 In the last few lecture, we discussed about the response of structures for the random excitation using the spectral analysis technique. And we had seen there that the input for the random vibration analysis of structures using spectral analysis technique is the power spectral density function of excitation and the cross power spectral density function between two excitations. If you recall, we discussed in seismic input that there are many types of seismic inputs for analyzing structures. The time history records is one with which we can obtain the response for deterministic analysis of the structures for earthquake that is we assume that the given earthquake is a deterministic quantity. And we analyze the structure and this analysis is called the deterministic analysis of the structures for earthquake. Then comes the analysis of the structures for the random vibration where we use the power spectral density function as input. For the deterministic analysis, not only the time history of the earthquake is an input, it could be also the Fourier spectrum of the earthquake record that also can be used as an input for analyzing the structure. The next type of input which we discussed at length is the response spectrum of earthquake. And we had seen that how a design response spectrum for earthquake is obtained and the different kinds of response spectrums that are available in the literature. Out of these different kinds of response spectrum, the response spectrum of acceleration, better it is called response spectrum of pseudo acceleration that has been adopted as an input for the response spectrum method of analysis for structures. In any earthquake code prescribing the code provisions for the analysis and design of structures for earthquake, a response spectrum that is the pseudo acceleration response spectrum is specified. And generally most of the structures are analyzed using that response spectrum which is specified in the code of practice. The response spectrum method of analysis has become a very very popular method of analysis for earthquake engineers for quite a few reasons. For example, the response spectrum method provides a technique for performing an equivalent static lateral load analysis. Now, that is the most attractive part of this analysis because it is very suitable for the designers. They do not want to perform a dynamic analysis at the design stage. In the design stage, it is better to have a equivalent static analysis so that the earthquake forces can be combined with other static loads or the responses produced due to other static loads. And a combined critical member force can be obtained for which the member can be designed. So, these equivalent static lateral load analysis which is performed for in the response spectrum method of analysis has become very attractive to earthquake engineers. Secondly, it allows a clear understanding of the contributions of different modes of vibration. So, that is another thing which a designer would like to understand. That is as the structure vibrates in different modes under an irregular time history of excitation, then obviously one would like to understand what are the different contributions of the different modes of vibration in the structure. And once they understand that, then one can consider how many modes they would take into account for finding out a good estimate of the response of the structure for which they would perform the design. Further, this also gives a physical understanding of the behaviour of the structure when it is undergoing a vibration under earthquake. Then it offers a simplified method for finding the design forces for structural members for earthquake. So, this is so because the forces are computed using a static load analysis. Therefore, the method is a simplified method and that a designer could consider in obtaining the design of the structures. The dynamic analysis part of the structure obviously is more complicated compared to a static analysis. And therefore, when this entire concept of response spectrum method of analysis was developed to find a simplified equivalent static load lateral load analysis, then the method really became a very popular method. It is also useful for approximate evaluation of seismic reliability analysis. Many a time one has to obtain the seismic reliability of structures and in place of doing a rather complex dynamic analysis, one can perform a simplified equivalent static lateral load analysis for structures and using the results of that one can also obtain a seismic reliability analysis of structures. The concept of equivalent lateral force for earthquake is a unique concept because it converts a dynamic analysis partly to dynamic and partly to static analysis for finding maximum stresses. It is partly dynamic and partly static because of the reason that in the first part of the analysis, one has to find out the more shapes and frequencies of the structure and in finding out the more shapes and frequencies of the structure, one has to perform a dynamic analysis of the system or rather you can say one has to solve a Eigen value problem. Once the more shapes and frequencies of the structures are known, then rest of the analysis is a static analysis and therefore, it is called partly a dynamic analysis partly a static analysis. The most important thing in the seismic design is the maximum stresses that is developed in the members so that the members can be designed for those forces. The designers really are not interested in the time history of the stresses because the maximum stress that is of direct interest to them and these method