AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE SPECTRUM METHOD OF SEISMIC ANALYSIS

ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 11, NO. 3 (010) PAGES 31-334 AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE SPECTRUM METHOD OF SEISMIC ANALYSIS M. Dhleep a*, N.P. Shahul Hameed b and S. Nagan c a Department of Cvl Engneerng, SCMS School of Engneerng and Technology, Karukutty, Ernakulam-68358, Inda b Department of Cvl Engneerng, Al-Ameen Engneerng College, Shornur-6791, Inda c Department of Cvl Engneerng, Thagarajar College of Engneerng, Madura-65015, Inda ABSTRACT In response spectrum method of sesmc analyss of rregular and complex structures, the response contrbutons of modes up to rgd frequency are consdered for the analyss and the effect of truncated hgh frequency modes are taken nto account usng sutable mssng mass correcton methods. In ths paper, an alternate method usng a cut-off frequency less than rgd frequency for the truncaton of modes s proposed. The proposed method s valdated wth the help of numercal examples by evaluatng the response of structures for El Centro (1940) and Frul (1976) earthquake ground motons. Keywords: Mssng mass; rgd frequency; resdual mode; hgh frequency modes 1. INTRODUCTION Response Spectrum method s the most attractve and wdely used method for the sesmc analyss of structures. The accuracy of the response spectrum method s generally good provded that no sgnfcant modes have been truncated. The practcal dffcultes assocated wth the analyss of large real lfe structural models leads to the truncaton of hgh frequency modes. Generally, the propertes of the frst few lower modes are calculated for ther use n the analyss and the hgh frequency modes are gnored. In some cases, the error due to the truncaton of modes can be too large to be gnored, especally n the calculaton of response of stff and rregular structural systems. Moreover, some response quanttes, whch have a sgnfcant contrbuton from the hgh frequency modes n even not so stff structures, may also be senstve to ths mode truncaton error. In order to check the accuracy of the calculated response when hgh frequency modes are truncated n a modal superposton, typcal buldng codes specfy that the results can be consdered accurate f about 90% of the total structural mass partcpates n the number of modes consdered [1-3]. The nnety * E-mal address of the correspondng author: kavyad@bsnl.n (M. Dhleep)

3 M. Dhleep, N.P. Shahul Hameed and S. Nagan percent crtera even when satsfed may not always result n correct responses n all the elements of an rregular structure [4]. Therefore, for the sesmc analyss of rregular buldngs usng response spectrum method, all the modes up to rgd frequency has to be consdered and mssng mass correcton beyond rgd frequency [4, 5]. Technques have been developed to take the effect of the mssng mass contaned n the uncalculated hgh frequency modes nto account. A comparson between the varous mssng mass correcton methods shows that resdual mode method s superor to other methods [6-8]. The method approxmates the perodc part of the response along wth the rgd part of the response. Further the resdual mode can be ncluded as modal propertes of an addtonal mode n the dynamc analyss. The ablty of the resdual mode to approxmate the perodc part of the response s not addressed. The modal responses, havng a frequency less than rgd frequency also have a rgd content, whch gradually dmnshes and becomes zero as the frequency further goes down [5,9]. Ths paper makes an attempt to conduct sesmc analyss of structures by truncatng the modes at a key frequency havng frequency less than rgd frequency and applyng a correcton usng resdual mode to take the response contrbutons of truncated modes nto account.. MODE SUPERPOSITION METHOD The equatons of moton for an N degrees of freedom(dof), vscously damped system wth classcal dampng can be wrtten as MU CU KU MU (1) b u g where M, C and K are the mass, dampng and stffness matrces, respectvely; U s the dsplacement vector; U b s the statc dsplacement vector when the base of the structure undergoes a unt deflecton n the drecton of earthquake; and u g s the ground acceleraton. The dsplacement U of the system can be expressed as the superposton of modal contrbutons U, N 1 N U U X () where are determned from the general egen value problem K M, X = modal coordnates and s the frequency of the th mode. By usng Eq.(), (1) can be transformed to a system of uncoupled equatons n modal coordnates, 1 g X X X Γ u, =1, N (3) T where, MUb, s called the modal partcpaton factor and s the dampng rato for the th mode. In response spectrum method, the contrbuton of th mode to nodal

AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE... 33 dsplacements U s gven by U X Γ S (4) D S D s the maxmum relatve dsplacement of a sngle degree of freedom system where havng frequency and dampng. The modal response U can be dvded nto two parts: the rgd part U r and the damped perodc part U p. The modal response U can be expressed as [5, 9], U r p ( U ) ( U ) (5) 3. RIGID FREQUENCY AND HIGHER MODES OF VIBRATION The mnmum frequency beyond whch the curves for varous dampng ratos have same values of spectral acceleraton s defned as the rgd frequency [10, 11]. Ths defnton of rgd frequency ntroduces consstency wth the physcal behavor, but can be prone to ndvdual judgment dependng upon a partcular record and the senstvty of ts spectral acceleratons to the dampng rato n a regon near the rgd frequency. The prescrpton of rgd frequency as 33 Hz, as gven by varous sesmc buldng codes can also ntroduce naccuraces n evaluatng the structural response usng response spectrum method [10, 1]. The responses n all the hgh frequency modes havng frequency equal to or greater than rgd frequency are n phase to each other and the rgd response always combnes algebracally. In ths regon, the structural frequency s suffcently hgher than the domnant frequences of the nput force and the spectral acceleraton becomes equal to the peak ground acceleraton, often referred to as the zero perod acceleraton (ZPA). At hgh frequences, the perodc part of the response becomes neglgble and only the rgd part of response remans. Moreover the perod of a hgh frequency mode s very short, so the response n such a mode s essentally statc than dynamc. Consder the frst n modes of a N degrees of freedom system, havng frequences less than rgd frequency and let the response n these n modes be U, and the response n the remanng modes be U.Then, o n U U X ; o U 1 n 1 N U X (6) U o n1 For hgh frequency modes, Eqs. (3), (6) and (7) gves N n1 U U (7) where MU o CU o KU o MU bou g ; (8)

34 M. Dhleep, N.P. Shahul Hameed and S. Nagan n U U (9) bo b Snce the resdual response n hgh frequency modes are pseudo statc, we can neglect the terms U and U n Eq. (8), therefore 1 KU o MU bou g (10) The response of a hgh frequency mode s essentally statc and could be determned by statc analyss usng Eq. (10) nstead of dynamc analyss. 4. MODAL RESPONSE COMBINATION The peak value of the total response R s estmated by combnng the peak modal response of ndvdual modes usng modfed double sum equaton [9,11,1] s gven by, N 1 N R R ε j R R (11) 1 where R s the maxmum modal response n th mode, and j s the modfed correlaton factor defned as, j j j 1 j j j 1 (1) where s the rgd response coeffcent n the th mode and j s the correlaton coeffcent of the damped perodc part of modal responses, gven by the well known complete quadratc combnaton (CQC) rule. For damped perodc modes, α = 0, and modfed double sum equaton reduces to CQC and for j = 0, modfed double sum method reduces to square root of sum of squares (SRSS). Eqs. (11) and (1) nclude the effect of rgd response of hgh frequency modes n the modfed correlaton coeffcent j. The rgd response coeffcent α s defned as [9, 11], td x (t)u (t)dt g 0 (13) x ug t dσ σ where x (t) s the acceleraton response, x σ and are the standard devatons of (t) u σ g x

AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE... 35 and u g (t) respectvely and t d s the duraton of responses. The value of α gradually reduces from one to zero, from a key frequency f to another key frequency f 1 [5, 9, 11]. The key frequency f s the lowest frequency at whch the rgd response coeffcent becomes 1 and the key frequency f 1 s the hghest frequency at whch the rgd response coeffcent becomes zero. An approxmate equaton for α can be represented by a straght lne between the two key frequences f 1 and f on a sem logarthmc graph, s gven by [5, 9], ln f f 1, 0 1 ln f f1 (14) where f s the modal frequency n hertz and the key frequences f 1 and f can be expressed as, S f, Hz. (15) 1 A max S V max where f r ( f ) / 3, Hz. (16) 1 f S A max = maxmum spectral acceleraton, S V max = maxmum spectral velocty and r and f = rgd frequency 5. RESIDUAL MODE METHOD In ths method, the nerta effect of modes havng frequences greater than the rgd frequency s lumped nto a mssng mass term whch yelds the resdual response. Snce the resdual response n hgh frequency modes are pseudo statc, for rgd modes, we can neglect the terms U and U n Eq. (1), and therefore for hgh frequency modes, where KU o MU (17) n 1 bou g U U (18) bo b U o = response n hgh frequency modes and n = number of modes of a N degrees of freedom system havng frequences less than rgd frequency. The term u g n Eq. (17) s a scalar and can be scaled out from the equaton. The soluton of resultng equaton yelds a T vector o, whch s normalzed such that o M o 1. The fcttous frequency ω 0 correspondng to the resdual mode s gven by,

