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:

• phenomena that can be compared to static events: constant wind flow, swell, rotary machines.

• transitory phenomena: collision, explosion, earthquakes.  

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:

• Stationary:
  - Complex Fourier Transform (FT),
  - Power Spectral Density (PSD),
  - Oscillator response spectrum (ORS).

• Transitory:
  - Load-displacement curve, speed or acceleration expressed as a function of time,
  - Efforts or pressures are expressed as a function of time.

Two big families of analysis can be considered:

• The modal analysis, which enables to identify the natural frequencies and the associated modes of a structure. This data is useful to characterize:
  - The stationary loading response applied using a method of spectral response,
  - A temporal response using the Duhamel integration of each loading curve corresponding to the modal responses.
  - A transfer function convolved to the signal expressed in terms of the frequencies to deliver an FT or PSD response.
  - The time-dependent dynamics that enables to compute the structure’s transitory dynamic response to any temporal vibration. The resolution can be conducted using schemes of time integration, which can be explicit or implicit. 

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

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

• M: the mass matrix expressed at the nodes,

• C: the damping matrix expressed at the nodes,

• K: the stiffness matrix expressed at the nodes,

• q: the vector containing the nodal displacements,

• q': the vector containing the nodal velocities,

• q": the vector containing the nodal accelerations. 

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