In the field of
solid mechanics, torsion is the twisting of an object due to an applied
torque. 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 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. 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 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 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
radians.
* ''G'' is the shear modulus, also called the
modulus of rigidity, and is usually given in
gigapascals (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 ''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 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 can be calculated with the following formula:
:
The torque carried by the shaft is related to the
power 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 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:
:
After substitution of the ''
torsion constant'', the following expression is obtained:
:
The
diameter 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 stresses 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 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 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
*
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)