 In the previous lecture, we were discussing about DVA spectrum and we had seen that the response spectrum can be plotted in a tripotide plot and can be idealized as a series of straight lines representing all the three spectrums that is the displacement, velocity and acceleration spectrums. We continue with this response spectrum for different earthquakes that were recorded all over the world. For them, the response spectrums were obtained and plotted in a tripotide plot and similar exercise was done to represent it by a series of straight lines and it was seen that the most of the earthquakes represented the same kind of pattern, only difference was between the values of TA, TB, TC, TD, TE and TF that were obtained and they were different for different earthquakes. So, this observation led to a general shape of the response spectrum for earthquake and therefore, that shape was utilized for constructing what is known as the design response spectrum. The design response spectrum obviously is the response spectrum that using which we design all the structure for future earthquake. Now, an exercise was carried out to show that the different earthquake records provide the same kind of idealized response spectrum on a tripotide plot. For Parkfield earthquake and for Elson to earthquake, these exercise were carried out and one can see that the idealized response spectrum for the two earthquake, this is the Elson to earthquake and this was the Parkfield earthquake. When they were idealized with the help of straight line, they nearly had the same kind of feature and only thing that differed was the TA T values, T values are different points and these T values were compared in this table. For the Parkfield earthquake and for the Elson to earthquake, one can see that TA value that basically was obtained for Parkfield and for the Elcentro, it was 0.03 and this was 0.041, for Parkfield it was 0.134 that is TB and it was 0.125 for Elcentro, for TC it was 0.436 and 0.349 for the Elcentro and 4.12 was for TD for Parkfield and for Elcentro it was 3.135, TC, TE and TF they were more or less the same for the two earthquakes especially the TF. So, thus these two earthquake records revealed that there could be difference between the values of TA, TB, TC, TD, TETF but the nature of the response spectrum can be approximated by a series of straight lines having the characteristics that we discussed before. Now let us come to the design response spectrum. The design response spectrum is the response spectrum which is used for designing all the structures for future earthquakes which are not known. Therefore, it is a difficult task, what generally is done for obtaining the quantities for future earthquake is that we try to see the trend of the parameter that we are looking for of that parameter in the previous earthquakes. So, therefore, once we were convinced from the results of the different response spectrums drawn on the tipotide plot that the shape of the response spectrum can be idealized by a series of straight lines between TA, TB, TC, TD, TE etc. We said that the future earthquake also will show up similar kind of trend and therefore, design response spectrum was developed on that basis. However, the design response spectrum should satisfy some other requirements for example, the spectrum should be as smooth as possible and therefore, the idealizing the response spectrum by a series of straight line is quite rational. If the response spectrum or the design response spectrum is not smooth enough and it changes rapidly with time period, then there could be a sudden change in the value of spectral acceleration for a small difference of time period. Now, it is quite expected that the time period of a structure in reality could be different than the time period that we theoretically calculate. Therefore, if there is a large change in the spectral value or the spectral acceleration for a small change in time period, then there could be a lot of difference between the theoretical spectral acceleration that is considered in the design and the actual spectral acceleration that the structure experiences at the time of earthquake. So, in order to eliminate that the design response spectrum were considered to be as smooth as possible. Then design spectrum should be representative of spectra of past ground motions that we explained before. Next two response spectra should be considered to cater to variations and design philosophy. Now, the design philosophy of the earthquake resistant design is that we design the structure for a design response spectrum and see that that structure can survive in the extreme earthquake condition. That is there will not be a complete collapse of that structure in the severe earthquake, it can have a huge amount of damage. So, this is the current design philosophy and therefore, we have to have two level response spectrum. One is the design response spectrum other is the extreme response spectrum. And next is that the response spectrum should be normalized with respect to peak ground acceleration because if you are wanting to provide or prescribe a design response spectrum that should be prescribed in the form of the shape. But the actual response spectrum would be obtained after we multiply that shape with the peak ground acceleration which may differ from place to place. Next let us see how we can construct a design spectrum. So, for that we follow certain steps. First the expected peak ground acceleration values for design and maximum probable earthquakes are derived for