Torsion (mechanics)

In the field of solid mechanics, torsion is the twisting of an object due to an applied torque.^[1]^[2] Torsion could be defined as strain^[3]^[4] or angular deformation,^[5] and is measured by the angle a chosen section is rotated from its equilibrium position.^[6] The resulting stress (torsional shear stress) 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 metres (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.^[7] For shafts of uniform cross-section unrestrained against warping, the torsion-related physical properties are expressed as:

T={\frac {J_{\text{T}}}{r}}\tau ={\frac {J_{\text{T}}}{\ell }}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.^[8]
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 radians.
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_TG is called the torsional rigidity w_T.

Properties

The shear stress at a point within a shaft is:

\tau _{\varphi _{z}}(r)={Tr \over J_{\text{T}}}

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 {T\ell }{GJ_{\text{T}}}}.

Sample calculation

Calculation of the steam turbine shaft radius for a turboset:

Assumptions:

Power carried by the shaft is 1000 MW; this is typical for a large nuclear power 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 maximal torque is:

T_{\max }={\frac {{\tau }_{\max }J_{\text{zz}}}{r}}

After substitution of the torsion constant, the following expression is obtained:

D=\left({\frac {16T_{\max }}{\pi {\tau }_{\max }}}\right)^{1/3}

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.

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.^[9]

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.

Torsional Resonator (example application)

A torsional resonator is an analytical system that takes advantage of torsional motion to provide insights into elastic/viscoelastic behavior of fiber materials. In a typical setup, a torsional resonator consists of a fiber fixed at one end with a rigid-material rod attached at the other end of the fiber.^[10] The motion of the rod is limited to rotation about the fiber which introduces torsional deformation to the fiber. The deformation of the fiber can be characterized to provide information about the material's energy dissipation and its relative viscoelasticity.

The fiber behaves as a spring, with the following equation describing its behavior of motion:^[11]

$T=K_{t}\theta +I{d^{2}\theta \over dt^{2}}$

where T is torque on the resonator, θ is rotation angle, K_t is torsional stiffness, and I is moment of inertia of the system (dependent on geometry of rod).

Assuming that the fiber is cylindrical, its torsional stiffness can be defined as:^[11]

$K_{t}\equiv T/\theta ={\frac {\pi Gd^{4}}{32l}}$

where d is the diameter of the fiber, l is the length of the fiber, and G is the shear modulus of the fiber.

Since the motion of the rod causes the fiber to experience torsional oscillation, its resonant angular frequency can also be determined:

$\omega _{n}={\sqrt {K_{t}/I}}$

Depending on the elasticity of the fiber, the solution to the behavior of motion can be determined. Assuming a completely elastic fiber, the solution is relatively simple. For a completely elastic material, applied stress/deformation does not cause a change in the shape of the object. The object restores its original shape even after deformation occurs because its energy is completely conserved.^[12]In this setup, the fiber has zero torque and the solution can be calculated as: ^[11]

$\theta =\theta _{0}cos(\omega _{n}t)$

However, most research that utilizes torsional resonators study fibers with viscoelastic character. The complex behavior, with some elastic and some viscous behavior means that additional viscosity considerations must be introduced into the equations. For a material demonstrating viscous behavior, the object does not restores its original shape after deformation. This is because some of its energy is lost in the form of dissipation.This makes the solution slightly different with theta being the real term solution to the following equation:

$\theta =\theta _{0}Realexp(i\omega _{n}t)$

Imaginary behavior is introduced from consideration of the dynamic modulus (G*) in place of G, the complex torsional stiffness (K_t*) in place of K_t, and complex resonant frequency in place of the resonant frequency. These complex terms account for both elastic and viscous material, where the real part describes elastic behavior and the imaginary term describes viscous (damping) behavior.^[11]

For example, the complex modulus G* has two terms:

$G^{*}=G'+iG''$ ^[13]

where G' is the shear storage modulus and G'' is the shear loss modulus.

The dependence on cosine and the real behavior of the angular frequency leads to oscillatory behavior, where the amplitude of the oscillations become lower and lower with time (as energy is dissipated due to viscous components of the fiber).

An example of the utility of the torsional resonator system was a study conducted by Valtorta and Mazza.^[14] Their study used a torsional resonator device to measure the viscoelastic properties of soft tissue. While their setup differs slightly from the one described above (they use an elastic fiber fixed to a soft tissue material and assess the fiber's response to understand the viscoelastic behavior of the soft tissue), the behavior of the fiber in response to viscoelastic elements provides a useful way to assess energy dissipation behavior in biological materials. Through the use of a torsional resonator device, they are able to characterize the complex shear modulus of the tissue and assess its relative elastic and viscous behaviors.

