Does the square-cube law let a carbon frame carry this?
Can it be built?
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 = 1.13 | n = 1.00 | n = 0.90 | n = 0.80 | n = 0.67 |
|---|---|---|---|---|---|
| 5,953 m³ | 33% | 52% | 74% | 105% ✕ | 167% ✕ |
| 15,803 m³ | 37% | 52% | 67% | 86% | 121% ✕ |
| 37,458 m³ | 42% | 52% | 61% | 72% | 90% |
| 80,000 m³ | 46% | 52% | 57% | 62% | 70% |
| 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 1.13 at R² = 0.94, 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. Three of these eight entries were wrong in the first version of this table, all in the direction that flattered the historical fleet.
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. The steps are the gondola, the engines and the fin roots.
Bending moment along the hull
The primary structural output, and what every laminate schedule downstream gets sized against. Warm above the line is hogging, ends down and middle up; cool below is sagging. This ship does both: it hogs forward of the gondola and sags aft of it, and it does so in still air at exact global equilibrium, because buoyancy and weight are never distributed the same way.
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. 822 MPa compressive against 1450 MPa on the datasheet: 59 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 | Section | Plies | Allowable | Reserve | Frame mass |
|---|---|---|---|---|---|---|
| 16 | 10 m | 151 × 0.9 mm | 4 | 43 MPa | 1.5× | 2,560 kg |
| 16chosen | 8 m | 151 × 0.9 mm | 4 | 68 MPa | 2.4× | 2,560 kg |
| 24 | 8 m | 151 × 0.9 mm | 4 | 68 MPa | 3.6× | 3,840 kg |
| 24 | 6 m | 151 × 0.9 mm | 4 | 120 MPa | 6.4× | 3,840 kg |
| 32 | 6 m | 151 × 0.9 mm | 4 | 120 MPa | 8.5× | 5,120 kg |
Two routes to the frame mass, and they do not agree
Sizing the members from the gust moment gives 2560 kg; scaling the Hindenburg's framework share gives 5443 kg. The bottom-up figure is 2.1 times LIGHTER, and that is the expected direction rather than a contradiction: the members come out at minimum gauge, so the sizing is a floor, and the factor of 2.1 is everything an idealised tube sizing leaves out. Local loads at every cell attachment and cover fitting. Wire bracing and its terminations. The fact that a real airship longitudinal is a LATTICE of small tubes rather than one large one, because a 151 mm single tube at four plies cannot be handled, drilled or joined. Handling and assembly loads, which for a structure this light are frequently larger than the flight loads. The model keeps BOTH numbers and uses the heavier one.