Polar-Zonohedron
PU-Foam Dome Studio — Structural Calculation Package

Calculations performed in this package.

1. Panel plate bending — screening envelope (all 9 panel types) + refined FE verdict
2. Adhesive joint tension and shear at panel bonds
3. Base-ring membrane compression
4. Local panel buckling (classical SS plate)
5. Global buckling — FE eigenvalue on the shell mid-surface (snow + wind uplift)
6. Global shell snap-through (classical spherical cap)
7. Foundation bearing pressure
8. Net wind uplift / anchorage demand (design input for the EOR's anchor design)
9. Foundation reactions for the geotechnical designer
10. Deflection vs IBC Table 1604.3

Two bracketing site envelopes are computed throughout: baseline (V = 115 mph Exp. C, p_g = 30 psf) and severe (V = 160 mph Exp. D, p_g = 100 psf). Factor of safety: 2.5 on strength allowables, 3.0 on buckling. Both envelopes are certified in full — including 160 mph Exposure D wind and the 100 psf severe ground snow. Door-buck material recorded: African teak, exceeding the modeled structural-lumber floor (Sheets 10, 13); build QC verifies installed members match. Excluded scope is listed on Sheet 14.

Governing codes in force.

Structural geometry and lab-tested allowables.

3.1 Geometry (measured from as-assembled structural CAD)

3.2 Material — Zomes PU foam (240 kg/m³), certificate QSW26030006

3.3 Allowables at FoS = 2.5 (strength) / 3.0 (buckling)

FoS basis: SIP-industry analog (APA Y510L); project decision for EOR confirmation — no code-prescribed value exists for primary-structure foam buildings.

Design loads per ASCE 7-22.

4.1 Dead load

D = ρ g t = 240 × 9.807 × 0.0762 = 0.179 kPa; total W_D = 13.44 kN.

4.2 Snow load (§7.4.4, domed roof)

4.3 Wind load (§26.10, §27, §30.4; h = 3.97 m)

Basis note. Coefficients are the ASCE 7 domed-roof analogy with enclosed internal pressure (GC_pi = ±0.18); no code figure covers the faceted zonohedron literally. Quantified sensitivities (review report § V.4): partially-enclosed classification (GC_pi = ±0.55, +13.5 % net) leaves every check passing (worst refined bending 0.78); the refined severe strength check tolerates a +44 % combined-coefficient premium before reaching unity. A shape-specific ASCE 7 Ch. 31 wind-tunnel / validated-CFD basis is the product-line follow-up (Sheet 14).

ASCE 7-22 §2.4 ASD combinations applied.

D · D+L · D+S (balanced & unbalanced) · D+0.6W (inward & uplift) · D+0.75(S+0.6W) · 0.6D+0.6W · (screening: 1.2D+1.6S). Governing strength combination: 0.6D + W_uplift with C&C peak suction → net outward reference pressure 3.72 kPa baseline / 8.88 kPa severe. Governing stability case: balanced snow.

Panel plate bending — screening envelope and refined FE.

6.1 Screening envelope (Timoshenko SS rectangle on panel diagonals)

σ = β(b/a) · q · a² / t², β per Timoshenko Table 8 (ν = 0.3), a = short diagonal. Checked for all nine panel types under all combinations. Worst rows at severe (q = 8.88 kPa): type 1 σ = 0.956 MPa (D/C = 1.10); type 2 0.951 (1.10); type 3 0.916 (1.06). Baseline worst: 0.401 MPa (D/C = 0.46). The severe screening rows exceed 1.0 and do not govern the verdict — the envelope spans the panel's diagonal bounding rectangle (2.0× the true rhombus area) and its over-prediction on these elongated panels is measured at 49 % (§6.2). Where the envelope passes (all baseline rows; severe types 4–9) no refined analysis is needed.

6.2 Refined FE verdict (severe, same case)

Issued verdict: PASS, D/C = 0.69 (worst of the refined tiers). The two refined models bracket the edge restraint from opposite directions and agree within the bracket; both are mesh-converged. Full method narrative: review report §§ S2, S6, S7.

Joint shear and tension at panel-to-panel bond.

Demand model: worst-panel uplift shared equally by its four edge bonds (conservative — corners are stiffer): F_edge = q·A/4 = 8.88 kPa × 1.008 m² / 4 = 2.24 kN; bond area = 1011 mm × 76.2 mm.

Base-ring membrane compression.

Severe D+S vertical reaction: W = 13.44 + 3.352 × 35.2 = 131.3 kN, shared by 11 base panels, resolved along the panel plane at the measured 40.3° tilt from vertical: σ = (131.3/11) / cos 40.3° / (76.2 × 1011 mm²) = 0.203 MPa vs 0.988 allowable → D/C = 0.21 PASS (baseline 0.08).

Classical SS plate buckling, k = 4.

σ_cr = k π² E / [12(1−ν²)] · (t/b)² on the worst base panel → allowable σ_cr/3.0 = 0.291 MPa vs the Sheet-8 demand 0.203 MPa → D/C = 0.70 PASS severe (0.26 baseline).

FE eigenvalue buckling on the shell mid-surface.

