Polar-Zonohedron
Polyurethane-Foam Dome Limit-State Calculation Package

Limit-state verification of a 73-panel PU-foam dome.

This calculation package documents the structural verification of a 5.63 m × 3.82 m polar-zonohedron rigid polyurethane-foam dome under ASCE 7-22 loading. The structure satisfies all short-term limit-state checks at the project's chosen factor of safety; one panel/load-case combination requires the Engineer of Record's discretion, and four long-term effects fall outside the scope of finite-element analysis and are flagged for separate testing.

Plain-language summary

Written for the project owner, the AHJ plan-checker, and other non-structural-engineering readers. The formal engineering synopsis follows in § 1.2.

This document analyses the structural strength of a small dome-shaped building made out of stiff polyurethane foam. The dome is roughly 5.6 m (18 ft) wide and 3.8 m (12 ft) tall, assembled from 73 diamond-shaped foam panels glued together at their edges with adhesive and mechanical fasteners. The question we set out to answer is simple: is it strong enough?

How we checked

We checked it four different ways. The first and most trusted is hand calculation — pencil-and-paper engineering math that has been used to design buildings for over a century. The other three are computer simulations using established structural-engineering software (CalculiX and OpenSees, both widely used in industry, plus an in-house tool we wrote for cross-verification). Every method was run against two different weather envelopes: a typical CONUS site (115 mph winds, 30 psf design snow) and a severe-weather site (160 mph winds, 100 psf snow) that brackets hurricane-coast and heavy- snow conditions.

What we found

The dome is comfortably strong under any weather it will realistically see. Most parts of the structure carry well under a quarter of their full capacity. When we ramped up the snow load in our most demanding computer simulation, the foam didn't begin to permanently deform until we reached fifty times the design snow — that's the equivalent of roughly 3.5 metres of ponded snow piled on the dome.

One panel sits very close to its limit under hurricane-grade wind suction at a severe-weather site. On the largest equatorial panel, in the worst-case combination of wind direction and pressure peak, the demand reaches 99 % of the panel's bending capacity. That's still a pass at the project's chosen factor of safety, but only just. Reinforcement options (a thicker panel, an internal stiffener rib, or restricting the design to lower-wind sites) are offered for the engineer of record's review in § X. The more realistic full-dome computer simulation, which accounts for the panel sharing load with its neighbours through the joints, gives a much lower demand (61 %) for the same case.

The glued joints turned out to be safer than expected. The lab measured them as 5–10 times weaker than the parent foam, which led us to expect they would govern the design. They don't. The dome's geometry loads the joints mostly in shear along their own plane (where the parent foam is strong) rather than in peeling tension (where the joint is weak). The worst joint demand is about 53 % of capacity — comfortable margin.

What this analysis can't tell you

Some questions can't be answered by short-term mechanical analysis. Foam slowly deforms under sustained load ("creep"), embrittles under ultraviolet light, and softens above about 60 °C. None of that is captured in this report — those effects need separate laboratory testing on this specific batch of foam, and they're listed as required follow-up tests in § XI. We've estimated the long-term sag of the apex at roughly 17–20 mm (less than an inch) over fifty years using published creep data for similar foams, but that's an indicative bound, not a certification for a multi-decade service life.

Several other items are also out of scope of this report and must be addressed separately: fire resistance and life-safety code compliance; the detailing around door and window openings; the foundation curb where the dome meets the ground (which has separate moisture and freeze-thaw concerns); construction-phase loads; and the exterior non-bonded fibre-cement skin (excluded by the project owner's instruction).

The bottom line

The structure passes every short-term strength check at the project's chosen factor of safety. A licensed structural engineer of record must review and stamp the analysis before construction; this document is prepared for that engineer, with all the supporting calculations, code citations, and raw data they need to make the call and submit the package to the permitting agency. Long-term durability beyond the first few years requires additional laboratory testing on this specific batch of foam, which the engineer should commission as a condition of permit issuance.

Synopsis

The structure satisfies every short-term limit-state check at the project-default factor of safety FoS = 2.5 (SIP industry analogue per APA Y510L; see § IV.3). The worst hand-calculated demand-to-capacity ratio is D/C = 0.99 — Timoshenko plate bending of the largest rhombic panel under load combination 0.6D + Wuplift at the severe-site envelope (160 mph, Exposure D, components & cladding peak suction). The same case in the full-dome CalculiX S3 shell analysis returns D/C = 0.61 because real load-sharing through joints is captured. Joint demand-to-capacity is at most 0.53 (hand calc) or 0.51 (per-triangle FE traction recovery) under the most adverse uplift case; joints are not the governing limit state despite being the weakest material property in the laboratory test certificate.

This revision (Rev. 1) folds in the OpenSees-track work completed 2026-05-04/05: a three-pipeline linear-tet cross-check (CalculiX, OpenSees, scikit-fem), a four-shell-solver investigation with a smooth-cap discriminator and a mesh-refinement convergence study, a multi-solver per-panel D/C envelope tool, and a homogenized weakest-link envelope used as a supporting argument. None of this work alters the controlling check. The new evidence confirms that CalculiX and OpenSees agree to within ~5 % on the linear-tet model; that all four shell formulations agree within ±7 % on the smooth equivalent geometry but disagree by 2–4× on the faceted CAD owing to a documented kink-line singularity at zero-thickness panel-to-panel joints (not a solver bug); and that the buckling result passes at the project FoS = 3.0 on the shell formulation. A synthesis suitable for permit-package review is documented at reports/2026-05-05--permit-appendix.md.

Long-term behaviour — creep, ultraviolet degradation, thermal effects, and cyclic-wind fatigue — is outside the scope of finite-element analysis on file and outside the scope of the laboratory test certificate. Section IX applies a literature-grounded Findley power-law creep model to bound the 50-year deflection; Section XII identifies the additional testing required before the EOR can certify the structure for a multi-decade service life.

Verdict matrix

Two ASCE 7 site presets are evaluated. Baseline is the typical CONUS envelope (115 mph wind, 30 psf ground snow, Exposure C). Severe is the high-wind/high-snow envelope (160 mph, 100 psf, Exposure D) — roughly 1.5× baseline pressures. Every limit state is checked under every ASCE 7-22 load combination; only the worst case is shown here.

