# 2026-05-04 — Creep analysis, Zomes PU foam zonohedron dome **Structure**: 5.6 m diameter × 3.8 m tall polar zonohedron, 73 rhombic panels of 76.2 mm (3 in) high-density polyurethane foam (240 kg/m³). **Material data**: ASTM-tested at Nanjing Guocai (QSW26030006), May 2026 — **short-term only**, no creep data on file. This document complements [`2026-05-04--findings.md`](2026-05-04--findings.md), which flagged creep as the largest unaddressed risk (Finding 6). It collects the literature-based assumptions we can defensibly use *today*, quantifies expected long-term behaviour, applies them to the existing FE results, and identifies what additional testing would be needed before a licensed engineer could stamp the structure for a multi-decade service life. **Sections 11–12 record the actual outcomes** of running the analysis on the existing dome mesh — they are the load-bearing results; sections 1–10 are the framework that produced them. --- ## TL;DR Creep is the gradual deformation of the foam under sustained load. The dome's sustained stress (dead load + typical snow) is **~2–3 % of short-term compressive yield** — well inside the linear viscoelastic regime where rigid PU foam creep is mild and predictable. Using a literature-grounded Findley power-law model with a 50-year creep coefficient **φ ≈ 1.5**, the apex sags by an estimated **17 mm at 50 years under sustained gravity** (vs. 6.9 mm elastic on day one), rising to **~20 mm with sustained design snow** — at the L/240 to L/360 serviceability boundary for a 5.6 m dome. **A new marginal joint failure mode emerged**: severe-site wind C&C peak joint tension reaches D/C = 1.024 under conservative blanket knockdown of joint allowables — the second flagged severe-site issue alongside Finding 2's worst-panel bending. **A new mesh-quality finding emerged**: the modulus-substitution FE re-solve produces non-physical per-node displacement ratios (0.03×–6.0× when E is halved, instead of the exact 2.0× linear elasticity guarantees), so the theoretical `(1+φ)·δ_day1` is the trustworthy long-term predictor on this mesh until NEXT_STEPS item 1 is closed. **Still required for an actual stamp**: 1000-hour parent-foam creep test, 1000-hour joint creep test, DMA Tg measurement, ponding-instability analysis, and a clean re-meshing of the assembly. --- ## Plain-English summary Polyurethane foam is a plastic, and plastics flow slowly under sustained load — they "creep." The dome's panels carry their own weight every minute of every day for the next 20+ years, and the foam will gradually deform under that load even though the load is well below what would break it. **The good news**: stress levels in this dome are tiny — about 2–3 % of the foam's short-term strength — so creep is in its mild, predictable regime. **The headline number**: the dome's apex sags about **7 mm on day one** and is estimated to sag an **additional ~10 mm over 50 years** from creep, for a total of about **17 mm at long term** under gravity alone — or **about 20 mm with sustained snow included**. That's about three quarters of an inch — a serviceability concern, not a safety one, and right at the boundary of what civil engineers consider acceptable for a roof of this span. **A new joint concern came out of the analysis**: when we apply a 50 % knockdown to joint capacity (industry practice for adhesive-bonded joints under sustained load) and then check the worst gust load, the joint demand-to-capacity ratio is **just barely over 1.0** at the severe-weather site. This is a marginal flag, not a definitive failure — it depends on whether you believe gust loads see fully creep-degraded joint strength (conservative view) or whether fast-rate loads see a stiffer effective adhesive (less conservative view, but defensible). Either way it joins the worst-panel-bending issue from the original findings as a thing that needs an engineer's call. **A new caveat about the FE machinery**: when we tried to verify the 50-year deflection by re-running the FE with a reduced "long-term" modulus, the answers came back **non-physical** — different nodes scaled by anywhere from 0.03× to 6× when the modulus was halved, instead of the uniform 2× that linear elasticity requires. This is the **same mesh-quality issue that the original findings flagged for buckling** (Finding 4), now showing up in creep too. The fix is to rebuild the merged mesh; in the meantime, the simple `(1 + φ) × day-1` multiplication is the trustworthy long-term deflection number. **The real risks** (unchanged from the original framing): 1. Hot summer roof temperatures (60+ °C) speed creep up by 5–10×. 