36 M. Dhleep, N.P. Shahul Hameed and S. Nagan T ω o K o o (19) The resdual modal vector o and the correspondng frequency o can be drectly ncluded as modal propertes of an addtonal mode n the dynamc analyss. The system can be analyzed for n + 1 modes, where the contrbutons of hgh frequency modes are taken nto account by the resdual mode. Resdual mode method takes both rgd and damped perodc part of the response correspondng to the resdual mode nto account and thus provdes an approxmate dynamc correcton. 6. PROPOSED METHODOLOGY The modal responses, havng frequency less than rgd frequency also have a rgd content. There s a transton from peak ground acceleraton to rgd spectral acceleraton as the frequency goes down from rgd frequency f r to the key frequency f. Further there s a transton from rgd spectral acceleraton to amplfed perodc spectral acceleraton from key frequency f to key frequency f 1. At key frequency f 1 the rgd content reduces to zero. In md-frequency regon between f 1 to f r the responses consst of two parts; the damped perodc part and the rgd part. The md-frequency regon can be further dvded nto two sub-regons; regon between, f 1 and f and the regon between f and f r. In the regon between f and f r of the spectrum, the modal responses move n phase wth the ground moton. The damped perodc porton of response n ths regon s neglgble and not consdered n the dervaton of rgd response coeffcent. The varaton of rgd response coeffcent n the md frequency regon for El Centro(1940) and Frul(1976) earthquake ground motons for 5% dampng are shown n Fgures 1 and respectvely. In the regon between f and f r, the perodc part of the response s neglgble. Fgure 1. Varaton of correlaton coeffcent for El Centro, 1940 earthquake

AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE... 37 Fgure. Varaton of correlaton coeffcent for Frul, Italy, 1976 earthquake In all mssng mass correcton methods other than resdual mode method, the damped perodc part of the response of the truncated modes s neglected. Resdual mode method tres to approxmate the damped perodc part of the response along wth the rgd part. The resdual mode method ncludes the dynamcs of the resdual mode and can be ncluded as an addtonal mode n the modal analyss. Therefore modal analyss can be conducted by truncatng the modes havng frequences hgher than the key frequency f, nstead of rgd frequency f r and applyng mssng mass correcton usng a resdual mode to take the effect of truncated modes nto account. Table 1 show that the value of f calculated usng Eq. (16) s less than the actual value of f. The value of f calculated usng the approxmate expresson (Eq. (16)) vares wth the actual value of f. The modal response combnaton rule usng modfed correlaton coeffcent s senstve to the value of f. The value f depends on the value of f r and f 1 (Eq. (16)). Furthermore the fxng up of rgd frequency can be prone to ndvdual judgment and can be lead to the use of dfferent rgd frequences by dfferent engneers. Table 1: Comparson of the value of f, Hz Sl. No. Earthquake f r, Hz f, Hz (Numercal method) f, Hz (Equaton (16)) 1 El Centro, 1940 33 7.59 Frul, Italy, 1976 15 1 10.3 The resdual mode approxmates both perodc and rgd part of the response n the zone between f and f r dependng on the modal mass contrbutons of the modes. The proposed

