Forensic analysis for the effect of Fourier series terms on the response of a structure to a periodic force

Fourier series is a powerful tool for analysing the dynamic response of structures under periodic loading, though selecting the optimal number of terms is challenging. This study aims to develop a stabilisation chart showing structural response for different numbers of Fourier terms, aiding both design and forensic engineers in failure analysis. The dynamic solution involves two stages: modelling the excitation and representing the response. The infinite series can be truncated at various points, with stabilisation charts generated analytically or numerically for both odd and even functions. Results show that accurately modelling periodic loading with enough harmonic terms improves convergence, which occurs faster with even-type loads. The periodicity ratio also affects the convergence rate. To ensure accurate modelling and reliable failure investigations, engineers must select a sufficient number of series terms.