the region. Then peak values of ground velocity and displacements are obtained using these empirical relationship that is if anyone of the quantities generate the ground acceleration, maximum ground acceleration if it is given then with the help of that one can find out the peak ground displacement and peak ground velocity using these values of this constant C and C 1 and C 2 has this. So, once we have the specified values of the maximum ground acceleration and from that once we have obtained the maximum ground displacement and maximum ground velocity then in a tripartite plot we can have a baseline plot in the 4 way log paper. That means that baseline plot will simply draw or show the peak ground acceleration, peak ground velocity and peak ground displacement. Then as we discussed before obtain the value obtain the b c d and c d segments using these multipliers that is the maximum ground acceleration is multiplied by alpha a then maximum ground displacement is multiplied by alpha d and maximum ground velocity is multiplied by alpha v. These alpha a alpha d and alpha v they are available for different soil condition and also for the design earthquake level and the extreme earthquake level in various literature and they are obtained from the past earthquake records. Since C and D points are fixed the T c is known and once T c is known then with respect to T c other values T a, T b, T d, T e, T f etcetera can be obtained from the again the experience that people had from the previous earthquake records. Now generally this empirical relationship is used in obtaining the values of T b, T a, T e etcetera for example they are all defined in terms of T c and T e and T f it has been seen that T f is generally in the ranges in this range that is 30 to 35 second and T e generally ranges between 10 to 15 second. So now let us look at how we can obtain using the steps the response spectrum in a tripartite plot. This is the base line, base line means the ground maximum ground acceleration that is plotted over here and it is parallel to the S D axis and we measure the acceleration along this line. Then this is the maximum ground displacement line which is parallel to acceleration line and the ground displacement S D not ground displacement S D is measured along this line. So this is the line showing the maximum ground displacement this is a line showing the ground maximum acceleration and obviously this is the line which shows the maximum ground velocity. So this is the base line that we obtain. Then as we have described before this segment that is from C to B between B to C this segment basically is parallel to this line as we discussed before and that is equal to alpha A times the ground acceleration. So therefore, we draw a line this line parallel to this line and by this value being equal to alpha A multiplied by u double dot x g. Similarly, this line is parallel to this line and this is equal to alpha D times the maximum ground displacement and this line is parallel to this line and this value is equal to alpha V times the maximum ground velocity. So once we have these lines plotted then one can join these lines to get the idealized response spectrum described with the help of a series of straight lines. Now if we wish to plot the spectral acceleration versus time period in a ordinary paper that means they are not log scale but ordinary scale and normalize with respect to the gravity that is g that S A by g plot can be shown to be of this type that means this shape has emerged from the shape of the response spectrum that was observed in a tripotide plot. So in almost all codes we find the spectral acceleration plot versus time is of this particular shape. Now a design spectrum can be obtained for 50th percentile at 84.1 percentile in a tripotide plot. 50th percentile means the design spectrum and 84.1 percentile that is the mean value plus one standard deviation that gives say 84.1 percentile and that is taken as the response spectrum for the extreme earthquake. So we can have these two earthquakes and for these earthquakes one can construct a design spectrum given the following quantities. For example the T A was given as 1 by 33 second, T B was 1 by 8 second, T E was 10 second and T F was 33 seconds and the alpha A alpha D and alpha V values that this multiplying factors they can be taken from a standard textbooks on earthquake engineering. For example in this case alpha A was 2.17, alpha V was 1.65 and alpha D was 1.39 against the these values in the bracket they denote for the 84.1 percentile and these one for the 50th percentile and the damping was considered 5 percent. So what we did is that this ground acceleration, peak ground acceleration was given and with the help of that we obtained the peak ground velocity and peak ground displacement using the empirical relationship that we have shown before. And then we plotted the two spectrum this was the base line that is showing the peak ground velocity, peak ground acceleration and peak ground displacement and then multiplying by alpha A, alpha D, alpha V values we obtained these segments of the straight line and joined and obtained the response spectrum in a tripartite plot and the T A, T B, T C, T D, T E values were specified. Therefore, once we have of the T A, T B, T C, T D, T E values specified and alpha A and alpha V and alpha D values are given then one can construct a design spectrum. Generally, if the T C value may be given and from the T C value one can obtain the other values that is T A, T B, etc. using the approximate