⛰️ 10. Summit bridge

The lightest concrete bridge across the chasm at the summit, carved out of a block of bricks by Topology Optimisation.

The lightest bridge. A block of concrete fills the chasm at the summit, pinned to the rock at both ends. Topology Optimisation keeps a quarter of it, only where the forces flow, and finds an arch.

GHsummit-start.ghGrasshopper file · download

GHsummit.ghGrasshopper file · download

Start from summit-start: the block, its concrete, the rocks and the walkway are there, and the steps below add the rest. summit is the finished definition: it runs as it opens, in about a second. The same bridge is the last area of the Playground, where you build it a piece at a time and watch what each wrong piece would carve.

What you will learn

• How Topology Optimisation works: a block of solid elements, the design domain, of which it keeps the share you allow, where it makes the structure stiffest.
• What Keep and VolumeFraction do, and why the deck has to be kept.
• Why supports shape the result: rock that pushes back gets an arch, a roller gets a tie.

The block

The chasm is 12 m across. The block that fills it is 4 m deep and 1.2 m wide, in C30/37 concrete (E = 33 000 MPa, 2.5 t/m³), cut into 30 × 10 bricks of 0.4 m: 300 bricks, 144 t. Topology Optimisation runs without a license on models of up to 500 elements. A walkway of 5 kN/m² stands on its top row, the deck. Alpaca4d works in metres and kilonewtons (Units).

Steps

1. The block. Open summit-start. Plan is the deck's plan, 12 × 1.2 m, and Plan mesh cuts it into Along faces, 30. Heights copies it up to 11 levels, 0.4 m apart, and Mesh Series to Brick makes a layer of bricks between each level and the next: 10 layers need 11 levels. It makes them from the bottom up, so the deck's 30 bricks are the last.
2. Concrete. Split List cuts the bricks at Deck starts, 30 × 9 = 270: the first 270 into the SSP Brick named Domain, the last 30 into the one named Deck. Both take the C30/37 nD material. The name in ElementId is how Topology Optimisation will find the deck.
3. The rocks. Two Supports, Rock A at X = 0 and Rock B at X = 12, each at both edges across, at the bottom of the block. Free frees Rx, Ry and Rz, which a brick's node does not have anyway; Tx, Ty and Tz are held at both, so the rock pushes the bridge back in.
4. The walkway. 5 kN/m² × 12 m × 1.2 m = 72 kN, shared by the 62 nodes of the deck's top that Deconstruct Mesh gives: 1.16 kN each, down, as Point Loads in a Load Pattern.
5. Assemble. Into Assemble Model: the Domain and the Deck into Elements, both rocks into Supports, the Load Pattern into LoadPatterns. Hold Shift to wire a second source into the same input.
6. What it keeps. A panel Keep holding Deck, and a slider Share kept at 0.25.
7. Topology Optimisation. Wire the model into Topology Optimisation's AlpacaModel, Keep into Keep, Share kept into VolumeFraction, a toggle Run set to True into Run, and a slider Step into Step. It runs: 40 iterations, about a second. Drag Step back to watch it again.

Check it by hand

What it keeps. VolumeFraction counts only the bricks it designs; the ones in Keep come on top. A brick is 0.4 × 1.2 × 0.4 = 0.192 m³, 0.48 t:

$$30\times 0.48+0.25\times 270\times 0.48=14.4+32.4=46.8\text{t}$$

of the block's 144 t. The design that comes out, the 90 bricks at a density of 0.5 or more, is 43.2 t.

The push on the rock. An arch carrying w = 5 × 1.2 = 6 kN/m over L = 12 m, rising f = 3.8 m from the foot of the rock to the middle of the deck, pushes each wall out with

$$H=\frac{w{L}^2}{8f}=\frac{6\times {12}^2}{8\times 3.8}=28.4\text{kN}$$

OpenSees gives 26.8 kN on the design: the deck carries some of the walkway itself. Group 8 of summit.gh analyses the design once the run has stopped, and colours it by Von Mises stress. Its 90 bricks are past the 50 elements Run Analysis and Model View take without a license, but a design that came out of Topology Optimisation is analysed and drawn free up to the 500 the optimisation took.

Try

• Free Rock B along X: wire Free into its Tx. Nothing pushes the ends together any more, so the optimiser ties them with a row of bricks along the bottom: a bowstring, 3.6 times as flexible, with its tie in tension.
• Hold it at the top: wire Depth into the Z of both feet. The bridge hangs between the tops of the walls, and the optimiser finds a hanging shape, 2.3 times as flexible. Concrete wants pushing.
• Carry the alpaca alone: one point at the middle of the deck instead of the walkway. It finds two struts meeting under that one spot.
• Change Share kept: 0.4 makes a heavier, stiffer arch; 1 keeps the whole block.

The usual mistakes

You see	Why	Fix
Point load at location '0,0,4' is not part of the model!, and Deck orange	Up wired straight into Heights' Count: 10 levels make 9 layers, so the block is 3.6 m deep and has no deck.	n layers need n + 1 levels: Up + 1.
Without a license Topology Optimisation takes models of up to 500 elements	Too fine: 60 × 20 bricks is 1 200.	0.4 m bricks, 30 × 10, show the shape.
The walkway crumbles at both ends	Keep is empty, so the deck is designed too.	Keep Deck.
"Dek" matched none of the 300 elements that have an ElementId	A misspelt name in Keep. Case does not matter.	Spell it as in ElementId.
The walkway is gone, the block below solid	The Deck and Domain names swapped, so Keep holds the block below.	Deck on the top row.
No part of the design is held still by the supports, so AlpacaModel is empty.	Share kept too small: 0.05 cannot reach from the deck to the rock.	0.25.
Set Run to true to start the optimisation.	Run is False.	A toggle set to True into Run.
A tie along the bottom	Rock B on a roller: nothing pushes the ends together.	Hold Tx at both rocks.

The components

• Mesh Series to Brick · SSP Brick · nD · Support · Point Load · Load Pattern · Assemble · Topology Optimisation · Run Analysis · Model View

See also

• Find a structure with topology optimisation
• Topology Bridge, the same idea in steel, 16 m long
• 9. Wall with a window: the stress lines that a topology follows