15. DYNAMIC ANALYSIS USING RESPONSE SPECTRUM SEISMIC LOADING

Transcription

1 From Static ad Dyamic of Structures by Ed Wilso 1 Modified July 13, DYNAMIC ANALYSIS USING RESPONSE SPECTRUM SEISMIC LOADING Before the Existece of Iexpesive Persoal Computers, the Respose Spectrum Method was the Stadard Approach for Liear Seismic Aalysis 15.1 INTRODUCTION The first versio of this chapter o the Respose Spectrum Method was writte over twety five years ago with the purpose of improvig the accuracy of this approximate liear method for the seismic loadig ad to provide guidelies for its practical uses. At that time, the author believed the method would be used for oly a few more years ad would soo be replaced by the more accurate, flexible ad simple time-history seismic respose aalysis method for both liear ad oliear aalysis of complex structures. The recet icrease i the speed ad capacity of persoal computers has made it practical to ru may time-history aalyses i a short period of time. The Curret Speed of a $1, Persoal Computer is 56, times faster tha the $1,, Computer of 1963 (See Appedix H). Also, i 1963 oly a few recorded time-history earthquake records existed. At the preset time, we have thousads of earthquake records available. Therefore, it is o loger ecessary to use this very approximate ad iaccurate spectrum method of aalysis. Ufortuately, the use of more accurate aalysis methods has ot occurred. I fact, the use of the respose spectrum method ad other simplified static methods appears to have icreased ad the use of the more accurate time-history aalysis methods has bee reduced durig the past several years. Also, may structural egieers believe a Equivalet Static Aalysis like a pushover aalysis is more accurate tha a rigorous oliear aalysis i which dyamic equilibrium is satisfied at each time step. Also, may structural egieers cotiue to claim they ca solve oliear problems by creatig oliear spectra. It appears they have forgotte that earthquake displacemets are applied at the base of the structure ad are the propagated upward ito the structure to cause the lateral displacemets i the superstructure. Fially, a example will be give where a log bridge structure is retrofitted to resist realistic groud displacemets that are differet at each foudatio. The retrofitted structure was desiged

2 15- STATIC AND DYNAMIC ANALYSIS to behave oliearly to dissipate eergy at a fiite umber of elemets that would be easily repairable or replaced after a major earthquake. 15. THE HISTORY OF THE METHOD I the EERI Oral History Series, writte i 1997, Professor George W. Houser summarized the early history of the creatio of a spectrum from a recorded earthquake acceleratio record. He stated The calculated spectrum curves of recorded groud acceleratio characterize the groud motio i a way that is very sigificat to egieers. I do ot kow who first called it a respose spectrum, but ufortuately the term leads people to thik that the spectrum characterizes the buildig s motio, rather tha the groud s motio. Nevertheless, respose spectrum has become stadard termiology. I the 5 page EERI documet he ever idicated he used the Respose Spectrum Method for his research or professioal work. I his cosultig work he always used basic wave propagatio theory or the time history aalysis method DEFINITION OF A RESPONSE SPECTRUM For three-dimesioal seismic motio, the typical modal Equatio (13.6) is rewritte as: & y (t) +ζ ω y(t) & + ω y(t) = p u(t) & +p u(t) & +p u& (t) (15.1) x gx y gy z gz