2. Sustained snow that sags the dome could create a ponding feedback loop where sag collects more snow, which causes more sag. 3. Joints likely creep faster than the parent foam (adhesive layers always do) — and we now have a marginal D/C to back this up. 4. UV embrittlement over decades changes the load-bearing cross-section. We can write down a defensible assumption set today using literature on rigid PU foam, but the dome will need actual long-term creep tests on this specific foam batch — ideally a 1000-hour Findley fit at service stress and elevated temperature — before an engineer can sign off on a 20-year service life. --- ## 1. What is creep, and why does it matter for this dome? Creep is **time-dependent strain under sustained stress**. For a polymer like PU foam, even a stress well below the short-term failure load will cause continuous, slow deformation — strain that increases without bound (in principle), but at a decreasing rate. For this dome, three properties of creep matter: 1. **It accumulates over the service life.** A short-term FE analysis tells you the deflection on day 1. Creep tells you the deflection on day 7,300 (20 years) under the same load. 2. **It is dominated by sustained loads, not peak loads.** Wind gusts contribute almost nothing — the load is on for seconds. Snow that stays on for weeks contributes meaningfully. **Self-weight, which is on continuously, dominates.** 3. **It is highly temperature-sensitive.** Roof temperatures vary wildly across a year. Creep rate at 60 °C is roughly an order of magnitude higher than at 23 °C. The lab tests at Nanjing Guocai measured strength at a strain rate of 5 mm/min — a few seconds of loading. Those tests revealed nothing about behaviour at strain rates of ~10⁻¹⁰ /s (which is what 20 years of sustained loading looks like). **Creep is a different physical regime that requires different tests.** --- ## 2. The standard model: Findley's power law For rigid PU foam, the workhorse creep model is **Findley's power law** (Findley & Khosla 1955, refined for foams by Huang & Gibson 1990, extensively validated for rigid PU in the insulation and sandwich- panel industries): $$\varepsilon(t) = \varepsilon_0 + m \cdot t^{\,n}$$ where | Symbol | Meaning | Typical value, rigid PU | |---|---|---| | ε₀ | Instantaneous elastic strain at load application | σ / E | | m | Creep amplitude coefficient | function of σ, T, ρ | | t | Time under load | seconds, hours, or years | | n | Creep exponent | **0.20–0.30** | The exponent n is remarkably consistent across rigid PU formulations — the literature converges around 0.25 within ±20 %. The amplitude coefficient m is what you actually need lab data for. **Why power law and not exponential?** Polymer creep does not asymptote to a finite value at low temperature; it approaches a finite value only as t → ∞ via the very slow t^{0.25} approach. For practical purposes (50 years), this is functionally an asymptote. **Practical use**: an experimental fit of (m, n) at one stress and temperature can be extrapolated to longer times *with confidence* (the log-log plot is straight) but only **modestly** to higher stresses or temperatures. --- ## 3. Empirical regularities for rigid PU foam The following are the load-bearing assumptions in any creep analysis, drawn from the rigid-PU-foam literature (insulation industry, sandwich panel testing, NASA foam reports, structural insulated panel certifications): ### 3.1 Stress linearity threshold Below approximately **25–30 % of short-term compressive yield σ_y**, creep is roughly linear viscoelastic: doubling the stress doubles the creep strain at a given time. Above ~30 %, creep becomes increasingly nonlinear: | σ / σ_y | Creep multiplier vs. linear extrapolation | |---|---| | 10 % | 1.0 (linear) | | 25 % | 1.0–1.2 | | 40 % | 2–3× | | 50 % | 3–5× | | 60 % | 5–10×, may not stabilise | | > 70 % | tertiary creep, runaway possible | For this dome, with σ_y = 2.47 MPa, the linearity threshold is **~0.62 MPa**. The CalculiX p99 von Mises stress under design loads is ~0.06 MPa — a factor of 10 below threshold. **We are very comfortably in the linear regime** under design conditions. ### 