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1. Differentiate Static analysis and Dynamic analysis with proper representing pictures (30 marks) AIM:- To Differentiate Between Static analysis and Dynamic analysis Ans:- Static Analysis:- In Static analysis we analysis static load means “a force that has constant size,…
Dinesh Kumar
updated on 12 Jul 2023
1. Differentiate Static analysis and Dynamic analysis with proper representing pictures (30 marks)
AIM:-
To Differentiate Between Static analysis and Dynamic analysis
Ans:-
Static Analysis:-
Dynamic Analysis:-
Example:
Structural analysis deals with the change in behaviour of a physical member under the applied loads. The nature of response of the system is completely dependent on the way the load is applied to the component. If the load applied slowly, the inertial effect defined on the basis of Newton’s first law of motion will be neglected and this is called as a static analysis. In static analysis, a single output value of solution is obtained for the system in the form of displacement, reaction forces etc. In a broader sense, static forces are considered as constant loads acting on the structure. A simple example of static load is a block of iron laying on a surface, a stationary truck etc.
Representation of static and dynamic loads
Dynamic means time varying, and as the name suggests, the applied loads vary with time and thus induce time varying responses (displacements, velocities, accelerations, reaction forces, stresses etc). One of the most notable difference in dynamic analysis is the explicit consideration of inertial forces developed in the structure when it is excited by the time varying loads. Due to its time-varying characteristics, dynamic analysis is more realistic in nature to the actual occurring event but it also makes the modelling more computationally demanding. Some of the examples of dynamic loads are hammer striking an iron block , a moving truck on a bridge etc.
2. Explain the following with relevant force-displacement graph ?
AIM:-
To determine Forece-displacement graph for different- different behaviors.
Ans:-
Figure:2.1
Force-Displacement graph for Elastic behavior.
Figure:2.2
Figure:2.3
Force-Displacement Graph for Inelastic behavior.
Figure:2.4
The normalized yield strength fy* of an inelastic system is defined as follows:
Figure:2.5
If the normalized yield strength of a system is less than unity, the system will yield and deform into the inelastic range e.g., fy0.5 implies that the yield strength of the system is one half of the minimum strength required for the system to remain elastic during the ground Motion.
Force-Displacement Graph for Plastic Behavior:
Figure:2.6
Force-Displacement Graph for Non-Linera Inelastic behavior:
3. Explain Mass, Stiffness & Damping components in the equation of motion (30 marks)
AIM:-
To explain the Mass, Stiffness and Damping components in the equation of motion.
ANS:-
The single degree of freedom system can be viewed as a combination of 3 pure independent system
The stiffness component:-
The Damping component:-
The mass component:-
Now, the external force p(t) is appilied to the complete system. They may therefore visualized among the 3 components of the structure. That is inertial, stiffness and damping force.
4. Provide relationship between Natural period (Tn) & Natural frequency (f) and provide their definitions (25
marks)
AIM:-
To determine the relationship between Natural period and Natrual frequency.
ANS:-
Natural period (Tn):-
Natural frequency (f):-
5. Explain in detail about Response spectrum & its Graph (25 marks)
AIM:- To Explain the Response spectrum and its graph.
Ans:-
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 text below is separated into three parts:
In most cases, the engineer performing a response spectrum analysis is presented with a given design response spectrum, in which case the two first parts can be considered as background material.
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).
Consider a mass-spring-damper system attached to a moving base. The foundation has a given movement,.
The SDOF system.
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
It can be seen that the support movement acts as a forcing term and that the solution depends only on the two parametersand , but not on the individual values of m, c, and k.
Instead of using the absolute displacement as the degree of freedom, it is possible to choose the relative displacement between the mass and the base, . This is actually a frame transformation where the oscillator is studied in a coordinate system attached to the base. As in any accelerating frame, there will be inertial forces. The equation of motion can be stated as(1)
Thus, the support acceleration appears as a gravity-like load. There are two advantages with this representation:
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 .
These are all relative spectra. It is possible to do a similar definition of the absolute spectra, by instead using the absolute displacement .
Sometimes, a distinction is made between the positive and negative spectra, so that
and similarly for velocity and acceleration spectra.
The velocity and acceleration response spectra are often approximated by
Such spectra are called pseudovelocity and pseudoacceleration spectra.
For a system without damping, the pseudoacceleration spectrum based on the relative displacement is actually equal to the absolute acceleration spectrum. This can be seen from the undamped equation of motion,
Thus,
The maximum absolute value of the relative displacement must thus occur at the same time as the maximum absolute value of the absolute acceleration. The scale factor between the two is . For systems with low damping, this relation will still be approximately true. Since most mechanical systems have a low damping (often 2% to 5%), it is customary to assume that the spectra for the absolute acceleration and the pseudoacceleration are the same.
Another common way of describing the damping in this context is by the Q factor (quality factor). The relation to the damping ratio is given by
The equation can be solved by a pure numerical time stepping, but there may be better ways of doing it. Ifis given as a number of points in an accelerogram, then it is natural to assume that the acceleration has a linear variation in time between those points. So, for each interval between two measurements, say fromto, the equation of motion for the oscillator is
Deformation Response Spectrum:-
Pseudo-Acceleration Response Spectrum:-
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