22. SEISMIC ANALYSIS USING DISPLACEMENT LOADING. Direct use of Earthquake Ground Displacement in a Dynamic Analysis has Inherent Numerical Errors

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1 22. SEISIC ANALYSIS USING DISPLACEENT LOADING Direct e of Earthqake Grond Diplacement in a Dynamic Analyi ha Inherent Nmerical Error 22.1 INTRODUCTION { XE "Diplacement Seimic Loading" }ot eimic trctral analye are aed on the relative-diplacement formlation where the ae acceleration are ed a the aic loading. Hence, experience with the direct e of aolte earthqake diplacement loading acting at the ae of the trctre ha een limited. Several new type of nmerical error aociated with the e of aolte eimic diplacement loading are identified. Thoe error are inherent in all method of dynamic analyi and are directly aociated with the application of diplacement loading. { XE "lti-spport Earthqake otion" }It i poile for the majority of eimic analye of trctre to e the grond acceleration a the aic inpt, and the trctral diplacement prodced are relative to the aolte grond diplacement. In the cae of mlti-pport inpt motion, it i neceary to formlate the prolem in term of the aolte grond motion at the different pport. However, the earthqake engineering profeion ha not etalihed analyi gideline to minimize the error aociated with that type of analyi. In thi chapter, it will e hown that everal new type of nmerical error can e eaily introdced if aolte diplacement are ed a the aic loading.

2 22-2 STATIC AND DYNAIC ANALYSIS { XE "Bridge Analyi" }A typical long-pan ridge trctre i hown in Figre Different motion may exit at pier ecae of local ite condition or the time delay in the horizontal propagation of the earthqake motion in the rock. Therefore, everal hndred different diplacement record may e neceary to define the aic loading on the trctre. SOFT ROCK or SOIL HARD ROCK Figre 22.1 Long Bridge Strctre With lti-spport Inpt Diplacement The engineer/analyt mt e aware that diplacement loading i ignificantly different from acceleration loading with repect to the following poile error: 1. The acceleration are linear fnction within a time increment and an exact oltion i normally ed to olve the eqilirim eqation. On the other hand, diplacement derived from a linear acceleration fnction are a cic fnction within each increment; therefore, a maller time increment i reqired, or a higher order oltion method mt e ed. 2. { XE "Relative Diplacement" }The patial ditrition of the load in the relative diplacement formlation i directly proportional to the ma; and the 90 percent modal ma-participation rle can e ed to enre that the relt are accrate. In the cae of ae diplacement inpt, however, the modal ma-participation factor cannot e ed to etimate poile error. For aolte diplacement loading, concentrated force are applied at the joint near the fixed ae of the trctre; therefore, a large nmer of highfreqency mode are excited. Hence, alternative error etimation mt e introdced and a very large nmer of mode may e reqired.

3 DISPLACEENT LOADING If the ame damping i ed for acceleration and diplacement analye, different relt are otained. Thi i ecae, for the ame damping ratio, the effective damping aociated with the higher freqency repone i larger when diplacement inpt i pecified (ee Tale 19.1). Alo, if ma proportional damping i ed, additional damping i introdced ecae of the rigid ody motion of the trctre. The dynamic eqilirim eqation for aolte eimic diplacement type of loading are derived. The different type of error that are commonly introdced are illtrated y an analyi of a imple hear-wall trctre EQUILIBRIU EQUATIONS FOR DISPLACEENT INPUT For a lmped-ma ytem, the dynamic eqilirim eqation in term of the nknown joint diplacement within the pertrctre and the pecified aolte diplacement at the ae joint can e written a: 0 0 && && C + C C C & & K + K K K 0 = R (22.1) The ma, damping and tiffne matrice aociated with thoe diplacement are pecified y ij, Cij, and K ij. Note that the force R aociated with the pecified diplacement are nknown and can e calclated after ha een evalated. Therefore, from Eqation (22.1) the eqilirim eqation for the pertrctre only, with pecified aolte diplacement at the ae joint, can e written a: & + C & + K = K C & (22.2) The damping load C & can e nmerically evalated if the damping matrix i pecified. However, the damping matrix i normally not defined. Therefore, thoe damping force are normally neglected and the aolte eqilirim eqation are written in the following form:

