 In the previous lecture, we discussed about how a design response spectrum is constructed. Then we discussed about the design earthquakes. Then we discussed on the site-specific spectrum, then uniform hazard spectrum and then we discussed about how one can obtain the artificial ground motion from the response spectrum and the power spectral density function of the ground motion. The generation of this artificial ground motions are extremely important because many a time when we have to perform a non-linear analysis, we cannot use the response spectrum or power spectral density function as input. In that case, we have to provide inputs as the time histories of ground motion. To obtain the power spectral, to obtain the time histories corresponding to a specified power spectral density function or to obtain a time history of ground motion corresponding to a target response spectrum, there are many standard programs which are available these days and one can use those standard programs. The methodology, how these programs work, that what we have discussed in the previous lecture. Now, in a region where we have a number of ground motion record available and other earthquake data, measured data, then one can have a description of the seismic input as power spectral density function or a Fourier spectrum or a response spectrum or one can provide the possible peak ground acceleration and many other seismic parameters that are used as input for analyzing the structures or performing any kind of seismic analysis of structures. However, there are many places where the recorded data are not sufficient and therefore, these kind of exercise cannot be done. Also, wherever we have many data collected over a long period of time, then from those data the researchers wanted to find out some empirical equations for describing the power spectral density function or for describing the response spectrum or for describing the Fourier spectrum or the impact relationship for duration of earthquake or different kinds of attenuation laws. And these exercise have been done in the past and people came out with different kinds of empirical formulas which can be used for predicting those seismic parameters. So, today's discussion is based on that, that how we can obtain different kinds of seismic parameter which will be used for future earthquake or in other words how to predict those seismic input parameters. Now, the seismic input parameters are obtained from the past earthquake data in different regions and there are several predictive laws or predictive equations and one has to choose the most appropriate one for the region in question. And this selection of the selection of the predictive relationships, they require a careful consideration of the geology of the region, geographical condition and geotechnical condition that is the soil condition of a particular region. And one has to see what are the similar conditions existing for other regions for which the predictive equations are available. Then one can use those predictive equations for predicting seismic input parameters. In many cases some kind of adjustments are made based on the geological, geographical and geotechnical parameters for the region and with those modifications those predictive relationships can be used. Now, the seismic input parameters and ground motion parameters are generally directly available from the recorded data as I told you and one uses those empirical equations for predictive purposes. The predictive relationships are generally expressed in the form of a function of magnitude, the epicentral distance and any other important parameter. For example, it could be pig ground acceleration, it could be intensity or it could be any other quantity of interest. They are developed based on certain consideration. The equation 2.40 shows the function or the predictive equation and its form, it is generally a function of magnitude, epicentral distance and the any other important parameter and that is what I told before. The important thing is that in obtaining this equation, it is generally assumed that the all the parameters are log normally distributed. Now, this assumption has come because of the reason that these parameters which are recorded in different regions were statistically analyzed and it was found that most of these parameters tend to follow a log normal distribution. Secondly, in obtaining the attenuation law, it was observed that decrease in wave amplitude with distance be as an inverse relationship. Then the energy absorption due to material damping causes amplitudes decrease exponentially and this was also seen from the previous earthquake data and the epicentral distance are that what is generally we consider is always greater than on the actual epicentral distance because the earthquake source moves in a from one point to the other in a line source or in area source. Therefore, it is very difficult to precisely talk about an epicentral distance. Therefore, the epicentral distance that is considered in these equations are always greater than the actual epicentral distance. The mean value of the parameter is obtained from the equation or the predictive relationships and there is a standard deviation which is specified that is the mean value is provided in terms of log value and a standard deviation also is in terms of sigma ln of the parameter. The probability of exceedance is given by this simple relationship that is probability of any seismic input parameter being greater than equal to a specified value is equal to 1 minus fp where the p is defined by this that is ln y minus ln y bar divided by sigma ln y ln y bar is the log of the value specified value and ln y bar is the mean value of the parameter and sigma ln y is the standard deviation. So, this normalized quantity is taken as the value of p and fp obviously