3.2 Density dependence For the polymer matrix that does the creeping, creep compliance scales roughly as ρ⁻^k where k ≈ 2.5–3 (Gibson & Ashby cellular solids scaling). Going from a typical 32 kg/m³ insulation foam to 240 kg/m³ HD foam reduces creep compliance by a factor of order **(240/32)^2.5 ≈ 150×**. This is why HD foam is structurally usable at all — and it's why literature on low-density insulation foam creep is *qualitative* guidance for this dome rather than directly applicable. ### 3.3 Temperature dependence Rigid PU has a glass transition Tg in the range **80–110 °C** depending on formulation. Below Tg, creep rate roughly doubles every **7–10 °C** (Arrhenius-like behaviour). | Temperature | Creep rate vs. 23 °C | |---|---| | 23 °C (lab) | 1.0× (reference) | | 40 °C (mild summer) | ~3× | | 50 °C (warm roof) | ~6× | | 60 °C (hot roof, direct sun) | ~12× | | 70 °C (extreme roof temp) | ~25× | | > 80 °C | approaching Tg, uncontrolled | **This is the dominant creep risk for this dome.** A black or dark- colored roof in direct sun easily reaches 60–70 °C internal temperatures. The lab tests were at 23 °C, in the dark. **The cement skin previously excluded from the analysis has thermal/UV protective value that would matter here.** ### 3.4 Moisture dependence Closed-cell HD PU absorbs ≤ 2 % moisture by volume and is largely unaffected. Surface layers degraded by UV become more hygroscopic. **Moisture is a coupling effect with UV, not a primary driver.** ### 3.5 50-year creep coefficient φ Industry codes (DIN 53291, EN 14509 sandwich panels) report a "creep coefficient" defined as $$\varphi = \frac{\varepsilon_{\text{creep,total}}}{\varepsilon_{\text{elastic}}}$$ i.e., total creep strain divided by initial elastic strain, evaluated at the design service life. | Sustained σ / σ_y | Typical 50-yr φ, rigid PU at 23 °C | |---|---| | 5 % | 0.5–1.0 | | 10 % | 0.5–1.2 | | 25 % | 1.0–2.0 | | 40 % | 2.0–4.0 (uncertain) | | > 50 % | not safe to assume finite asymptote | **For this dome at ~3 % of σ_y**, a conservative estimate is **φ ≈ 1.5**. This bakes in some allowance for non-ideal field conditions (variable temperature, occasional excursions in load). --- ## 4. Defensible assumption set for this dome Given: - σ_y (compressive parent foam) = 2.47 MPa - E (instantaneous parent foam) = 70.8 MPa, ν = 0.30 - p99 von Mises under design load (CalculiX) ≈ 0.05–0.06 MPa - Sustained stress ratio σ_sustained / σ_y ≈ 2–3 % - Service life target: 50 years (typical building stamp horizon) - Install climate: not specified — **assume CONUS continental, with occasional roof excursions to 50 °C** **Recommended assumption set:** | Parameter | Symbol | Value | Source / justification | |---|---|---|---| | Findley exponent | n | 0.25 | Mid-range rigid PU literature | | Linear viscoelastic threshold | σ_lin | 0.25 σ_y = 0.62 MPa | Industry rule of thumb | | 50-yr creep coefficient (parent foam, sustained service stress, mean climate temperature) | φ_parent | 1.5 | Conservative for σ < 5 % σ_y | | Effective long-term modulus | E_∞ | E / (1 + φ) ≈ 28 MPa | Direct from φ | | Joint creep coefficient (assumed) | φ_joint | 3.0 | 2× parent; adhesive layers always creep faster | | Effective long-term joint stiffness | E_∞,joint | E_joint / (1 + φ_joint) | Use whatever joint stiffness is in the FE | | Temperature derate factor (creep rate) | k_T | 2× per +10 °C above 23 °C | Arrhenius shorthand | | Long-term allowable sustained stress | σ_LT,allow | 0.5 × σ_short-term,allow | DIN 53291, EN 14509 sandwich panel practice | | Sustained loads to apply | — | 1.0 D + 0.3 S | Dead load + 30 % of design snow (long-duration component) | | Wind contribution to creep | — | **0** | Wind is short-duration, does not creep | **Headline implication**: in principle, any existing elastic FE run for sustained loads (gravity, gravity + sustained snow) can be converted to a 50-year creep result by **substituting E_∞ ≈ 28 MPa for E**, and the deflection should scale by (1 + φ) = 2.5×. **In practice, on the current merged mesh, this substitution does not behave linearly** — see §11.1 for empirical results. The theoretical multiplication `(1 + φ) × δ_day1` is the trustworthy long-term predictor on this mesh; the modulus-substituted FE re-solve is not. Stresses are unchanged at first order regardless (creep is mostly strain redistribution, not stress redistribution, in a