3 RESPONSE SPECTRUM ANALYSIS 15-3 Where, the three Mode Participatio Factors are defied by T Mi pi = - φ i which i is equal to x, y or z. Two major problems must be solved to obtai a approximate respose spectrum solutio to this equatio. First, for each directio of groud motio, maximum peak forces ad displacemets must be estimated. Secod, after the respose for the three orthogoal directios has bee solved, it is ecessary to estimate the maximum respose from the three compoets of earthquake motio actig at the same time. This sectio addresses the modal combiatio problem from oe compoet of motio oly. The separate problem of combiig the results from motio i three orthogoal directios will be discussed later i this chapter. For iput i oe directio oly, Equatio (15.1) is writte as: & y (t) +ζ ω y(t) & + ω y(t) =p u& (t) (15.) i g Give a specified groud motio u& (t) g, dampig value ad assumig p i = 1., it is possible to solve Equatio (15.) at various values of ω ad plot a curve of the maximum peak respose. For this acceleratio iput, the curve is by defiitio the displacemet respose spectrum for the earthquake motio. A differet curve will exist for each differet value of dampig. Structural egieers that use a respose spectrum curve must uderstad the time the maximum peak value y (ω ) MAX occurs has bee lost i the creatio of the curve ad ca be sigificatly differet for each value of ω. Therefore, combiig resposes from differet modes will always be approximate. Ad, the iteral member forces calculated are ot i static equilibrium. A plot of ω y (ω) MAX is defied as the pseudo-velocity spectrum ad a plot of ω y ( ω) MAX is defied as the pseudo-acceleratio spectrum. The three curves displacemet respose spectrum, pseudo-velocity spectrum, ad pseudoacceleratio spectrum are ormally plotted as oe curve o special log paper. However, the pseudo-values have miimum physical sigificace ad are ot a essetial part of a respose spectrum aalysis. The true values for maximum velocity ad acceleratio must be calculated from the solutio of Equatio (15.). There is a mathematical relatioship, however, betwee the pseudo-acceleratio spectrum ad the total acceleratio spectrum. The total acceleratio of the uit mass, sigle degree-of-freedom system, govered by Equatio (15.), is give by: u &( t) = & y( t) + u& ( t) (15.3) T g Equatio (15.) ca be solved for y& (t) ad substituted ito Equatio (15.3) to yield: u& ( t) T = ω y( t) ξωy& ( t) (15.4) Therefore, for the special case of zero dampig, the total acceleratio of the system is equal to ω y( t). For this reaso, the displacemet respose spectrum curve is ormally ot plotted as

4 15-4 STATIC AND DYNAMIC ANALYSIS modal displacemet y (ω) MAX versus ω. It is stadard to preset the curve i terms of S( ω ) versus a period T i secods, where: S ( ω ) = ω y( ω) ad a MAX T = π (15.5a ad 15.5b) ω The pseudo-acceleratio spectrum curve, S (ω) a, has the uits of acceleratio versus period that has some physical sigificace for zero dampig oly. It is apparet that all respose spectrum curves represet the properties of the earthquake at a specific site ad are ot a fuctio of the properties of the structural system. After a estimatio is made of the liear viscous dampig properties of the structure, a specific respose spectrum curve is selected CALCULATION OF MODAL RESPONSE The maximum modal displacemet for a structural model ca ow be calculated for a typical mode with period T ad correspodig spectrum respose value S( ω ). The maximum modal respose associated with period T is give by: y( T ) MAX S( ω ) = (15.6) ω The maximum modal displacemet respose of the structural model is calculated from: u = y( ) φ (15.7) T MAX The correspodig iteral modal forces, f k, are calculated from stadard matrix structural aalysis usig the same equatios as required i static aalysis CREATION OF RESPONSE SPECTRUM CURVES A te-secod segmet of the Loma Prieta earthquake motios recorded o a soft site i the Sa Fracisco Bay Area is show i Figure 15.1a. The record has bee corrected usig a iterative algorithm for zero displacemet, velocity ad acceleratio at the begiig ad ed of the tesecod record.

5 RESPONSE SPECTRUM ANALYSIS TIME - secods Typical Earthquake Groud Acceleratio - Percet of Gravity From the recorded earthquake acceleratios show i Figure 15.1a, the resultig groud displacemets are calculated by umerical itegratig the record twice ad the plot of the results groud displacemets is show i Figure 15.1b. The Figure 15.1b Typical Earthquake Groud Displacemets Iches respose spectrum curves for the relative displacemet with respect zero groud displacemet ad pseudo-acceleratio are calculated from Equatios 15. ad 15.6 are summarized below i Figure 15.a ad 15.b.

