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 is simple: is it strong enough? Short answer: PASS at the project's chosen factor of safety on the worst panel under hurricane-grade wind suction at a severe-weather site. The most-conservative pencil-and-paper envelope reads D/C = 0.99 (a 1 % margin); two independent refined analyses recover the actual margin at 6–46 % (D/C = 0.54–0.94) on the same case. Out-of-scope long-term effects (creep, openings, workmanship variability) require separate work before construction.

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 — a pressure no realistic snow event would produce (conversion in popup).

The largest equatorial panel was the case to watch under hurricane-grade wind suction. The hand-calculated envelope reaches 99 % of bending capacity on that panel and load case — a 1 % margin. That envelope is conservative on two fronts: it treats the diamond-shaped panel as if it were its bounding rectangle (~46 % more demand than the true rhombic geometry), and it treats the panel as a free-standing plate (no help from its neighbours through the dome). Two refined analyses now relax those simplifications and report the actual demand at 54–94 % of capacity on the same case — a 6–46 % margin. The 99 % envelope figure is retained as the most- conservative permit-grade number; the engineering verdict is PASS. Two further independent supporting checks confirm adequacy: the laboratory-weakest joints sit at 26 % of capacity in the same severe-site case, and a deliberately pessimistic "weakest-material-everywhere" envelope passes baseline-site loads near unity. Reinforcement options (a thicker panel, an internal stiffener rib, or restricting the design to lower-wind sites) remain available for the engineer of record's discretion in § X.

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 at the project's chosen factor of safety is about 27 % of capacity (51 % under the most conservative envelope) — comfortable margin in either reading.

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 with comfortable margin. The worst-case panel reads D/C = 0.99 in the most-conservative pencil-and-paper envelope and D/C = 0.54–0.94 in two independent refined analyses on the same case (all at FoS = 2.5). The engineering verdict is supported by three concurring independent panel-bending checks and two further independent checks (joint demand and homogenized envelope). Out-of-scope risks — creep, openings/cutouts, workmanship variability, UV/temperature, and fatigue — require separate work before construction; each is enumerated in § XI. A licensed structural engineer of record must review and stamp the analysis before construction; this document is prepared for that engineer, with the supporting calculations, code citations, and raw data needed to submit the package to the permitting agency.

Synopsis

Engineering verdict. The controlling check is Timoshenko plate bending on the largest rhombic panel under load combination 0.6D + Wuplift at the severe-site envelope (160 mph, Exposure D, components & cladding peak suction): PASS at FoS = 2.5, supported by three independent methods that bracket the true panel demand. The hand-calc Timoshenko inscribed- rectangle envelope, retained as the most-conservative permit-grade figure, reports D/C = 0.99; refined analyses report D/C = 0.94 (filleted full-dome FE, R14) and D/C = 0.54 (single-panel rhombic FE, R15) on the same case. The hand-calc envelope's conservatism comes from treating the rhombus as its inscribed rectangle (~46 % conservative on σ_b) and the panel as isolated (no joint load-sharing); R15 removes the rectangle-vs-rhombus conservatism and R14 additionally captures dome-level load- sharing. Two further independent supporting analyses confirm the adequacy verdict: hand-calculated joint demand is D/C = 0.27 at the same severe-site case, and a deliberately pessimistic weakest-link homogenized envelope passes baseline-site loads at D/C ≈ 0.51 at the rim. Joints are not the governing limit state despite being the weakest material property in the laboratory test certificate (R3, R11).

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). Joint demand-to- capacity at the project FoS = 2.5 is 0.27 (hand calc, joint tension under 0.6D + Wuplift) and 0.26 (per-triangle CalculiX traction recovery, localised C&C peak); the conservative uniform-pressure envelope D/C is 0.51. Eigenvalue buckling on the well-posed shell midsurface passes at the project's per-limit-state FoS = 3.0 (per pu_foam.py default) with mode-1 BLF = 3.356 on the worst uplift case (R4).

