Structural dynamics

Aim: The main aim of this challenge is to gain the knowledge about the concept of structural dynamics which helps to understand the behaviour of the structure when it is subjected to the seismic force. The various terms that are related to the dynamic analysis are also learnt

Q1.

Static Analysis:

• A static analysis can only be performed if the system being simulated does not depend on time, and if the loads being applied are constant.
• Static analyses don’t consider inertia.
• In a static analysis, only the stiffness matrix of the FEA model is solved.
• Static analyses require less computational time.

Dynamic analysis:

• In a dynamic analysis, the system itself, the load application, or both might change with time.
• In a dynamic analysis, the inertial loads developed by the system due to acceleration are taken into account.
• In a dynamic analysis, in addition to the stiffness matrix, the mass matrix (and damping matrix, if not zero) is solved as well.
• Dynamic analyses require more computational time.

Q2.

Elastic behaviour:

It is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal.

Inelastic behaviour

The term, “inelastic” is used to define the material response in relation to the stress-strain diagram that is non-linear and that retains a permanent strain or returns to an unstrained state on complete unloading.

Plastic behaviour

It is the ability of a solid material to undergo permanent deformation, a non-reversible change of shape in response to applied forces.

Non-linear Inelastic behaviour

Materials have elastic behaviour usually only upto a certain point after that deformation are plastic and relation in this is not linaer i.e. non linear elsatic behaviour.

Q3.

Stiffness component:

The frame without damping or mass. The external force fs on the stiffness component is related to displacement u by fs = ku, if the system is linearly elastic.

Damping component:

The frame with its damping property with no stiffness or mass. The external force on the damping component is related to velocity by fd=cv

Mass component:

The roof mass without the stiffness or damping of the frame. The external force fi on the mass component is related to the acceleration by fi = mű

mü + fD + fs = p(t) → mü + cὺ + ku = p(t)

Q4.

Natural Period Tn of a building is the time taken by it to undergo one complete cycle of oscillation. It is an inherent property of a building controlled by its mass m and stiffness k. These three quantities are related by its units are seconds (s).

Tn = 2 π √ (m/k)

Natural frequency:

The reciprocal (1/Tn) of natural period of a building is called the Natural Frequency fn; its unit is Hertz (Hz).

Q5.

A response spectrum is a graphical plot of the frequency of an oscillator and its damping. The response spectrum plot represents the peak or steady-state response (velocity, displacement, or acceleration) of a series of oscillators of varying natural frequency. A response spectrum is a function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. The response spectrum is a function of the natural frequency of the oscillator and of its damping. Thus, it is not a direct representation of the frequency content of the excitation, but rather of the effect that the signal has on a postulated system with a single degree of freedom (SDOF).

Explanation about the response spectrum curve

• • A response spectrum is a function of frequency or period, showing the peak response of a simple harmonic oscillator that is subjected to a transient event. The response spectrum is a function of the natural frequency of the oscillator and of its damping. Thus, it is not a direct representation of the frequency content of the excitation (as in a Fourier transform), but rather of the effect that the signal has on a postulated system with a single degree of freedom (SDOF).
  • Response spectrum analysis is a method to estimate the structural response to short, nondeterministic, transient dynamic events. Examples of such events are earthquakes and shocks. Since the exact time history of the load is not known, it is difficult to perform a time-dependent analysis. Due to the short length of the event, it cannot be considered as an ergodic ("stationary") process, so a random response approach is not applicable either.
  • The response spectrum method is based on a special type of mode superposition. The idea is to provide an input that gives a limit to how much an eigenmode having a certain natural frequency and damping can be excited by an event of this type.

  The equation of motion for the mass can, if there are no external forces, be written as

  

  Dividing by the mass, and using customary notation,

  

  Here, the undamped natural (angular) frequency is

  

  and the damping ratio is

  

  For given values of , , and , this equation can be solved for a sufficiently long time. The displacement, velocity, and acceleration response spectra are defined as the maximum values caused by the acceleration history .

  

  

  

  

Conclusion:

From the challenge carried out it can be concluded that the concept of behaviour of a structure when subjected to the seismic force is learnt. The concept of analysing a structure by converting the dynamic force to the seismic force is also learnt.