Does the square-cube law let a carbon frame carry this?
Will it hold together?
The mass fraction against every rigid airship with published figures, the buckling allowables that actually size the frame, and the gust case that turns out to govern rather than the static one.
Can it be built?
Empty weight scaled from the Hindenburg, across the range of structural scaling exponents the historical record cannot distinguish between. This is deliberately a family of curves: one curve would be a claim the evidence does not support, and the two ends disagree about whether bigger ships are better or worse.
| Envelope volume | n = 1.06 | n = 1.00 | n = 0.90 | n = 0.80 | n = 0.67 |
|---|---|---|---|---|---|
| 6,000 m³ | 42% | 52% | 74% | 104% ✕ | 167% ✕ |
| 14,417 m³ | 44% | 52% | 67% | 88% | 124% ✕ |
| 34,641 m³ | 47% | 52% | 62% | 74% | 93% |
| 83,236 m³ | 49% | 52% | 57% | 62% | 69% |
| 200,000 m³ | 52% | 52% | 52% | 52% | 52% |
✕ marks a hull that cannot lift its own empty weight. All exponents agree at 200,000 m³ because that is the Hindenburg, where the scaling is anchored.
Undecided, and the record cannot settle it
Fitting all eight rigids with published figures gives an exponent of 1.0603029053889061 at R² = 0.9579764518567386, which would mean the baseline closes comfortably and that mass fraction gets worse with size, not better. Restrict to the five best-sourced ships, whose volumes span only 1.41 to 1, and the fit collapses to 0.16 at R² = 0.45. The scatter from gas choice, structural material and national design philosophy is about 30 percentage points, which swamps any size trend over that range.
At the theoretical square-cube value the baseline ship cannot lift its own empty weight. A model that quietly picked the favourable end would report a comfortable design where the truth is a coin flip.
| Every rigid with published figures | Gas | Structure | Empty weight fraction |
|---|---|---|---|
| R-38 / ZR-21921 | hydrogen | duralumin | 45.0% |
| LZ-129 Hindenburg1936 | hydrogen | duralumin | 51.8% |
| LZ-126 / USS Los Angeles1924 | helium | duralumin | 59.0% |
| USS Macon / ZRS-51933 | helium | duralumin | 60.1% |
| USS Akron / ZRS-41931 | helium | duralumin | 62.1% |
| USS Shenandoah / ZR-11923 | helium | duralumin | 63.1% |
| R1001929 | hydrogen | duralumin | 67.4% |
| R1011929 | hydrogen | stainless steel | 76.9% |
Structural material moves the fraction by 9.5 points at constant size, year and specification: R100 in duralumin against R101 in stainless steel, both built to the same Air Ministry requirement in the same year. That is larger than any size effect in the dataset, which is why the material choice decides the mass fraction and the hull size does not.
Diagnostics
The curves the design actually turns on. Shear and bending moment are drawn as two charts sharing an axis rather than one chart with two scales, because newtons and newton metres are not comparable heights and putting them on one plot invites a reading that means nothing.
Power required against airspeed
Drag goes as the square of speed, so power goes as the CUBE. Doubling cruise speed costs eight times the power, and on a vehicle whose energy comes from a fixed area of sunlight that single fact shapes the whole mission concept. This ship is slow because being fast is unaffordable, not because it cannot be made faster.
Hours of station keeping per day, against wind
How long the daily solar budget can hold position against a given wind. There is a speed above which the ship cannot hold station at all and must drift, and finding it is one of the most operationally important numbers the model produces.
Shear force along the hull
Buoyancy is distributed in proportion to cross-sectional area and weight is distributed wherever the heavy things are. Those two do not match, and the running difference is shear. Every step is a real item: the arrangement hangs 31 discrete masses on this girder, and the ship is trimmed with water to neutral buoyancy before it is loaded, because the lift margin is not spare capacity in flight.
Bending moment along the hull
Warm above the line is hogging, ends down and middle up; cool below is sagging. This ship does both, in still air at exact global equilibrium, because buoyancy and weight are never distributed the same way. But the still-air case is NOT what sizes the girder, and the panel below says what does.
What actually sizes the girder
An aeroplane's gust case gets worse as it flies faster. An airship's gets worse as it SLOWS, because the load is the Munk moment and incidence from a vertical gust is atan(w/V). This vehicle spends its life at station-keeping speed, which is the worst place to be.
The static case is 0.35 MN m and the gust case is 1.26 MN m, so the gust sizes the girder. A 7.5 m/s vertical gust at 8 m/s of forward speed is 43 degrees of incidence, and the Munk moment peaks at 45. An airship's gust case gets WORSE as it slows down, which is the reverse of an aeroplane's and is why station-keeping is the structural design condition.
The frame, member by member
Everywhere else the frame mass is a scaling estimate: the Hindenburg's framework share of empty weight, corrected for carbon. That sizes a concept, and it is not a structure. This sizes the actual members against the gust moment and the buckling allowable, and then compares the two.
The laminate you can actually lay up
47 percent fibre volume, 3.4 percent voids, woven fabric, vacuum bagged. 408 MPa compressive and 64 GPa, against 570 MPa for the measured laminate this is scaled from: 62 percent of what a prepreg autoclave would give. Every one of those knockdowns is in the flattering direction if you skip it, and a buckling-critical frame is sized by exactly the properties they hit hardest.
Leaving the vacuum bag off costs a further 26% of compressive strength. The bag is not optional, and this is the number that says so.
| Longitudinals | Bay | a/s | Section | Plies | Allowable | Reserve | Frame mass |
|---|---|---|---|---|---|---|---|
| 16 | 6 m | 1.29 | 151 × 0.9 mm | 4 | 75 MPa | 1.7× | 6,169 kg |
| 16chosen | 8 m | 1.73 | 189 × 1.2 mm | 5 | 66 MPa | 2.3× | 9,639 kg |
| 24 | 4 m | 1.29 | 151 × 0.9 mm | 4 | 166 MPa | 5.6× | 9,253 kg |
| 24 | 5.4 m | 1.75 | 151 × 0.9 mm | 4 | 92 MPa | 3.1× | 9,253 kg |
| 32 | 4 m | 1.73 | 151 × 0.9 mm | 4 | 166 MPa | 7.5× | 12,337 kg |
Two routes to the frame mass, and they do not agree
Sizing the members from the gust moment gives 9639 kg; scaling Akron's measured framework share gives 4137 kg. A ratio of 2.33, and the bottom-up figure is HEAVIER, which is the wrong direction for an idealised sizing. THE LIKELY CAUSE IS THE SECTION, and it is the one thing this sizing cannot model. A real airship longitudinal is a LATTICE GIRDER: a triangular or square arrangement of small chords with diagonal bracing between them, and its depth comes from the lattice geometry rather than from a tube wall. This calculation models it as ONE LARGE TUBE, and at the four ply minimum practical laminate a single 151 mm tube carries far more material than four 30 mm chords of the same overall depth. Ebner (NACA TM 872) describes the lattice arrangement and it is what every rigid airship used. So the honest reading is that the bottom-up figure is an over-estimate of a structure nobody would build that way, and the historical share is the better number until somebody builds a bay and weighs it. The model uses the historical share, and carries the difference as margin rather than banking it.