What is section modulus?

Section modulus is the single property of a beam's cross-section that sets how much bending stress it develops under load. It is written S (sometimes W or Zel in European texts) and defined as

S = I / c

where I is the area moment of inertia about the centroidal (neutral) axis and c is the distance from that neutral axis to the extreme fibre — the outermost surface of the section, where the bending stress is highest. Section modulus has units of length cubed (mm³ or in³). The bigger S is, the lower the stress for a given bending moment, and the stronger the beam.

Why section modulus sets the stress

When a beam bends, one face stretches and the opposite face compresses. The stress varies linearly across the depth, zero at the neutral axis and largest at the extreme fibre a distance c away. The peak bending stress is simply the applied bending moment divided by the section modulus:

σ = M / S

That is the whole reason the property exists. Once you know the bending moment M at a point along the beam, dividing by S gives the maximum stress there directly — no need to carry I and c separately. Keep that stress below the material's allowable value (yield strength divided by your factor of safety) and the section is strong enough. A larger S means the same moment produces less stress, which is exactly what "a stronger beam" means in bending.

Section modulus vs moment of inertia

This is the distinction that trips people up most, so it is worth stating plainly: I and S answer two different questions.

• Moment of inertia I (mm⁴) governs deflection — stiffness. Beam deflection is inversely proportional to I: for a given load and span, δ ∝ 1/I. A section with a large I resists bending out of shape. This is why you check I when a part feels too bouncy or sags too far, even though it is nowhere near breaking.
• Section modulus S (mm³) governs stress — strength. Peak stress is M/S, so S resists overstressing the material. You check S when you care whether the part will yield or fracture.

They are linked through S = I / c, but they are not interchangeable. A deep, thin section can be very stiff (high I) yet have a perfectly ordinary S; conversely two sections can share the same S and still deflect quite differently. In a real design you typically size for stress using S, then verify that deflection stays acceptable using I. The section modulus calculator reports both I and S for the same cross-section so you can do exactly that.

Formulas for common sections

For simple shapes the area moment of inertia and section modulus have closed-form expressions. The depth measured in the plane of bending dominates every one of them.

Solid rectangle (width b, depth h)

I = b · h³ / 12   S = b · h² / 6   c = h / 2

Solid round bar (diameter d)

I = π · d⁴ / 64   S = π · d³ / 32   c = d / 2

Hollow round tube (outer D, inner d)

I = (D⁴ − d⁴) · π / 64   S = I / (D / 2)

Notice how the depth term carries the load. In the rectangle, depth appears as h² in S and h³ in I; in the round bar it is d³ and d⁴. Doubling the depth multiplies the section modulus by four and the moment of inertia by eight, while doubling the width only doubles both. The practical takeaway: orient the deep dimension in the plane of bending. A joist stood on edge is dramatically stronger and stiffer than the same joist laid flat — same material, same weight, just rotated.

Worked example — a 50 × 100 mm rectangle

Take a solid rectangular bar 50 mm wide and 100 mm deep (b = 50, h = 100), oriented with the 100 mm dimension vertical so it bends about the strong axis. First the moment of inertia:

I = 50 · 100³ / 12 = 4.17 × 10⁶ mm⁴

Then the section modulus, with c = h/2 = 50 mm:

S = 50 · 100² / 6 = 83 333 mm³

Now apply a bending moment of M = 5 kN·m = 5 × 10⁶ N·mm. The peak bending stress is:

σ = M / S = 5 × 10⁶ / 83 333 = 60 MPa

So this section carries the load at 60 MPa — comfortably below the yield strength of structural steel (around 250 MPa), giving a factor of safety of roughly four against yield. Lay the same bar flat (100 mm wide, 50 mm deep) and S drops to 100 · 50² / 6 ≈ 41 667 mm³, exactly half, so the stress doubles to about 120 MPa for the same moment. Same bar, same steel — orientation alone halved the strength.

Elastic vs plastic section modulus

The S defined above is the elastic section modulus: it assumes stress varies linearly across the depth and the extreme fibre is the first point to reach yield. It is the right property for everyday allowable-stress design, where you keep the whole section safely below yield. There is also a plastic section modulus, usually written Z, which assumes the entire cross-section has yielded and reached a uniform stress. Z is larger than S and is used in limit-state, plastic-design and ultimate-capacity checks (the ratio Z/S is the shape factor — about 1.5 for a solid rectangle). Unless a code explicitly calls for plastic design, the elastic S = I / c is what you want, and it is what this calculator returns.

Common mistakes

• Confusing I with S. Plugging the moment of inertia (mm⁴) into σ = M/S gives a nonsense stress that is far too low and has the wrong units. Stress uses the section modulus; deflection uses the moment of inertia.
• Using the wrong axis. A rectangle has a different I and S about each principal axis. Always compute about the axis the bending moment acts on — the neutral axis is perpendicular to the load, through the centroid.
• Orienting a joist flat. Because depth dominates, laying a beam on its side throws away most of its capacity. Keep the deep dimension in the plane of bending unless you have a specific reason not to.

Once you have S and the resulting stress, the deflection side of the problem belongs to I. For a round shaft used as a beam, the shaft deflection calculator takes the same geometry through to a deflection figure, and for hollow box sections the square tube deflection calculator does the same for rectangular HSS. Use them alongside the section modulus calculator to size for both strength and stiffness in one pass.

Frequently asked questions

What is the difference between section modulus and moment of inertia?

They do different jobs. The area moment of inertia I (units mm⁴) governs stiffness — how much a beam deflects, since δ ∝ 1/I. The section modulus S = I / c (units mm³) governs strength — the bending stress, since σ = M / S. A beam can be stiff but weak, or strong but flexible, so you usually check both.

What is the section modulus formula for a rectangle?

For a solid rectangle of width b and depth h bending about the horizontal axis, I = b·h³/12 and the elastic section modulus is S = b·h²/6, with c = h/2. The depth h is squared in S and cubed in I, so the deep dimension should run in the plane of bending.

Is this the elastic or the plastic section modulus?

The calculator returns the elastic section modulus S = I / c, used for normal design where the material stays below yield. The plastic section modulus Z is a separate, larger value used in limit-state and ultimate-strength checks once the whole cross-section has yielded.

Last reviewed: 2026-05-29.