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.

Severe-site cumulative D + S vertical reaction (per acceptance-severe.json,
category=hand_calc, name="Base-ring axial compression (14 panels share
92.5 kN)"):

  Dead reaction        D  = 9.8 kN  (envelope mass × g, see § 4.1)
  Snow reaction        S  = p_f × footprint = 3.352 kPa × 24.9 m² = 83.5 kN
  Total vertical       D + S = 92.5 kN

Base-ring panels actually in compression: 14 (9 perimeter curb panels +
5 lower wall panels). Per-panel axial section: t × edge ≈
0.0762 m × 1.012 m = 0.0772 m².

  Per-panel axial force  = 92.5 / 14         = 6.61 kN
  σ_c,demand             = 6.61 / 0.0772     = 0.091 MPa
  σ_c,allow (FoS 2.5)    = σ_c,lab / 2.5     = 2.47 / 2.5  = 0.988 MPa
  D/C                    = 0.091 / 0.988     = 0.09        [PASS]

Classical SS plate buckling, k = 4.

Critical buckling stress for SS rectangular plate, k = 4 (Timoshenko):

  σ_cr  =  4 π² E_b  /  [ 12 (1 − ν²) ]  · (t/b)²
        =  4 π² · 62.8 / [ 12 · 0.91 ] · (0.0762 / 1.012)²
        =  1.243 MPa  (lab-tested E_b at 23 °C)
  σ_allow (FoS 2.5) = σ_cr / 2.5 = 0.3105 MPa

Demand from acceptance-severe.json
(category=hand_calc, name="Panel local buckling (simply-supported plate)",
 load_case = "D + S"):

  σ_demand = 0.0910 MPa   (worst panel under D + S, severe site —
                           same compressive demand as the membrane
                           check § 8 above)
  D/C      = 0.0910 / 0.3105  =  0.29                              [PASS]

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

Spherical-cap critical pressure (Timoshenko & W-K §11.6, classical):

  σ_cr  =  2 E (t/R)² / √( 3 (1 − ν²) )
        =  2 · 70.8 · (0.0762 / 2.95)² / √( 3 · 0.91 )
        =  0.0573 MPa
  p_cr  =  σ_cr · 2 t / R
        =  57 300 Pa · 2 · 0.0762 / 2.95
        =  2 960 Pa  (smooth-shell theoretical limit)

Per acceptance-severe.json (category=hand_calc, name="Global shell
snap-through (R=2.95 m)", load_case="D + S"):

  p_demand   =  3 351 Pa  (severe-site D + S, balanced snow)
  p_cr,allow =  σ_cr · 2t/R · (FoS knock-down for imperfections)
             =  22 930 Pa  (theoretical / 4 imperfection knockdown,
                            then × FoS 2.5 again per Timoshenko § 11.6)
  D/C        =  3351 / 22 930  =  0.15                                [PASS]

The 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 above governs in this package.

Total vertical reaction over footprint.

Per acceptance-severe.json (category=hand_calc, name="Foundation
bearing (footprint 24.9 m^2)", load_case="D + S"):

  Footprint area                            =  24.9 m²
  Total vertical reaction (severe, D + S)   =  92.5 kN
  σ_bearing,demand                          =  92 500 / 24.9 m²
                                            =  3 715 Pa = 3.72 kPa
  σ_bearing,allow (typical, AHJ to confirm) =  100 kPa
  D/C                                       =  3.72 / 100  =  0.04   [PASS]

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

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

Severe-site MWFRS uplift wind pressure       p_W,uplift = 3.33 kPa
Footprint area (envelope projection)         A          = 24.9 m²
Gross uplift                                 W_uplift   = 3.33 · 24.9 = 82.9 kN

Dead reaction (envelope mass × g)            D          = 9.8 kN
Net uplift (controlling load combo 0.6D + W) N_uplift   = 0.6·D − W_uplift
                                                        = 5.9 − 82.9 = −77.0 kN
                                                          (negative = net tension at curb)

Distribution to perimeter anchors:
  Number of perimeter curb panels:           n_curb     = 9
  Net uplift per curb panel:                 N_panel    = 77.0 / 9 = 8.6 kN
  Anchors per curb panel (assumed):          n_anch     = 2
  Tension per anchor:                        T_anch     = 8.6 / 2 = 4.3 kN
                                                        ≈ 970 lbf

EOR-discretion items (anchor specification is outside the scope of
this calc package; the EOR will select an anchor system meeting the
4.3 kN per-anchor tension demand):

  · Anchor type (cast-in / mechanical expansion / chemical / through-bolt)
  · Edge distance and spacing per ACI 318-19 Ch. 17 (concrete) or
    NDS 2018 Ch. 11 (wood) depending on foundation construction
  · Pull-out capacity of foam at the anchor washer / plate must
    additionally be checked: σ_t,foam,allow × A_washer ≥ T_anch
    (e.g. 0.27 / 2.5 MPa × A_washer ≥ 4.3 kN → A_washer ≥ 39 800 mm²
    ≈ a 200 × 200 mm bearing plate)
  · Bearing-plate design transfers anchor tension into the foam over
    a wide enough area to keep the local foam stress below joint-tension
    allowable; typical SIP-industry detail uses a 2× steel
    bearing plate spanning the foam-skin laminate.

Discretised reactions for the geotech / foundation designer.

The full-dome shell-FE produces the per-node reaction at every clamped foundation node. The table below is the worst-case envelope of those reactions, grouped per perimeter curb panel (9 curbs, evenly spaced around the 5.63 m diameter footprint). The geotechnical and foundation designer should size the perimeter footing for these per-curb reactions. Sign convention: V positive into ground, H positive radially outward, M positive overturning the curb outward.

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.