What is cross section plan?

What cross section has the greatest second moment of area per cross sectional area?

• I know its probably something like an I beam, but different I beams have different second moment of areas per cross sectional area, so has it been established what the ratio of side lengths are of the cross section with the greatest second moment of area per cross sectional area?

• Answer:

  The second moment of area is i = integral(r^2)da where da is an element of area at distance r from the reference axis. This axis will usually be the axis of symmetry of the member under consideration - the neutral axis of an I-beam for example. i will be greatest when r is greatest. This means that the shape having the greatest second moment of area per cross sectional area will be one where all of the cross-sectional area of the member is concentrated at a large radius. If the cross-section is required to be symmetrical, such a shape could be a pair of areas one on each side of the axis at equal large distances from the axis. In such an arrangement i increases with the square of the radius without limit. To hold these elements in position a radial web is required, and this is the explanation of the shape of an I-beam. Alternatively, the elements of area could be distributed around a circle of large radius; this would lead to a large-diameter tubular member. The tubular arrangement is superior in that no web is required (i.e. none of the cross-sectional area is at small radius), and it would have equal stiffness in all directions. But it would be inferior if only forces in one direction (and its opposite) are to be resisted; in this case a bilaterally-symmetric arrangement (like an I-beam) would be better. These considerations remain valid whatever the total area; the rule is "always contrive to get as much as possible of the cross sectional area at the greatest radius". In practice other considerations will need to be taken into consideration - such as the ability of the member to resist twisting or buckling. The requirement that the beam should not buckle under load explains the considerable thickness of the web of practical I-beams; to maximise second moment of area per cross sectional area the web should be very thin, but this would lead to an impractical design. So maximising this parameter is not actually a practical or desirable requirement.