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Model summary

Single vertical spring represents the compressible subsoil beneath the raft. Eref = 15 MPa (pre‑excavation secant modulus). pref = pre‑excavation overburden at z = h_exc + 0.5 m (i.e., γ′·(h_exc + 0.5)). Excavation unloads the spring to p₁, construction applies the gross foundation pressure Q so p₂ = p₁ + Q. Stress law E(p) = E_ref·(p/pref)ⁿ with n = 0.5. Recompression multiplier K_r models stiffer reload branch. Disturbance factor φ reduces baseline stiffness where appropriate.

Governing expressions

• pref = γ′ · (h_exc + 0.5)


• p₁ = γ′ · 0.5 (post‑excavation confinement at z = h + 0.5; enforce p′_min as needed)


• p₂ = p₁ + Q (gross)


• Eref* = φ · Eref (apply disturbance factor φ, 0.5–1.0)


• E_unloaded = Eref* · (p₁/pref)ⁿ


• E_loaded = Eref* · (p₂/pref)ⁿ


• E_recomp = K_r · E_loaded (K_r ≈ 1.0–1.5 to model stiffer reload branch)

Numerical illustration (γ′ = 20 kN/m³, Eref = 15 MPa, n = 0.5, Q = 100 kPa)

Common: p₁ = 10 kPa (γ′·0.5; enforce p′_min = 10 kPa). Choose K_r = 1.2. Two disturbance cases: φ = 1.0 (no disturbance) and φ = 0.7 (30% reduction).

Physical interpretation (summation of effects)

• Decompression: excavation reduces p → E drops (E_unloaded ≪ Eref).


• Gross re‑confinement: applying the gross load Q raises p to p₂ and restores stiffness to E_loaded; if p₂>pref E_loaded can exceed Eref.


• Recompression: real soils often reload on a stiffer branch; K_r models this (E_recomp = K_r·E_loaded).


• Disturbance: excavation can reduce Eref (φ<1) and offset recompression benefits.


• Overall: choose E_design from E_unloaded, E_loaded, E_recomp with conservative choices for φ, K_r, and n; compute settlements with layered methods or distributed springs.

Practical implementation steps

1. Measure Eref in situ at z_ref = h + 0.5 where possible.


2. Set pref = γ′·(h + 0.5). Choose φ to reflect expected disturbance (e.g., 1.0, 0.7, 0.5).


3. Compute p₁ (post‑cut) and p₂ (post‑cut + gross Q). Enforce p′_min (5–20 kPa) to avoid zero stiffness.


4. Compute E_unloaded, E_loaded, and E_recomp (choose n = 0.4–0.6; K_r ≈ 1.0–1.5).


5. For settlement use E_design (best estimate) and run sensitivity cases with reduced φ and K_r.


6. Where critical, calibrate φ and K_r with plate tests and use staged FEM for final design.

Critical appraisal and cautions

• The model captures the primary physics: unloading reduces stiffness; gross loading can re‑constrain and increase stiffness; reloading can be stiffer than virgin loading; disturbance reduces all moduli.
• Using pref tied to the pre‑excavation overburden at the key depth makes Eref empirically meaningful and isolates excavation influence.
• The approach is simple, transparent, and easy to implement for rapid design iterations and sensitivity studies.

• Scalar stress law ignores stress anisotropy and directional stiffness changes; excavation changes horizontal stresses too, and stiffness is tensorial in reality.
• The power‑law scaling and choice of n are empirical; values vary with site, density and testing method — calibrate where possible.
• Recompression multiplier Kr is a crude model of hysteresis; real unloading/reloading behaviour can be nonlinear, path dependent, and irreversible. Kr should be calibrated to unload/reload lab tests or plate tests.
• Disturbance factor φ is also empirical and spatially variable; treating it as uniform risks masking local weak spots.
• Applying gross Q entirely at the shallow evaluation depth is conservative for that depth but may misrepresent vertical load distribution; use influence factors or layered elastic solutions for more realistic load partitioning.
• For cohesive soils the undrained immediate stiffness and pore pressure effects dominate, so this effective‑stress based scaling must be adapted for undrained modulus and short‑term behaviour.

• Always run sensitivity analyses for φ (e.g., 1.0, 0.7, 0.5) and Kr (1.0–1.5) and for alternative n (0.4–0.6).
• Use layered models (multiple springs) under the raft to capture depth variation and load spreading; calibrate with one or two plate/load tests after excavation if possible.
• Where excavation is large (p1 very small) or q large relative to pref, prefer staged numerical modelling with an unloading/reloading capable constitutive law.

• For preliminary design use the schema above to produce E_unloaded, E_loaded and E_recomp profiles and then perform settlement calculations with conservative and best‑estimate parametric runs.
• For final design calibrate φ and Kr with field tests and perform staged FEM analysis to capture anisotropy, arching and stress path effects.
• Avoid relying on a single scalar modulus; present a band of predicted settlements with clear assumptions about disturbance and recompression.