6 15-6 STATIC AND DYNAMIC ANALYSIS Percet Dampig 5. Percet Dampig PERIOD - Secods Figure 15.a Relative Displacemet Spectrum y (ω) MAX - Iches PERIOD - Secods 1. Percet Dampig 5. Percet Dampig Figure 15.b Pseudo-Acceleratio Spectrum, a = Percet of Gravity MAX S ω y( ω) The maximum groud acceleratio for the earthquake defied by Figure 15.1a is.1 percet of gravity at.9 secods. It is importat to ote that the pseudo-acceleratio spectrum show i Figure 15.b has the same value for a very short period system. This is because of the physical fact that a very rigid structure moves as a rigid body ad the relative displacemets withi the structure are equal to zero, as idicated by Figure 15.a. Also, the behavior of a rigid structure is ot a fuctio of the viscous dampig value. The maximum groud displacemet show i Figure 15.1b is iches at 1.97 secods. For log period systems, the mass of the oe-degree-of-freedom structure does ot move sigificatly

7 RESPONSE SPECTRUM ANALYSIS 15-7 ad has approximately zero absolute displacemet. Therefore, the relative displacemet spectrum curves show i Figure 15.a will coverge to 11.6 iches for log periods ad all values of dampig. This type of real physical behavior is fudametal to the desig of base isolated structures. The relative displacemet spectrum, Figure 15.a, ad the absolute acceleratio spectrum, Figure 15.b, have physical sigificace. However, the maximum relative displacemet is directly proportioal to the maximum forces developed i the structure. For that earthquake, the maximum relative displacemet is 18.9 iches at a period of 1.6 secods for 1 percet dampig ad 16. iches at a period of 4 secods for 5 percet dampig. It is importat to ote the sigificat differece betwee 1 ad 5 percet dampig for this typical soft site record. Figure 15.b, the absolute acceleratio spectrum, idicates maximum values at a period of.64 secods for both values of dampig. Also, the multiplicatio by ω teds to completely elimiate the iformatio cotaied i the log period rage. Because most structural failures durig recet earthquakes have bee associated with soft sites, perhaps we should cosider usig the relative displacemet spectrum as the fudametal form for selectig a desig earthquake. The highfrequecy, short-period part of the curve should always be defied by: y ( ω) MAX T = u& g MAX / ω or y( T) = u& MAX gmax 4π (15.8) where u& gmax is the peak groud acceleratio THE SRSS ad CQC METHOD OF MODAL COMBINATION The most coservative method that is used to estimate a peak value of displacemet or force withi a structure is to use the sum of the absolute of the modal respose values. This approach assumes that the maximum modal values for all modes occur at the same poit i time. Aother very commo approach is to use the Square Root of the Sum of the Squares, SRSS, o the maximum modal values to estimate the values of displacemet or forces. The SRSS method assumes that all of the maximum modal values are statistically idepedet. For threedimesioal structures i which a large umber of frequecies are almost idetical, this assumptio is ot justified. However, it was used util the mid eighties. The modal combiatio by the Complete Quadratic Combiatio, CQC, method [1] that was first published i Oe purpose of this chapter is to explai by example the advatages of usig the CQC method ad illustrate the potetial problems i the use of the SRSS method of modal combiatio. The CQC method is based o radom vibratio theories ad has foud wide acceptace by mechaical egieers for the predictio of fatigue failure i metal structures. However, some structural egieers have foud that it over-estimates the member forces i most civil egieerig structures. The peak value of a typical force ca ow be estimated from the maximum modal values usig the CQC method with the applicatio of the followig double summatio equatio:

8 15-8 STATIC AND DYNAMIC ANALYSIS ρ F f f (15.9) = m m m where f is the modal force associated with mode. The double summatio is coducted over all modes. Similar equatios ca be applied to ode displacemets, relative displacemets ad base shears ad overturig momets. The cross-modal coefficiets, ρ m, for the CQC method with costat dampig are: 3/ 8ζ (1 + r) r ρ m = (15.1) (1 r ) + 4ζ r(1 + r) where r = ω / ω m ad must be equal to or less tha 1.. It is importat to ote that the crossmodal coefficiet array is symmetric ad all terms are positive NUMERICAL EXAMPLE OF MODAL COMBINATION The problems associated with usig the absolute sum ad the SRSS of modal combiatio ca be illustrated by their applicatio to the four-story buildig show i Figure The buildig is symmetrical; however, the ceter of mass of all floors is located 5 iches from the geometric ceter of the buildig. Figure 15.3 A Simple Three-Dimesioal Buildig Example The directio of the applied earthquake motio, a table of atural frequecies ad the pricipal directio of the mode shape are summarized i Figure 15.4.

