A circular hollow section with an inner-to-outer diameter ratio k = 0.7 resists torsion 2.92 times more effectively than a solid shaft of identical weight, and the advantage climbs to 9.53 times at k = 0.9 [S3]. The gain comes directly from the polar moment of inertia equation J = π(D⁴ − d⁴)/32: every kilogram of steel at the centre of a solid bar contributes zero torsional stress, so relocating that mass to the outer fibre multiplies stiffness without adding weight.
Open sections (I, C, Z, T) do not follow the same closed-form pattern because the polar moment splits into a St. Venant torsional constant plus a warping torsion term, and shear flow loops around the open profile instead of circulating through a closed cell [S1]. For structural steel design under twisting loads, the practical decision matrix reduces to one question: can the section behave as a closed cell, or is warping restraint available?
Torsional Constant J: Closed Cell vs Open Profile
For a closed thin-walled section such as rectangular hollow section (RHS), the AISC Steel Design Guide 9 treatment expresses the St. Venant constant as J = 4·A² / ∮(ds/t) for a single-cell tube, where A is the area enclosed by the shear-flow path and ds/t is the wall slenderness integral around the median line [S1]. For a square hollow section (SHS) of side b and wall thickness t, this reduces to roughly J ≈ 2·b³·t, a value that scales with the cube of the outer dimension rather than the fourth power of a solid round.
Open sections invert the pattern. The St. Venant torsional constant for an I-shape equals J = (1/3)·Σ(b·t³), summing the third powers of flange width times flange thickness plus web thickness times web length cubed, divided by three [S1]. Because the dominant term is t³, doubling the flange thickness from 10 mm to 20 mm quadruples J, but doubling the flange width from 200 mm to 400 mm leaves J essentially unchanged. Hollow sections scale with the dimension to the third or fourth power, which is why a 200×200×8 SHS outperforms a 200 mm wide-flange of equal mass in pure torsion by roughly an order of magnitude [S3].
Warping Torsion: The Open-Section Penalty
Open cross-sections cannot carry St. Venant torsion efficiently, so the resisting mechanism shifts to warping torsion governed by the warping constant Cw, the polar moment of inertia about the shear centre, and the St. Venant constant acting in series [S1]. The total twist per unit length is the sum of St. Venant rate (T/GJ) and warping rate (T·sinh(kL) terms) in differential form, and the result is that an unrestrained I-beam twists far more under a given torque than a closed tube of equal weight.
AISC DG-9 quantifies the practical consequence: shapes of open cross-section tend to warp under torsional loading, and the warping normal stresses on I-, C-, and Z-sections follow σw = E·Cw·θ″·ω, where ω is the sectorial coordinate measured from the shear centre [S1]. For a simply supported I-beam of length L, lateral-torsional buckling intervenes before the St. Venant capacity is reached, and the critical moment Mcr scales with √(E·Iy·Cw) over L². Continuous lateral-torsional bracing of the compression flange can suppress warping, but every unbraced span pays the open-section tax in either deflection, buckling, or both.
Hollow Section Geometry: k, D/t, and the J-Ratio Table
For circular hollow sections at constant mass, the J-ratio of hollow-to-solid torsional stiffness follows J_ratio = (1 + k²)/(1 − k²), where k is the inner-to-outer diameter ratio [S3]. The table of values is the single most useful decision aid for sizing a tubular shaft against a solid bar: at k = 0.5 the gain is 1.67×, at k = 0.7 it is 2.92×, at k = 0.8 it reaches 4.56×, and at k = 0.95 the section is 19.5× stiffer than a solid bar of equal mass, but the D/t ratio has climbed to 40 and local buckling risk is classified as very high [S3].
Engineers should treat the right-hand column of that table as a hard limit, not a soft preference. Once the diameter-to-thickness ratio passes roughly 20, ring-stiffened or filled sections become mandatory for primary load paths. The 2025 study on S700 high-strength steel RHS stub columns confirms that yield strength ≥ 460 MPa changes the design philosophy but not the topology: HSS with high strength-to-weight ratio, high torsional strength, and enhanced durability still relies on the same closed-cell shear flow path, and the test programme validates FE models that predict torsional capacity within Eurocode 3 tolerance bands [S4].
Selection Criteria: When Open Sections Still Win
Open sections are not obsolete. They win on four criteria that hollow sections cannot match. First, beam-column efficiency in pure bending: an I-shape concentrates material in flanges away from the neutral axis, giving a high Ix per kilogram for uniaxial bending about the strong axis. Second, connection simplicity: bolted end-plate and double-angle connections to I-beams are stock items, while SHS/RHS cap-plate and through-bolt detailing requires slotted holes or welded plates. Third, fabrication cost: roll-forming and flame-cutting open sections is cheaper per metre than induction-bending or press-bending closed tubes for non-standard geometries. Fourth, inspectability: the inside of a hollow section is invisible to routine visual inspection, which matters for corrosion-prone service such as offshore splash zones. [S5]
For the cases where torsion dominates the design, the same article summarises: place material away from the neutral axis, providing higher torsional stiffness compared to open sections like I-beams, and resist twisting very well as a closed loop, vital for bridges or cranes where loads move and change direction [S5]. Cold-formed SHS, RHS, and CHS are the default pick for space frames, edge beams, grillages, and gantry stanchions where the load path includes a significant twisting component [S2].
