3D Consolidation Calculator — Mandel-Cryer Effect
Terzaghi's theory for 1D consolidation assumes that excess pore pressure always decreases monotonically with time. For localised three-dimensional loads (footings, raft slabs, finite-width fills), Mandel (1953) and Cryer (1963) demonstrated that pore pressure can rise above the applied load during the initial moments before dissipating. This counterintuitive effect is critical in foundations on soft clays: underestimating the initial pore pressure can compromise stability and generate unforeseen short-term settlements. This calculator provides the Mandel-Cryer amplification factor and the time to reach peak pressure.
What is it and when is it applied?
When a circular footing or a fill column loads a saturated clay layer, initially the pore pressure responds elastically and equals the applied load (Skempton B ≈ 1 response in saturated soils). The clay then begins to drain at the lateral edges, where total stress decreases faster than at the centre, generating a load transfer towards the interior that transiently raises the pore pressure at the central axis above the initial value. This peak is called the Mandel-Cryer effect. Applicable to footings and rafts on low-permeability clays (Cv < 2 m²/year), finite-width embankments on soft deposits, earth dams during initial impoundment, and deep foundation analysis in marine clays.
Applied formulas
Pore pressure amplification factor (Cryer 1963, cylinder):
Δu_max / Δσ = 1 + f(ν, r/a), where ν = undrained Poisson's ratio, r/a = normalised radial distance
For saturated clay (ν = 0.5): maximum Δu/Δσ ≈ 1.32 at the centre of the cylinder
3D time factor: T3D = Cv·t / a², with a = radius of the equivalent circular load
Time to peak pressure: T3D_peak ≈ 0.08-0.15 depending on geometry and ν
Post-peak decay (Mandel approximation):
Δu(t) / Δσ = 1.32 · exp(−(T3D − T3D_peak)/τ), with τ ≈ 0.25
3D degree of consolidation: U3D = 1 − Δu_avg/Δσ, with integration of the spatial distribution
Time to 90% 3D consolidation: t90 = 1.2·a²/Cv (vs t90 1D = 0.848·H²/Cv), faster due to radial drainage
Calculation example
| Parameter | Value |
|---|---|
| Effective load radius a | 7.5 m |
| Uniform load Δσ from full silo | 120 kPa |
| Soft clay thickness H | 12 m |
| Coefficient of consolidation Cv | 0.8 m²/year |
| Undrained Poisson's ratio νu | 0.5 (saturated) |
| Permeability anisotropy Kh/Kv | 2 (typical) |
Initial response time (T3D peak): t_peak = 0.10 · a²/Cv = 0.10 · 7.5² / 0.8 = 7.03 years. Peak pore pressure at centre (νu = 0.5, a/H = 0.625 → ratio not negligible but not infinite): factor ≈ 1.25. Δu_max = 1.25 · 120 = 150 kPa. This peak pressure is 25% higher than the applied load, occurring approximately 7 years after the start of permanent silo loading. Comparison with simple 1D prediction: Terzaghi 1D predicts Δu_max = Δσ = 120 kPa immediately, then decreasing. Underestimates the peak by ~30 kPa. Degree of consolidation after 15 years: T3D = 0.8·15/56.25 = 0.213. Interpolating in Cryer tables for a circular cylinder: U3D ≈ 65%. Settlement at 15 years (if S∞ = 35 cm from consolidation): S(15 years) = 0.65 · 35 = 22.8 cm. Time to 90%: t90 = 1.2 · 7.5² / 0.8 = 84.4 years. The project would require preloading with radial drains to accelerate consolidation to a reasonable timeframe (see radial consolidation calculator).
Result: Δu_max = 150 kPa (1.25·Δσ) at 7 years · U3D(15 years) ≈ 65% · t90 = 84 years without drains.
Interpretation of results
The Mandel-Cryer effect implies that the stability of the silo at 5-10 years may be more critical than immediately after loading: the persistent excess pore pressure reduces the effective strength of the soil (τ = c' + σ'·tan φ'). It is recommended to verify the stability FS with Δu_max instead of the initial Δu. To accelerate dissipation, vertical drains are used (see radial consolidation) or the design incorporates a higher temporary surcharge to achieve consolidation before operation. Ignoring the Mandel-Cryer effect in soft clays with low Cv is one of the classic errors in geotechnical studies and explains late failures of silos and tanks on marine clays.
Reference standards
- BS EN 1997-1 (Eurocode 7) — Appendix on undrained analysis in saturated clays
- Biot, M.A. (1941). General theory of three-dimensional consolidation
- Mandel, J. (1953). Consolidation des sols (étude mathématique)
- Cryer, C.W. (1963). A comparison of the three-dimensional consolidation theories of Biot and Terzaghi
- Verruijt, A. (2013). Theory and problems of poroelasticity, ch. 5
- BS EN 1997-1 (Eurocode 7) — Appendix on undrained analysis in saturated clays
- BS 1377-7 (+ BS EN ISO 17892-10) — Consolidated-undrained triaxial test (to obtain effective parameters)
Frequently asked questions
Why does pore pressure rise above the load?
Because the three-dimensional load is not uniformly distributed: the edges drain faster and transfer the load-bearing responsibility towards the centre. Since saturated clay is nearly incompressible in volume (ν = 0.5), the only way to accept more load is by raising the water pressure. This phenomenon disappears in infinite 1D loading (Terzaghi) because there are no lateral edges.
When is the effect significant?
For finite-dimension loads with a/H < 2 and Cv < 2 m²/year. In typical building footings the effect exists but dissipates within a few months. In silos, tanks, wide embankments (a/H > 5) the effect is negligible as it approaches 1D. In railway foundations on soft clays and port embankments it does matter and should be modelled with software (PLAXIS with Soft Soil Creep model).
How do I account for it in practice?
In simplified design: increase Δu by 20-30% over the value predicted by Terzaghi 1D and verify stability with that value at the critical time. For important projects, finite element analysis with flow-deformation coupling (full Biot). Software: PLAXIS 2D/3D, Settle3D, FLAC.
Does it affect final settlements?
No. The total settlement S∞ is the same (it depends only on the compression index Cc and the applied load). What changes is the temporal evolution: in the early years S(t) is lower than predicted by Terzaghi 1D because the increased pore pressure delays load transfer to the soil skeleton. It then accelerates and eventually coincides.