 everyone. We have seen so far in this course how to obtain response of a system subject to either well-defined analytical forces like harmonic forces and we also discussed if there is an arbitrary excitation how to get the solution and in the previous lecture we saw that if the response or if the loading cannot be expressed as a closed form representation like pt equal to some function then how to employ numerical methods to obtain the solution. Now once we are equipped with this tool we are in a position to analyze a system or a single degree of freedom system subject to earthquake excitation and that would we would present a just a we will discuss in briefly there is a whole field of earthquake engineering which we may which you may do in our advanced course but in this basically course of structural dynamics I'll just giving you introduction the concept of how to obtain response of single degree of freedom system subject to arbitrary or random excitation like earthquake excitation and also discuss the concept of peak response and basically response spectra which is very famously used for analysis and design of the structure subject to earthquake loading. Okay so let us get started earthquake response of single degree of freedom system and as we have previously done we would be focusing on linearly elastic system. Okay so let me write down the equation of motion that we have been using till now for linear elastic single degree of freedom system. Okay now finding response to seismic excitation or earthquake excitation is one of the major field where you can see the application of structural dynamics. Okay and basically the problem becomes in that case as we have previously discussed so if I consider this frame representation okay of a single degree of freedom system so let us say initially the system was there and then there was a ground movement let us say this is u g of t and then the relative deformation of structure which let us say okay so this is the relative deformation u so that my total deformation is u t of t which is sum of u g t and then u t okay and this is how it looks like. Alright so basically the problem statement for these cases becomes okay in this case p t is effectively represented using mass times u g of t okay so I can substitute it here alright so remember that this basically this expression basically comes from assuming that the total acceleration is nothing but relative acceleration of the mass plus the ground acceleration okay and then we substitute in here okay the equation of motion is actually written terms of total acceleration however with relative velocity and displacement equal to 0 considering that there is no external force okay however when I substitute this expression here okay then I can rearrange this and I can get it okay get this expression right here okay so again you can write this one in terms of function of t as well okay and this effectively becomes mass times u g t and this basically is the effective force that we say is acting on this mass here okay in this case alright so let us do one more thing let us divide this whole equation here by m so that I can get my expression as c by m okay plus k by m u of t and that is equal to minus u g t which is the ground acceleration okay this can be further written as okay this expression c by m is nothing but 2 zeta omega n times u dot of t and k by m can be written as omega n square u of t okay this is my expression here okay now this ground motions are basically measured using seismograph okay and we have discussed this the working principle of a seismograph in a previous chapter the seismograph is nothing but in its very elementary form it is a spring mass tamper system okay so it is a single degree of freedom system okay the spring mass tamper elements okay and the way this seismograph actually works is that we utilize frequencies for this seismograph which are quite high okay such that okay omega by omega n where omega is the excitation frequency and omega n is the natural frequency of the seismograph alright this should be equal to 0.5 okay and we introduce very high damping okay 60 to 70 percent in the system if that happens then my seismograph can be calibrated to measure various type of ground motion with different frequencies okay so this seismograph actually measure the ground acceleration which is also many times you would see being referred as strong motion okay we are only concerned about the strong motion because unless the ground acceleration is above a certain threshold it is not going to do any damage to a structure so if you consider magnitude of earthquake let us say less than 4 typically it does not lead to any kind of damage in the structure okay so in terms of strong motion what we do the seismographs are actually located at various distances from the source of an earthquake okay and this seismographs are lying there for a long period of time 20 years 30 years like that okay and you can imagine these earthquakes need not to be very frequent okay it might come once in a lifetime let us say once in a 50 year or 100 year okay and the idea is that you cannot continuously be measuring the acceleration due to very small earthquakes like less than 4 okay so these are these accelerographs are actually triggered when the acceleration is above a certain threshold or when it is calibrated to be triggered with some other mechanism so what do we do these accelerographs would give you a strong motion and then it would also report you the strong motion duration of an earthquake ground shaking again it is based on the same principle that you are only bothered about the ground motions okay which are going to have any realistic effect on the structure so strong duration there are different way to characterize the duration of a ground strong of a strong motion okay there is a area sent in city there is like you know threshold at the max with respect to maximum PG certain kind of thing but basically just remember that these do not continuously measure but it measures above a certain