How to use this calculator
- Choose the cross-section. Pick a solid rectangle or round bar, a round or box tube, or an I-beam, channel, tee or angle. The relevant dimension fields appear.
- Enter the dimensions. Enter the width and height (or diameter, plus wall thickness for tubes, or depth, flange and web thicknesses for the rolled sections). Height is the depth in the plane of bending.
- Read I, S, A and c. Read the area moment of inertia I, section modulus S = I/c, cross-section area A and the neutral-axis distance c.
- Check bending stress. Use σ = M / S to turn a bending moment into peak stress, or M = σ·S for the moment capacity.
How it works
The area moment of inertia I (the second moment of
area) describes how a cross-section resists bending: the further the material sits
from the bending axis, the more it contributes. For a solid rectangle of width
b and depth h,
I = b·h³ / 12
and the distance from the neutral axis to the outer fibre is c = h / 2.
The section modulus is the moment of inertia divided by that fibre
distance:
S = I / c
It is the property that ties straight into bending stress, σ = M / S:
a larger section modulus carries a larger bending moment for the same stress. For
the rectangle, S = b·h² / 6.
Because I grows with the cube of the depth h,
depth is by far the most effective dimension — doubling h multiplies
I by eight. All values here are taken about the horizontal centroidal
(neutral) axis.
The same idea extends to built-up shapes. For an I-beam or
channel the web-height void is subtracted from the outer
bf · d rectangle; for a tee or angle
the centroid no longer sits at mid-depth, so the calculator locates it and sums each
part’s own inertia plus A·d² — the parallel-axis theorem — then takes
c to the farthest fibre, giving the governing section modulus.
Worked example
Verified against the live calculator
A solid 50 × 100 mm rectangle (50 wide, 100 deep) has
I = 50 × 100³ / 12 = 4,166,667 mm⁴,
c = 100 / 2 = 50 mm, and so
S = 4,166,667 / 50 = 83,333 mm³ (equivalently
50 × 100² / 6). The area is 5,000 mm². A solid
Ø50 mm round bar, by contrast, has
I = π × 50⁴ / 64 ≈ 306,796 mm⁴ and
S = π × 50³ / 32 ≈ 12,272 mm³ — far less, because its
depth is only 50 mm. Those are the numbers the calculator shows for these inputs.
Frequently asked questions
What is the section modulus and how is it calculated?
The section modulus S measures a cross-section’s resistance to bending. It is the area moment of inertia divided by the distance from the neutral axis to the outermost fibre: S = I / c. For a solid rectangle b wide and h deep, I = b·h³/12 and c = h/2, so S = b·h²/6. A 50 × 100 mm rectangle has S = 50·100²/6 ≈ 83,333 mm³.
What is the difference between section modulus S and moment of inertia I?
The moment of inertia I (also called the second moment of area) describes how the area is distributed about the bending axis and governs deflection: more I means less deflection. The section modulus S = I / c relates directly to stress: bending stress is σ = M / S, so a larger S carries more bending moment before the material yields. I is in mm⁴ (or in⁴); S is in mm³ (or in³).
Why does depth matter more than width for a beam?
For a rectangle I = b·h³/12, so the moment of inertia grows with the cube of the depth h but only linearly with the width b. Doubling the depth multiplies I by eight and S by four, while doubling the width only doubles each. That is why beams and joists are taller than they are wide and why you stand a board on edge.
How do I use section modulus to check bending stress?
Bending stress is σ = M / S, where M is the bending moment. Rearranged, the moment the section can carry at a given allowable stress is M = σ·S. So compute S here, then divide your applied moment by S to get the peak stress, or multiply S by the material’s allowable stress to get the moment capacity.
What axis is the section modulus calculated about?
About the horizontal centroidal (neutral) axis — bending in the vertical plane, the most common case. For the doubly-symmetric sections (rectangle, round, the tubes, the I-beam, and a channel about its strong axis) the neutral axis sits at mid-depth, so c is half the depth and S is the same on the tension and compression sides. A tee or angle is singly symmetric: the calculator locates the centroid, applies the parallel-axis theorem, and reports c to the farther fibre — the governing (smaller) section modulus.
Does this work for tubes and in imperial units?
Yes. For a round tube the bore is hollow, so I = π(OD⁴ − ID⁴)/64 with ID = OD − 2t; for a rectangular (box) tube the inner void is subtracted. Toggle SI / Imperial in the header to enter dimensions in mm or inches and read I in cm⁴ / in⁴ and S in cm³ / in³.
Can it calculate the section modulus of an I-beam, channel, tee or angle?
Yes. Choose I-beam / wide flange, channel, tee or angle and enter the depth, flange or leg widths, and thicknesses. The I-beam and channel use I = [bf·d³ − (bf − tw)·(d − 2·tf)³]/12 about the strong (horizontal) axis; the tee and angle are singly symmetric, so the tool finds the centroid, applies the parallel-axis theorem, and takes c to the extreme fibre. Values are sharp-corner idealisations — a rolled section’s root fillets add a few percent to I.
Method & assumptions
- Bending is about the horizontal centroidal (neutral) axis, in the vertical plane.
- The rectangle, round, tubes, I-beam and channel are symmetric about the bending axis — the neutral axis is at mid-depth and
cis half the depth, so the section modulus is equal top and bottom. The tee and angle are not: the centroid is located andcis taken to the extreme fibre, soS = I/cis the governing (smaller) value. - Tube, I-beam and channel inner voids use
outer − 2t(ord − 2·tf), clamped at zero — a wall or flange thicker than the half-size is treated as solid. - For an angle, values are about the horizontal geometric axis; an angle’s principal axes are rotated, so it also bends sideways unless restrained against twisting.
- For a built-up wood beam where span, LVL plies, line loads and deflection limits matter, use the LVL beam calculator instead of only calculating
IandS. - These are pure geometric section properties with sharp corners — no root fillets (so rolled I-beams, channels and angles read a few percent low), stress concentrations, or material strength. Combine
Swith the allowable stress viaσ = M / S.