| Case | Mode | Inputs | Output |
|---|---|---|---|
| 1 | I/H approximation | Iₓₑₐₐₖ = 8.50×106 mm4, hₒ = 260 mm | Cw ≈ 1.4365×1011 mm6 |
| 2 | Segment method | 4 segments: (L,t,Ω) = (120,10,4200), (120,10,4200), (260,6,1800), (260,6,1800) | Cw ≈ 4.5202×1011 mm6 |
Warping torsion constant (Cw)
The warping torsion constant measures resistance to non-uniform torsion. It appears when warping is restrained and axial stresses develop to carry part of the applied torque.
Mode 1: Doubly-symmetric I/H approximation
- Iₓₑₐₐₖ is the weak-axis second moment of area (unit4).
- hₒ is the distance between flange centroids (unit).
- Cw is reported in unit6.
Mode 2: Thin-walled open section (segment method)
- Lᵢ is segment midline length (unit).
- tᵢ is local thickness (unit).
- Ωᵢ is the sectorial (omega) coordinate at that segment (unit2).
- This approximates the area integral Cw = ∫ Ω2 dA using dA ≈ t ds.
Engineering note: Many thin-walled closed shapes have small warping deformation, so Cw is often taken near zero. Always follow your governing code and section property source.
- Select a calculation mode that matches your section data.
- Choose the length unit you will use for all inputs.
- Enter the required values:
- I/H mode: provide Iₓₑₐₐₖ and hₒ.
- Segment mode: fill one or more segment rows with L, t, and Ω.
- Press Calculate. Your result appears above the form.
- Use Download CSV or Download PDF to export the result.
1) What the warping torsion constant describes
The warping torsion constant, Cw, is a section property that captures stiffness against non-uniform torsion. When an open section twists, parts of the cross-section try to warp out of plane. If warping is restrained by end connections, stiffeners, or continuity, axial stresses develop and increase the effective torsional resistance.
2) Why warping matters for open sections
Open shapes such as I, channel, tee, and angle sections can warp significantly compared with closed tubes. Under restraint, warping stresses may govern serviceability, fatigue, or connection design. In many problems, St. Venant torsion (J) and warping torsion (Cw) both contribute, so you need reliable section properties to model the full response.
3) Where Cw appears in engineering calculations
Cw is used in advanced beam torsion solutions, lateral-torsional buckling formulations, and frame analysis with warping restraint. In steel design workflows, Cw is commonly paired with J, shear modulus, and warping boundary conditions to compute bimoment, warping normal stress, and twist along the member.
4) Unit scale and quick reasonableness checks
Cw has units of length6, so values can look very large in small units. For example, using the built-in sample I/H values Iweak = 8.50×106 mm4 and ho = 260 mm gives Cw ≈ 1.44×1011 mm6. Doubling ho increases Cw by a factor of four.
5) Using the I/H approximation wisely
The I/H mode applies Cw = Iweak·ho2/4 for doubly-symmetric I or H sections. It is fast and useful for preliminary work when section tables provide Iweak. Estimate ho as the distance between flange centroids, not the overall depth.
6) Thin-walled segment method and omega data
The segment mode approximates Cw ≈ Σ(t·L·Ω2) using thickness times midline length as an area element. The key input is Ω, the sectorial coordinate from an omega diagram. Ω is typically obtained from specialized references or software that computes sectorial properties for thin-walled open sections.
7) Typical data sources for inputs
For standard rolled shapes, use manufacturer or code-approved tables for Iweak and geometric dimensions to estimate ho. For custom built-up sections, CAD or section-property software can provide Iweak, while thin-walled theory tools can generate Ω values along the midline for the segment approach.
8) Limitations and modeling tips
This calculator supports two common approximations and is best for open sections where warping can be relevant. For thick or solid sections, or when local distortions dominate, rely on validated section-property methods. Always pair Cw with realistic warping boundary conditions; a free-warping end and a fully restrained end can lead to very different stresses.
1) What does the warping torsion constant represent?
It measures resistance to non-uniform torsion when warping is restrained. A higher Cw usually means less warping deformation and lower warping stress for the same restraint.
2) What are the units of Cw?
Cw has units of length6. If you enter millimeters, the output is mm6; if you enter inches, it is in6. The panel also reports m6 for reference.
3) When should I use the I/H approximation?
Use it for doubly-symmetric I or H sections when you know the weak-axis inertia and the flange centroid spacing. It is a fast estimate that works best for standard rolled shapes.
4) What is the omega value in the segment method?
Omega (Ω) is the sectorial coordinate from an omega diagram of the section. It can come from thin-walled theory, hand methods, or software that computes sectorial properties.
5) Can I use the segment method for thick-walled shapes?
The sum assumes thin walls so dA ≈ t·ds. For thick plates or solid sections, use tabulated properties, validated section-property software, or finite element analysis.
6) Why is Cw sometimes taken as zero?
Many thin-walled closed sections warp very little, so warping restraint contributes little to the torsional response. Some workflows treat Cw as negligible for these shapes.
7) Is this output suitable for final design?
Use it as an engineering aid for comparisons and checks. For final design, confirm Cw from authoritative tables or validated software and apply your governing code provisions.