Reviewer questions, answered up front

The five questions a reviewing engineer is most likely to ask, with the supporting analyses cross-referenced.

Q1. Is the structure adequate on day one under code loads?

Yes. Every ASCE 7-22 strength-design and component-and-cladding load combination is satisfied at FoS = 2.5 for both site envelopes (baseline CONUS and severe). Under the baseline envelope the worst limit-state D/C is 0.66 (panel plate bending); most other checks are below 0.20. CalculiX Drucker–Prager nonlinear runs maintain convergence to approximately 50× design snow pressure. See Result R1.

Q2. What is the governing limit state, and how close is it to capacity?

Hand-calculated Timoshenko plate bending of the largest equatorial rhombus (1012 mm × 1012 mm) under load combination 0.6D + Wuplift at the severe-site components-and-cladding peak suction (GC_p = −2.6). D/C = 0.99 at FoS = 2.5, D/C = 0.61 at the same combination in the full-dome CalculiX S3 shell analysis (which captures joint load-sharing). The hand-calc envelope is the conservative bound; reconciliation discussion in Result R2.

Q3. Are the laboratory-weakest joints the governing element?

No. Joint tension capacity is 0.27 MPa — five to ten times below parent-foam allowables. The polar-zonohedron geometry, however, transfers panel-to-panel pressure primarily as in-plane shear within the joint's own plane (parent shear allowable 0.41 MPa) rather than as peeling tension across it. Worst joint D/C = 0.26 (hand calc) and 0.51 (per-triangle CalculiX traction recovery, conservative envelope). See Result R3.

Q4. Do the independent solvers agree?

On the linear-tet model, yes: CalculiX 2.23 (industry-validated) and OpenSees 3.8 (independent open-source) agree to within 1–5 % on displacement and within 1 % on gravity-only von-Mises stress on the same merged tetrahedral mesh. Pressure-load p99 stresses diverge 20–60 % owing to known surface-pressure lumping differences between the two formulations. The in-house scikit-fem branch disagrees with both reference solvers by 5–6 orders of magnitude on stress; the bug has been isolated to a units / force-magnitude error and does not affect any published D/C value (which uses Timoshenko stress composed with FE deflection). See Result R5 and Result R7.

On the shell formulation, the answer is more nuanced and is now thoroughly documented. Four independent shell solvers (CalculiX S3, in-house CST + DKT, OpenSees ShellMITC4, OpenSees ShellDKGT) were run on the same dome midsurface; on the faceted production geometry their apex deflections spread by 2–4×. A smooth-cap discriminator test ran the same four solvers on a smooth spherical cap of the same overall geometry and found all four agree within ±7 % on the smooth equivalent. A subsequent mesh-refinement convergence study showed that all four solvers drift or diverge with refinement on the faceted dome — none of them has a stable limit. The diagnosis is that adjacent panels are joined at a kink (a zero-thickness, infinite-stiffness fold), which is a singular feature in shell theory where bending stress diverges logarithmically as the mesh resolves it. This is a geometric property of the CAD, not a solver bug. The hand-calc envelope is independent of the shell-FE deflection number and remains the controlling check. See Result R10 and Result R12.

Q5. What remains for the EOR to resolve before issuance for construction?

Four items, enumerated formally in § XI: (1) accept or reinforce the worst-panel/severe-uplift case (R2), with reinforcement options provided in § X; (2) commission the outstanding long-term tests (creep per ASTM D2990, DMA T_g, UV exposure per ASTM G155, fatigue), or accept the design-life risk in writing; (3) confirm or override the per-limit-state safety factors in § IV.3; (4) review and approve the field workmanship-and-QC programme in § X.3.

Project data and scope of this calculation package.

Identifying information for the project, the structure under analysis, and the limits of what this report is intended to certify.

Project data

Scope of this report

This calculation package documents the limit-state structural verification of the primary panelised dome envelope under gravity, snow, and wind loading per ASCE 7-22. It is intended to be the engineering basis of design submitted with the building permit application, subject to review, revision as required, and stamping by the Engineer of Record.

The following items are within the scope of this report:

• Plate-bending capacity of the rigid PU foam panels under all ASCE 7-22 dead/snow/wind combinations.
• Joint capacity (shear and tension) at every panel-to-panel interface, evaluated by hand calculation and by per-triangle traction recovery from the CalculiX shell finite-element analysis.
• Membrane (in-plane compressive) capacity at the foundation curb interface.
• Local panel buckling and global shell snap-through at design pressure.
• Foundation bearing pressure on a typical 100 kPa allowable soil bearing.
• Short-term elastic deflection serviceability at the apex and equator.
• Indicative long-term creep deflection on a Findley power-law assumption with a literature-grounded creep coefficient (full creep certification deferred — see § XI).

The following items are excluded from this report:

• Long-term creep, UV, thermal, moisture, and fatigue effects beyond the indicative serviceability bound — these require specimen-level testing on this foam batch (see § XI).
• The non-bonded exterior fibre-cement skin, treated as non-structural per the project owner's instruction.
• Stress concentrations around door, window, and ventilation cutouts — the un-cut rhombus is used as the conservative envelope; cut-panel detailing to be addressed in the architectural/engineering shop drawings.
• Foundation curb panels in soil contact and below-grade moisture / freeze-thaw / radon-pathway considerations — out of scope of structural FE; a separate civil/geotechnical review is required.
• Fire-resistance, life-safety, egress, and code occupancy classification — these are governed by IBC chapters not addressed herein.
• Mechanical, electrical, and plumbing penetrations of the panel envelope.
• Construction-phase loads (lifting, transport, temporary bracing).

Governing codes and material-test standards.

Every load value, allowable stress, and safety factor in this report is traced to a published code or standard. The corresponding clause is cited inline at point of use.

Geometry, panel inventory, and lab-tested allowables.

The dome is a 9-fold polar zonohedron of rigid 240 kg/m³ polyurethane foam. Geometry derives from the Rhino structural-full-office.obj model (assets/Office Files/); material allowables derive from laboratory test certificate QSW26030006 (assets/QSW26030006 (Zomes PU).pdf).

