Structural dynamics / tutorial
Choosing a time step, and proving it
Stability, accuracy and the demonstration that the step is adequate.
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 notesA single-degree-of-freedom model with a period of 0.45 s is to be integrated. Two schemes are available: the explicit central-difference method and Newmark with constant average acceleration. Choose a step for each, and quantify what the choice costs.
- Natural period Tₙ = 0.45 s
- Newmark constant average acceleration, γ = ½, β = ¼
Practice
What is the largest stable time step for the explicit central-difference method, in seconds?
Practice
Using the accuracy rule Δt ≤ Tₙ/20, what step should be used, in seconds?
Practice
At Δt/Tₙ = 0.1, what period elongation does constant average acceleration introduce, as a percentage?
Practice
Halving the step to Δt/Tₙ = 0.05, what is the elongation, as a percentage?
Report the convergence, not the assertion. 'Halving the time step changed the peak displacement by 0.3%' is a check a reviewer can weigh; 'the analysis converged' is not.