Seismic Analysis Of Structures
THEORY
Civil engineering is known to be the mother of all engineering branches. Since the era of Indus valley civilization, civil engineering techniques have been burgeoning.
In the 21st century world, we get to see perfect examples of the limit of civil engineering. In fact there is no limit, there is no end to these innovations. Today what we call civil engineering marvels, are actually the result of spotless construction skills hybridized with the knowledge of safeguarding the structures from natural calamities(earthquake, tsunami, tornado, floods, etc).
Of all the natural calamities, earthquakes tend to be the most devastating. In order to protect our structures from the rage of earthquakes, we need to design them in such a way that our structures can take the horizontal ground acceleration and the combination of several irregular excitations caused by it without any damage.
This situation calls for a Frequency Domain Analysis of structures. Here we take into account the ground acceleration produced by any earthquake and analyze the reaction of the structure to it.
The above problem can be easily solved if we convert we analyze the ground acceleration obtained in frequency domain. So we convert the data from time domain to frequency domain. This procedure can be simplified with the help of a tool in MATLAB known as ‘fft’. it corresponds to Fast Fourier Transform. This command converts the (ground acceleration vs time) i.e.,[input] to (real no. vs frequency) + (complex no. vs frequency).
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After the above operation we can obtain the plots of (phase angle vs frequency) & (magnitude of ‘fft’ vs frequency).
Now in order to get the desired output (preferably displacement), we have to multiply a factor (frequency response factor or transfer function) with the input.
The equation of motion for the system can be written as:
mx + cx + kx = p(t)
p(t) is Fourier synthesized using FFT. If p(w) is the FFT of p(t), then the frequency contents x(t) of the response x(t) can be written as:
x(w) = h(w) . p(w)
in which, h(w) = (-mw^2 +k +icw)^(-1)
where, h(w) is the frequency response factor or transfer function
Frequency Domain Analysis (FDA) Steps
Ground acceleration vs time( input)
fft
ff
(complex + real components) vs frequency
Frequency response function (transfer function) is found out
Output (in frequency domain) = (transfer function)*(fft output)
output (displacement) in frequency domain
ifft
Output in frequency domain is finally converted to time domain
Advantages of FDA
In any single degree of freedom (SDOF) system subjected to an arbitrary loading, analysis in the frequency domain is more convenient than analysis in the time domain.
Furthermore when the equation of motion contains parameters which might be frequency dependent, such as ‘stiffness’ or ‘damping’ , the frequency domain approach is much superior to the time domain approach.
Frequency domain analysis gains edge over time domain taking into account its compatibility with computer programs (MATLAB).
Programs like MATLAB have inbuild functions like ‘fft’, ‘ifft’ which makes the task a lot easier.
NUMERICAL EXAMPLE
We analyze three structures subjected to earthquake ground acceleration with the following properties:-
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Structure A
K = 63000 N/m
M = 100 kg
C = 100
Structure B
K = 4000 N/m
M = 100 kg
C = 25
Structure C
K = 630 N/m
M = 100 kg
C = 10
………………………………………………………….
Damping ratio (ξ) = 2% = 0.02
RESULTS
Structure A
Displacement = 0.007 mm
Structure B
Displacement = 0.08 mm
Structure C
Displacement = 2 mm
Stucture C
Conclusion
The study confirms the existence of three kinds of structures. The response of the most flexible among the two oscillators amplifies in the low range of the frequency spectrum (flexible structures);
Whereas, the response of the most stiff among the two oscillators amplifies at the upper range of the frequency spectrum (stiff structures).
Structures which vibrate on the intermediate frequency range are known as (intermediate structures).
The [(transfer function)2 vs frequency] plot gives us an idea about the resonant frequency of a structure. This plot is frequently used study resonance in structures.
