Concrete House Printing, Plan and Section

What is modelled

A gantry printer builds the walls of a small house (an 8 x 6 m ring with a partition, door and window openings) bead by bead from a printable mortar. The question the page answers is the one that decides whether 3D concrete printing works: does the fresh stack of layers carry its own weight while the lower layers are still soft? You choose the print speed, layer height, bead width, wall thickness, the mix (initial yield stress, build-up rate, stiffness) and an optional pause between layers, and watch the stress in every layer against its strength.

Equations

• Structural build-up: the static yield stress of a layer grows with rest time, τy(t) = τ0 + A t (A in Pa/s, Roussel). The elastic modulus grows too: E(t) = E0 + Ek t.
• Plastic collapse: the layer that carries m layers above (including itself) feels σ = ρ g h m. It yields when σ ≥ √3 τy(age) (von Mises compressive strength of a yield-stress material, unconfined).
• With a constant time T per layer this gives a closed form: the stack is stable for ever if √3 A T ≥ ρ g h (critical layer time Tc = ρ g h / (√3 A)), otherwise it fails when more than jmax = (√3 τ0 + √3 A T/2 − ρ g h) / (ρ g h − √3 A T) layers lie above a layer. Without build-up (A = 0) the limit is H = √3 τ0 / (ρ g).
• Time per layer T = nb · path / v + pause, flow rate Q = w h v (the nozzle prints one bead at a time, a wall of n_b beads takes n_b passes). Elastic squashing of the soft stack is ε = σ / E(age) per layer.
• Buckling of a slender wall under its own weight (Greenhill): Hcr = (7.837 E b2 / (12 ρ g))1/3 with b the wall thickness; the page uses the stiffness of the layer one third of the way down from the top as an effective modulus.

Checks run in the page

• Print ledger: pump flow integrated over the schedule equals the geometric volume of the wall; total time equals path x layers / speed.
• Static limit: brute-force stack check against N = floor(√3 τ₀ / (ρ g h)) + 1 for four layer heights and yield stresses.
• Thixotropic stacks: brute-force failure layer against the closed form j_max for four layer times, and the critical-layer-time boundary (stable 5 % above, failing 10 % below).
• The buckling constant 7.837 recovered by solving the column ODE with Runge-Kutta shooting.

Honest simplifications

• The mortar is treated as a linear-growth yield-stress material at constant density; real build-up is nonlinear (a fast early flocculation, then hydration), depends on temperature, accelerators and shear history, and the values here are typical of published mixes, not a specific product.
• Collapse uses the unconfined uniaxial strength of the lower layer; in a real wall lateral confinement, bead interlock, the wall geometry (corners, partition) and the width-to-height of each bead raise the strength, while eccentric loading, interlayer weak planes and the vibration of the gantry lower it. Buckling ignores corners and cross walls, which stiffen real houses a lot.
• Layer ages use the middle of each layer's deposition time; the ends of a layer are older or younger than that. Openings are skipped by omitting beads; the lintels are placed by hand and not simulated mechanically. The roof, reinforcement, shrinkage, nozzle travel moves and pump start-up are not modelled. The collapse picture is schematic.
• Wall thickness is the number of beads times the bead width; zig-zag or cellular infill used by real printers is not modelled. Not a structural design tool.

• Start with Stable print (slow nozzle) and press play: every layer waits long enough, so the capacity bar stays green.
• Pick Too fast: layers still wet, wall slumps: the bottom layers cannot carry the weight and the wall collapses plastically.
• Add a 90 s pause between layers (third preset): the same fast nozzle now works, because each layer gains yield stress while waiting.
• Pick Thin single-bead wall buckles: the wall bows out before the mortar crushes, as the Greenhill criterion predicts.