Stage 6 / settlement

Stage 6 settlement.

The settlement route integrates vertical strain along the CPT column under the stress increase imposed by a footing. Stress is built from the Boussinesq solution for strip or rectangular geometry; stiffness is the interpreted Hardening Soil oedometric modulus at the mean stress level; optional Terzaghi time-rate interpretation is available for fine-grained layers.

Scope and intent

1. Problem class and the meaning of the computed settlement

The settlement route is a centreline constrained-modulus integration. It evaluates stress increase beneath the loaded area, updates Eoed at the mean effective stress in each sublayer, and integrates vertical strain over depth to yield a single scalar settlement S at the evaluation point.

qnet = qgross − σv(Df)
S = Σ ΔSi, ΔSi = Δσv,i/Eoed,i · Δzi
Primary quantities
qgross, qnet
Gross and net foundation pressure [kPa].
σv(Df)
In-situ total vertical stress at foundation depth [kPa].
Df
Foundation embedment depth [m].
S
Total settlement at the evaluation point [m].

Net load

2. Net pressure definition and physical meaning

The soil below the foundation is pre-loaded by the weight of excavated material that the structure displaces. Only the additional stress above that pre-existing state contributes to strain. The net foundation pressure is therefore the quantity that drives the Boussinesq integral; the full qgross drives the bearing-capacity check, not the settlement integration.

qnet = qgross − γ Df (dry column)
qnet = qgross − [γ zw + γsat (Df − zw)] + γw(Df − zw) (partially submerged)
  • Net pressure is the quantity that drives settlement. Gross pressure drives bearing capacity.
  • For deep foundations with effective-stress relief larger than qgross, qnet can be negative — the app flags this as a heave condition rather than a settlement.

Stress increase

3. Boussinesq solutions for strip and rectangular geometry

Vertical stress beneath a surface load is obtained from the Boussinesq half-space solution. The app provides three routes: an exact closed form for the strip centreline, a Newmark / Fadum four-quadrant superposition for rectangular areas, and a simplified 2:1 distribution for quick screens.

Strip centreline: Δσv(z) = (qnet/π)[2α + sin(2α)]
α = atan(B/(2z)) (half-angle subtended by the strip from the depth point)
Corner of rectangle B × L at depth z (Fadum / Newmark):
Δσv = (qnet/(2π))[ (m n (m² + n² + 2))/((m² + 1)(n² + 1)√(m² + n² + 1)) + atan(m n / √(m² + n² + 1)) ]
m = B/z, n = L/z
Centre of rectangle: Δσv,centre(z) = 4 Iz(B/2, L/2, z) qnet (four-quadrant superposition of the corner formula)
2:1 approximation: Δσv(z) = qnet B L / [(B + z)(L + z)]
Notation
α
Strip-foundation geometric angle [rad].
Δσv(z)
Vertical stress increase at depth z [kPa].
B, L
Foundation width and length [m].
m, n
Influence-chart ratios B/z and L/z [-].
Iz
Newmark/Fadum corner influence factor (the bracketed quantity above divided by qnet).
  • The strip closed form is exact at the centreline; off-centre points require superposition of two half-strips.
  • The four-quadrant superposition gives Δσv at the centre of a rectangle. Δσv at other plan points requires decomposing the loaded area into rectangles that share a corner at the target.
  • The 2:1 rule over-predicts near the surface and under-predicts at depth relative to Boussinesq; it is offered as a sensitivity check, not as the primary route.
  • Boussinesq assumes a homogeneous isotropic elastic half-space. Layered stiffness contrasts shift the Δσv(z) profile; Westergaard would be closer for strongly layered fabrics but is not in the current route.

Integration

4. Constrained-modulus integration of vertical strain

For each sublayer the app forms the mean effective stress between the in-situ and loaded state, evaluates the stress-dependent oedometric modulus at that level, and computes the vertical strain increment through the one-dimensional constitutive relation Δεv = Δσv/Eoed.

σ′v,f,i = σ′v,0,i + Δσv,i
σ′mean,i = 0.5 (σ′v,0,i + σ′v,f,i)
Eoed,i = Eoed,ref[(c′ cot φ′ + σ′mean,i) / (c′ cot φ′ + pref)]m
Δεv,i = Δσv,i / Eoed,i
ΔSi = Δεv,i Δzi, S = Σ ΔSi
Notation
σ′v,0,i, σ′v,f,i
Initial and final effective vertical stress in sublayer i [kPa].
σ′mean,i
Mean effective stress used for stiffness evaluation [kPa].
Eoed,ref, m, pref
Hardening Soil reference oedometric stiffness [kPa], stress exponent [-], and reference pressure [kPa].
c′, φ′
Effective cohesion and friction angle [kPa, °]. Used in the Janbu-style pressure offset c′ cot φ′.
Δεv,i
Vertical strain increment in sublayer i [-].
Δzi
Sublayer thickness [m].
Engineering meaning. The settlement result is directly tied to the interpretation workflow: Eoed,ref, m, c′, and φ′ all come from the interpreted CPT layer model. Changes to the Stage 4 / Stage 5 interpretation propagate immediately to the settlement output.