statically determinate-ish system). --- ## 5. Quantitative estimates for this dome ### 5.1 Long-term apex deflection — final estimate | Quantity | Value | Method | |---|---|---| | Day-1 elastic max ‖u‖ (gravity) | **6.89 mm** | CalculiX, E = 70.8 MPa | | Creep increment over 50 years | **+10.3 mm** | φ × δ_day1 = 1.5 × 6.89 | | 50-year max ‖u‖ (gravity only) | **17.2 mm** | (1 + φ) × δ_day1 | | 50-year max ‖u‖ + sustained snow @ 30 % | **~20 mm** | scaled load addition | | Dome diameter | 5,600 mm | — | | Long-term sag / diameter | **L/280** | 5600 / 20 | Civil engineering serviceability limits for roofs run L/240 to L/360 (corresponding to 23 mm and 16 mm respectively at this span). **The 20 mm long-term estimate is between L/280 and L/329** — within the acceptable band for a non-precision interior, but not by a wide margin. An engineer of record would want this number explicitly disclosed. The empirical CalculiX re-solve at E_∞ = 28.32 MPa returned 20.3 mm (max), 16.7 mm (p99), and 10.1 mm (p95) — see §11.1. These do **not** satisfy linear-elastic 1/E scaling and should not be used as the long-term deflection prediction. ### 5.2 Long-term stress under creep Stresses do not change much from creep alone in this geometry, because the dome is largely shell-membrane-dominated. The main mechanism by which creep raises stress is **redistribution toward stiffer load paths** — and in this dome, all load paths are nominally the same material with the same φ, so there's nothing to redistribute toward. **Exception**: joint adhesive creeping faster than parent foam (φ_joint = 3 vs. φ_parent = 1.5) would gradually transfer load from joints to panel-spanning paths. This is **stabilising** for the joint stress problem (joints unload over time) and **destabilising** for the worst-panel bending problem (Finding 2): if joints creep, panels carry more load in isolation, and the conservative hand-calc envelope (which already failed for severe-site uplift) becomes more representative. ### 5.3 Buckling under creep Creep reduces the *effective* stiffness over time. Eigenvalue buckling loads scale linearly with stiffness, so buckling capacity at 50 years is roughly **(1 + φ)⁻¹ ≈ 0.4×** the day-1 elastic value. Hand-calc shell buckling D/C is currently 0.05 — even after a 0.4× knockdown, that's 0.13. **Comfortable.** The CalculiX eigenvalue results are mesh-quality-limited (Finding 4) so we can't translate them through this scaling meaningfully yet. --- ## 6. Risks that break the assumption set The φ = 1.5 number is only good if the structure stays in its assumed regime. Each of the following invalidates it: ### 6.1 Temperature excursions A roof reaching 60 °C for 4 hours/day during summer effectively *advances* the creep clock by 12× during those hours. Naively weighted, that contributes more creep than the other 20 hours combined. Over 50 years, summer afternoons alone could double the expected creep deflection. **Mitigation**: light-coloured surface (reflective coating), the excluded cement skin (provides both thermal mass and reflectance), ventilation, or shade. ### 6.2 Sustained snow and ponding instability The serviceability case is dead load + sustained snow (~30 % of design snow over weeks). The dangerous case is **ponding feedback**: snow sags the dome, the sag holds more snow, which sags it more. PU foam domes are particularly vulnerable because (a) creep makes the deflection grow over the load duration even at constant snow, and (b) the dome geometry has favourable curvature for water collection at the apex if it sags inward. This is not a "creep makes things 20 % worse" scenario — it is a **stability** scenario where creep can take a marginal load case past a tipping point. ASCE 7 explicitly addresses ponding for flat roofs; the dome equivalent is not standardised and would need a custom analysis. ### 6.3 Joint adhesive creep Adhesive layers nearly always creep faster than the bulk substrate. The 0.27 MPa joint tension allowable was measured in short-term tests; the *sustained* joint tension allowable is plausibly **0.10–0.15 MPa** — half the lab number. The good news from Finding 3: the dome doesn't load joints in tension much. Worst joint D/C = 0.51 in short-term FE. **The 0.5× knockdown re-check has now been done** (see §11.2): under conservative blanket knockdown, severe-site **wind C&C peak joint tension reaches D/C = 1.024** — a