Response Spectra
Response Spectra
1. Introduction
In the context of seismic analysis and design of structures, various earthquake data may be required depending upon the nature of analysis being carried out. The most common way to describe a ground motion or earthquake data is a time history record. Time history is a record of time vs. acceleration values at the respective time. The records can also be of velocity, or displacement. Generally, the directly measured quantity is the acceleration and the other parameters are the derived quantities.
2. Response Spectra of an Earthquake
A response spectrum is simply a plot of the peak or steady-state response (displacement, velocity or acceleration) of a series of oscillators of varying natural frequency, that are forced into motion by the same base vibration or shock. The resulting plot can then be used to pick off the response of any linear system, given its natural frequency of oscillation. The response spectrum of an earthquake is the most favored seismic input for earthquake engineers. There are a number of response spectra that are defined for representing the ground motion, such as, displacement response spectrum, pseudo velocity response spectrum, absolute acceleration response spectrum, and energy spectrum.
2.1 Displacement Response Spectrum
A displacement response spectrum is the plot of maximum displacement of a single degree of freedom system (SDOF) to a particular ground motion as a function of the natural frequency and damping ratio of the SDOF. Derivation of the displacement response spectrum forms the basis for deriving other spectra.
The time response of a linear SDOF system to a given ground acceleration is governed by the differential equation:
[2.1.1]
Where m is mass of SDOF, u is the displacement relative to the ground, is the ground acceleration; c, k are damping and stiffness of SDOF respectively. Eq. 2.1.1 can be written as follows:
[2.1.2]
Where ωn = 2π/Tn is the natural frequency and Tn is the natural period of vibration.
To get the response of SDOF system, equation (2.1.2) is integrated w.r.t time for a given damping, for a given seismic input once natural period of vibration is known.
If the response for different SDOF systems is calculated , maintaining the damping ratio and varying the natural period of vibration, and represent the maximum value of each response against the natural period of vibration, we get the Displacement Response Spectrum of the given seismic data for a particular damping ratio.
2.2 Pseudo Velocity and Pseudo Acceleration Response Spectra
The pseudo-velocity and pseudo-acceleration are given by:
[2.2.1]
[2.2.2]
The respective plots are called pseudo-velocity and pseudo-acceleration response spectra.
3. Code ResponseSpectra
A CODE is written in MATLAB to plot the response spectra for a given seismic input data. In the present code EL-centro earthquake data for one minute is taken and its various response spectra are plotted. Eq. (2.1.2) is solved for response using ode45 present in MATLAB. The relative damping is taken as 2%. Following are the various plots obtained using the code “ResponseSpectra.m”.
Figure 1: EL-Centro Accelerogram
Figure 2: Response of SDOF system with 2% damping and natural period of 5s to El Centro ground motion
3: Displacement Response Spectrum of EL-Centro Earthquake for 2% damping
Figure 4: Displacement Response Spectrum of EL-Centro Earthquake for 2% damping
Figure 5: Pseudo Velocity Response Spectrum of EL-Centro Earthquake for 2% damping
Figure 6: Pseudo Velocity Response Spectrum of EL-Centro Earthquake for 2% damping
Figure 7: Pseudo Acceleration Response Spectrum of EL-Centro Earthquake for 2% damping
Figure 8: Acceleration Response Spectrum of EL-Centro Earthquake for 2% damping
The code ResponseSpectra can be obtained from the following link:
Response Spectra:
http://www.mathworks.com/matlabcentral/fileexchange/32913-response-spectra