 In the previous two lectures, we discussed about the response spectrum method of analysis and try to explain to you how the response spectrum method of analysis is developed from the principles of the normal mode theory, the equivalent static loads which are calculated in each mode of vibration, how they are developed. Then we solved a problem for single point excitation system to illustrate the method of analysis for a single point excitation system. At the end we started the extension of the method for obtaining the response spectrum method of an or applying the response spectrum method of analysis for multi-support excitation system. In that we started saying about the assumptions that are additionally used for extension extending the response spectrum method of analysis for multi-support excitation. Those assumptions are the assumptions that the future earthquake is represented by an averaged smooth response spectrum and a power spectral density function that is obtained from an ensemble of time histories. Then lack of correlation between ground motions at two points that is at the two supports, two different supports where the excitations are caused due to earthquake is represented by a coherence function. Many expressions for the coherence functions were discussed in seismic input. One can choose any one of those coherence functions. The third assumption is the most important assumption that is the peak factor in each mode of vibration and the peak factor for the total responses are assumed to be the same. We have come across the term peak factor in relation to the seismic inputs for structures for the case of random ground excitation. The peak factor is a factor which is multiplied with the standard deviation or the RMS value of the ground motion in order to obtain the peak value of the ground motion better it is called expected peak value of the ground motion. Then expression for that was given in seismic input. So in each mode of vibration similarly if we wish to find out the peak value of the response then one should multiply a peak factor with the standard deviation of the response for the response for that mode of vibration. This peak factor is obtained from the power spectral density function of the response in that mode. Once we get the power spectral density functions for the responses in different modes of vibration from that one can calculate not only the standard deviation of the modal response but also the peak factors associated with that. Multiplying the peak factors with the standard deviation of the response one can get the peak value of the response. Thus the peak factors are involved in each mode of vibration that is also a peak factor for the total response in the structural coordinate system. This assumption means that all the peak factors in different modes of vibration the peak factor which is associated with the total response of the system all of them are assumed to be the same therefore if we find out the peak factor for any mode of vibration then that peak factor can be utilized throughout the analysis. Next what is required is a relationship between the ground displacement spectrum and the power spectral density function of response. This relationship is required because we wish to find out also the power spectral density function of the ground motion from the specified response spectrum. That is necessary for obtaining the coherence function the cross power spectral density function between the two responses which are required in the derivation of this particular analysis. Mean peak value of any response quantity R consists of two portions. First part is the pseudo static response due to the displacements of the support. The second part is the dynamic response of the structure with respect to supports. The first part is obtained by applying unit displacement at each support successively finding out the responses to non support degrees of freedom. So, that gives the pseudo static response due to the displacements of the support. They are some coefficients and these coefficients are stored in the program so that this coefficient later on can be multiplied with the ground displacements that is taking place at different supports. The second part straight away comes from the response analysis of the structure or the dynamic response analysis of the structure with respect to the support. Now, using normal mode theory the uncoupled dynamic equation of motion for each mode of vibration can be written by equations like equation 5.18 where we have the ith modal equation providing the ith generalized displacement z i. There is a term beta k i which is the mode participation factor. This we discussed in previously when we are solving the problem in time domain. There we have seen that if it is a case of multi support excitation then there is not an unique value of the mode participation factor like single point excitation system. But the mode participation factor in a particular mode is associated also with the support. So, for the case support excitation there is a mode participation factor in the ith mode that is called beta k i and that is given by phi i t m r k divided by phi i t m phi i r k is the column vector or the kth column vector of the r matrix that is the influence coefficient matrix that we multiply with the mass matrix then multiply to the x double dot g to find out the earthquake force or in other words for multi degree freedom system with multi support excitation on the right hand side we write minus m r then x double dot g. So, r k is the kth column of that r matrix. If the response of the single degree freedom oscillator to a specific excitation for a particular support is u double dot k for that if the response is z bar k i then one can write down the response or generate the response in the ith mode is equal to the summation of beta k i multiplied by z bar