, and 55 kW applied at C. Using G = 83 GPa, find the maximum shearing stress and the angle of rotation of gear A relative to gear C.
 

315 Power Applied and Removed at Shaft Gears

Problem 315
A 5-m steel shaft rotating at 2 Hz has 70 kW applied at a gear that is 2 m from the left end where 20 kW are removed. At the right end, 30 kW are removed and another 20 kW leaves the shaft at 1.5 m from the right end. (a) Find the uniform shaft diameter so that the shearing stress will not exceed 60 MPa. (b) If a uniform shaft diameter of 100 mm is specified, determine the angle by which one end of the shaft lags behind the other end. Use G = 83 GPa.
 

316 Permissible Torque That Can Be Applied to a Compound Shaft

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.
 

Figure P-316

Solution to Problem 317 Torsion

Problem 317
A hollow bronze shaft of 3 in. outer diameter and 2 in. inner diameter is slipped over a solid steel shaft 2 in. in diameter and of the same length as the hollow shaft. The two shafts are then fastened rigidly together at their ends. For bronze, G = 6 × 106 psi, and for steel, G = 12 × 106 psi. What torque can be applied to the composite shaft without exceeding a shearing stress of 8000 psi in the bronze or 12 ksi in the steel?
 

Solution to Problem 318 Torsion

Problem 318
A solid aluminum shaft 2 in. in diameter is subjected to two torques as shown in Fig. P-318. Determine the maximum shearing stress in each segment and the angle of rotation of the free end. Use G = 4 × 106 psi.
 

318-solid-aluminum-shaft.jpg

 

Derivation of the Double Angle Formulas

The Double Angle Formulas can be derived from Sum of Two Angles listed below:
$\sin (A + B) = \sin A \, \cos B + \cos A \, \sin B$   →   Equation (1)

$\cos (A + B) = \cos A \, \cos B - \sin A \, \sin B$   →   Equation (2)

$\tan (A + B) = \dfrac{\tan A + \tan B}{1 - \tan A \, \tan B}$   →   Equation (3)
 

Solution to Problem 319 Torsion

Problem 319
The compound shaft shown in Fig. P-319 is attached to rigid supports. For the bronze segment AB, the diameter is 75 mm, τ ≤ 60 MPa, and G = 35 GPa. For the steel segment BC, the diameter is 50 mm, τ ≤ 80 MPa, and G = 83 GPa. If a = 2 m and b = 1.5 m, compute the maximum torque T that can be applied.
 

Solution to Problem 320 Torsion

Problem 320
In Prob. 319, determine the ratio of lengths b/a so that each material will be stressed to its permissible limit. What torque T is required?
 

Solution 320

Solution to Problem 321 Torsion

Problem 321
A torque T is applied, as shown in Fig. P-321, to a solid shaft with built-in ends. Prove that the resisting torques at the walls are T1 = Tb/L and T2 = Ta/L. How would these values be changed if the shaft were hollow?
 

Figure P-321

 

Solution to Problem 322 Torsion

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 to Problem 323 Torsion

Problem 323
A shaft composed of segments AC, CD, and DB is fastened to rigid supports and loaded as shown in Fig. P-323. For bronze, G = 35 GPa; aluminum, G = 28 GPa, and for steel, G = 83 GPa. Determine the maximum shearing stress developed in each segment.
 

Solution to Problem 324 Torsion

Problem 324
The compound shaft shown in Fig. P-324 is attached to rigid supports. For the bronze segment AB, the maximum shearing stress is limited to 8000 psi and for the steel segment BC, it is limited to 12 ksi. Determine the diameters of each segment so that each material will be simultaneously stressed to its permissible limit when a torque T = 12 kip·ft is applied. For bronze, G = 6 × 106 psi and for steel, G = 12 × 106 psi.
 

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