Channel / C Section Properties Calculator

Enter the depth, flange width, web thickness and flange thickness of a parallel-flange channel and get its area, centroid, second moments, section moduli, radii of gyration, torsion constant and shear centre, drawn to scale.

Section dimensions

Enter the dimensions in the selected unit — the drawing and the results update as you type. The drawing uses centroidal coordinates: the bold gridlines are the x and y axes through the centroid C.

Length unit

h — Depth mm b — Flange width mm t_w — Web thickness mm t_f — Flange thickness mm

Section

Section with its dimensions
Centroid C, centroidal axes x and y
Principal axes 1 and 2, angle θ_p
Shear centre S

Results

Results in

Area & centroid

Cross-sectional areaA1,348mm²

Centroid from the left edgex̄16.872mm

Centroid from the bottom edgeȳ50mm

Second moments of area (centroidal axes)

About the x axisI_x2,070,100mm⁴

About the y axisI_y330,560mm⁴

Product of inertiaI_xy0mm⁴

Polar moment of areaI_x + I_yI_p2,400,700mm⁴

Principal axes

Maximum principal momentI₁2,070,100mm⁴

Minimum principal momentI₂330,560mm⁴

Angle from x to axis 1 (CCW)axis 1 = x, axis 2 = yθ_p0.00°

Section moduli

Elastic section modulus about xS_x41,402mm³

Elastic section modulus about y, left fibreS_y,left19,592mm³

Elastic section modulus about y, right fibreS_y,right9,978.5mm³

Plastic section modulus about xZ_x49,221mm³

Plastic section modulus about yZ_y17,934mm³

Radii of gyration

About the x axisr_x39.188mm

About the y axisr_y15.66mm

Distances to the extreme fibres

Centroid to top / bottom fibrec_y50mm

Centroid to left fibrec_left16.872mm

Centroid to right fibrec_right33.128mm

Torsion & shear centre

Torsion constantThin-walled open-section approximation, J ≈ Σ b·t³/3 (no fillets).J26,447mm⁴

Shear centre offset from the centroid, along xThin-walled theory: e = 3·b′²·t_f / (h′·t_w + 6·b′·t_f) from the web mid-line, with b′ and h′ measured between mid-lines. The shear centre lies outside the section, behind the web.e_x-32.993mm

Channels are the workhorse of purlins, rails, machine frames and anything that needs a flat back to bolt against. About the strong axis the channel behaves like an I beam with the flanges cut short: the centroid sits at mid-depth, and I_x is again a difference of rectangles. About the weak axis the centroid is close to the web, and the toe fibre is much farther away than the back of the web — hence two different section moduli S_y.

What sets the channel apart is its shear centre, which lies outside the section, behind the web, at a distance e = 3b′²t_f/(h′t_w + 6b′t_f) from the web mid-line in thin-walled theory. A load applied through the web — as most are — twists the channel as well as bending it; the drawing marks S so you can see how far off it is.

How the channel / c section properties are calculated

Area: A = 2·b·t_f + (h − 2·t_f)·t_w
Centroid from the back of the web: x̄ = [2·t_f·b²/2 + (h − 2·t_f)·t_w²/2] / A
Second moment about x: I_x = [b·h³ − (b − t_w)·(h − 2·t_f)³] / 12
Second moment about the centroidal y axis: I_y = 2·[t_f·b³/12 + b·t_f·(b/2 − x̄)²] + (h − 2·t_f)·t_w³/12 + (h − 2·t_f)·t_w·(x̄ − t_w/2)²
Shear centre from the web mid-line (thin-walled): e = 3·b′²·t_f / (h′·t_w + 6·b′·t_f), b′ = b − t_w/2, h′ = h − t_f
Torsion constant (thin-walled): J ≈ [2·b·t_f³ + (h − 2·t_f)·t_w³] / 3

Principal moments and axes follow from I_x, I_y and I_xy with I_1,2 = (I_x + I_y)/2 ± √[((I_x − I_y)/2)² + I_xy²] and tan 2θ_p = −2·I_xy/(I_x − I_y); the radii of gyration are r = √(I/A) and the elastic section moduli S = I/c for each extreme fibre. Definitions of every property are in the glossary on the section properties overview.

Assumptions. Parallel flanges of constant thickness, sharp corners, no root radius. Rolled UPN channels have tapered flanges and fillets; the results are within a few percent for I_x and closer for parallel-flange (UPE / PFC) sections. Torsion constant and shear centre use thin-walled theory.

Other sections

Frequently asked questions

Why is the shear centre of a channel outside the section?

When a channel bends about its strong axis, the shear flow in the two flanges runs in opposite directions and forms a couple. For the resultant of the shear stresses to be a single force with no twisting moment, that force must act behind the web, at the distance e — for a 100 × 50 × 6 × 8.5 channel about 16 mm behind the back face. Loads applied at the web therefore produce torsion in addition to bending.

Which S_y do I use for weak-axis bending?

S_y,left (at the back of the web) and S_y,right (at the flange toes) differ by a factor of about two for typical proportions, because the centroid sits near the web. The flange toes are the critical fibre for a given moment; use the smaller value unless you have checked the stress at both.

How do the results compare with UPN / UPE tables?

For a UPE (parallel flanges) the difference is the root and toe radii only — a few percent on A and I_x. For a UPN the tapered flanges (8% slope) change the flange mass distribution: entering the mean flange thickness gets I_x within about 5%, but I_y, J and the shear centre can differ more. Use catalogue values for the final check of a rolled section.

References & further reading

Shear centre — Wikipedia — the point a transverse load must pass through to bend a beam without twisting it.
Second moment of area — Wikipedia — definition, parallel-axis theorem and the sign convention for the product of inertia.
Torsion constant — Wikipedia — why J differs from the polar moment for non-circular sections, with the thin-walled formulas.

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