provides directly the maximum stresses which are induced in different members due to earthquake. Equivalent lateral force for an earthquake is defined as a set of lateral force which will produce the same peak response as that obtained by dynamic analysis of structures. This is the most attractive part of the equivalent lateral load analysis for earthquake because we construct a set of equivalent static lateral load in each mode of vibration and this set of lateral load when is applied to the structure, it provides a response of the structure that response happens to be equal to the maximum dynamic response that the structure would have undergone if it were vibrating only in that mode. So, if it is assumed that the structure is vibrating only in one mode then this equivalent lateral force can straight away provide us the maximum response the structure would undergo in that particular mode of vibration. Obviously, the equivalence is restricted to a single mode of vibration. The response spectrum method of analysis is developed using the following steps. First a modal analysis of the structure is carried out to obtain mode shapes, frequencies and modal participation factor. All of you have done this exercise when you have undergone a course in dynamic analysis of structures. The mode shapes and frequencies they are very routinely obtained for any given structures provided the mass matrix and the stiffness matrix corresponding to the dynamic degrees of freedom unknown. And once we have the mode shapes and frequencies then one can find out the mode participation factor for each mode that we will see little later. Then using the acceleration response spectrum which is given in a code and equivalent static load is derived which will provide the same maximum response as that obtained in each mode of vibration that I have explained earlier. The maximum modal responses are combined then to find total maximum response of the structure. The first step that is the finding of the mode shapes and frequencies of the structure that is a dynamic analysis while the second step is a static analysis because we obtain an equivalent static load and carry out a static analysis. The first two steps do not have any approximation that is we do not compromise any accuracy in this first two steps and they are derived from the they can be derived from the first principles. While the third step that is when we are combining the maximum responses in different modes of vibration then we make some kind of approximations. As a result of that the response spectrum analysis is called an approximate analysis but applications show that it provides mostly a good estimate of peak responses. So the response spectrum method of analysis although it is called an approximate analysis because of the combination of the contributions of the different modes yet the method is found to be a very useful and it can provide or estimate the peak responses. Quite satisfactorily the method as such is developed for single point single component excitation for classically damped linear systems. However with additional approximations it has been extended for multi point multi component excitations and for nonclassically damped systems. So first what we will do is that we will look into how we come into the response spectrum method of analysis or develop the required equations for obtaining the response using the response spectrums of earthquake. For single point single component excitations and for classically damped linear system that is where the damping of the structure is constructed out of the modal damping ratio and it is defined for one type of material that means if the structure is constructed out of one type of material for which the damping ratio is specified then we can obtain or derive the required equations for the response spectrum method of analysis. When we encounter a structure which is not classically damped systems like a structure in which two kinds of materials are used for example steel and concrete in that case it no more remains a classically damped system. Similarly if we consider the soil structure interaction in a problem then also the system does not remain a classically damped system. Also for the multi support excitations and multi component excitations these response spectrum method of analysis cannot be directly derived without making some kind of additional approximations and we will look into this later. Let us take the equation of the motion for a multi degree of freedom system under single point excitation. The equation which is shown in 5.1 it is not very clear it will be m x double dot plus c x dot plus k x is equal to minus m i x double dot g and if we use the normal mode theory which all of you are acquainted with an uncoupled sets of equation can be obtained which take the form of z i double dot plus twice zeta i into omega i into z i dot plus omega i square z i is equal to minus lambda i into x double dot g that is the ith modal equation and the number of modes that is considered is equal to m. So, therefore, i varies from 1 to m now each one of these modal equations how it is derived using normal mode theory is known to all of you the z i is the ith generalized coordinate for the ith mode lambda i is the mode participation factor for the ith mode and omega i is the natural frequency for the ith mode. The lambda i that is the mode participation factor is obtained using this formula that is phi i t m i divided by phi i t m phi i where phi i t is the transpose of the ith mode and i is an influence coefficient vector generally consisting of 1 1 1 1 or 1 0 1 0 1 0 which we