Civil Engineering

Steel Section Properties Calculator

Compute the geometric properties of a steel section from its dimensions. Enter an I-section, channel, equal angle, square hollow section or circular hollow section and this calculator returns the cross-sectional area, the second moment of area about both axes, the elastic and plastic section moduli, both radii of gyration, the weight per metre, the IS 800 section classification and the design moment capacity.

IS 800:2007 Table 2 classification · Cl.8.2.1.2 moment capacity · γ_M0 = 1.10

📏 Section Dimensions

Inputs convert when you switch. Results are always reported in SI — see the note under the results.

Section Type

Overall Depth H (mm)

Flange Width B_f (mm)

Flange Thickness T_f (mm)

Web Thickness T_w (mm)

Yield Stress f_y (MPa)

📊 Section Properties

Press Calculate Properties to compute the section.

ƒ Governing Formulae

I-section and channel

h_w = H − 2T_f
A = 2B_fT_f + h_wT_w
I_xx = [B_fH³ − (B_f−T_w)h_w³] / 12
I_yy,I = 2(T_fB_f³/12) + h_wT_w³/12
Z_px = 2[B_fT_f(h_w/2 + T_f/2) + T_w(h_w/2)(h_w/4)]

For a channel, I_yy is taken about the true centroid, which sits away from the web because the section is singly symmetric.

Equal angle — principal axes

A = Lt + (L−t)t   ȳ = z̄ by symmetry
I_xy = Σ A_i(y_i − ȳ)(z_i − z̄)
I_max = I_xx + |I_xy|  (u-u)
I_min = I_xx − |I_xy|  (v-v, governs buckling)

Hollow sections

SHS A = B² − B_i², I = (B⁴ − B_i⁴)/12, Z_p = (B³ − B_i³)/4
CHS A = π(D² − D_i²)/4, I = π(D⁴ − D_i⁴)/64, Z_p = (D³ − D_i³)/6

Derived properties

Z = I / y_max r = √(I / A)
weight = A × 7850 / 10⁶ kg/m
ε = √(250 / f_y)
M_c = Z_px f_y / γ_M0, γ_M0 = 1.10

H, B_f — overall depth and flange width

T_f, T_w — flange and web thickness

h_w — clear web depth between flanges

I_xy — product of inertia about the centroidal leg axes

Z — elastic section modulus

Z_px — plastic section modulus about the major axis

ε — yield strength ratio used to scale the classification limits

M_c — design plastic moment capacity of the bare section

Root radii are not included. Every section here is built from rectangles with sharp corners, so the area comes out a little below the published value for a rolled section — typically by half a percent to two percent, and the inertia by a similar margin. For an ISMB 300 this calculation gives 5536 mm² against a published 5626 mm². Use published table values for a final design and this calculator for built-up plate girders, non-standard sections and quick comparisons.

M_c is the bare plastic capacity, not a design answer. It assumes the compression flange is fully restrained. There is no check on lateral torsional buckling, which usually governs an unrestrained beam and can reduce capacity by half or more, and no reduction for high coincident shear. For a Class 4 slender section the plastic modulus is not attainable at all and the value shown is unconservative — IS 800 requires an effective section for those.

Classification is computed for I-sections only. Channels, angles and hollow sections report "N/A" and are not classified here, so a slender SHS or CHS will not be flagged. Check those against IS 800 Table 2 yourself.

📝 Worked Example 1 — ISMB 300 I-Section

Given

An ISMB 300 to nominal dimensions: overall depth 300 mm, flange width 140 mm, flange thickness 12.4 mm, web thickness 7.5 mm, in E250 steel with f_y = 250 MPa.

Step 1 — area

h_w = 300 − 2 × 12.4 = 275.2 mm
A = 2 × 140 × 12.4 + 275.2 × 7.5 = 3472 + 2064 = 5536 mm²

Step 2 — second moments of area

I_xx = [140 × 300³ − (140 − 7.5) × 275.2³] / 12 = 84.866 × 10⁶ mm⁴
I_yy = 2(12.4 × 140³/12) + 275.2 × 7.5³/12 = 5.681 × 10⁶ mm⁴
The minor axis carries only 6.7 percent of the major-axis stiffness, which is why an I-section must be restrained laterally.

Step 3 — moduli and radii of gyration

Z_xx = I_xx / 150 = 565.78 × 10³ mm³
Z_yy = I_yy / 70 = 81.15 × 10³ mm³
r_x = √(84.866 × 10⁶ / 5536) = 123.81 mm
r_y = √(5.681 × 10⁶ / 5536) = 32.03 mm

Step 4 — plastic modulus and shape factor

Z_px = 2[140 × 12.4 × (137.6 + 6.2) + 7.5 × 137.6 × 68.8] = 641.277 × 10³ mm³
Shape factor = 641.277 / 565.78 = 1.13, the usual value for a rolled I-section.

