and Sabah Moussaoui a Accepted 10 June 2013, Available online 16 June 2013, Vol.3, No.2 (June 2013)

Transcription

1 Research Article International Journal of Current Engineering and Technology ISSN INPRESSCO. All Rights Reserved. Available at Coparison of Static Pushover Analysis in the Case of Sall and Large Deforation with Tie History Analysis using Flexibility-Based Model for an Existing Structure Mourad Belgasia a* and Sabah Moussaoui a a Departent of Civil Engineering Setif University,19000 Algeria Accepted 10 June 2013, Available online 16 June 2013, Vol.3, No.2 (June 2013) Abstract This work presents the nonlinear ethods of analyses for seisic design of structures. The first ethod is the nonlinear pushover procedure, which is based on the N2 ethod. The second ethod is the classical nonlinear tie history analysis. The objective of this paper is to ake a coparative study of an existing reinforced concrete building in Bonefro, Italy between static nonlinear analysis and tie history analysis using flexibility-based finite eleent, and the sensitivity of the tie history analyses to the seisic paraeters through Z_Soil; an engineering software based on the finite-eleent ethod. Keywords: flexibility-based finite eleent, seisic engineering and design, push-over, tie history, nonlinear analyses, sensitivity to seisic paraeters. 1. Introduction 1 Most studies to date concerned with the non-linear analysis of reinforced concrete frae structures are based on finite eleent odels which are derived with the stiffness ethod. The work done by (S.Kaba & al, 1984) (C.Zeris & al, 1998, 1991) have deonstrated the advantage of flexibility-based odels, but have failed to give a clear and convincing way of deterining the eleent resisting forces fro the given displaceents. This difficulty arises when the flexibility-based finite eleent is ipleented in a non-linear analysis progra based on the direct stiffness ethod. In this case, the solution of the global equilibriu equations yields the displaceents of the structural degrees of freedo. During the state deterination phase the resisting forces of all eleents in the structure need to be deterined. In a flexibility-based eleent, there are no deforation interpolation functions to relate the deforations along the eleent to the end displaceents, therefore, the process is not straightforward and is not well developed in flexibility-based odels proposed to date. This fact has led to soe confusion in the nuerical ipleentation of previous odels. To overcoe this proble (V.Ciapi & al 1984) proposed a consistent flexibility-based ethod for forulating frae eber odels. This ethod was refined and applied to the developent of two types of eleents in (F.Taucer & al, 1991) (E.Spacone, 1994) *Corresponding author: Mourad.Belgasi, Sabah.Moussaoui is PhD student conducted a large nuber of static and dynaic siulations of sall structures with these eleents with great success. The procedure is general in scope and applies to any section aterial behaviour. Modern seisic design codes allow engineers to use either linear or nonlinear analyses to copute design forces and design displaceents. In particular, (Eurocode 8, 2003) contains four ethods of analysis: siplified static analysis, odal analysis, nonlinear pushover analysis and nonlinear tie-history analysis. These ethods refer to the design and analysis of fraed structures, ainly buildings and bridges. The two nonlinear ethods require advanced odels and advanced nonlinear procedures in order to be fully applicable by design engineers. This paper gives a coparaison of static nonlinear pushover analysis which is based on the N2 ethod developed by (P.Fajfar, 2005, 1999, 2002) and classical nonlinear tie history analysis of existing reinforced concrete frae structures in Italy using flexibility-based finite eleent. Displaceent-based and force-based eleents are used in this study. The forer is a classical two-node, displaceent-based, Euler-Bernoulli frae eleent. The later is a two-node, force-based, Euler Bernoulli frae eleent. The ain advantage of the second eleent is that it is exact within the relevant frae eleent theory. This iplies that one eleent per frae eber (bea or colun) is used in preparing the frae esh, thus leading to a reduction of the global nuber of degrees of freedo. The coplete theory for the force-based eleent can be found in (S. Antoniou, 2004) (M.N. Aydinoglu & al, 2004). In this paper four 655