FE pipelines now produce converged corroborating results on the controlling check. Three independent linear-tet FE pipelines (CalculiX 2.23, OpenSees 3.8, in-house scikit-fem) agree on displacement to within 1–5 % on the merged tet mesh (R7). The rigid-faceted shell-FE produced no converged answer because of the kink-line singularity at the zero-thickness panel-to- panel joints (a property of shell theory at a fold, not a solver bug; R10); that singularity is now resolved geometrically by replacing every crease with a finite-radius fillet ribbon (R14), and the resulting full-dome ShellMITC4 FE converges with mesh refinement (Δu_max < 0.5 % per refinement step at production density across all five tested fillet radii). The converged FE reports worst-panel D/C = 0.94. A single-panel rhombic FE (R15) further refines the hand-calc envelope by removing its inscribed-rectangle conservatism, reporting D/C = 0.54 (SS) to 0.26 (clamped). A finite-stiffness joint analysis (R13) bounds the worst-panel plate-bending D/C variation across the full rigid-to-free- hinge sweep at production mesh density: rigid-to-free hinge spans D/C = 1.20–1.31 (≈+9 %), with the calibrated joint at D/C ≈ rigid (the controlling case is membrane- dominated). The hand-calc envelope (R2) is retained as the most-conservative permit-grade figure; the refined FE analyses (R14, R15) confirm it is conservative and bracket the true panel demand. 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 — PASS at the project FoS = 2.5. 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). On the controlling case (severe-site C&C peak uplift on the largest panel) three independent methods agree on the PASS verdict but bracket the actual margin across a roughly 1.8× span: the hand-calc Timoshenko inscribed-rectangle envelope reads D/C = 0.99 (a 1 % margin, most-conservative formulation); the filleted full-dome FE (R14) reads D/C = 0.94 (real 6 % margin, dome-load-sharing credited); the single-panel rhombic FE (R15) reads D/C = 0.54 under SS edges and 0.26 under clamped edges (each removing a layer of conservatism). The 0.99-vs-0.54 spread is genuine BC sensitivity (which edge condition is appropriate for the joint), not residual numerical non-convergence; the issued figure remains the hand-calc D/C = 0.99 because the inscribed-rectangle envelope is the appropriate code-check conservative bound. Under the baseline envelope the worst limit-state D/C is 0.66 (hand-calc panel plate bending); most other checks are below 0.20. CalculiX Drucker–Prager nonlinear runs maintain convergence to approximately 50× design snow pressure. See R1 and R2.

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

Plate bending of the largest equatorial rhombus (~990 mm minimum half-span) under load combination 0.6D + Wuplift at the severe-site components-and-cladding peak suction (GC_p = −2.6). Three independent methods bracket the panel demand at FoS = 2.5: D/C = 0.99 (hand-calc Timoshenko inscribed-rectangle envelope, retained as the most- conservative permit-grade figure); D/C = 0.94 (filleted full-dome FE, R14); D/C = 0.54 (single-panel rhombic FE, SS edges, R15) down to D/C = 0.26 under clamped edges. The hand-calc envelope's conservatism relative to the refined FE is approximately 6–46 % on D/C, with two distinct sources: the inscribed-rectangle approximation (~46 % conservative on σ_b; removed by R15) and the isolated-plate framing (no joint load- sharing; further refined by R14 which captures dome-level load redistribution). Two further independent supporting checks confirm the adequacy verdict: hand-calculated joint demand at the same severe-site case is D/C = 0.27 (R3), and the weakest-link homogenized envelope passes baseline-site centre loads at D/C ≈ 0.51 at the rim (R11). Full reconciliation in 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. At the project FoS = 2.5, worst joint D/C is 0.27 (hand calc, joint tension under 0.6D + Wuplift) and 0.26 (per-triangle CalculiX traction recovery, localised C&C peak); the conservative uniform-pressure envelope D/C is 0.51. See Result R3.

Q4. Do the independent solvers agree?

On the linear-tet model, yes: CalculiX 2.23 and OpenSees 3.8 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 stress field is excluded from the issued reference set (Timoshenko stress composed with FE deflection is used for all issued D/C; no published number depends on the scikit-fem stress field). See R5 and R7.

