Application · Consolidation and embankments

Settlement against time, with the pore pressure that drives it.

How much will it settle is half a question. The other half is when — and whether the embankment is stable before the answer arrives. KATAI 2D solves Biot’s coupled equations with an elastoplastic soil skeleton, so excess pore pressure, effective stress, settlement and strength evolve together through the construction programme. It reproduces the Terzaghi solution to within 0.0061 on the degree of consolidation, and it can be asked how long rather than only told.

Install KATAI 2D See the verified numbers

The engineering problem

On soft clay the load arrives faster than the water can leave. The soil skeleton takes up the load only as the excess pore pressure dissipates, so strength is lowest when the fill has just been placed and the factor of safety rises with time. Staged construction exists to exploit that: build a lift, wait, build the next. The wait is the design.

One-dimensional consolidation theory gives the time scale for a uniform layer under a uniform load. It does not give the lateral drainage under an embankment toe, the stress redistribution as the foundation yields, the secondary compression that continues after the pore pressure has gone, or the stability of an intermediate stage. A coupled analysis gives all four from one model.

The formulation

∇ ⁣⋅ ⁣(σ′−p I)+ρ g=0,∂εv∂t+∇ ⁣⋅ ⁣(−kγw ∇pexcess)=0\nabla\!\cdot\!\bigl(\boldsymbol{\sigma}' - p\,\mathbf{I}\bigr) + \rho\,\mathbf{g} = \mathbf{0}, \qquad \frac{\partial \varepsilon_v}{\partial t} + \nabla\!\cdot\!\Bigl(-\frac{\mathbf{k}}{\gamma_w}\,\nabla p_{\mathrm{excess}}\Bigr) = 0Biot consolidation: equilibrium of the two-phase medium, and continuity of the pore fluid with Darcy flow (incompressible water; tension-positive stress, pressure-positive pore pressure).
U(Tv)=1−∑m=0∞2M2 e−M2Tv,M=(2m+1)π2,Tv=cv tHdr2,cv=k EoedγwU(T_v) = 1 - \sum_{m=0}^{\infty} \frac{2}{M^2}\,e^{-M^2 T_v}, \qquad M = \frac{(2m+1)\pi}{2}, \qquad T_v = \frac{c_v\,t}{H_{dr}^2}, \quad c_v = \frac{k\,E_{\mathrm{oed}}}{\gamma_w}Terzaghi’s one-dimensional solution — the oracle of cases KV-CON-001 and KV-CON-002.
εvc(t)=μ∗ ln⁡ ⁣(1+tτ),τ=1 day\varepsilon_v^{c}(t) = \mu^{*}\,\ln\!\Bigl(1 + \frac{t}{\tau}\Bigr), \qquad \tau = 1\ \text{day}Creep on the normal-consolidation line in the Soft Soil Creep model — the oracle of case KV-CST-010.

The full statement of what is solved, element by element and model by model, is on the theory and constitutive models pages.

What KATAI 2D does

  • Coupled, not sequential. Displacement and excess pore pressure are solved in one system each time step; the skeleton may be any of the constitutive models, plastic or not.
  • Undrained loading, then consolidation. A plastic phase with Undrained (A), (B) or (C) materials generates the excess pore pressure; a consolidation phase dissipates it.
  • Stop on a state, not only on a time. until_degree=90 or until_excess_pore=1.0 lets the phase choose its own steps and report the time it took.
  • Soft Soil and Soft Soil Creep for normally consolidated clays and peat: logarithmic compression, a pre-consolidation memory, and secondary compression that continues at constant effective stress.
  • Structures stay in. Walls, rafts, anchors and geogrids, with their interfaces, remain active while the ground consolidates.
  • Stability at any stage. A Safety phase after a consolidation phase uses the pore pressures that are actually there.

Verified

The numbers, with their sources.

Each row is a case in the generated verification matrix, asserted by an automated test on every build of release 0.9.0.