Geometry & panel inventory

The 70 structural panels segregate into 9 unique rhombus types by edge length and acute angle, identified by PCA fitting of the as-built OBJ vertex cloud (tools/extract_panels_from_obj.py, tools/fit_panels.py). Dimensions are cited in millimetres; areas account for actual rhombic geometry, not the inscribed rectangle.

Geometry summary

• Footprint diameter (mean of the two horizontal axes): 5.63 m (5.755 m × 5.505 m bounding).
• Apex height: 3.82 m (150.25 in).
• Equivalent spherical-cap radius (used in the membrane proxy): R = (D²/4 + H²)/(2H) = 2.95 m.
• Panel thickness, all panels: 76.2 mm (3.0 in), confirmed in writing by the project owner.
• Panel-to-panel joints: adhesive plus mechanical fasteners (joint detail and fastener spec to be specified by the EOR — see § X.3).
• Symmetry: N = 9 polar zonohedron with three latitudinal rings.

Material properties (laboratory-tested)

All material values below are means over 5–6 specimens per test, ASTM-traceable, performed at 23 °C / 50 % RH at Nanjing Guocai Testing Co., Ltd. (test certificate QSW26030006, May 2026). The canonical encoding lives at src/zomestruct/material/pu_foam.py.

Source: ↓ Download the original laboratory test certificate (PDF, 1.0 MB) — QSW26030006. The PDF is the authoritative document; the table below is its digest.

The joint allowables (highlighted) are the lowest material properties on file by an order of magnitude. The geometry of the polar zonohedron loads the joints predominantly in in-plane shear rather than peeling tension; the consequences of this for limit-state governance are documented in Result R3.

Allowables & safety factors

Allowable stresses are computed as laboratory ultimate ÷ FoS. The factor of safety used throughout this report is the project default FoS = 2.5, traceable to APA Y510L industry practice for structural insulated panels of similar cellular construction. The Engineer of Record is requested to confirm or override these factors per per-limit-state engineering judgement; the underlying ultimate values are unchanged in either case.

ASCE 7-22 derivation, baseline and severe envelopes.

Two site envelopes are evaluated. Baseline represents a typical CONUS site (Risk Cat II, 115 mph wind, 30 psf ground snow, Exposure C). Severe represents a high-wind / high-snow envelope (160 mph, 100 psf, Exposure D) approximately 1.5× baseline pressures. Every load is derived directly from ASCE 7-22 with the governing clause cited at point of use.

Dead load (D)

Dead load is the foam panel self-weight only; the non-bonded fibre-cement skin is excluded per the project owner's instruction (and is conservatively excluded from snow / wind sharing as well). Taking ρ = 240 kg/m³ and t = 76.2 mm:

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 weight of the dome envelope (70 structural panels × mean area 0.78 m²) is approximately 9.8 kN (≈ 1000 kgf), distributed as a foundation reaction over the nine perimeter curb panels.

Snow load (S) — ASCE 7-22 Ch. 7

Roof snow load on the dome is computed per ASCE 7-22 §7.3 (balanced) and §7.6.1 (unbalanced for curved roofs). The flat-roof snow load p_f is taken as the envelope; the slope factor C_s for a foam surface (slippery, unobstructed) is conservatively retained at 1.0 because the dome's average slope at the equator drops below the threshold where slope reduction is permitted, while the apex zone collects.

Wind load (W) — ASCE 7-22 Ch. 27 / 30

Velocity pressure follows ASCE 7-22 §26.10 and is evaluated at roof mean height. The dome is treated as a domed roof per §27.3 for MWFRS pressures and per §30.4 for components and cladding. Internal pressure is taken as the enclosed-building envelope GC_pi = ±0.18 (Table 26.13-1).

Pressure coefficients applied to the dome surface envelope:

• MWFRS roof (§27.3 / Fig. 27.3-2) — windward zone GC_p ≈ +0.4 (inward), top GC_p ≈ −0.7, leeward GC_p ≈ −0.5.
• MWFRS uplift envelope — net design pressure ≈ −0.99q_z at top zone, applied as uniform suction across the cap (conservative).
• Components & Cladding (§30.4 / Fig. 30.4-7, Tbl. 30.4-1) — peak negative GC_p = −2.6 at corner zones, peak positive GC_p = +1.5 (inward) at the same zones; tributary area for a single rhombic panel ≤ 1 m² is below the size-effect roll-off threshold.
• C&C peak suction applied uniformly — used as the conservative envelope in the per-panel checks (R2, R3) since real C&C peaks are localised; the dome-level FE applies the spatial GC_p distribution from Fig. 27.3-2.

Load combinations (ASCE 7-22 §2.4)

Allowable-Stress-Design combinations from ASCE 7-22 §2.4.1 are applied. (LRFD combinations from §2.3.1 produce the same governing case at higher D/C and are retained in the source analysis files for cross-check.) Roof live load L_r and rain load R are taken as zero for the dome geometry.

ASCE 7-22 §2.4.1, 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 evaluated:                Wind cases evaluated:
  S_balanced (uniform p_f)              W_uplift (MWFRS top zone)
  S_unb (peak windward depletion)       W_inward (MWFRS windward)
                                        W_CC_peak (C&C envelope)

Combinations dominated by uplift (C5, C7) with W = W_CC,peak govern the worst-panel plate-bending check. Combinations dominated by snow (C3 with S_unb) govern the compressive checks. Seismic (E) is not governing for this light-weight structure but is retained in the analysis source for completeness; D/C values are < 0.10.

Four independent analysis routes — and what they cross-check.

The structural verification combines a closed-form hand calculation backbone, two independent shell-FE solvers, a volume-FE collapse analysis, and a third-solver cross-check. Each method is validated either against a published analytical solution or against an industry-validated reference solver.

Hand calculation (the verification backbone)

All headline limit-state D/C ratios are evaluated by hand-calc first; the FE results either confirm or relax these envelopes, but never tighten them in the issued numbers. The hand-calc module lives at src/zomestruct/checks/structure.py.

