RESPONSE SPECTRUM METHOD
1
SEISMIC RECORDS
The ground displacements recorded by a seismometer located
directly above a fault that ruptured during the 1985 Mw = 8.1,
Michaocan, Mexico earthquake.
Ground motion histories (North South directions) for El
Centro (1940)
“How to do seismic analysis using finite elements”
Phil Cooper, Philip Hoby, Nawal Prinja
2
SEISMIC RECORDS
CALIFORNIA ZONE 2
G
The earthquake is a non-stationary process
and the methods of Random Vibrations can
not be used.
Here, each graph contains 6145 data points
Time Response analysis is impractical.
time [s]
Special analysis method called Response
Spectrum Method has been developed to
CALIFORNIA ZONE 4
analyze long duration non stationary
G
processes like the earthquake or pyrotechnic
shock events.
time [s]
3
RESPONSE SPECTRA
Each data point on the response spectrum curve
represents the peak response from a time history
analysis of the earthquake applied to 1-DOF
oscillator system. The ordinate defines the natural
period of the oscillator, and the abscissa the peak
response. Repeating the procedure for a great many
frequencies defines a continuous curve for an
assumed level of damping. The figure shows the
response of 2% damped system tuned to 1Hz and
2Hz and the transfer of the peak calculated
responses to the response spectrum graph.
“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja
4
RESPONSE SPECTRA
Response spectra are typically presented for a damping ratio of 5% that is considered to be typical for buildings. This
does not mean that a damping ration of 5% is appropriate for any given analysis.
When response spectra are used as an input, a single smoothed spectrum derived from several events or several
response spectra from different events are used. The use of smoothed response spectra implies the use of several
earthquake records.
A given shock response spectrum does not have a unique corresponding time history.
“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja
5
RESPONSE SPECTRA
There are a number of approximate methods for scaling 5% damped spectra for other damping levels.
Given two consistent spectra of different damping values, the spectrum for an intermediate damping values
can be calculated based on interpolation between spectral amplitude and natural logarithm of damping ratio.
“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja
6
CONCEPT OF THE RESPONSE SPECTRUM
The concept of the Response Spectrum Method is based on the following observations:
1
A system vibrating in resonance can be described as a single degree of freedom harmonic oscillator system with certain
equivalent mass, equivalent stiffness and damping.
2
A response of a system with more than one resonant frequency can be represented as a combination of responses of
harmonic oscillators, each harmonic oscillator corresponding to particular resonant frequency. This is the basis of the modal
superposition method.
3
If the excitation frequency is equal to one of structure resonance frequencies, then the system response to that excitation is
controlled only by system damping. Mass does not matter, stiffness does not matter, they have no impact on system’s
response. The only thing controlling your system response is its damping!
Imagine two vastly different systems: A harmonic oscillator and a bridge. Those two systems have two important things in
common: the natural frequency and damping. Now imagine that both the harmonic oscillator and the bridge are excited by the
same excitation that happens to have the same frequency as the resonance frequency of both the harmonic oscillator and the
bridge. The response (e.g. the maximum displacement) will be the same for the harmonic oscillator and the bridge!
4
Say you agree that structure dynamic response can be adequately described by using the modal superposition method based
on four modes. Then rather than studying the response of the actual structure, you can study the response of four harmonic
oscillators with natural frequencies corresponding to those four modes and damping the same as damping of the structure
you study. In particular, if you study the structure response to seismic excitation, then rather than testing the actual structure,
you can just subject those four oscillators to the earthquake acceleration time history and record e.g. the maximum
acceleration of each oscillator as a function of oscillator’s frequency. This way you will build the response spectrum curve.
5
If the natural frequencies of the examined structure fall “in-between” frequencies of your oscillators, the structure response
can be interpolated. So if you know structure natural frequencies and you also know the acceleration response spectrum
curve you have just constructed by examining the response on harmonic oscillators, you can find out by interpolation, the
maximum acceleration of the structure corresponding to each mode. Double integration will give you the maximum
displacement.
6
Notice that the information of interactions between modes has been lost in the above process. Therefore, it must be re-built
using the SRSS or the Absolute Sum method, as described further.
