Comparing layouts before sizing

The measure

For a structure working in axial load, the area a member needs is roughly proportional to the force it carries. Its volume is area times length, so it is proportional to force times length. Sum that over the structure:

Structural volume = Σ |F|·L

A layout with a smaller sum needs less material. No section sizes are involved, so it can be computed the moment an analysis runs — and it can compare layouts that would need completely different sections.

The invariant

The measure would be merely convenient if that were all. What makes it a theorem is this: split the sum into tension and compression parts, and take the difference:

Σ F·L over tension members − Σ |F|·L over compression members = constant

For a given set of loads at given positions and supports at given positions, that difference does not change however you rearrange the structure between them. It equals the work the external forces — applied loads and reactions — do on their own positions.

So two layouts with different structural volumes must have the same difference. A tool that shows both makes the theorem checkable rather than quotable, and if the difference is not constant across your comparison, the comparison is not fair — something about the loads or the supports has changed.

Making the comparison fair

The invariant needs the same loads at the same points and the same reactions at the same points. That second requirement is easy to miss.

If the supports are statically indeterminate, the reactions themselves depend on the layout — so the external work changes and the constant stops being one. The fix is to use statically determinate supports in the comparison: a pin and a roller. Then the reactions follow from statics and are identical for every layout.

This course's bracing comparison does exactly that, and the constant comes out at zero — because for a horizontal load applied at a point with determinate supports on a level line, every external force has zero work on its own position. That in turn means the tension and compression load paths must be exactly equal in every layout, which is a strong and checkable claim.

What it omits

Every one of these has reversed a ranking somewhere:

• Buckling. A compression member must also be stable, so its real volume exceeds the load path. Compare like with like — or design the members.
• Connections. Not in the number at all. In timber, connections routinely drive member size, and the ranking can invert.
• Fabrication, transport and erection. Absent.
• Repetition. A layout with twenty distinct members scores the same as one with three, all else equal.

The library returns these caveats alongside every result, so an interface cannot display the number without them. That is deliberate: a measure this convenient gets over-trusted, and the omissions are where the over-trust bites.