threshold okay and each of these okay each of these accelerograph which are located at different location they can measure depending upon what kind of accelerograph okay in one direction two direction or three direction okay so okay two horizontal direction and what one vertical direction okay so typically any earthquake would have three components although some of the old accelerograph would give you used to give only two components okay so usually any earthquake would have three components of course there are additional research which are now characterizes six components of including rotational okay but we are not bothered about that okay this is beyond the scope of this course so remember that accelerograph would provide you basically the acceleration history which would be in terms of okay either time acceleration okay so time and then acceleration which might be expressed let us say in terms of units g okay so something like let us say like that if it is an interval of let us is 0.02 second uniform interval okay it would be something like this okay okay so this is just the fictitious value so it would give you basically discretized value of the acceleration at different time instance okay now in the previous chapter we have seen that given any arbitrary excitation how to find out the response of a system okay to any arbitrary excitation PT and what do we do basically we use numerical methods okay so depending upon the whether the system is linear or non-linear okay we basically discuss three methods one was interpolation excitation interpolation method okay other one was central difference method and we also discussed new marks methods okay so any of these methods can be utilized to get the response of the single degree of freedom system subject to this ground excitation okay now depending upon okay depending upon what is the structure there could be different response quantities of of interest alright so for example it could be of absolute values of the response quantity like u dot of t the total acceleration in the system okay or it could perhaps be total velocity let us say or total displacement okay or it could be relative acceleration relative velocity okay not here okay relative velocity and relative displacement now usually what happens for a structural engineer okay the total velocity is not of concern to us okay what is concerned to us is basically the total acceleration okay and I will talk about where do we actually utilize that okay but also total displacement is also concerned to us okay and of course the relative displacement which is basically the most important parameter okay if you want to analyze a structure okay to find out the internal forces in the system okay now the idea is how do we apply the knowledge of structural dynamics to earthquake engineering okay so let us say somehow I am able to find out these response quantities okay once these response quantities are found out okay then we can utilize this to get the forces and moments in a structural member because in the end remember as a structural engineer your goal is to find out basically forces on a beam or a column okay so depending upon let us say this is axial load this is shear force and this is M because once you have these forces in the members you can perhaps go ahead and design the structural member for those forces okay so utilizing the knowledge of a structural dynamics we are going to find out the values of these parameters okay and then from that we are going to find out the internal forces now if you remember the internal force in any structure okay it depends only on relative deformation okay it only depends on relative deformation it does not depend on velocity or it does not depend on the acceleration okay it only depends on the relative deformation so if we somehow able to find out this parameter UT using some sort of numerical methods or any other method then I would be able to find out the forces in the member and one of the simplest example would be let us say I have a cantilever column here okay and under the action of let us say UGT okay it has certain deformation of course it will be some value like this but let us say relative deformation is U of t now you know that if the deformation of U of t the force let us say V here would be nothing but okay I can write it as 3EI by L cube times U of t okay so if you know the displacement okay knowing the structural property and geometrical material properties of the structural member you can find out or relate it to the forces in the member and then you can design the member to sustain those forces okay so that is what we are going to learn in this chapter okay so let us see how do we do that now I have already written down an alternative form of the equation of motion okay so we said that our equation of motion can be written as okay like this plus omega n square U t equal to negative of ground acceleration. Now if you look at carefully at this equation can I say my displacement U depends only on two factors here which are those factors basically zeta and omega n because if there are no I mean then if you look at this equation there except these two parameters there are no other parameters okay that are involved in this equation so basically U or the displacement depends only on omega n or you can also call this like t n basically the time period and the damping in the system only these two parameters alright so I can write down my U t as a function of of course the time t okay and then the time period of the system and zeta okay which is the damping in the system alright and similarly the acceleration okay which is related to U and velocity also depends on only these three parameters so given any ground motion okay given any ground motion U g of t alright my response U t only depends on the time period and damping in the system okay and for this case remember it does not matter whether the system is heavy or lighter or whether it is flexible or rigid okay two different system with different masses and stiffness would have