2 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) applications are presented. The first, second and the third one is the study of a SDOF proble under step axial loading, a cantilever bea with bending loading and the earthquake response of 2D odel for an existing reinforced concrete frae structures in linear and nonlinear case using flexibility-based and displaceent based forulation odel in order to copare between the two. The forth one is to ake a coparative analysis of 3D odel of an existing reinforced concrete building between static nonlinear analysis and tie history analysis using flexibility-based finite eleent according to (Eurocode 8, 2003), and the sensitivity study of the tie history analyses to the seisic paraeters. 2. Flexibility based forulation The proposed forulation is flexibility based and uses force interpolation functions for the bending oent variation that depend on the transverse displaceents and strictly satisfy equilibriu in the defored configuration. The derivation of the governing equations is substantially ore involved than for stiffness-based eleents (A.Neuenhofer & al, 1998). Nonetheless, the eleent offers significant advantages over existing stiffness-based approaches, since no discretization error occurs and all governing equations are satisfied exactly. Consequently, fewer eleents are needed to yield results of coparable accuracy. This is deonstrated with the analysis of several siple exaple structures by coparing the results fro flexibility and stiffness-based eleents. The use of flexibility instead of stiffness is otivated by the fact that the dynaically easured flexibility atrix is doinated by the lowest odes of a structure, which can be easily easured, while the dynaically easured stiffness atrix is doinated by the highest odes of the structure, which are hard if not ipossible to easure (E.Reynders & al, 2010). The plane frae finite eleent odels are based on the Euler-Bernoulli bea theory (K.D.Hjelstad & al,2002) for geoetrically nonlinear behavior (H.RValipour & al, 2010). In this case, the governing variables are the axial and transverse displaceent fields u(x) and w(x), respectively, of the eleent reference axis that give rise to deforation fields [ ] [ ] (1) Where is the axial strain at the reference axis and is the curvature, with the prie denoting differentiation with respect to x. Displaceents and strains are assued to be sall. The nonlinear axial strain-displaceent relation in Equation (1) fors the basis for the proposed geoetrically nonlinear forulation. The corresponding stress resultants or internal force fields are Where N(x) is the axial force and M(x) the bending oent. It is assued that the section constitutive relation (3) With [ ] Where EA is the axial and EI the flexural rigidity 2.1. Flexibility ethod If equilibriu is considered in the defored eleent configuration in Fig. 1, the relation between nodal forces in the syste without rigid body odes and internal forces f(x) is [ ] Where [ ] [ (4) ], (5) Is the atrix of displaceent-dependent force interpolation functions. Since shear deforations are neglected, the shear force does not appear in Equation (4), but it can be deterined a posteriori fro the equilibriu condition (see Figure 1) Fig. 1 Equilibriu in defored configuration The weak for of the copatibility condition in Equation ( 1), (7) Leads to three copatibility equations for the frae eleent without rigid body odes. one for the axial displaceent and two for the end rotations and in Fig. 1. The latter are identical with the linear case. The forer becoes [ [ ] ] (8) After integrating the preceding expression by parts and accounting for the boundary ters. The copatibility condition for reduces to (6) [ ] (2) (9) 656