values that was given by this relationship. Once you know this relationship approximate relationship one can obtain the values of T B and T A, T E and T F generally lie in this range. We can we have discussed how one can obtain a design response spectrum using that idealized segmented or idealized series of straight lines and plotted in a tripartite plot. Now, we come to the different levels of earthquake that is described in the earthquake literature. Design earthquake that generally specify certain value of peak ground acceleration and from that peak ground acceleration one can also estimate the expected peak ground acceleration for the extreme earthquake. Now, in the literature we get different terminologies for the design earthquake. For example, one is M C E that is maximum credible earthquake. There is a largest earthquake that we expect from a source then S S E that is safely shut down earthquake that is used in a nuclear power plant design. Other terms which denotes similar levels are credible safety level maximum etc. and they denote the upper limits for the two level concept that means they denote the earthquake PGA values for the extreme condition. The lower level that is the design PGA value is called OBE that is the ordinary base earthquake and other terminologies which are used or the similar terminologies which are used for ordinary base earthquake is operating level probable design and strength level earthquakes. And generally the OBE that is the design usual design earthquake that is the OBE that is the ordinary base earthquake is approximately equal to half of the safe shut down earthquake used in the nuclear power plant design. Next we come to the site specific spectra. The site specific spectra is called a site dependent response spectrum and they are used for designing specialty structures for a particular place and that particular place may have certain characteristics in terms of geology and geography that is the design response spectrum that we talk of that design response spectrum may not be valid for that particular site because of the geological and geographical conditions of the site. Now for that what we do is that we collect as many number of earthquake data possible for that particular site that is in and around that site. Sometimes if the collected data is not sufficient then we augment the earthquake data by collecting the data for similar geographical and geological regions. Once we have collected enough number of earthquake records in and around that site after augmentation then we scale the earthquake records and also modify it for the soil condition. Now the scaling is necessary because different earthquake records that have been collected they may be valid for certain epicentral distances, certain peak ground acceleration or certain magnitude of earthquake. Whereas the site specific spectrum that may have to be constructed for a given epicentral distance, for a given magnitude of earthquake and for a given soil condition. Therefore, all the earthquake records that have been collected they must be properly scaled so that it reflects the epicentral distance and the magnitude or the peak ground acceleration for which the site specific spectrum have to be constructed. And also the site specific spectrum should be compatible with the soil condition. And when most of the cases the records that have been collected these records are collected on certain particular sites whose soil conditions may not be same as the soil condition that is existing for the site. Therefore, certain techniques are used for scaling and after the scaling we obtain the required response spectrum or site specific spectrum. The scaling depends upon the data that are available. In fact, if the data are given in the form of the time history records then these time history records are furious synthesized and then the frequency contents of the ground motion of the each record they are normalized to provide a common base or common magnitude of earthquake for which the site specific spectrum is to be constructed. Also it is normalized so that epicentral distance of the particular site for which you are constructing the site specific spectrum that epicentral distance is also reflected in the scaling process. After the response or the earthquake records have been scaled properly then and modified for the local soil condition then a response spectrum corresponding to each one of those modified records of the ground motions are considered for obtaining the response spectrum from each one of them. And these response spectrums are then averaged and smoothed in order to obtain the site specific spectrum. The effect of appropriate soil condition generally is incorporated by deconvolution and convolution as shown in this figure. In most of the cases the ground records that are available they may be on the surface of a particular ground and that ground may be having a particular soil property. And the soil property that is existing for the site is say different that is at point t say for example, we are interested in obtaining the site specific spectrum and say at point a we have the some ground record that is available. Then in order to get the appropriate soil condition in the site specific spectrum we do a deconvolution that is from the ground recorded ground motion at station A we obtain the ground motion at the bedrock by a deconvolution process. And once we get the bedrock motion and then we assume that the bedrock motion that is available at point D that will be same as that which is