![Twisted bar](https://upload.wikimedia.org/wikipedia/commons/4/4f/Twisted_bar.png)
In the field of
solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ot ...
, 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 th ...
. Torsion is expressed in either the
pascal
Pascal, Pascal's or PASCAL may refer to:
People and fictional characters
* Pascal (given name), including a list of people with the name
* Pascal (surname), including a list of people and fictional characters with the name
** Blaise Pascal, Fren ...
(Pa), an
SI unit for newtons per square metre, or in
pounds per square inch
The pound per square inch or, more accurately, pound-force per square inch (symbol: lbf/in2; abbreviation: psi) is a unit of pressure or of stress based on avoirdupois units. It is the pressure resulting from a force of one pound-force applied to ...
(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 applie ...
s (N·m) or
foot-pound force
The foot-pound force (symbol: ft⋅lbf, ft⋅lbf, or ft⋅lb ) is a unit of Mechanical work, work or energy in the English Engineering Units, engineering and Foot–pound–second_system#force, gravitational systems in United States customary ...
(ft·lbf). In sections perpendicular to the torque axis, the resultant
shear stress
Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
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:
:
where:
* ''T'' is the applied torque or moment of torsion in Nm.
*
(tau) is the maximum shear stress at the outer surface
* ''J''
T is the
torsion constant The torsion constant is a geometrical property of a bar's cross-section which is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear-elastic bar. The torsion constant, together w ...
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
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
(FEA) is the best method. Other calculation methods include
membrane analogy The elastic membrane analogy, also known as the soap-film analogy, was first published by pioneering aerodynamicist Ludwig Prandtl in 1903.
It describes the stress distribution on a long bar in torsion. The cross section of the bar is constant al ...
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 that c ...
s.
* ''G'' is the shear modulus, also called the
modulus of rigidity
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackrel ...
, and is usually given in
gigapascal
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined as ...
s (GPa),
lbf/in2 (psi), or lbf/ft
2 or in ISO units N/mm
2.
* The product ''J''
T''G'' is called the
torsional rigidity
Stiffness is the extent to which an object resists deformation in response to an applied force.
The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is.
Calculations
The stiffness, k, of a bo ...
''w''
T.
Properties
The shear stress at a point within a shaft is:
:
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
In solid mechanics, a stress concentration (also called a stress raiser or a stress riser) is a location in an object where the stress is significantly greater than the surrounding region. Stress concentrations occur when there are irregularitie ...
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:
:
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 turbin ...
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 b ...
plant.
*
Yield stress
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
of the steel used to make the shaft (''τ''
yield) is: 250 × 10
6 N/m
2.
* 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
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
can be calculated with the following formula:
:
The torque carried by the shaft is related to the
power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
by the following equation:
:
The angular frequency is therefore 314.16
rad
RAD or Rad may refer to:
People
* Robert Anthony Rad Dougall (born 1951), South African former racing driver
* Rad Hourani, Canadian fashion designer and artist
* Nickname of Leonardus Rad Kortenhorst (1886–1963), Dutch politician
* Radley R ...
/
s and the torque 3.1831 × 10
6 N·m.
The maximal torque is:
:
After substitution of the ''
torsion constant The torsion constant is a geometrical property of a bar's cross-section which is involved in the relationship between angle of twist and applied torque along the axis of the bar, for a homogeneous linear-elastic bar. The torsion constant, together w ...
'', the following expression is obtained:
:
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 for ...
is 40 cm. If one adds a
factor of safety
In engineering, a factor of safety (FoS), also known as (and used interchangeably with) safety factor (SF), expresses how much stronger a system is than it needs to be for an intended load. Safety factors are often calculated using detailed analy ...
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 stress
In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that completely ...
es via
Mohr's circle
Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.
Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineer ...
. 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
A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Bre ...
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
The following is a list of second moments of area of some shapes. The second moment of area, also known as area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with respect to an arbitrary axis ...
*
Saint-Venant's theorem
*
Second moment of area
The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...
*
Structural rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.
Definitions
Rigidity is the property of a structure ...
*
Torque tester A torque tester is used as a quality control device to test or calibrate torque controlled tools. This includes electronic torque wrenches, click torque wrenches, dial torque wrenches, electric screwdrivers, air screwdrivers, pulse tools, cordless s ...
*
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
A torsion spring is a spring that works by twisting its end along its axis; that is, a flexible elastic object that stores mechanical energy when it is twisted. When it is twisted, it exerts a torque in the opposite direction, proportional ...
or -bar
*
Torsional vibration
Torsional vibration is angular vibration of an object—commonly a shaft along its axis of rotation. Torsional vibration is often a concern in power transmission systems using rotating shafts or couplings where it can cause failures if not contr ...
References
External links
*
* {{wikibooks inline, Solid Mechanics
Mechanics
Torque
Moment (physics)