9 RESPONSE SPECTRUM ANALYSIS 15-9 Figure 15.4 Frequecies ad Approximate Directios of Mode Shapes Oe otes the closeess of the frequecies that is typical of most three-dimesioal buildig structures that are desiged to resist earthquakes from both directios equally. Because of the small mass eccetricity, which is ormal i real structures, the fudametal mode shape has x, y, as well as torsio compoets. Therefore, the model represets a very commo three-dimesioal buildig system. Also, ote that there is ot a mode shape i a particular give directio, as is implied i may buildig codes ad some text books o elemetary dyamics. The buildig was subjected to oe compoet of the Taft 195 earthquake. A exact time history aalysis usig all 1 modes ad a respose spectrum aalysis were coducted. The maximum modal base shears i the four frames for the first five modes are show i Figure Figure 15.6 summarizes the maximum base shears i each of the four frames usig differet methods. The time history base shears, Figure 15.6a, are exact. The SRSS method, Figure 15.6b, produces base shears that uder-estimate the exact values i the directio of the loads by approximately 3 percet ad over-estimate the base shears ormal to the loads by a factor of 1. The sum of the absolute values, Figure 15.6c, grossly over-estimates all results. The CQC method, Figure 15.6d, produces very realistic values that are close to the exact time history solutio.

10 15-1 STATIC AND DYNAMIC ANALYSIS Figure 15.5 The Base Shears I Each Frame For First Five Modes Usig The Exact Time History Method Figure 15.6 Compariso of Modal Combiatio Methods The modal cross-correlatio coefficiets for this buildig are summarized i Table It is of importace to ote the existece of the relatively large off-diagoal terms that idicate which modes are coupled.

11 RESPONSE SPECTRUM ANALYSIS Table 15.1 Modal Cross-Correlatio Coefficiets -ζ = 5. Mode ω (rad/sec) If oe otes the sigs of the modal base shears show i Figure 15.3, it is apparet how the applicatio of the CQC method allows the sum of the base shears i the directio of the exteral motio to be added directly. I additio, the sum of the base shears, ormal to the exteral motio, ted to cacel. The ability of the CQC method to recogize the relative sig of the terms i the modal respose is the key to the elimiatio of errors i the SRSS method DESIGN SPECTRA Desig spectra are ot ueve curves as show i Figure 15. because they are iteded to be the average of may earthquakes. At the preset time, may buildig codes specify desig spectra i the form show i Figure Normallized Pseudo Acceleratio PERIOD - Secods Figure 15.7 Typical Desig Spectrum The Uiform Buildig Code has defied specific equatios for each rage of the spectrum curve for four differet soil types. For major structures, it is ow commo practice to develop a sitedepedet desig spectrum that icludes the effect of local soil coditios ad distace to the earest faults.

12 15-1 STATIC AND DYNAMIC ANALYSIS 15.9 ORTHOGONAL EFFECTS IN SPECTRAL ANALYSIS A well-desiged structure should be capable of equally resistig earthquake motios from all possible directios. Oe optio i existig desig codes for buildigs ad bridges requires that members be desiged for "1 percet of the prescribed seismic forces i oe directio plus 3 percet of the prescribed forces i the perpedicular directio." However, they give o idicatio o how the directios are to be determied for complex structures. For structures that are rectagular ad have clearly defied pricipal directios, these "percetage" rules yield approximately the same results as the SRSS method. For complex three-dimesioal structures, such as o-rectagular buildigs, curved bridges, arch dams or pipig systems, the directio of the earthquake that produces the maximum stresses i a particular member or at a specified poit is ot apparet. For time history iput, it is possible to perform a large umber of dyamic aalyses at various agles of iput to check all poits for the critical earthquake directios. At the preset time, however, due to the low-cost of high-speed persoal computers several realistic three-dimesioal earthquake records ca be cosidered i a additioal few miute. Ad, the computer program will produce the maximum demad/capacity ratios for all member i a few additioal miutes. It is reasoable to assume that motios that take place durig a earthquake have oe pricipal directio []. Or, durig a fiite period of time whe maximum groud acceleratio occurs, a pricipal directio exists. For most structures, this directio is ot kow ad for most geographical locatios caot be estimated. Therefore, the oly ratioal earthquake desig criterio is that the structure must resist a earthquake of a give magitude from ay possible directio. I additio to the motio i the pricipal directio, a probability exists that motios ormal to that directio will occur simultaeously. I additio, because of the complex ature of three-dimesioal wave propagatio, it is valid to assume that these ormal motios are statistically idepedet. Based o those assumptios, a statemet of the desig criterio is "a structure must resist a major earthquake motio of magitude S 1 for all possible agles θ ad at the same poit i time resist earthquake motios of magitude S at 9 o to the agle θ." These motios are show schematically i Figure Basic Equatios for Calculatio of Spectral Forces The stated desig criterio implies that a large umber of differet aalyses must be coducted to determie the maximum desig forces ad stresses. It will be show i this sectio that maximum values for all members ca be exactly evaluated from oe computer ru i which two global dyamic motios are applied. Furthermore, the maximum member forces calculated are ivariat with respect to the selectio system.