References

^ Escudier, Marcel (2019). A dictionary of mechanical engineering. New Yoek, NY: Oxford University Press. ISBN 978-0-19-883210-2.
^ Rennie, Richard; Law, Jonathan, eds. (2019-03-21), "A Dictionary of Physics", A Dictionary of Physics, Oxford University Press, doi:10.1093/acref/9780198821472.001.0001/acref-9780198821472, ISBN 978-0-19-882147-2, retrieved 2024-11-29
^ Horner, Joseph Gregory (1960). Dictionary of terms used in the theory and practice of mechanical engineering. London: Technical Press.
^ Weld, Le Roy D. (1937). Glossary of physics. New York London: McGraw-Hill Book Company, Inc.
^ Lindsay, Robert Bruce (1943). Student's handbook of elementary physics. New York, N.Y.: The Dryden press.
^ Michels, Walter C. (1956). The International dictionary of physics and electronics. Princeton, N.J.: Van Nostrand.
^ Seaburg, Paul; Carter, Charles (1997). Torsional Analysis of Structural Steel Members. American Institute of Steel Construction. p. 3.
^ Case and Chilver "Strength of Materials and Structures
^ Fakouri Hasanabadi, M.; Kokabi, A.H.; Faghihi-Sani, M.A.; Groß-Barsnick, S.M.; Malzbender, J. (October 2018). "Room- and high-temperature torsional shear strength of solid oxide fuel/electrolysis cell sealing material". Ceramics International. 45 (2): 2219–2225. doi:10.1016/j.ceramint.2018.10.134. ISSN 0272-8842. S2CID 139371841.
^ Heinisch, Martin; Niedermayer, Alexander; Dufour, Isabelle; Jakoby, Bernhard. "Concept Studies of Torsional Resonators for Viscosity and Mass Density Sensing Applications" (PDF). Procedia Engineering. 87: 1198–1201 – via ScienceDirect.
^ ^a ^b ^c ^d Shull, Ken (May 5, 2025). "Mechanical Properties of Materials, Northwestern University" (PDF).
^ Barber, J. R. (1992). Elasticity (4th ed.). Kluwer Academics Publishers. ISBN 978-3-031-15214-6.
^ Meyers, Mare; Chawala, Krishan (2009). Mechanical Behavior of Materials (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 978-0-511-45557-5.
^ Valtorta, Davide; Mazza, Edoardo (October 2005). "Dynamic measurement of soft tissue viscoelastic properties with a torsional resonator device". Medical Image Analysis. 9 (5): 481–490 – via Science Direct.

External links

The dictionary definition of torsion at Wiktionary
Solid Mechanics at Wikibooks

[1] Escudier, Marcel (2019). A dictionary of mechanical engineering. New Yoek, NY: Oxford University Press. ISBN 978-0-19-883210-2.

[2] Rennie, Richard; Law, Jonathan, eds. (2019-03-21), "A Dictionary of Physics", A Dictionary of Physics, Oxford University Press, doi:10.1093/acref/9780198821472.001.0001/acref-9780198821472, ISBN 978-0-19-882147-2, retrieved 2024-11-29

[3] Horner, Joseph Gregory (1960). Dictionary of terms used in the theory and practice of mechanical engineering. London: Technical Press.

[4] Weld, Le Roy D. (1937). Glossary of physics. New York London: McGraw-Hill Book Company, Inc.

[5] Lindsay, Robert Bruce (1943). Student's handbook of elementary physics. New York, N.Y.: The Dryden press.

[6] Michels, Walter C. (1956). The International dictionary of physics and electronics. Princeton, N.J.: Van Nostrand.

[7] Seaburg, Paul; Carter, Charles (1997). Torsional Analysis of Structural Steel Members. American Institute of Steel Construction. p. 3.

[8] Case and Chilver "Strength of Materials and Structures

[9] Fakouri Hasanabadi, M.; Kokabi, A.H.; Faghihi-Sani, M.A.; Groß-Barsnick, S.M.; Malzbender, J. (October 2018). "Room- and high-temperature torsional shear strength of solid oxide fuel/electrolysis cell sealing material". Ceramics International. 45 (2): 2219–2225. doi:10.1016/j.ceramint.2018.10.134. ISSN 0272-8842. S2CID 139371841.

[10] Heinisch, Martin; Niedermayer, Alexander; Dufour, Isabelle; Jakoby, Bernhard. "Concept Studies of Torsional Resonators for Viscosity and Mass Density Sensing Applications" (PDF). Procedia Engineering. 87: 1198–1201 – via ScienceDirect.

[:0-11] Shull, Ken (May 5, 2025). "Mechanical Properties of Materials, Northwestern University" (PDF).

[12] Barber, J. R. (1992). Elasticity (4th ed.). Kluwer Academics Publishers. ISBN 978-3-031-15214-6.

[13] Meyers, Mare; Chawala, Krishan (2009). Mechanical Behavior of Materials (2nd ed.). Cambridge, UK: Cambridge University Press. ISBN 978-0-511-45557-5.

[14] Valtorta, Davide; Mazza, Edoardo (October 2005). "Dynamic measurement of soft tissue viscoelastic properties with a torsional resonator device". Medical Image Analysis. 9 (5): 481–490 – via Science Direct.

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