Section Properties and Structural Profiles
Area, second moment and section modulus, why shape matters, standard tubes, angles, channels, I-beams and extrusions.
Section properties
The strength and stiffness of a beam depend on its cross-section through a few numbers. The area A carries axial load and sets the weight. The second moment of area I (mm⁴, also called moment of inertia) sets the stiffness in bending: deflection is inversely proportional to E I. The section modulus S = I ÷ c (mm³) sets the bending stress: σ = M ÷ S. The radius of gyration r = √(I ÷ A) governs column buckling (see Columns and Buckling), and the polar moment J torsion (Torsion, Shear and Combined Loads). All are measured about an axis through the centroid (the neutral axis of bending), and a section has different values about its two axes.
Formulas for the basic shapes
| Shape | A | I (about the horizontal centroidal axis) | S = I ÷ c | r |
|---|---|---|---|---|
| Rectangle b × h | b h | b h³ ÷ 12 | b h² ÷ 6 | h ÷ √12 = 0.289 h |
| Hollow rectangle (B × H outside, b × h inside) | B H − b h | (B H³ − b h³) ÷ 12 | I ÷ (H ÷ 2) | √(I ÷ A) |
| Solid circle, diameter d | π d² ÷ 4 | π d⁴ ÷ 64 | π d³ ÷ 32 | d ÷ 4 |
| Hollow circle (D outside, d inside) | π (D² − d²) ÷ 4 | π (D⁴ − d⁴) ÷ 64 | π (D⁴ − d⁴) ÷ (32 D) | √(D² + d²) ÷ 4 |
| Triangle (base b, height h), about the base | b h ÷ 2 | b h³ ÷ 12 (about the base); b h³ ÷ 36 (about the centroid) | — | — |
| Thin-walled tube (mean radius R, wall t) | 2 π R t | π R³ t | π R² t | 0.707 R |
For a rectangle the orientation matters: the stiffness depends on the cube of the depth (in the direction of the load). A 20 × 10 mm bar loaded across its 10 mm face has I = 1 667 mm⁴; on edge (20 mm deep) 6 667 mm⁴: four times stiffer for the same material (see the example in Beam Bending and Deflection).
Composite sections and the parallel-axis theorem
Real profiles are built from simple shapes. To find the properties: (1) locate the centroid, y̅ = Σ(Ai yi) ÷ ΣAi; (2) find the second moment of each part about the centroidal axis of the whole with the parallel-axis theorem I = Ic + A d², with Ic the part’s own moment about its own centroid and d the distance between the two axes; (3) add them up: I = Σ(Ic,i + Ai di²).
Worked example: a T-section
- A T-section has a flange of 60 × 10 mm on top of a web 10 × 50 mm (total height 60 mm), measured from the bottom of the web. Flange: A1 = 600 mm², centroid at y1 = 55 mm. Web: A2 = 500 mm², y2 = 25 mm.
- Centroid: y̅ = (600 × 55 + 500 × 25) ÷ 1 100 = 41.36 mm above the bottom.
- Second moment: flange Ic = 60 × 10³ ÷ 12 = 5 000 and A d² = 600 × (13.64)² = 111 570; web Ic = 10 × 50³ ÷ 12 = 104 167 and A d² = 500 × (-16.36)² = 133 884. Total I = 354 621 mm⁴.
- Section moduli: bottom (tension side for a downward load) S = I ÷ y̅ = 8 573 mm³; top S = I ÷ (60 − 41.36) = 19 028 mm³. The two differ because the neutral axis is not at mid-height: the stress is larger at the side farther from the axis.
Why shape matters
Bending stress is highest at the outside and zero at the middle, so the material near the neutral axis is working hardly at all. Moving material outwards (a tube, an I-beam, a box) increases I and S greatly for the same weight. The same area of steel can be a rod, a flat bar or a tube; they differ hugely in stiffness.
Worked example: a rod against a tube of the same weight
Take 2 500 mm² of steel. As a solid round bar the diameter is 56.4 mm and I = π d⁴ ÷ 64 = 497 359 mm⁴. As a tube of outside diameter 80 mm with the same area the bore is 56.7 mm (wall 11.6 mm) and I = π (80⁴ − 56.7⁴) ÷ 64 = 1 502 641 mm⁴: 3.0 times stiffer, and the same weight. For an I-beam of equal area the factor can exceed 20. The price: thin walls can buckle locally and tubes are harder to connect.