Step 5 — classification and capacity

ε = √(250/250) = 1.0
Flange outstand ratio B_f/2T_f = 140 / 24.8 = 5.65, against the Class 1 limit of 9.4ε
Web ratio h_w/T_w = 275.2 / 7.5 = 36.7, against the Class 1 limit of 84ε
Both are comfortably inside, so the section is Class 1 — Plastic.
M_c = 641.277 × 10³ × 250 / (1.10 × 10⁶) = 145.74 kN·m
Weight = 5536 × 7850 / 10⁶ = 43.46 kg/m

Area 5536 mm² · Weight 43.46 kg/m

I_xx 84.866 × 10⁶ mm⁴ · I_yy 5.681 × 10⁶ mm⁴

Z_xx 565.78 × 10³ mm³ · Z_px 641.277 × 10³ mm³

r_x 123.81 mm · r_y 32.03 mm

Classification Class 1 — Plastic · M_c 145.74 kN·m

These are the calculator's default inputs. Press Calculate Properties without changing anything and you should get exactly these figures back. Published tables give A = 5626 mm² and I_xx = 8603.6 cm⁴ for ISMB 300; the small difference is the root fillet radius, which this calculation omits.

🔄 Worked Example 2 — Why an Angle Buckles Diagonally

Given

An ISA 100 × 100 × 10 equal angle used as a strut. Leg length 100 mm, thickness 10 mm.

Step 1 — area and centroid

A = 100 × 10 + 90 × 10 = 1900 mm²
The centroid sits 28.68 mm from the back of each leg, published as 28.4 mm.

Step 2 — inertia about the legs

About the horizontal leg axis, I_xx = 1.800 × 10⁶ mm⁴, and by symmetry I_yy is identical. If you stopped here you would conclude the angle is equally stiff in both directions, and both radii of gyration come to 30.8 mm.

Step 3 — and then the product of inertia

An equal angle is symmetric about its diagonal, not about its legs, so I_xy is not zero:
I_xy = −1.066 × 10⁶ mm⁴
I_max = 1.800 + 1.066 = 2.866 × 10⁶ mm⁴ about u-u
I_min = 1.800 − 1.066 = 0.734 × 10⁶ mm⁴ about v-v
Published values are 2.80 and 0.729 × 10⁶ mm⁴.

r about the legs 30.8 mm — the number that looks right and is not

r_max (u-u) 38.84 mm, published 38.4

r_min (v-v) 19.66 mm, published 19.4 — this is the one that governs

Using 30.8 instead of 19.66 overstates the radius of gyration by 57 percent

What that costs you

Buckling capacity depends on slenderness KL/r, and the reduction factor falls away roughly with the square of it in the elastic range. A 2.0 m strut of this angle has KL/r = 65 on the leg axis but 102 on the true weak axis. Feed the wrong figure into a column check and you will believe the strut has a comfortable margin when it is close to its limit. This is the single most common error made with angle struts, and it is why every published angle table lists r_vv separately.

Switch the section type to Equal angle and press Calculate to reproduce these figures. The results panel adds a note explaining that I_xx and I_yy are principal values for this section type, because the axis convention differs from the other four.

📐 Inputs, Units and Accepted Ranges

Input	Applies to	SI unit	Imperial unit	Accepted
Overall depth H	I-section, channel	mm	in	> 2T_f
Flange width B_f	I-section, channel	mm	in	> T_w
Flange thickness T_f	I-section, channel	mm	in	> 0
Web thickness T_w	I-section, channel	mm	in	> 0, < B_f
Leg length L	Equal angle	mm	in	> t
Leg thickness t	Equal angle	mm	in	> 0, < L
Outer size B	SHS	mm	in	> 2t
Outer diameter D	CHS	mm	in	> 2t
Wall thickness t	SHS, CHS	mm	in	> 0, < half the outer size
Yield stress f_y	All	MPa	MPa	250 – 410

📚 Reference Tables

IS 800 Table 2 — classification limits, scaled by ε = √(250/f_y)

Class	Flange outstand B_f/2T_f	Web h_w/T_w	Behaviour
1 — Plastic	≤ 9.4ε	≤ 84ε	Forms a plastic hinge and rotates
2 — Compact	≤ 10.5ε	≤ 105ε	Reaches M_p, limited rotation
3 — Semi-compact	≤ 15.7ε	≤ 126ε	Reaches first yield only
4 — Slender	> 15.7ε	> 126ε	Buckles locally before yield

ε and the resulting limits by steel grade

Grade	f_y, MPa	ε	Class 1 flange limit	Class 1 web limit
Fe410 / E250	250	1.000	9.40	84.0
E300	300	0.913	8.58	76.7
Fe490 / E350	345	0.851	8.00	71.5
Fe540 / E410	410	0.781	7.34	65.6

Higher grade steel tightens the classification limits, so the same geometry can be Class 1 in E250 and Class 2 or 3 in E410. Strength gained at the material is partly given back at the section.