4 22-4 STATIC AND DYNAIC ANALYSIS && + C & + K = K = J j= 1 f ( t) j j (22.3) Each independent diplacement record j (t) i aociated with the pace fnction fj that i the negative vale of the j th colmn in the tiffne matrix. The total nmer of diplacement record i J, each aociated with a pecific diplacement degree of freedom. { XE "Cic Diplacement Fnction" }For the pecial cae of a rigid-ae trctre, a grop of joint at the ae are jected to the following three component of diplacement, velocitie and acceleration. K x ( t) = y ( t), ( t) z & x ( t) & = & y ( t), and & ( t) z && x ( t) & = && y ( t) (22.4) && ( t) z The exact relationhip etween diplacement, velocitie and acceleration i preented in Appendix J. The following change of variale i now poile: + I = r xyz, r xyz & = & + I &, and & = && r + I xyz& (22.5) The matrix I xyz = [ IxI yiz ] and ha three colmn. The firt colmn ha nit vale aociated with the x diplacement, the econd colmn ha nit vale aociated with the y diplacement, and the third colmn ha nit vale aociated with the z diplacement. Therefore, the new diplacement r are relative to the pecified aolte ae diplacement. Eqation (22.2) can now e rewritten in term of the relative diplacement and the pecified ae diplacement: && r + C & r + K I && xyz r = [ C I + C ] & [ K I + K xyz xyz ] (22.6) The force [ K I xyz + K ] aociated with the rigid ody diplacement of the trctre are zero. Becae the phyical damping matrix i almot impoile to

5 DISPLACEENT LOADING 22-5 define, the damping force on the right-hand ide of the eqation are normally neglected. Hence, the three-dimenional dynamic eqilirim eqation, in term of relative diplacement, are normally written in the following approximate form: && r + C & r + K r = = x I xyz x && I && ( t) I && ( t) I && ( t) y y z z (22.7) Note that the patial ditrition of the loading in the relative formlation i proportional to the directional mae. { XE "Higher ode Damping" }It mt e noted that in the aolte diplacement formlation, the tiffne matrix K only ha term aociated with the joint adjacent to the ae node where the diplacement are applied. Therefore, the only load, f j, acting on the trctre are point load acting at a limited nmer of joint. Thi type of patial ditrition of point load excite the high freqency mode of the ytem a the diplacement are propagated within the trctre. Hence, the phyical ehavior of the analyi model i very different if diplacement are applied rather than if the ma time the acceleration i ed a the loading. Therefore, the compter program er mt ndertand that oth approache are approximate for non-zero damping. If the complete damping matrix i pecified and the damping term on the righthand ide of Eqation (22.2 and 22.6) are inclded, an exact oltion of oth the aolte and relative formlation will prodce identical oltion USE OF PSEUDO-STATIC DISPLACEENTS An alternate formlation, which i retricted to linear prolem, i poile for mlti pport diplacement loading that involve the e of pedo-tatic diplacement, which are defined a: p 1 = K K = T (22.8) The following change of variale i now introdced:

6 22-6 STATIC AND DYNAIC ANALYSIS = + p = + T, & = & + T& and && = && + T& (22.9) The tittion of Eqation (9) into Eqation (2) yield the following et of eqilirim eqation: && + C & + K = K C & C T&& T& K T (22.10) Hence Eqation (22.10) can e written in the following implified form: & + C & + K = T&& [ C + C T] & (22.11) Eqation (22.11) i exact if the damping term are inclded on the right-hand ide of the eqation. However, thee damping term are normally not defined and are neglected. Hence, different relt will e otained from thi formlation when compared to the aolte diplacement formlation. The pedo-tatic diplacement cannot e extended to nonlinear prolem; therefore, it cannot e conidered a general method that can e ed for all trctral ytem SOLUTION OF DYNAIC EQUILIBRIU EQUATIONS The aolte diplacement formlation, Eqation (22.3), and the relative formlation, Eqation (22.7), can e written in the following generic form: J & ( t) + C& ( t) + K( t) = f g ( t) (22.12) j= 1 j j any different method can e ed to olve the dynamic eqilirim eqation formlated in term of aolte or relative diplacement. The direct incremental nmerical integration can e ed to olve thee eqation. However, ecae of taility prolem, large damping i often introdced in the higher mode, and only an approximate oltion that i a fnction of the ize of the time tep ed i otained. The freqency domain oltion ing the Fat-Forier-Tranform, FFT, method alo prodce an approximate oltion. Therefore, the error identified in thi paper exit for all method of dynamic repone analyi. Only the mode perpoition method, for oth linear acceleration and cic

