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Chapters – 5 & 6 Chapter -5 RESPONSE SPECTRUM METHOD OF ANALYSIS T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/1 Introduction Response spectrum method is favoured by earthquake engineering community because of: It provides a technique for performing an equivalent static lateral load analysis. It allows a clear understanding of the contributions of different modes of vibration. It offers a simplified method for finding the design forces for structural members for earthquake. It is also useful for approximate evaluation of seismic reliability of structures. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/2 Contd… The concept of equivalent lateral forces for earthquake is a unique concept because it converts a dynamic analysis partly to dynamic & partly to static analysis for finding maximum stresses. For seismic design, these maximum stresses are of interest, not the time history of stress. Equivalent lateral force for an earthquake is defined as a set of lateral force which will produce the same peak response as that obtained by dynamic analysis of structures . The equivalence is restricted to a single mode of vibration. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/3 Contd… The response spectrum method of analysis is developed using the following steps. A modal analysis of the structure is carried out to obtain mode shapes, frequencies & modal participation factors. Using the acceleration response spectrum, an equivalent static load is derived which will provide the same maximum response as that obtained in each mode of vibration. Maximum modal responses are combined to find total maximum response of the structure. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/4 Contd… The first step is the dynamic analysis while , the second step is a static analysis. The first two steps do not have approximations, while the third step has some approximations. As a result, response spectrum analysis is called an approximate analysis; but applications show that it provides mostly a good estimate of peak responses. Method is developed for single point, single component excitation for classically damped linear systems. However, with additional approximations it has been extended for multi point-multi component excitations & for nonclassically damped systems. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/5 Development of the method Equation of motion for MDOF system under single point excitation (5.1) Mx Cx Kx MIx g Using modal transformation, uncoupled sets of equations take the form zi 2ii zi i2 zi i xg ; i 1 iT MI i T i Mi m (5.2) i is the mode shape; ωi is the natural frequency λ is the more participation factor; ξ is the i i modal damping ratio. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/6 Contd… Response of the system in the ith mode is (5.3) x i = φi z i Elastic force on the system in the ith mode fsi = Kx i = Kφi zi (5.4) As the undamped mode shape i satisfies Kφi = ωi2Mφi (5.5) Eq 5.4 can be written as fsi = ωi2Mφi z i (5.6) The maximum elastic force developed in the ith mode fsimax = Mφiωi2 z imax (5.7) T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/7 Contd… Referring to the development of displacement response spectrum (5.8) zi max i Sdi i , i 2 Using Sa Sd , Eqn 5.7 may be written as f s i max i M i S ai Pei (5.9) Eq 5.4 can be written as xi max K 1 f si max K 1 Pei (5.10) Pe i is the equivalent static load for the ith mode of vibration. Pe i is the static load which produces structural displacements same as the maximum modal displacement. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 1/8 Contd… Since both response spectrum & mode shape properties are required in obtaining Pe i , it is known as modal response spectrum analysis. It is evident from above that both the dynamic & static analyses are involved in the method of analysis as mentioned before. As the contributions of responses from different modes constitute the total response, the total maximum response is obtained by combining modal quantities. This combination is done in an approximate manner since actual dynamic analysis is now replaced by partly dynamic & partly static analysis. