Running a linear time-history analysis

Two routes to the same answer

Direct integration. Step the coupled equations Mü + Cu̇ + Ku = −Mιüg forward in time, solving a linear system at each step. The effective stiffness is factorised once and reused, so the cost is one factorisation plus one back-substitution per step.

Modal superposition. Transform to modal coordinates, integrate each modal SDOF equation independently, and add the contributions. The cost is the eigensolution once, then a scalar integration per mode per step.

With all modes retained and the same time step, these give the SAME answer to machine precision. They are algebraically identical.

The course library asserts exactly this: on a three-storey building the two agree to 1e-6 at every step. Any difference between two real analyses is therefore attributable to a specific decision, and there are only four.

The four choices

How many modes. Modal superposition normally truncates. Direct integration never does — it carries every mode implicitly, including the very high ones that carry no mass and contribute nothing but numerical noise.

The time step. Must resolve the highest mode that matters. Module 8 covered this and it remains the commonest error.

The integration scheme. Constant average acceleration by default. γ > 1/2 introduces numerical damping, which is sometimes exactly what is wanted in a direct integration to quieten the high modes that modal truncation would have removed.

The damping model. Modal damping ratios in a modal analysis; a damping MATRIX in a direct integration. Rayleigh damping delivers the target at only two frequencies, so the two analyses will differ unless the matrix was built to match the modal ratios.

That last point catches people out. A modal analysis with 5% in every mode and a direct integration with Rayleigh damping anchored at two frequencies are not the same problem, and they will not give the same answer.

Which to use

Modal when the structure is linear, the damping is classical, and interpretability matters — the modal contributions can be inspected separately, which direct integration cannot offer.

Direct when the damping is non-classical (base isolation, local dampers, soil–structure interaction), when the structure yields, or when the model is small enough that the eigensolution is not worth the trouble.

The outputs, and the order in which they converge

Relative displacement. Dominated by the first mode. Converges first, with the fewest modes and the coarsest step.

Inter-storey drift. A difference between two displacements, so the common part cancels and what remains is more sensitive to the higher modes. Converges after displacement.

Floor acceleration. The second derivative, dominated by the higher modes. Converges last, and by a wide margin.

Base shear. Sum of the storey inertia forces. Governed by effective mass, so it converges roughly with the mass criterion.

Member actions. Recovered from the displacements at each step, so they follow displacement — but a member near a mode's node can be dominated by a higher mode and behave like acceleration.

Check convergence on the quantity you are going to use. Displacement agreeing between two runs says nothing about whether the floor accelerations have converged.

Why one record is not enough

A single ground motion is one sample from a random process. Two records with the same magnitude, distance and site class can produce peak responses differing by a factor of two, because the detailed phasing of a rupture is not predictable.

This is not a deficiency in the analysis. It is a property of earthquakes, and it is why design standards require several records rather than one — typically seven, with the MEAN response used, or three, with the MAXIMUM used. The trade is deliberate: more records permit the average, which is less conservative and better founded.

Reporting a result from one record is reporting a sample, not an estimate, and the sample-to-sample scatter is large enough that the distinction matters.

Selecting and scaling

Records are chosen to be consistent with the hazard being designed for — magnitude, distance, site class, and where relevant near-fault characteristics. They are then scaled so that their spectra are consistent with the target over the period range of interest.

Two cautions worth carrying:

Scaling changes amplitude and not frequency content. A record scaled up by 3 has three times the acceleration and exactly the same spectral SHAPE. If its shape is wrong for the site, scaling does not fix it.

Spectrum-compatible records are broader-band than reality. A synthetic record matched to a smooth target at every period simultaneously matches better than any real earthquake does, and can over-drive a structure responding in several modes at once.

Numerical artefacts to recognise

High-frequency noise in direct integration. The very high modes of a finite-element model are discretisation artefacts rather than structure. Direct integration carries them; modal truncation removes them. A little numerical damping — γ = 0.6 — is the standard remedy, applied deliberately.

Baseline drift in integrated quantities. If velocity or displacement is obtained by integrating an acceleration output, small errors accumulate quadratically. The same problem as record processing, with the same remedy.

Spikes at record boundaries. A record that does not begin and end at zero acceleration imposes a step, which excites everything. Records should be tapered.

Apparent response before the strong motion arrives. Usually the initial acceleration set to zero instead of computed from equilibrium — Module 8's mistake, showing up here.