k i because z bar k i is the response in the ith mode produced due to the kth support excitation. So, if we sum up for all the support excitation then we get the value of the z i in a particular mode of vibration. So, this z i is the dynamic response of the system with respect to the support the total response is given by the equation 5.21 where the response r t is given by this equation that is a k into u k t summation over the supports and then this is the 5 bar i z i t that is the mode shape coefficient corresponding to the response quantity of interest. So, this response quantity of interest need not be displacement could be bending moment shear force or any other quantity of interest. So, the 5 bar i k will be the mode shape coefficient for that response then we multiply that coefficient with z i t and sum it over all the modes. So, that gives the response of that particular response of the system and that is the dynamic response with respect to the support. This part is the pseudo static response part and as I explained before this a k coefficient that is the responses that is obtained at the non support degrees of freedom due to displacement applied at each support that is stored and that is the coefficient a k. So, that a k is multiplied by the ground motion u k to get the pseudo static component of the response at all non support degrees of freedom. This a k coefficient could be also obtained for any response quantity of interest if this r t is a response other than the displacement then we use those values of a k which we have obtained for the response quantity of interest substituting z i t from this equation we obtain finally equation 5.22 and these 5.22 equation can be written in a compact form in matrix notation that is r t becomes equal to a t u t plus 5 b t z bar t where 5 v z bar are vectors of size m into s say the number of support is 3 and the number of modes that we consider is 2. In that case the 5 b vector would typically look like this that is the 5 bar 1 beta 1 1 5 bar 1 beta 2 1 5 bar 1 beta 3 1. So, this beta 1 1 beta 2 1 and beta 3 1 are the mode participation factors coming from the 3 modes for mode 1 then we have 5 bar 2 beta 1 2 5 bar 2 beta 2 2 and 5 bar 2 beta 3 2 they are the again the mode participation factors coming from supports 1 2 and 3 for the second mode. Similarly, the z vector consists of z bar 1 1 z bar 2 1 and z bar 3 1 that is the responses coming from the support 1 to mode 1 the z bar 2 1 indicates the response that is coming from the second support 2 mode 1 z bar 3 similarly is the response that is coming to the mode 1 from support 3. The last 3 terms that is z bar 1 2 z bar 2 2 and z bar 3 2 they are similarly the responses the modal responses that is coming for the second mode from the 3 supports then we assume that RT, UT and z bar T to be random processes the PSDF of RT then is given by equation 5.25 and this equation is shown over here the reason for assuming RT, UT and z bar T to be random process is that the excitation is an excitation which is assumed to be a response to be an excitation coming from an ensemble of time history of records that ensemble of time history of record is characterized by a stationary random process as well as a smooth response spectrum is obtained for that ensemble time history of records and therefore we could relate the response spectrum of displacement with the power spectral density function of the ground acceleration and a relationship of that type was shown previously and that relationship is used to obtain the power spectral density function of the ground acceleration from a given response spectrum of displacement. So once we assume that RT, UT and z bar T are the quantities which are random quantities or random processes then one can write down straight away the power spectral density function of the response by this equation and the basis of this equation has been discussed before while discussing the spectral analysis of response for a specified ground for a specified power spectral density function of ground excitation. phi AT, SUA that comes from the first part of the equation that is AT, UT the second part phi BT S z bar z bar phi beta that is coming from the second part of the equation and these two parts are the cross terms that is the cross power spectral density function that exists between U and z bar and z bar and U. So once we have this expression for SRR then one can perform an integration of this power spectral density functions after we integrate this power spectral density function then one can get the standard deviation of these quantities those standard deviations are multiplied by the peak factor in order to get the mean peak values of the responses and from that mean peak value of the responses one can get the mean peak value of the response quantity of interest. So once we do that then the expected maximum value of the response quantity of interest that is E max RT that can be written finally in this form BT LUB plus BT LU z bar phi BD plus phi T BD L z bar phi BD plus phi T BD L z bar U B where B and phi BD are the vectors like this that is B vector is equal to A1 into UP1 A2 into UP2 and so on. So if there are S number of S supports then it will go up to A S into UPS phi T beta D is given by this expression phi bar 1 beta 11 multiplied by D11 then phi bar 1 beta 21 D21 and that is how it goes up to phi bar 1 beta S1 DS1 then for the last series of terms will be phi bar M beta 11 D1 M and the last term of the series will be phi bar M beta S1 multiplied by DSM. Now let me let us explain the terms of this BT and phi T BD A1 A2 A3 are the coefficients of the response or the pseudo static coefficient of the response quantity that we have discussed before UP1 is the peak ground displacement for the support excitation 1, UP2 is the peak ground displacement corresponding to support excitation 