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Section properties of an I-section, from its dimensions

Area, second moment of area about both axes, elastic and plastic section modulus, radius of gyration and shape factor of a doubly symmetric I-section built from flanges and a web. The same lines work for any plate girder or welded section.

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Section properties of an I-section calculated in Equation Pad.

The calculation

b := 150 mm
t_f := 10.7 mm
h := 300 mm
t_w := 7.1 mm
h_w := h - 2*t_f = mm
A_s := 2*b*t_f + h_w*t_w = mm^2
I_y := (b*h^3 - (b - t_w)*h_w^3)/12 = cm^4
I_z := (2*t_f*b^3 + h_w*t_w^3)/12 = cm^4
W_el := I_y/(h/2) = cm^3
W_pl := b*t_f*(h - t_f) + t_w*h_w^2/4 = cm^3
i_y := sqrt(I_y/A_s) = mm
i_z := sqrt(I_z/A_s) = mm
shape := W_pl/W_el =

Results for these inputs

  • Area 5188 mm².
  • Iy = 7999 cm⁴ and Iz = 602.7 cm⁴.
  • Elastic section modulus 533.3 cm³, plastic section modulus 602.1 cm³, shape factor 1.13.
  • Radii of gyration 124.2 mm and 34.1 mm.

How it works

The second moment of area is the outer rectangle minus the two voids beside the web, bh³/12 − (b − tw)hw³/12. The plastic modulus adds the first moments of the flanges and the web about the centre line. Root radii are left out, so values are slightly lower than for a rolled section.

These calculations use classical textbook mechanics, not a design code. For design, use the method, factors and checks that apply to your work and have the result checked by a competent person.

Other shapes

For T-sections, angles and built-up sections, find the centroid with y_c := (A_1*y_1 + A_2*y_2)/(A_1 + A_2) and use the parallel axis rule I := I_1 + A_1*(y_1 - y_c)^2 + …. Keep a table of plates in a spreadsheet and sum it in the maths.

Questions

What is the difference between elastic and plastic section modulus?

The elastic modulus W_el = I/(h/2) gives the moment at first yield; the plastic modulus W_pl is the first moment of area of both halves and gives the fully plastic moment. Their ratio is the shape factor.

Can I calculate the properties of any section?

Yes: write the area and second moment of each part, the centroid, and add them with the parallel axis rule. Units are handled for you.

Also used for

Equation Pad runs in your browser, with nothing to install, and we never see or store your calculations. Results must be checked by a competent person before they are relied on; see the disclaimer.

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