Structural Constraint Analysis (Phase 6)¶

Date: 2026-03-28 (updated from 2026-03-27) Model version: habitat-constraints 0.1.0 (Phase 6) Constraints: 13 total — Phase 3's 9 + hoop stress, cylinder length, rotational stability, spin-up energy Experiment script: models/habitat_constraints/experiments/run_phase6_analysis.py

Summary¶

Adding three structural constraints reveals that material choice, wall thickness, and cylinder geometry are now the dominant design drivers. The rotational stability constraint (\(L < 1.3r\) for single cylinders) is far more restrictive than bending mode resonance, reducing maximum length by 5–8× at typical radii.

Key update (2026-03-28): Wall thickness is now a tunable design parameter across the full stack (API, solver, UI). The feasible radius band is not fixed at ~997m — it widens substantially with thicker walls or stronger materials:

Wall thickness	Material	Feasible radius band	Band width
\(t = 0.2\) m	HS steel	982–1,000 m	~18 m
\(t = 0.5\) m	HS steel	982–2,100 m	~1,100 m
\(t = 1.0\) m	HS steel	982–3,100 m	~2,100 m
\(t = 0.2\) m	CFRP	982–3,200+ m	~2,200+ m

The interactive demo now shows green feasible-range indicators on sliders for radius, wall thickness, cylinder length, and atmosphere pressure, allowing real-time exploration of the design trade space.

Phase 3 → Phase 6 Comparison¶

Metric	Phase 3 (9 constraints)	Phase 6 (12 constraints)
Min radius at 1g	982 m	982 m (unchanged)
Max radius at 1g	9,177 m (rim speed)	~1,000 m (steel, \(t = 0.2\) m)
Binding lower	Cross-coupling	Cross-coupling (unchanged)
Binding upper	Rim speed	Hoop stress
Max cylinder length	Unbounded	\(L < 1.3r\) (single)
O'Neill feasible?	Yes (all 9 pass)	No (hoop stress fails)

Critical insight: the Phase 3 feasible band was optimistic because it ignored structural material limits. With steel at 0.2m wall thickness, the hoop stress constraint (\(\sigma = \rho \omega^2 r^2 + Pr/t\)) is dominated by the pressure term (\(Pr/t\)), which alone is 497 MPa at \(r = 982\) m. This leaves almost no margin for the rotational stress component.

Experiment Results¶

1. Full 12-Constraint Scorecard¶

Design	\(r\) (m)	\(L\) (m)	Paired?	Result	Failing
Min viable (single)	982	1,276	No	PASS	—
Min viable (old \(L\))	982	2,000	No	FAIL	rotational stability
Kalpana One	250	325	No	FAIL	cross-coupling
O'Neill (paired)	3,200	32,000	Yes	FAIL	hoop stress
O'Neill (single)	3,200	32,000	No	FAIL	hoop stress, rot. stability

Key observations: - Our previous "minimum viable" design at \(L = 2{,}000\) m now fails rotational stability (\(L/r = 2.04 > 1.3\)). The corrected maximum length is \(L = 1{,}276\) m for a single cylinder. - O'Neill's design fails hoop stress with our default steel — it would require CFRP or a much thicker wall. - Kalpana One's \(r = 250\) m fails cross-coupling regardless (too small).

2. Length Limit Comparison¶

Rotational stability is always the binding length constraint for single cylinders. Bending mode resonance only matters for counter-rotating pairs.

\(r\) (m)	Rotational stability	Bending mode	Ratio
250	325 m	1,322 m	4.1×
982	1,277 m	7,311 m	5.7×
2,000	2,600 m	17,788 m	6.8×
3,200	4,160 m	32,010 m	7.7×
5,000	6,500 m	55,920 m	8.6×

The ratio grows with radius because rotational stability scales as \(L \propto r\) while bending mode scales as \(L \propto r^{5/4}\).

3. Material Comparison (Hoop Stress)¶

At \(r = 982\) m, \(g = 1.0\), \(t = 0.2\) m, \(P = 101.3\) kPa:

Material	\(\sigma_y\) (MPa)	\(\sigma_{\text{hoop}}\) (MPa)	Margin	Status
Structural steel (A36)	400	574	−187%	FAIL
High-strength steel (4340)	1,200	574	+4%	Barely pass
Titanium Ti-6Al-4V	900	541	−20%	FAIL
CFRP	3,500	512	+71%	PASS

Pressure dominates: at \(t = 0.2\) m, the pressure term \(\sigma_p = Pr/t = 101.3 \times 982 / 0.2 = 497\) MPa is ~87% of total hoop stress. The rotational component (\(\rho g r = 76\) MPa for steel) is secondary. Increasing wall thickness is the most direct way to reduce \(\sigma_p\), but it adds mass.