available at point B. And assuming these two ground bedrock motions to be same then we do a forward wave propagation problem or carry out a forward propagation problem at point D to point T and consider the appropriate soil condition while performing these convolution technique. Now this is explained with the help of a particular problem here in place of the ground time history records we consider that there is a site over here and there are three stations where basically the data have been collected and at these three stations the epicentral distance is given and the shear wave velocities are given. These shear wave velocities and epicentral distances are different for the three stations. The shear wave velocity which is given basically indicates the soil condition for the particular station. Now in place of the ground history or time history of ground motion for these stations it is given in the form of the power spectral density function of the ground motion or power spectral density function of the ground acceleration for these three stations. With the help of that information and the particular attenuation law that is valid for the region we perform the calculations in order to obtain the site specific spectrum at this particular site and in the in that process we will see how we can do the scaling for the magnitude of the ground acceleration and also do the scaling for the epicentral distance. And finally, consider the appropriate soil condition that is existing at the site. The basis of the problem will be much more clear when we will cover the spectral analysis technique for solving a structures response to random ground motion. But for the time being the problem is explained with the help of the some accepted formula in order to show you how the scaling is done. Now for the site one the power spectral density function of the ground acceleration is given by this for site two the this is the power spectral density function of the ground acceleration which is shown and this is for the site three the power spectral density function which is given. Now these power spectral density functions are available for the surface of the ground and as I told you if we have the information available on the surface of the ground then many a time we have to do a deconvolution technique to obtain the corresponding quantity at the rock bed level. And then only we take that particular rock bed level information and we assume that the same condition exist at the rock bed of the particular site. So, what we do we first convert this power spectral density function which is given at the surface at each site to the corresponding power spectral density function which will exist at the rock bed level just below site one site two and site three. And for that we use these particular formula which will be discussed in the in chapter four where we will be discussing the response analysis of structures for random ground motion. And there we will show that the if the power spectral density function is given for a particular system then and if it is an input to that particular system then output power spectral density function can be obtained by multiplying the given power spectral density function with the transfer function square of the system. So, here the power spectral density function at the surface that basically is obtained as a function of the product of power spectral density function at the rock bed multiplied by the transfer function square that is a forward problem or in other words forward wave propagation problem that is from the rock bed we are going to the surface. Now, if this equation holds good then from this equation we can work back what is the power spectral density function at the rock bed that will be equal to power spectral density function at the surface divided by the transfer function square. And the transfer function square is given in terms of the depth of the soil and the shear wave velocity and the damping of the system and this is obtained for a different frequencies. Now, using these transfer function square for the specified values of the V s for the three sides for the three sides we have seen that we have different shear wave velocities and the soil layer depth is given as 20 meter xi is taken as 5 percent. And then for each omega value we calculated the values of transfer function square and we divided the surface power spectral density function of acceleration by the transfer function square that gave us the power spectral density function of the ground acceleration at the rock bed level just below site one. Similarly, the value of the power spectral density function at the rock bed level for site two is given over here and the power spectral density function of the ground acceleration at the rock bed level at site three is shown over here. So, once we got this power spectral density function then there are certain method to calculate the peak ground acceleration for the site at the rock bed level from the rock bed power spectral density function. Similarly, for the site two we can find out the peak ground acceleration at the rock bed level from the power spectral density function of the ground acceleration that we have obtained at the rock bed level. And for site three also we did the same kind of calculation in order to obtain the peak ground acceleration at the rock bed level from the power spectral density function that is obtained for the site three. Now, once we know the peak ground accelerations for the three sites at the rock bed level and we use their