7
GENERATING RESPONSE SPECTRUM
The Response Spectrum model is a series of Single Degree of Freedom (SDOF) harmonic oscillators with
different natural frequencies subjected to a particular earthquake ground motion. Each SDOF system
has a unique time history response to a given base input. The shock response spectrum is the peak
absolute acceleration of each SDOF oscillator to the history base input.
The SDOF oscillators are arranged in order of ascending natural frequency. The oscillator on the far left is
the most compliant one; and the oscillator on the far right is the most stiff system.
We subjected the oscillators to a half sine shock. Note that:
1.
The most compliant system is "isolated" from the shock pulse.
2.
The stiffer systems almost follow the input pulse exactly. They have very little reverberation.
3.
Some of the middle systems actually reach the highest amplitude due to a transient resonance effect.
8
Base
excitation
GENERATING RESPONSE SPECTRUM
1
0
0
50
100
150
200
250
300
350
400
time
Response of SDOF oscillators to half-sine pulse. Courtesy of Vibrationdata
9
2.0
vertegi4
yg
GENERATING RESPONSE SPECTRUM
1.5
G [ m / s2 ]
1.0
Base input
0.5
0.0
yg
-0.5
-1.0
-1.5
-2.0
0
10
20
time [ s ]
30
40
time
Step 1 A synthesized waveform;
typical acceleration time history
Step 2 A series of SDOF harmonic oscillators is subjected to the
excitation as defined by the waveform from step 1.
10
Maximum
acceleration
1
0.1
0.01
0.001
0.1
1
10
100
Frequency [Hz]
Step 3 Response spectrum curve ready to be used as input in
the Dynamic Shock analysis. It is usually plotted in log scale.
10
USING RESPONSE SPECTRUM FOR FEA
Maximum
acceleration
10
1
0.1
0.01
0.001
0.1
1
10
100
Frequency [Hz]
Response spectrum methods is ubiquitous in earthquake engineering, with very widespread support in general
purpose FE systems, and almost universal application. It is important to remember that it is a response spectrum:
i.e. once you know the natural frequencies of the structure, in simple cases domminated by a single mode, the
respoinse can be simply read from a graph. FE implementation of the procedure merely automate the interpolation,
and add post-processing to combine effects of mutliple modes and shaking directions.
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USING RESPONSE SPECTRUM FOR FEA
Step 1
Modal analysis of the analyzed structure determines modes, frequencies and mass participation
factors. Structure is thus represented by a series of SDOF oscillators.
Step 2
he acceleration response for each mode (for each SDOF oscillator) is interpolated using the
response spectrum curve (at the resonant frequency of corresponding mode).
Step 3
The complete response of each mode is found based on the acceleration response found in step 2
Step 8
An assumption is made to find the combined response of all SDOF oscillators because the time at
which oscillator (each mode) reached its peak value was discarded when the response was
calculated. There is no precise way of combining the modes to find the total response.
12
USING RESPONSE SPECTRUM FOR FEA
An assumption has to be made to find the resultant structural response since the time at which the mode
reached its peak value was discarded when the response was calculated. There is no precise way of
combining the modes to find the total response.
SRSS
The Square Root of the Sum of the Squares
Tends to underestimate the response if modes are spaced out closely. Closely spaced modes are typical for
structures with high torsional stiffness and in structures with identical members. Modes are closely spaced if,
for example, there are repetitive elements in the structure. Modes are considered closely spaced if:
CF 
fi  fi 1
 0.1
fi 1
Absolute Sum The sum of the absolute values of the contributions of each mode
Tends to overestimate the response and can be very conservative.
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TABLE
Model file
table.sldprt
Model type
beams and shells
Material
STEEL
Supports
fixed to all four legs
Objectives
•Response Spectrum analysis using Response Spectrum
curve
Fixed support to four legs
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TABLE
Mode 1
6.5Hz
Mode 1
6.7Hz
The first two modes of vibration
15
TABLE
Results of modal analysis up to 61Hz
16
TABLE
Shock response spectrum curve California zone 4; source: Bellcore
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TABLE
Maximum displacement
18
TABLE
Maximum displacement
Maximum stress
Maximum stress
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