Torsion (mechanics)

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 (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.^[1] For shafts of uniform cross-section unrestrained against warping, the torsion is:

${\displaystyle T={\frac {J_{\text{T}}}{r}}\tau ={\frac {J_{\text{T}}}{\ell }}G\varphi }$

where:

• T is the applied torque or moment of torsion in Nm.
• ${\displaystyle \tau }$ (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.^[2]
• 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.