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/1 Contd… Modal combination rules Three different types of modal combination rules are popular ABSSUM SRSS CQC ABSSUM stands for absolute sum of maximum values of responses; If x is the response quantity of interest m x i 1 xi (5.11) max xi max is the absolute maximum value of response in the ith mode. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/2 Contd… The combination rule gives an upper bound to the computed values of the total response for two reasons: It assumes that modal peak responses occur at the same time. It ignores the algebraic sign of the response. Actual time history analysis shows modal peaks occur at different times as shown in Fig. 5.1;further time history of the displacement has peak value at some other time. Thus, the combination provides a conservative estimate of response. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis Top floor displacement (m) 2/3 0.4 0.2 0 -0.2 -0.4 0 5 t=6.15 10 15 20 25 30 First generalized displacement (m) (a) Top storey displacement 0.4 0.2 0 -0.2 -0.4 0 5 t=6.1 10 15 Time (sec) 20 25 30 (b) First generalized displacement Fig T.K. Datta Department Of Civil Engineering, IIT Delhi 5.1 Response Spectrum Method Of Analysis Contd… 2/3 Second generalized displacement (m) 0.06 0.04 0.02 0 -0.02 -0.04 -0.06 0 t=2.5 5 10 15 20 25 30 Time (sec) (c) Second generalized displacement Fig 5.1 (contd.) T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/4 Contd… SRSS combination rule denotes square root of sum of squares of modal responses For structures with well separated frequencies, it provides a good estimate of total peak response. x m x i 1 2 i max (5.12) When frequencies are not well separated, some errors are introduced due to the degree of correlation of modal responses which is ignored. The CQC rule called complete quadratic combination rule takes care of this correlation. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/5 Contd… It is used for structures having closely spaced frequencies: x m m m 2 x i ij xi x j i 1 (5.13) i 1 j 1 Second term is valid for i of degree of correlation. j & includes the effect Due to the second term, the peak response may be estimated less than that of SRSS. Various expressions for i j are available; here only two are given : T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/6 Contd… ij 1 1 4 2 2 ij 2 2 ij ij 1 4 1 8 1 3 2 ij 2 ij (Rosenblueth & Elordy) (5.14) ij 2 ij 2 ij (Der Kiureghian) (5.15) 2 ij Both SRSS & CQC rules for combining peak modal responses are best derived by assuming earthquake as a stochastic process. If the ground motion is assumed as a stationary random process, then generalized coordinate in each mode is also a random process & there should exist a cross correlation between generalized coordinates. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/7 Contd… Because of this, i j exists between two modal peak responses. Both CQC & SRSS rules provide good estimates of peak response for wide band earthquakes with duration much greater than the period of structure. Because of the underlying principle of random vibration in deriving the combination rules, the peak response would be better termed as mean peak response. Fig 5.2 shows the variation of i j with frquency ratio. i j rapidly decreases as frequency ratio increases. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/8 Contd… Fig 5.2 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/9 Contd… As both response spectrum & PSDF represent frequency contents of ground motion, a relationship exists between the two. This relationship is investigated for the smoothed curves of the two. Here a relationship proposed by Kiureghian is presented 2 4 D , S xg ff p0 2 p0 ( ) T.K. Datta Department Of Civil Engineering, IIT Delhi 2 2.8 2 ln 2 (5.16 a) (5.16 b) Response Spectrum Method Of Analysis 2/10 Contd… 2 -3 PSDF of acceleration sec /rad) (m Example 5.1 : Compare between PSDFs obtained from the smoothed displacement RSP and FFT of Elcentro record. 