Truncation

5. Practical truncation and the "depth of influence"

The integral is taken to a finite depth zmax. Truncation choice matters because the constrained-modulus form weights deeper, stiffer, lower-stress layers lightly but not negligibly over many sublayers. The app exposes three truncation rules so the engineer can match the question being asked.

(a) Δσv(zmax) ≤ 0.1 σ′v,0(zmax) — classical "10% of in-situ" rule
(b) Δσv(zmax) ≤ 0.10 qnet — load-scaled rule (Lunne default)
(c) zmax = zCPT,bottom — cap at the sounding depth
  • Rule (a) is the most engineering-defensible for deep loaded zones below stiff layers.
  • Rule (b) is the default Lunne / Robertson CPT practice; it converges faster in weak soils where σ′v,0 is small.
  • Rule (c) trades physical meaning for auditability: the integration stops where the interpretation stops.
  • The app reports the truncation rule alongside S so the number is not interpretable without its provenance.

Corrections

6. Rigid-footing and finite-layer corrections

The centreline integral over-estimates settlement for a rigid footing because the contact pressure is not uniform but bowl-shaped. It can over- or under-estimate for a flexible footing at points off-centre. Two corrections are retained in the app as switchable options for comparison with closed-form elastic estimates.

Rigid-footing reduction (Skempton–Bjerrum style): Srigid ≈ κr Scentreline, κr ≈ 0.80 for rectangles, 0.85 for strips
Finite-layer correction (Christian–Carrier): Sfinite = μ0 μ1 qnet B / Es
Notation
κr
Rigidity reduction coefficient. Derived by equating the rigid-footing closed-form elastic displacement to the flexible-footing centreline displacement.
μ0, μ1
Christian–Carrier embedment and finite-layer factors from the chart. Used as sensitivity comparators, not as the primary estimate.
  • The rigid-footing reduction applies to the centreline integration only. For off-centre points the flexible-footing integral is more direct.
  • The Christian–Carrier form assumes uniform elastic stiffness over the layer; it is a comparator in layered CPT profiles, not a replacement for the constrained-modulus integral.

Time rate

7. Terzaghi consolidation and the time factor

For saturated fine-grained layers the total settlement develops over time as excess pore pressure dissipates. The app exposes the one-dimensional Terzaghi route as a companion output. It does not replace transient consolidation; it converts the total S to a U(t) percentage so the engineer can compare early-life and long-term behaviour.

∂u/∂t = cv ∂²u/∂z² (1D consolidation)
cv = k Eoed/γw
Tv = cv t / Hd²
U(Tv) ≈ √(4 Tv/π) for U < 0.6
U(Tv) ≈ 1 − (8/π²) exp(−π²Tv/4) for U > 0.6
S(t) = U(Tv(t)) · S∞
Notation
cv
Coefficient of consolidation [m²/s].
Hd
Longest drainage path in the consolidating layer [m]. Half-layer thickness for double drainage, full layer for single drainage.
Tv
Dimensionless time factor [-].
U(Tv)
Average degree of consolidation [-].
  • Layers are identified as consolidating (fine-grained) from the Stage 3 classification and the interpreted conductivity.
  • The consolidation integration is one-dimensional; radial drainage (wick drains) is outside the current route.
  • For overlapping fine-grained layers, the app reports both single-layer U(t) and a conservative envelope using the largest Hd.

Limitations

8. Assumptions and boundaries of validity

  • Centreline integration by default; differential settlement across a footing is not produced unless the engineer repeats the calculation at multiple plan positions.
  • Boussinesq (homogeneous isotropic half-space). Strong stiffness layering biases the Δσv(z) shape; Westergaard and Fröhlich are alternative transforms and are not in the current route.
  • Creep (secondary compression) is not integrated; the result is the primary-consolidation estimate.
  • Soft clay at high stress level sits outside the Janbu-style constitutive form; the app warns when the mean stress exceeds a configurable limit relative to the yield stress.
  • Time-rate output is a 1D comparison; true 3D consolidation with radial drainage requires a dedicated transient model.
  • The deeper audit trail remains in the full technical specification.

References

References

  • Boussinesq, J. (1885). Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques. Gauthier-Villars.
  • Newmark, N. M. (1935). Simplified computation of vertical pressures in elastic foundations. Univ. of Illinois Eng. Exp. Sta., Circ. 24.
  • Fadum, R. E. (1948). Influence values for estimating stresses in elastic foundations. Proc. 2nd ICSMFE, Vol. 3, 77–84.
  • Terzaghi, K. & Peck, R. B. (1967). Soil Mechanics in Engineering Practice. Wiley.
  • Janbu, N. (1963). Soil compressibility as determined by oedometer and triaxial tests. Proc. Eur. Conf. on Soil Mech., Wiesbaden.
  • Christian, J. T. & Carrier, W. D. (1978). Janbu, Bjerrum and Kjaernsli's chart reinterpreted. Can. Geotech. J. 15(1), 123–128.
  • Lunne, T., Robertson, P. K. & Powell, J. J. M. (1997). Cone Penetration Testing in Geotechnical Practice. Blackie Academic & Professional.