Dynamic amplification in structural design

Dynamic amplification is a phenomenon that is both physically intuitive and mathematically poorly understood. For example, the effects of earthquakes, in terms of dynamic amplification, can be easily understood by reasoning about a simple spring. Anyone who shakes a flexible spring with a mass attached to it will immediately understand that the more violent the shaking, the more the mass will swing. Conversely, if the spring is replaced by a rigid tube, the mass will no longer sway and will follow the imposed jolt, however explosive it may be.

This is a trivial illustration of dynamic amplification: the first case corresponds to a flexible oscillator, and the second to a rigid oscillator without dynamic amplification (the mass tends to follow the imposed motion of the base in its entirety). We'll see in particular that this characteristic of rigid oscillators is reflected in the asymptotic behavior of seismic spectra, the latter being a graphical representation of dynamic amplification. This asymptote coincides with the maximum acceleration of the imposed motion.

This simple example is the key to understanding linear dynamic calculations. Indeed, any structure can be interpreted as the superposition of an infinite number of oscillators, of which only a few play a major role in the overall behavior. These oscillators correspond in fact to the structure's natural vibrational modes, and are referred to in the literature as modal oscillators.

After a first part describing the general objectives of the document, and the associated notations, we present in the second part the theoretical reminders relative to the classical resolution of a linear dynamic problem, through the writing of the solution on the orthogonal basis of the eigenmodes.

These theoretical foundations will make it possible to quantify dynamic amplification (whether for a time-dependent force or a time-dependent acceleration), and thus to define a numerical criterion differentiating flexible and rigid modes. This criterion will subsequently be used in the modal extraction phase, as only soft modes are sought. In fact, most calculation software incorporates a static correction function to ensure that rigid modes are taken into account when determining the solution.

Time resolution is the only computational technique that provides the exact theoretical response of the system under analysis. However, this type of analysis requires the calculation of a complete solution for each time step (there can be tens of thousands of them) and the knowledge of precise loading data. For this reason, many engineers prefer to deploy less "costly" techniques such as spectral analysis (monospectral or multispectral), or pseudo-static analysis, the...