Use the tabs in the calculator to compute the Saint-Venant (St. Venant) torsional constant, J, for common shapes (rectangles and circular sections). For rectangles, use Rectangle (Thin) only for flat bars (one side much smaller than the other) and use Rectangle (Accurate) for general rectangles.

St Venant Torsional Constant Calculator

Rectangle (Thin)
Rectangle (Accurate)
Solid Circle
Hollow Circle (Tube)

Thin-rectangle (flat bar) approximation. Enter the long side as Width (b) and the short side / thickness as Height (t). Enter any 2 values to calculate the missing variable.

Related Calculators

St Venant Torsional Constant Formula

The following formulas are commonly used to approximate the Saint-Venant (St. Venant) torsional constant (J) for a solid rectangular cross-section. Let a be the longer side and b be the shorter side (so a ≥ b).

\begin{aligned}
J &\approx \frac{1}{3}\,a\,b^{3}\quad(\text{thin rectangle / flat bar, } b/a \lesssim 0.1)\\
J &\approx a\,b^{3}\left[\frac{1}{3}-0.21\frac{b}{a}\left(1-\frac{b^{4}}{12a^{4}}\right)\right]\quad(\text{Cowper approximation})
\end{aligned}

Variables:

  • J is the St Venant (Saint-Venant) torsional constant (units of length4)
  • a is the longer side of the rectangle
  • b is the shorter side (often called the thickness)

Note: The simple J ≈ (1/3)ab3 expression is a thin-rectangle approximation and becomes inaccurate for near-square rectangles. For general rectangles, use the Cowper approximation (or the calculator’s Rectangle (Accurate) tab).

What is the St Venant Torsional Constant?

The St Venant torsional constant is a cross-section property used in Saint-Venant torsion to relate torque to twist (for example, in the common relation T = GJ(θ/L)). It depends on the shape and size of the cross-section and is especially important for non-circular sections, where the shear stress distribution is non-uniform and warping occurs.

How to Calculate the St Venant Torsional Constant?

The following steps outline how to calculate the St Venant torsional constant for a rectangular cross-section.

  1. Measure both side lengths of the rectangle.
  2. Identify the longer side as a and the shorter side as b (thickness), so a ≥ b.
  3. If the rectangle is thin (a flat bar where b/a is small), you can use J ≈ (1/3)ab3. Otherwise, use the Cowper approximation (or the calculator’s Rectangle (Accurate) tab).
  4. Enter your dimensions into the calculator to compute J in your preferred units.

Example Problem :

Use the following variables as an example problem to test your knowledge.

Long side (a) = 100 mm

Short side / thickness (b) = 10 mm