Shape, size and form

Shape: the interior optimum

Take a parallel-chord truss and sweep its depth.

Shallow. Chord force is moment over depth, so a shallow truss has large chord forces. Chord material dominates and the structural volume is high.

Deep. Chord forces fall, but the diagonals and verticals get longer and the total length of web material grows. Eventually the web dominates.

Somewhere between the two is a minimum — a genuine interior optimum, not a boundary one. Finding it is the simplest honest demonstration of shape optimisation there is, and it makes a point worth generalising: the existence of an interior optimum means there is a real trade-off, and a design pushed to either extreme is worse than one in the middle.

The deflection has an interior optimum too, and it is at a different depth. That is worth pausing on, because the intuition 'deeper is always stiffer' is only true if you re-size the members as you go. At constant member area it is false: past some depth the diagonals become long enough that their own axial flexibility outweighs what the deeper lever arm saves, and the truss gets softer again.

For the 24 m truss in the lab, at constant area, volume is least at about 5 m and deflection is least at about 8 m. So depth chosen for material and depth chosen for stiffness are different numbers, and which governs depends on whether the design is strength-controlled or deflection-controlled.

Size: the snowball

For a statically determinate structure, sizing is easy. The forces come from equilibrium and do not depend on the sections, so one pass gives the answer: for each member, area = force / permissible stress. Every member ends fully stressed.

For a statically indeterminate structure it is not, and the reason is the sentence from Module 9: stiff members attract load.

Size a member up to carry its force, and it becomes stiffer, so it attracts more force, so it needs to be larger still. Meanwhile its neighbours are relieved and can shrink, which relieves them further. The design chases itself, and it may converge, oscillate, or wander.

So an indeterminate size optimisation is a loop: solve, resize, solve again, until the areas stop changing. The library reports whether it converged, and if it did not, it says so rather than returning its last guess as a design.

Fully stressed is not optimal, and not robust

Two separate points that get conflated.

Fully stressed is not always minimum weight. In an indeterminate structure, deliberately softening one member can shed load onto a more efficient path and reduce the total. A fully-stressed design is a fixed point of the resizing loop, not necessarily the minimum.

Fully stressed is, by construction, exactly critical everywhere. Every member is at 100 % utilisation, so any member failing has no reserve anywhere to redistribute into. That is the opposite of robustness, and a fully-stressed optimisation will deliver it enthusiastically unless a constraint prevents it.

Optimisation gives you what you asked for. If robustness matters, it has to be a constraint, because it is not going to emerge from minimising weight.

Discrete catalogues

Real sections come from a catalogue. Rounding a continuous optimum up to the next available size costs material, and the loss is larger when the catalogue is coarse — utilisations end up scattered below 1 rather than sitting at it.

That is not a defect to be minimised away. Fewer distinct sizes means cheaper fabrication, simpler procurement and fewer opportunities to erect the wrong member. A design with three section sizes at 70 % average utilisation is often better than one with twelve at 95 %, and no weight-based objective will ever say so.

Form-finding

A different idea. In shape optimisation you choose the geometry and evaluate it. In form-finding you specify the forces and let equilibrium produce the geometry.

The force-density method is the clean version. Fix, for every cable, the ratio q = F/L of its force to its length. That single choice turns a nonlinear problem into a linear one, and the equilibrium shape falls out of a linear solve.

What comes back is a shape you did not draw. A hanging net under gravity with uniform force densities gives a catenary-like surface; raise the densities and it flattens; vary them across the net and it warps. The shape is an output, and that is the whole distinction:

In modelling you draw a shape and ask what it does. In form-finding you state what you want it to do and it tells you the shape.

Which is why the funicular forms — catenaries, hanging chains, inverted arches — are not aesthetic choices. They are the answers to an equilibrium problem, and a structure built on one carries its load in pure axial force with no bending at all.