Torsion (mechanics) - Citizendia

In solid mechanics, torsion is the twisting of an object due to an applied torque. Solid mechanics is the branch of Mechanics, Physics, and Mathematics that concerns the behavior of solid matter under external actions (e A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about In circular sections, the resultant shearing stress is perpendicular to the radius. A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material

For solid or hollow shafts of uniform circular cross-section and constant wall thickness, the torsion relations are:

$\frac{T}{J} = \frac{\tau}{R} = \frac{G\phi}{l}$

The shear stress at a point on a shaft is:

$\tau_{\phi_{z}} = {T r \over J}$

Note that the highest shear stress is at the point where the radius is maximum, the surface of the shaft. 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 its service life.

The angle of twist can be found by using:

$\phi_{} = {T l \over JG}$

Where:

r is the distance from the center of rotation
Φ is the angle of twist in radians. The radian is a unit of plane Angle, equal to 180/ π degrees, or about 57
T is the torque (N·m or ft·lbf). Newton metre is the unit of moment ( Torque) in the SI system The foot-pound force, or simply foot-pound (symbol ft·lbf or ft·lb) is a unit of work or Energy (a scalar
l is the length of the object the torque is being applied to or over.
G is the shear modulus or more commonly the modulus of rigidity and is usually given in gigapascals (GPa), lbf/in² (psi), or lbf/ft². In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear The pound per square inch or more accurately pound-force per square inch (symbol psi or lbf/in² or lbf/in²) is a unit of
J is the torsion constant for the section . The torsion constant is a geometrical property of a beam's cross-section which determines the relationship between angle of twist and applied torque It is identical to the polar moment of inertia for a round shaft or concentric tube only. Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects with an invariant circular Cross-section 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, if the section is not re-entrant. For thick walled tubes of arbitrary shape there is no simple solution, FEA may be the best method.
GJ is together called as torsional rigidity.

Polar moment of inertia

Main article: Polar moment of inertia

The polar moment of inertia for a solid shaft is:

$J = {\pi \over 2} r^4$

Where r is the radius of the object. Polar moment of inertia is a quantity used to predict an object's ability to resist torsion, in objects (or segments of objects with an invariant circular Cross-section

The polar moment of inertia for a pipe is:

$J = {\pi \over 2} (r_{o}^4 - r_{i}^4)$

Where the o and i subscripts stand for the outer and inner radius of the pipe. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers

For a thin cylinder

J = 2π R³ t

Where R is the average of the outer and inner radius and t is the wall thickness.

Failure mode

The shear stress in the shaft may be resolved into principal stresses via Mohr's circle. Mohr's circle is a graphical representation of any 2-D stress state proposed in 1892 by Christian Otto Mohr. 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.

Polar moment of inertia

Failure mode

See also