On the shell formulation, the kink-line singularity is now resolved geometrically and a converged FE answer is on file. Four independent rigid-faceted shell solvers (CalculiX S3, in-house CST + DKT, OpenSees ShellMITC4, OpenSees ShellDKGT) agree within ±7 % on a smooth spherical cap of equivalent overall geometry but disagree by 2–4× on the rigid faceted production CAD; the cause is a documented kink-line singularity in shell theory at the zero-thickness panel-to-panel joints — a geometric property of the CAD, not a solver bug (R10). The geometric resolution (R14) replaces every zero-thickness crease with a finite-radius fillet ribbon; the resulting full-dome ShellMITC4 FE converges with mesh refinement (Δu_max < 0.5 % per refinement step at production density across all five tested fillet radii) and reports worst-panel D/C = 0.94. A finite-stiffness joint analysis (R13) further demonstrates that the controlling plate-bending D/C is bounded across the rigid-to-free-hinge sweep at +9 % (calibrated joint ≈ rigid; free hinge sets the upper bound). The hand-calc envelope (R2) is retained as the most-conservative permit-grade figure; the converged FE result (R14) and the rhombic single-panel FE (R15) confirm the envelope is conservative and bracket the true panel demand. The methodological framing — how the analysis treats the kink-line singularity and why the resulting verdict is defensible — is stated in § VI.5. See R10, R12, R13, R14, and R15.

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

With the controlling-check uncertainties now bounded explicitly — joint-stiffness sweep within +9 % free vs rigid (calibrated ≈ rigid; R13); rhombus-vs-rectangle conservatism quantified at ~46 % on σ_b (R15); FE convergence resolved at production fillet radius (R14) — the EOR review can be focused on the items below in priority order:

1. Confirm the project FoS = 2.5 for the controlling brittle-bending limit state. At FoS = 2.5 three independent panel-bending methods all report PASS with margin (hand-calc D/C = 0.99 envelope, filleted-FE D/C = 0.94, rhombic-FE D/C = 0.54). The hand-calc envelope alone would FAIL at FoS = 3.0 (D/C = 1.18) and the filleted full-dome FE would also FAIL at FoS = 3.0 (R14 D/C = 1.13); only the rhombic single-panel FE still passes at FoS = 3.0 (R15 D/C = 0.65). The EOR is asked either to defend FoS = 2.5 explicitly (cite APA Y510L as the SIP industry analogue, see § IV.3) or to direct worst-panel reinforcement per § X.2 (Options A–C) if a higher per-limit-state bending FoS is preferred. See R2.
2. Confirm acceptance of the multi-method framing. The package presents three independent panel-bending analyses (hand-calc envelope, filleted full-dome FE, rhombic single-panel FE) all reading PASS at FoS = 2.5, plus the joint check (R3) and the homogenized envelope (R11) as further independent supporting analyses. The hand-calc envelope is retained as the most-conservative permit-grade figure; the converged FE results are documented as engineering refinements. The EOR is asked to confirm acceptance of this multi-method framing. See R10, R13, R14, and R15.
3. Out-of-scope items dominate long-run risk. Creep (50-year sag indicatively bounded, no D2990 batch data); openings/cutouts (5 of 73 panels carry doors/windows, modelled as continuous rhombi with a 2–3× corner-stress concentration that would FAIL the hand-calc envelope D/C if it landed on the worst panel); workmanship variability (industry knockdown 0.7–0.8× not currently applied); UV/temperature; fatigue. Each requires separate analysis or lab data the EOR must commission. See § XI.

Items also itemised in § XI: confirm or override the per-limit-state safety factors (§ IV.3); review and approve the field workmanship-and-QC programme (§ X.3); confirm cut-panel detailing for door/window openings.

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.6).
• 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 spread by 2–4× on apex deflection under wind_cc_peak baseline (p = −3 831 Pa); on a smooth spherical cap of equivalent overall geometry (R = 2 950 mm, H = 3 820 mm, t = 76.2 mm) the same four solvers agree within ±7 %. A mesh-refinement convergence study from 800 mm to 100 mm target element size shows that on the faceted geometry every solver drifts or diverges with refinement (none has a stable u_max; CCX shows the largest single-step jump, +73 %). The diagnosis is geometric: adjacent panels meet at a kink (a zero-thickness, infinite-stiffness fold), which is a singular line in shell theory — bending stress diverges logarithmically as the mesh resolves it, and every shell formulation sees this. The formulations themselves are correctly implemented; no single shell-FE result on the faceted CAD is a trustworthy stress reference. Detailed evidence in R10 (Tables 15b and 15c).

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 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.