marginal failure. Under duration-aware knockdown (only sustained loads see degraded capacity), all cases pass. This is a **second flagged severe-site issue** alongside Finding 2's worst-panel bending, and the resolution depends on which knockdown interpretation is accepted by the engineer of record. ### 6.4 UV embrittlement UV does not directly cause creep, but it removes ~0.5–2 mm of effective material from the outer surface over decades. For a 76.2 mm panel, a 1 mm loss of effective skin is a ~3 % stiffness loss — negligible at short-term, but multiplied by (1 + φ_remaining) it amplifies the creep deflection by a few percent. Bigger concern: the embrittled outer layer cracks, which provides moisture ingress paths, which accelerates everything. **Mitigation**: opaque skin (cement, paint, etc.) — the original design intent. Excluding the skin from analysis was deliberate but note that the skin is not just decorative. ### 6.5 Workmanship variability Lab-fabricated foam has uniform density, uniform cure, no voids. Field-poured or shop-laminated foam typically has 10–20 % density variability and occasional voids. Creep scales steeply with local density (ρ⁻²·⁵), so a 20 % low-density region could creep ~2× faster locally. This shows up as **local bulging** near voids rather than uniform sag, and would be visible to inspection. --- ## 7. What testing would close the gap? To replace assumptions with data, in priority order: 1. **1000-hour parent-foam compressive creep test** at σ = 0.1 MPa, T = 23 °C and T = 50 °C, on the same foam batch as the dome. Output: Findley (m, n) at two temperatures. Cost: small, ~$3–5 k at a polymer testing lab. **Highest-value single test.** 2. **1000-hour joint creep test** in tension and shear at design joint stress. Output: φ_joint, validates or invalidates the 2× parent assumption. Cost: similar. 3. **DMA (dynamic mechanical analysis) sweep** to measure Tg directly and confirm temperature-rate behaviour. Cost: very small, hours, any polymer lab. 4. **Outdoor exposure of a witness panel** for 12+ months, instrumented for deflection. Real-world creep + UV + temperature + moisture coupling. Cost: depends on installation; weeks to set up. 5. **Long-term ponding test** — load a representative panel with a water-bag at design snow load and monitor deflection for 90+ days. Output: ponding instability check. Tests 1–3 are the minimum to make a defensible 50-year stamp possible. Tests 4–5 are what make it *correct*. --- ## 8. Confidence summary | Aspect | Confidence | Why | |---|---|---| | Sustained stress is in linear viscoelastic regime | **High** | 2–3 % of σ_y is a factor of 10 below threshold | | Findley n ≈ 0.25 | **High** | Strong literature consensus across rigid PU foams | | φ_parent ≈ 1.5 at 23 °C | **Medium-high** | Standard assumption, conservative for our stress level | | 50-yr apex sag ≈ 17 mm gravity / 20 mm with sustained snow | **Medium** | (1+φ)·δ_day1 with δ_day1 = 6.89 mm from CalculiX | | Effective long-term E ≈ 28 MPa | **Medium** (derivation) / **Low** (FE re-solve) | φ-derivation is sound; modulus-substitution FE on the merged mesh produces non-physical 0.03–6× per-node ratios — see §11.1 | | Joint creep behaviour at sustained load | **Low** (assumption) / **Quantified marginal** (consequence) | φ_joint = 3 still assumed; consequence is wind C&C peak D/C = 1.024 under blanket knockdown — §11.2 | | Temperature-history effect on creep | **Low** | Need actual roof T(t) data for the install location | | Ponding instability | **Low** | Specialised analysis, not yet performed | | UV / moisture coupling | **Very low** | No data, no testing planned | | Modulus-substitution as an FE technique on this mesh | **Failed** | New finding: per-node displacement ratios non-uniform due to merged-mesh near-singularity — §11.1 | --- ## 9. Action items Status legend: ✅ done, 🟡 in-progress, ⬜ not started. These are mirrored in [`../NEXT_STEPS.md`](../NEXT_STEPS.md) item 7. 1. ⬜ **Commission 1000-hour parent-foam creep test** at 0.1 MPa, 23 °C and 50 °C, same batch as dome. Highest-value single test. 2. ⬜ **Commission joint creep test** in tension and shear at design joint stress. 3. ⬜ **DMA sweep** to confirm Tg directly. 4. ✅ **Re-run joint D/C check** with 0.5× sustained-load knockdown on joint allowables. Done in [`tools/ccx_creep.py`](../tools/ccx_creep.py); results in §11.2. **New finding**: severe-site wind C&C peak joint tension at D/C = 1.024 under conservative blanket knockdown. 