38 M. Dhleep, N.P. Shahul Hameed and S. Nagan method helps to reduce the error due to the approxmaton nvolved n the calculaton of f usng Eq. (16) and the use of dfferent f values by dfferent engneers. The method reduces the cut-off frequency from rgd frequency to key frequency f, havng frequency less than rgd frequency. The proposed method smplfes the sesmc analyss of structures havng sgnfcant contrbutons from hgher modes. The followng numercal examples wll confrm ths pont. 7. NUMERICAL EXAMPLES To evaluate the accuracy of the proposed method numercally, a response spectrum analyss usng El Centro (1940) earthquake ground moton s performed on the followng numercal examples. Numercal example 3 s further analyzed usng Frul (1976) earthquake ground moton. The key frequences f 1, f and rgd frequency f r for El Centro (1940) ground moton and Frul (1976) earthquake ground moton are shown n Table 1. Equatons (15) and (16) are used to evaluate the values of f 1, and f. The varaton of rgd response coeffcent for fve percentage dampng wth respect to frequency s shown n Fgures 1 and. For the analyss, hgher modes havng frequency more than key frequency f are truncated and mssng mass correcton s appled usng resdual mode method. The error s calculated by comparng the peak modal responses calculated usng the proposed method, wth the peak modal responses calculated usng modal analyss wth all modes. 7.1 Numercal example 1 Consder a two DOF system wth story stffness and the floor masses as shown n Fgure 3. The frequences, of ths system are 14.84Hz and 5.33Hz respectvely and the modal dampng s fve percentage. The mass partcpaton percentage for dfferent modes for ths system s gven n Table. El Centro (1940) ground moton s used for earthquake exctaton. The frequency of second mode s greater than key frequency f and less than the rgd frequency. Therefore, the response of second mode contans a rgd as well as a damped perodc part. For the analyss, second mode s truncated and mssng mass correcton s appled usng resdual mode method. The frequency correspondng to the resdual mode vector s 5.33Hz, same as the frequency of the second mode. The results of modal analyss wth all modes and resdual mode method are same (Table 3). The resdual mode method approxmates the damped perodc part along wth the rgd part of the response correspondng to the resdual mode. Fgure 3. -DOF system, numercal example 1

AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE... 39 Table : Frequences, dampng rato and mass partcpaton percentage for the DOF system shown n Fgure 3 Varable Includng all modes Truncated modes Mode 1 Mode Mode 1 Resdual mode Frequency (Hz) 14.84 5.33 14.84 5.33 Dampng rato 0.05 0.05 0.05 0.05 Mass Partcpaton % 4.0 58.0 4.0 58.0 Table 3: Sprng force (N) of DOF system, shown n Fgure 3 Analyss Element 1 Element Modal analyss (All modes) 11.58.84 Resdual mode method (cut off frequency f ) Error 11.58.84 7. Numercal example A 5 DOF model of a structure wth a relatvely stff base supportng a flexble tower shown n Fgure 4 s consdered for analyss usng the proposed method. The story stffness and the floor masses of the system are gven n Fgure 4. The frequences, modal dampng and the mass partcpaton percentage for dfferent modes for ths system are gven n Table 4. El Centro (1940) ground moton s used for earthquake exctaton. The thrd, fourth and ffth modes havng frequency hgher than key frequency f are truncated for the analyss and mssng mass correcton s appled usng resdual mode method. The frequency correspondng to the resdual mode vector s 3.9Hz. The results are compared wth the peak modal responses calculated usng modal analyss wth all modes. The error nvolved n the calculaton of sprng force usng each method s shown n Table 5. The error nvolved n the calculaton of response n the elements s less than 1 percentage. Fgure 4. 5-DOF system, numercal example