similar displacement if the ratio are such that it gives you the same value of the time period t n okay so yet you have to keep in mind okay so depending upon these two parameter t n okay the response to a ground motion might differ okay so basically if you consider let me consider first two cases like in a two series of cases here okay so in the first case I am going to consider systems in which the time period is actually varying but the damping is same okay and in the second case I am going to consider systems in which time period is same but damping is actually varying alright so let us see what happens so for any and I am just drawing like in an approximate representation so for any ground motion U g of t alright so for any ground motion U g of t okay let us say the first system is t n is equal to 0.5 second zeta is equal to 2 percent second one is t n is equal to 1 second zeta is equal to again 2 percent okay and third one is t n is equal to 2 second and zeta is equal to 2 percent okay now in the second case let us say t n is same at 2 second however zeta is first is equal to 0 in the second case again t n is same but zeta is 2 percent okay and in the third case let us say t n is again I mean it is same but let us say zeta is equal to 5 percent okay so what happens okay typically if you given a ground motion okay this would look like something like this okay alright as you increase the time period okay typically it would look like something like this now if you further increase the time period to look like something like this okay and this is the direct consequence of increasing the time period which is evident from the response if you consider the response here okay first the amplitude is smaller and the second basically the time period of the response is smaller okay but as you go to 1 second and 2 second you can see that the time period of the response is also increasing as well as the amplitude is also increasing okay now let us go to the second case in which it is somehow somewhat similar to this case okay so let me just draw it like this okay as I increase the damping so this was with the 0 damping as I increase the damping what you will see okay while the this remains same but the amplitude actually decreases okay and with the further increase in damping this further decreases okay alright so these are the typical characteristic okay and you know you can go ahead and perhaps obtain the response of a any single degree of freedom system subject to a selected ground motion with different time period and damping and these type of characteristics would be obtained alright so two things are to be noted here first that the response of a ground motion or response of a linear elastic single degree of freedom system can be completely characterized using the time period of that structure at the damping of the structure okay so this you have to keep in mind now let us come to the response quantities that are of interest to us okay so we want to look at response quantities that are of interest to us and we said that it is the relative displacement okay or it could be total displacement and the total acceleration now relative displacement as I have previously mentioned leads to the internal stresses or basically forces and internal stresses in the structural member in a structural member of a structure okay then the total displacement might be important for cases where you have buildings that are adjacent to each other okay so that you need to find out what is the separation between build these buildings that need to be provided so there is no pounding damage because what happens when you have structures of different time period shaking with respect to us shaking due to the ground excitation okay at this point due to the displacement of the structure at the top of it this structure might leads to might go ahead and like you know basically impact this structure at the point of contact it might lead to the damage to the structure so here the relative displacement is not the important factor but for pounding you need to find out what is the effect of the ground displacement plus the relative displacement because that is what would be the response the responsible displacement here that might lead to okay making decision regarding the separation of this building okay similarly for this one it would be ug t plus u of t okay so let us know this is u 1 of t and u 2 of t so you need to find out the total basically displacement here in this case if you want to assess the pounding damage so relative displacement is important in cases where you want to find out internal forces and stresses total displacement becomes important if you have to consider the pounding damage okay and total acceleration becomes important if you have some sensitive equipment okay sensitive equipment inside a structure okay so whether if there is sensitive whether it is kept on floor or whether it is connected to the ceiling these equipments are actually sensitive to the acceleration or total acceleration so there it becomes important okay so and of course like you know apart from that the forces in the members okay which of course is related to the relative deformation so basically these are the typical response quantities that are of interest to us okay so the question becomes okay the question becomes I have this equation of motion let us say okay how do I go from this equation of motion to the forces finding out the internal forces in the structure okay and what we are going to do here is basically introduce the concept of equivalent static force okay let us see what is an equivalent static force now as I have told you there is internal forces or stresses in the structure depends only on the displacement okay not on the acceleration so let us say I am able to solve this equation and find out the value of u of t okay with any method any numerical method let us say this is my structure and I am able to find out the value of u of t okay once that is found out the internal forces in the structure can be