discussed in connection with the response analysis of structures to the deterministic ground motions. So, using these formula one can obtain the value of lambda i and once we have the lambda i then one can solve each one of these modal equations in time and then the responses for the original structural coordinate system that is the coordinate system. This x i can be obtained by using equation 5.3 that is we multiply the phi i with z i where phi i is the ith mode shape and z i is the time history of the generalized coordinate for the ith mode and this x i is also a time history of displacement in structural coordinate. And the value of x i that I get is the response of the structure if it vibrates only in the ith mode. So, in this fashion one can find out the responses in any mode of vibration and once we obtain the vibrations in different modes of vibration then we sum up the time histories of these responses to obtain the final response of the structure and the summation is carried out for m number of modes that we consider for the analysis. Now, say if we wish to find out the maximum value of the force that we have in the that is generated in the structure during its vibration in the ith mode and we call it as f s i then f s i can be obtained by simply multiplying the stiffness matrix of the structure with x i that is the displacement that is taking place in the ith mode of vibration. And this can be written as k into phi i into z i which is given in or rather using equation 5.3. Now, as the undamped mode shape phi i satisfies the following equation that is k phi i is equal to omega i square m phi i giving rise to the what is known as the eigenvalue problem then one can substitute for k phi i and can write down the f s i that is the force which is attracted in the ith mode of vibration equal to omega i square m phi i into z i. Since, z i is a single quantity or the time history of a single quantity one can find out a maximum value of z i and then the f s i would be the maximum force that is developed onto the structure in the ith mode and this is called as f s i max equal to m phi i into omega i square into z i max. This z i max is the maximum displacement in the generalized coordinate in the ith mode. Now, let us go back to the development of the displacement response spectrum. If you recall the displacement response spectrum for an earthquake is obtained by analyzing the single degree of freedom system with a specified damping and a frequency for a given record of earthquake and finding out the maximum displacement and this maximum displacement that we get for the single degree of freedom system is plotted against t i which is equal to 2 pi by omega i and this way one can have a plot of the maximum displacement of the oscillator against time period t which is called the displacement response spectrum and these displacement response spectrum is obtained for a given damping ratio z i. Therefore, if we use that definition of the displacement response spectrum then z i max that is obtained is nothing but the is nothing but is equal to lambda i into s d i where s d i is the displacement response spectrum ordinate at time t i or period t i is equal to 2 pi by omega i and is valid for a damping given damping ratio z i. It is to be multiplied by lambda i because originally the equation that we had is this equation that is on the right hand side of this single degree of freedom equation it is not x double dot g it is multiplied by lambda i therefore, lambda i comes into picture here. So, z i max can be written as lambda i into s d i therefore, for a given earthquake if we know the displacement response spectrum then corresponding to the time period omega 2 pi by omega i and the damping ratio z i one can obtain the value of s d i from the displacement response spectrum curve and that gets multiplied by lambda i and that gives you the value of z i max then we know the relationship between the displacement response spectrum and the pseudo acceleration response spectrum loosely called the acceleration response spectrum and that is s i is equal to omega square into s d then that equation now can be written as f s i max is equal to lambda i into m into phi i into s i that is we substitute in place of omega i square into s d the value of s i in place of omega i square into s d the value of s a i and this quantity is called p e i that is the equivalent static load and this equivalent static load produces a maximum displacement of the structure if it were vibrating only in the ith mode. Therefore, this load p e i is an equivalent static load and if this equivalent static load is applied on to the structure statically it will produce a displacement in the structure which would be equal to the maximum displacement that the structure would have if it were vibrating only in the ith mode then once we obtain the value of p e i using the formula 5.9 then the maximum the things that are required for us are s a i that is the spectral acceleration that must be defined and that is generally given in all course of practice phi i that is the mode shapes in the ith mode and the mode participation factor lambda i that can be obtained using the formula that are shown before. So, with the help of this now one can obtain p e i and once you obtain p e i then one can obtain the maximum displacement that the structure would have if it were vibrating only in the ith mode and the analysis that we perform in equation 5.10 is a purely a static analysis. So, therefore, we see that in finding out the equation 5.9 we step by step follow the normal mode theory come to a single degree of freedom