330 M. Dhleep, N.P. Shahul Hameed and S. Nagan Table 4: Frequences, dampng rato and mass partcpaton percentage for 5 DOF system, shown n Fgure 4 Includng all modes Mode number Frequency (Hz) Dampng rato Mass partcpaton % 1 6.17 0.05 63.3 17.61 0.05 13.7 3 5.89 0.05 16.01 4 30.69 0.05 6.89 5 35.39 0.05 0.48 Table 5: Sprng force (kn) of 5 DOF system shown n Fgure 4 Analyss Element 1 Element Element 3 Element 4 Element 5 Modal Analyss (All modes) 1146.89 887.06 73.9 55.36 75.19 Resdual Mode method (cut r off frequency f ) Error 1146.89 887.06 73.9 55.36 75.19 Resdual Mode method (cut off frequency f ) Error 1143.1 0.33 % 883.9 0.4 % 739.68 0.9% 54.7 0.1% 71.94 0.71% 7. Numercal example 3 A 5 DOF model of a structure wth a relatvely stff base supportng a flexble tower shown n Fgure 5 s consdered for analyss usng the proposed method. The story stffness and the floor masses of the system are gven n Fgure 5. The frequences, modal dampng and the mass partcpaton percentage for dfferent modes for ths system are gven n Table 6. El Centro (1940) ground moton s used for earthquake exctaton. The thrd, fourth and ffth modes havng frequences hgher than key frequency f are truncated for the analyss and mssng mass correcton s appled usng resdual mode method. The frequency correspondng to the resdual mode vector s 7.61Hz. The results are compared wth the peak modal responses calculated usng modal analyss wth all modes. The error nvolved n the calculaton of sprng force usng each method wth respect to modal analyss wth all modes s shown n Table 7. The error nvolved n the calculaton of response usng resdual mode method s neglgble.

AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE... 331 Fgure 5. 5-DOF system, numercal example 3 Table 6: Frequences, dampng rato and mass partcpaton percentage for the 5 DOF system shown n Fgure 5 Mode Number Includng all modes Frequency (Hz) Dampng rato Mass partcpaton % 1 5.71 0.05 40.55 16.44 0.05 6.1 3 5.1 0.05 4.51 4 30.95 0.05 8.00 5 33.3 0.05 40.73 Table 7: Sprng force (kn) of 5 DOF system, shown n Fgure 5 for El Centro (1940) Analyss Element 1 Element Element 3 Element 4 Element 5 Modal Analyss (All modes) 907.3 600.35 50.90 36.58 190.3 Resdual Mode method (cut r off frequency f ) Error 907.3 600.35 50.90 36.58 0.0% 190.3 0.0% Resdual Mode method (cut off frequency f ) Error 897.31 1.10 % 597.5 0.51 % 507.69 0.95 % 36.36 0.06 % 188.5 1.09%

33 M. Dhleep, N.P. Shahul Hameed and S. Nagan The 5 DOF system s further analyzed usng Frul (1976) ground moton. The second, thrd, fourth and ffth modes havng frequences hgher than key frequency f are truncated for the analyss and mssng mass correcton s appled usng resdual mode method. The results are compared wth the peak modal responses calculated usng modal analyss wth all modes. The error nvolved n the calculaton of sprng force usng each method wth respect to modal analyss wth all modes s shown n Table 8. The error nvolved n the calculaton of response usng resdual mode method s less than 10 percentage. Table 8: Sprng force (N) of 5 DOF system, shown n Fgure 5 for Frul (1976) Analyss Element 1 Element Element 3 Element 4 Element 5 Modal analyss (All modes) 98.39 54. 43.93 31.0 16.06 Resdual mode method (cut off frequency f r ) Error 98.35 0.04 % 51.6 5.45 % 44.90.0 % 33.4 7.15% 17.66 9.96% Resdual mode method (cut off frequency f ) Error 98.35 0.04 % 51.6 5.45 % 44.90.0 % 33.4 7.15% 17.66 9.96% The mass partcpatng n the truncated modes n numercal examples 1, and 3 are 58 percentage, 53.4 percentage and 3.38 percentage respectvely. In all the above examples, the modes up to.59hz s consdered for the analyss usng El Centro (1940) earthquake ground moton, nstead of 33 Hz and mssng mass correcton usng resdual mode s appled beyond 3Hz. Numercal example 3 s further analyzed for Frul (1976) earthquake ground moton. All the modes up to 10.3Hz are consdered for the analyss usng Frul (1976) earthquake ground moton, nstead of 15 Hz and mssng mass correcton usng resdual mode s appled beyond 10.3Hz. The examples show that the error nvolved n the calculaton of response usng the proposed method s less than 10 percentage. Therefore structures havng sgnfcant contrbuton from hgh frequency modes can be analyzed by truncatng modes above key frequency f and applyng mssng mass correcton usng resdual mode to take the contrbuton of the truncated modes nto account. The resdual response correspondng to the resdual mode can be combned wth the modal response of other modes usng the modal response combnaton rules. The proposed method smplfes the analyss procedure for the sesmc analyss of rregular buldng structures havng sgnfcant contrbutons from hgher modes. 8. CONCLUSIONS Response spectrum method of sesmc analyss gves accurate results when the modal responses are calculated for all the modes. It s mpractcal to calculate all the modes for systems wth large degrees of freedom. Therefore the response contrbutons of modes up to