follows a normal distribution and there are standard charts which are available for standard normal variate and from that one can easily find out the value of fp and hence probability of exceedance of certain parameter can be obtained easily. Many predictive relationships laws and empirical equations exist most widely used ones are given in the text book and the ones which are widely used are described here now and let us see one by one those predictive relationship that are widely used and are given in the book. For example, the peak horizontal acceleration and peak horizontal velocity they were proposed by Asteva to be an exponential function like this exponential function of magnitude m and one can see that the r the epicentral distance is not is taken as r plus 25 that is what we discussed before that means the epicentral distance is increased by some value. Then the peak horizontal velocity that is again an exponential function of magnitude m and one can see that here the epicentral distance also is increased by r plus 25. Some value so from these two equations given a particular value of magnitude m and an epicentral distance one can find out what is the peak horizontal acceleration and what is the peak horizontal velocity. The Campbell gave another predictive law for the peak horizontal acceleration and that was from the data set that he analyzed and there the p h a is given in terms of the g unit here I forgot to mention that this p h a is given in terms of centimeter per second square and obviously this is in centimeter per second. So, log of p h a in g unit is again a linear function of magnitude m and the ln of the epicentral distance r and we see that the epicentral distance r is again increased by certain quantity. TORO obtained another expression for peak horizontal acceleration in g unit and his equation relates the moment magnitude m w with the p h a value and the epicentral distance the sigma r value is given by this that means the r m here that is the mean value of the r and there is a standard deviation which is specified for the epicentral distance r and they are given for this particular bounds over here. And the standard deviation for the p h a is given by this that is sigma square m plus sigma square r. So, we assume here in this particular equation of TORO assumed that the sigma value of p h a will be related to the sigma value of m and sigma value of r that is r is considered here again as a random variable like magnitude m. Next there was an attempt to relate the peak horizontal velocity of the ground to the with the intensity of earthquake and if you remember I said that the intensity of earthquake is a subjective measure whereas p h v is something which we expect that we should be able to measure but this subjective quantity has been used to find out a quantity which can measure. So, this is the equation that Rosenbrot observed that the p h v maintains with the intensity of earthquake. Similarly, the Cornell obtained a relationship between the intensity of earthquake and the magnitude of earthquake where the magnitude of earthquake again is a quantity which can be measured whereas intensity of earthquake is a subjective one, but from the recorded data of the earthquake and the kinds of damages and destruction that are observed the people have tried to equate i with m using this particular equation and this Cornell's equation is widely used in relating i with m where r is the epicentral distance and h is the focal depth. Then Gutenberg gave a very important relationship between the energy released and the energy released the magnitude of earthquake. So, for a given magnitude of earthquake one can assess what was the energy released and this equation is widely used in the literature in relating the magnitude of earthquake with the energy release. Boren and Boer provided another empirical equation for the peak horizontal velocity and that is a function of magnitude of earthquake. This magnitude of earthquake is the local magnitude and the epicentral distance r and we can see that it is not simply one r, but r is increased by some value. Here the J 1, J 2, J 3, J 4, J 5 and J 6 they are the constants which are variable in the sense that it may vary from region to region and one can get a particular set of value for this constants while using this equation for a particular region. Duration of earthquake also has been attempted for prediction. So, the duration of earthquake key which people observed could be related to the magnitude of earthquake and the epicentral distance and this is a exponential function of magnitude. So, with the help of this the duration of earthquake can be predicted for a particular region if enough data is not available about the duration of earthquake. Next is the Fourier amplitude spectra that we discussed in the previously and Fourier amplitude spectra provides the frequency contents of the ground motion and as an important predictive parameter for performing a frequency domain analysis of structures and this is given with the help of this equation and one can see here the constants which are involved over here is f c and that is the frequency which is called the critical frequency or cut-off frequency and then f is a usual frequency. So, the amplitude Fourier amplitude spectrum is expressed in terms of the frequency obviously it can be converted to time period because frequency bears an inverse relationship with time period and these f c is related to the moment magnitude m 0 and shear wave velocity and the constant c over here is dependent on the shear wave velocity with the help of this relationship. Now, these constants that means the value of the shear wave velocity f and delta sigma, so these values may differ from the side to side and using the specific combination of values for a particular side one can obtain a