 In the previous lecture, we started discussing on the seismic inputs that is the inputs which are necessary for analyzing the structure for earthquakes. Now, there are various forms of seismic inputs and the one which is to be used depends upon the problem at hand. Out of the different inputs, the time history records are the most direct and the easy seismic inputs. There are 3 components of ground motions which can be given as an input to a structure. Apart from that, there is a torsional component of ground motion and a rotational component of ground motion which can be obtained from the ground motions measured in the 3 directions. The frequency contents of the ground motion is an extremely important thing in the earthquake literature. Looking at the frequency contents of the ground motion, one can understand the characteristics of the earthquake and can take a decision about what kind of structure with its dynamic properties should be designed. The frequency contents of ground motion is also very important for frequency domain analysis of structures for earthquake. The frequency contents are obtained using the Fourier synthesis technique that we described in the previous lecture. Using the FMT algorithm, these days the frequency contents can be very easily obtained and a Fourier spectrum can be constructed for given ground motion. When we looked at the power spectral density function of ground motion, the description of the power spectral density function of ground motion is required for the random vibration analysis of structures for earthquake forces. In fact, in that case, the ground motions are modeled as a stochastic process and the power spectral density function forms one of the important inputs for analyzing the structure or considering the earthquake as a random process. And we continued with this and seen the definition of power spectral density function assuming the ground motion to be a stationary ergodic process. From the power spectral density function of the ground motion, one can obtain the peak ground acceleration using the relationship that is shown over here, that is in the equation 2.19c and this requires the knowledge of the duration of the ground motion, the capital omega that is this symbol and lambda 0. Lambda 0 is the mean square value of the ground motion or in other words, it is the area under the power spectral density function curve and the capital omega is equal to the square root of the second moment of the power spectral density function and divided by the 0th moment or the mean square value of the ground motion and taking a square root of that. Now, once we have the peak ground acceleration that is the knowledge of the peak ground acceleration, then this can be utilized for analyzing the structure for different levels of the peak ground acceleration. The predominant frequency or the period is where the PSDF or Fourier spectrum peaks. Another input is needed for probabilistic dynamic analysis of spatially long structures that have multi support excitations if exact results. So, in order to take into account the effect of this phase lag, the cross power spectral density function or cross correlation function is inputted along with the power spectral density function of the ground motion. Now, this time lag or lack of correlation between excitation at different supports is represented generally by a coherence function and a cross power spectral density function. The cross power spectral density function between two excitation which is needed for the analysis of such structures is given by these two equations. That is if X1 and X2 are the two points, then we say that there exists a cross power spectral density function between the ground excitation at these two points and it is represented by SX1, SX2 and that can be written as SX1 to the power half that is the power spectral density function at point X1, SX2 to the power half that is the power spectral density function to the power half at point 2 and they are multiplied by a coherence function which is a function of X1, X2 and omega. In fact, this coherence function is the correlation represent the correlation between the two ground excitations at points X1 and X2. For the case of a ground motion which is moving in a particular direction and if we the points are aligned in that, then the power spectral density function at X1 and power spectral density function at X2 they are the same or in other words in the sense of mean square value these two ground excitations are the same. And therefore, this product turns out to be the same as SX that is the unique value of the power spectral density function of the ground motion and it gets multiplied by this coherence function. Coherence function generally is a exponentially decaying function with the distance r and the value omega the greater the distance between the two points we expect that there will be less correlation between the two ground motion and therefore, a function is chosen which is exponentially decaying. So, this kind of function is multiplied with SX to get a cross power spectral density function which represent the correlation between the two ground motion in frequency domain. For a very small value of the distance between X1 and X2 and for small value of frequency we can see that the coherence function approaches towards unity that is we get a case of perfect correlation in that case the cross power spectral density function between the two points become same as the power spectral density function of the ground motion. Now, more discussion on the cross power spectral density function will be taken up in chapter 4 which we will discuss later. Then another interesting thing was observed for the ground motion although we have assumed that the ground motion is a stationary random process in which the mean square value along the ensemble does not change with time or is invariant with time, but in reality it is not so. It is found that the mean square value does change with time and therefore, we have to account for this particular pattern of the ground motion in modeling the ground motion as a