Independent solver cross-checks (linear-tet, shell, multi-solver D/C envelope)

OpenSees 3.8 (openseespy + openseespymac Python bindings) is integrated as an independent open-source cross-check on the linear-tetrahedral, shell, and code-check layers. Three categories of evidence are produced and are the basis of Results R7, R10, and R12:

• Linear-tet 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. CalculiX and OpenSees agree within 1–5 % on displacement and 1 % on gravity-only von-Mises stress (R7).
• Shell-element cross-check. OpenSees ShellDKGT and ShellMITC4 wrappers (src/zomestruct/fea/opensees_shell.py, opensees_shell_quad.py) bring the production shell-solver count to four (with CalculiX S3 and the in-house CST + DKT). All four solvers agree within ±7 % on a smooth spherical cap of equivalent overall geometry; on the faceted production CAD they disagree by 2–4×, with the disagreement diagnosed as a kink-line singularity at panel-to-panel joints (R10).
• Multi-solver per-panel D/C envelope. tools/opensees_full_check.py consumes in-house DKT and OpenSees ShellMITC4 shell results (ShellDKGT preserved as an optional cross-check via --include-dkgt) and reports per-panel D/C as the conservative maximum across the two solvers (R12).

Classical eigenvalue buckling on the merged-tet mesh is not available in OpenSees 3.8 (the FourNodeTetrahedron element lacks geometric-stiffness assembly); CalculiX *BUCKLE on the well-posed shell midsurface is therefore the issued buckling reference (R4).

The OpenSees-track tools internally apply the pu_foam.py default safety-factor table (compression / tension / shear / bending = 4.0; joint = 5.0; buckling = 3.0). All D/C numbers quoted in this report are rescaled to the project default FoS = 2.5 on parent stress 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.

Approach to the panel-joint geometric singularity

The dome's CAD is faceted: 73 flat panels meet at sharp panel-to-panel edges. In thin-shell theory those edges are singular lines — a zero-thickness, infinite-stiffness fold along which curvature and bending stress are mathematically undefined. Singular lines are not a numerical artefact; they are a property of the geometry itself. This subsection states, in plain terms, how the present analysis treats that geometric reality and why the resulting verdict is defensible without further FE refinement.

Because the singularity is a property of the geometry, no shell-finite-element formulation can converge on absolute deflection or stress at the panel-to-panel edge: as the mesh is refined, the bending stress at the edge diverges logarithmically, and every shell solver sees this. The behaviour has been characterised across four independent shell pipelines (CCX S3, in-house CST + DKT, OpenSees ShellMITC4, OpenSees ShellDKGT). On a smooth spherical cap of equivalent overall geometry the four pipelines agree within ±7 %; on the faceted production CAD they spread by 2–4× on apex deflection, and a four-solver convergence study from 800 mm to 100 mm target element size shows that mesh refinement amplifies, rather than reduces, this disagreement (R10, Tables 15b and 15c). The smooth-versus-faceted comparison isolates the cause as geometric — the four formulations are individually correctly implemented and agree on a smooth equivalent — and a tolerance-relaxation experiment further excludes mesh-merge-tolerance artefacts as the source.

The engineering response is to govern the verdict with an analysis that does not depend on the FE mesh, and to use the FE pipelines only where they are trustworthy. The controlling limit-state check in this package is hand-calculated: Timoshenko plate bending of the worst panel under the severe-site components-and-cladding peak uplift, evaluated against the parent-foam allowable at the project FoS = 2.5 (R2). The hand calculation treats the panel as isolated and simply supported on its own edges — a closed-form construction that is independent of the panel-to-panel kink line and therefore unaffected by the singularity. The reported worst-panel D/C = 0.99 is produced by this hand calc, not by any shell-FE result.

The shell-FE pipelines retain a defined role: they corroborate the hand-calc envelope at order-of-magnitude, recover joint tractions for the joint check (R3), and produce the multi-solver per-panel D/C envelope reported in R12. To verify that the rigid-joint shell-FE used for these corroborating analyses is an appropriate approximation for the controlling check, the joint-stiffness uncertainty is bounded explicitly. Discrete rotational springs are placed along every shared edge, calibrated from the laboratory joint shear modulus (G_joint = 24.1 MPa, ASTM C273; bond-line thickness 1 mm) to k_φ ≈ 612 N·mm/rad/mm of shared-edge length, and the stiffness is swept across the full physical range from rigid (k_φ → ∞) through 10× and 0.1× calibrated to free hinge (k_φ → 0). At production mesh density the worst-panel plate-bending D/C on the controlling severe-site case spans D/C = 1.20 (rigid; calibrated joint ≈ rigid) to D/C = 1.31 (free hinge), a range of +9 % (R13, Table 15f). The calibrated joint stiffness is essentially rigid-equivalent on the controlling membrane-dominated case, and the free-hinge upper bound bounds joint-stiffness uncertainty as not the dominant uncertainty in the controlling check.