Fourier amplitude spectrum by using this equation. Next people also try to obtain the empirical relationships for the response spectrum or velocity response spectrum SV. So, this is time period dependent, so log SVT is given by this equation and one can see that it is related to the m w magnitude and epicentral distance r and a number of constants. So, these constants again vary from region to region and by substituting appropriate values of this constant for a particular region one can get an estimate of the SV that is the pseudo velocity spectrum ordinates at different time period t. Similarly, this equation is used for obtaining the acceleration response spectrum ordinate for different period t and this is related to the magnitude again magnitude of earthquake and epicentral distance r. In this AT, BT and CT they are again side dependent and one has to find out these constants AT, BT, CT and in some cases these AT, BT, CT they are plotted the plots are available and these plots are with respect to the period t. Another expression for the pseudo velocity spectrum ordinate SV that was obtained by this and here one can see that for different period given period t one can calculate the value of the pseudo velocity spectrum. So, therefore, this in fact will be a function of t and this is also dependent upon the surface magnitude m s and the epicentral distance r. Next relationship, predictive relationships were obtained for power spectral density function of ground motion and there are several such expressions. Which are available now one can see that this is expressed in terms of the ratio of omega by omega g where omega is the frequency against which we plot the power spectral density of ground motion. Omega g is the predominant frequency of the ground through which the seismic waves passes from the rock bed to the surface. So, for that soil medium one can find out the predominant frequency of the ground and g is the again damping associated with the soil. The concept here is that the ground motion while traveling from the rock bed to the surface get modified due to the soil condition and depending upon the soil predominant frequency and damping it takes a shape of the power spectral density function that is the shape there is a change in shape between the power spectral density function which is recorded or which is obtained at the rock bed and which is obtained at the surface. Now this is again another what we call expression empirical equation for the power spectral density function. These values are instead of omega g etcetera they are specified values that is for a particular class of the soil condition this is valid. There is another equation which is of this type where these constants are specified so they are valid for a particular region. The general type of the power spectral density function equation was provided by Klauff and Penjin. Here the concept is that the power spectral density function that exists at the rock bed level that gets filtered through 2 filtering medium and the frequency response functions square or absolute value of the frequency response function square are given by this equation and by this equation. So this is for filter 1 and this is for the filter 2 and the predominant frequency response for the filter 1 is omega g and for filter 2 it is omega f. Similarly the damping constant for the 2 filters they vary. Now this was a modification of the power spectral density function expression given by this equation. Now here the form of the equation is of this type that means there is only one particular filter existing and the rock bed power spectral density function gets modified through one filter that means the entire soil is considered as one filter and that obtained that gave a relationship between the surface power spectral density function and the rock bed power spectral density function. Now this was modified to these 2 filter concept of power spectral density function because of the reason that this is not able to provide an adequate or correct value of the power spectral density function of displacement at the 0 frequency whereas this expression when we use the double filter they can provide the correct value or some finite value to the power spectral density function of displacement at 0 frequency. Next we come to the coherence function that we discussed before. The coherence functions they are used for obtaining the cross power spectral density function of the ground motion between 2 points. So we work out say the power cross power spectral density function between 2 points 1 and 2 as S x 1 S x 2 to be is equal to S x 1 to the power half into S x 2 to the power half multiplied by a coherence function that was a function of omega and the distance between the 2 points r. So this is how a cross power spectral density function is used between 2 points that is computed. S x 1 and S x 2 are the power spectral density function of the ground motion may be power spectral density function of acceleration at these 2 points and for a homogeneous field we generally assume that the power spectral density function of ground acceleration at these 2 points are the same as a result of that one can write down the cross power spectral density function as S x of this coherence function. So if one knows or if the power spectral density function of the ground motion in terms of acceleration or displacement they are given then one can find out the cross power spectral density function provided one knows the coherence function. So different forms of the coherence function that is given in the literature here this is a coherence function and one can see that this is a exponentially again decaying function with a gamma function that is the distance between the 2 points and is a function of omega and this function is more precisely written over here this is also a