2 and so on. In these expression phi bar 1 represents the first mode shape coefficient for the response quantity of interest that we can obtain or that we have discussed before also beta 11 is the mode participation factor that is coming from the support 1 to mode 1, D11 is a displacement response spectrum ordinate corresponding to the first time period and the first support excitation that is we are assuming that the response spectrums at support 1, support 2, support 3 and so on these response spectrums as if are different then D11 represents the response spectrum ordinate for time period for the first mode and for the response spectrum that we have obtained or that is given for the support excitation 1. So similarly beta 21 is the mode participation factor that is coming to the first mode from the second support and D21 is the displacement response spectrum for the first for the first mode time period coming from the support excitation 2. So that way these terms basically can be easily explained and can be known that is again for example phi bar 1 here is the mode shape coefficient for the response quantity of interest in the first mode and beta S1 represents the mode participation factor which is coming to the mode 1 coming from support S and DS1 represents the displacement response spectrum ordinate for the time period for mode 1 coming from the support excitation at S and that is how we are writing Dij and this Dij means that the response spectrum ordinate coming to the or corresponding to the time period j and its contribution coming from the support i. If we assume that Dij that is the support excitations for all the supports have the same power spectral density function and they have the same response spectrum or displacement response spectrum then Dij simply becomes equal to D i omega j xi j which means that for a given response spectrum which will remain constant for all the supports for that the response spectrum ordinate corresponding to the time period which will be the 2 pi by omega j and for the damping ratio xi j. So, for that we take the displacement response spectrum ordinate and thus D11, D21, DS1, D1M etc all of them they basically become simplified and we have for each mode we have a one particular value of the displacement response spectrum ordinate. The elements of the correlation matrices L U, L U Z bar and L Z Z bar are given in equations 5.28, 5.29 and 5.30. U i, U j are the elements of L U matrix and it equal to 1 by sigma U i into sigma U j and integration of the S U i U j D omega in which sigma U i is the standard deviation of the displacement or ground displacement at support i and sigma U j is the standard deviation of the ground displacement at support j. S U i U j is the cross pore spectral density function between the ground displacement at support i and support j. L U i Z bar k j are the elements of L U Z bar matrix and they are equal to 1 by sigma U i into sigma Z bar k j in which sigma U i is the standard deviation of the ground displacement at the ith support and the sigma Z bar k j is the generalized standard deviation of the generalized displacement for mode j coming from support k that is this Z bar is obtained for the jth mode using beta k j that is the mode participation factor beta k j that we had obtained before that is equation 5.19. Then within the integration we have h j star which is the complex conjugate of the frequency response function for the jth mode and S U i Z bar S U i U double dot k that is equal to the cross pore spectral density function between the displacement at the ith support and the acceleration at the kth support. The L Z bar k i Z bar L j they are the elements of L Z bar Z bar matrix and are equal to 1 by sigma Z bar or k i into sigma Z bar L j where sigma Z bar k i is the standard deviation of the generalized displacement for the ith mode and the kth support that is which is obtained using beta k i. Similarly, sigma Z bar L j is the standard deviation of the generalized displacement for the jth mode from the lth support. Within the integration we have h i h j star h i is a complex frequency response function for the ith mode h j is the complex conjugate of the frequency response function for the jth mode and S U bar k L and U double dot k and U double dot L they are the cross pore spectral density function between the accelerations at the kth support and the lth support. Now, with this equations defined and after we integrate these equations the elements of the matrices can be obtained. In those equations what we have to find out is S U i U k which we have to find out is S U i U k which is equal to 1 by omega square S U i to the power half S U k to the power half and coherence function i k that is the it takes care of the partial correlation between the ground displacements at the ith support and the kth support. Since the power spectral density function of the ground displacements for all the supports are the same therefore, S U i to the power half S U k to the power half the multiplication of these two turns out to be S U double dot G that is the power spectral density S U G that is the power spectral density of the ground specified ground displacement. Then we have S U i U double dot S U i U j that is equal to 1 by omega to the power 4 into S U i to the power half S U j to the power half multiplied by S U i to the power half divided by coherence function i comma j and which again turns out to be equal to coherence i j divided by omega to the power 4 into S U double dot G. I am sorry in the previous equation it was S U i U double dot k. So, that is given as coherence i k divided by omega square into S U double dot G. And finally, we have S U double dot k and U U double dot l that turns out to be simply coherence k l multiplied by S U double dot G. So, in terms of S U double dot G and a coherence function defined by S U double