Section Shape Comparison: Circle, Square, Rectangle, Tube and Box in Bending

How this calculator works

The bending stress is the moment divided by the section modulus. For a rectangle of width b and depth h the section modulus is b h squared over 6, and for a solid circle it is pi d cubed over 32.

With the same area the shape with more material far from the centre line has the larger section modulus and carries more moment. A solid square carries about 18 percent more than a solid circle. A deep rectangle, a tube or a box carries much more.

Formulas used

• Z_{rect} = \dfrac{b h^2}{6}
• Z_{circle} = \dfrac{\pi d^3}{32}
• I_{rect} = \dfrac{b h^3}{12}
• I_{circle} = \dfrac{\pi d^4}{64}
• \sigma = \dfrac{M}{Z}

Worked example

With the default numbers in the calculator above:

1. Solid circle: 17.63 cm³ section modulus, 283.59 MPa, diameter 56.42 mm, 100.00 % of the circle
2. Solid square: 20.83 cm³ section modulus, 240.00 MPa, side 50.00 mm, 118.16 % of the circle
3. Solid rectangle (on its deep side): 29.46 cm³ section modulus, 169.71 MPa, 35.36 wide by 70.71 mm deep, 167.11 % of the circle
4. Hollow circle (tube): 48.19 cm³ section modulus, 103.75 MPa, outside 94.03 mm, wall 9.40 mm, 273.33 % of the circle
5. Hollow square (box): 56.94 cm³ section modulus, 87.80 MPa, outside 83.33 mm, wall 8.33 mm, 322.98 % of the circle

Change the numbers in the calculator to see the full working, every step, for your own case.

A second worked example

A different case, with these numbers: Cross-section area (the same for every shape) 3600 mm²; Bending moment 12 kN·m; Rectangle: depth divided by width 3; Hollow shapes: wall thickness divided by outside size 0.08.

1. Solid circle: 30.47 cm³ section modulus, 393.88 MPa, diameter 67.70 mm, 100.00 % of the circle
2. Solid square: 36.00 cm³ section modulus, 333.33 MPa, side 60.00 mm, 118.16 % of the circle
3. Solid rectangle (on its deep side): 62.35 cm³ section modulus, 192.45 MPa, 34.64 wide by 103.92 mm deep, 204.67 % of the circle
4. Hollow circle (tube): 95.77 cm³ section modulus, 125.30 MPa, outside 124.78 mm, wall 9.98 mm, 314.35 % of the circle
5. Hollow square (box): 113.16 cm³ section modulus, 106.04 MPa, outside 110.58 mm, wall 8.85 mm, 371.44 % of the circle

Common questions

Related calculators

• Section Properties Calculator: Area, Second Moment and Section Modulus
• Beam Calculator: Reactions, Bending Moment, Stress and Deflection
• Plastic Section Modulus and Shape Factor Calculator

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Standard strength of materials texts (second moment of area and section modulus).

← All engineering calculators