The section modulus calculator above returns the geometric properties of a cross section: the second moment of area I, the elastic section modulus S, the plastic modulus Z, the cross-sectional area and the radius of gyration. It covers solid rectangles, solid circles, hollow circular and rectangular tubes, and symmetric I-sections, all about the strong horizontal axis.
One thing has to be said before any of the numbers mean anything, and Arb Digital publishes it prominently rather than in a footnote. Section modulus is a property of shape alone. It knows nothing about the material, so it is not a strength, not a capacity, and not a load rating. A steel section and a cardboard section of identical dimensions have identical section moduli and wildly different abilities to carry anything. This page is a preliminary and teaching tool. It is not a structural design, it is not stamped, and it does not replace a licensed engineer.
What This Section Modulus Calculator Does
It integrates the geometry. The second moment of area I measures how the material is distributed about the bending axis: the further from the axis, the more it contributes, and the contribution goes as the square of the distance. The elastic section modulus S is I divided by the distance from the neutral axis to the extreme fibre, and it is the quantity that converts a bending moment into an extreme-fibre stress. The plastic modulus Z is the corresponding quantity for a fully yielded section, used in limit-state design.
The radius of gyration r is √(I ÷ A), and it is the property that governs buckling rather than bending — a slender column with a small radius of gyration buckles at a load far below its material strength. Area A is included because it is what governs axial and shear demand, and because it is the honest measure of how much material a section uses.
All results are about the horizontal centroidal axis of the shape as drawn. For a rectangle standing on its narrow edge that is the strong axis; lay the same rectangle flat and the properties change dramatically, which is why joists are installed on edge. To get the weak-axis properties, swap b and d.
How to Use It
- Pick a shape and unit system. Section properties carry units to the third and fourth power, so a unit mix-up is not a small error — one inch to the fourth is 41,623 millimetres to the fourth.
- Enter actual dimensions, not nominal ones. A nominal two by six is 1.5 by 5.5 inches, and using 2 by 6 overstates S by about 45 percent.
- For hollow sections, enter the outside dimension and the wall thickness. The tool derives the inside dimensions and subtracts the void.
- For an I-section, b is the flange width and d is the overall depth. Flange and web thicknesses complete the shape. Root fillets are ignored.
- Read the geometry, then go elsewhere for capacity. Converting S into an allowable moment needs a material design value and a code procedure.
The Formula / How It's Calculated
For a solid rectangle of width b and depth d, I = b d³ ÷ 12 and S = b d² ÷ 6, since the extreme fibre distance c is d/2. The plastic modulus is Z = b d² ÷ 4. For a solid circle of diameter D, I = π D⁴ ÷ 64, S = π D³ ÷ 32 and Z = D³ ÷ 6.
Hollow sections are the outer shape minus the inner void, because both share the same centroidal axis: I = (π ÷ 64)(D⁴ − d⁴) for a pipe, and I = (B D³ − b d³) ÷ 12 for a box. A symmetric I-section is treated as the full bounding rectangle minus the two side voids: I = (b d³ − (b − tw)(d − 2tf)³) ÷ 12.
Worked example with the loaded defaults. A 1.5 by 5.5 inch rectangle on edge gives I = 1.5 × 5.5³ ÷ 12 = 1.5 × 166.375 ÷ 12 = 20.797 in⁴. S = 20.797 ÷ 2.75 = 7.5625 in³, which matches b d² ÷ 6 = 1.5 × 30.25 ÷ 6 exactly. The area is 8.25 in², the radius of gyration is √(20.797 ÷ 8.25) = 1.588 in, and the plastic modulus is 1.5 × 30.25 ÷ 4 = 11.344 in³. The ratio Z ÷ S is exactly 1.5 for any rectangle, which is the shape factor.
Why Depth Beats Width Every Time
Section modulus goes with the square of depth and only the first power of width. Double the width of a rectangle and S doubles. Double the depth and S quadruples. That single fact explains most of structural shape design: it is why joists are deep and thin rather than square, why I-beams put material in flanges far from the neutral axis and only a thin web in between, and why turning a joist on its side is catastrophic rather than merely unhelpful.
Put numbers on it. A 1.5 by 5.5 rectangle on edge has S = 7.5625 in³. The same piece laid flat has S = 5.5 × 1.5² ÷ 6 = 2.0625 in³, less than a third. Nothing about the material changed; only the distribution of it relative to the bending axis did. The same reasoning explains why a hollow tube is so efficient: removing material from near the neutral axis, where it contributes least, costs very little stiffness and saves a great deal of weight.
The efficiency has a limit, and it is stability. Make the web too thin and it buckles locally before the section reaches its bending capacity; make the compression flange too slender and it buckles the same way. That is why steel design codes classify sections by the slenderness of their elements and reduce the usable capacity for slender ones. Geometry gives you S; whether the section can actually develop it is a separate check.
Elastic S and Plastic Z Are Different Questions
The elastic section modulus assumes stress varies linearly across the depth and the extreme fibre is the first to reach the limit. The plastic modulus assumes the whole section has yielded, with a uniform stress block above and below the neutral axis. Because Z is always larger than S, a section that has reached first yield still has reserve before a plastic hinge forms.