High-Strength Steel Hollow Sections: 2025 Test Data
The 2025 S700 RHS stub-column study fills a long-standing gap in the literature on torsion plus axial compression for high-strength tubes. Yield strength 460 MPa or more makes HSS suitable for bridges, high-risk buildings, and earthquake-resistant design, and the experimental programme included residual-stress measurement, tensile coupon testing, and FE validation against Eurocode 3 provisions [S4]. Lateral-torsional buckling behaviour of HSS-RHS beam-columns is captured by strain-energy formulations that confirm Eurocode 3 provisions effectively predict LTB behaviour, giving designers a defensible code path for closed-section bending plus torsion in S460 and S700 grades [S4].
The same study notes that RHS plays an important role in hollow steel sections because they provide a combination of strength, aesthetic appeal, and efficiency, which makes them a better choice in different engineering and construction applications, with versatility and user friendliness making them more attractive than the other cross-sectional shapes [S4]. The relevance to torsion is indirect: closed-cell shear flow benefits identically from higher yield, but the elastic stiffness gain in torsion comes from geometry (J), not from material (G), so a 50% increase in yield buys roughly 0% increase in torsional stiffness and the full 50% in torsional strength.
Comparison Matrix: Open vs Hollow on Five Decision Criteria
Lining the two families up against the criteria that drive a real specification yields the following matrix. On torsional stiffness at equal mass, open sections score low (St. Venant J dominated by Σbt³, warping uncontrolled without bracing) and hollow sections score high (J scales as A²/(ds/t) for closed cells) [S1][S3]. On bending efficiency about the strong axis, open sections win (flanges far from neutral axis) and hollow sections are competitive but not dominant. On connection cost, open sections win (stock end-plates, double angles) and hollow sections require welded caps, slotted holes, or through-bolts with backing plates. On corrosion inspection, open sections win (fully visible) and hollow sections require UT, borescope, or coupon retrieval.
On weight per metre at equal torsional stiffness, the comparison is decisive: for a target J of 1×10⁵ mm⁴, a 200×200×8 SHS weighs around 47 kg/m while a 400-grade I-section of equivalent J weighs roughly 80–100 kg/m, a 1.7–2.1× weight penalty for the open profile [S1][S3]. This single ratio is why industrial cranes, gantry girders, and sign-support masts have shifted almost entirely to SHS/RHS/CHS over the last two decades. For a deeper look at how material grade selection drives different decisions in adjacent tooling applications, the NADCA #207-2025 H13 acceptance criteria breakdown is a useful parallel read on specification discipline.
Failure Modes and Limits on k
Three failure modes set the upper bound on how far the hollow-section advantage can be pushed. First, local buckling of the wall under axial compression or bending compression, governed by the b/t or D/t ratio and the steel grade; EN 1993-1-1 Class 3 limits for SHS are roughly b/t ≤ 42·ε for outstand flanges and 67·ε for internal compression parts, where ε = √(235/fy) [S4]. Second, lateral-torsional buckling of the whole member in bending, which for closed sections is far less sensitive than for open sections because of the high St. Venant J, but still requires LTB checks for slender spans [S4]. Third, fatigue at welded connections, where SHS/RHS branch joints concentrate stress at the saddle and crown points and require thicker walls or smoother weld profiles to meet fatigue classes.
Wall-thinning economics also matter. At k = 0.95, the 19.5× torsional gain over a solid bar sounds attractive, but the wall is only 2.5% of the diameter, and a single corrosion pit through the wall is a leak path, not a strength concern. Most industrial specifications cap practical k at 0.85 to 0.90, balancing stiffness gain against fabrication handling and in-service damage tolerance. For applications where twist under load is a serviceability limit (think hydraulic actuator response time vs flow rate calculations, where shaft compliance couples into valve dynamics), the J-ratio table is the right starting point and the k = 0.7 to 0.8 range is the sweet spot for the most designs.
Standards and Sourcing
For European projects, EN 1993-1-1 (EC3) governs the strength and stability checks for both open and hollow sections, and EN 1993-1-14 covers the additional rules for cold-formed members. North American projects work to AISC 360, with the Steel Design Guide 9 [S1] providing the explicit torsional-analysis framework that ties warping torsion into LRFD and ASD checks. Material supply follows EN 10025 for hot-finished S275/S355/S460 hollow sections, EN 10210 for hot-finished structural hollow sections, and EN 10219 for cold-formed welded structural hollow sections, with high-strength S700 variants now in production for the grades discussed in the 2025 Nature study [S4]. For a foundational look at how stainless steel grades and carbon steel families differ in available HSS products, the spec encyclopaedia covers the trade-offs in weight, weldability, and cost that drive grade selection.
For procurement teams comparing mill quotes against stockist offers, the choice of section family is the bigger cost lever than the choice of grade. A 200×200×8 S275 SHS typically undercuts a 400 mm UB of equivalent torsional stiffness by 30–45% on delivered cost in 2026 because the closed-section supply chain is shorter and the tonnage is lower. The 2026 sourcing signal worth tracking: more stockists are publishing k-ratio tables in their cut-sheet calculators, which means the stiffness advantage of hollow sections is becoming a quote-line item rather than a calculated afterthought [S3][S5].
For the relevant spec sheets and selection criteria, see steel section.