3 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) Thus the coplete set of governing equations, the flexibility-based is [ ] (10) F F 0 With [ ] [ ] (11) In geoetrically nonlinear flexibility-based analysis the evaluation of the flexibility atrix F requires special attention. This is because both b(x) and b*(x) depend on the transverse displaceents w(x). Starting fro the governing equation in Equation (10) we obtain [ [ [ ] ] ] (12) Where is the section flexibility atrix. Since Equation (12) is derived fro Equation (10), which is based on a variational principle, it consists of three contributions. The first ter represents the "direct" change in the eleent end displaceents caused by a change in the eleent end forces for fixed displaceents w(x). The second and third ters are of indirect nature accounting for the change in caused by the change in w(x), which in turn originates fro a change in Fig. 2 Applied step load 3.1.1Axial loading (J.M.Biggs 1964) Linear case Equation of otion : Mu Ku F 0 u u (1 cos t) u st F0 K The exact solution is st with u For this case the axiu displaceent is ax 2. Figure 3 copares the results obtained using displaceent- and force-based (flexibility based) eleent. As expected the exact solution is obtained using a single eleent in both cases. Newark s algorith with 0.5 and 0.25 is used. t 3. Nonlinear and linear Coparison between flexibility and displaceent base forulation Advanced odels for frae analysis are used. In particular, nonlinear beas with fiber section odels are available. The cross section is divided into fibers and the constitutive law of each fiber is assigned fro the constitutive law library available in software. Both displaceent-based and force-based forulations are available. Force-based eleents (E. Spacone & al, 1996) are exact within the classical Euler-Bernoulli bea theory. As for geoetric linearities, these are considered in the general fraework of the progra and thus follow a corotational approach (Th. Zierann & al, 2008). 3.1 SDOF test proble We consider a step load (F = 0 for t < 0 and F = F 0 for t 0) applied to SDOF nonlinear oscillator Figure 2. The oscillator, odeled as a bar, has the following characteristics: stiffness K =1 N/ (odulus of elasticity E =1 N/ 2, section A=1 2, length L = 1 ), ass M =1 kg, force F 0 = 1. Fig. 3 Coparison between displaceent-based and force-based forulation in the evolution of displaceent in tie, Δt = Nonlinear case Nonliniarity is characterized by a yield stress 2 yield 1.5 N/. Elastic-perfectly plastic behavior is assued. The equation of otion is M u F u F 0 There are there are 03 cases to consider in the response u uyield case 1) 657

4 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) Fyield 1.5 N uyield 1.5 Since,. The tie corresponding to yield can be coputed by setting : yield uyield l 1.5 ust(1 cos tyield ) E All values of the nonlinear three cases are checked in Figure. 4 and Figure. 5. We can see that the results are identical for both displaceent- and force-based eleent and agree to the theory. A single eleent was used in both cases. with t yield ust arccos( 0.5) s 1. We obtain uyield u u case 2) yield After yielding the internal force is A, thus the equation of otion is Mu Fyield F0 Mu F0 Fyield 0.5 Mu Fyield F0 Mu F0 Fyield 0.5 Fig. 4 Displaceent response for nonlinear Biggs s exaple, Δt = 0.1 Mu F F Mu ( F F ) t c 0 yield 0 yield 1 with t t tyield F F yield M u t c1t c2 t t tyield with The initial conditions are used to deterine c 1 and c 2. u( tyield ) ust(1 cos t) 1.5 u( tyield ) sin( tyield ) c and c this case is true till u 0. Based on the previous equations: u F0 Fyield ( t tyield ) t t t thus t yield s F0 Fyield u t c1t c2 (1.732) 0.866(1.732) u 2.25 case 3) t s The displaceent is Fig. 5 Force response for nonlinear Biggs s exaple, Δt = Bending loading A cantilever bea is used in this second exaple. The bea has the following characteristics: L= 1, rectangular section b=h=0,3, E= kn/ 2, =0.16, f t =f c =3000 kn/ 2. The cantilever has a 750kg luped ass at the free end and is loaded at the free end with a transverse step load of 15N applied at tie t=0 (Figure. 6 and Figure. 7). The cantilever response in ters of tip displaceent (in the transverse direction) and shear force at the fixed end is shown in Fig. 8 and Fig. 9, respectively. Fyield F0 Fyield F0 u ( u ) cost with t t t K K u ( ) 0.5cos( t ) for t 7s t s and u 1.25 In this last case the aplitude of vibration will reain u below. Fig. 6 odel of a cantilever bea discretised with one eleent (force based eleent) 658