13 RESPONSE SPECTRUM ANALYSIS S 9 S 1 θ Pla View Figure 15.8 Defiitio of Earthquake Spectra Iput Figure 15.8 idicates that the basic iput spectra S1 ad S are applied at a arbitrary agle θ. At some typical poit withi the structure, a force, stress or displacemet F is produced by this iput. To simplify the aalysis, it will be assumed that the mior iput spectrum is some fractio of the major iput spectrum. Or: S = a S1 (15.11) where a is a umber betwee ad 1.. Recetly, Meu ad Der Kiureghia [3] preseted the CQC3 method for the combiatio of the effects of orthogoal spectrum. The fudametal CQC3 equatio for the estimatio of a peak value is: F = [ F + (1 a + a F9 ) F 9 (1 a )( F si θcos θ+ F z 9 1 F ] )si θ (15.1) where, F f ρ f (15.13) = m m m F9 f9 ρ f9 (15.14) = m m m F 9 = f ρ f9 (15.15) m m m F f ρ f (15.16) Z = m z m zm

14 15-14 STATIC AND DYNAMIC ANALYSIS i which f ad f 9 are the modal values produced by 1 percet of the lateral spectrum applied at ad 9 degrees respectively, ad f z is the modal respose from the vertical spectrum that ca be differet from the lateral spectrum. It is importat to ote that for equal spectra a =1, the value F is ot a fuctio of θ ad the selectio of the aalysis referece system is arbitrary. Or: MAX + F F 9 z F = F + (15.17) This idicates that it is possible to coduct oly oe aalysis with ay referece system, ad the resultig structure will have all members that are desiged to equally resist earthquake motios from all possible directios. This method is acceptable by most buildig codes The Geeral CQC3 Method of modal combiatio For a =1, the CQC3 method reduces to the SRSS method. However, this ca be over coservative because real groud motios of equal value i all directios have ot bee recorded. Normally, the value of θ i Equatio (15.1) is ot kow; therefore, it is ecessary to calculate the critical agle that produces the maximum respose. Differetiatio of Equatio (15.1) ad settig the results to zero yields: θ cr 1 = ta 1 F 9 [ F F 9 ] (15.18) Two roots exist for Equatio (15.17) that must be checked i order that the followig equatio is maximum: F MAX (1 a = [ F ) F + a 9 F9 si θ (1 a cr cos θ )( F cr z + F F9 1 ] )si θ cr (15.19) At the preset time, o specific guidelies have bee suggested for the value of a. Referece [3] preseted a example with values a betwee.5 ad Examples of Three-Dimesioal Spectra Aalyses The previously preseted theory clearly idicates that the CQC combiatio rule, with a equal to 1., is idetical to the SRSS method ad produces results for all structural systems that are ot a fuctio of the referece system used by the egieer. Oe example will be preseted to show the advatages of the method. Figure 15.9 illustrates a very simple oe-story structure that was selected to compare the results of the 1/3 percetage rules with the SRSS rule.