random process. So, this is usually done by considering the earthquake to be a evolutionary stationary process meaning that the mean square value of the process changes with time in a particular fashion and the way it changes is a function called the modulating function modulating function which modulates the stationary process with time that is the mean square value changing with time. One such modulating function is shown in this figure later on we will see at the end of this lecture series on inputs that we have several empirical relationship or empirical equations representing different types of modulating function that are used in modeling the earthquake as a evolutionary stationary process. In the case where we consider the earthquake as a evolutionary stationary process the equation 2.2 to represents the evolutionary power spectral density function of the ground motion. Here the parameter t comes into picture because the power spectral density function changes with time and we obtain it by simply multiplying the modulating function square with the power spectral density function which we obtain assuming the earthquake to be a stationary random process. So, in order to describe the evolutionary power spectral density function of ground motion we must know the modulating function as well as the power spectral density function of the ground motion. So, these 2 inputs are provided for analyzing the structure. Now, from the collection of records various predictive relationships for cross power spectral density function Fourier spectrum modulating functions have been derived and they are given later or rather at the end of this lecture series on seismic input. Now, let us come to the next seismic input which is the most favored seismic input for earthquake engineers and this input has been codified and people use for the seismic design of structures for the design response spectrum and the basis of the response spectrum and the design response spectrum are the subject of discussion here. Response spectrum of earthquake is most favored seismic input for earthquake engineers there are a number of response spectra used to define ground motion displacement, pseudo velocity, absolute acceleration and energy. So, one can obtain the response spectrum for each one of these parameter the spectra show the frequency contents of ground motion, but not directly as Fourier spectrum does. Displacement spectrum out of them forms the basis for deriving other spectrum. The displacement response spectrum is defined as the plot of maximum displacement of an SDF system to a particular earthquake as a function of the frequency and damping ratio. I assume that all of you have a preliminary knowledge about structural dynamics there you have solved a single degree freedom system having a natural Fourier and a damping ratio to support excitation and for that support excitation you can obtain the relative displacement of the ground motion and that can be obtained by various method. For example, one of the method that is widely used specially in earthquake engineering to obtain the response of the single degree freedom system to a support excitation is using the Duhamel integral. This is the Duhamel integral which is shown in equation 2.23 where we obtain the relative displacement of ground motion using the recorded ground acceleration and using this equation. Now we can see that this equation contains a term which is some sin omega d t minus tau and other term is a exponentially decaying function which gets multiplied with this. The basis of the Duhamel integral I just want to mention so that you can recall it that if we give an impulse of x double dot g tau into d tau then the response at a time t will be equal to the free vibration that takes place due to the impulse. The impulse in fact provides a initial condition to the single degree of freedom system that initial condition is that the system is at zero displacement at that point if we assume but there is some velocity which is imparted at that particular point of time because of the impulse. Then for the elapsed time that is t minus tau elapsed time the single degree of freedom system vibrates as a free vibration problem or vibrates as a damped free system. Now using that concept we are able to obtain the x t using this integration. The maximum value of the displacement we can call it as S d is written as S v a parameter divided by omega n it results from this equation where S v is nothing but this equation that is this integration and the maximum value of that. That means we perform this integration at every time t and take a maximum value of that that constitute the value of S v and that S v divided by omega n and from that we get the value of the maximum displacement. Now this maximum displacement for different values of omega can be plotted for a given damping ratio. At the maximum value of displacement all of us know that the kinetic energy would be equal to zero because the maximum displacement condition is equivalent to the zero velocity condition and therefore the entire energy is the static energy that is half into k into S d square S d is the maximum displacement. So the other part of the energy that is the kinetic energy that turns out to be zero at the position of the maximum displacement. Now if we say that we want to express this energy as an equivalent velocity of the system then the equivalent velocity for the system can be obtained by equating this energy with half m v square where m v is the equivalent velocity square and from this one can get an expression for the equivalent velocity which is equal to omega n times S d that is the frequency of the oscillator multiplied by maximum displacement. Now if we look at the previous relationship that we have written X m to be is equal to S v by omega n then we see that the this X dot equivalent is nothing but S v itself. So we can write down X dot equivalent is equal to S v this velocity is called pseudo velocity and is different from the actual maximum velocity. The plots of the S d that is the maximum displacement and the S v that is the