What is the Torsion Constant?

rafters

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
 torsional constant for circle  $$J = \frac{1}{2}\pi r^4$$
 torsional constant for ellipse  $$J = \frac{\pi a^3\cdot b^3}{a^2 + b^2}$$
 torsional constant for square  $$J = 2.25a^4$$
 torsional constant for W or HP section $$J = \frac{2bt^3 + d’w^3}{3}$$
$$d’ = d – t$$
 torsional constant for channel section  $$J = \frac{2b’t^3 + d’w^3}{3}$$
$$b’ = b – \frac{w}{2}$$
$$d’ = d – t$$
 torsional constant for L section $$J = \frac{(b’ + d’)t^3}{3}$$
$$b’ = b – \frac{t}{2}$$
$$d’ = d – \frac{t}{2}$$
 torsional constant for structural T $$J = \frac{bt^3 + d’w^3}{3}$$
$$d’ = d-\frac{t}{2}$$
 torsional constant for mono-symmetric wide flange shapes $$J = \frac{b_1 t_1^3 + b_2 t_2^3 + d’w^3}{3}$$
$$d’ = d – \frac{t_1 + t_2}{2}$$

Standard Shapes

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

About Bernie Roseke, P.Eng., PMP

Bernie Roseke, P.Eng., PMP, is the president of Roseke Engineering. As a bridge engineer and project manager, he manages projects ranging from small, local bridges to multi-million dollar projects. He is also the technical brains behind ProjectEngineer, the online project management system for engineers. He is a licensed professional engineer, certified project manager, and six sigma black belt. He lives in Lethbridge, Alberta, Canada, with his wife and two kids.

View all posts by Bernie Roseke, P.Eng., PMP

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