Torsion | Mechanics of Materials

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TORSION
is the twisting of an object caused by a moment acting about the object's longitudinal axis.
Torsion
Consider a bar rigidly clamped at one end and twisted at the other end by a torque (twisting moment) T = Fd applied in a plane perpendicular to the axis of the bar. Such a bar is in torsion. An alternative representation of the torque is the curved arrow shown in the figure. Occasionally a number of couples act along the length of a shaft. In that case it is convenient to introduce a new quantity, the twisting moment, which for any section along the bar is defined to be the algebraic sum of the moments of the applied couples that lie to one side of the section in question. The choice of side is arbitrary.

Polar Moment of Inertia
A mathematical property of the geometry of the cross section which occurs in the study of the stresses set up in a circular shaft subject to torsion is the polar moment of inertia J, defined in a statics course. For a hollow circular shaft of outer diameter Do with a concentric circular hole of diameter Di the polar moment of inertia of the cross-sectional area is given by: The polar moment of inertia for a solid shaft is obtained by setting Di = 0. Occasionally it is convenient to rewrite the above equation in the form.
This last form is useful in numerical evaluation of J in those cases where the difference (Do – Di) is small. Let us derive an expression relating the applied twisting moment acting on a shaft of circular cross section and the shearing stress at any point in the shaft. The shaft is shown loaded by the two torques T in static equilibrium. One fundamental assumption is that a plane section of the shaft normal to its axis before loads are applied remains plane and normal to the axis after loading. This may be verified experimentally for circular shafts, but this assumption is not valid for shafts of noncircular cross section.
A generator on the surface of the shaft, denoted by O1A, deforms into the configuration O1B after torsion has occurred. The angle between these configurations is denoted by a. By definition, the shearing strain g on the surface of the shaft is:
where the angle a is assumed to be small. From the geometry of the figure,
But since a diameter of the shaft prior to loading is assumed to remain a diameter after torsion has been applied, the shearing strain at a general distance p from the center of the shaft may likewise be written Ypo/L. Consequently the shearing strains of the longitudinal fibers vary linearly as the distances from the center of the shaft.

Since we are concerned only with the linear range where the shearing stress is proportional to shearing strain, it is evident that the shearing stresses of the longitudinal fibers vary linearly as the distances from the center of the shaft. Obviously the distribution of shearing stresses is symmetric around the geometric axis of the shaft. They have the appearance shown. For equilibrium, the sum of the moments of these distributed shearing forces over the entire circular cross section is equal to the torque T. Thus we have where dA represents the area of the shaded element shown. However, the shearing stress varies as the distance from the axis; hence where the subscripts on the shearing stress denote the distances of the element from the axis of the shaft.
Consequently we may write.

Torsional Shearing Stress
So, for either a solid or a hollow circular shaft subject to a twisting moment T the torsional shearing stress tat a distance p from the center of the shaft is written as.

Shearing Strain
The amount of twist of a shaft is often of interest. Let us determine the angle of twist of a shaft subjected to a torque T. The ratio of the shear stress to the shear strain y is called the shear modulus and, is given by.

Example (1):

If a twisting moment of 1100 Nm is impressed upon a 4.4-cm-diameter shaft, what is the maximum shearing stress developed? Also, what is the angle of twist in a 150-cm length of the shaft? The material is steel for which G = 85 GPa.

Example (2):
A hollow 3-m-long steel shaft must transmit a torque of 25 kNm. The total angle of twist in this
length is not to exceed 2.5° and the allowable shearing stress is 90 MPa. Determine the inside and outside diameters of the shaft if G = 85 GPa.

Example (3):
Consider two solid circular shafts connected by 5-cm- and 25-cm-pitch-diameter gears Find the angular rotation of D, the right end of one shaft, with respect to A, the left end of the other, caused by the torque of 280 Nm applied at D. The left shaft is steel for which G = 80 GPa and the right is brass for which G = 33 GPa.