Standard profiles
| Profile | Shape | Good for | Notes |
|---|---|---|---|
| Square and rectangular hollow section (SHS, RHS) | Closed box | Frames, tables, columns, beams in two directions | Stiff in torsion; easy to weld; closed so it does not trap water if sealed |
| Circular hollow section (CHS, pipe) | Tube | Columns, rails, shafts, handrails, roll cages | Equal stiffness in all directions; the best for compression and torsion |
| I-beam, H-beam, wide flange (IPE, HEA, W) | Two flanges and a web | Beams and girders in one direction | The most efficient in bending about the strong axis; weak in torsion and sideways |
| Channel (UPN, C) | Open C shape | Frames, purlins, rails | Easy to bolt to; twists under an off-center load |
| Angle (L) | Two legs at 90° | Bracing, brackets, light frames, stiffeners | Unsymmetric: bends in a skew direction; load through the shear center |
| Tee (T) | Flange and stem | Stiffeners, light beams | Cut from an I-beam |
| Flat bar, plate | Rectangle | Gussets, brackets, base plates | Stiff only on edge; buckles easily |
| Aluminum T-slot extrusion | Square with slots | Machine frames, workstations, 3D printers | Easy assembly; lower stiffness than steel (E = 70 GPa) and slot-nut joints are flexible |
| Z-section, cold-formed sections | Thin-walled shapes | Light roofs and walls | Light and cheap; local buckling governs |
Sizes and properties
Properties computed from the formulas for common steel hollow sections (mass at 7.85 g/cm³; the real standard sections have rounded corners, so published values are a little different):
| Square tube B × B × t (mm) | Area (mm²) | I (cm⁴, ×10⁴ mm⁴) | S (mm³) | Mass (kg/m) |
|---|---|---|---|---|
| 20 × 20 × 2 | 144 | 0.8 | 787 | 1.13 |
| 30 × 30 × 2 | 224 | 2.9 | 1 961 | 1.76 |
| 40 × 40 × 3 | 444 | 10.2 | 5 099 | 3.49 |
| 50 × 50 × 3 | 564 | 20.8 | 8 340 | 4.43 |
| 60 × 60 × 4 | 896 | 47.1 | 15 690 | 7.03 |
| 80 × 80 × 5 | 1 500 | 141.3 | 35 313 | 11.78 |
| Round tube D × t (mm) | Area (mm²) | I (mm⁴) | S (mm³) | Mass (kg/m) |
|---|---|---|---|---|
| 25.0 × 2.0 | 145 | 9 628 | 770 | 1.13 |
| 33.7 × 3.2 | 307 | 36 047 | 2 139 | 2.41 |
| 48.3 × 3.2 | 453 | 115 857 | 4 797 | 3.56 |
| 60.3 × 3.6 | 641 | 258 737 | 8 582 | 5.03 |
| 88.9 × 4.0 | 1 067 | 963 398 | 21 674 | 8.38 |
| Rolled steel section | Area (cm²) | I strong axis (cm⁴) | W strong axis (cm³) | Mass (kg/m) |
|---|---|---|---|---|
| IPE 100 | 10.3 | 171 | 34.2 | 8.1 |
| IPE 200 | 28.5 | 1 943 | 194 | 22.4 |
| IPE 300 | 53.8 | 8 356 | 557 | 42.2 |
| HEA 100 | 21.2 | 349 | 72.8 | 16.7 |
| UPN 100 (channel) | 13.5 | 206 | 41.2 | 10.6 |
| L 50 × 50 × 5 (angle) | 4.80 | 11.0 | 3.05 | 3.77 |
| L 100 × 100 × 10 (angle) | 19.2 | 177 | 24.7 | 15.1 |
The rolled-section values are from common European sections (EN 10365, typical catalog values; the weak-axis and torsion properties are much lower). Always take the final numbers from the supplier’s or the standard’s table for the exact section.
Selecting a profile
Worked example: a 2 m beam carrying 2 kN at mid-span
- The maximum moment is M = P L ÷ 4 = 2 000 × 2 000 ÷ 4 = 1 000 000 N·mm. With an allowable stress of 140 MPa (S235 steel with a factor of 1.7) the section modulus must be at least S = M ÷ σ = 1 000 000 ÷ 140 = 7 143 mm³.
- For a deflection limit of L ÷ 360 = 5.6 mm: I = P L³ ÷ (48 E δ) = 2 000 × 2 000³ ÷ (48 × 210 000 × 5.6) = 285 714 mm⁴ (28.6 cm⁴).
