Torsional Constant Calculator

How to use the torsional constant calculator

What this calculator does

This calculator estimates the Saint-Venant torsional constant, K, for eight common cross-section families. K is a geometric property with units of length to the fourth power. It is used in the elastic twist relationship φ = TL/(KG), where torque T, member length L, and shear modulus G are separate inputs to a full deformation analysis. The calculator does not determine allowable load, torsional stress, warping stress, buckling resistance, code compliance, or a safety factor. For non-circular and open sections, those checks may require a more complete structural or finite-element model. The LibreTexts treatment of shear and torsion explains why non-circular sections warp and why their torsional constant is not generally the same as a polar second moment of area.

When to use it

Use the calculator to compare candidate shaft or beam geometries, check a hand calculation before entering K into an angle-of-twist equation, estimate the effect of changing a wall thickness, or prepare a preliminary model for a solid or thin-walled section. It is also useful for documenting the geometric assumptions behind an I-section approximation before a detailed design review.

How to calculate

1. The calculator opens with a ready-to-use demonstration: a 10 mm by 5 mm solid rectangle. Its result and a validated example XLSX workbook are available immediately.
2. Choose the required Section. Only the dimensions used by that profile remain visible.
3. Select the Length unit. Existing dimensions are converted to preserve the same physical size. Enter dimensions with a decimal point; standard comma thousands grouping is accepted, while decimal-comma and scientific notation are rejected to avoid ambiguity.
4. Read Torsional constant (K), the SI equivalent, method classification, formula, accuracy note, interpretation, and the Calculation details table. Results update as you type.
5. Select Download Excel to export the current validated model. Reset clears the demonstration and all dimension fields; export then remains unavailable until a complete valid section is entered again.

Input guide

Section is required and selects Circle, Solid ellipse, Hollow ellipse, Thin-walled ellipse, Solid square, Solid rectangle, Thin-walled rectangle, or I-section. Length unit is required and supports mm, cm, m, in, and ft. Do not mix units within one section. Radius (r) is a positive circle radius or a nonnegative I-section fillet radius; 25 mm is a realistic shaft example, and increasing it raises circular K with the fourth power. Long side / width (a) is a positive length: the longer rectangle side, square side, outer ellipse semi-axis, thin-walled rectangle width, or flange width. For rectangle and I-section flange calculations, a must be at least b. Short side / thickness (b) is the shorter rectangle side, outer ellipse semi-axis, thin-walled rectangle height, or flange thickness. Confusing full ellipse diameters with semi-axes produces a large fourth-power error.

Inner-to-outer ratio (q) is required only for a hollow ellipse. Enter a decimal strictly between 0 and 1, such as 0.70, where q = a₀/a = b₀/b; the inner and outer ellipses must be geometrically similar. A higher q removes more material and lowers K. Wall thickness (t) is required for a thin-walled ellipse or the horizontal walls of a thin rectangle. It must be positive and small enough to leave positive median dimensions. Vertical-wall thickness (t₁) is required for a thin rectangle and must be less than b. Web height (c) and Web width (d) are required for an I-section. Web width (d) must also satisfy d < 2(b + r), the validity condition for the empirical fillet correction. Common mistakes include entering the overall I-section depth instead of clear web height, using a flange width smaller than flange thickness, and treating a thick closed tube as thin-walled.

Output guide

Torsional constant (K) is the primary geometric result in the selected length unit raised to the fourth power. A high value means more geometric resistance to uniform Saint-Venant twist; zero is not a valid result for the supported positive-area sections. SI equivalent converts the same result to m⁴, following the fourth-power unit relationship described in NIST's recommended SI practice for engineering quantities. Section model confirms the active geometry. Method identifies whether the expression is exact under the idealized assumptions, a thin-wall relation, or an approximation. Accuracy / scope states the main limitation for the selected formula. Formula displays the equation used. Interpretation reminds you that K alone is not torsional rigidity KG. The Calculation details table exposes intermediate terms, derived ratios, or component contributions; its rows and workbook values come from the same model as the headline result.

Worked example

For the startup solid rectangle, a = 10 mm and b = 5 mm. The calculator evaluates K = ab³/3 – 0.21b⁴ + 0.0175b⁸/a⁴. The three contributions are 416.6667 mm⁴, – 131.2500 mm⁴, and +0.6836 mm⁴. Their sum is 286.1003 mm⁴, equivalent to 2.861003 × 10⁻¹⁰ m⁴. Changing both dimensions by the same scale factor changes K by that factor to the fourth power. For example, doubling both sides multiplies K by 16.