7 DISPLACEENT LOADING 22-7 diplacement load, can e ed to prodce an exact oltion. Thi approach i preented in Chapter NUERICAL EXAPLE Example Strctre The prolem aociated with the e of aolte diplacement a direct inpt to a dynamic analyi prolem can e illtrated y the nmerical example hown in Figre z 20@15 =300 Propertie: Thickne = 2.0 ft Width =20.0 ft I = 27,648,000 in 4 E = 4,000 ki W = 20 kip /tory x = 20/g = kip-ec 2 /in yy = kip-ec 2 -in Total a = 400 /g Typical Story Height h = 15 ft = 180 in. Typical Story Load && (t) x A. 20 Story Shear Wall With Story a && (t) B. Bae Acceleration Load Relative Formlation Firt Story Load 12EI h ( ) t 3 Firt Story oment 6EI ( t) 2 h C. Diplacement Load Aolte Formlation Figre 22.2 Comparion of Relative and Aolte Diplacement Seimic Analyi Neglecting hear and axial deformation, the model of the trctre ha forty diplacement degree of freedom, one tranlation and one rotation at each joint. The rotational mae at the node have een inclded; therefore, forty mode of viration exit. Note that load aociated with the pecification of the aolte ae diplacement are concentrated force at the joint near the ae of the trctre. The exact period of viration for thee imple cantilever trctre are mmarized in Tale 22.1 in addition to the ma, tatic and dynamic load-

8 22-8 STATIC AND DYNAIC ANALYSIS participation factor. The derivation of ma-participation factor, taticparticipation factor, and dynamic-participation factor are given in Chapter 13. Tale 22.1 Period and Participation Factor for Exact Eigenvector ode Nmer Period (Second) Cmlative Sm of Cmlative Sm of a Participation Load Participation Factor Factor Bae Diplacement Loading X-Direction (Percentage) (Percentage) Static Dynamic

9 DISPLACEENT LOADING 22-9 It i important to note that only for mode are reqired to captre over 90 percent of the ma in the x-direction. However, for diplacement loading, 21 eigenvector are reqired to captre the tatic repone of the trctre and the kinetic energy nder rigid-ody motion. Note that the period of the 21 th mode i econd, or approximately 50 cycle per econd. However, thi high freqency repone i eential o that the aolte ae diplacement i accrately propagated into the trctre Earthqake Loading The acceleration, velocity and diplacement ae motion aociated with an idealized near-field earthqake are hown in Figre The motion have een elected to e imple and realitic o that thi prolem can e eaily olved ing different dynamic analyi program. &&( t) g 0.50 g 0.50 g ACCELERATION Time 1.00 g &( t) g 0.1 Sec. VELOCITY in./ec ( t) g DISPLACEENT 3.22 inche Figre 22.3 Idealized Near-Field Earthqake otion Effect of Time Step Size for Zero Damping { XE "Time Step Size" }To illtrate the ignificant difference etween acceleration and diplacement loading, thi prolem will e olved ing all forty eigenvector, zero damping and three different integration time tep. The

10 22-10 STATIC AND DYNAIC ANALYSIS aolte top diplacement, ae hear and moment at the econd level are mmarized in Tale In addition, the maximm inpt energy and kinetic energy in the model are mmarized. Tale 22.2 Comparion of Acceleration and Diplacement Load (40 Eigenvale 0.0 Damping Ratio) 20 (Inche) V 2 (Kip) 2 (K - In.) ENERGY (Inpt To odel) K-ENERGY (Within odel) Linear Acceleration Load Linear Diplacement Load t = 0.01 t = t = t = t = t = For linear acceleration load, all relt are exact regardle of the ize of the time tep ecae the integration algorithm i aed on the exact oltion for a linear fnction. The minor difference in relt i ecae ome maximm vale occr within the larger time tep relt. However, ing the ame linear integration algorithm for diplacement load prodce error ecae diplacement are cic fnction within each time tep (Appendix J). Therefore, the larger the time tep, the larger the error. For linear diplacement load, the maximm diplacement at the top of the trctre and the moment t at the econd level appear to e inenitive to the ize of the time tep. However, the force near the top of the trctre and the hear at the econd level can have ignificant error ecae of large integration time tep. For a time tep of 0.01 econd, the maximm hear of kip occr at econd; wherea, the exact vale for the ame time tep i kip