exponentially decaying function with the shear wave velocity coming to picture. So using this into this over here one can get a coherence function which will be a multiplication of 2 exponentially decaying function. A very popular coherence function which is used in many cases is given by this expression. So this is again exponentially decaying function with a constant c specified r is the distance between the 2 points and omega is the frequency and v is the shear wave velocity of the earthquake. So this particular equation is a real quantity it does not have any imagination. So this component and as a result of that provided we know the value of r that is the distance between the point x 1 and x 2 then one can easily obtain a value of the coherence function only in terms of omega because v s will be specified c would be specified then one can get the coherence function only as a function of omega and s x is also given as a function of omega therefore s x 1, s x 2 can be easily expressed for each frequency that you consider in our analysis. This is another coherence function that is reported in the literature. This is in the form of a harmonic function that is a cosine function and an exponential function that is with the distance this exponentially decays and this form of the coherence function was obtained from the recorded data in tau i 1 and an exercise was done there with the available data to find out different forms of the coherence function and once at the time of data happen to coincide with this kind of empirical equation. The general form of the coherence function represented by this equation is given over here where we can see that this is a exponential function of a imaginary quantity i. So therefore one can write it in the form of a real part cosine and an imaginary part which will be a sine. So this particular form of the coherence function is a complex quantity becomes a complex quantity in the form of a plus i b and then s x 1, s x 2 no more remains a real quantity but a complex quantity. But in the problems of probabilistic analysis of structures using spectral analysis there is absolutely no problem in tackling s x 1, s x 2 as a complex function. In fact it is logical that the cross power spectral density function terms would be a complex function or a complex quantity rather than a real quantity. However if we use this particular form of the equation which is a special form of this one then we get the cross power spectral density function as the real quantity. Then we have a number of expressions for the modulating function and the modulating function that we talked of in the previous lecture that modulates a stationary process to a uniformly modulated non-stationary process what we call uniformly modulated power spectral density function which is also known as the evolutionary power spectral density function. And there we had seen that the power spectral density function x x of the earthquake is multiplied by a modulating function h t square and this gives the value of the power spectral density function as a function of omega and t both. So various forms of the modulating function that were observed from the recorded data. This is a very simple one which is a modulating function of this type that is a rectangular modulating function then one can have a modulating function which is of a type like this that is not exactly trapezoidal type because these two are non-linear curves but it has a form of a trapezoidal function then one can have an exponential function as these are for the exponential function rather this is an exponential function rather this is an exponential function and with the help of this modulating function one can obtain a power spectral density function which is a uniformly modulated at power spectral density function used for obtaining the response of structures for eventually power spectral density function of earthquake. So we see that a number of predictive relationship that exists in the literature and one can use any one of these predictive relationships for performing the seismic analysis of structures mostly in regions where we do not have enough earthquake data we look for these predictive relationship to predict the future form of the future earthquake that will be given as an input for the analysis. Now in using this predictive relationship one has to take care of the local geological, geotechnical and geographical conditions and many a time the constants of the empirical equations are adjusted for these conditions. So the predictive relationship as such are very useful in obtaining the seismic response of structures for a variety of cases and we will see later that for the random vibration analysis of structures using the using the spectral analysis technique we use these equations of the power spectral density functions or various forms of power spectral density functions that I have shown you and out of that the double filtered power spectral density function is widely used for obtaining the response of the structure to a specified power spectral density function and this double filtered power spectral density function is obtained with the concept that the power spectral density function of ground motion that exists at the rock bed gets filtered through two filters. And the concept of two filters were used for finding out a finite or a reasonable value of the power spectral density function of displacement at zero frequency. If you use the single filter for obtaining the power spectral density function or at the ground surface given the power spectral density function of ground motion at the rock bed then we get a situation get into a situation where the power spectral density of displacement at zero frequency remains undefined. Similarly the response spectrum ordinates for future earthquakes can be used and the analysis