8.14  Lab instruction: the lightest bridge

“It would be well if engineering were less generally thought of, and even defined, as the art of constructing. In a certain important sense it is rather the art of not constructing; or, to define it rudely but not inaptly, it is the art of doing that well with one dollar, which any bungler can do with two after a fashion.”

— A. M. Wellington, The Economic Theory of the Location of Railways, 1887

A footbridge has to span 10 m and carry 5 kN at each of three deck nodes. Any number of bridges will do that. We are after the lightest one that does it without yielding, without sagging past the limit, and without buckling. The solver is written for you, so the work here is design: where the nodes go, which bars connect them, and how thick each bar needs to be.

The brief

GIVEN                                    MUST SATISFY
  span      10 000 mm                      |sigma| <= 150 MPa
  supports  pin at (0,0)                    |u|    <= 10 mm
            roller at (10000,0)             compression: |N| <= N_cr
  deck      nodes at x = 2500, 5000, 7500
  load      5 kN down at each deck node   MINIMISE
  steel     E = 210 000 MPa                 mass = rho * sum(A_e L_e)
            rho = 7.85e-6 kg/mm^3
  sections  square, side 10 15 20 25 30 40 mm

FREE: any extra nodes, any bar pattern, any section per bar.

Buckling is Euler’s, with pinned ends: \(N_{cr} = \pi^2 E I / L^2\), and \(I = a^4/12\) for a square bar of side \(a\). It applies to bars in compression only. A bridge of uniform section that satisfies all three weighs 380.6 kg, so that is the bungler’s two dollars. Beating 120 kg takes thought. Beating 100 kg takes a good idea.

Workflow

  1. Sketch the bridge on paper and number the nodes. Do this before you open anything.
  2. Write down your prediction: which bar carries the largest force, which of the three requirements will be the one that stops you, and roughly what the bridge will weigh.
  3. Open the notebook at mechanics.ju.se/marimo/trussAnalysis, download it, and fill in the two tables and the boundary conditions. Node \(i\) owns two degrees of freedom, \(2i-1\) for its \(x\) and \(2i\) for its \(y\); you decide which of them the supports hold.
  4. Run the two investigations in the notebook and write down what you found.
  5. Improve the bridge. Keep the three designs you learned most from.
  6. Upload your .py file, the photograph of your sketch, your prediction, and your log.

Investigations

  1. Fix the roller so that it cannot slide either. What happens to the displacement, what happens to the force in the bottom chord, and what happens to the mass? Why do real bridges keep the roller?
  2. Make the truss deeper, then deeper again. The mass falls and then rises. Say why it falls, say why it rises, and say what buckling does to the depth that wins.

What you hand in

The notebook, the sketch, the prediction you wrote before computing, a log of three designs with the mass of each and the requirement that stopped it, and one sentence for every real decision you made.

Your bridge appears at mechanics.ju.se/bridge2026 under your group number. The ranking is recomputed from your tables on our machine, so editing the solver changes nothing but your own screen.

The defence

The grade is the defence, not the mass. You present at the computer, any member of the group may be asked any question, and at least one question will be a change we make on the spot for you to answer before you run it.

Using AI

Use it if you want to. Write one line saying what you used it for. It will not be standing next to you at the defence.