dot defined to take care of the partial correlation between the ground motions. One can obtain these terms S U i S U double S U i U double dot k S U i U j and S U double dot k U double dot l. Now, for a single train of seismic wave d i j becomes simply the value of the displacement response spectrum for the earthquake which is given in this equation 5.27 c. There d i j is equal to becomes equal to d i omega i z i i for the single train of earthquake that is it has a unique displacement spectrum. And S U double dot G can be obtained from the specified displacement spectrum from the relationship that we have discussed before. So, for a given displacement spectrum of an earthquake and an additionally specified coherence function to take care of the partial correlation between the ground motions. One can now obtain all the terms necessary to calculate the mean peak value of the response of the system. Thus for a multi support excitation case one can find out the values of the expected peak value of the responses using the same response spectrum method of analysis. Only thing that is required over here is that in addition to defining the displacement response function we have to define a coherence function and a relationship that equates the acceleration of power spectral density function in terms of the displacement response spectrum of the earthquake. Now, if only the relative peak displacement is required then the third term of equation 5.26 that is required. So, the third term only gives the relative displacement, but if we take all the terms in equation 5.26 then this will give the absolute displacement. The steps for obtaining the response spectrum method of response spectrum analysis for multi support excitation using MATLAB is given in the book. In the MATLAB one can develop the method that is described over here the first step would be to obtain a matrix R which is constructed as we have discussed before. Then from the R matrix one can take out the R k vector that is for the k th support and use it for finding out the mode participation factor for the mode for the k th support. Then beta k i is that is a beta k i that is obtained and this beta k i is obtained for all the supports from 1 to s and for all the modes 1 to m. Next phi t beta phi t b d they are obtained from the equations and in obtaining phi t beta d we require the ordinal equation for the units of the displacement response spectrum at different modes corresponding to the time periods of those modes. We also require the mode shape coefficient for the response quantity of interest as well as we require the mode participation factor for a particular mode coming from a particular support. So all these quantities are known so therefore phi t beta d and phi t beta can be constructed. Once we have these 2 vectors next vector that is required is the A vector that is the influence coefficient vector which comes from the pseudo static analysis and that can be stored for a particular response the pseudo static vector and for the displacement the pseudo static vector they could be different. Next is to construct a vector B and these vector B we have shown in expression 5.27 these vector B requires the peak ground displacement at different supports but this peak ground displacement at different supports are assumed to be the same because we have the same train of earthquake moving through all the supports this can be obtained from the specified power spectral density function and for the support excitation and then converting it to the power spectral density function for displacement the area under that curve give the standard deviation and that standard deviation can be multiplied by a peak factor in order to obtain the peak value of the or expected value of the ground displacement. The matrices L U U, L U U Z bar and L Z Z bar they required basically the integration of the cross power spectral density function multiplied by the frequency response function for each mode which was shown in equation 5.28 to 5.30 and in that we have seen that we require also the standard deviation of the generalized displacement in a particular mode for the excitation or the kth excitation or the excitation at the kth excitation kth support. So that standard deviation can be obtained by assuming the power spectral density function or not assuming by obtaining the spectral analysis for the specified power spectral density function of the ground motion and from that one can get the power spectral density function of the generalized displacement the area under the curve will give the standard deviation. The cross power spectral density function that is required between the displacement at one support with the acceleration at the other support the cross power spectral density function between the displacements at two supports and the cross power spectral density function between the acceleration at two supports which are required in those integrations they are obtained again by a set of equation which can be finally represented in the form of a coherence function and the specified power spectral density function of the ground acceleration coherence function is additionally provided for this problem. So one can develop a program in easily in the MATLAB for finding out the response or the mean peak response for systems having multi support excitation and subjected to a set of ground motions which can be represented with the help of an averaged response spectrum which can be again converted to a ground acceleration of spectrum through an empirical relationship. We illustrate the method with the help of a problem this is a problem if we recall is a problem for a three support frame that is example 3.8 we had a 2.8 we had a 2 story frame with 3 supports the non support degrees of freedom are 2 and therefore the mode shapes are 2 by 2 mode shape matrix and since we are wanting