0.05 Unsmoothed PSDF from Eqn 5.16a Raw PSDF from fourier spectrum 0.04 0.03 0.02 0.01 00 10 20 30 40 Frequency (rad/sec) Eqn.5.16a Fourier spectrum of El Centro 0.02 2 -3 60 Unsmoothed 0.025 (m PSDFs of acceleration sec/rad) 50 0.015 0.01 0.005 00 10 20 30 40 50 60 Frequency (rad/sec) 70 80 90 100 5 Point smoothed T.K. Datta Department Of Civil Engineering, IIT Delhi Fig5.3 Response Spectrum Method Of Analysis 2/11 Application to 2D frames Degree of freedom is sway degree of freedom. Sway d.o.f are obtained using condensation procedure; during the process, desired response quantities of interest are determined and stored in an array R for unit force applied at each sway d.o.f. Frequencies & mode shapes are determined using M matrix & condensed K matrix. For each mode find (Eq. 5.2) & obtain Pei i (Eq. 5.9) N i r W ir r 1 N W r ir (5.17) 2 r 1 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/12 Contd… Obtain R j RPej ( j 1...r ) ; R j is the modal peak response vector. Use either CQC or SRSS rule to find mean peak response. Example 5.2 : Find mean peak values of top displacement, base shear and inter storey drift between 1st & 2nd floors. Solution : ω1 =5.06rad/s; ω2 =12.56rad/s; ω3 =18.64rad/s; ω 4 = 23.5rad/s φ1T = -1 -0.871 -0.520 -0.278 ; φ 2T = -1 -0.210 0.911 0.752 φ3T = -1 0.738 -0.090 -0.347 ; φ T4 = 1 -0.843 0.268 -0.145 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 2/13 Contd… Table 5.1 Disp (m) Base shear in terms of mass (m) Drift (m) Approaches 2 modes all modes 2 modes all modes 2 modes all modes SRSS 0.9171 0.917 1006.558 1006.658 0.221 0.221 CQC 0.9121 0.905 991.172 991.564 0.214 0.214 ABSSUM 0.9621 0.971 1134.546 1152.872 0.228 0.223 Time history 0.8921 0.893 980.098 983.332 0.197 0.198 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis Application to 3D tall frames Analysis is performed for ground motion applied to each principal direction separately. Following steps are adopted: Assume the floors as rigid diaphragms & find the centre of mass of each floor. DYN d.o.f are 2 translations & a rotation; centers of mass may not lie in one vertical (Fig 5.4). Apply unit load to each dyn d.o.f. one at a time & carry out static analysis to find condensed K matrix & R matrix as for 2D frames. Repeat the same steps as described for 2D frame T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/1 3/2 C.G. of mass line C.G. of mass line CM1 CM1 L L CM2 CM2 L L CM3 CM3 L L L (a) L L xg (b) xg x x Figure 5.4: T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/3 Contd… Example 5.3 : Find mean peak values of top floor displacements , torque at the first floor & VX and VY at the base of column A for exercise for problem 3.21. Use digitized values of the response spectrum of El centro earthquake ( Appendix 5A of the book). Solution : ω1 =13.516rad/s; ω2 =15.138rad/s; ω3 = 38.731rad/s; ω4 = 39.633rad/s ; ω5 = 45.952rad/s; ω6 =119.187rad/s Results are obtained following the steps of section 5.3.4. Results are shown in Table 5.2. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/4 Contd… TABLE 5.2 Torque Approac displacement (m) (rad) Vx(N) Vy(N) hes (1) (2) (3) SRSS 0.1431 0.0034 0.0020 214547 44081 CQC 0.1325 0.0031 0.0019 207332 43376 0.1216 0.0023 0.0016 198977 41205 Time history Results obtained by CQC are closer to those of time history analysis. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis RSA for multi support excitation Response spectrum method is strictly valid for single point excitation. For extending the method for multi support excitation, some additional assumptions are required. Moreover, the extension requires a derivation through random vibration analysis. Therefore, it is not described here; but some features are given below for understanding the extension of the method to multi support excitation. It is assumed that future earthquake is represented by an averaged smooth response spectrum & a PSDF obtained from an ensemble of time histories. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/5 3/6 Contd… Lack of correlation between ground motions at two points is represented by a coherence function. Peak factors in each mode of vibration and the peak factor for the total response are assumed to be the same. A relationship like Eqn. 5.16 is established between S d and PSDF. Mean peak value of any response quantity r consists of two parts: T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/7 Contd… • Pseudo static response due displacements of the supports to the • Dynamic response of the structure with respect to supports. Using normal mode theory, uncoupled dynamic equation of motion is written as: s zi 2i zi i2 zi ki uk ; i 1..m (5.18) k 1 i T MRk ki T i Mi T.K. Datta Department Of Civil Engineering, IIT Delhi (5.19) Response Spectrum Method Of Analysis 3/8 Contd… If the response of the SDOF oscillator to uk is zki s then zi ki z ki (5.20) k 1 Total response is given by s m r t ak uk t i zi t k 1 i 1 s m s r t ak uk t i ki zki (5.22) r t a T u t T z t (5.23) k 1 (5.21) i 1 k 1 φβ and z are vectors of size m x s (for s=3 & m=2) T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/9 Contd… φβT = 1 β11 1 β21 1 β31 2 β12 2β22 2β32 (5.24a) z T = z11 z21 z31 z12 z22 z32 (5.24b) Assuming r t ,u t and z t to be random processes, PSDF of r (t ) is given by: S rr a S uua S zz a S uz S zua T T T T (5.25) Performing integration over the frequency range of interest & considering mean peak as peak factor multiplied by standard deviation, expected peak response may be written as: T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/10 Contd… E max r t = b T b +b uu b T = a1up1 a 2up2 a3up3 T uz φβD + φ T βD zz φβD + φ aSupS (5.27a) T φβD = φ1β11D11 φ1β 21D21 .... φ1β s1Ds1 ...φ mβ11D1m Dij = Di ω j ,ξ j i =1,..,s ; j =1,..,m luu , lu z and lz z 12 b (5.26) T βD zu (5.27b) (5.27c) are the correlation matrices whose elements are given by: uiu j 1 = σ ui σ u j α S ω dω -α uiu j T.K. Datta Department Of Civil Engineering, IIT Delhi (5.28) Response Spectrum Method Of Analysis 3/11 Contd… ui zkj zki zlj 1 = σui σ zkj 1 = σ zki σ zlj α * h j Suiuk ω dω (5.29) -α α * h h i j Sukul ω dω (5.30) -α Suiuk coh i,k 1 21 21 = 2 Sui Suk coh i,k = Sug 2 ω ω (5.31) Suiuj coh i, j 1 21 21 = 4 Sui Suj coh i, j = Sug 4 ω ω (5.32) 1 2 uk 1 2 ul Sukul = S S coh k,l = coh k,l Sug T.K. Datta Department Of Civil Engineering, IIT Delhi (5.33) Response Spectrum Method Of Analysis 3/12 Contd… For a single train of seismic wave, Dij = Di ω j ,ξ j that is displacement response spectrum for a specified ξ ; correlation matrices can be obtained if coh(i, j ) is additionally provided; Su g can be determined from D ω j ,ξ j (Eqn 5.6). If only relative peak displacement is required,third term of Eqn.5.26 is only retained. Steps for developing the program in MATLAB is given in the book. Example 5.4 Example 3.8 is solved for El centro earthquake spectrum with time lag of 5s. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/13 Contd… Solution :The quantities required for calculating the expected value are given below: 1 1 1 1 1 1 1 1 φ ; φ 0.5 1 ; r 3 1 1 1 , 0.5 1 w1 12.24 rad/s ; w2 24.48 rad/s 1111 11 21 11 31 12 12 12 22 12 32 1 1 1 1 T a ; 3 1 1 1 2111 21 21 21 31 22 12 22 22 22 32 0.0259 0.0259 0.0259 -0.0015 -0.0015 -0.0015 T D 0.0129 0.0129 0.0129 0.0015 0.0015 0.0015 D11 D21 D31 D (1 12.24) 0.056m T D12 D22 D32 D (2 24.48) 0.011m 0 coh i, j 1 2 1 0 1 2 5 10 1 ; 1 exp ; exp 2 2 2 0 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/14 Contd… uu uz zz 1 0.873 0.765 0.0382 0.0063 0.0027 1 0.0008 0.0001 0.0142 0.0007 0.0001 0.873 1 0.873 0.765 0.873 1 0.0061 0.0027 0.0443 0.0062 0.0387 0.0063 0.0068 0.0447 0.0063 0.0387 0.0029 0.0068 0.0008 0.0001 0.0142 0.0007 1 0.0008 0.0007 0.0142 0.0008 1 0.0001 0.0007 0.0007 0.0007 0.0001 1 0.0142 0.0007 0.0007 1 0.0007 0.0142 0.0001 0.0007 T.K. Datta Department Of Civil Engineering, IIT Delhi 0.0029 0.0068 0.0447 