determined by considering the response of this letters a single degree of freedom system subject to an equivalent static force which is equal to the lateral stiffness k of the systems times u of t okay if you apply this load okay and find out subject to this load of course it would be an static analysis that is why it is called equivalent static analysis that you first do the dynamic analysis and then you find out at each whatever time and time instant of interest to us you find out what is the displacement okay or at each every at every time step you find out the u t value okay and then the response to a system can be found out considering the force under a action of a statically applied force which is equal to k times u t okay and you apply that what you would find out the response in the structure is similar if you had considered the dynamic analysis and found out the same thing okay and this can be observed from the equation of motion like this if you consider the terms of the total value total value of the absolute value of the acceleration okay remember we had this equation here plus k u of t okay so somebody might ask a question why is this external force is k times u t and not m times u t double dot because in the end the external force is basically the total acceleration that the experience that the structure is experiencing right mass times the total acceleration so why not this force now if you consider this equation of motion here okay that question can be rephrased if I write it like this okay m u t is equal to minus c u t and minus k u of t okay so the question becomes why I am neglecting this because if I am considering k u t it is effectively that I am considering the mass times total acceleration if I neglect this damping term here okay now the reason that I am considering only k u t as an equivalent static force is because any type of damping mechanism let us say it is due to viscous damping mechanism if I am assuming the nature of the energy dissipation to be viscous damping okay so if there is some energy dissipation at the joints or any other mechanism those kind of forces actually do not lead to any kind of internal stresses in the member mostly you know but I mean there are like you know if you go into non-linear analysis of course non-linear systems are equated as equivalent linear system and in that case the damping becomes important however for this case for equivalent static analysis we can say that the response u t okay if I apply this force k u of t okay how much would be the response in the system let us find that out okay I have done the dynamic analysis okay using this equation these equations so basically I have solved this and found out the u t in the system now let us say if I apply the f s t equal to k u of t how much is the displacement in the system okay and this is a static analysis so for this case can I say that if I apply this force the displacement in the system this displacement here would be the lateral stiffness of the system total force divided by the lateral stiffness of the system which is u of t okay so whether I do the dynamic analysis and find out u t in the system or I apply a static equivalent static force of k u t and find out the response in the system the response in terms of deformation would still be u of t okay so that is why we consider a equivalent static force okay and this is very central this concept of equivalent static forces central to earthquake resistant design okay and it is basically adopted in many of the design codes earthquake design codes around the world okay all right so I hope that concept is clear to you okay so let us see once I have the k u of t what would be the base shear in this structure okay so base shear if I consider due to this applied force k u of t and now remember that once I apply once I find out u t okay I apply the equivalent static force k u of t then I am again now I am doing basically static analysis so I am interested in finding out what is the base shear and basically what is the base moment because once these quantities are known then I can distribute the total base shear okay in the structure or the moment and the structure can be designed for that okay so base shear in this case if you consider the free body diagram subject to this equivalent static force can I say my base shear would be nothing but whatever the equivalent static forces and that is equal to k u of t and my moment would be nothing but whatever the FSTS times the height of the structure okay which I can also write as base shear times the height of the structure so let me write it again okay let me write this this equation for the base shear and moment again okay and then see what do we get okay so basically saying subject to equivalent static force of k u of t okay I am finding out my base shear as okay my base shear as same as FST okay which is equal to k u of t now one more thing can be done here if you look at this expression here k can be written as m omega n square okay remember k is nothing but m omega n square so I can substitute it here and write it as m omega n square u of t okay now if you consider this expression here m omega n square u of t this quantity here has units of acceleration and basically we write this expression as m a times t where at is referred to as pseudo acceleration okay so it is referred to as pseudo acceleration alright now this acceleration is actually not the actual acceleration of the system it has units of the acceleration but remember the actual acceleration of the system is u t of t and this a t is not equal to u t of t okay and that is evident from this equation here the equation that we have been considering okay so if I write down in terms of total acceleration remember my equation of motion becomes this plus omega n square u of t okay and my total acceleration is actually minus omega n square u of t which is actually the a of t here okay minus 2 zeta omega n u of t okay so I have this term here which differentiates the total acceleration from the pseudo acceleration