Application · Seismic dynamics

Site response in the time domain, with boundaries that let the waves leave.

KATAI 2D integrates the equations of motion of a soil deposit — and of the structures in it — through an earthquake record. Outgoing waves are absorbed at a compliant base and at free-field lateral boundaries instead of being reflected back into the model, damping is calibrated at two frequencies, and the surface motion comes out as a time history and as a response spectrum. Resonance of a soil column is reproduced to −0.6% in displacement and −0.2% in acceleration.

Install KATAI 2D See the verified numbers

The engineering problem

A finite element mesh has edges; the ground does not. Waves that reach a fixed or free boundary are reflected and return to the region of interest as energy that should have been radiated away, and the computed response is then a property of the mesh. Every dynamic geotechnical analysis is, first, a boundary-condition problem.

The second problem is the input. Applying a recorded motion at a rigid base assumes that the recorded motion is the motion of rigid bedrock; applying it as an incident wave at a compliant base assumes it is the outcrop motion. The two give different answers on the same site, and the difference is physical.

The formulation

M u¨+C u˙+fint(u)=− M r ag(t),C=α M+β K\mathbf{M}\,\ddot{\mathbf{u}} + \mathbf{C}\,\dot{\mathbf{u}} + \mathbf{f}_{\mathrm{int}}(\mathbf{u}) = -\,\mathbf{M}\,\mathbf{r}\,a_g(t), \qquad \mathbf{C} = \alpha\,\mathbf{M} + \beta\,\mathbf{K}Equations of motion under base excitation, with Rayleigh damping fitted to the target ratio at two frequencies.
tn=ρ Vp u˙n,ts=ρ Vs u˙st_n = \rho\,V_p\,\dot{u}_n, \qquad t_s = \rho\,V_s\,\dot{u}_sViscous (Lysmer–Kuhlemeyer) boundary tractions that absorb normally incident P and S waves.
f1=Vs4H,∣asurf∣res=2Aπ ξf_1 = \frac{V_s}{4H}, \qquad |a_{\mathrm{surf}}|_{\mathrm{res}} = \frac{2A}{\pi\,\xi}Fundamental frequency of a uniform layer on rigid rock, and its surface acceleration at resonance for base amplitude A and damping ratio ξ — the oracle of case KV-DYN-002.

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

  • Input motions: harmonic, Ricker pulse, or a stored accelerogram. The record is stored inside the project file, so a run is reproducible from that file alone.
  • Compliant base: viscous dashpots absorb the downgoing wave and the motion is applied as the upgoing incident wave.
  • Free-field lateral boundaries in place of rollers, so a two-dimensional model behaves at its sides like the layered site it is cut from.
  • Rayleigh damping specified as a target ratio at two frequencies.
  • Linear or nonlinear. By default the shaking is solved as a linear dynamic problem; with nonlinear=True the internal force is re-evaluated by Newton iteration at every step, so the soil can yield and interfaces can slip during the shaking.
  • Response spectra of any computed motion, and structural forces through the shaking.

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
Soil column driven at f1 = Vs/4H — peak surface |u| and |a| damped SH closed form (Kramer 1996, ch. 7) −0.6% / −0.2% KV-DYN-002
El Centro 1940 NS — identity of the shipped record published PGA ≈ 0.319 g at t ≈ 2 s 0.31882 g at 2.02 s KV-DYN-001
El Centro 5%-damped spectrum — peak amplification published band: Sa/PGA 2.0–3.5 2.87× at 0.19 s  ·  Sa(3 s) 0.118 g KV-DYN-003
Two-layer site with a compliant base, record carried in the .k2d file the verified path’s surface response reproduced within 5% KV-DYN-003
Time-step refinement at resonance — convergence onto the closed form Richardson extrapolation extrapolates onto the closed form KV-NUM-011

The accelerogram is not a file on somebody’s disk. A digitisation of the El Centro 1940 north–south record, with its provenance, is checked into the repository and travels inside the benchmark project file, so the published peak ground acceleration and spectral shape are asserted on the same data the analysis uses.

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

In Python

The analysis as a script.

prj.initial(procedure="k0")

prj.phases.dynamic("El Centro", duration=31.2, steps=3120,
                   wave="record", record=accel, record_dt=0.02,   # m/s2
                   damping=0.05, rayleigh=(1.0, 10.0),            # 5 % at 1 and 10 Hz
                   compliant_base=True, free_field=True)

job = prj.run()

Units in a dynamic phase are seconds and m/s². nonlinear=True integrates the constitutive model during the shaking and is much slower.

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.

  • No liquefaction model. Cyclic pore-pressure generation and the associated loss of strength are not in this release.
  • Viscous boundaries absorb normally incident waves exactly and obliquely incident waves approximately; keep them at a sensible distance from the region of interest.
  • Rayleigh damping is frequency dependent: it equals the target ratio only at the two chosen frequencies.
  • A linear dynamic phase reports motion, not the stress change it implies; the run says so (K2D-A001).

Questions

Is this an equivalent-linear (SHAKE-type) or a time-domain analysis?

Time domain. The equations of motion are integrated step by step with Newmark’s method, in two dimensions, with structures in the model if there are any. Soil nonlinearity enters through the constitutive model when the phase is run with nonlinear=True, not through iterated strain-compatible moduli.

What boundary conditions are available for earthquake analysis?

A compliant base (viscous dashpots, with the motion applied as the incident wave) and free-field lateral boundaries, in addition to plain fixed and roller boundaries. The first two are what keep the response from being a property of the mesh.

Can it assess liquefaction?

No. There is no cyclic pore-pressure or liquefaction model in the release, and the site says so wherever the question comes up.

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

Excavations & retaining walls

Walls, prestressed anchors, struts and interfaces through a real construction sequence.

beam moments +0.00% vs theory 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