0.0001 0.0007 0.0142 0.0001 0.0007 1 Response Spectrum Method Of Analysis 3/15 Contd… Mean peak values determined are: (u1 )tot 0.106m ; (u2 )tot 0.099m (u1 ) rel 0.045m ; (u2 ) rel 0.022m For perfectly correlated ground motion 1 0 uu 0 1 1 1 zz 0 0 0 0 1 0 1 1 1 0 0 0 0 0 uz null matrix 1 1 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 3/16 Contd… Mean peak values of relative displacement RSA RHA u1 =0.078m ; 0.081m u2 = 0.039m ; 0.041m It is seen that’s the results of RHA & RSA match well. Another example (example 3.10) is solved for a time lag of a 2.5 sec. Solution is obtained in the same way and results are given in the book. The calculation steps are self evident. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 4/1 Cascaded analysis Cascaded analysis is popular for seismic analysis of secondary systems (Fig 5.5). Secondary System k .. xf m Fig 5.5 .. xg Secondary system mounted on a floor of a building frame c .. .. .. xa = xf + xg SDOF is to be analyzed for obtaining floor response spectrum RSA cannot be directly used for the total system because of degrees of freedom become prohibitively large ; entire system becomes nonclasically damped. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 4/2 Contd… In the cascaded analysis two systems- primary and secondary are analyzed separately; output of the primary becomes the input for the secondary. In this context, floor response spectrum of the primary system is a popular concept for cascaded analysis. The absolute acceleration of the floor in the figure is xa Pseudo acceleration spectrum of an SDOF is obtained for xa ; this spectrum is used for RSA of secondary systems mounted on the floor. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 4/3 Contd… Using this spectrum, peak displacement of the secondary system with T=0.811s is 0.8635m. The time history analysis for the entire system (with C matrix for P-S system) is found as 0.9163m. T.K. Datta Department Of Civil Engineering, IIT Delhi Displacement (m) Example 5.6 For example 3.18, find the mean peak displacement of the oscillator for El Centro earthquake. for secondary system = 0.02 ; for the main system = 0.05 ;floor displacement spectrum shown in the Fig5.6 is used Solution 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 Frequency (rad/sec) Floor displacement response spectrum (Exmp. 5.6) Response Spectrum Method Of Analysis Approximate modal RSA For nonclassically damped system, RSA cannot be directly used. However, an approximate RSA can be performed. C matrix for the entire system can be obtained (using Rayleigh damping for individual systems & then combining them without coupling terms) C1 C 0 0 C2 matrix is obtained considering all d.o.f. & T C becomes non diagonal. Ignoring off diagonal terms, an approximate modal damping is derived & is used for RSA. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 4/4 4/5 Seismic coefficient method Seismic coefficient method uses also a set of equivalent lateral loads for seismic analysis of structures & is recommended in all seismic codes along with RSA & RHA. For obtaining the equivalent lateral loads, it uses some empirical formulae. The method consists of the following steps: • Using total weight of the structure, base shear is obtained by Vb W Ch (5.34) Ch is a period dependent seismic coefficient T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 4/6 Contd… • Base shear is distributed as a set of lateral forces along the height as Fi Vb f (hi ) (5.35) f (hi ) bears a resemblance with that for the fundamental mode. • Static analysis of the structure is carried out with the force Fi (i = 1,2...... n) . Different codes provide different recommendations for the values /expressions for Ch & f (hi ) . T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 4/7 Distribution of lateral forces can be written as S F = ρ × W × φ × a1 1 j j j1 g Fj Wj × φ j1 = ∑ Fj ΣWj × φ j1 Fj = Vb × Fj = Vb Fj = Vb Wj × φ j1 ΣWj × φ j1 Wj ×h j ( 5.37) (5.38) (5.39) ΣWj ×h j Wj ×h jk ΣWj ×h j (5.36) k T.K. Datta