• Plate bending — Timoshenko & Woinowsky-Krieger Table 8 (simply-supported rectangular plate, uniform pressure). β coefficient interpolated from b/a ∈ [1.0, ∞) at ν = 0.30. Rhombic panels are treated as the inscribed rectangle on their two diagonals, which is conservative for skinny rhombi (rhombus area > rectangle area means the inscribed rectangle carries the same total load on a smaller span — under uniform pressure σ scales with b², so the inscribed rectangle gives a higher demand).
• Joint shear & tension — equal-edge sharing of net pressure over panel area; bond area = edge × thickness. Compared to lab joint allowables.
• Membrane base compression — total dead + snow vertical load distributed over base-ring panels; section = thickness × edge.
• Local panel buckling — classical k = 4 SS plate buckling, σ_cr = 4π²E / [12(1−ν²)] · (t/b)².
• Global shell snap-through — Timoshenko spherical-cap, σ_cr = 2E (t/R)² / √(3(1−ν²)) at R = 2.95 m.
• Foundation bearing — total vertical reaction divided by footprint, vs. a typical 100 kPa allowable (AHJ-confirmed at issuance).

Shell FE — four formulations, smooth-cap discriminator, convergence study

Four independent shell-element implementations operate on the same dedup'd mid-surface mesh (89 unique corners, 4 639 nodes, 8 960 triangles for the triangular formulations; 319 nodes / 280 quads for the production MITC4 sweep, with mesh-refinement variants up to 4 480 quads in the convergence study). All four are run from the same Python driver and consume the same loads and clamped foundation boundary condition.

• CalculiX S3 (src/zomestruct/fea/calculix_shell.py): the mid-surface mesh exported as a CalculiX 2.23 *ELEMENT, TYPE=S3 deck. Industry-validated thin-shell Mindlin–Reissner triangle. Foundation BC *BOUNDARY 1, 6 on every node with y < 200 mm; pressure as *DLOAD ESHELL P p with normal-direction sign correction in the exporter.
• In-house pure-Python CST + DKT (src/zomestruct/fea/shell_element.py, shell_solver.py): 18 × 18 element stiffness with 6 DOFs per node. Membrane via constant-strain triangle, plate bending via the Discrete Kirchhoff Triangle (Batoz, Bathe & Ho, 1980), drilling DOF via 1 % penalty stiffness. Validated to within ~12–15 % of the Timoshenko square-plate solution at the validated mesh density (§ VI.5).
• OpenSees ShellMITC4 (src/zomestruct/fea/opensees_shell_quad.py): 4-node Mindlin–Reissner quad with MITC interpolation, run via the openseespy Python bindings. Mesh quad-converted from the same midsurface; degenerate quads tolerated at 3-node corners.
• OpenSees ShellDKGT (src/zomestruct/fea/opensees_shell.py): OpenSees' own Discrete Kirchhoff Triangle. Same midsurface input as the in-house DKT.

On the faceted production geometry, the four shell solvers do not agree to within engineering tolerance on apex deflection: under wind_cc_peak baseline (p = −3 831 Pa) they read 12.78 mm (CCX), 29.47 mm (in-house DKT), 29.83 mm (MITC4), and 52.42 mm (DKGT) — a 2–4× spread. The cause is geometric: adjacent panels are joined at a kink (a zero-thickness, infinite-stiffness fold), which is a singular line in shell theory. Two purpose-built experiments establish this:

• Smooth-cap discriminator (tools/smooth_cap_4way_compare.py; report reports/2026-05-04--smooth-cap-4way.md). All four solvers are run on a smooth spherical cap of the same overall geometry (R = 2 950 mm, H = 3 820 mm, t = 76.2 mm) under the same load. All four agree within ±7 %: 2.588 mm (CCX), 2.581 mm (in-house DKT), 2.766 mm (MITC4), 2.628 mm (DKGT). The single change of geometry from smooth to faceted — keeping every formulation, mesh generator, and load identical — produces the entire disagreement. The four formulations are individually correctly implemented.
• Mesh-refinement convergence study (tools/shell_convergence_study.py; report reports/2026-05-05--shell-convergence-study.md). All four solvers refined from 800 mm to 100 mm target element size on the faceted dome. Result: every solver drifts or diverges with refinement — none has a stable u_max. CCX in particular reads 5.47 → 7.90 → 12.93 → 22.33 mm (a +73 % single-step jump at the finest mesh, the largest of any solver). The "CCX-converged-everyone-else-overshoots" reading from Rev. 0 was an artefact of all four solvers being compared at a single point on each one's divergence curve.

Both experiments together establish that no single shell-FE result on the faceted CAD is a trustworthy stress number. Issued D/C ratios in this package therefore lean on the hand-calc Timoshenko envelope (§ VI.1) for governance; the shell-FE results are retained for order-of-magnitude corroboration, for the joint-traction recovery in R3, and for the multi-solver D/C envelope tool described in R12. The eigenvalue-buckling calculation runs on the well-posed shell formulation rather than on the merged-tet mesh — see R4.

CalculiX volume FE (C3D4) — collapse analysis

CalculiX 2.23 (Homebrew formula calculix-ccx) on the merged volume tetrahedral mesh reports/full_dome_merged.vtu: 20 187 nodes, 57 725 C3D4 (linear tet) elements, foundation BC encastre on y < 50 mm. Used for two purposes:

• Linear elastic cross-check — gravity, snow, wind uplift, baseline + severe. Confirms that p99 von-Mises stress is at most an order of magnitude below allowable everywhere on the envelope at design load.
• Drucker-Prager nonlinear collapse — yield surface calibrated from σ_c = 2.47 MPa and σ_t = 0.27 MPa (β = 67.45°, ψ = 33.73°, d = 0.487 MPa, perfect plasticity). Pressure ramped at 1×, 5×, 20×, 30×, 50×, 100× design; collapse is bracketed by the highest converging step. Result: collapse ≥ 50× design snow at severe site, ≥ 60× design uplift at severe site, with first-yield safety factor ≥ 36×.

The volume FE is not used for the issued D/C numbers because (a) linear tetrahedra shear-lock in pure plate bending and (b) the merged-mesh approach produces sliver tets at joint-merge points that contaminate peak-stress recovery (see Result R5). It is used only for the Drucker-Prager collapse-load bracket, where the issue is material yielding through the bulk and slivers do not invalidate the result.

