Saint Venant's Approximation

Saint Venant represented the torsion constant $J$ of a solid section by a function relating the already known characteristic values of a cross section thus

$$\begin{aligned} J &= f \left(A,I\right) \\ &= \frac{1}{K}\frac{A^4}{I} \end{aligned}$$

For narrow rectangles $\left( b_{\max} \times b \right)$ (i.e. thin plates):

$$J = \frac{1}{36}\frac{A^{4}}{I}$$

which reduces to

$$J = \frac{1}{3}b^{3}b_{\max}$$

For circular sections

$$J = \frac{\pi r^{4}}{2}$$

which is the polar moment of inertia.

It can be shown that for members composed of thin rectangles, the torsion constant is equal to the sum of the $J$-values of component rectangles, except when the section is closed or ‘hollow’