- Both are met by the 60.3 × 3.6 round tube: S = 8 582 mm³ and I = 258 737 mm⁴ (25.9 cm⁴), at 5.03 kg/m; a 40 × 40 × 3 square tube (I = 10.2 cm⁴) is near the limit and a 50 × 50 × 3 (I = 20.8 cm⁴) is comfortable. Choose the stiffness first: it usually governs for a long beam.
- Check the others: the self-weight, local bearing at the load, the connections, and the lateral stability for open sections.
Closed and open sections, torsion and local buckling
- Torsion: a closed tube resists twist enormously better than an open section of the same weight, because the shear flows around the closed loop. An open channel or angle twists easily and fails unless the load goes through its shear center. Use closed sections for loads that twist (see Torsion, Shear and Combined Loads).
- Local buckling: the thin walls of a tube or the flange of an I-beam can buckle in compression before the member as a whole fails. Keep the width-to-thickness ratio b ÷ t below the code limit (about 40 for a steel box wall in bending, 15 to 20 for the outstanding flange of an I-beam).
- Lateral-torsional buckling of a deep I-beam: the compression flange buckles sideways and twists the beam. Prevent it with bracing every few meters, or choose a wide-flange or closed section.
- Web crippling and bearing: a concentrated load on a thin web can crush it: use a bearing plate or a stiffener.
- Welding and connections: holes reduce the net area and cause stress concentrations (see Bolted and Welded Joints and Fatigue and Stress Concentration); welding distorts thin sections.
Steel, aluminum and other materials
| Material | E (GPa) | Density (kg/m³) | E ÷ density | Comments |
|---|---|---|---|---|
| Structural steel (S235, S355) | 210 | 7 850 | 26.8 | Cheap, stiff, weldable; the default |
| Aluminum 6061 and 6063 | 69 | 2 700 | 25.6 | One third the weight and stiffness: the same deflection needs a section about 1.45 times deeper for the same mass; corrosion-resistant, easy to extrude |
| Stainless steel (304) | 193 | 8 000 | 24.1 | Corrosion-resistant; costlier; strength similar to mild steel but higher on cold working |
| Wood (pine, along the grain) | 10 to 12 | 500 | 20 to 24 | Efficient for its cost; creep and moisture |
| Carbon-fiber composite tube | 70 to 130 (in the fiber direction) | 1 500 | 50 to 90 | Very stiff for the weight; costly, directional and sensitive to impact |
| Plastics (nylon, ABS) | 2 to 3 | 1 100 | 2 | Use ribs and large sections; creep |
For the same weight, the specific stiffness E ÷ ρ of steel, aluminum and titanium is almost identical, so the lighter metal wins only by using a bigger section (an aluminum tube of larger diameter and thinner wall): the advantage shows up when the stiffness is geometric. See Material Properties for more data.
Related
- Structures and Strength of Materials: the overview of this guide.
- Beam Bending and Deflection and Columns and Buckling: using I and S for beams and columns.
- Torsion, Shear and Combined Loads: torsion of closed and open sections.
- Material Properties: the properties of steel, aluminum and plastics.
- CNC Machining and 3D Printing for making the parts.
FAQ
What is the difference between I and S?
I (second moment of area) governs deflection; S = I ÷ c (section modulus) governs bending stress. A deeper section increases both, but the stress depends on the distance c to the outer fiber.
Why do I-beams have a different I about the two axes?
The strong axis has material far from the neutral axis in the flanges, giving a high I; the weak axis (across the flanges) has the flanges near the axis and so a low I. Orient the beam so that the load acts in the strong direction and brace it against sideways movement.
Why is a tube better than a bar?
It puts the material where it carries most of the stress, away from the neutral axis, so it is much stiffer and stronger for the same weight, and it resists torsion in every direction.
What does the section modulus mean for choosing a profile?
Divide the maximum bending moment by the allowable stress to find the smallest S that is acceptable, then pick a section from the table with at least that S; check the deflection with its I.
References
- Beer, F. P., Johnston, E. R. and DeWolf, J. T., Mechanics of Materials, McGraw-Hill: centroids and moments of area.
- EN 10365 (hot-rolled steel sections), EN 10219 and EN 10210 (hollow sections); AISC Steel Construction Manual (US sections).
- The Aluminum Association, Aluminum Design Manual; Machinery’s Handbook (section properties).
Last reviewed: October 2026. Section properties are computed from the nominal dimensions; use the supplier’s tables for exact values and the applicable code for structural use.