11 DISPLACEENT LOADING and occr at econd. A time-hitory plot of oth hear force i how in Figre Linear Acceleration Load, or Cic Diplacement Load - Zero Damping - 40 ode Linear Diplacement Load - Zero Damping - 40 ode SHEAR - Kip TIE - Second Figre 22.4 Shear at Second Level V. Time With t = Second and Zero Damping The error relting from the e of large time tep are not large in thi example ecae the loading i a imple fnction that doe not contain high freqencie. However, the athor ha had experience with other trctre, ing real earthqake diplacement loading, where the error are over 100 percent ing a time tep of 0.01 econd. The error aociated with the e of large time tep in a mode perpoition analyi can e eliminated for linear elatic trctre ing the new exact integration algorithm preented in Chapter 13. An examination of the inpt and kinetic energy clearly indicate that there i a major mathematical difference etween acceleration load (relative diplacement formlation) and diplacement load (aolte diplacement formlation). In the relative diplacement formlation, a relatively mall amont

12 22-12 STATIC AND DYNAIC ANALYSIS of energy, 340 k-in, i pplied to the mathematical model; wherea the point load aociated with the aolte formlation applied near the ae of the trctre impart over 1,000,000 k-in of energy to the model. Alo, the maximm kinetic energy (proportional to the m of ma time velocity qared) within the model i 340 k-in for the relative formlation compared to 164 kip-in for the aolte formlation. The relt clearly indicate that error are introdced if large time tep are ed with the linear diplacement approximation within each time tep. The patial load ditrition i ignificantly different etween the relative and diplacement formlation. For linear acceleration load, large time tep can e ed. However, very mall time tep, econd, are reqired for aolte diplacement loading to otain accrate relt. However, if modal perpoition i ed, the new cic diplacement load approximation prodce relt identical to thoe otained ing linear acceleration load for zero damping Earthqake Analyi with Finite Damping It i very important to ndertand that the relt prodced from a mathematical compter model may e ignificantly different from the ehavior of the real phyical trctre. The ehavior of a real trctre will atify the aic law of phyic, wherea the compter model will atify the law of mathematic after certain amption have een made. The introdction of claical linear vico damping will illtrate thi prolem. Tale 22.3 mmarize elective relt of an analyi of the trctre hown in Figre 22.2 for oth zero and five percent damping for all freqencie. The time tep ed for thi tdy i econd; hence, for linear acceleration load and cic diplacement load, exact relt (within three ignificant figre) are prodced. The relt clearly indicate that 5 percent damping prodce different relt for acceleration and diplacement loading. The top diplacement and the moment near the ae are very cloe. However, the hear at the econd level and the moment at the tenth level are ignificantly different. The hear at the econd level v. time for diplacement loading are plotted in Figre 22.5 for 5 percent damping.

13 DISPLACEENT LOADING Tale Comparion of Acceleration and Diplacement Load for Different Damping (40 Eigenvale, Second Time Step) Linear Acceleration Load Cic Diplacement Load ξ = 0.00 ξ = ξ = ξ = (Inch) V 2 (Kip) (K-in.) Linear Acceleration Load - 5 Percent Damping - 40 ode Cic Diplacement Load - 5 Percent Damping - 40 ode 60 SHEAR - Kip TIE-Second Figre 22.5 Shear at Second Level V. Time De To Cic Diplacement Loading. (40 Eigenvale t = Second)

14 22-14 STATIC AND DYNAIC ANALYSIS The relt hown in Figre 22.5 are phyically impoile for a real trctre ecae the addition of 5 percent damping to an ndamped trctre hold not increae the maximm hear from kip to kip. The reaon for thi violation of the fndamental law of phyic i the invalid amption of an orthogonal damping matrix reqired to prodce claical damping. Claical damping alway ha a ma-proportional damping component, a phyically illtrated in Figre 22.6, which cae external velocity-dependent force to act on the trctre. For the relative diplacement formlation, the force are proportional to the relative velocitie. Wherea for the cae of the application of ae diplacement, the external force i proportional to the aolte velocity. Hence, for a rigid trctre, large external damping force can e developed ecae of rigid ody diplacement at the ae of the trctre. Thi i the reaon that the hear force increae a the damping i increaed, a hown in Figre For the cae of a very flexile (or ae iolated) trctre, the relative diplacement formlation will prodce large error in the hear force ecae the external force at a level will e carried direct y the dah-pot at that level. Therefore, neither formlation i phyically correct. r m & & x C I x & x = 0 C I x & x 0 = + x r x. RELATIVE DISPLACEENT FORULATION ABSOLUTE DISPLACEENT FORULATION Figre 22.6 Example to Illtrate a-proportional Component in Claical Damping.

