Time History Analysis for Seismic Retrofit Projects

Response spectrum analysis, as covered elsewhere on this site, deliberately discards everything about a structure's response except each mode's peak value. Most of the time that's a reasonable trade — faster analysis, less input data, adequate accuracy for the decision at hand. Time history analysis makes the opposite trade: it keeps everything, solving the structure's actual equation of motion second by second against a real or realistic ground acceleration record, at the cost of substantially more computational effort, more input data, and more engineering judgment about how to select and interpret the results. This article covers what that method actually does, how ground motions get selected for it, and — just as important — when it's genuinely required versus when RSA remains the better tool.

What Time History Analysis Actually Does

Rather than transforming the problem into the period domain the way RSA does, time history analysis solves the structure's equation of motion directly against a ground acceleration record, step by step through time — commonly via a numerical direct-integration scheme. At each small time step, the method computes the structure's displacement, velocity, and acceleration based on the ground motion at that instant and the structure's state at the previous step, then advances forward.

The result isn't a single peak value per mode — it's a complete response history for every quantity of interest: displacement, member force, or hinge rotation, all as continuous functions of time across the full duration of shaking. This preserves exactly the information RSA discards — the sequence in which different peak demands occur, how many cycles of significant response the structure experiences, and (in a nonlinear analysis) the actual cumulative path of inelastic deformation a member follows, not just its single largest excursion.

Selecting and Scaling Ground Motions

A time history analysis is only as representative as the ground motion records driving it, which makes record selection a genuinely consequential engineering decision rather than a formality. Records are typically drawn from real recorded earthquakes with similar magnitude, distance, and site conditions to the project's own seismic hazard characterization, then scaled or spectrally matched so that their response spectra reasonably envelope or match the project's target design spectrum over the period range of interest. Where suitable real records are scarce for the target hazard level, synthetic or spectrally matched records generated to fit the target spectrum directly are also used, with the trade-off that they may not fully preserve the natural variability of real recorded motions.

Because any single record's response can vary substantially from another's even at similar intensity, codes require multiple records — commonly a minimum of several ground motions (often eleven under contemporary ASCE 7 and ASCE 41 provisions, or as few as three to seven under older or alternative procedures, with the response taken as an envelope rather than an average when fewer records are used) — specifically so the design result reflects a statistically reasonable range of response rather than one record's particular characteristics.

Linear vs. Nonlinear Time History

Linear time history analysis solves the equation of motion assuming the structure stays elastic throughout — useful where response timing or higher-mode participation matters but genuine yielding isn't expected. It's less common than the nonlinear version for retrofit work, since most retrofit evaluations exist because the structure is expected to behave inelastically.

Nonlinear time history analysis (NLTHA) models the structure's actual hysteretic behavior at each inelastic element — the force-deformation relationship a plastic hinge or base isolator follows through repeated cycles of loading, unloading, and reloading, including strength and stiffness degradation where the element model represents it. This captures a genuinely different picture than a pushover (monotonic, one direction) or RSA (linear by definition): the actual cumulative demand a member experiences across a realistic sequence of cyclic loading, including whether earlier cycles have already degraded its capacity before a later, larger cycle arrives. This is why NLTHA is the standard, often required, method for base-isolated buildings — an isolator's hysteretic behavior and cycle-to-cycle interaction is central to its performance in a way a linear spectral method can't represent.

When Time History Analysis Is Actually Required

Despite its added rigor, NLTHA isn't automatically the "better" choice for every project — it demands more analysis time, more defensible ground motion selection, more careful interpretation (including judgment on parameters like Rayleigh damping that influence results without one universally correct value), and more opportunity for modeling error to go undetected across a large volume of output. In practice it's generally reserved for: base-isolated and other structures with significant hysteretic or cyclic-degrading behavior a monotonic pushover can't represent; genuinely irregular or tall structures where higher-mode effects and true response timing matter enough that RSA's statistical combination isn't adequate; and performance-based projects where demonstrating performance beyond a linear or pushover-based method is specifically required.

For a large share of conventional retrofit projects — a regular, low-to-mid-rise building with a well-understood lateral system — a properly executed nonlinear pushover per ASCE 41 remains an adequate and far more practical tool, and reaching for NLTHA where it isn't required adds cost without a proportional gain in reliability.

Practical Application: Why an Isolated Building Needed More Than a Spectrum

An illustrative, composite case: a base isolation retrofit for a mid-rise office building, using lead-rubber bearings as covered in our base isolation engineering article, is initially evaluated using response spectrum analysis with the isolators represented by their effective (linearized) stiffness and damping — a reasonable first pass, but one that assumes the isolator's actual bilinear hysteretic behavior can be adequately approximated by a single equivalent value.

Because the project falls under a code-mandated performance verification requirement for isolated structures above a certain scale, the design team runs a full nonlinear time history analysis using eleven ground motion pairs scaled to the site-specific target spectrum, modeling each isolator's actual bilinear force-displacement relationship rather than its linearized equivalent. The results show peak isolator displacement roughly 15–20 percent larger than the linearized RSA had estimated at several motions — traced to the linearized model's inability to capture how the lead core's yielding actually interacts with a realistic, irregular sequence of displacement cycles.

Moat clearance and isolator displacement capacity are revised upward based on the NLTHA envelope rather than the initial linearized estimate — a change the RSA-only approach would never have flagged, and precisely the category of result that time history analysis exists to capture where a linear spectral method cannot.

Common Mistakes

Using too few ground motion records, or records poorly matched to the site's actual hazard. A time history result is only as representative as its input records — an inadequate suite can look precise without being representative.

Assuming NLTHA is always more accurate and therefore always right. For a conventional, regular structure, its added complexity can introduce more practical risk of undetected modeling error than a well-executed, code-standard pushover.

Linearizing isolator or damper behavior for a final check rather than just a preliminary one. An equivalent-linear approximation is a reasonable start, but hysteretic devices are specifically nonlinear — a project requiring isolator-level verification generally needs the actual nonlinear behavior modeled before finalizing displacement and clearance requirements.

References & Standards

1. ASCE/SEI 7, Minimum Design Loads and Associated Criteria for Buildings and Other Structures — Chapter 16, Nonlinear Response History Procedures, American Society of Civil Engineers.
2. ASCE/SEI 41-17, Seismic Evaluation and Retrofit of Existing Buildings, American Society of Civil Engineers.
3. Chopra, A.K., Dynamics of Structures: Theory and Applications to Earthquake Engineering, Pearson.
4. PEER/ATC-72, Modeling and Acceptance Criteria for Seismic Design and Analysis of Tall Buildings, Applied Technology Council.