SDOF anchor for THA time integration (Newmark average-acceleration, dt convergence)

• Production integrator Newmark 0.5 0.25 + Path -dt -prependZero ground motion
• §1: independent closed-form SDOF recurrence (undetermined coefficients), cross-checked vs RK4 pre-run
• §1: O(dt²) convergence order measured directly, not assumed
• §2: SM5 Bathe-Wilson 9-storey frame reused verbatim (production writeNodes/writeElements/computeNodeMassMap)
• §2: Rayleigh a0/a1 calibrated LIVE from this model’s own eigen output, not reused

THA (time-history analysis) has no third-party published example in STRIX’s scope the way CSI 1-020/1-026 served the RSA/Pushover benchmarks — CSI 1-022’s ground-motion record was never digitized (devplan §12.12, explicit user decision), and a full nonlinear-hinge THA is already covered by SH1 (element mechanics) + SP1 (pushover driver). TH1 instead isolates and checks the ONE thing SH1/SP1 do not: the time-integration scheme itself, in two complementary stages. §1 (SDOF anchor) drives a single isolated spring-mass (same isolation strategy as SH1’s bare-hinge rig) with a resonant sine pulse through STRIX’s production Newmark integrator, and compares it — at six halvings of dt — against a closed-form solution re-derived from scratch (undetermined coefficients on the piecewise-linear-interpolated forcing OpenSees’s own Path -dt assumes) and cross-validated against an independent RK4 integrator BEFORE any OpenSees run (SH1 lesson: hand-derivation alone is not trustworthy — devplan §12.11/§12.13). §2 (MDOF self-convergence) then exercises a REAL multi-mode structure — SM5’s already-validated Bathe & Wilson frame — and checks that STRIX’s two independent production damping formulations, modalDampingQ and Rayleigh rayleigh a0 a1 0 0, agree with each other when both are calibrated to the same 3% ratio at the structure’s own two dominant frequencies (calibrated live from this run’s own eigen call, not reused from SM5’s approximate shear-rigid eigenvalues — devplan §12.14). No external reference exists for a full MDOF THA, so §2 is graded as bounded self-consistency (gap must shrink and floor out with dt, target <0.5% at the finest dt), one notch looser than §1’s convergence-to-truth proof.

1. Peak relative displacement, zeta=5%, dt=0.003125s (finest)max|u(t)|, SDOF recurrence for piecewise-linear-interpolated forcing (undetermined coefficients), zeta=5%= 200.786209 mm
2. Peak relative displacement, zeta=0% (undamped), dt=0.003125s (finest)max|u(t)|, SDOF recurrence for piecewise-linear-interpolated forcing (undetermined coefficients), zeta=0%= 397.874575 mm
3. MDOF §2 — peak roof Ux |modal-rayleigh| relative diff %, dt=0.0125s (finest)|peakModal - peakRayleigh| / max(|peakModal|,|peakRayleigh|) * 100, both formulations calibrated to 3% at this model's own two dominant eigenfrequencies (mirrored from mdof.quantities below for §5/§6 rendering — see mdof block for the full MDOF model description)= 0 %

Reference is NOT transcribed from any external table — it is the exact solution of the damped SDOF equation of motion (ü + 2*zeta*omega*u̇ + omega²*u = -üg(t)) for the SAME piecewise-linear-interpolated ground-motion samples OpenSees's `Path -dt` assumes, re-derived from scratch via undetermined coefficients (free-vibration term + forced response to a linear-in-tau interval force, zero IC) in benchmarks/dynamic/TH1_sdofAnchor.ts. De-risked (SH1 lesson, devplan §12.11, 'hand-derivation isn't trustworthy alone') two ways BEFORE any OpenSees run: (1) an internal algebraic identity (d/dtau[unit-ramp response] == unit-step response, a basic LTI fact not assumed anywhere in the derivation) falls out exactly; (2) internal/TH1_closedform_selfcheck.ts cross-validates the closed form against an independent fixed-fine-substep RK4 integrator (completely different numerical method, shares no formula) to ~1e-11% before the Tcl runner was ever written. Because both STRIX and the closed form consume the identical discretized (piecewise-linear) excitation, any difference between them is pure Newmark time-integration error — which the average-acceleration method (unconditionally stable, 2nd-order accurate) must shrink as O(dt²) as dt -> 0.

§1 (zeta=0/5%, first 12 rows): as dt halves six times (T/10 → T/320), the peak-displacement error against the independent closed form shrinks by almost exactly 4× each step — a measured convergence order of 2.000 at the finest dt pairs for both damping ratios, textbook confirmation of the average-acceleration Newmark method’s theoretical O(dt²) accuracy. The finest-dt error itself is a few thousandths of a percent, orders of magnitude inside the 0.01% gate. §2 (last 5 rows, "mdof dt=…"): because modalDampingQ and Rayleigh are two genuinely different numerical formulations (not one exact answer vs an approximation), their gap does not vanish to zero — it shrinks with dt and floors out at a small residual from the few off-resonance modes (3–9) each scheme treats slightly differently. At the finest swept dt (0.0125s) the gap is 0.00026%, about 1,900× inside the 0.5% self-consistency gate. (Note: the MDOF excitation amplitude a0 was reduced 10× from the value first used in devplan §12.14 purely for report-card readability — the model is linear-elastic, so this rescales the roof peak displacement proportionally without moving the gating diff% at all, which is a pure ratio of two amplitudes.)

1. Newmark, N. M. (1959) — A Method of Computation for Structural Dynamics, ASCE Journal of the Engineering Mechanics Division 85(3) (average-acceleration method, γ=0.5, β=0.25, unconditionally stable, O(dt²) accurate — the closed-form comparison target for §1’s convergence order).
2. Chopra, A. K. — Dynamics of Structures (SDOF piecewise-linear-interpolated-excitation recurrence form, re-derived independently in benchmarks/dynamic/TH1_sdofAnchor.ts, not transcribed from any table).
3. SM5 (devplan §12.1, refsolutions/SM5.json) — Bathe & Wilson ten-bay nine-storey frame, buildBatheWilsonModel, reused verbatim for §2 (already validated <0.001% against published eigenvalues).
4. OpenSees — integrator Newmark, modalDampingQ, rayleigh, timeSeries Path -dt -prependZero, UniformExcitation; STRIX production buildThaTcl (TclBuilder.ts, integration/damping/ground-motion wiring reused conceptually, not the full recovery-ladder pipeline — see benchmarks/dynamic/TH1_sdofAnchor.ts and TH1_mdofSelfConvergence.ts file headers).

Engine

v1.0.6 (tcl)

Run date

2026-07-11

Record

records/TH1.json

Evidence archive

verif-evidence-eng1.0.2-win-x64.zip · TH1/

sha256

(pending publish)