Structural Dynamics / Linear time-history analysis
Running a linear time-history analysis
Two methods that must agree, four decisions that make them differ, and the outputs that converge at quite different rates.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Checks needed for project use
- A dynamics reviewer must confirm excitation models, damping, modal assumptions and any comfort or serviceability interpretation. Small displacement alone is not an acceptance criterion.
- Independent validation of the associated interactive implementation, input domain and project applicability remains required; this review checks lesson context and selected calculations, not design certification.
- Verify record count, selection, scaling, pairing and response statistic against the current governing procedure; check damping equivalence and integration convergence.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy this matters
A response-spectrum analysis gives peaks. That is often enough, and it is cheap.
A time-history analysis gives the whole response: when each peak occurs, in what order, how many large cycles there were, and what the structure was doing at the instant a particular member reached its worst. It also handles cases a spectrum cannot — non-classical damping, and any structure that yields.
It costs more, it needs records selected and scaled, and it presents more opportunities to get a plausible answer that is wrong. This lesson is about doing it defensibly.
By the end of this lesson you should be able to
- Set up direct and modal time-history analyses and confirm they agree
- Attribute any difference to the specific choice that caused it
- Select a time step and demonstrate convergence on the governing quantity
- Extract each response quantity correctly
What you should already know
- Newmark integration and convergence (Module 8)
- Modal decoupling (Module 12)
- Base excitation (Module 13)
Direct and modal integration
Direct integration solves the coupled equations Mü+Cu̇+Ku=−Mιüg. For constant linear coefficients and step size, the effective stiffness can be factorised once and reused.
With classical damping, a complete modal transformation produces independent scalar equations. Direct and modal results should agree within numerical precision when they use the same mass, stiffness, damping, initial conditions, input representation and integration scheme. For non-classical damping, transformed modal equations remain coupled.
Explain differences before comparing results
Check mode truncation, time step, scheme, damping, initial conditions, constraints and input interpolation. Uniform modal5%damping is generally not the same as a Rayleigh matrix calibrated at two frequencies.
Direct integration retains all coordinates in the discretised model. Some high modes may be poorly represented; others are physically relevant. Establish which frequencies affect the requested response through convergence and input-bandwidth checks. Numerical damping can suppress high-frequency response, so document the parameters and verify that useful response is not removed.
Choose the method for the question
Independent modal integration is useful for linear systems with classical damping and a justified mode set. Direct integration or coupled modal methods can handle non-classical linear damping. Yielding requires a suitable nonlinear formulation.
Check each required output
Displacement often converges sooner than drift or floor acceleration, but no fixed ordering applies to every model and load. Local member actions can depend on modes carrying little global effective mass. Check time-step and mode-count convergence for every quantity used, including base shear, drift and equipment acceleration. A modal-mass percentage alone does not prove that any particular response has converged.
Try it
A linear time-history analysis, with the choices exposed
The same building and the same record, analysed the ways an engineer would actually choose between.
Record
Method
Δt = 0.010 s — that is T₁/60
Integration scheme
- This analysis
- Converged reference (direct, all modes, full step)
Show the numbers behind this plot
- First period T₁
- 0.598 s
- Δt used
- 0.01 s
- Steps in the analysis
- 2001
- Peak roof displacement
- 35.3 mm
- Error against the reference
- 0.00%
- Peak inter-storey drift
- 0.232%
- Peak base shear
- 1240 kN
- As a fraction of the weight
- 0.060
T₁/60
at storey 1
Direct integration at Δt = T₁/60 is converged to 0.00%.
Two things this workspace is designed to show
- Coarsen the time step and watch the roof displacement drift out of phase rather than change in amplitude. That is the signature of the average-acceleration scheme, and it is why a plot that 'looks right' can still be wrong.
- The time step must resolve the HIGHEST mode you care about, not the first. A step of T₁/50 sounds generous until you notice it is only T₆/5.
- Displacement converges long before drift does, and drift long before acceleration. Check convergence on the quantity you are going to use.