The ratio Z ÷ S is the shape factor, and it is a property of the shape alone: exactly 1.5 for a rectangle, about 1.7 for a solid circle, and typically around 1.1 to 1.2 for a wide-flange I-section, because an I-section already concentrates its material at the extremes. Which modulus a design uses depends on the material and the code. Timber design is normally carried out in the elastic range using S. Steel design in the limit-state method uses Z for compact sections and S where the section is not compact enough to reach a plastic hinge. Do not mix them.
Turning S Into Something Structural
The stress at the extreme fibre from a bending moment is σ = M ÷ S. Rearranged, the section modulus required to keep the stress within an allowable value is Srequired = M ÷ σallowable. That is genuinely useful for preliminary sizing, and it is also where people go wrong, because the allowable stress is not a number you can look up casually. It depends on the material, the grade, the load duration, the moisture condition, the temperature, the size of the member, whether it is laterally braced, and the adjustment factors the governing code applies.
This page publishes no allowable stresses and no design values, for exactly that reason. Timber design values and their adjustment factors are set out in the National Design Specification for Wood Construction, the ANSI-approved standard published by the American Wood Council. Steel section properties and the design procedures that use them are published by the American Institute of Steel Construction in the Steel Construction Manual. Local code governs which edition applies, and amendments differ by state, province and country.
Bending is also not the only check. A beam that satisfies bending can still fail in shear, can deflect beyond a serviceability limit such as L/360 or L/240, can buckle laterally if it is not braced, or can crush at its bearing points. Our beam deflection calculator and beam load calculator address the deflection and load side, and the column buckling calculator uses the radius of gyration this page produces.
Composite, Built-Up and Unsymmetric Sections
The shapes here are single-material and symmetric about the bending axis, which keeps the neutral axis at mid-depth. Two situations break that. An unsymmetric section such as a tee or a channel bent about its weak axis has a neutral axis that is not at mid-depth, so the extreme fibre distances differ top and bottom, and there are two different section moduli — the smaller one governs. A built-up section of two different materials, such as a flitch beam of timber and steel, needs the transformed section method, where one material's area is scaled by the ratio of the elastic moduli before the properties are computed.
Built-up timber beams have a further practical catch: three separate joists nailed together are not the same as a solid member of the combined width unless they are connected well enough to act compositely. If they slide relative to each other, each carries its own share and the assembly is much less stiff than the arithmetic suggests. How they must be fastened to act together is specified in the design standard. For plain area work without any of this, the cross-sectional area calculator is simpler, and the factor of safety calculator covers the margin side.
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Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Reading S as a capacity — it is geometry only, identical for steel and for balsa of the same dimensions.
- Using nominal timber sizes — a nominal two by six is 1.5 by 5.5 inches, and using the nominal figures overstates S by roughly 45 percent.
- Mixing units — properties are to the third and fourth power, so an inch-millimetre slip is an error of four or five orders of magnitude.
- Using the wrong axis — a member laid flat instead of on edge loses most of its section modulus, and the tool reports only the axis as drawn.
- Mixing elastic and plastic methods — S belongs to elastic design and Z to plastic design, and the code decides which one applies.
Related Free Tools From Arb Digital
Pair this with the beam deflection calculator for serviceability, the beam load calculator for reactions and moments, the column buckling calculator for slenderness, the wood beam span calculator for timber members and the cross-sectional area calculator for area alone. The full free online tools hub lists every calculator we publish.
Frequently Asked Questions
No. It is a purely geometric property of the cross section. A steel bar and a plastic bar of identical dimensions have identical section moduli. Strength requires a material design value applied through a code procedure.
The second moment of area I describes how material is distributed about the bending axis and governs stiffness and deflection. The elastic section modulus S is I divided by the distance to the extreme fibre, and it converts a bending moment into an extreme-fibre stress.
Using the actual size of 1.5 by 5.5 inches, S is 1.5 times 5.5 squared divided by 6, which is 7.5625 cubic inches. The second moment of area is 20.797 inches to the fourth.
Because section modulus varies with the square of depth and only linearly with width. Doubling the width doubles S, while doubling the depth quadruples it. That is why beams are deep and thin rather than square.
Z applies to limit-state design where the section is allowed to yield fully and form a plastic hinge, which is common in steel design for compact sections. Elastic design, including most timber design, uses S. The governing code decides.
The ratio of plastic to elastic modulus, Z divided by S. It is exactly 1.5 for a rectangle, around 1.7 for a solid circle, and typically 1.1 to 1.2 for a wide-flange I-section because that shape already concentrates material at the extremes.
Not directly. Those are unsymmetric about the bending axis, so the neutral axis is not at mid-depth and there are two different section moduli, with the smaller one governing. They need a centroid calculation first.
Not on its own. You would need an allowable or design stress for the material and grade, all applicable adjustment factors, plus separate checks on shear, deflection, lateral stability and bearing. That work belongs to a licensed engineer.
This tool computes geometric section properties for education and preliminary work only. It is not a structural design, it is not stamped, and member selection must be carried out by a licensed engineer in accordance with local code.