CalculiX *BUCKLE, S3 shells on the 94-panel corner-network mid-surface, carried through a three-level mesh-refinement study (6,161 / 24,353 / 96,833 nodes — each level a genuine 4× element refinement), foundation band encastre, pressure-only reference load. The mode-1 factors decrease from above with a stable geometric ratio (0.407, both sites) and the issued basis is the extrapolated limit of the sequence — a lower bound, since the sharp-crease idealization under-represents the real 76 mm glued-joint stiffness. Criterion: mode-1 factor ≥ 3.0.

¹ Suction puts the dome membrane in tension — wind uplift is not a credible eigenvalue-buckling mode; the computed factors are local crease-band values (above 3.0 at every level of every model) and the strength sheets govern the uplift case. The bare-shell mode-1 eigenvector is a local panel-band mode at the door bay — exactly where that model omits the framing — which motivates the credited rows: with the jambs and header reinstated, the mode moves into the adjacent panel field on both credited bases. Issued basis = recorded African-teak bucks credited at a structural-lumber floor: extrapolated 4.08 pressure-only / 3.92 with gravity (−4.1 % measured) vs required 3.0 — PASS with ≈30 % margin. Member-thickness sensitivity: +1.7 % (the frame acts by in-plane bracing, not member bending). The recorded species exceeds the modeled floor by 17–53 %; build QC verifies installed members match the record. Classical spherical-cap snap-through cross-check (R = 3.39 m): q_cr = 2E(t/R)²/√(3(1−ν²)) = 43.3 kPa → allowable 14.4 kPa vs the 3.35 kPa severe balanced-snow reference → D/C = 0.23 — the FE mode governs because it resolves the local door-bay geometry the classical formula cannot.

Foundation bearing pressure.

Severe D+S: 131.3 kN / 35.2 m² = 3.7 kPa vs 100 kPa fair-residential-soil presumptive → D/C = 0.04 PASS (baseline 0.01). Site-specific allowable bearing to be confirmed by the geotechnical designer.

Severe-site wind uplift exceeds dead weight; perimeter anchors required.

Net uplift (conservative vertical envelope): U = |W_MWFRS| × A − 0.6 W_D.

The anchor / hold-down design to deliver this capacity is an EOR deliverable and is not part of this package. This sheet quantifies the demand so that design can proceed. The panel-to-curb / curb-to-foundation connection detail is a Zomes-provided design input (drawings to accompany this package) that the EOR checks and adopts rather than originates; the foundation itself is designed to a geotechnical report or CBC §1806 presumptive values. A permit should not be issued without the anchorage design.

Discretised reactions for the geotech / foundation designer.

IBC Tbl. 1604.3 yardsticks, span = 6.69 m.

Indicative 50-year creep bound (literature Findley, φ = 1.5): gravity-only apex sag ≈ 2 mm; batch creep testing per the follow-up program before multi-decade certification.

Summary of compliance & rated envelope.

13.1 Worst-case summary

13.2 Certified site envelope

Wind: 160 mph, Exposure D. Ground snow: 100 psf (the full severe envelope) — all checks at full factors of safety, governing stability margin 3.92 vs 3.0 on the recorded African-teak door-buck basis (Sheet 10). Conditions: (a) build-QC verification that the installed jambs/header match the recorded species (African teak; published MOE ≈ 9.4–12.2 GPa vs the modeled 8,000 MPa floor — any structural lumber also satisfies it); (b) should a future unit substitute a non-structural buck material, the foam-strength fallback basis rates 96 psf ground snow and an EOR-sized rib on the door-bay flanking panels recovers the full envelope. Sites beyond the severe envelope are outside this package's analyzed range.

Not included in this package.

1. Anchor / hold-down design (demand quantified, Sheet 11A) — EOR deliverable, permit prerequisite; connection detail is a Zomes-provided design input
2. Sliding / overturning at foundation interface; foundation and soils design (geotech report or CBC §1806 presumptive values)
3. Fire resistance and life-safety compliance — fire-protection engineer / code consultant scope with product fire testing (foam requires code-compliant thermal barrier)
4. Shape-specific wind pressures — dome-analogy basis disclosed with quantified sensitivities (Sheet 4 note); one-time ASCE 7 Ch. 31 wind-tunnel / validated-CFD study is the product-line closure
5. Enclosure classification — confirm enclosed vs partially enclosed from the door/glazing design per ASCE 7-22 §26.2 (partially-enclosed sensitivity passes all checks)
6. Seismic site overlay — bounding no-credit screening shows wind governs this 13.4 kN structure (review report § XII); site-specific parameters (geotechnical investigation typically required in California) confirm per project
7. Door / window opening local detailing and hardware attachment
8. Long-term material effects: creep certification, UV, temperature, moisture, fatigue (follow-up test program defined in the review report § XII)
9. Construction-phase loads and erection engineering
10. Exterior non-bonded fibre-cement skin (owner exclusion)
11. Site-specific loads (topography, drift, soil) — confirm at permit time

Companion narrative with methodology, validation evidence, and deep-linked sources for every value in this package: ZS-2026-002 Engineering Review Report.