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| [[File:Twisted bar.png|thumbnail|120px|right|Torsion of a square section bar]]
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| In [[solid mechanics]], '''torsion''' is the twisting of an object due to an applied [[torque]]. It is expressed in [[newton metre]]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.
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| For shafts of uniform cross-section the torsion is:
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| :<math> T = \frac{J_T}{r} \tau= \frac{J_T}{\ell} G \theta</math>
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| where:
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| * ''T'' is the applied torque or moment of torsion in Nm.
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| * <math>\tau</math> is the maximum shear stress at the outer surface
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| * ''J<sub>T</sub>'' is the [[torsion constant]] for the section. It is identical to the [[second moment of area]] ''J<sub>zz</sub>'' for concentric circular tubes or round solid shafts only. For other shapes, ''J'' must be determined by other means. For solid shafts, the [[membrane analogy]] is useful, and for thin-walled tubes of arbitrary shape, the shear flow approximation is fairly good,<ref>Case and Chilver "Strength of Materials and Structures</ref> if the section is not re-entrant. For thick-walled tubes of arbitrary shape, there is no simple solution, and [[finite element method|finite element analysis]] (FEA) may be the best method.
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| * ''r'' is the distance between the rotational axis and the farthest point in the section (at the outer surface).
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| * ''ℓ'' is the length of the object the torque is being applied to or over.
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| * ''θ'' is the angle of twist in [[radian]]s.
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| * ''G'' is the shear modulus or more commonly the [[modulus of rigidity]] and is usually given in [[gigapascal]]s (GPa), [[pounds per square inch|lbf/in<sup>2</sup>]] (psi), or lbf/ft<sup>2</sup>.<!-- Addition of psi units and correction of psf units.--~~~~ -->
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| * The product ''J<sub>T</sub> G'' is called the [[torsional rigidity]] ''w<sub>T</sub>''.
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| ==Properties==
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| The shear stress at a point within a shaft is:
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| :<math> \tau_{\varphi_{z}} = {T r \over J_T} </math>
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| 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.
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| The angle of twist can be found by using:
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| :<math> \varphi_{} = \frac{T \ell}{G J_T}. </math>
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| ==Sample calculation==
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| [[Image:Dampfturbine Laeufer01.jpg|thumb|250px|The rotor of a modern [[steam turbine]]]]
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| Calculation of the [[steam turbine]] shaft radius for a turboset:
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| Assumptions:
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| * Power carried by the shaft is 1000 [[megawatts|MW]]; this is typical for a large [[nuclear power]] plant.
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| * [[Yield stress]] of the steel used to make the shaft (''τ''<sub>yield</sub>) is: 250 × 10<sup>6</sup> N/m².
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| * Electricity has a frequency of 50 [[Hertz|Hz]]; this is the typical frequency in Europe. In North America, the frequency is 60 Hz.
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| The [[angular frequency]] can be calculated with the following formula:
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| : <math>\omega=2 \pi f</math>
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| The torque carried by the shaft is related to the [[Power (physics)|power]] by the following equation:
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| : <math>P=T \omega</math>
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| The angular frequency is therefore 314.16 [[radian|rad]]/[[second|s]] and the torque 3.1831 × 10<sup>6</sup> [[Newton metre|N·m]].
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| The maximal torque is:
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| : <math>T_\max = \frac{ { \tau }_\max J_{zz} }{r}</math> | |
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| After substitution of the ''polar moment of inertia'', the following expression is obtained:
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| : <math> D = (\frac{16 T_\max}{\pi {\tau}_\max})^{1/3}</math>
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| The [[diameter]] 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.
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| ==Failure mode==
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| The shear stress in the shaft may be resolved into [[principal stress]]es 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.
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| 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.
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| ==See also==
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| * [[Structural rigidity]]
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| * [[Torsion siege engine]]
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| * [[Torsion spring]] or -bar
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| * [[Torsional vibration]]
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| * [[Torque tester]]
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| * [[Saint-Venant's theorem]]
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| * [[List of moments of inertia]]
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| == References ==
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| <references />
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| == External links ==
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| * [http://www.amesweb.info/Torsion/TorsionalStressCalculator.aspx Online engineering calculator for torsion of shaft]
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| {{Wiktionary|torsion}}
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| {{wikibooks|Solid Mechanics}}
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| [[Category:Mechanics]]
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