Calculate the torsional constant (K) for common structural cross-sections. Select a section type — solid circular, solid elliptical, hollow circular, solid rectangular, or thin-walled circular — enter the relevant dimensions, and get the torsional constant in mm⁴. Used in the angle of twist formula ϕ = TL / (KG) for non-circular beam analysis.
Torsional Constant (K)
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Polar Moment of Inertia (J)
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K / J Ratio
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Section Type
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The torsional constant K is a geometric property of a cross-section that describes its resistance to torsion (twisting). It appears in the angle of twist formula ϕ = TL / (KG), where T is torque, L is length, and G is the shear modulus. For circular sections K equals the polar moment of inertia J, but for non-circular sections K is always less than J.
The polar moment of inertia J assumes that plane sections remain plane during twisting — a condition only true for circular cross-sections. The torsional constant K accounts for warping in non-circular sections and is generally smaller than J. For solid circular sections, K = J = πd⁴/32.
The torsional constant K has units of length to the fourth power — mm⁴ in SI (millimeters) or in⁴ in imperial units. This is the same dimensional unit as the polar moment of inertia and the second moment of area.
For a solid circular cross-section with diameter d, the torsional constant is K = πd⁴/32. This equals the polar moment of inertia because circular sections do not warp under torsion — they remain plane.
For a solid rectangle with longer side b and shorter side h, an approximate formula is K ≈ b·h³·[1/3 − 0.21·(h/b)·(1 − h⁴/(12b⁴))], where b ≥ h. This expression uses a series-based correction factor to account for warping at the corners.
For open thin-walled sections like I-beams, the torsional constant is approximated as K ≈ (1/3)·Σ(b·t³), where each b is the length and t is the thickness of each constituent plate (flanges and web). This shows that I-beams have relatively low torsional stiffness compared to closed box sections.
The torsional constant directly controls how much a beam twists under an applied torque. A higher K means the member is stiffer in torsion and will experience less angular rotation. It is critical for designing beams subjected to eccentric loads, curved girders, and any member with torsional loading.
For a hollow circular section with outer diameter d and inner diameter d_i, the torsional constant is K = π(d⁴ − d_i⁴)/32. Hollow sections are much more efficient in torsion than solid sections of the same weight because the material is placed farther from the center axis.