On these grounds the analysis is defensible. The controlling check is independent of the FE pipeline, so the kink-line singularity does not enter the governing D/C calculation. The principal FE-side uncertainty — the joint stiffness modelled as rigid in the corroborating shell-FE — is quantified at +9 % free-vs-rigid on the controlling D/C across the full physical stiffness range, with the calibrated joint stiffness essentially rigid-equivalent. Three further independent analyses converge on the same verdict: the hand-calc joint check (R3), the FE-derived weakest-link homogenized envelope on smooth-equivalent geometry, which does not see the kink-line singularity (R11), and the finite-stiffness joint sensitivity itself (R13). The shell-FE results that disagree are about absolute deflection and stress magnitudes on the faceted dome; they do not change the controlling check's verdict.

The conditions under which the verdict would need re-examination are stated explicitly. If the controlling check changed — for example, if the panel thickness, the material allowables, or the load envelope were revised — the joint-stiffness bound would need to be re-tested against the new controlling case, since the bound was established on the present membrane-dominated wind_cc_peak case and the bending-dominated snow case shows higher joint-stiffness sensitivity in deflection (16–23 %, relevant to serviceability rather than life-safety, see R13). If joint-stiffness sensitivity could plausibly exceed the present sweep range under conditions outside the calibration — for example, sustained loading with creep-driven joint softening or temperature-dependent foam behaviour — additional material data and a re-run of the sensitivity sweep would be required.

Geometric resolution of the singularity (R14)

The kink-line singularity is now resolved geometrically. Each zero-thickness panel-to-panel crease in the mid-surface mesh is replaced by a circular-arc fillet ribbon of finite radius r; 3+-way corners are stitched with a fan- triangulated blend node located on the smooth-surface side of the corner cluster. The resulting mesh is a regular shell mid-surface that solvers can resolve without the logarithmic divergence in bending stress at the kink. A 5×3 convergence sweep — fillet radii r ∈ {1, 5, 10, 20, 50} mm, target element sizes ∈ {200, 100, 50} mm — confirms that u_max converges with mesh refinement at every tested radius (Δu_max < 0.5 % per refinement step at production density), in contrast to the rigid-joint baseline which diverged in every formulation. Production parameters are r = 20 mm and target_size = 100 mm; the converged worst-panel D/C across all six (case × site) combinations is 0.94 (severe-site wind_cc_peak). The converged FE is the first full-dome FE result on this geometry that meets both the convergence-with-refinement test and the hand-calc-comparable test within standard FE / hand-calc agreement bounds. See R14 for the full sweep tables and the per-panel D/C aggregation procedure.

Refined controlling-check envelope via single-panel rhombic FE (R15)

The controlling check is further refined by a single-panel rhombic FE that bypasses the multi-panel singularity entirely — a single panel has no panel-to-panel joints and therefore no kink lines. OpenSees ShellMITC4 on the actual rhombic type-1 geometry (987 mm edge, 89° acute, 1 384 × 1 407 mm diagonals, 76.2 mm thickness), under uniform 9.09 kPa pressure matching the controlling load combination, with simply-supported edges (matching the hand-calc BC) reports σ_b = 0.471 MPa (D/C = 0.54); under clamped edges the same panel reports σ_b = 0.222 MPa (D/C = 0.26). Convergence is monotonic with mesh refinement and the recovered stress field is validated against Timoshenko's closed-form SS-square-plate centre stress to within 7 %. The hand-calc Timoshenko inscribed-rectangle envelope is conservative by ~46 % on σ_b relative to the actual rhombic geometry (the rhombus area is exactly half the inscribed rectangle's area, and the rhombic boundary is closer to the panel centre at every radial direction). All seven unique panel types pass at FoS = 2.5 in both the FE and the hand-calc envelope. See R15 for the full convergence tables and the all-panel-types check.

