🏗️ 建築——深基坑監測 即時模型

依據：數字背後的模型

模擬器使用的每一個關係式及其來源。標示為假設的常數是示意校準值。

Clough 與 O'Rourke 的系統剛度：牆的抗彎剛度除以無支撐跨度的四次方。

S = EI / (γw · h⁴), h = depth − lowest strut level (unsupported height)[3][5]

每挖一米都會增加變形，增加速度隨系統剛度下降而上升；預加力可減少變形，而深於停工點的開挖面則會持續蠕變。

Δδh = ΔH · ρ(S) · (1 − 0.8·preload); ρ(%) = clip(2.3 − 0.7·log10 S, 0.12, 1.6); creep 2.5 mm/d per m of h above 3.5 m (berm −2 m)[3][5][10]僅為教學用的示意量級，並非任何真實工地的數值。

牆後地面沉降：約為牆體變形的 0.75 倍，在 0.75 H 範圍內持平，至 2 H 處減至零（軟至中等黏土的包絡線）。

δv,max = 0.75·δh,max; δv(d) = δv,max for d ≤ 0.75H, × (2 − d/H)/1.25 to 0 at d = 2H[3][4]

留在坑底以下的土體重量，必須超過含水層向上推的承壓水壓力。

FS = γt·(z_aq − H) / (γw·(z_aq − z_wt − s_in)); required ≥ 1.2[10]

Thiem 穩定流：水位降深隨與水井距離的對數而減小；回灌井則帶來相反的效果。

s(r) = Q/(2πT)·ln(R/r) − Q_r/(2πT)·ln(R/r_r) (steady radial flow, superposition; heads lag the steady state with τ = 1.5 d); Q_r ≤ 0.55·Q[9]僅為教學用的示意量級，並非任何真實工地的數值。

太沙基（Terzaghi）一維固結：降低的水壓在數日內向黏土擴散，有效應力上升，黏土便被壓縮。

∂u/∂t = c_v ∂²u/∂z²; Δσ' = −Δu; settlement = Σ m_v·Δσ'·Δz (unloading: 0.1·m_v)[8]

角變形：樓宇各處沉降之差除以樓宇長度，並與 Bjerrum 限值比較。

β = (s_near − s_far) / L; 1/500 = 2 ‰ (limit where cracking is not acceptable), 1/300 = 3.3 ‰ (first cracking)[6][7]

Peck 表觀土壓力作用於最底一道橫撐的受荷高度上；開挖面越深，它承受的越多。

P = 0.3·γ·H · (half span above + h) · 5 m ≥ preload; shown as % of design load 2400 kN[2]僅為教學用的示意量級，並非任何真實工地的數值。

儀器讀數帶有量度誤差——要看趨勢，不要只看單一數值。

reading = true value + N(0, σ); σ = 0.5 mm (settlement points), 1.0 mm (inclinometer)[1]僅為教學用的示意量級，並非任何真實工地的數值。

模型使用的常數

final depth 15 m · struts at 1, 4, 7, 10, 13 m, 5 m apart along the wall, crew starts 0.5 m below a level and needs 2 days · over-excavation allowance 4 m unsupported · wall 0.8 m concrete (EI = 1.28 GN·m²/m per m) · γ = 19 kN/m³ · building 12 m long, near edge 6 m behind the wall · water table 1 m · aquifer T = 300 m²/d, R = 400 m, excavation radius 12 m, recharge line 4 m behind the wall (effective radius 3 m) · clay 5 m above the aquifer, m_v = 0.2 /MPa, c_v = 0.15 m²/d · ground factors lognormal (σ 0.10 wall, 0.15 m_v, 0.20 c_v) · base unworkable below FS 1.05 · recharge capped at 55 % of the water pumped (supply from the discharge, well clogging)僅為教學用的示意量級，並非任何真實工地的數值。

隨機性：帶種子的 mulberry32 產生器；所用分布包括均勻分布、指數分布（反 CDF）、常態分布（Box–Muller）及泊松分布（Knuth）。種子會顯示出來，並可分享。

來源

1. R. B. Peck — Advantages and limitations of the observational method in applied soil mechanics (Ninth Rankine Lecture) — Géotechnique 19(2), 171–187, 1969
2. R. B. Peck — Deep excavations and tunneling in soft ground (apparent earth-pressure diagrams for braced cuts) — Proc. 7th ICSMFE, Mexico City, State-of-the-Art Volume, 225–290, 1969
3. G. W. Clough, T. D. O'Rourke — Construction induced movements of in-situ walls (system stiffness EI/(γw·h⁴), settlement envelopes) — ASCE Geotechnical Special Publication 25, Design and Performance of Earth Retaining Structures, 439–470, 1990
4. A. I. Mana, G. W. Clough — Prediction of movements for braced cuts in clay — Journal of the Geotechnical Engineering Division 107(GT6), ASCE, 1981
5. A. Gens — Deep excavations: empirical methods for movements; procedures to reduce movements (early bracing, preloading, avoiding over-excavation) — NZ Geotechnical Society lecture notes, 2010
6. L. Bjerrum — Allowable settlement of structures (angular distortion 1/500, 1/300, 1/150); summarised in NGI report "Skadegrenser" — Proc. European Conf. SMFE Wiesbaden, Vol. II, 135; NGI, 1963
7. M. D. Boscardin, E. J. Cording — Building response to excavation-induced settlement — Journal of Geotechnical Engineering 115(1), 1–21, 1989
8. Evolution of primary consolidation settlement with time — Terzaghi 1-D consolidation, Tv = c_v·t/H_dr², U–Tv relation (Fundamentals of Foundation Engineering, §4.8) — University of Newcastle (open textbook)
9. Steady-state pumping: the Thiem equation s = Q/(2πT)·ln(R/r); superposition of wells — University at Buffalo, Aquifer hydraulics notes
10. C.-Y. Ou — Deep Excavation: Theory and Practice (upheaval / uplift check against a confined aquifer, dewatering, strut preload) — Taylor & Francis, 2006