here okay and if a system has zero damping or when the velocity is zero so when this is zero or when the velocity is zero then of course my total acceleration okay or the absolute acceleration is the pseudo acceleration but in general that is not the case so you have to keep in mind that distinction okay alright if that is the case then I can say I can either apply k times u t as equivalent static force or I can also apply mass times acceleration the pseudo acceleration where a t is basically omega n square u of t okay now if that is known okay then my base shear can be again written as v v of t is equal to equivalent static force this can be written as either k u of t or m a of t okay and my base shear is f s of t times h or I can also write as v v of t times h okay now remember what I have done here okay for this single degree of freedom system okay I found out at any time instant what is the u t then I applied the equivalent force f s t equal to k of u t or m of a t and subject to this force I found out what are the what are the internal forces for example in this case the total base shear and the total base moment okay so from the dynamic analysis okay for this seismic excitation I found out what is the base shear and moment and once that is known I can go ahead and do the seismic resistant design or earthquake resistance design okay earthquake resistant design of these structural members all right okay once that is known okay it becomes typically for a designer it becomes very difficult and remember here we are only considering single degree of freedom system but in reality any structure is multiple degree of freedom system okay and you know I mean it is computationally expensive to do response or to like you numerically analyze the system and find out the response at each and every time step okay so what do we do and also you know that if a structure is experiencing several magnitudes of forces typically we design a structural member for peak forces okay so if somehow I can find out the peak values of the response quantities okay I don't have to go ahead and do the numerical analysis every time okay and I have told you previously that if a ground motion is there okay then the response of a system to this ground motion depends only on these two parameters T and n zeta okay so what we are going to do now we are going to introduce concept of response spectrum and then see how this concept of response spectrum can be utilized okay to come up with a method to efficiently find out the response of a of any structure okay so response spectrum is nothing but peak response and this definition is basically we are defining with respect to seismic excitation so peak response of a single degree of freedom system okay alright so this is the peak response of single degree of freedom system and of course we are considering linear elastic here okay because there are also non-linear response spectrum but we are only considering linear response spectrum okay so this is peak response of single degree of freedom system I should not say a okay all let us say peak response of all single degree of freedom system for a particular ground motion okay for a particular ground motion and let us see how is that useful so let us say that peak response could be anything it could be displacement it could be velocity okay or it could be let us say acceleration okay now let us say if I have okay a plot of peak response let us say this is u naught okay with some arbitrary function I do not know what that is I do not know the nature of it let us say it is any arbitrary okay function and let us say this is Tn here okay all single degree of freedom system means that systems with different values of Tn here okay so basically what do we do okay if you go back to the graph that we had considered here okay this one just go ahead yeah so remember depending upon the value of Tn it is 0.5 or 1 second or 2 second I basically got different value of the peak response isn't it okay and similarly different value of damping also give me different value of peak response so now let us consider a single value of damping and different value of time period I can perhaps plot the peak responses on this system here for all possible values of Tn okay and this is specific for a ground motion so each ground motion will have a response spectrum okay so each ground motion would have a response spectrum for particular value of zeta okay let us say it is 5 percent now if that is derived and I am designing a certain system for which let us say Tn is let us say 0.75 second okay then I can come here okay I can come here and I can find out subject to 0.75 second what is the response of this system so what is the value of u0 okay so I would be basically finding out what is the value of u0 or the peak displacement once the peak displacement is known I can again basically adopt the same procedure where I am going to apply the peak value of the equivalent static force which is k times u0 or m times let us say peak value of pseudo force which is m times a okay and subject to this force I can find out what is the base shear and what is the base moment okay so if the peak values are known that is what our goal is okay for design of any structural number so this is how we connect the idea of structural dynamics using response spectrum to design a system for any seismic excitation using the peak responses okay so we are considering remember we are considering peak responses so I can write down okay that my peak response for any system okay so peak response in time u0 Tn and zeta would be basically I would find out what is the maximum response of this quantity here is okay so this is for a particular Tn similarly I am going to find out the velocity okay and again the maximum over Tn and zeta and then total acceleration here okay again this is this okay so this would be Tn and zeta all right so using this we can vary the value of Tn okay and we can find out different response spectrum for example this would give me okay this would give me if I do