pseudo velocity over the full range of frequency and a damping ratio are called the displacement and pseudo velocity response spectrum. A closely related spectrum called pseudo acceleration spectrum also called spectral acceleration is defined as S a equal to omega n square S d. Now one can see that the omega n multiplied by S d that is called the S v and if it is further multiplied by omega n that is then we get the definition of the spectral acceleration. Now the spectral acceleration is the most important quantity in determining the maximum force that comes on to the structure during earthquake and the definition of spectral acceleration is consistent with that. So one can plot now the S a versus omega n for given damping ratio. So we can have we can plot 3 response spectrum that is the displacement response spectrum, velocity response spectrum and the spectral acceleration response spectrum or called spectral acceleration for different values omega n. In fact for the plots that are available they are the omega it is not plotted against omega n but they are plotted against the time period that is 2 pi by omega n giving the value of T n. So the plots are available against the period T n. Now if you look at the definition of the maximum force developed in the spring then one can show that the maximum force in the spring is equal to mass times the spectral acceleration that is obtained. Now this can be proved in this particular fashion that is k by m is equal to omega n square that is known to everyone then k becomes equal to m into omega n square. Now f max that is the maximum force in the spring that will be given by the spring constant k multiplied by S d and if we substitute for k over here that is m omega n square into S d that becomes the value of the maximum force. Now omega n square multiplied by S d that is nothing but the spectral acceleration S a that we had seen here from this equation S a is nothing but omega n square into S d. So therefore the f max becomes equal to mass times the spectral acceleration. Now that is a very interesting thing and the spectral acceleration therefore had become a very useful quantity for defining the maximum force in the structure. What we need to know for finding out the maximum force in the structure developed due to earthquake is the quantity mass of the structure which are easier to obtain than the stiffness. Now this observation shows the importance of the spectral acceleration. While displacement response spectrum is the plot of maximum displacement plots of pseudo velocity and acceleration spectrums are not so that is the pseudo velocity is not equal to the maximum velocity. The pseudo acceleration or spectral acceleration is also not the maximum relative acceleration. These three response spectra provide directly some physically meaningful quantities namely displacement response spectrum provides the information about the maximum deformation that takes place in the single degree freedom system. Pseudo velocity spectrum provides the peak value of strain energy in the system and pseudo velocity or pseudo acceleration spectrum provides the peak force in the system. Now let us come to energy spectrum. Energy spectrum is the plot of the energy of the system over a full range of frequency for a specified damping ratio. So the energy is expressed in this particular fashion that is written as square root of 2 times E t divided by m and the maximum value of this is obtained at every instant of time t. For example, if you look at the definition of energy at any time t, this is equal to half into k x t square plus half into m x dot t square and this dividing the entire thing by m and taking these two onto this side then we can write down twice E t divided by m is equal to omega n square k by m will be equal to omega n square into x t square plus x dot t square. So therefore, for a given single degree freedom system at every instant of time t, if we know the displacement and the velocity then easily we can obtain this quantity twice E t by m and the square root of that is obtained for every instant of time t and the maximum value of that is obtained and that maximum value is plotted against omega n and for a given damping ratio psi. The plot in fact also can be obtained for the time period rather than omega n. Now, for the damping ratio to be 0, it can be easily shown that the value of this maximum or this quantity root over twice E t by m turns out to be of this form and if we compare this equation with the amplitude of the earthquake amplitude or Fourier amplitude of the earthquake record then we can see that there is a similarity between this equation and the equation that is given over here. So, we generally say that the Fourier spectrum and the energy spectrum have similar form. However, Fourier amplitude spectrum may be viewed as a measure of the total energy at the end of t is equal to t of an undamped single degree of freedom system. Now, we take an example and draw the spectrums for the L centro acceleration for psi is equal to 0.05. Now, the equations that had been shown earlier using those equation one can obtain the values of S d, S v, S a and the maximum energy and can plot them against not omega n, but 2 pi by omega n that is the period for a damping ratio of 0.05 and these plots are shown in these figures this is the energy spectrum that is the maximum energy against the time period. That means, we change the oscillator time period every time we calculate the maximum value of the energy and that is how by varying the time period of the oscillator we can obtain the value different values of the maximum energy and plot them for a particular damping ratio. The same thing is done for the Fourier amplitude spectrum where the Fourier amplitude can be obtained against frequency or we can obtain also in the form of time period whereas, the response spectrum that is the acceleration response spectrum which is plotted over here that is plotted against time period. Now, if we compare the peaks of these energy spectrum Fourier spectrum and the acceleration spectrum the if these where the peaks