Shape factor Z_p/Z — the reserve between first yield and full plasticity

Section	Shape factor	Comment
I-section, major axis	1.13 – 1.15	Material already at the extreme fibres
I-section, minor axis	≈ 1.5	Behaves like a pair of rectangles
Channel, major axis	≈ 1.17	Similar to an I-section
SHS	≈ 1.19	Depends on wall slenderness
CHS	≈ 1.33	Thin wall approaches 1.27
Solid rectangle	1.50	Exactly bd²/4 over bd²/6
Solid circle	1.70	The highest of the common shapes

Equal angle principal properties — computed here against published ISA values

ISA size	A, mm²	r_vv computed, mm	r_vv published, mm	Difference
50 × 50 × 6	564	9.79	9.7	+0.9 %
65 × 65 × 6	744	12.80	12.7	+0.8 %
75 × 75 × 8	1136	14.72	14.5	+1.5 %
90 × 90 × 10	1700	17.65	17.5	+0.9 %
100 × 100 × 10	1900	19.66	19.4	+1.3 %
130 × 130 × 12	2976	25.61	25.5	+0.4 %

The consistent small overestimate is the root radius, which rounds the heel and slightly reduces the true radius of gyration. Within about 1.5 percent across the range.

Weight per metre from area — steel at 7850 kg/m³

Area, mm²	Weight, kg/m	Roughly equivalent to
1000	7.85	ISA 75 × 75 × 8
2000	15.70	ISA 100 × 100 × 10
4000	31.40	ISMC 250 / CHS 168 × 8
5500	43.18	ISMB 300
9000	70.65	ISMB 450
13000	102.05	ISMB 550

❓ Frequently Asked Questions

What is the difference between elastic and plastic section modulus?

Elastic section modulus is the second moment of area divided by the distance to the extreme fibre, and it gives the moment at which the outermost fibre first reaches yield. Plastic section modulus is the first moment of area of the two halves of the section about the plastic neutral axis, and it gives the moment at which the whole section has yielded. The ratio between them is the shape factor, about 1.15 for a rolled I-section bent about its major axis and 1.5 for a solid rectangle.

How do you classify a steel section as per IS 800?

Classification compares the width to thickness ratio of each compression element against limits that scale with the square root of 250 divided by the yield stress. A Class 1 plastic section can form a plastic hinge and rotate; Class 2 compact can reach the plastic moment but not rotate; Class 3 semi-compact can reach first yield only; and Class 4 slender buckles locally before yield and needs an effective section. This calculator classifies I-sections from the flange outstand and web depth ratios.

Why does an angle buckle about an axis that is not horizontal or vertical?

An equal angle is symmetric about its diagonal, not about its horizontal and vertical axes, so its principal axes are rotated forty five degrees from the leg directions. The minimum second moment of area occurs about the axis running along the diagonal through the heel, conventionally called the v-v axis, and that is the axis a strut will buckle about. For an ISA 100 by 100 by 10 the radius of gyration about the legs is 30.5 millimetres but about the weak principal axis only 19.4, so using the wrong one overestimates buckling capacity substantially.

How do you calculate the weight per metre of a steel section?

Multiply the cross-sectional area in square millimetres by the density of steel, 7850 kilograms per cubic metre, and divide by one million to get kilograms per metre. A section of 5536 square millimetres therefore weighs 43.5 kilograms per metre. Published tables for rolled sections give slightly higher figures than a calculation from nominal dimensions because they include the root fillet radii, which this calculator ignores.

What is the shape factor of a steel section?

Shape factor is the ratio of plastic to elastic section modulus, and it measures how much extra moment a section carries between first yield and full plasticity. A rolled I-section bent about its major axis has a shape factor of about 1.13 to 1.15 because most of its material already sits near the extreme fibres. A solid rectangle reaches 1.5 and a solid circle 1.7, since material near the neutral axis contributes little elastically but fully once plastic.

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Steel Section Properties Calculator — multicalci.com. Properties are computed from nominal dimensions with sharp corners; root fillet radii are not included, so results run slightly below published table values. Moment capacity assumes full lateral restraint and no coincident shear. Classification is provided for I-sections only. Results are indicative and must be verified against IS 808 tables and IS 800 for design.