OpenSees track (Phases 1–4)

OpenSees 3.8 (openseespy + openseespymac Python bindings) was integrated as an independent open-source cross-check across four phases. The full work is captured in reports/2026-05-04--findings.md (Findings 7 and 9) and reports/2026-05-05--permit-appendix.md; the phase-by-phase summary follows.

• Phase 1 — linear-tet wrapper and three-pipeline cross-check. A FourNodeTetrahedron wrapper (src/zomestruct/fea/opensees_tet.py, tools/opensees_crosscheck.py) runs the same merged tet mesh as CalculiX and scikit-fem under the same clamp NSET and loads. Result: CalculiX and OpenSees agree within 1–5 % on displacement and 1 % on gravity-only von-Mises stress; scikit-fem disagrees by 5–6 orders of magnitude on stress, isolating the bug to the scikit-fem branch. See R7.
• Phase 2 — shell wrappers and the smooth-cap discriminator. ShellDKGT and ShellMITC4 wrappers (src/zomestruct/fea/opensees_shell.py, opensees_shell_quad.py) bring the production shell-solver count to four. The smooth-cap discriminator test established that the 2–4× faceted-dome disagreement among the four solvers is caused by the kink-line singularity at panel-to-panel joints, not by any formulation difference. See R10.
• Phase 3 — buckling cross-check (with documented limitation). OpenSees 3.8's FourNodeTetrahedron lacks geometric-stiffness assembly, so classical eigenvalue buckling on the tet mesh is not available in OpenSees. CalculiX *BUCKLE on the well-posed shell formulation is therefore the buckling reference; the merged-tet buckling investigation was retired as documented in R4 after a tolerance-relaxation experiment (reports/2026-05-05--relaxed-tol-buckling.md) confirmed the underlying mesh defect is geometric, not parameter-tunable.
• Phase 4 — multi-solver code-check envelope tool. tools/opensees_full_check.py consumes in-house DKT and OpenSees ShellMITC4 results (ShellDKGT excluded by default per the smooth-cap-test outlier finding) and reports per-panel D/C as the conservative max across the two solvers. Default safety factors from src/zomestruct/material/pu_foam.py (FoS = 4 on parent stress, FoS = 5 on joints, FoS = 3 on buckling); the project's chosen FoS = 2.5 on parent stress is applied at the report-tier rescaling and is documented in § IV.3. See R12.

A note on the FoS = 5 / FoS = 3 figures referenced above: the OpenSees-track tools run with the pu_foam.py default safety-factor table (compression / tension / shear / bending = 4.0; joint = 5.0; buckling = 3.0). The project default for parent stress in the issued package is FoS = 2.5; D/C numbers in this report are quoted at FoS = 2.5 unless explicitly noted otherwise. The buckling FoS = 3.0 is retained as the per-limit-state choice and is the basis of the buckling PASS in R4; an earlier internal note that used FoS = 5 on buckling has been corrected.

Software validation evidence

Each solver path includes a published-analytical-solution validation case retained in tests/ and run as part of the package selftest (python -m zomestruct selftest).

Software stack and exact versions:

Python                   3.12.8
numpy                    2.4.3
scipy                    (matching numpy)
gmsh                     4.15.2     (Python API + OCC kernel)
ezdxf                    1.4.3
meshio                   5.3.5
scikit-fem               12.0.1
CalculiX (ccx)           2.23       (Homebrew calculix-ccx)
OpenSeesPy               3.8        (Python bindings; openseespymac 3.8.0.0)
ParaView                 latest     (post-processing only, .vtu)

Eight numbered results, with reconciliation between methods.

Each result is presented with its limit-state status, the analysis methods that contributed, the full computation, and download links to the underlying artefacts. Click any underlined value for the popup with the math, source files, and reproduction command.

Per-panel-type and per-combination check matrix.

The summary matrix below tabulates the worst-case demand-to-capacity ratio for each unique panel type (§ IV.1) under each ASCE 7-22 combination (§ V.4), across both site envelopes. All values at FoS = 2.5.

Per-panel · per-combination matrix

Type 1 (the 9 equatorial near-square rhombi, 1.025 m² each) governs every plate-bending check in the matrix below; demand scales approximately as the square of the panel's smaller diagonal under uniform pressure. Smaller panel types reach D/C ≤ 0.5 in the worst combination; complete numerical records for every panel are retained in reports/acceptance-severe.json.

Joint demand summary

Joint demand has been evaluated by hand calculation (§ VI.1) and by per-triangle traction recovery from the CalculiX shell FE (§ VI.2), 12 216 joint triangles. Hand-calc is conservative; FE recovers actual loading. Both methods PASS at FoS = 2.5.

Short-term deflection and indicative long-term creep.

Serviceability is evaluated for short-term elastic deflection (FE-resolved) and for long-term sustained-load creep (literature-grounded Findley power law). Full creep certification for a multi-decade service life requires specimen-level testing on this foam batch, deferred to § XI.

Short-term deflection

Serviceability deflection criteria from IBC Tbl. 1604.3 are L/360 for floor live load, L/240 for roof live or snow, and L/180 for roof under wind. For a 5.63 m span dome, these translate to 15.6 mm, 23.5 mm, and 31.3 mm respectively.

The C&C peak case is a uniform-application envelope and not a realistic deflection condition (real C&C peaks are localised over a few panels). Under realistic spatial GC_p distribution the apex deflection is approximately the MWFRS value (10.9 mm). The EOR is requested to confirm acceptance of this envelope or to direct the spatially-resolved C&C analysis.

Long-term creep — indicative bound

Rigid PU foam under sustained load creeps. The dome's sustained stress (dead + typical snow) is approximately 2–3 % of short-term compressive yield, well within the linear viscoelastic regime where creep is mild and predictable. A Findley power-law model fitted from rigid PU foam literature (Findley, Lai & Onaran 1976) gives a 50-year creep coefficient of approximately φ = 1.5.