occur if we find out then we see that the energy spectrum peaks around 0.55 second Fourier spectrum peaks around 0.58 second whereas, the acceleration spectrum peaks around 0.51 second that means, the ground motion has a predominant frequency content around a period of 0.5 to 0.6 within this range of the time period we get the maximum frequency content of the ground motion. Now we come to what is known as the DVA spectrum or the combined displacement velocity acceleration spectrums mind you displacement response spectrum is denotes the maximum displacement, but the velocity and acceleration spectrum are actually the pseudo velocity and pseudo acceleration spectrum they do not represent the maximum value of those quantities. Now, DVA spectrum forms the basis for defining the shape of the design spectrum that is used in different codes all over the world. Now, all the three spectrums are useful in defining the design response spectrum which we will discuss later a combined plot of the three spectrum is desirable and we constructed because of the relationship that exist between the three spectrums. For example, here we define the maximum displacement is equal to s v divided by omega n and therefore, by taking log we can write that log s d is equal to log s v minus log omega n s a the spectral acceleration can be written as omega n square into s d equal to omega n into s v. So, if we take a log of these then we get log s a is equal to log s v plus log omega n. Now, these two equations are shown over here and by looking at these two equation one can see there is these two equation represent two straight lines on a log paper and these two straight lines will be orthogonal to each other because one is minus other is plus and these two lines will be an inclined lines in a plot where log s v is the vertical axis and log of omega n will be the horizontal axis. So, taking advantage of these two relationship we are able to plot all the three quantities that is the s d s v and s a in a single plot being the logarithmic relationship or in other words a special kind of log or a graph in which we have four ordinates one is the horizontal axis in the log scale other is a vertical axis in the log scale and two other scales that is which will be inclined lines perpendicular to each other and also defined in a log scale. Now, these four axis define the quantities that is s v s d and s a for a given omega n value the plot also can be obtained not against omega n, but against t n that is the period in that case the two inclined lines will get interchanged that is the s a line would become the s d line and s d line will become s a line if we plot it against t n. Now, while plotting the response all the three response spectrum in this particular graph papers and which is known as a tripartite plot we need some limiting condition in order to define the spectrum. So, these limiting conditions are obtained by setting t tending to 0 in one case and in other case t tending to infinity and in fact it can be easily shown that limit of the maximum displacement as t tends to infinity becomes the maximum ground displacement itself. Similarly, the limit of the spectral acceleration as t tends to 0 becomes the maximum ground acceleration. Now, this can be easily realized from these two figures when t tends to infinity the frequency of the system becomes or tends to 0 and there those kinds of systems are called very flexible system. Once a system was shown for the earthquake measurement equipment that is there was a bracket and from that bracket a bob mass hangs with the help of a flexible stream and there was a drum and as the ground motion takes place in this direction then the bob goes on tracing the ground motion on this chart paper. It was possible because the flexible stream cannot pass on the vibration from this point to this point as a result of this the bob remains stationary whereas these entire bracket started moving like this. So, we can see that and that plot was this relative displacement plot came over here. Therefore, we can see that for a very flexible system the relative displacement of the system becomes equal to the ground displacement itself because the plot which you got is the plot of the ground displacement on the chart paper. On the other hand when t tends to 0 the frequency of the system tends to infinity or this is called what is as a rigid system the example of this is that we put a rigid block on a on the ground and if there is a ground acceleration here then the acceleration that is passed on to the top of the rigid block is the ground acceleration itself because the this entire rigid block moves along with the what you call the ground motion and there is no relative displacement of the rigid block with respect to the ground because it is very rigid and since there is no relative displacement and there is no relative acceleration and therefore the acceleration which is experienced at the top of the rigid block is the ground acceleration itself. So, therefore the following physical conditions emerge that is the limit t tending to infinity for SD becomes equal to UG max and limit t tending to 0 for SA becomes equal to the ground acceleration. So, now we show the plot this is the log SV that is the SV is in the log scale in the vertical axis and this is the log of not omega n but log of t n which is shown in the horizontal axis and this inclined line is one axis and this inclined line is another axis. Now, this axis corresponds to the acceleration in the logarithmic scale and this axis shows the displacement in the logarithmic scale again. Now, when we we can plot the SV versus t n in the logarithmic scale and if we make a plot let us say this plot looks like this for a particular ground record in this figure the ground record taken is the L centre earthquake and the dotted line is the averaged that means we have averaged that zigzag portion and more or less a smooth curve has been obtained for the L centre earthquake. Now, this plot which is the plot of log SV versus log t provides also the information about SA and SD that is if we measure