5 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) Fig. 7 odel of a cantilever bea discretised with one 20 eleents (displaceent based eleent) Fig. 10 2D frae studied Fig. 8 Response in ters of tip displaceent of cantilever bea, Δt = 0.1 Fig. 11 Top floor response to El Centro accelerogra Fig. 9 Response in ters of shear force at fixed end of cantilever bea, Δt = 0.1 We can deduce after studying these two exaples, that there is a clear convergence between the values given by the theory either in linear or nonlinear cases with those given by the software used and especially with the flexibility-based finite eleent Response of 2D odel of Bonefro building to ground acceleration The response of a single 2D frae is studies. The side frae analyzed is shown in Figure 10, The response of the frae of to the El Centro accelerogra is shown in the following Figure 11 shows the 2D frae top (third) floor response to the El Centro earthquake using displaceent based and flexibility base eleent. The sae displaceent is plotted for the Hollister and Friuli accelerogras in figure 12 and figure 13. Fig. 12 Top floor response to El Hollister accelerogra After satisfactory coparison of flexibility and displaceent based forulation, only flexibility-based was used in the following study which is a coparative study of a fully 3D frae odel of an existing reinforced concrete building between static nonlinear analysis and tie history analysis using flexibility-based finite eleent 659

6 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) corresponding to the given Acceleration Displaceent Response Spectru (ADRS spectru). Fig. 13 Top floor response to Fruili accelerogra 4. Push-Over Approach The push-over approach is a nonlinear static ethod described in (Eurocode 8, 2003). Its ipleentation in Z_SOIL.PC is described in detail in (A.Urbański & al, 2007). In short, force distribution (unitary or odal) is applied to the structure and onotonously increased. A capacity curve is obtained, drawing the total shear force at the base of the structure with respect to the top displaceent (in our case, the bonefro building). This curve is then expressed for an equivalent single degree of freedo oscillator and it is bi-linearized, giving birth to the so-called capacity spectru (Figure 14). Fig14.Capacity spectru The seisic action depends on the type of the structure, soil conditions and the zone of application. It is expressed as an acceleration-displaceent response spectru (ADRS), or deand spectru. The superposition of both the capacity and deand spectra leads to obtaining target displaceent for the single degree of freedo oscillator, and finally to target displaceent for the real structure see Figure 15. This target displaceent represents the axial horizontal displaceent which will be experienced by the structure during an earthquake Fig.15 Deand spectru and target displaceent 5. Nonlinear Frae Models in Z_Soil - version of the new Z_Soil progra, (M.Belgasia & al, 2007), (A.Urbański & al, 2007), which includes full dynaic capabilities. The first is a classical two-node, displaceent-based, Euler-Bernoulli frae eleent. The second is a two-node, force-based, Euler Bernoulli frae eleent. The ain advantage of the second eleent is that it is exact within the relevant frae eleent theory. This iplies that one eleent per frae eber (bea or colun see Figure. 17) is used in preparing the frae esh, thus leading to a reduction of the global nuber of degrees of freedo. The coplete theory on the forcebased eleent is found in (E. Spacone & al, 1996). For describing the section response, both eleents use a fibre discretization. Fibre sections autoatically account for oent-axial load interaction. In the present ipleentation, siple uniaxial constitutive laws have been ipleented for concrete and steel. Enhanceents to these laws are planned for future developents. Both eleents include both aterial and geoetric nonlinearities. Material nonlinearities derive fro the fibre nonlinear constitutive laws. Geoetric nonlinearities are included in the fraework of the co-rotational forulation. Geoetric nonlinearities are iportant for analyses carried out up to the collapse liit state. 6. Application The nonlinear response of the 3D odel of an existing building is presented. The building is a residential twostorey reinforced concrete building in Bonefro, Italy. It is representative of typical residential building construction in Italy in the 1970 s and 1980 s. The building is shown in Figure Bonefro building odeling and aterial properties A general 3D odel is presented in Figure 17. The following aterial properties are used for the fiber section (see table 1) 660