to find out the expected peak value of the displacements then fibre also becomes the same matrix the R matrix for the multi support excitation case was computed before for this problem and R is given by this matrix the 3 frequencies R 2 frequencies R omega 1 and omega 2. So they are also given 80 that is the pseudo static coefficients at the ground acceleration non support degrees of freedom coming from all the three excitations they are same as R therefore 80 is same as R this is the phi t beta matrix and the phi t beta matrix will have these sets of this is the first set fibre 11 beta 11 fibre 11 beta 21 fibre 21 11 beta 31 all these things I have been explained before and fibre 1 2 fibre 2 2 or beta 21 beta 1 2 etcetera they can be all obtained the phi t beta d matrix that is computed over here so it requires as I told you before the ordinates of the response distance displacement response spectrum then phi terms and the terms for the mode participation factor for a particular mode the displacement response spectrum which is assumed to be the same for all the 3 supports because it is the same train of earthquake that is passing through all the 3 supports that is obtained for the time periods corresponding to these 2 frequencies and they are 0.056 and 0.011 coherence function or matrix can be obtained easily these will not be 0 this will be 111 and rho 1 and rho 2 will be minus 5 omega by 2 pi and minus 10 omega by 2 pi because it is assumed that there is a 5 second time lag between the supports so between the first support and the second support this is the this will be the value of rho 1 we have computed it also before also and for first support and the third support the rho 2 value will be equal to minus 10 omega 2 pi so plugging in the values of rho 1 rho 2 etcetera in this coherence matrix we get the complete coherence matrix. LUU LUU Z bar matrix they are again computed that is the that will integrations were performed and we finally obtained each of these elements of those matrices and constructed this matrix LUU similarly the matrix LUU Z bar that was constructed and LZ Z bar matrix also was constructed with the help of these matrices finally we obtained the mean peak values of the displacements that is for the displacement 1 the mean peak value was obtained as 0.106 and for the second non support degree of freedom it was 0.099 meter when we obtained the relative displacement for the response 1 then the relative displacement mean peak value of the displacement came out to be 0.045 and for the second non support degree of freedom the mean peak value of the displacement came as 0.022 note that we can obtain both the mean peak values of the total response and the mean peak value of the relative displacement the mean peak value of the total displacement means the relative displacement plus the pseudo static component they are taken together and for that we obtained the mean peak values and the relative displacements they come purely from the modal displacement Z bar multiplied by the mode shape factor the general expression that we had shown in the development of the method they are if we retain only the this term then we get the if we retain only the third term and ignore all other terms then we get the straight away the relative displacements or the mean peak value of the relative displacements so that is what was done over here to get the value of the mean peak value of the relative displacements we also solve the problem assuming that the ground motions are perfectly correlated in that case i u u i z bar z bar and i u z bar they take this particular form using those we obtain the values of the mean peak relative displacement by response spectrum method of analysis for u 1 and u 2 they turn out to be like this for the time history analysis for the same problem and for the same L centro earthquake record for which the power spectral density function and the corresponding displacement response spectrum were used that gave us a values which are 0.081 and 0.041 and we can see that these 2 values match quite well thus in spite of the different assumptions that have been considered in the development of the response spectrum method of analysis for multi support excitation case we see that the results that we obtain from the response spectrum method of analysis compare quite well with the time history analysis we explain the same method with the help of another example and in that if you recall we solved a problem which has a which was a beam problem resting on 3 spring supports that is the problem this problem this was a beam which is resting on a soil soil is replaced by the beam which is the spring and dash dash supports this has a spring constant case case and case and the damping constants are C s for the soil the structural damping for the structure is taken as 5 percent these are the masses which are lumped at the 3 degrees of freedom translations are the 3 degrees of freedom rotation is condensed out there is a time lag which is assumed between the supports the time lag is 2.5 second C s value is given as 0.6 meter and case value is given as 48 this is not meter m and this is case is equal to 48 m the structure damping matrix is obtained as C is equal to alpha m plus beta k the C bar matrix that is the damping matrix considering the soil damping that turns out to be this note that once we add the soil damping to the structural damping the damping becomes a non classical damping so this problem is a problem of non classical damping therefore to use the response spectrum method of analysis for this becomes really difficult however by making an approximation that is by diagonalizing the damping matrix that is the modal damping matrix or in other words phi T C bar phi for that the diagonal terms are ignored and with the help of that we solve the same or this problem and obtain the responses.