15 DISPLACEENT LOADING Thee inconitent damping amption are inherent in all method of linear and nonlinear dynamic analyi that e claical damping or ma-proportional damping. For mot application, thi damping-indced error may e mall; however, the engineer/analyt ha the reponiility to evalate, ing imple linear model, the magnitde of thoe error for each different trctre and earthqake loading The Effect of ode Trncation { XE "ode Trncation" }The mot important difference etween the e of relative and aolte diplacement formlation i that higher freqencie are excited y ae diplacement loading. Solving the ame trctre ing a different nmer of mode can identify thi error. If zero damping i ed, the eqation of motion can e evalated exactly for oth relative and aolte diplacement formlation and the error aociated with mode-trncation only can e iolated. Selective diplacement and memer force for oth formlation are mmarized in Tale Tale 22.4 ode-trncation Relt - Exact Integration for Second Time Step Zero Damping Nmer of ode Linear Acceleration Load Cic Diplacement Load 20 V V ,400 81, ,580-1,441, , ,500 81, , , , ,500 81, ,180-4,576,000 78, ,500 81, , , , ,35 149,500 81, , , , ,500 81, ,500 81,720 The relt hown in Tale 22.4 clearly indicate that only a few mode are reqired to otain a converged oltion ing the relative diplacement formlation. However, the relt ing the aolte diplacement formlation are almot nelievale. The reaon for thi i that the comptational model and

16 22-16 STATIC AND DYNAIC ANALYSIS the real trctre are reqired to propagate the high freqencie excited y the ae diplacement loading into the trctre. The diplacement at the top of the trctre, which i dominated y the firt mode, i inenitive to the high freqency wave propagation effect. However, the hear and moment force within the trctre will have ignificant error if all the freqencie are not preent in the analyi. Tale 22.5 mmarize elective diplacement and memer force for oth formlation for 5 percent damping. Tale 22.5 ode-trncation Error - Exact Integration for Second Time Step 5 % Damping Nmer of ode Linear Acceleration Load Cic Diplacement Load 20 V V ,300 77, ,153 1,439, , ,300-64, , , , ,300-64, ,170-4,573,000 77, , , , , , ,800 80,480 The relt hown in Tale 22.5 indicate that the addition of modal damping doe not ignificantly change the fndamental ehavior of the comptational model. It i apparent that a large nmer of high freqencie mt e inclded in the analyi if the comptational model i to accrately predict force in the real trctre. It i of coniderale interet, however, that mode trncation for thi prolem prodce erroneoly large force that are difficlt to interpret. To explain thoe error, it i neceary to examine the individal mode hape. For example, the 21t mode i a lateral diplacement at the econd level only, with all other mode diplacement near zero. Thi i a very important mode ecae a concentrated force aociated with the ae diplacement loading i applied at the econd level. Hence, the addition of that mode to the analyi increae the ending moment at the econd level to 4,573,000 and decreae the moment at the 10 th level to 77,650. Additional mode are then reqired to redce the internal force at the econd level.

17 DISPLACEENT LOADING USE OF LOAD DEPENDENT RITZ VECTORS In Tale 22.6 the relt of an analyi ing different nmer of Load Dependent Ritz vector i mmarized. In addition, ma, tatic and dynamic participation factor are preented. Tale 22.6 Relt Uing LDR Vector- t = Cic Diplacement Loading Damping = 5 % Nmer of Vector 20 V a, Static and Dynamic Load-Participation ,100 80, ,700 80, ,800 80, , ,800 80, The e of LDR vector virtally eliminate all prolem aociated with the e of the exact eigenvector. The reaon for thi improved accracy i that each et of LDR vector contain the tatic repone of the ytem. To illtrate thi, the fndamental propertie of a et of even LDR vector are mmarized in Tale Tale 22.7 Period and Participation Factor for LDR Vector Vector Nmer Approximate Period (Second) Cmlative Sm of Cmlative Sm of Load Participation Factor a Participation Bae Diplacement Loading Factor (Percentage) X-Direction (Percentage) Static Dynamic