Benchmark Reference KATAI 2D Case
Terzaghi column — U(Tv) at Tv = 0.2 / 0.4 / 0.6 / 0.9, from the checked-in file Terzaghi (1943) series −1.2% … −0.4% KV-CON-002
Phase asked to stop at 90% degree of consolidation — the time it stops closed form: 14.566 day 14.776 day  ·  +1.45% KV-CON-003
Time-step refinement of the same column — observed order of the scheme backward Euler: 1 0.985 and 0.992  ·  GCI band ±0.25% KV-NUM-010
Undrained (A) confined column — settlement and effective stress Skempton 1D closed form +0.0% / +0.0% KV-CST-001
Soft Soil oedometer — primary loading, swelling, reloading λ* and κ* logarithmic laws −0.01% / −1.01% / −1.63% KV-CST-009
Soft Soil Creep column — settlement after 100 days at constant load μ* ln(1 + t/τ) = 0.018460 m +1.81% KV-CST-010
Wall and raft at the end of consolidation the same model solved drained settlement 0.0450102 vs 0.0450113 m KV-STR-006

The second row carries a distinction worth knowing before you use any program’s “degree of consolidation”. The stop criterion is a pressure ratio — the largest excess pore pressure left, over the largest the stage generated. The settlement ratio reaches 90% about a fifth of the time earlier (11.98 days against 14.57 on this column). The two are different quantities with the same name; KATAI 2D documents which one it uses and reports the settlement ratio at the stop — 93.9% here.

The complete record, with every tolerance band, is on the verification page.

In Python

The analysis as a script.

clay = prj.materials.soft_soil_creep("Soft clay", lam_star=0.10, kap_star=0.02,
                                     mu_star=0.004, c=1.0, phi=25.0,
                                     gamma=16.0, drainage="undrained_a", k=1.0e-4)

prj.initial(procedure="k0")
prj.phases.plastic("First lift", activate=[lift1])

# wait a fixed time ...
prj.phases.consolidation("30 days", duration=30.0, steps=100)
# ... or until the excess pore pressure has dissipated to a target,
#     and let the phase report how long that took
prj.phases.consolidation("Until 90 %", until_degree=90)

prj.phases.safety("FoS before the second lift")

A consolidation phase that ends on a target reports its duration in res.consol_time[-1]. The parameters are illustrative, and lift1 is the handle of a fill region created earlier in the script.

The whole scripting surface is documented on the Python API page.

Stated limits

What this analysis does not do, in the same place as what it does. A limit you know about is an engineering judgement; one you find out about later is an error.

  • Time integration is first order (backward Euler): accurate and unconditionally stable, with a measured order of 0.99. Halving the time step halves the error; the grid-convergence band of a run can be computed and is ±0.25% on the reference column.
  • Small-strain formulation. Very large settlements relative to layer thickness (several tens of percent strain) call for an updated-mesh analysis, which this release does not have.
  • Vertical drains are modelled by their effect on drainage boundaries or permeability, not as individual three-dimensional elements.

Questions

Does KATAI 2D do true coupled consolidation or just Terzaghi theory?

Coupled Biot consolidation: displacements and excess pore pressures are unknowns of the same system, solved together each time step, with an elastoplastic soil skeleton. Terzaghi’s solution is used as a verification oracle (reproduced within 0.0061 on U), not as the method.

Can it model creep (secondary compression)?

Yes, with the Soft Soil Creep model: the volumetric creep rate depends on the ratio of the current equivalent stress to the pre-consolidation stress, and on the normal-consolidation line it integrates to μ* ln(1 + t/τ). The verification case reproduces that law to +1.81% after 100 days.

What does “90% degree of consolidation” mean in a stop criterion?

A pressure ratio: the maximum excess pore pressure remaining is 10% of the maximum the stage generated. It is not the settlement ratio, which reaches 90% earlier. KATAI 2D reports both at the stop.

Related

Other analyses on the same engine.

Application

Slope stability

Factor of safety by φ–c reduction; the failure mechanism is found, not assumed.

FoS 1.384 vs 1.380 charted Read more → Application

Groundwater flow & seepage

Confined and free-surface flow, wells, drains and barriers — coupled to the soil.

dam discharge +1.02% Read more → Application

Foundations & bearing capacity

Strip, circular and piled foundations — settlement, load–displacement, limit load.

Prandtl N_c: +0.6% Read more →

Install it, run a published benchmark, check the number.

KATAI 2D is free under Apache‑2.0, installs in one line and ships the input files of the benchmarks it is verified against. The fastest way to judge it is to rerun one.

Install KATAI 2D See the verification record Source on GitHub