7 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) In our case b/d = 6.67 & h/t = 2.5 thus b'/b = so b'= Fig. 18 Rectangular section for floor beas 6.2 Pushover response and target displaceent of 3D odel Fig.16 Two-storey bonefro building used for analyses The design spectru for the building was obtained fro (Eurocode 8, 2003) using the local soil properties and the peak ground acceleration given by the new Italian seisic ap. The building is regular in height but is irregular in plan because of the eccentric position of the staircase. The pushover analysis of the building in the x direction is shown in Figure. 19. The deterination of the target displaceent is shown in Figure. 20 the target displaceent of single degree of freedo (SDOF) is 0.06 in order to have the displaceent of ulti degree of freedo (MDOF) we ust ultiply the results of SDOF by the ass participation factor which is equal for 3D odal to 1.28 thus the MDOF results Fig.17 3D odel Concrete:E=2.7e7 KN/ 2 Steel E=2.1e8 KN/ 2 υ=0.16 υ=0.2 f t =300 KN/ 2 f t = f c = KN/ 2 f c =3000KN/ 2 The following reinforceent details are used: Table 1. The reinforceent details of eleents eleents & caracteristics eleent b() h() Reinforceent 2Ф16 top; 2Ф16 botto; 2 Ф 14 iddle eleent Ф16 top ; 2Ф16 botto eleent Ф14 top ; 4Ф14 botto eleent Ф14 top ; 6Ф14 botto eleent Ф14 top ; 2Ф14 botto eleent Ф14 top ; 2Ф14 botto For the floor beas geoetry, an equivalent rectangular bea is used. The bea diensions are found iposing the sae inertia of the T-bea (see Figure 18). The results are : Fig. 19 Pushover response of 3D odel with odal load distribution in x direction In order to perfor the tie-history analyses, and to copare the results with those obtained with the pushover analysis, the first step is to select a set of spectrucopatible ground otions. In this application, three artificially generated ground otions are used. The ground otions are generated using a coputer progra based on the theory presented in (F.Sabetta & al, 1996). The ost iportant input paraeters are the epicentral distance, the agnitude and the type of soil. In this case, the Magnitude was set to 6.02, the epicentral distance to 22.8 k and the soil type to shallow. The results are three ground otions 661

8 displaceent() Sa/g Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) 0.5 Plastic Deand Spectru 0.4 Elastic Deand Spectru Elastic Response Curvilinear Response Spectru 0.3 Bilinear Response Spectru Sd () Fig. 20 Target displaceent for pushover response whose response spectra are shown in Figure 21. Figure. 22 copares the ean spectru to the three separate spectra. The tie history responses in ters of top floor center of ass displaceent are shown in Figure.23, Figure. 24 and Figure. 25. Fig. 22 Verification that three generated ground otions spectra are close to ean spectru 6.3 Nonlinear tie history of 3D odel Fig. 21 Generated accelerogra spectru vs EC8 spectru A tie history analysis of the 3D odel is presented with the three accelerogras of the previous section. The results for input ground otion applied in one direction only, are copared for pushover and tie-history analysis. Stiffness proportional Rayleigh daping is prescribed, with 5% daping at 2 Hertz. The resulting values for 0,008. The top- Rayleigh daping are 0 and floor response to the three accelerogras is shown in Figures 23, 24, E E E E E E E E E-02 Z-soil -2.00E E E E E E E E E E E E E E E E E+01 tie(s) Fig. 23 Response of Bonefro building to ground otion 1 applied in the x direction 662