AN ALTERNATE CUT-OFF FREQUENCY FOR THE RESPONSE... 333 rgd frequency s ncluded n the analyss and mssng mass correcton technques are appled to account for the response contrbutons of uncalculated hgh frequency modes. Ths paper proposes an alternate method for the sesmc analyss of buldng structures n whch the modes can be truncated at a cut off frequency f, less than the rgd frequency and applyng a correcton usng a resdual mode to take the response contrbutons of uncalculated hgher modes nto account. The resdual mode approxmates the damped perodc part also, along wth the rgd part of the response of the uncalculated modes. Ths method helps to nullfy the approxmaton nvolved n the calculaton of f. The resdual mode can be ncluded as an addtonal mode n the analyss and the correspondng response can be combned accordng to the modal response combnaton rules. The proposed method s smpler to mplement and can be used n the sesmc analyss of rregular buldng structures. REFERENCES 1. Internatonal Assocaton for Earthquake Engneerng. Regulatons for Sesmc Desgn - A World Lst, Compled By Internatonal Assocaton for Earthquake Engneerng, Tokyo, Japan, 1996.. Internatonal Assocaton for Earthquake Engneerng. Regulatons for Sesmc Desgn - A World Lst1996- Supplement, Compled By Internatonal Assocaton for Earthquake Engneerng, Tokyo, Japan, 000. 3. IS 1893 (Part 1): 00. Crtera for Earthquake Resstant Desgn of Structures, Part 1 General Provsons and Buldngs, Bureau of Indan Standards, New Delh, 00. 4. Dhleep M, Bose PR. Sesmc analyss of rregular buldngs: mssng mass effect, Journal of Structural Engneerng, No. 5, 35(008-009) 359-65. 5. USNRC. Combnng modal responses and spatal components n sesmc response analyss, Regulatory gude 1.9, Offce of nuclear regulatory research, US Nuclear Regulatory Commsson R, 006. 6. Dhleep M, Bose PR. A crtque of varous mssng mass correcton methods for sesmc analyss of structures, Proceedngs of the Tenth Asa Pacfc Conference on Structural Engneerng and Constructon, Bangkok, Vol. 3, 006, pp. 3-8. 7. Dhleep M, Bose PR. A Comparatve study of mssng mass correcton methods for response spectrum method of sesmc analyss, Computers and Structures, 86(008) 087-94. 8. Dckens JM, Nakagawa JM, Wttbrodt MJ. A crtque of mode acceleraton and modal truncaton augmentaton methods for modal response analyss, Computers and Structures, 6(1997) 985-98. 9. Gupta AK. Response Spectrum Method n Sesmc Analyss and Desgn of Structures, Blackwell Scentfc Publcatons, Boston, 199. 10. Dhleep M, Bose PR. Effect of hgh frequency modes n sesmc analyss of buldngs, Proceedngs of the Internatonal Conference on Cvl Engneerng n the New Mllennum: Opportuntes and Challenges, Howrah, 007, pp. 1-6. 11. Gupta AK, Hassan T, Gupta A. Correlaton coeffcents for modal response

334 M. Dhleep, N.P. Shahul Hameed and S. Nagan combnaton of non-classcally damped systems, Nuclear Engneerng and Desgn, 165(1996) 67-80. 1. Dhleep M, Bose PR. Combnaton of hgh frequency modes n sesmc analyss of rregular buldngs, Frst Internatonal Conference of European Asan Cvl Engneerng. Forum, Jakarta, Indonesa, 007, C37-C44. 13. www.peer.bekely.edu/ products/ strong-ground- moton-db.html. Accessed between 10/07/008 and 10/10/008