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 nonlinearities, 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:
 Loaddisplacement 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 timedependent 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 smalltime 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 
Loading representation 
Quantities available 

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):

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 timedependent analysis, the displacements q(t), the velocities q'(t), and the accelerations q"(t) at the nodes are calculated for each timestep 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.4 Particularities of a seismic analysis
B.4 Particularities of a seismic analysis
No Comments