15 RESPONSE SPECTRUM ANALYSIS Y 3 Typical Colum: 3 3 X = Y = ft. 4 X =Y=16.65 ft. Sym. 4 I = 1ft 4 I33 = ft E = 3 k/ft L = 1ft MTOP =.5k - sec / ft X =1ft. X =15ft. X Total Mass: M = 1. k - sec / ft Ceter of Mass: x = 16.6 y = Figure 15.9 Three-Dimesioal Bridge Structure Note that the masses are ot at the geometric ceter of the structure. The structure has two traslatios ad oe rotatioal degrees-of-freedom located at the ceter of mass. The colums, which are subjected to bedig about the local ad 3 axes, are pied at the top where they are coected to a i-plae rigid diaphragm. The periods ad ormalized base shear forces associated with the mode shapes are summarized i Table 15.. Because the structure has a plae of symmetry at.5 degrees, the secod mode has o torsio ad has a ormalized base shear at.5 degrees with the x-axis. Because of this symmetry, it is apparet that colums 1 ad 3 (or colums ad 4) should be desiged for the same forces. Mode Table 15. Periods ad Normalized Base Shear Period (Secods) X-Force Y-Force Directio of- Base Shear- (Degrees) Torsio

16 15-16 STATIC AND DYNAMIC ANALYSIS The displacemet respose spectrum used i the spectra aalysis is give i Table Mode Table 15.3 Participatig Masses ad Respose Spectrum Used Period (Secods) X-Mass Y-Mass Spectral Value Used for Aalysis The momets about the local ad 3 axes at the base of each of the four colums for the spectrum applied separately at. ad 9 degrees are summarized i Tables 15.4 ad 15.5 ad are compared to the 1/3 rule. Table 15.4 Momets About -Axes SRSS vs. 1/3 Rule Member M M9 = MSRSS M + M9 M1/3 Error(%) Member M M9 Table 15.5 Momets About 3-Axes SRSS vs. 1/3 Rule = MSRSS M + M9 M1/3 Error(%) For this example, the maximum forces do ot vary sigificatly betwee the two methods. However, it does illustrate that the 1/3 combiatio method produces momets that are ot symmetric, whereas the SRSS combiatio method produces logical ad symmetric momets. For example, member 4 would be over-desiged by 3.4 percet about the local -axis ad uderdesiged by 7.8 percet about the local 3-axis usig the 1/3 combiatio rule.

17 RESPONSE SPECTRUM ANALYSIS Demad/Capacity Calculatios for 3D Frame Elemets I order to satisfy various buildig codes specify that all oe-dimesioal compressio members withi a structure satisfies the followig Demad/Capacity Ratio at all poits i time: R( t ) = P( t ) M ( t )C ± φcpcr P( t ) φbm c( 1 P e M 3( t )C3 ± P( t ) ) φbm c3( 1 P e3 ) 1. (15.) Where the forces actig o the frame elemet cross-sectio at time t are P t), M ( t) ad M ( ) ( 3 t (icludig the static forces prior to the applicatio of the dyamic loads). The empirical costats are code ad material depedet ad are ormally defied as φ c ad φ b = Resistace factors C ad C 3 = Momet reductio factors M c ad c3 M = Momet capacity P cr = Axial load Capacity P e ad e3 P = Euler buckig load capacity about the ad 3 axis with effective legth approximated. For each time-history seismic aalysis, P t), M ( t) ad M ( ) at every cross-sectio of all ( 3 t members ca be easily calculated as a fuctio of time. Therefore, the maximum Demad/Capacity Ratio, R ( t ) for all load coditios, ca be accurately calculated ad idetified by ay moder computer aalysis desig program i a fractio of a secod. All of the CSI series of programs have this capability built ito their iteractive post-processig programs. For each respose spectrum aalysis, however, the value of P ( t), M( t) ad M3( t) caot be calculated accurately sice oly positive values of P, M ad M3 are produced. These are peak maximum values have a very low probability of occurrig at the same time. Therefore, the Demad/Capacity Ratios are always sigificatly greater tha those produced by a time-history aalysis. The author, actig as a cosultat o the retrofit of the Sa Mateo Bridge after the Loma Prieta Earthquake, has had sigificat experiece with the problem of calculatig Demad/Capacity Ratios usig the respose spectra method. The seismologists ad geotechical egieers created two differet sets of three-dimesioal groud motios. They geerated both ear ad far field motios from both the Hayward ad Sa Adreas faults. The, they averaged the various groud motios ad produced three-dimesioal desig spectra to be used to desig the retrofit of the Bridge.