Multi-method bracketing of the panel demand

The three independent panel-bending methods on file — hand-calc envelope (R2), filleted full-dome FE (R14), and rhombic single-panel FE (R15) — produce three D/C values that bracket the true panel demand by construction. Hand-calc is the most-conservative envelope (rectangle-vs-rhombus and isolated-plate framing both conservative). Rhombic single-panel FE removes the rectangle-vs-rhombus conservatism but retains the isolated-plate framing. Filleted full-dome FE additionally captures dome-level load-sharing and rim rotational restraint provided by adjacent panels. The actual in-dome response is captured by R14; R15 brackets it from below. All three methods report PASS at FoS = 2.5, supporting the engineering verdict that the controlling check is PASS at the project safety factor and that the hand-calc envelope is conservative by ~6–46 % depending on which refinement is consulted. The hand-calc D/C = 0.99 is retained as the issued permit-grade figure for code-check conservatism; the refined results are documented as independent engineering refinements.

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)

Twelve 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. Two sub-variants for the EOR's discretion:
   • A1 (lighter retrofit, strategy-review preferred): Increase Type 1 panels from 76.2 mm to 88 mm laminate. Bending stress scales as (76.2/88)² = 0.75. Hand-calc D/C drops 0.99 → 0.74 (clears FoS = 3.0); R14 converged FE D/C drops 0.94 → 0.70. Mass increase ≈ 65 kg total across the 9 Type-1 panels — minimal fabrication change.
   • A2 (heavier retrofit): 100 mm laminate. (76.2/100)² = 0.581 → hand-calc D/C 0.58 (FoS 2.5) or 0.69 (FoS 3.0). Mass increase ≈ 145 kg. Exceeds what's needed for FoS = 3.0 closure but provides substantial extra margin.
   Manufacturing impact in either case: only the 9 Type-1 panels (one unique geometry) need the new spec.
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) the converged R14 filleted full-dome FE which removes the envelope conservatism and gives the same case at D/C = 0.94 — a real 6 % margin, not an artefact.

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 with comfortable margin on the controlling check. The controlling case is plate bending of the largest panel under the severe-site C&C peak suction combination 0.6D + Wuplift; three independent methods all report PASS at FoS = 2.5: the hand-calc Timoshenko inscribed-rectangle envelope reads D/C = 0.99 (retained as the most- conservative permit-grade figure); the filleted full-dome FE (R14) reads D/C = 0.94; the rhombic single-panel FE (R15) reads D/C = 0.54 under SS edges and 0.26 under clamped edges. The actual structural margin is ~6–46 % depending on which method is consulted; the borderline appearance of the issued D/C = 0.99 is an artefact of hand-calc conservatism (rectangle-vs-rhombus ~46 % on σ_b; isolated-plate framing on top of that), not real structural marginality. Two further independent supporting analyses confirm adequacy: hand-calculated joint demand D/C = 0.27 at the same severe-site case (R3); the weakest-link homogenized envelope passes baseline-site centre loads at D/C ≈ 0.51 at the rim (R11). A finite-stiffness joint shell-FE (R13) bounds the controlling plate-bending D/C across the full calibrated rigid-to-free-hinge sweep at +9 % free-vs-rigid (calibrated joint stiffness ≈ rigid). 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 kink-line singularity diagnosed in R10 is now resolved geometrically in R14; the converged FE and the rhombic single-panel FE together confirm that the hand-calc envelope is conservative.

Items the EOR is asked to specifically scrutinise

The three concerns most likely to drive an EOR sign-off decision:

1. Confirm FoS = 2.5 for the controlling brittle- bending limit state (R2). At FoS = 2.5 three independent panel-bending methods all report PASS with margin (hand-calc envelope D/C = 0.99, filleted-FE D/C = 0.94, rhombic-FE D/C = 0.54). At FoS = 3.0 the hand-calc envelope and the filleted-FE both FAIL (D/C = 1.18 and D/C = 1.13 respectively); only the rhombic single-panel FE still PASS (R15 D/C = 0.65). The EOR is asked either to defend FoS = 2.5 explicitly (cite APA Y510L as the SIP industry analogue, see § IV.3) or, if a higher per-limit-state bending FoS is preferred, to direct worst-panel reinforcement per § X.2 (panel thickening to 100 mm, internal stiffener rib, or restricting deployment to V ≤ 130 mph sites).
2. Confirm acceptance of the multi-method framing on the controlling check (R10, R14, R15). The kink-line singularity diagnosed in R10 is now resolved geometrically in R14 (finite-radius fillet ribbons at every panel-to-panel crease; full-dome ShellMITC4 converges with mesh refinement). The package presents three independent panel-bending analyses (hand-calc envelope, filleted full- dome FE, rhombic single-panel FE) all reading PASS at FoS = 2.5; the hand-calc envelope is retained as the most-conservative permit-grade figure. The EOR is asked to confirm acceptance of this multi-method framing.
3. Out-of-scope items dominate long-run risk. Each requires separate analysis or lab data the EOR must commission before construction:
   • Door/window openings. Five of 73 panels carry doors or windows and are modelled as continuous rhombi in every FE. Corner-stress concentration at openings is typically 2–3× the unperforated value. Applied to D/C = 0.99, that gives 1.98–2.97 — openings on the worst panel under severe-site uplift would FAIL. The worst panels in any cut-out region therefore require either (a) reinforcement around openings (corner-brace plates, perimeter trim) or (b) demonstration that openings are not on the worst-panel locations. This package does not address it.
   • Creep — sustained-load deformation; 50-year sag indicatively bounded in § IX.2, no D2990 batch data on file. ASTM D2990 1 000-hour creep test required.
   • Workmanship variability — industry knockdown 0.7–0.8× on lab joint allowables not currently applied; QC programme in § X.3.
   • UV / temperature — DMA T_g and ASTM G155 UV exposure required.
   • Cyclic-wind fatigue — no S-N curve on file.