Findley power-law creep:
    ε(t) = ε_0 + ε_t · t^n
    φ(t) = ε_t · t^n / ε_0    (creep coefficient)

    For rigid PU foam at service stress (≤ 5% σ_c):
        ε_t ≈ 0.18 ε_0,    n ≈ 0.25,    50 yr ≈ 4.4 × 10^5 hours
        φ(50 yr) ≈ 0.18 · (4.4e5)^0.25 ≈ 1.5

Apex sustained-load deflection at 50 yr:
    δ_long-term  =  (1 + φ) · δ_day-1
                 =  2.5 × 6.9 mm  ≈  17 mm  (gravity only)
                 =  2.5 × 8.0 mm  ≈  20 mm  (with sustained design snow)

The 50-year apex deflection bound (~ 20 mm under sustained gravity + design snow) is at the boundary of L/240 to L/360 serviceability for a 5.63 m span. This is a serviceability concern, not a strength concern.

Long-term certification beyond this indicative bound requires the additional specimen-level tests enumerated in § XI. The full Findley analysis, including the modulus-substitution FE re-solve and the ponding-instability sensitivity, is documented in reports/2026-05-04--creep-analysis.md with backing data at reports/creep-results-severe.json.

Summary of compliance, with specific design recommendations.

The structure satisfies all short-term limit-state checks at FoS = 2.5; one panel/load-case combination requires either the EOR's acceptance of the FoS = 2.5 margin or one of the reinforcement options below; long-term effects are addressed in § XI.

Statement of compliance (short-term)

At the project default FoS = 2.5, the dome envelope satisfies every ASCE 7-22 strength-design and component-and-cladding load combination evaluated under both the baseline (Risk Cat II, V = 115 mph, p_g = 30 psf, Exposure C) and severe (V = 160 mph, p_g = 100 psf, Exposure D) site envelopes. The worst-case demand-to-capacity ratio is 0.99 at the largest panel under code-extreme wind suction; the next-worst limit state runs at D/C 0.66.

Design options for the EOR's discretion (R2)

The 0.99 D/C result on the largest panel under severe-site C&C peak suction (R2) is at the edge of acceptance even at the project FoS = 2.5; it would not pass at the more conservative FoS = 3.0 sometimes called for on brittle- bending of primary members. The EOR may select any of the following options (or accept R2 as-is at FoS = 2.5):

1. Option A — Thicken the largest panels. Increase Type 1 panels from 76.2 mm to 100 mm laminate. Bending stress scales as (t_old / t_new)² = (76.2/100)² = 0.581. Resulting D/C drops from 0.99 to approximately 0.58 (FoS 2.5) or 0.69 (FoS 3.0), with comfortable margin in either case. Manufacturing impact: 9 panels of one unique geometry; mass increase ≈ 145 kg total.
2. Option B — Internal stiffener rib. Bond a single internal foam (or composite) rib aligned with the long diagonal of each Type-1 panel. Rib width and depth to be sized by the EOR; analytical bending capacity of the composite section approximately doubles for a rib half the panel thickness. Aesthetic impact on interior; QC challenge on adhesive line through-thickness.
3. Option C — Geographic restriction. Restrict deployment to sites with V ≤ 130 mph. At V = 130 mph the q_z-driven C&C peak suction reduces by (130/160)² = 0.66, putting the D/C at approximately 0.65 (FoS 2.5) or 0.78 (FoS 3.0). The "severe" preset analysed in this report is no longer governing. Permitted deployment envelope to be documented in the design intent statement.
4. Option D — Accept R2 at FoS = 2.5. The EOR signs the package as-is, citing the conservative composition of (a) hand-calc envelope using the inscribed rectangle's smaller dimension, (b) C&C peak applied uniformly across the entire panel surface, and (c) full-dome shell-FE confirmation that real load-sharing reduces the same case to D/C 0.61.

Field workmanship and QC programme (R3, R6)

The joint capacity values used throughout this report are laboratory means on carefully fabricated specimens (5–6 coupons per test, ASTM-traceable, lab QSW26030006). Field joints in production will inevitably show greater variance. The following QC items should be incorporated by the EOR or the field-construction QC plan:

• Adhesive specification shall match or exceed the product used in the lab joint specimens. Substitution requires re-testing.
• Surface preparation at every joint bond-line: dry, dust-free, and within the adhesive manufacturer's specified surface-roughness window.
• Cure conditions for the adhesive: log ambient temperature and humidity at lay-up; restrict bond- line work to the manufacturer's specified ranges.
• Witness-coupon testing. Manufacture one ASTM C273 joint-shear coupon per production run, kept in the same conditions as the field joints, tested per § III; require > 0.85× lab mean shear strength to accept the run.
• Mechanical fastener pattern per the EOR's joint detail; fastener torque to be specified and sample-checked.
• Visual inspection of every assembled joint for adhesive squeeze-out, voids, and surface damage.

Inspection and maintenance schedule

The following maintenance-and-inspection schedule should be embodied in the project's Operations & Maintenance manual:

• Annual visual inspection for surface damage (UV-induced surface degradation, impact damage), joint squeeze-out, and any visible deflection at the apex. Sounding-mallet check on every joint accessible from the interior.
• Five-year deflection measurement. Triangulated apex elevation, baseline established at commissioning; trend monitored against the indicative creep bound in § IX.2. Apex sag exceeding 25 mm to be reported.
• Post-storm inspection after any 50-yr- return wind or snow event. Specifically check perimeter foam at curb interface and the apex zone.
• Re-application of UV protective coating per the coating-manufacturer schedule (out of scope of the structural design but interlocked with the serviceability envelope).

What the package establishes, and what the EOR must specifically scrutinise.

This section consolidates the controlling verdict and lists the items the Engineer of Record is asked to scrutinise before stamping. It is an at-a-glance synthesis; full supporting detail is in § I, § VII, and § XI.

Controlling verdict

The structure satisfies every short-term limit-state check at the project default FoS = 2.5. The controlling case is hand-calculated Timoshenko plate bending on the largest panel under the severe-site C&C peak suction combination 0.6D + Wuplift: worst-case D/C = 0.99 — borderline PASS. Eigenvalue buckling on the well-posed shell formulation passes at the project's per-limit-state FoS = 3.0 with worst BLF = 3.356 (D/C = 0.89 on uplift) and BLF = 4.886 (D/C = 0.61 on snow). Joints are not the governing limit state. The four-shell-solver disagreement on the faceted CAD is a documented kink-line singularity at panel-to-panel joints and does not undermine the controlling check.