that means if I take a point like this then for this particular point the SV value will be read from this axis SA value will be read from this axis and SD value will be read from a line parallel to this axis. So, any particular point reveals three quantities SA, SV and SD therefore, in one single plot we get all the three response spectrum. So, this is called the tripartite plot of the response spectrum in a special log graph paper. Now, what has been done is that the the graph that we get or that is a response spectrum that the smooth response spectrum that we get for a particular earthquake in this case is L centre earthquake this has been further idealized by a number of straight lines which are represented by the dotted lines over here and we obtain the response spectrum for a number of earthquakes and plotted them in the tripartite plot and those response spectrums show similar kind of trend and to understand that trend let us look at the different segments. For example, this is a segment which is designated by TA time period TA that is below this time period TA we have a segment then we define another time period TB between TA to TB we have another segment then from TB to TC we have got another segment then from TC to TD we have another segment and from TD to TE we have another segment and from TE to TF we have got another segment and beyond TF we have another segment. So, these segments are represented with the help of these time periods TA, TB, TC, TD, TE and TF. Now, if we plot a different earthquake spectrums in a tripartite plot and idealize them by straight line then we will get similar kind of form only the TA value, TB value, TC value these values will be different for different earthquakes. Now, each one of these segment represent certain thing that is what we are going to discuss now the straight lines below A and between points B and C are parallel to SD axis that is this line below A and between this point and this point if we look at these two lines these two lines are parallel that is what observe parallel to the line which is called the SD line. So, that is parallel to SD those below F and between D and D are parallel to SA axis that is this line and these two lines are parallel to the this acceleration or SA line below A shows constant value of SA which is equal to the ground acceleration and below F shows constant SD which is equal to the ground displacement that is what we discussed before that is in they give the limiting conditions that is here the limiting condition that value is shown as the acceleration that means the we measure along this axis therefore, it shows a constant acceleration and this shows a constant displacement which is equal to the ground displacement that is how the specified ground acceleration and specified ground displacement their maximum values are utilized in plotting this curve. In between these between B and C constants SA is equal to alpha times U double dot G max and between D to E that SD is equal to alpha D times U double dot or U G max that is this one this line and this line if we consider since this line and this line and parallel which is now logical is that these value that means this parallel line this will indicate a particular value of acceleration constant acceleration this constant acceleration will be equal to the constant acceleration or the ground acceleration that is shown over here the maximum ground acceleration which is which represent this constant portion this multiplied by some factor gives the constants acceleration over this inclined line. Similarly the maximum ground displacement which is represented by this segment and this portion of the segment which is parallel to this shows a displacement or maximum displacement which is equal to some multiplication factor multiplied by the maximum ground displacement. So, that is what is written over here the SA in that segment is equal to some multiplication factor alpha A into the maximum ground acceleration and on the other side the it is the maximum displacement is shown or the SD is shown as the some multiplication factor multiplied by the maximum ground acceleration. Now the left of C is directly related to maximum acceleration because in this plot we can see that on the left of C everything that is there that is parallel to the the SD axis representing the acceleration and beyond this point everything that is there that represent the displacement because there these lines are parallel to the acceleration axis representing the displacement. So the intermediate portion CD is directly related to maximum velocity of ground motion and most sensitive to damping ratio that is the horizontal portion it is obviously is most related to the velocity and this is also found to be very sensitive to the damping ratio that means if we change the damping constant we will find that in this range the response spectrum is varying widely. So, now let me summarize over here what we discussed. We discussed the cross power spectral density function that exists between two points along the direction of the wave propagation these cross power spectral density function represents the lag time lag or the lack of correlation between the ground motions at these two points and this cross power spectral density function is defined with the help of a coherence function which is exponentially dequeuing function of the distance between the two points and frequency. Then we obtained the then we started discussing on the response spectrum. So we have seen that there are three types of response spectrum acceleration response spectrum velocity response spectrum displacement response spectrum and also there is energy spectrum all of them can be plotted using a single degree freedom system and for a particular earthquake we can go on changing the period of the single degree freedom system and obtain these quantities and can plot it. Then we discussed about the DVS spectrum that is the tripartite plot where all the three spectrums can be plotted and can be seen with the help of one particular graph and the characteristics of the idealized DVS spectrum.