Computational Engineering / Structural optimisation
Shape, size and form
Moving the nodes, choosing the members, and letting equilibrium choose the geometry.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A computational engineering reviewer must check algorithms, convergence claims, statistical assumptions and numerical examples beyond the cases explicitly recorded.
- Verify the four bracing-layout force tables and actual stability in the lab; simplified rejection of mechanisms is not a design check.
- Confirm truss depth optima against independently reproduced engine calculations.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
Topology decides the load path and is decided first. Everything after that is refinement — but the refinements have distinct characters, and two of them contain traps worth knowing about before you meet them on a project.
By the end of this lesson you should be able to
- Find an interior optimum by sweeping a shape variable
- Explain the snowball in an indeterminate size optimisation
- Say why a fully-stressed design is not the same as a good design
- Explain what force density does and why the shape is an output
Shape
Changing truss depth changes chord forces, web lengths and stiffness. A particular layout may have interior minima for a load-path score and for deflection, at different depths. Establish them by a suitably resolved sweep over the feasible range; do not assume that every truss has the same pattern or that a sampled minimum is global.
Sizing
For a determinate axial structure with fixed loads and only a stress limit, A = |F|/σ gives a one-pass continuous sizing estimate. Buckling, deflection, minimum sizes, self-weight and discrete sections require further checks. In an indeterminate structure, changing stiffness can redistribute forces, so a resizing procedure must repeatedly analyse and check convergence. A fully stressed fixed point does not necessarily minimise mass.
Stress utilisation of 1.0 means the chosen check is active for that load case. It does not by itself measure alternate load paths, ductility, design reserve or robustness. Include the relevant robustness requirements and assess them directly.
Catalogue sections
Discrete sizing can add material relative to a continuous optimum. Grouping sizes may improve procurement and fabrication, but the trade-off depends on the project. If it matters, represent it explicitly in the objectives or constraints.
Form-finding
Form-finding solves for an equilibrium geometry under stated boundary and force conditions. In the basic force-density method, prescribed q = F/L and prescribed nodal loads give linear coordinate equations when the boundary conditions and force-density matrix are suitable. Geometry-dependent loading or other constraints can require iteration.
A funicular form can carry its associated idealised loading through axial action. Other load patterns, imperfect geometry, finite bending stiffness and connection effects can introduce bending, so the form-found shape still needs structural design checks.
Try it
Truss shape optimiser
Sweep the depth. Both material and stiffness have interior optima — at different depths.
2 m
- Least-material depth
- 5 m
- Volume there
- 13024 kN·m
- Inspecting depth
- 2 m
- Volume there
- 19120 kN·m
- Deflection there
- 38.90 mm
47 % above the minimum
least at about 8 m, deeper than the volume optimum
The volume curve has an interior minimum: shallow costs chord force, deep costs diagonal length. So does the deflection curve, at a different depth: at constant member area a very deep truss softens again, because its diagonals have become long. Two objectives, two different depths, from the same structure.
Reading it
- Chord force is moment over depth, so the shallow end is dominated by chord material.
- Diagonals lengthen in proportion to depth, so the deep end is dominated by web material — and by web flexibility, which is why the deflection turns round too.
- An interior optimum means a genuine trade-off, and a design pushed to either extreme is worse than one in the middle.
- If the truss is deflection-controlled, the material optimum is not the answer — and no single-objective optimiser will mention that. 'Deeper is always stiffer' is only true if you re-size the members as you go.
What this shows: Two objectives give two different answers from the same sweep — and which one governs depends on whether the design is strength-controlled or deflection-controlled.
Try it
Size optimiser
A fully-stressed design, iterated. Add a deflection limit or a discrete catalogue and watch what each costs.
Structure
The indeterminate one has a duplicated diagonal and both supports held horizontally.
Sections available
Zero means no deflection constraint — stress alone governs.
- Converged?
- yes
- Iterations
- 2
- Governing constraint
- stress
- Total volume
- 8200 cm³
- Mass
- 0.064 t
- Mean utilisation
- 1.000
- Deflection
- —
exactly critical everywhere — no reserve anywhere
| Member | Area (cm²) | Utilisation |
|---|---|---|
| AC | 5.00 | 1.000 |
| BC | 5.00 | 1.000 |
| AB | 4.00 | 1.000 |
Every member is at 100 % utilisation. The optimiser did exactly what it was told — and the result has no reserve anywhere, so a single member failing has nothing to redistribute into. Robustness is not an emergent property of minimising weight; if it matters it has to be a constraint.
Try this
- Determinate, stress only: converges in one or two passes with every member fully stressed. The forces do not depend on the sizes.
- Switch to indeterminate: sizing changes the forces, so the loop has to run several times. That is the snowball.
- Add a deflection limit: the governing constraint changes and the volume rises. Stress-only optimisation can violate serviceability without saying so.
- Switch to the catalogue: utilisations scatter, volume rises, and fabrication gets simpler.
What this shows: Sizing changes the forces, so one pass is never the answer in an indeterminate structure — and a fully-stressed design is exactly critical everywhere, which is not the same as robust.
Try it
Form-finding explorer
Set a force density for each cable and the equilibrium shape falls out of a linear solve. The shape is an output.
Fixing this ratio is what turns a nonlinear problem into a linear one.
Raise the force density on the left half only and watch the shape warp.
- Solved?
- yes
- Maximum sag
- 2.2500 m
- Sag / span
- 0.2250
- Symmetric?
- yes
Uniform force densities and uniform loads give a symmetric hanging shape. Double q and the sag halves exactly — the relationship is linear, which is the whole trick.
Why the shape is not an aesthetic choice
- Fixing q = F/L for every cable makes the equilibrium equations linear in the coordinates, so the shape comes from a single linear solve.
- The result is funicular: it carries its load in pure axial tension, with no bending anywhere.
- Invert it and you have the funicular arch for that loading — which is why hanging models were used to design masonry vaults long before anyone could solve the equations.
- Set the load to zero and the cable becomes straight: with no transverse load there is nothing for it to hang under.
What this shows: 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.
Worked example
Finding the best truss depth
Given
- A 24 m Warren truss with 8 panels, carrying 40 kN at each of 7 interior nodes
- Depth swept from 0.5 m to 12 m
- Structural volume computed by load path at each depth
Find
The depth that minimises material, and whether it is the depth to build
Practice
A truss carries a midspan moment of 960 kN·m. Compute the chord force at 2 m depth and at 6 m depth, and give the ratio of the shallow value to the deep one.
Practice
A member carries 450 kN and the permissible stress is 200 000 kN/m². What area does a fully-stressed design give, in mm²?
Check yourself
A size optimisation returns a fully-stressed design with every member at 100 % utilisation. What is the most important thing to say about it?
Check yourself
A truss's least-material depth and its least-deflection depth are different. What does that tell you?
Check yourself
What makes form finding different from shape optimisation?
Summary
- A truss depth sweep has an interior optimum for material AND one for stiffness, at different depths
- Sizing an indeterminate structure is a loop, because stiff members attract load
- Fully stressed does not necessarily mean minimum mass; assess robustness separately
- A coarse catalogue costs material and buys fabrication simplicity
- In form-finding the shape is an output of equilibrium, not something drawn