Items the EOR is asked to specifically scrutinise

1. Borderline R2 result (worst-panel uplift). Accept FoS = 2.5 on the worst-panel bending limit state, or direct one of the reinforcement options enumerated in § X.2 (panel thickening, internal stiffener rib, or geographic restriction).
2. Severe-site uplift and C&C peak envelope. Confirm whether the localised C&C peak, the uniform C&C envelope, or a spatially-resolved C&C distribution is the appropriate basis for the issued check; the package preserves all three for the EOR's review.
3. Foundation curb / geotechnical bearing. The hand-calc bearing check uses a generic 100 kPa allowable; a site-specific geotechnical capacity is required for the actual site (§ XI item 5).
4. Long-term creep and durability. Commission ASTM D2990 1 000-hour creep on this foam batch; DMA T_g; ASTM G155 UV exposure; joint- creep variant of ASTM D1623 — or accept the residual life-cycle risk in writing (§ XI item 1).
5. Field workmanship and joint QC programme. Confirm the witness-coupon-and-inspection regime in § X.3; decide whether to bake an industry knockdown (0.7–0.8×) into joint allowables given the field QC programme.
6. Buckling FoS. Confirm FoS = 3.0 as appropriate for this material and geometry; if a higher FoS is preferred (e.g. IBC §1604.3 reference of 5.0), the uplift case becomes marginal and Option A or C in § X.2 would apply.
7. Cement skin and openings. Confirm structural exclusion of the non-bonded fibre-cement skin for permit purposes; confirm cut-panel detailing (corner-brace plates, perimeter trim) for door and window openings (§ XI items 3 and 4).

Permit-appendix synthesis

A self-contained 9-section synthesis of the analysis package prepared for stamp-review purposes is at reports/2026-05-05--permit-appendix.md. It consolidates the controlling check, the supporting evidence from the OpenSees track (Phases 1–4 plus the smooth-cap and convergence experiments and the homogenized envelope), the out-of-scope risks, the methodology and reproducibility record, and the same EOR-checklist items above. It is intended to give the reviewing engineer a single reading path through the work.

The boundary of what this calculation package certifies.

Acceptance of this package by the Engineer of Record is understood to be subject to the following enumerated limitations. Each is either (a) an item the EOR is asked to address before issuance for construction, or (b) a residual limitation that the EOR records in writing as accepted residual risk.

1. Long-term material behaviour. The laboratory test certificate is short-term only (5 mm/min strain rate). Creep, UV embrittlement, thermal softening, and cyclic-wind fatigue are not certified by this package. Required additional tests on this foam batch:
   • ASTM D2990 1 000-hour creep at service stress (5 % σ_c) at 23 °C and at the maximum design service temperature;
   • 1 000-hour joint-creep test on the joint geometry (ASTM D1623 long-term variant);
   • DMA T_g (dynamic mechanical analysis glass-transition) measurement, ASTM E1640;
   • ASTM G155 UV exposure with periodic strength sampling, on parent and joint specimens;
   • Wind-cycle fatigue test if the project is in a cyclic-wind-dominated environment (S-N curve to 10⁶ cycles).
2. Merged-tet pipeline retired for buckling (R4, R5, R10). The 17/57 725 reoriented tets and the kink-line panel joints make the merged volume mesh unsuitable for eigenvalue buckling: the lowest λ is not deterministic and is traced to mesh artefacts rather than physical instability. A follow-up tolerance-relaxation experiment (reports/2026-05-05--relaxed-tol-buckling.md) confirmed the underlying defect is geometric rather than a meshing-parameter issue, and a separate high-resolution OBJ remesh attempt (reports/2026-05-05--tet-buckling-on-highres-obj.md) produced gmsh PLC errors at the default 30 mm tolerance. The shell-formulation eigenvalue analysis via ccx_buckle.py on the well-posed midsurface (R4) is therefore the issued buckling reference. The merged-tet model is retained only for cross-checking static stress fields away from joints.
3. Cement skin excluded structurally. The screwed-on, non-bonded fibre-cement skin is treated as non-structural per the project owner's instruction. If the skin's structural contribution is to be credited (additional stiffness, UV/thermal protection), the assembly requires independent assembly-level testing and a re-issue of this package.
4. Door / window cutouts. Not modelled in the FE; the un-cut rhombus is used as the conservative envelope. Panels with cutouts are expected to show 2–3× peak stress concentration locally at corners. Cut-panel detailing and reinforcement (corner-brace plates, perimeter trim) to be addressed by the EOR in the architectural shop drawings.
5. Foundation curb panels at grade. Out of scope of structural FE. The 9 perimeter curb-strip panels are in soil contact; PU foam at grade has separate moisture, freeze-thaw, and radon-pathway concerns that fall under civil/geotechnical review. The structural model treats this ring as a fully clamped foundation BC.
6. Joint stiffness in FE. Treated as a perfect bond. Lab-measured G_joint = 24.1 MPa (~76 % of parent G); a cohesive-zone joint model is the next refinement. Issued numbers are conservative for joint capacity (the perfect-bond assumption increases panel-level load-sharing beyond what a real joint provides).
7. Construction-phase loads. Lifting, transport, temporary bracing, and partial-completion wind loads are not addressed. The construction sequence and any temporary bracing scheme are the responsibility of the contractor's engineer.
8. Fire resistance, life safety, MEP penetrations, IBC occupancy classification. Out of scope. PU foam flame-spread classification is governed by IBC chapters not addressed herein and depends on the surface treatment of the finished envelope.
9. The R2 worst-panel result at FoS = 2.5 is an EOR discretion item per § X.2; one of Options A–D is to be selected before issuance for construction.
10. The marginal creep-with-knockdown joint result (§ IX.2) sits at D/C ≈ 1.02 under a conservative sustained-load knockdown. EOR discretion: either accept the marginal value with the knockdown rationale, or commission the joint-creep test that closes the question.

Acceptance of items 1, 9, and 10 (and any other items the EOR elects to defer to residual-risk acceptance) is documented in writing on the EOR Approval form (§ XII).

Final review, disposition, and signature.

Affixing the EOR's wet seal and signature below constitutes acceptance of the analysis as the engineering basis of design, subject to the limitations enumerated in § XI. Disposition of EOR-discretion items (Options A–D, accepted residual risks) is to be recorded on this page.