equation and for that single degree of freedom equation one can find out the maximum displacement of the structure if it were vibrating only in that particular mode. So, this is the main derivation that goes behind the development of the response vector method of analysis and once you obtain the value of p e i then the entire problem is converted to a static problem that is p e i is applied statically on to the structure and we can not only obtain the response or the displacement of the structure, but also can find out bending moment shear force or any other quantity of interest for a particular mode. And for this reason we call p e i as the equivalent static load for the ith mode of vibration. Since both response spectrum and mode shape properties are required in obtaining p e i it is also called modal response spectrum analysis and it is evident from the above discussion that both dynamic and static analysis are involved in the method of analysis using response spectrum. So, up to this part there is no approximation it can be straight away derived using the normal mode theory and one can find out an equivalent lateral load for each mode of vibration. Then one has to combine these modes or the modal responses obtained in each mode of vibration and these modal responses are the maximum responses that would have occurred if the structure are vibrating in that particular mode and then we have to combine these responses in order to find out the total response of the structure. Now in this part we make an approximation and there are different combination rules that is prescribed and we adopt one or the other combination rule and these combination rules are somewhat approximate. The reason for this will be explained shortly. The modal combination rules that we have are absum, SRSS and CQC. Absum stands for absolute sum of maximum values of responses say if x is the response quantity of interest say displacement or bending moment or whatever with the response quantity of interest then we sum up the absolute maximum value of the response quantity in each mode of vibration and the resulting quantity is equal to the desired value of the or desired maximum value of the response. Now this is an approximate one because of two reasons. Number one it assumes that modal peak responses occur at the same time that means if we have the plot of the z i that is the generalized displacement at the ith mode and i is equal to 1 to m that way one can have m number of such plots. If we look at these plots we will find that the maximum value of z that is the generalized coordinate in a particular mode this occurs say at a time t i and this t i is not same for all the modes that is for i is equal to 1 to m we will get m numbers of time at which the peak occurs and these times will not be the same. So, if the peak of the generalized responses in each mode of vibration do not occur at the same time then the addition of the absolute value of the maximum responses that clearly do not give the actual peak value of the response. Secondly it ignores the algebraic sign of the response that would be made clear with the help of figure 5.1 and say this is the figure said we are interested in a particular top story of displacement in a multi-story building and when we combine the effect of all the modes and find out the time history of displacement using normal mode theory then the maximum displacement that takes place say is at t is equal to 6.15. Now, if we consider the first generalized displacement then the first generalized displacement we find that the maximum value is occurring at t is equal to 6.1 and at that time the value is negative not positive. So, therefore, we can see that the if we sum up the absolute values then it is ignoring the sign. Similarly, second generalized displacement here we see that the time at which the maximum displacement that is occurring is equal to 2.5. So, there is a difference between the times at which the maximum response is at maximum generalized displacement in each mode of vibration that takes place, but also the signs could be different that is it could be sometimes plus or sometimes minus and the total response or displacement time history has a different kind of plot where the maximum or peak displacement occurs at a different time. Thus, the summation that we have considered in equation 5.11 is an approximate one. Now, this value of the first generalized displacement of the response that we get from absent generally is conservative because it is firstly ignoring the sign and then also it is assuming that the responses in each generalized coordinate occur at the same time. This rule is the SRS combination rule which denotes square root of sum of squares of modal responses. Now, the idea behind this is that the maximum responses that occur in each mode they are occurring at different times and these quantities that is the maximum response in each mode in each mode of vibration there is no relationship between them. And each one of them is assumed to be a independent random variable. Now, if you are wanting to combine n number of independent random variables then the best way to combine them is by taking the square of those values and then add them together and take a square root of that. So, that is what is known as SRSS combination rule and 5.1 to explain that that is the maximum displacement is equal to the summation of x i max square and the summation extends from i is equal to 1 to m and then we take a square root of this summation. Now, this particular formula is valid when frequencies are well separated that is they are not close enough