for displacement a deformation response spectrum okay deformation response spectrum okay this would give me velocity response spectrum okay and the third one would give me acceleration response spectrum all right so these three response spectrum are of interest to us okay and given these three response spectrum I can go ahead and I can find out the response now let us see okay for any ground motion let us say if I have a ground motion which has a very random response something like this okay now depending upon the value of okay value of the time period of the structure as I have shown you before okay so let us say this is time period 0.5 second okay let us say this is time period okay one second so that it is little bit spaced apart then the above the amplitude is also more and then I have the third one okay which is Tn equal to one second now each of these would give me certain peak responses okay let us say this is u10 let us say this is u20 and let us say this is u30 okay so I can if I plot this the displacement response spectrum what will happen okay if this is my displacement u0 and this is time period these would correspond to Tn this is 0.5 second here and this is u10 value then this would correspond to one second okay and let us say this is one here not actually this is one second and this is two seconds here okay and this is actually two second and each of these correspond to peak responses that we have obtained okay u20 and u30 so this is how the displacement response look like okay and similarly we can go ahead and we can find out the basically the velocity response spectrum and the acceleration response spectrum however however if you remember when I wrote down the equivalent static force okay is actually k times the relative velocity which is equal to mass times sudo acceleration here this is not the total acceleration which I have written here okay so it would be useful for me okay if I can get the response spectrum for sudo acceleration okay so if I can somehow get the response spectrum for at and remember at is related to ut by how omega n square times ut so if I can get the value of u0 let us say now I am start to represent this as d okay the peak value as d okay I can find out the peak value of the sudo acceleration as multiplying this with omega n square times t which I can also write it as 2 pi by Tn times t so if my displacement response is obtained okay then I can go ahead and I can obtain the acceleration response similarly I can also define a new parameter called sudo velocity okay I can define sudo velocity as v equal to omega times ut okay remember there is only single power omega n times ut now this is vt here now if you look at it this quantity here has units of velocity however my vt is not actually the relative velocity in the system okay so do not again get confused so that is why again I refer to this as sudo velocity and not the absolute or the relative velocity okay so I can also define sudo velocity okay so I can get the sudo velocity spectrum as well so instead of finding out velocity response spectrum and acceleration response spectrum I am more interested in finding out the sudo acceleration response spectrum okay because my equivalent static force is actually a function of sudo acceleration not the absolute acceleration and if the equivalent static force is a function of sudo acceleration then of course my base shear and moment would also be a function of that and you know because of this relationship okay I can write down the relationship between basically a by omega n is equal to v times omega n d okay and this describes the relationship between relative displacement remember displacement is relative okay it is not sudo it is the actual relative displacement in the system and velocity is sudo velocity and acceleration is sudo acceleration these are related by these expressions here okay now you as I previously told you okay that for any system the sudo acceleration is actually not equal to the total acceleration in the system okay so the peak value is also not equal to the peak value of the acceleration okay in the system okay however if zeta is very small okay let us say 2 percent 1 percent or 5 percent okay these values are quite close actually okay and we can go ahead and plot this okay for different value of damping okay at different for different value of tn as well okay for all realistic purposes these are quite close okay all right so now I know that if I know one of either a v or d I can find out the other two okay because these are okay related by this expression here okay all right now knowing that okay that I have acceleration response vector which typically something looks like this so let us say this is d here and this is tn okay I can also plot the velocity a sudo velocity response vector which would okay look like something like this and then acceleration sudo acceleration spectrum which looks like this so this is v and this is a the horizontal axis is tn okay once one of these are obtained then it becomes very easy for me to find out the peak response in the system isn't it so look at if I look at this here okay remember the peak response is basically would be obtained subject to the equivalent static force okay which is either k times u0 or let us write this k times d or I can also write this as mass times sudo acceleration so either k times d or mass times sudo acceleration subject to that I can find out the peak value of the base shear which is fso times the height of this structure okay and mass as whatever the base shear times the height of the structure okay there is no height here this is this is simply base shear is the applied equivalent static force okay and we utilize this knowledge now if you look at the expression for the base shear here okay basically we get as mass times the peak acceleration and if I write down mass as weight divided by g okay this can also be write down as a divided by g now this quantity here which is the peak sudo acceleration divided by g okay times the weight of the structure