At \(r = 3{,}200\) m, only CFRP survives (4.6% margin). No material in our set can handle \(r = 5{,}000\) m at this wall thickness.

4. Single vs. Counter-Rotating Pairs¶

Counter-rotating pairs unlock dramatically more usable space:

\(r\) (m)	Mode	\(L_{\max}\) (m)	Land area (km²)	Pop. capacity
982	Single	1,276	3.9	98,000
982	Paired	7,306	22.5	563,000
2,000	Single	2,594	16.3	407,000
2,000	Paired	17,786	111.8	2,794,000
3,200	Single	4,156	41.8	1,045,000
3,200	Paired	31,995	321.6	8,041,000

Counter-rotating pairs provide 5.7–7.7× more length and proportionally more population capacity. This explains why O'Neill chose paired cylinders — a single cylinder at \(r = 3{,}200\) m can only be 4.2 km long, not 32 km.

5. Radius Sweep — Wall Thickness as Design Lever¶

The feasible radius band is not a single point. It widens as wall thickness increases (reducing the pressure term \(Pr/t\)):

\(t\) (m)	\(\sigma_p\) at 982 m (MPa)	Feasible \(r_{\min}\)	Feasible \(r_{\max}\)
0.2	497	982 m	~1,000 m
0.5	199	982 m	~2,100 m
1.0	99	982 m	~3,100 m
2.0	50	982 m	~5,000+ m

Lower binding: cross-coupling (unchanged from Phase 3)
Upper binding: hoop stress — but now controllable via \(t\)
Atmosphere pressure also affects the band: half-atmosphere (50 kPa) roughly halves \(\sigma_p\), equivalent to doubling \(t\)

The interactive demo's /api/feasible_ranges endpoint computes these ranges dynamically, and green bars on the sliders show viable values in real time.

Design Implications¶

1. Our Minimum Viable Cylinder Must Be Shorter¶

The previous default of \(L = 2{,}000\) m at \(r = 982\) m is infeasible for a single cylinder. The corrected design:

Parameter	Old	New (single)	New (paired)
\(r\)	982 m	982 m	982 m
\(L\)	2,000 m	1,276 m	7,306 m
\(L/D\)	1.02	0.65	3.72
Land area	6.2 km²	3.9 km²	22.5 km²

2. O'Neill Requires Advanced Materials¶

O'Neill's Island Three (\(r = 3{,}200\) m) fails hoop stress with any steel at \(t = 0.2\) m. Feasible paths: - CFRP hull: passes with 4.6% margin (but CFRP at this scale is speculative) - Thicker steel walls: \(t \geq 0.4\) m would halve \(\sigma_p\) to ~280 MPa, but doubles hull mass - Half-atmosphere (50 kPa): halves \(\sigma_p\), at the cost of 40% O₂ (fire risk)

3. Counter-Rotating Pairs Are Not Optional¶

For any cylinder longer than \(1.3r\), counter-rotating pairs are a structural requirement, not a luxury. O'Neill understood this — it's why his design specifies paired cylinders. Our model now captures this constraint explicitly.

What Changed in the Model¶

Component	Change
`HumanAssumptions`	Added `max_length_to_radius_ratio` (1.3), `counter_rotating_pair` (False)
`RotationalStabilityConstraint`	New — enforces \(L < 1.3r\) (single) or \(L < 10r\) (paired)
`CylinderLengthConstraint`	Existing — \(L < C \cdot r^{5/4}\) (bending mode)
`HoopStressConstraint`	Existing — \(\sigma_{\text{hoop}} \cdot \text{FoS} \leq \sigma_y\)
`SpinUpEnergyConstraint`	New — spin-up time \(\leq\) max at available power
Total constraints	9 → 13

Next Steps¶

~~Wall thickness as a design variable~~ — DONE (2026-03-28). Wall thickness slider in UI, passed through API/sweep/solver.
~~Update the interactive demo~~ — DONE (2026-03-28). All 12 constraints shown, green feasible-range indicators on sliders.
Material selection in the API — expose material presets (steel, titanium, CFRP) rather than raw \(\sigma_y\) and \(\rho\)
CFRP as default material — documented in structural_engineering.md §3.2; needs API integration
~~Spin-up energy budget~~ — DONE (2026-03-29). Constraint SpinUpEnergyConstraint computes \(E = \frac{1}{2} I \omega^2\) and spin-up time at available power. Reference design: 279 TJ, ~8 hours at 10 GW. O'Neill class: 94 PJ, ~109 days at 10 GW.
Half-atmosphere reproduction safety — literature review (§3.1 of literature_review_structural.md) shows likely safe, but animal studies needed before committing to colony design