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The following table from Ghali and Neville (1978) gives values of the torsion constant (J) for various cross-sectional shapes.

other-sections-JJ=πr42J = \frac{\pi r^{4}}{2}
other-sections-JJ=0.1406b4J = 0.1406b_4
other-sections-JJ=π(r14r24)2J = \frac{\pi\left( {r_{1}}^{4} -{r_{2}}^{4} \right)}{2}
other-sections-JJ=[130.21bc(1b412c4)]J = \left\lbrack \frac{1}{3} - 0.21\frac{b}{c}\left( 1 - \frac{b^{4}}{12c^{4}} \right)\right\rbrack
other-sections-JJ=3b480J =\frac{\sqrt{3}b^{4}}{80}
other-sections-JJ=4a21tdsJ =\frac{4a^{2}}{\int_{}^{}{\frac{1}{t}ds}} where aa is the area enclosed by a line through the centre of the thickness and the integral is carried out over the circumference
other-sections-JJ=2t1t2(b1t2)2(b2t1)2b1t2+b2t1t22t12J = \frac{2t_{1}t_{2}\left( b_{1} - t_{2} \right)^{2}\left( b_{2} - t_{1} \right)^{2}}{b_{1}t_{2} + b_{2}t_{1} - {t_{2}}^{2} - {t_{1}}^{2}}
other-sections-JJ=b1t13+b2t23+b3t333J = \frac{b_{1}{t_{1}}^{3} +b_{2}{t_{2}}^{3} + b_{3}{t_{3}}^{3}}{3}
other-sections-JJ=b1t13+b2t233J = \frac{b_{1}{t_{1}}^{3} + b_{2}{t_{2}}^{3}}{3}
other-sections-JJ=13biti3J = \frac{1}{3}\sum_{}^{}{b_{i}{t_{i}}^{3}}

All the above assumes that the material is linear elastic.