18 22-18 STATIC AND DYNAIC ANALYSIS The firt ix LDR vector are almot identical to the exact eigenvector mmarized in Tale However, the eventh vector, which i a linear comination of the remaining eigenvector, contain the high freqency repone of the ytem. The period aociated with thi vector i over 400 cycle per econd; however, it i the mot important vector in the analyi of a trctre jected to ae diplacement loading SOLUTION USING STEP-BY-STEP INTEGRATION { XE "Step By Step Integration" }The ame prolem i olved ing direct integration y the trapezoidal rle, which ha no nmerical damping and theoretically conerve energy. However, to olve the trctre with zero damping, a very mall time tep wold e reqired. It i almot impoile to pecify contant modal damping ing direct integration method. A tandard method to add energy diipation to a direct integration method i to add Rayleigh damping, in which only damping ratio can e pecified at two freqencie. For thi example 5 percent damping can e pecified for the lowet freqency and at 30 cycle per econd. Selective relt are mmarized in Tale 22.8 for oth acceleration and diplacement loading. Tale 22.8 Comparion of Relt Uing Contant odal Damping and the Trapezoidal Rle and Rayleigh Damping (0.005 Second Time Step) Trapezoidal Rle Uing Rayleigh Damping Acceleration Loading Exact Soltion Uing Contant, odal Damping ξ = Trapezoidal Rle Uing Rayleigh Damping Diplacement Loading Exact Soltion Uing Contant, odal Damping ξ = (Inch) V 2 (Kip) (k in.) (K-in.) ,210@

19 DISPLACEENT LOADING It i apparent that the e of Rayleigh damping for acceleration loading prodce a very good approximation of the exact oltion ing contant modal damping. However, for diplacement loading, the e of Rayleigh damping, in which the high freqencie are highly damped and ome lower freqencie are nder damped, prodce larger error. A plot of the hear at the econd level ing the different method i hown in Figre It i not clear if the error are caed y the Rayleigh damping approximation or y the e of a large time tep. It i apparent that error aociated with the nrealitic damping of the high freqencie excited y diplacement loading are preent in all tep-y-tep integration method. It i a property of the mathematical model and i not aociated with the method of oltion of the eqilirim eqation Exact Cic Diplacement Soltion - 5 % odal Damping Step By Step Soltion - Rayleigh Damping SHEAR - Kip TIE - Second Figre 22.7 Comparion of Step-By-Step Soltion Uing the Trapezoidal Rle and Rayleigh Damping with Exact Soltion (0.005 econd time-tep and 5% damping) The effective damping in the high freqencie, ing diplacement loading and Rayleigh damping, can e o large that the e of large nmerical integration time tep prodce almot the ame relt a ing mall time tep. However,

20 22-20 STATIC AND DYNAIC ANALYSIS the accracy of the relt cannot e jtified ing thi argment, ecae the form of the Rayleigh damping ed in the compter model i phyically impoile within a real trctre. In addition, the e of a nmerical integration method that prodce nmerical energy diipation in the higher mode may prodce nrealitic relt when compared to an exact oltion ing diplacement loading SUARY Several new orce of nmerical error aociated with the direct application of earthqake diplacement loading have een identified. Thoe prolem are mmarized a follow: 1. Diplacement loading i fndamentally different from acceleration loading ecae a larger nmer of mode are excited. Hence, a very mall time tep i reqired to define the diplacement record and to integrate the dynamic eqilirim eqation. A large time tep, ch a 0.01 econd, can cae ignificant npredictale error. 2. The effective damping aociated with diplacement loading i larger than that for acceleration loading. The e of ma proportional damping, inherent in Rayleigh and claical modal damping, cannot e phyically jtified. 3. Small error in maximm diplacement do not garantee mall error in memer force. 4. The 90 percent ma participation rle, which i ed to etimate error for acceleration loading, doe not apply to diplacement loading. A larger nmer of mode are reqired to accrately predict memer force for aolte diplacement loading. 5. For diplacement loading, mode trncation in the mode perpoition method may cae large error in the internal memer force. The following nmerical method can e ed to minimize thoe error:

21 DISPLACEENT LOADING A new integration algorithm aed on cic diplacement within each time tep allow the e of larger time tep. 2. To otain accrate relt, the tatic load-participation factor mt e very cloe to 100 percent. 3. The e of LDR vector will ignificantly redce the nmer of vector reqired to prodce accrate relt for diplacement loading. 4. The example prolem illtrate that the error can e ignificant if diplacement loading i applied aed on the ame rle ed for acceleration loading. However, additional tdie on different type of trctre, ch a ridge tower, mt e condcted. Alo, more reearch i reqired to eliminate or jtify the difference in relt prodced y the relative and aolte diplacement formlation for nonzero modal damping. Finally, the tate-of-the-art e of claical modal damping and Rayleigh damping contain ma proportional damping that i phyically impoile. Therefore, the development of a new mathematical energy diipation model i reqired if modern compter program are to e ed to accrately imlate the tre dynamic ehavior of real trctre jected to diplacement loading.

22 22-22 STATIC AND DYNAIC ANALYSIS