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In the field of solid mechanics, torsion is the twisting of an object due to an applied
torque In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of t ...
. Torsion is expressed in either the pascal (Pa), an SI unit for newtons per square metre, or in pounds per square inch (psi) while torque is expressed in
newton metre The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton appli ...
s (N·m) or foot-pound force (ft·lbf). In sections perpendicular to the torque axis, the resultant shear stress in this section is perpendicular to the radius. In non-circular cross-sections, twisting is accompanied by a distortion called warping, in which transverse sections do not remain plane. For shafts of uniform cross-section unrestrained against warping, the torsion is: : T = \frac \tau= \frac G \varphi where: * ''T'' is the applied torque or moment of torsion in Nm. * \tau (tau) is the maximum shear stress at the outer surface * ''J''T is the torsion constant for the section. For circular rods, and tubes with constant wall thickness, it is equal to the polar moment of inertia of the section, but for other shapes, or split sections, it can be much less. For more accuracy, finite element analysis (FEA) is the best method. Other calculation methods include membrane analogy and shear flow approximation. * ''r'' is the perpendicular distance between the rotational axis and the farthest point in the section (at the outer surface). * ''ℓ'' is the length of the object to or over which the torque is being applied. * ''φ'' (phi) is the angle of twist in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s. * ''G'' is the shear modulus, also called the modulus of rigidity, and is usually given in gigapascals (GPa), lbf/in2 (psi), or lbf/ft2 or in ISO units N/mm2. * The product ''J''T''G'' is called the torsional rigidity ''w''T.


Properties

The shear stress at a point within a shaft is: : \tau_ = Note that the highest shear stress occurs on the surface of the shaft, where the radius is maximum. High stresses at the surface may be compounded by stress concentrations such as rough spots. Thus, shafts for use in high torsion are polished to a fine surface finish to reduce the maximum stress in the shaft and increase their service life. The angle of twist can be found by using: : \varphi_ = \frac.


Sample calculation

Calculation of the
steam turbine A steam turbine is a machine that extracts thermal energy from pressurized steam and uses it to do mechanical work on a rotating output shaft. Its modern manifestation was invented by Charles Parsons in 1884. Fabrication of a modern steam turb ...
shaft radius for a turboset: Assumptions: * Power carried by the shaft is 1000 MW; this is typical for a large
nuclear power Nuclear power is the use of nuclear reactions to produce electricity. Nuclear power can be obtained from nuclear fission, nuclear decay and nuclear fusion reactions. Presently, the vast majority of electricity from nuclear power is produced ...
plant. * Yield stress of the steel used to make the shaft (''τ''yield) is: 250 × 106 N/m2. * Electricity has a frequency of 50 Hz; this is the typical frequency in Europe. In North America, the frequency is 60 Hz. The angular frequency can be calculated with the following formula: : \omega=2 \pi f The torque carried by the shaft is related to the power by the following equation: : P=T \omega The angular frequency is therefore 314.16 rad/ s and the torque 3.1831 × 106
N·m The newton-metre (also newton metre or newton meter; symbol N⋅m or N m) is the unit of torque (also called ) in the International System of Units (SI). One newton-metre is equal to the torque resulting from a force of one newton applied ...
. The maximal torque is: : T_\max = \frac After substitution of the '' torsion constant'', the following expression is obtained: : D = \left(\frac\right)^ The
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...
is 40 cm. If one adds a factor of safety of 5 and re-calculates the radius with the maximum stress equal to the ''yield stress/5'', the result is a diameter of 69 cm, the approximate size of a turboset shaft in a nuclear power plant.


Failure mode

The shear stress in the shaft may be resolved into principal stresses via Mohr's circle. If the shaft is loaded only in torsion, then one of the principal stresses will be in tension and the other in compression. These stresses are oriented at a 45-degree helical angle around the shaft. If the shaft is made of brittle material, then the shaft will fail by a crack initiating at the surface and propagating through to the core of the shaft, fracturing in a 45-degree angle helical shape. This is often demonstrated by twisting a piece of blackboard chalk between one's fingers. In the case of thin hollow shafts, a twisting buckling mode can result from excessive torsional load, with wrinkles forming at 45° to the shaft axis.


See also

* List of area moments of inertia *
Saint-Venant's theorem In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle.E. Makai, A proof of Saint-Venant' ...
* Second moment of area * Structural rigidity * Torque tester *
Torsion siege engine A torsion siege engine is a type of siege engine that utilizes torsion to launch projectiles. They were initially developed by the ancient Macedonians, specifically Philip II of Macedon and Alexander the Great, and used through the Middle Ages u ...
* Torsion spring or -bar * Torsional vibration


References


External links

* * {{wikibooks inline, Solid Mechanics Mechanics Torque Moment (physics)