Structural Dynamics / module 15
Linear time-history analysis
The full answer, instant by instant — and the four choices an analyst makes that decide whether it is worth having.
What you will understand
- Compare direct integration with modal superposition on the same model
- Choose a time step from the modes that matter, and demonstrate convergence
- Extract drift, floor acceleration, base shear and member actions correctly
- Explain why several records are required rather than one
- Recognise the numerical artefacts a time-history analysis can produce
- Say when a time-history analysis is worth its cost over a spectrum analysis
Lessons
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.
Module summary
- Direct and modal integration are algebraically identical when all modes are kept
- Every difference between two analyses comes from a choice, not from the methods disagreeing
- Choose the step from the frequency content relevant to the requested response, including local modes with little global effective mass
- Acceleration and drift often need more refinement than displacement; verify each response
- A spectrum analysis gives peaks; a time history gives peaks AND their timing and sequence
- One record is one sample of a random process — codes require several for good reason
- Results should be reported as an envelope or a mean over the record set, never from one
- Time-history analysis is worth its cost when timing, sequence or non-classical damping matter
Teaching basis and review status
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 notesCheck your understanding
Practice
A structure has T₁ = 2.0 s and its highest retained mode has T₁₀ = 0.11 s. Using Δt ≤ T/20 on the governing mode, what step should be used, in seconds?
Practice
A building of 11 000 tonnes has a peak base shear of 19.4 MN. What is V/W?
Check yourself
Two analyses use identical physical models, input records, initial conditions and damping; only the numerical refinement differs. Peak drifts differ by0.4% and floor accelerations by38%. What should be concluded?
Practice
A 25 s record sampled at 0.01 s must be analysed at Δt = 0.002 s. By what integer factor must it be interpolated to a finer integration grid?
Check yourself
Why do design standards require several ground-motion records rather than one?