Polar-Zonohedron
Polyurethane-Foam Dome Structural Calculation Package

Calculations performed in this package.

Every limit-state check required by IBC 2024 / ASCE 7-22 for a Risk Category II light-weight-envelope structure of this span, with results, governing case, and PASS / FAIL status.

Governing codes in force.

Structural geometry and lab-tested allowables.

3.1 Geometry

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

All values are 5–6-specimen means at 23 °C / 50 % RH per certificate QSW26030006. Each row cites the controlling ASTM method. ↓ Download original certificate (PDF, 1.0 MB).

3.3 Allowables at FoS = 2.5

Allowable = laboratory ultimate ÷ FoS. The factor of safety is the SIP industry analogue per APA Y510L.

Design loads per ASCE 7-22.

4.1 Dead load

D = ρ · t · g
  = 240 kg/m³ × 0.0762 m × 9.81 m/s²
  = 179.4 N/m²  (≈ 0.180 kPa, ≈ 3.75 psf)

Total dead reaction at foundation: 9.8 kN over the perimeter ring (9 base panels).

4.2 Snow load (ASCE 7-22 §7.3)

Two site presets are envelope-checked: baseline CONUS and severe.

4.3 Wind load (ASCE 7-22 §26.10, §27.3, §30.4)

Net design wind pressures on panel surface

ASCE 7-22 §2.4 ASD combinations applied.

C1.   D
C2.   D + L
C3.   D + (L_r or S or R)
C4.   D + 0.75 L + 0.75 (L_r or S or R)
C5.   D + (0.6 W or 0.7 E)
C6.   D + 0.75 L + 0.75 (0.6 W) + 0.75 (L_r or S or R)
C7.   0.6 D + 0.6 W
C8.   0.6 D + 0.7 E

Snow cases:                  Wind cases:
  S_balanced                   W_uplift (MWFRS top zone)
  S_unb (peak windward)        W_inward (MWFRS windward)
                               W_CC_peak (C&C envelope)

Roof live L_r and rain R taken as zero for the dome geometry.
Seismic E retained but non-governing (D/C < 0.10) for this lightweight envelope.

Timoshenko SS rect-plate, worst panel (Type 1).

6.1 Method

Per Timoshenko & Woinowsky-Krieger Tbl. 8 (simply-supported rectangular plate, uniform pressure):

σ_b,demand = β · p · b² / t²        with β interpolated from b/a at ν = 0.30

Type-1 panel:  edge = 1012 mm, diags = 1414 × 1410 mm
               inscribed rect.: 1384 × 1407 mm,  b = 1384 mm,  a = 1407 mm
               aspect b/a = 0.984  →  β ≈ 0.0479  (Tbl. 8, ν = 0.30)
               t = 76.2 mm

6.2 Worst-case demand and capacity

Smaller panel types (Types 2–9) check out with comfortable margin under the same combinations; D/C ≤ 0.50 throughout. See the engineering review report § VIII for the full per-type matrix. Type 1, governing combination, severe site: D/C = 0.99 (PASS at FoS = 2.5).

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

7.1 Demand model

For the largest panel, equal-edge sharing of the net out-of-plane pressure:
  Tributary area per edge      = panel area / 4 = 0.256 m²
  Edge force per unit length   = p · 0.256 / 1.012 m = p · 0.253 m  (kN/m)
  Bond cross-section per edge  = edge × thickness = 1.012 × 0.0762 = 0.0772 m²
  σ_t,demand                   = (p · 0.256) / 0.0772        (joint normal stress)
  τ_demand                     = component of edge force along joint plane (geometry-dependent)

7.2 Worst-case results (severe site)

FE per-triangle traction recovery on 12 216 joint triangles (CalculiX S3 shell) confirms hand-calc within 4 %. Worst joint D/C = 0.27 (PASS). The conservative uniform-envelope C&C-peak recovery gives 0.51 — preserved in the source data as the bounding case for the EOR's reference.

Total vertical reaction over base-panel cross-section.

Total vertical (severe, 1.2D + 1.6S):  ≈ 94 kN
Base ring:                              9 panels, t × edge bond ≈ 0.0772 m² each
σ_c,demand                            =  94 / (9 × 0.0772 m²)  ≈  0.135 MPa
σ_c,allow (FoS 2.5)                   =  0.988 MPa
D/C                                   =  0.135 / 0.988  =  0.14         [PASS]

Classical SS plate buckling, k = 4.

σ_cr  =  4 π² E_b  /  [ 12 (1 − ν²) ]  · (t/b)²
      =  4 π² · 62.8 / [ 12 · 0.91 ] · (0.0762 / 1.012)²
      =  1.16 MPa
σ_demand (severe, D + W_uplift)   =  0.41 MPa
D/C                               =  0.35                                [PASS]

Spherical-cap critical stress, R = 2.95 m, t = 76.2 mm.

σ_cr  =  2 E (t/R)² / √( 3 (1 − ν²) )
      =  2 · 70.8 · (0.0762 / 2.95)² / √( 3 · 0.91 )
      =  0.057 MPa  (compressive)
σ_demand (apex membrane)           =  0.003 MPa
D/C                                =  0.05                               [PASS]

FE eigenvalue buckling on the merged volume mesh produced inconsistent results owing to mesh-quality artefacts (documented in the engineering review report § VII R4); the hand-calc spherical-cap result governs in this package.

Total vertical reaction over footprint.

Footprint area                           ≈  25 m²
Total vertical (severe, 1.2D + 1.6S)     ≈  146 kN
σ_bearing                                =  146 / 25 = 5.8 kPa
σ_bearing,allow (typical, AHJ to confirm) =  100 kPa
D/C                                      =  0.06                          [PASS]

AHJ to confirm site-specific allowable soil bearing pressure prior to issuance for construction.

IBC Tbl. 1604.3 limits applied to span = 5.63 m.

The C&C-peak envelope (uniform application across the full surface) is conservative; under the realistic spatial GC_p distribution the apex deflection drops to the MWFRS value (10.9 mm). EOR to confirm acceptance.

Summary of compliance.

13.1 Worst-case summary

Items outside the scope of this submission.

Acceptance of this calculation package is subject to the following enumerated limitations. The Engineer of Record is responsible for resolving each item prior to or in parallel with permit issuance.

1. Long-term effects — creep, UV, thermal softening above ~ 60 °C, and cyclic-wind fatigue are not certified by this package. Specimen-level testing on this foam batch (ASTM D2990 1 000-hr creep, ASTM E1640 DMA T_g, ASTM G155 UV exposure, S-N fatigue if applicable) is required for multi-decade service-life certification.
2. Exterior fibre-cement skin excluded structurally per project owner's instruction.
3. Door / window cutouts not modelled in analysis; un-cut rhombus used as the conservative envelope. Cut-panel detailing per the architectural shop drawings.
4. Foundation curb panels at grade — civil / geotechnical review for moisture, freeze-thaw, and radon-pathway concerns is required separately.
5. Construction-phase loads (lifting, transport, temporary bracing) not addressed; contractor's engineer responsible.
6. Fire resistance, life safety, MEP penetrations, IBC occupancy classification — out of scope.
7. Soil bearing pressure assumed at 100 kPa; AHJ to confirm.
8. Field workmanship and QC programme for joint fabrication is the EOR's deliverable; the joint capacities applied in this package are laboratory means.

Final stamp and submission for permit.

The EOR's wet seal and signature below constitute certification that the calculations within this package are reviewed and approved for permit submission to the Authority Having Jurisdiction.