9 Sa/g displaceent() shear force (kn) displaceent() Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) 3.50E E E E E E E E+00 Z-Soil -5.00E E E E E E E E E E E E E E E E E+01 tie(s) Fig. 24 Response of Bonefro building to ground otion 2 applied in the x direction 6.00E E E-02 Fig. 26 Base shear at selected top-floor displaceents fro tie history analyses: coparison with pushover curve pushover curve (large deforation) in 3D odel with acceleration which follows the first ode 1.60E E E E E E E E E E-02 Z-Soil 1.40E E E E E E E E E E E E E E E E E E+01 tie(s) Fig. 25 Response of Bonefro building to ground otion 3 applied in the x direction The three responses show a residual displaceent at the end of the tie histories, indicating a nonlinear response in parts of the building. The residual displaceent is larger under ground otion 1. The axiu displaceents due to the three ground otions are , , eters. Because only three ground otions are used, the design displaceent is Figure 26 shows on the sae plot the pushover curve and the based shear easured during the tie history analysis at given displaceents, 0.01, 0.02 and Because during a cyclic analysis the top displaceent reaches at several instances a given value, several points correspond to a given displaceent. Several points correspond to a given displaceent, the displaceent of 0,04 is attained at several instances. It is interesting to note fro Figure 26 that in the tie history analysis the axiu based shear is reached even for sall displaceents due to the cylcic nature of the response. This figure points out how the pushover curve gives different inforation copare to the three tie history. 6.4 Sensitivity of response to large deforation The pushover analyses are repeated using the large deforation analysis. The results are shown in the following figures 27. Fig. 27 Pushover response to large deforation of 3D odel with odal load distribution in x direction Fig E E E E E E E E E-01 displaceent () Plastic Deand Spectru Elastic Deand Spectru Elastic Response Curvilinear Response Spectru Bilinear Response Spectru Sd () The pushover analysis to large deforation of the building in the x direction is shown in Figure 27. The deterination of the target displaceent is shown in Figure 28 the target displaceent of single degree fo freedo (SDOF) is in order to have the displaceent of ulti degree of freedo (MDOF) we ust ultiply the results of SDOF by the ass participation factor which is equal for 3D odal to 1.28 thus the MDOF results The pushover curve becoes strain softening after yielding, 663

10 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) due to the geoetric nonlinearities. The increase in target displaceent coparing with target displaceent curve with sall deforation (Figure 20) is equal to 3.1% for odal distribution. 7. Sensitivity to seisic paraeters The sensitivity of the tie history analyses to the seisic paraeters used in generating the accelerogras with the progra by (F.Sabetta & al, 1996) is presented here. The three paraeters agnitude, epicenter distance and soil type were changed. The results are shown in the following tables. The reference values are agnitude = 6.02, epicenter distance = 22.8 k, shallow soil type. These are the values used for generating the three earthquakes used for the tie history analyses (see table 2 (a), 2 (b), 2 (c)). Table 2 (a) Sensitivity to agnitude seisic paraeter Maxiu response The for static pushover analysis Maxiu response under earthquakes in x direction for Tie history analysis Magnitude= 5.72 Max = 0.06 Magnitude= 6.02 Max = 1.28* 0.06 Max= Max = Magnitude= 6.32 Max = Table 2 (b) Sensitivity to epicentre seisic paraeter Table 2 (c) Sensitivity to soil type seisic paraeter Maxiu response The for static pushover analysis Maxiu response under earthquakes in x direction for Tie history analysis Soil type shallow Max = 1.28* 0.06 Max= Max = Soil type deep Max = The results are suarized in copact for in Figure 29 The agnitude and epicenter results are noralized with respect to the reference values indicated in bold in the above tables. There sees to be a very high sensitivity to the earthquake agnitude, but further studies are needed. Noralized dax Noralized Magnitude or Epicenter Dist. Magnitude Epicenter shallow soil deep soil Epicenter distance=11.4k Epicenter distance=22.8k Epicenter distance=45.6k Fig 29 Sensitiviy of response (expressed in ters of d ax at top floor) Discussion of the results Maxiu response The for static pushover analysis Maxiu response under earthquake s in x direction for Tie history analysis Max = 1.28* 0.06 Max= Max = Max = Max = The coparison between pushover and dynaics gives difference of 8%. - The sensitivity to Magnitude, in dynaics a (+-5%) variation in reference agnitude value (6.02) yields a variation in ax displaceent of( +33% and 15%) (see table 2a). - The sensitivity to epicenter distance, in dynaics (via Sabetta progra) a (+100%, -50%) variation in reference agnitude value (22.8 K) yields a variation in ax displaceent of (+25%, -8%) (see table 2b). - The sensitivity to soil, in dynaics (via Sabetta progra): shallow and deep soil type the difference is about 8% (see table 2a). - The axiu base shear indicated by the pushover analysis is systeatically reached for alost any 664