18 15-18 STATIC AND DYNAMIC ANALYSIS The structural egieerig group that tried to use the desig spectra for the aalysis ad retrofit of the bridge foud that a large umber of members i the structure required retrofit. After a careful study of the maximum peak values of the member forces (especially the large peak axial forces), it was decided to ru ew time-history aalyses usig the basic time-history records that were used to create the desig spectra. After ruig all the time-history records, the maximum Demad/Capacity Ratios were reduced by approximately a factor of three compared to the desig spectra results. SAP9 was used for this project i approximately 1997 prior to the release of SAP. The major amout of mapower was i creatig the computer model of the structure for the respose spectra aalyses. The time-history post-processig calculatios of the maximum Demad/Capacity Ratios were calculated for all members i a series of simple overight computer rus. Therefore, there was ot a sigificat icrease i mapower or time to obtai very accurate results (which satisfied dyamic force equilibrium) as compared to the very poor results obtaied from use of spectra iput Example of a very Complex Seismic Retrofit Project After the completio of the retrofit desig for the Sa Mateo Bridge the same structural egieerig group started workig o the retrofit desig of the Richmod-Sa Rafael Bridge. This bridge was a very complex double-deck steel truss structure built i the mid ietee fifties. Also, sigificat oliear behavior was expected i the retrofitted structure sice the purpose of the retrofit was to prevet collapse. I additio, the time-history seismic displacemets were differet at each pier foudatio level. Clearly the respose spectrum method could ot possibly model the oliear behavior ad the multi-support displacemet iput. Durig the previous several years the author had developed a ew method to accurately solve a certai class of oliear problems usig the dyamic mode superpositio method plus additioal static modes associated with the oliear elemets (see Fast Noliear Aalysis - Chapter 18). This ew method FNA was implemeted i my persoal research ad developmet program, SADSAP, which had the same iput data format as SAP9. This author wated them to use this ew method for the project i order to evaluate the FNA method o a large ad sigificat project. They accepted my offer ad I agreed to modify the program to meet their eed ad to provide free cosultig services durig the life of the project. Needless to say, I was very busy for the ext few years attedig meetigs ad makig modificatios to the SADSAP program. The retrofitted structure was desiged to behave oliearly to dissipate eergy at a fiite umber of elemets that would be easily repairable or replaced after a major earthquake. The Demad/Capacity Ratios reflected the redistributio of forces due to oliear behavior as well as the opeig ad closig of all joits. Withi a year after the successful completio of the project, CSI released SAP, which is a Widows based program, added a improved versio of the FNA method of oliear aalysis ad may other valuable optios.

19 RESPONSE SPECTRUM ANALYSIS SUMMARY I this chapter it has bee illustrated that the respose spectrum method of dyamic aalysis is a very approximate method for structures over oe degree of freedom. The CQC method of modal combiatio ad the SRSS to combie the respose from ay two orthogoal directios is the theoretically correct approach to apply the approximate respose spectrum method. Also, this very approximate method ca ever be exteded to produce correct results for the oliear seismic aalysis of real structures. Durig the last twety years umerical methods for both liear ad oliear time-history dyamic aalysis have improved sigificatly. CSI has icreased the speed of all phases of structural aalysis by effectively usig the power of the ew multiprocessor chips. The use of 64 bit addressig capabilities has icreased the size ad complexity of the structures it ca ow solve o iexpesive persoal computers. Withi the last few years their programs have bee used for aalysis ad desig of the largest structures ever built REFERENCES 1. Wilso, E. L., A. Der Kiureghia ad E. R. Bayo "A Replacemet for the SRSS Method i Seismic Aalysis," Earthquake Egieerig ad Structural Dyamics. Vol. 9. pp. l87-l9.. Pezie, J., ad M. Watabe "Characteristics of 3-D Earthquake Groud Motios," Earthquake Egieerig ad Structural Dyamics. Vol. 3. pp Meu, C., ad A. Der Kiureghia A Replacemet for the 3 % Rule for Multicompoet Excitatio, Earthquake Spectra. Vol. 13, Number 1. February.