Chapter B. Structural Dynamics

Chapter B. Dynamics

For many applications such as seismic calculations, collisions, vibrations… it is necessary to consider the dynamic phenomena.  

The dynamic charges applied to a civil engineering structure belong to two different categories:

Concerning seismic motions, if they are theoretically considered as transitory, it is admissible to assimilate them as stationary phenomena while in their strong phase. For the cases in which one tries to model the structure with geometric or material non-linearities, stationary assumptions cannot be considered anymore. 

The means of representing the loading categories can then be distinguished as shown below:

Two big families of analysis can be considered:

The explicit schemes dictate the choice of very small-time steps. Thus, they are the most used to solve problems with small periods (like collision/impact problems.) On the contrary, implicit schemes allow us to use greater time steps and are therefore favorable to study problems occurring on wider time ranges. 

Examples of applications


Applications

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Modal

Vibrations analysis

FT

FT

PSD

PSD

Tracking of natural frequencies

ORS

Spectrum extrema of quantities of various interests

Implicit transitory

Seismic Study

Accelerations, velocities, forces, pressures, or displacements as a function of time

Quantities of diverse interests expressed throughout time

Weakening 

Explicit transitory

Fall of on object

Modeling of projectiles in contact, collisions

Quantities of diverse interests expressed throughout time

Plane crash

Accelerations, velocities, forces, pressures, or displacements as a function of time

Once the dynamic problem is discretized in finite elements, the resolution of the equilibrium equation can be written as shown below (cf. chapter 1):

In a modal analysis, the natural frequencies ω_i and the associated modes φ_i are used.

In the time-dependent analysis, the displacements q(t), the velocities q'(t), and the accelerations q"(t) at the nodes are calculated for each time-step t by direct integration of the equilibrium equations. 

The second approach has an advantage as it allows us to handle nonstationary solicitations.

B.1 Analysis based on modal search

B.1 Analysis based on modal search

B.2 Analysis based on a direct temporal integration

B.2 Analysis based on a direct temporal integration

B.3 Considering the damping

B.3 Considering the damping

B.4 Particularities of a seismic analysis

B.4 Particularities of a seismic analysis



Révision #4
Créé 18 September 2023 15:34:19 par Paul Terrasson Duvernon
Mis à jour 10 October 2023 09:09:04 par Paul Terrasson Duvernon