What is the Torsion Constant?

In structural steel design, the Torsion Constant, J,  represents the ability of the steel beam to resist torsion, i.e. twisting.  It’s units are mm⁴ or inches⁴.

Equation

The bending resistance formula, in which the torsional constant is used, is:

$$\theta = \frac{T\cdot L}{G\cdot J}$$
Where:
θ = Angle of Twist
T = Applied Torque (N·m or lb·ft)
L = Length of Beam (mm or in)
J = Torsional Constant (mm⁴ or in⁴)
G = Modulus of Rigidity (GPa or psi)

Values of J for various shapes

Shape	Torsional Constant
	$$J = \frac{1}{2}\pi r^4$$
	$$J = \frac{\pi a^3\cdot b^3}{a^2 + b^2}$$
	$$J = 2.25a^4$$
	$$J = \frac{2bt^3 + d’w^3}{3}$$ $$d’ = d – t$$
	$$J = \frac{2b’t^3 + d’w^3}{3}$$ $$b’ = b – \frac{w}{2}$$ $$d’ = d – t$$
	$$J = \frac{(b’ + d’)t^3}{3}$$ $$b’ = b – \frac{t}{2}$$ $$d’ = d – \frac{t}{2}$$
	$$J = \frac{bt^3 + d’w^3}{3}$$ $$d’ = d-\frac{t}{2}$$
	$$J = \frac{b_1 t_1^3 + b_2 t_2^3 + d’w^3}{3}$$ $$d’ = d – \frac{t_1 + t_2}{2}$$

Standard Shapes

Standard structural steel shapes, including the Torsional Constant, J, are found in the AISC Steel Construction Manual, section 1.