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Saint Venant's Approximation

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

J=f(A,I)=1KA4I\begin{aligned} J &= f \left(A,I\right) \\ &= \frac{1}{K}\frac{A^4}{I} \end{aligned}

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

J=136A4IJ = \frac{1}{36}\frac{A^{4}}{I}

which reduces to

J=13b3bmaxJ = \frac{1}{3}b^{3}b_{\max}

For circular sections

J=πr42J = \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 JJ-values of component rectangles, except when the section is closed or ‘hollow’