11 Mourad Belgasia et al International Journal of Current Engineering and Technology, Vol.3, No.2 (June 2013) axiu top displaceent, this is probably indicative of significant influence of 2 nd 3 rd odes. - Large deforation induces global softening and increase target displaceents in 3D pushover +3.1% for odal loading 0.062/ Conclusions The eleent forulation is based on force interpolation functions strictly satisfy eleent equilibriu and, thus, belongs to the category of flexibility-based eleents. The use of exact force interpolation functions in the eleent requires fewer eleents for the representation of the non-linear behaviour of a structure, and gives a good nuerical results without difficulties. We can now copare the pushover and the tie history analysis; in the case of tie history analysis the Eurocode propose to take the axiu response, if we apply to structure less than 7 earthquakes, and copare it to the target displaceent of ulti degree of freedo pushover odel. The axiu response to the three earthquakes is In pushover analysis study the target displaceent of a ulti degree of freedo is 0,077. We can say that the result of tie history and pushover analysis are very close to eath other. There are no doubt advantages in using nonlinear analyses vs using linear ethods. Most iportantly, nonlinear analyses allow designers to follow ore closely the nonlinear response of buildings and bridges to the design earthquakes corresponding to the ultiate and collapse liit states. The displaceent of 0,04 is attained at several instances. It is interesting to note fro figure26 that in the tie history analysis the axiu based shear is reached even for sall displaceents due to the cylcic nature of the response.figure26 points out how the pushover curve gives different inforation copare to the three tie history. Pushover analysis provides the axiu base shear one can expect for a given axiu target displaceent (corresponding to a given earthquake intensity). The tie histories provide not only the axiu values, but the entire history. For the exaple at hand, tha ax displaceent of the tie history is , and the axiu base shear is approxiately equal to 1500 kn, but during the tie history this value of the base shear can be reached at several instances and for different values of the top displaceent. On the pushover curve this translates into a single point that provides axiu displaceent and axiu base shear that can be expected for that given earthquake. References S. Kaba, and S.A. Mahin (1984), Refined odeling of reinforced concrete coluns for seisic analysis, Earthquake Engineering Research Center, University of California, Berkeley. EERC Report 84/03 C.A. Zeris, and S.A Mahin (1988), Analysis of reinforced concrete bea-coluns under uniaxial excitation, Journal of Structure Engineering, Vol 114, pp C.A. Zeris, and S.A Mahin (1991), Behavior of reinforced concrete structures subjected to biaxial excitation, Journal of Structure Engineering, Vol 117, pp V. Ciapi, and L. Carlesio (1986), A nonlinear bea eleent for seisic analysis of structures, Proceedings of the 8th European conference, Lisbon. F. F. Taucer, E. Spacone, and F.C Filippou (1991), A fiber beacolun eleent for seisic response analysis of reinforced concrete structures, EERC Report 91/17, Earthquake Engineering Research Center, University of California, Berkeley. E. Spacone (1994), Flexibility-based finite eleent odels for the nonlinear static and dynaic analysis of concrete frae structures, Ph.D. Dissertation, Departent of Civil Engineering, University of California, Berkeley. Eurocode 8 (2003), Design of Structures for Earthquake Resistance, European Coittee for Standardization. P. Fajfar, V. Kilar, D. Marusic, and I. 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