What this shows: Direct integration and modal superposition give the same answer when all modes are kept. Every difference you see comes from a choice you made — truncation, time step or integration scheme — not from the two methods disagreeing.
Worked example
Setting up a time-history analysis defensibly
Given
- A ten-storey frame; modal analysis gives T₁ = 1.18 s
- The sixth mode, at T₆ = 0.082 s, is the highest with meaningful effective mass; cumulative mass reaches 94% at mode 6
- Ground-motion record sampled at 0.02 s, duration 30 s
- The analysis is required to produce inter-storey drifts AND floor accelerations for equipment
Find
The analysis settings, and the evidence that they are adequate.
Assumptions
- Linear elastic response; classical damping assumed and to be verified
Predict first
A direct integration and a modal superposition of the same linear building give peak roof displacements of 148 mm and 131 mm. What is the most likely explanation?
Record sets
Ground motions with similar magnitude, distance and site conditions can produce different responses. Select a record set representative of the hazard and report the rule used to combine results. Record counts, pairing, scaling and use of a mean, maximum or other statistic depend on the current governing procedure. A universal seven-record mean or three-record maximum rule is not established here.
Selecting and scaling
Consider magnitude, distance, site conditions, usable frequency band, duration and relevant near-fault effects. Multiplying a record by a scale factor changes its amplitude, not its frequency content. Spectrum matching can alter other record properties, so check the method and suitability rather than assuming every matched record is either realistic or overly broad-band.
Numerical artefacts
Investigate high-frequency response against mesh, time-step and input convergence before calling it numerical noise. If numerical damping is introduced, specify both Newmark parameters and their stability/accuracy implications.
Review baseline correction when integrated velocity or displacement drifts. Document any filtering, padding or tapering; changing record ends changes the input and should not be automatic.
For unexpected response before the main excitation, inspect initial conditions, time alignment, filtering and the actual early input. Setting initial acceleration incorrectly can cause an initial transient, but it does not by itself explain response before a zero-input interval.
Worked example
Reading a set of results correctly
Given
- A twelve-storey building, storey height 3.4 m
- Peak roof displacement 214 mm relative to the ground
- Peak displacements at levels 7 and 8: 141 mm and 158 mm
- Peak floor acceleration at roof: 6.8 m/s²
- Seismic mass 8 400 tonnes; peak base shear 14.9 MN
Find
The drift, the base shear coefficient, and what each result does and does not establish.
Assumptions
- Linear elastic; single record, so these are one sample
Practice
A building has T₁ = 1.5 s and its highest significant mode has T₈ = 0.075 s. Using Δt ≤ T/20 on the mode that governs, what time step should be used, in seconds?
Practice
A 40 s record is analysed at Δt = 0.002 s. How many time steps does the analysis take?
Practice
A building of seismic mass 6 500 tonnes has a peak base shear of 11.2 MN. What is the base shear coefficient V/W?
Practice
Floors at 24.0 m and 20.6 m have peak displacements of 132 mm and 118 mm. What inter-storey drift would subtracting the peaks give, as a percentage?
Check yourself
Which of these quantities often requires the most careful high-frequency and time-step convergence check?
Check yourself
What does a time-history analysis give that a response spectrum cannot?
Worked example
What a time history gives that a spectrum does not
Given
- A response spectrum analysis gives a peak base shear of 162 kN
- A linear time history using a matched record gives 149 kN
Find
Why the two differ, and which to use
Worked example
Choosing the time step for a record
Given
- A ground motion record digitised at 0.02 s intervals
- The structure's shortest significant period is 0.10 s
Find
Whether the record's step is fine enough
Summary
- Direct and modal integration are identical with all modes, same step, same damping
- Any difference comes from truncation, time step, scheme or damping model
- Set the step from the highest mode that matters, not from T₁
- Convergence order depends on the model and input; check every response quantity used
- The peak of a difference is not the difference of the peaks — compute drift histories
- One record is one sample; codes require several and a stated combination rule
- Scaling changes amplitude, not frequency content — it cannot fix a wrong record