of structures can be carried out with the help of these predictive relationships given for the response spectrum of acceleration or velocity. There are cases where the response spectrum ordinates using the empirical relationship has been also used for obtaining the seismic hazard analysis of structures or seismic hazard analysis of regions in order to find out the hazard or the probability of exceedance of certain value of the response spectrum ordinates. Now here three examples are solved over here pH and pH are solved over here. They were calculated using different empirical equations and compared they were obtained for a magnitude of earthquake 7 and epicentral distance of 75 and 120. The comparison shows that the PHA calculated by different equations give different values for the same epicentral distance for example, Esteva his equation provided PHA of the order of 0.034, Campbell his equation provided a PHA of 0.056, then Bojorgina his equation provided 0.03 which is quite compatible with 0.034, then Toro is 0.072 and Trifunath his equation is given a value which is a wide departure from all other values. At 120 kilometer distance we can see that there is a similarity between Esteva and Bojorgina and this and this they have a good similarity Toro's and the Campbell and again there is a large departure of the PHA calculated by the equation of Tifunath. Similarly pH v was compared Esteva's equation provided 8.535 which is very high compared to the other two equations given by Joyner and Rosenbluth. At 120 kilometer distance they were more or less they are they are very much near to each other whereas this was very much different, then we compare the smooth normalised Fourier spectrum obtained from Elson to earthquake and that given by Maguier's equation and this was for the these specified values f max was taken as 10 hertz, f c was taken as 0.2 hertz, m w was taken as 7, r was taken as 100 kilometer, v s 1500 meter per second and we can see that the Fourier amplitude spectrum obtained by this equation and the Elson to earthquake they seem to compare quite well. So the expression that is given by Maguier's equation for the Fourier amplitude spectrum seems to provide a good approximation to the Fourier amplitude spectrum when the earthquake is broadbanded earthquake because we know that Elson to earthquake is a broadband earthquake. Then compare between the normalised spectrums obtained by IBC Euro 8, IS 1893 and that given by Bohr at all. So here the response spectrums which was normalised of course with PGA value. So the shape of this response spectrums were obtained for the various constants which are given in the Bohr's equation like B 1 to B 6, these constants were taken for table 3.9 given in the book G c is equal to 0 and PGA was taken as 0.35 g that is the normalised spectrums were normalised with respect to 0.35 g. And this is the comparison which we see we can see that Eurocode and IS code they are more or less the same. IBC code and the Bohr's they were IBC code was this and the Bohr's response spectrum is this. So at this region of course is fairly matching in this region and in fact in this following region it is more or less matching but there is a departure here, wide departure in the beginning of the time period. Then we compare the power spectral density functions of ground acceleration given by different expressions. For example, Hausner and Jennings, Newmark and Rosenblot, Kanai and Tajimi and Klauff and Penjin, Kanai Tajimi and Klauff and Penjin they are widely used for most of the analysis and we see that the there is a very good what we call matching between the Klauff and Penjin, Kanai Tajimi and Hausner Jennings whereas the Newmark and Rosenblot that tends to differ quite a bit from them. So, this shows that we have different kinds of spectrums given by different equations and which one is to be used depends upon the local condition but the most widely ones which are used in the literature they have a they are found to match with the Klauff and Penjin most of the power spectral density function that are obtained in the database and therefore the ones we generally accept for predicting the future earthquake can be taken for those or from those equations. In general the predictive relationships all the predictive relationship that I have shown that is for the peak ground acceleration, peak ground velocity, peak horizontal acceleration, peak horizontal velocity then response spectrum, Fourier spectrum and the power spectral density function including the modulating functions they are available in the literature some of them I have shown over here there are more and in the many websites these predictive relationships are given. So, analyzing a structure we use various forms of seismic inputs these seismic inputs if they are not directly given as a time history then one has to rely on these predictive relationships in order to describe the Fourier amplitude spectrum or the response spectrum or the power spectral density function of earthquake and attenuation relationships are quite extensively used in determining the peak ground acceleration or peak ground velocity of earthquake for a given magnitude and epicentral distance. So, these quantities are used in both probabilistic analysis of structures as well as the deterministic earthquake analysis of structures when we use the deterministic analysis of structures generally either we use response spectrum provided and these response spectrums are usually the design response spectrum specified in the code with the help of these design response spectrum one can perform a seismic response spectrum method of analysis which is an elegant method and very easy to follow and therefore widely used in earthquake engineering the main reason for this is that the entire earthquake analysis can be carried out statically.