Disposition of EOR-discretion items

Engineer-of-Record signature

Reproduction, raw data, references.

Every numerical value in this report is reproducible from the source code and input data archived alongside this calculation package. The raw FE artefacts and the laboratory test certificate are referenced here for completeness.

A — Reproduction commands

From the project root (/Users/teacher/…/zomestruct/) run any of the commands below. All numbers in this report are deterministic outputs of these tools; runtime is on the order of seconds to a few minutes.

# Geometry parsing
python3 tools/parse_panels.py                 # rhombus inventory across all DXFs
python3 tools/parse_assembly.py               # bbox + parts breakdown of full assembly OBJ
python3 tools/extract_panels_from_obj.py      # 1.5 GB OBJ → 82 .npz files
python3 tools/fit_panels.py                   # PCA per panel → 9 unique rhombus types

# Hand calc + Tier 1 (single panel) + Tier 2 (spherical-cap dome)
python3 tools/run_check.py severe             # ASCE 7 severe envelope
python3 tools/run_check.py baseline           # ASCE 7 baseline CONUS
python3 tools/run_full_analysis.py severe     # adds Tier 1 + Tier 2 FE
python3 tools/run_full_analysis.py baseline

# Acceptance harness (writes reports/acceptance-{site}.{json,txt})
python3 -m zomestruct test severe
python3 -m zomestruct test baseline
python3 -m zomestruct selftest                # 14 unit tests

# Shell FEA (in-house DKT + CalculiX S3) — issued reference
python3 tools/shell_validate_square.py        # validates DKT vs Timoshenko
python3 tools/shell_dome.py severe            # in-house shell on dome mid-surface
python3 tools/ccx_shell_dome.py severe wind_uplift_main
python3 tools/ccx_shell_dome.py severe wind_cc_peak
python3 tools/ccx_shell_dome.py severe snow_balanced

# CalculiX volume FEA — collapse analysis
python3 tools/ccx_crosscheck.py gravity baseline
python3 tools/ccx_crosscheck.py snow severe
python3 tools/ccx_crosscheck.py uplift severe
python3 tools/ccx_nonlinear.py snow severe 50.0    # 50× design snow
python3 tools/ccx_nonlinear.py uplift severe 60.0

# OpenSees three-way cross-check (R7)
python3 tools/opensees_crosscheck.py gravity baseline
python3 tools/opensees_crosscheck.py snow baseline
python3 tools/opensees_crosscheck.py uplift baseline

# Per-triangle joint traction recovery (R3)
python3 tools/ccx_joint_check.py wind_cc_peak severe
python3 tools/ccx_joint_check.py wind_uplift_main severe
python3 tools/ccx_joint_check.py snow_balanced severe

# OpenSees-track Phase 2/3/4 (R10, R11, R12, shell-formulation buckling)
python3 tools/smooth_cap_4way_compare.py            # four-shell-solver smooth-cap discriminator
python3 tools/shell_convergence_study.py            # mesh-refinement convergence study
python3 tools/tier2_homogenized_envelope.py         # weakest-link homogenized envelope (R11)
python3 tools/opensees_full_check.py baseline       # multi-solver per-panel D/C (R12)
python3 tools/opensees_full_check.py severe
python3 tools/ccx_buckle.py snow baseline           # shell-formulation eigenvalue buckling (R4)
python3 tools/ccx_buckle.py uplift baseline

B — Material test report digest

The full material test certificate (Nanjing Guocai Testing Co., Ltd., document QSW26030006, May 2026) is the laboratory basis of every allowable in this package and is bundled with this report:

• Download the lab test certificate (PDF, 1.0 MB) — QSW26030006, Nanjing Guocai Testing Co., Ltd., May 2026. 5–6 specimens per test, ASTM-traceable methods.
• The canonical digital encoding used throughout the analysis lives at src/zomestruct/material/pu_foam.py on GitHub; see § IV.2 for the values applied in this package.

C — Per-panel inventory

Per-panel surface meshes (extracted from the as-built OBJ, PCA-fitted, and dedup'd) are archived in reports/panels_npz/ — 82 NumPy archives, one per part-block of the source assembly. Indexing convention: panel_NNN.npz contains vertices, faces, centroid, normal, edge_length, acute_angle, type. The 70 structural panels group into the 9 unique types reported in Table 4.

D — Raw FE output index

All FE output files referenced in this report are retained in reports/. The principal artefacts:

E — References

1. American Society of Civil Engineers (2022). ASCE/SEI 7-22 — Minimum Design Loads and Associated Criteria for Buildings and Other Structures. Reston, VA.
2. International Code Council (2024). International Building Code, 2024 Edition.
3. ASTM International. ASTM D1621-16 (2023), D790-17, C273/C273M-20, D1623-17 (2023), D1622-20.
4. APA — The Engineered Wood Association. Y510L — Panel Design Specification, Structural Insulated Panels.
5. Timoshenko, S. P. & Woinowsky-Krieger, S. (1959). Theory of Plates and Shells, 2nd ed. McGraw-Hill, New York. (Plate-bending β table; spherical-cap snap-through.)
6. Batoz, J.-L., Bathe, K.-J. & Ho, L.-W. (1980). "A study of three-node triangular plate bending elements." Int. J. Numer. Meth. Engng. 15(12): 1771–1812. (DKT element formulation.)
7. Findley, W. N., Lai, J. S. & Onaran, K. (1976). Creep and Relaxation of Nonlinear Viscoelastic Materials. North-Holland, Amsterdam.
8. Drucker, D. C. & Prager, W. (1952). "Soil mechanics and plastic analysis or limit design." Quart. Appl. Math. 10(2): 157–165.
9. Dhondt, G. (2004). The Finite Element Method for Three-dimensional Thermomechanical Applications. John Wiley & Sons. (CalculiX theoretical reference.)
10. Mazzoni, S., McKenna, F., Scott, M. H., Fenves, G. L. et al. OpenSees Command Language Manual. University of California, Berkeley, PEER.
11. Nanjing Guocai Testing Co., Ltd. (2026). Test certificate QSW26030006: Zomes PU foam (240 kg/m³) — ASTM-traceable mechanical properties.

F — Glossary & symbols

Frequently used abbreviations and symbols.