RESPONSE SPECTRUM METHOD

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

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. Special analysis method called Response Spectrum Method has been developed to analyze long duration non stationary processes like the earthquake or pyrotechnic shock events. time [s] CALIFORNIA ZONE 4 G time [s]

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

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

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

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.

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: The most compliant system is "isolated" from the shock pulse.  The stiffer systems almost follow the input pulse exactly.  They have very little reverberation.  Some of the middle systems actually reach the highest amplitude due to a transient resonance effect.

GENERATING RESPONSE SPECTRUM Base excitation time Response of SDOF oscillators to half-sine pulse. Courtesy of Vibrationdata

GENERATING RESPONSE SPECTRUM acceleration Base input 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. Maximum acceleration 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.

USING RESPONSE SPECTRUM FOR FEA Maximum acceleration 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.

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.

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: Absolute Sum The sum of the absolute values of the contributions of each mode Tends to overestimate the response and can be very conservative.

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

The first two modes of vibration TABLE Mode 1 6.5Hz Mode 1 6.7Hz The first two modes of vibration

Results of modal analysis up to 61Hz TABLE Results of modal analysis up to 61Hz

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

TABLE Maximum displacement

TABLE Maximum displacement Maximum stress Maximum stress