okay this is also many times referred as seismic coefficient okay seismic coefficient okay so the initial seismic resistant design of the structure when the practice of earthquake engineering was not yet matured had not matured at that time what people simply used to do remember initially all the structure were used to design for vertical loads okay and earthquake is actually a lateral load okay so everybody knew that you have to design the structure for lateral loads to sustain earthquakes but nobody knew how much so basically initially they started the design as such that they are going to take certain percentage okay certain percentage of the seismic weight okay let us say 10% and apply it laterally and design the structure for that and even to this state that method is reasonably accurate even if you design a structure for let us say 10% or 20% even if you do sophisticated seismic analysis your basically results come out to be somewhat closer okay all right now let us get into finding out the peak responses okay back to peak responses and we have already discussed that if somehow I can get my U naught or D or let us say pseudo acceleration okay or let us say even V then I can go ahead and do the equivalent static analysis okay so let us say this is FSO is equal to k times U naught or k times D also equal to mass times acceleration the total base shear would be as I previously indicated FSO naught equal to k U naught times acceleration okay and basically the moment the base moment would be FSO times H okay and we would be following this procedure and depending upon you know what response is a spectrum is available to you you can directly get D or even if A is given let us say the pseudo acceleration is given or V is given you can find out the other parameter okay all right okay now let us get on to now we know how to find out the peak responses there is one challenge here okay challenges okay I have shown you that an acceleration response pseudo acceleration response spectrum looks like this okay so there are lot of non-uniformity if you look at a lot of zagged lines okay this is still like in a very smooth but actual response spectrum would look like something like this okay now if you measure okay if you measure the acceleration at the same side after certain number of years okay the same thing would now look like this okay and again after certain year it could again change okay so the question becomes if I have to design the structure okay so basically to design the structure I need to find out these peak forces and peak moments okay which one to choose okay remember I have told you this is at the same site only that it is measured at different instances of time okay and the response spectra would be quite different okay in terms of the peaks that you see here okay so the question becomes which response spectra should I use okay so actually what happens in these cases okay that just for if you select any response spectra just for a small change in the time period your forces would increase substantially okay and that is not realistic okay from the design perspective so what do we basically do we define or we develop something called design spectra from a set or let us say an ensemble of okay ensemble of several response spectra and how do we do that remember if you take the average of these response spectra here how would they look like if you look at this the average would look like okay little bit smoother compared to individual response spectra okay so this let us say this is the mean value of different response spectra at the same site okay now typically what do we design there are different like you know very sophisticated design procedures okay where you do seismic hazard analysis you know do statistical analysis to come up with but in very simple or layman term if we consider okay let us say mean plus standard deviation it would look like something like this okay so that it would consider okay most of these peaks were to be captured under mean plus standard deviation okay and then approximate that with set of straight lines straight lines or curves okay so that my response from a design spectra in reality looks like a much smoother curve which is the idealization of the or which is the idealization derived from the statistical analysis of several response spectra so this is my design spectra and this is what we use in design of structure not a single ground motion like this okay so design spectra basically obtained doing statistical analysis of several ground motions a several response spectra okay and then come up with some conservative estimate of the acceleration at different time period and then approximate that using okay curved lines or smoother straight lines okay so these lines basically would represent the response spectra at the site okay and this is what is utilized in the design of the structure okay all right so these are basically the concepts okay that are required to understand once you know what is the value okay so from here let us say this is the pseudo acceleration response spectra okay from here for a given structure of a given value of tn you can get what is the value of the peak acceleration once you know the peak acceleration from here okay you can go ahead and you can find out the equivalent static force that you need to apply on this structure okay and subject to that okay or you can also write it as kd where d is basically you know a by omega n square okay you can find out the peak base shear and peak moment okay and this is the typically the procedure that we use so this is a very simplified presentation of the concept of earthquake resistance analysis and design using response spectra of course there are like you know a much more detailed discussion of this in that would be presented in an earthquake engineering course but for this case this is how we relate or apply a concept of structural dynamics to earthquake resistance analysis and design of the structure all right okay with this I would like to conclude this lecture thank you