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Moments of Sections

A beam of uniform section, loaded in simple tension by a force F, carries a stress s = F/A where A is the area of the section. Its response is calculated from the appropriate constitutive equation. Here the important characteristic of the section is its area, A. For other modes of loading, higher moments of the area are involved. Those for various common sections are given below. They are defined as follows.

The second moment of area I measures the resistance of the section to bending about a horizontal axis (shown as a broken line). It is

Figure A1.1

where y is measured vertically and b(y) is the width of the section at y. The moment K measures the resistance of the section to twisting. It is equal to the polar moment J for circular sections, where

Figure A1.2

where r is measured radially from the centre of the circular section. For non-circular sections K is less than J. The section modulus Z = I/ym (where ym is the normal distance from the neutral axis of bending to the outer surface of the beam) determines the surface stress generated by a given bending moment, M:

Figure A1.3

Finally, the moment S, defined by

Figure A1.4

measures the resistance of the beam to fully-plastic bending. The fully plastic moment for a beam in bending is

Figure A1.5

where s_y is the yield strength.

Thin or slender shapes may buckle locally before they yield or fracture. It is this which sets a practical limit to the thinness of tube walls and webs.