5. ✅ **Substitute E_∞ = 28 MPa in the existing CalculiX gravity run**. Done. Result in §11.1. **New finding**: substitution does not behave linearly on the merged mesh — theoretical (1+φ)·δ_day1 is the trustworthy estimator until NEXT_STEPS item 1 is closed. 6. ⬜ **Commit to a roof colour / skin specification** before stamping. The thermal/UV protection of the cement skin is not optional for a 50-year service life; it should be in scope. 7. ⬜ **Ponding analysis** — model an iterative load-deflection-water- accumulation loop to find the snow load at which the system goes unstable. 8. ⬜ **Get a licensed structural engineer** to review these creep assumptions and either accept them or commission additional tests. Creep is the single biggest difference between "it stands on day 1" and "it stands for 50 years," and that judgement call is exactly what stamping is for. 9. ⬜ **Rebuild the assembly mesh** (NEXT_STEPS item 1) so that the modulus-substitution FE technique becomes usable. New action item surfaced by §11.1 — without a clean mesh, every FE-based long-term prediction inherits the same 0.03–6.0× per-node noise. --- ## 10. References The numbers in this report draw on the following bodies of literature and standards. None of these are specific to this dome; they are the generic engineering basis for rigid PU foam creep. - **Findley, W.N., Khosla, G.** (1955). Application of the superposition principle to creep behavior of plastics under compressive stress. *Trans. ASME*. — Original power-law model. - **Huang, J.S., Gibson, L.J.** (1991). Creep of polymer foams. *J. Materials Science* 26: 637–647. — Cellular-solids scaling. - **Gibson, L.J., Ashby, M.F.** (1997). *Cellular Solids: Structure and Properties*, 2nd ed. — Density-stiffness-creep scaling. - **DIN 53291** — Sandwich panel sustained-load creep test method. - **EN 14509** — Self-supporting double skin metal-faced insulating sandwich panels: long-term performance / creep coefficient definitions. - **ASTM D2990** — Standard test method for tensile, compressive, and flexural creep of plastics. - **PIMA Technical Bulletins** (Polyisocyanurate Insulation Manufacturers Association) — long-term thermal & mechanical performance of rigid foam. For lab testing of the specific foam batch, ASTM D2990 is the controlling standard. --- ## 11. Outcomes — implementation of action items 4, 5 The two analytical action items from Section 9 (joint knockdown and long-term modulus substitution) have been implemented in [`tools/ccx_creep.py`](../tools/ccx_creep.py). Raw results are at [`reports/creep-results-severe.json`](creep-results-severe.json). ### 11.1 Action 5 — long-term modulus substitution Re-ran the gravity-only assembly FE in CalculiX twice — once at the short-term modulus E = 70.8 MPa, once at the long-term modulus E_∞ = 28.32 MPa — on the existing `reports/full_dome_merged.vtu` mesh. | Quantity | E = 70.80 MPa (day-1) | E = 28.32 MPa (50-yr) | Empirical ratio | Theoretical ratio | |---|---|---|---|---| | max ‖u‖ | 6.886 mm | 20.329 mm | **2.95** | 2.500 | | p99 ‖u‖ | 6.138 mm | 16.711 mm | 2.72 | 2.500 | | p95 ‖u‖ | 5.183 mm | 10.116 mm | 1.95 | 2.500 | | p99 von Mises | 0.0472 MPa | 0.1292 MPa | 2.74 | **1.000 (E-invariant)** | **The empirical ratios disagree with linear-elastic scaling.** Under force-controlled linear elasticity, the displacement field scales exactly as 1/E and the stress field is exactly E-invariant. We see neither: per-node displacement ratios across the 19,453 nodes with non-trivial deflection have **median 2.02, std 1.05, range 0.03–6.0**. **Diagnosis**. The merged-mesh has near-singular regions from the node-merging step (Findings 4 and 5 of [`2026-05-04--findings.md`](2026-05-04--findings.md)). The global stiffness matrix is sufficiently ill-conditioned that SPOOLES finds particular solutions whose per-node values drift with E in non-physical ways. The same root cause that made the buckling eigenvalues unreliable (Finding 4) and the scikit-fem stress 90 %+ off CalculiX (Finding 5) is now contaminating the modulus- substitution test. **Implication for the 50-year deflection prediction**. The *theoretical* prediction `δ_∞ = (1 + φ) × δ_day1 = 2.5 × 6.89 mm = 17.2 mm` is more reliable than the *empirical* re-solve result (20.3 mm), because the theoretical version is a direct consequence of linear elasticity and holds independent of mesh quality. The empirical FE re-solve number should be treated as **upper-bound noise** until the merged mesh is fixed (NEXT_STEPS item 1). **Long-term apex deflection — final estimate**: ~17 mm under gravity alone at φ = 1.5. With sustained snow at 30 % design level the estimate rises to ~20 mm — at the L/240 to L/360 serviceability limit for a 5.6 m dome. **Marginally acceptable, validates the report's TL;DR.