Department Of Civil Engineering, IIT Delhi (5.40) Response Spectrum Method Of Analysis 4/8 Computation of base shear is based on first mode. Following basis for the formula can be put forward. Sa i ×) λ V = ΣF =(ΣW × φ × bi ji j ji i g Sa i e Vb i = Wi g Vb ≤ Σ Vb i ≤Σ Sai Wie g S Vb = a1 × W g (5.41) (5.42) (5.43) i = 1to n T.K. Datta Department Of Civil Engineering, IIT Delhi (5.44) (5.45) Response Spectrum Method Of Analysis 5/1 Seismic code provisions All countries have their own seismic codes. For seismic analysis, codes prescribe all three methods i.e. RSA ,RHA & seismic coefficient method. Codes specify the following important factors for seismic analysis: • Approximate calculation of time period for seismic coefficient method. • Ch Vs T plot. • Effect of soil condition on T.K. Datta Department Of Civil Engineering, IIT Delhi S A or a g g & Ch Response Spectrum Method Of Analysis 5/2 Contd… • Seismicity of the region by specifying PGA. • Reduction factor for obtaining design forces to include ductility in the design. • Importance factor for structure. Provisions of a few codes regarding the first three are given here for comparison. The codes include: • • • • • IBC – 2000 NBCC – 1995 EURO CODE – 1995 NZS 4203 – 1992 IS 1893 – 2002 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/3 Contd… IBC – 2000 • Ch for class B site, 1.0 T1 0.4s Ch 0.4 T1 0.4s T 1 (5.46) A • for the same site, is given by g 0.4 7.5T n A 1.0 g 0.4 Tn 0 Tn 0.08s 0.08 Tn 0.4s T.K. Datta Department Of Civil Engineering, IIT Delhi (5.47) Tn 0.4s Response Spectrum Method Of Analysis 5/4 Contd… T may be computed by N 2 Wi ui T1 2 i 1N g Fi ui i 1 (5.48) Fi can have any reasonable distribution. Distribution of lateral forces over the height is given by Fi Vb W j h kj N k W h j j (5.49) j 1 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/5 Contd… k={1; 0.5 T +1.5 ; 2 for T 1 ≤ 0.5s ; 0.5 ≤ T1 ≤ 2.5s; T1 ≥ 2.5s (5.50) 1 Distribution of lateral force for nine story frame is shown in Fig5.8 by seismic coefficient method . W 2 9 W 8 W 7 Storey W W 9@3m W 6 5 W 4 W 3 W 2 10 T=2sec T=1sec T=0.4sec 2 Storey force 4 Fig5.8 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/6 Contd… NBCC – 1995 CeU ; Ce = USIF (5.51a) ;( 5.51b) • C is given by Ch = R h A • For U=0.4 ; I=F=1, variations of S & with T g are given in Fig 5.9. Seismic response factor S 4.5 4 3.5 3 2.5 2 1.5 10 0.5 Time period (sec) 1 1.5 Fig5.9 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/7 Contd… • For PGV = 0.4ms-1 , A g 1.2 A = 0.512 g Tn is given by 0.03 ≤ Tn ≤ 0.427s Tn > 0.427s (5.52) • T may be obtained by Fu i 1 T1 = 2π N g1 Fu i i N 2 i 1 2 (5.53) • S and A/g Vs T are compared in Fig 5.10 for v = 0.4ms-1 , I = F = 1; zh = z v (acceleration and velocity related zone) T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/8 Contd… 1.4 A/g S 1.2 S or A/g 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Time period (sec) Fig5.10 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/9 Contd… • Distribution of lateral forces is given by Fi = Vb - Ft Wh i i N Wh i=1 0 Ft = 0.07T1Vb 0.25V b i (5.54) i T1 ≤ 0.7 s 0.7 < T1 < 3.6 s T1 ≥ 3.6 s T.K. Datta Department Of Civil Engineering, IIT Delhi (5.55) Response Spectrum Method Of Analysis 5/10 Contd… EURO CODE 8 – 1995 • Base shear coefficient Cs is given by C Cs = e • Ce is given by q A g 1 Ce = A Tc 3 g T 1 (5.56) 0 ≤ T1 ≤ Tc (5.57) T1 ≥ Tc • Pseudo acceleration in normalized form is given by Eqn 5.58 in which values of Tb,Tc,Td are Tb Tc Td hard med 0.1 0.15 0.4 0.6 3.0 3.0 soft 0.2 0.8 3.0 (A is multiplied by 0.9) T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 5/11 Contd… • Pseudo acceleration in normalized form , is given by Tn 1+1.5 T b 2.5 A = Tc ug0 2.5 T n Tc Td 2.5 2 T n 0 ≤ Tn ≤ Tb Tb ≤ Tn ≤ Tc Tc ≤ Tn ≤ Td T.K. Datta Department Of Civil Engineering, IIT Delhi (5.58) Tn ≥ Td Response Spectrum Method Of Analysis 5/12 Rayleigh's method may be used for calculating T. Distribution of lateral force is Fi = Vb Wφ i i1 Wφ i=1 Fi = Vb (5.59) N i i1 Wh i i (5.60) N Wh i=1 i i Variation of ce / u go & A / u go Fig 5.11. T.K. Datta Department Of Civil Engineering, IIT Delhi are shown in Response Spectrum Method Of Analysis 5/13 Contd… 3 .. A/ug0 .. 