because if they are close that means two frequencies in the two modes are very close to each other then it is expected that there is there will be a correlation between the responses as of the two modes and the if the frequencies are well separated for the two modes then we expect that the maximum values that we obtain in the two modes will not be that much correlated. So, therefore, this equation 5.12 is developed with an assumption that the maximum values in each mode of vibration are independent random variables and they are summed up with the help of SRSS rule. Now, the effect of the correlation between the two responses in two modes of vibration that can be taken care of by just modifying equation 5.12 and the rule that we use for that is called CQC rule or complete quadratic combination rule. The complete quadratic combination rule is given in equation 5.13 and we can see that it consists of two parts. First part is simply the SRSS rule that we have used before and the second part of the equation contains rho i j where rho i j is the correlation between the two modes of vibration and x i and x j are the maximum values of the responses in the ith and jth mode. Now, second mode or the second term is valid for i not equal to j when i is equal to j then it is perfectly correlated and the equation only then becomes only simply x i square. The second term in fact includes the degree of correlation that exist between the two modes and since the second term has the value rho i j and the second term has the value rho into x i into x j then this particular quantity could be plus or minus that is depending upon the value of x i and x j. So, the SRSS rule which ignores the second term can provide a higher value compared to the CQC rule or in other words the CQC rule quite often estimate less value than that provided by SRSS rule. Now, the various expressions for the CQC correlation between the two modes are available in the literature. We have taken here two such correlation coefficient expressions, one is given by Rosenbluth and LOD that is given in equation 5.14 where xi is the damping ratio. And beta i j is the ratio between the natural frequency in the ith mode and the jth mode. So you can see here these the two equations are slightly different in their form and using these equations one can obtain the value of rho i j that is required for finding the response using CQC rule. And to find out this rho i j what we should know is the ratio between the frequencies of the two modes which are known to us and the damping ratio xi which is specified. So, the CQC rule can be easily implemented in place of SRSS rule and that is done for cases where we find the two frequencies are close very close to each other in a particular structure. And one can compare as such the values of the responses obtained using SRSS rule and CQC rule and can compare them for any given problem. The SRSS and CQC rules for combining peak modal responses are best derived by assuming earthquake as a stochastic process and there are exclusive papers for that and using those principles one can show that the or one can justify the combination rule that we use for class or that we classically used for combining the modal responses in the response spectrum method of analysis. Now as both response spectrum and the power spectral density function represent frequency contents of ground motion a relationship exists between the two. So, that is a thing that one should take note of because the response spectrum of any earthquake is nothing but the maximum value of the response spectrum of the responses or the displacement response plotted against time period or it could be plotted against the frequency and from that one can obtain the response spectrum for acceleration simply by multiplying the response spectrum for displacement with omega square. So, this gives basically a distribution of the energy of the earthquake with frequency or indicates a distribution of the energy of earthquake and that is what is the major frequency component that exists in the earthquake. Similarly, power spectral density function since it is a plot of the variance or the mean square value of the process with frequency then this also provides an idea about the major frequency of the earthquake. The content of the earthquake process so therefore, there is a relationship or there is expected to have a relationship between the PSDF and the response spectrum of an earthquake and this has been established by many researchers and the relationship between the two for the smooth response spectrum and smooth power spectral density function have been given and one such relationship is shown over here and here in this we can see that the power spectral density function is related to the displacement response spectrum D and P 0 is called the peak factor and tau is the duration of the earthquake. So, if the duration of the earthquake is known and the frequency they are different frequencies omega one can obtain the relationship between the displacement response spectrum for a given value of xi with the power spectral density function of the acceleration on which is x s x double dot g. Now this derivation is obtained using what is known as the or is derived using the concept that the earthquake is modeled as a stationary random process and using that one can find out this particular expression. Apart from that we will require the cross post spectral density function between the two processes in an earthquake and these relationships would be required when we will extend the response spectrum method of analysis for multi support excitation.