How this calculator works
The torsion constant describes free-warping Saint-Venant torsion and equals the polar second moment only for circular sections. Solid rectangle results use a convergent odd-term series, including the square case. Thin closed sections use Bredt-Batho theory; the hollow-ellipse option integrates an approximate normal wall thickness between the supplied inner and outer ellipses. Thin uniform elliptical walls use an approximate mean ellipse. These elliptical approximations require thin walls and are not thick elliptical-tube solutions. The open I formula neglects fillets, flange-web interaction and restrained warping.
Formula
Circular J = pi(Ro^4-Ri^4)/2; solid ellipse J = pi a^3 b^3/(a^2+b^2); thin closed J = 4 Am^2/integral(ds/t); thin open J approximately sum(b t^3)/3.
Worked example
A solid circular shaft of radius 50 mm has J = 9.817477 x 10^-6 m4. A hollow shaft with outer radius 50 mm and inner radius 40 mm has J = 5.796238 x 10^-6 m4.
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Frequently asked questions
Can I use the polar second moment of a rectangle?
No. Non-circular warping makes its torsion constant different from the polar second moment.
How accurate is the rectangular series?
The calculator sums 100 odd terms. For ordinary aspect ratios the truncation error is very small.
Does J include restrained warping?
No. Open-section torsion with restrained warping also involves the warping constant and boundary conditions.