response spectrum.ppt

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RESPONSE SPECTRUM METHOD

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Ground motion histories (North South directions) for El Centro (1940)

“How to do seismic analysis using finite elements”

Phil Cooper, Philip Hoby, Nawal Prinja

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.

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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.

Special analysis method called Response

Spectrum Method has been developed to

analyze long duration non stationary

processes like the earthquake or pyrotechnic

shock events.

CALIFORNIA ZONE 2

CALIFORNIA ZONE 4

G

G

time [s]

time [s]

SEISMIC RECORDS

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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.

RESPONSE SPECTRA

“How to do seismic analysis using finite elements” Phil Cooper, Philip Hoby, Nawal Prinja

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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.

RESPONSE SPECTRA


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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.


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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. 2A 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.3If 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.6Notice 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.

CONCEPT OF THE RESPONSE SPECTRUM

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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.

GENERATING RESPONSE SPECTRUM

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GENERATING RESPONSE SPECTRUM

Response of SDOF oscillators to half-sine pulse. Courtesy of Vibrationdata

time

Ba

se

exc

itatio

n

10

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.

Step 3 Response spectrum curve ready to be used as input in the Dynamic Shock analysis. It is usually plotted in log scale.

acce

lera

tion

Frequency [Hz]

Maximum acceleration

gyBase input

gy GENERATING RESPONSE SPECTRUM

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USING RESPONSE SPECTRUM FOR FEA

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.

Frequency [Hz]

Maximum acceleration

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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.


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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:

Absolute Sum The sum of the absolute values of the contributions of each mode

Tends to overestimate the response and can be very conservative.

1

1

CF 0.1i i

i

f f

f


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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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The first two modes of vibration

Mode 1 6.5Hz Mode 1 6.7Hz

TABLE

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Results of modal analysis up to 61Hz

TABLE

17Shock response spectrum curve California zone 4; source: Bellcore

TABLE

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Maximum displacement

TABLE

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Maximum displacement Maximum stress

Maximum stress

TABLE

response spectrum.ppt

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