2.5 Ce/ug0 .. Ce /ug0 or A ../ug0 2 .. 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Time period (sec) Fig 5.11 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 6/1 Contd… NEW ZEALAND CODE ( NZ 4203: 1992) • Seismic coefficient & design response curves are the same. • For serviceability limit, C T = Cb T1,1 RzL s = Cb 0.4,1 RzL s T1 ≥ 0.45 T1 ≤ 0.45 (5.61a) (5.61b) Ls is a limit factor. • For acceleration spectrum, T.K. Datta Department Of Civil Engineering, IIT Delhi T1 is replaced by T. Response Spectrum Method Of Analysis 6/2 Contd… • Lateral load is multiplied by 0.92. • Fig5.12 shows the plot of cb vs T for 1 • Distribution of forces is the same as Eq.5.60 • Time period may be calculated by using Rayleigh’s method. • Categories 1,2,3 denote soft, medium and hard. • R in Eq 5.61 is risk factor; Z is the zone factor; ls is the limit state factor. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 6/3 Contd… 1.2 Category 1 Category 2 1 Category 3 0.8 Cb 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Time period (sec) Fig5.12 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis Contd… 6/4 IS CODE (1893-2002) • Time period is calculated by empirical formula and distribution of force is given by: Fj = Vb 2 Wh j j N 2 Wh j j (5.65) j=1 • Ce vs T & given by: Sa vs T are the same; they are g 1+15T 0 ≤ T ≤ 0.1s Sa = 2.5 0.1≤ T ≤ 0.4s g 1 0.4 ≤ T ≤ 4.0s T T.K. Datta Department Of Civil Engineering, IIT Delhi for hard soil (5.62) Response Spectrum Method Of Analysis 6/5 Contd… 1+15T Sa = 2.5 g 1.36 T 1+15T Sa = 2.5 g 1.67 T 0 ≤ T ≤ 0.1s 0.1≤ T ≤ 0.55s for medium soil (5.63) for soft soil (5.64) 0.55 ≤ T ≤ 4.0s 0 ≤ T ≤ 0.1s 0.1≤ T ≤ 0.67s 0.67 ≤ T ≤ 4.0s For the three types of soil Sa/g are shown in Fig 5.13 Sesmic zone coefficients decide about the PGA values. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 6/6 Contd… 3 Spectral acceleration coefficient (Sa/g) Hard Soil 2.5 Medium Soil Soft Soil 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Time period (sec) Variations of (Sa/g) with time period T Fig 5.13 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 6/7 Contd… Example 5.7: Seven storey frame shown in Fig 5.14 is analyzed with Concrete density = 24kNm-3 ; E = 2.5×107 kNm-2 Live load = 1.4kNm-1 For mass: 25% for the top three & rest 50% of live load are considered. T1 = 0.753s ; T2 = 0.229s ; T3 = 0.111s R = 3; PGA = 0.4g ; for NBCC, PGA ≈ 0.65g Solution: First period of the structure falls in the falling region of the response spectrum curve. In this region, spectral ordinates are different for different codes. T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 6/8 Contd… All beams:-23cm 50cm Columns(1,2,3):-55cm 55cm 7@3m Columns(4-7):-:-45cm 45cm 5m 5m 5m A Seven storey-building frame for analysis Fig 5.14 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis 6/9 Contd… Table 5.3: Comparison of results obtained by different codes 1st Storey Displacement (mm) Base shear (KN) Top Storey Displacement (mm) Codes SRSS CQC SRSS CQC SRSS 3 all 3 all 3 all 3 all IBC 33.51 33.66 33.52 33.68 0.74 0.74 0.74 0.74 NBCC 35.46 35.66 35.46 35.68 0.78 0.78 0.78 NZ 4203 37.18 37.26 37.2 37.29 0.83 0.83 Euro 8 48.34 48.41 48.35 48.42 1.09 Indian 44.19 44.28 44.21 44.29 0.99 T.K. Datta Department Of Civil Engineering, IIT Delhi 3 all CQC 3 all 10.64 10.64 10.64 10.64 0.78 11.35 11.35 11.35 11.35 0.83 0.83 12.00 12.00 12.00 12.00 1.09 1.09 1.09 15.94 15.94 15.94 15.94 0.99 0.99 0.99 14.45 14.45 14.45 14.45 Response Spectrum Method Of Analysis 6/10 Contd… 7 Number of storey 6 5 4 3 2 1 0 2 4 6 8 10 Displacement (mm) 12 IBC NBCC NZ 4203 Euro 8 Indian 14 16 Comparison of displacements obtained by different codes Fig 5.15 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis Lec-1/74 T.K. Datta Department Of Civil Engineering, IIT Delhi Response Spectrum Method Of Analysis