epicentral distance then using the epicentral distance and the peak ground acceleration one can calculate the magnitudes of earthquake for which these power spectral density functions were obtained for the sites one two and three at the surface level. So, these magnitudes of the three sites for which the ground motion or the ground motion power spectral density functions were available are now calculated that is m 1 is equal to 6.2 m 2 was found to be 5.8 and m 3 was found to be 7.3. So, these magnitudes were obtained by using a attenuation relationship given by Toro and if you recall most of the attenuation relationship they provide a peak ground acceleration as a function of the magnitude of earthquake and epicentral distance. Therefore, if peak ground acceleration is given and epicentral distance is given then from that formula one can find out the magnitudes of the earthquake. Now, it is specified that the for the site the site specific spectrum should correspond to a magnitude of earthquake which is equal to 7. Therefore, all the power spectral density functions which were obtained at the rock bed level for site one site two and site three must be scaled for the magnitude of earthquake that is these magnitude levels should be brought to the level of 7. So, for that what is done is again we go back to the attenuation relationship and in that relationship we put the value of the magnitude at 7 and epicentral distance as the epicentral distance of the site which is specified. So, once we provide these two information from there we get the peak ground acceleration which should exist at the rock bed level and for the site for which we are wanting to have the site specific spectrum. So, once we know that peak ground acceleration of the site then one can obtain the scaled power spectral density function at the rock bed level is equal to the peak ground acceleration square of the site and the peak ground acceleration square of the different locations that is location one two and three for which we obtained the magnitudes of earthquakes at m 1 m 2 m 3 for those locations the peak ground accelerations have already been calculated before. So, we know those peak ground acceleration. So, we modify the PSDFs of the ground acceleration at the rock bed level of the three locations in this particular way this is the unmodified that is the peak the power spectral density function of the ground acceleration at any particular location say location one at the rock bed level and that is now multiplied by p g a square s that is p g a that we obtain for the site for which you are wanting to draw the site specific spectrum and peak ground acceleration for that particular location. Similarly, for location two and three we can get a power spectral density function modified power spectral density function which will be valid for that particular site in question at the rock bed level. Thus we will get three scaled power spectral density function at the rock bed level of the site in question for which we are going to obtain the site specific spectrum. So, this scaling takes care of the scaling of magnitude or the peak ground acceleration also take care of the scaling of the epicentral distance. So, after we have done that then we consider the soil condition and for the particular site in question. So, the power spectral density function which we have obtained at the particular site coming from location one that we call as p s D F 1 and if we multiply that by the transfer function square then this will provide us a power spectral density function at the surface level. So, this is the power spectral density function that we have obtained at the rock bed level coming from location one after scaling and this will be the power spectral density function at the surface of the ground that is at the site itself. So, that way one can obtain the power spectral density function and for coming from location 2 to the site and similarly one can construct p s D F 3 that will be coming from the location 3 which will be considering and for obtaining the site specific spectrum here. Now, once we get these 3 power spectral density functions at the site then we generate the time histories of the ground motion or synthetically generate the time history of ground motion from the given power spectral density function and there are techniques for the for obtaining the synthetic ground motion or generate synthetic ground motion for a given power spectral density function. So, utilizing that one can obtain the power spectral or that obtain the time histories of the ground motion for the 3 p s D F or in other words we get 3 time histories of ground motions from these 3 time histories of ground motion we obtain the response spectrum 3 response spectrums and these 3 response spectrums are average and smoothened in order to obtain the final response spectrum which is the site specific spectrum and which is shown here in the last figure. So, this last figure shows the site specific spectrum which is an average and smooth spectrum coming from the 3 earthquake records that have been synthetically generated. So, the procedure that we adopt depends upon the given quantity that we have and if the given quantity is a quantity which is only the time histories then these time histories are to be scaled for the peak ground acceleration or magnitude of earthquake and a central distance through Fourier series analysis. If it is given in terms of the power spectral density function then the procedure that I have mentioned that can be obtained in order