** ### 11.2 Action 4 — joint allowable knockdown The joint stress field is exactly E-invariant in linear elasticity, so the existing severe-site joint tractions from [`acceptance-severe.json`](acceptance-severe.json) were re-used directly. No new FE runs needed. A 0.5× knockdown was applied to joint allowables and D/C tabulated. **Severe-site results**, p99 joint tractions × 0.5× knockdown: | Load case | Duration | Mode | Demand (kPa) | D/C short-term | D/C long-term | Verdict | |---|---|---|---|---|---|---| | snow_balanced | sustained | tension | 8.69 | 0.161 | **0.322** | PASS | | snow_balanced | sustained | shear | 13.95 | 0.170 | **0.340** | PASS | | wind_uplift_main | gust | tension | 17.57 | 0.325 | 0.651 | PASS | | wind_uplift_main | gust | shear | 22.96 | 0.280 | 0.560 | PASS | | wind_cc_peak | gust | tension | 27.65 | 0.512 | **1.024** | **FAIL** | | wind_cc_peak | gust | shear | 35.58 | 0.434 | 0.868 | PASS | **Two interpretations:** 1. **Conservative blanket knockdown** (apply 0.5× to all cases regardless of duration): wind C&C peak tension fails marginally at D/C 1.024. 2. **Duration-aware knockdown** (apply 0.5× only to sustained loads, full lab allowable to short-duration gusts since the gust loading is too brief to creep the joint): all cases pass with margin (worst is wind_cc_peak short-term at D/C 0.512). **Which is correct?** The defensible position depends on what "long-term capacity" means. If we believe joint adhesive *strength* is permanently degraded by years of sustained creep — even when a gust eventually arrives — then interpretation (1) is correct, and the dome has a marginal joint failure mode at 50-year-degraded state under severe-site C&C peak gusts. If we believe joint adhesive responds at higher effective stiffness/strength to fast-rate loads (a viscoelastic stiffening effect), then interpretation (2) applies and there's no problem. The literature is mixed. **DIN 53291 / EN 14509 are explicit about duration-dependent capacity** and would support interpretation (2) for properly-designed sandwich joints. But neither standard was written for an unreinforced foam-only dome at 240 kg/m³, so the extrapolation is an engineering judgement. **Recommendation**: treat interpretation (1) as the screening envelope. The marginal D/C of 1.024 under that interpretation is **not a structural failure** — it's a flag that the C&C peak / 50-yr joint behaviour is on the boundary and warrants either: - A targeted long-term joint test (Action 2 from Section 9), or - Reinforcing the joint at panel corners where the C&C peak load concentrates, or - Accepting that interpretation (2) governs and documenting that acceptance. ### 11.3 What this changes about the report's headline numbers Nothing structural changes. Specifically: - **TL;DR remains valid**: 50-year apex sag estimate is now ~17 mm (theoretical) rather than the originally written 15–20 mm range. Same ballpark. - **The L/240 serviceability boundary is real**: a 5.6 m dome with a 17 mm sag is at L/329 — fine for a non-precision interior, but not zero. The engineer of record needs to be aware. - **Joint capacity adds a new marginal case**: severe-site C&C peak joint tension at D/C 1.024 under conservative blanket knockdown. Previously the only flagged severe-site issue was worst-panel bending (Finding 2 of findings.md). This is a second one. - **The merged-mesh quality issue (Finding 4/5) is now load-bearing for creep too**: any modulus-substitution-based long-term FE on this mesh is unreliable. Theoretical extrapolation via (1 + φ) is the correct path until the mesh is rebuilt. This adds urgency to NEXT_STEPS item 1. ### 11.4 Reproducibility ```bash python3 tools/ccx_creep.py severe ``` writes `reports/creep-results-severe.json` and the four CalculiX artefacts at `reports/ccx/creep_gravity_*.{inp,frd,sta}`. Total runtime ~3 seconds (two static linear runs at 1 s each plus post-processing).