Calculate the St Venant torsional constant J from rectangle, solid circle, or hollow tube dimensions, or solve a missing size from J.

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.

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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