to get the site specific spectrum. Next we come to uniform hazard spectrum the uniform hazard spectrum is also used in many cases when we are calculating the hazard analysis that is seismic risk analysis of structures at a particular site and that may require the construction of a uniform hazard spectrum. Statistical analysis of available spectrum is performed to find distribution of peak ground acceleration and spectral ordinate at each period in order to obtain the uniform hazard spectrum. So, what is done is that if we have a number of the response spectrums which are available at a particular region then what we do is that the peak ground accelerations which we get from each one of these response spectrums these peak ground accelerations are taken and a pdf and cdf that is cumulative distribution function and the probability density function of the peak ground acceleration is obtained from the given data. Similarly for a particular ordinate of the response spectrum at a given time period that is taken as a random variable and if we have got say 50 response spectrums available for the site then we get for a particular time period 50 values of the ordinate of the response spectrum. So, we construct again a power spectrum probability density function and the cdf for that particular ordinate of the spectrum at a particular time period. The same thing can be done for all the time periods and we can have a distribution curve for a number of time periods and a pdf ground acceleration and once we get these probability density functions or the cdf cumulative distribution function then from these distributions the values of the spectral ordinates which specified probability of accidents are used to construct the uniform hazard spectrum. Alternatively one can obtain a seismic hazard analysis which can be carried out for spectral ordinates. We have explained the calculation of the seismic hazard curve for the peak ground acceleration. So, the same concept can be extended for the spectral ordinates at each time period for a given value of jai and once we have these hazard curve for the spectral ordinate at a particular time period then from that hazard curve again one can get the spectral ordinate corresponding to a certain probability of accidents and with the help of that data one can construct the uniform hazard spectrum. Next we come to the synthetic accelerograms. For many cases one may have to obtain response spectrum or power spectral density function which are compatible with the time history records or for a given response spectrum or a given power spectral density function one may have to calculate synthetically a compatible time history record. So, these time history records which are generated from the response spectrum or given response spectrum or given power spectral density function. These time history records if they are used for constructing back the response spectrum or power spectral density function they will be same as the ones from which they were generated. And these compatible time history records are many a time required for performing non-linear analysis of structures for a specified response spectrum or a specified power spectral density function. And as all of you know that the response spectrum method of analysis and power spectral density function method of analysis they are valid for the linear case. Therefore, if one has to perform in non-linear analysis then one has to obtain the compatible time history records. Now response spectrum compatible ground motion is generated by iteration to match a specified spectrum say its first is starts with the generation of a set of Gaussian random numbers and a set of iterations are performed. And these iterations are performed in such a way that at every iteration the calculated response spectrum is matched with the target response spectrum. Now in the case of the power spectral density function we obtain the compatible time history of ground motion by using equation 2.39 which is nothing but the concept that we have explained for the case of the Fourier series. And you can see here the any arbitrary function t can be written in the form of a Fourier series only difference over here is the values of phi i that is added now and a i that is to be obtained. So, a i basically can be obtained from the relationship that exists between C n and the ordinate of the power spectral density function multiplied by d omega. If you recall we obtain the power spectral density function from the Fourier spectrum in one particular problem and there we have just seen that how C n is related to s into d omega. So, once we know the ordinate of the power spectral density function then one can find out the value of C n and that C n is nothing but the a i values. The phi i values are some random phase angles which is uniformly distributed between 0 to 2 pi and we have got now many standard programs which can perform this that is they can generate the random spectral which is uniformly distributed between 0 to 2 pi. So, utilizing that one can generate a time history of ground acceleration or ground motion for a given power spectral density function. Many standard programs are these available or these days available in order to obtain the time history of ground motion which are compatible with the power spectral density function or a given response spectrum. Generation of the partially correlated ground motion at a number of points having the same PSDA we somewhat involved and however the methodology is given in the references of the book.