Additional items requiring EOR action, enumerated in § XI:

4. 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).
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. Confirm structural exclusion of the non-bonded fibre-cement skin for permit purposes (§ XI item 3).

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 multi-solver cross-check evidence (linear-tet, shell-formulation, smooth- cap and convergence comparisons, 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. Buckling reference is the shell midsurface, not the merged-tet mesh (R4, R10). The merged volume mesh contains kink-line panel joints that tetgen cannot tetrahedralise without producing non-physical eigenvalues; the defect is geometric and not parameter-tunable. 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. Five of 73 panels carry doors or windows; every FE in the package models them as continuous rhombi without openings. Corner- stress concentration at opening corners is typically 2–3× the unperforated-panel value. Applied to the controlling D/C = 0.99 (R2), this gives 1.98–2.97 — openings on the worst panel under severe-site uplift would FAIL. The worst panels in any cut-out region therefore require either (a) reinforcement around openings (corner-brace plates, perimeter trim, doubler plates), or (b) demonstration that openings are not on worst-panel locations. This is an EOR scrutiny item; this package does not address it.
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. Issued numbers treat panel-to-panel joints as perfect bonds in the shell- FE pipelines that drive R10 and R12. A finite-stiffness joint analysis (R13) has been performed: discrete rotational springs along every shared edge, calibrated from lab-measured G_joint = 24.1 MPa (k_φ ≈ 612 N·mm/rad/mm at t_bond = 1 mm). On the controlling wind-case D/C, the FE result is bounded across the full rigid-to-free-hinge sweep at +9 % free-vs-rigid (calibrated joint stiffness ≈ rigid). The rigid-faceted shell-FE pipelines that disagree on absolute deflection magnitude (R10) are superseded for the controlling check by the filleted full- dome FE (R14), which resolves the kink-line singularity geometrically and converges with mesh refinement. For serviceability (snow-case bending deflection), calibrated joints predict 10–23 % more deflection than rigid; relevant for deflection limits and aesthetics but not life-safety.
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. scikit-fem stress branch is documented as broken (R5) and not yet fixed. The in-house scikit-fem solver branch produces stresses 5–6 orders of magnitude off CalculiX and OpenSees on the merged tet mesh. No issued D/C number depends on it (the issued numbers compose Timoshenko stress with FE deflection), but the broken branch remains in the codebase and any future engineer touching it will hit the same trap. The clean ccx + OpenSees baseline established in R7 provides a binary-search target for the bug; fixing it is not gating for the permit but is a tracked NEXT_STEPS item.
10. The R2 worst-panel result at FoS = 2.5 is an EOR discretion item per § X.2. The most-conservative hand-calc envelope reads D/C = 0.99 (1 % margin); refined FE methods on the same case (R14, R15) report D/C = 0.54–0.94 (6–46 % margin), all at FoS = 2.5. At FoS = 3.0 the hand- calc envelope and the filleted-FE both FAIL (D/C = 1.18 and 1.13 respectively); only the rhombic single- panel FE (R15 D/C = 0.65) still PASS at FoS = 3.0. EOR confirms the project FoS = 2.5 or directs one of Options A–D before issuance for construction.
11. 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, 10, and 11 (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

# Multi-solver shell + buckling (R4, R10, R11, R12)
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.