OS MODELER - EXAMPLES OF APPLICATION Version 1.0. (Draft)

Transcription

1 OS MODELER - EXAMPLES OF APPLICATION Version 1.0 (Draft) Matjaž Dolšek February 2008

2 Content 1. Introduction Four-storey reinforced concrete frame designed according to EC Description of the structure Pseudo-dynamic tests on full scale specimen Mathematical modeling Gravity loads Beam effective width Moment-rotation relationship for plastic hinges Eigen value analysis Pushover analysis Nonlinear dynamic analysis and comparison with pseudo-dynamic test Incremental dynamic analysis Determination of the target displacement by N2 method Determination of the target displacement by SDOF-IDA approach Influence of some modeling uncertainty on the response of structure ICONS frame Description of the structure Pseudo-dynamic tests Mathematical modeling Pushover analysis for model H and model T Nonlinear dynamic analysis and comparison with pseudo-dynamic test References... 49

3 1. Introduction The objective of this report is to show the capability of the OS Modeler [Dolšek, 2008], which consists of a set of Matlab functions used for determination of the structural model, for performing nonlinear analyses by employing program OpenSees [PEER, 2007] and for post-processing of the analysis results. In the first example the pushover analysis, nonlinear dynamic analysis and IDA analysis is presented for the four-storey reinforced concrete frame building, which has been designed according to previous versions of Eurocodes 2 and 8 [Fardis (Ed.), 1996]. The results of the nonlinear dynamic analyses are compared with the results of the pseudo-dynamic tests, which were performed in the ELSA Laboratory in Ispra [Negro and Verzeletti, 1996, Negro et al., 1996]. The results of the pushover analysis are used to determine the target displacement for a defined seismic loading. The target displacement is determined according to the N2 method [Fajfar, 2000] and according to the SDOF- IDA approach. Further on the influence of some selected modeling uncertainties on the seismic response is presented. The four-storey frame, designed to reproduce the design practice in European countries about forty to fifty years ago [Carvalho and Coelho (Eds.), 2001], is the second presented example. In this case only the pushover analysis and the nonlinear dynamic analysis are performed. The reuslts of the nonlinear dynamic analysis are presented with the results of the pseudo-dynamic test [Carvalho and Coelho (Eds.), 2001]. 1

4 2. Four-storey reinforced concrete frame designed according to EC Description of the structure The elevation and plan of the four-storey reinforced concrete building, as well as the typical reinforcement of columns and beams are shown in Figure 2.1. The height of the bottom storey is 3.5 m. In other stories the height is reduced for 0.5 m. The building has two bays in each direction with the raster of 5 m in the X direction and with the raster of 4 and 6 m in the Y direction. This direction was also the direction of loading in the pseudo-dynamic test. The columns are 40/40 cm. The only exception is the column D (Figure 2.1), which is 45/45 cm. All beams have rectangular cross section with 30 cm width and 45 cm height. The slab has the thickness of 15 cm. Figure 2.1. The four-storey reinforced concrete frame building. The concrete C25/30 is used for this building and the B500 Tempcore reinforcing steel for which the characteristic yield strength is 500 MPa. Since the pseudo-dynamic test was performed for the studied building more information regarding material characteristics is available. In Table 2.1 the mean concrete strength and modulus of elasticity is presented. The mean concrete strength differs from 32 MPa to 56 MPa. The smallest strength corresponds to columns in third storey and the highest concrete strength corresponds to beams in first storey. Similarly, the modulus of elasticity varies from 28.5 GPa to 35.3 GPa. It should be emphasized that the material characteristics of concrete significantly differs 2

5 from the nominal material characteristics for C25/30, which are, according to Eurocode 2 [CEN, 2004], 33 MPa for mean concrete strength and 31 GPa for modulus of elasticity. The yield strength, ultimate strength and the corresponding deformation of reinforcing bars are presented in Table 2.2. The mean yield strength exceeds the characteristic yield strength for 10 to 20%, depending on the diameter of the reinforcing bar. The structure was designed according to previous versions of Eurocodes 2 and 8 [Fardis (ed.), 1996]. The design spectrum was defined based on the prescribed peak ground acceleration of 0.3 g, the soil type B, the ductility class high (DCH) and the behavior factor q=5. In addition to the self weight of the structure the 2 kn/m 2 of permanent load was assumed in order to represent floor finishing and partitions, and 2 kn/m 2 of live load was also adopted. The corresponded masses are 87 t, 86 t and 83 t for bottom, second and third, and top storey, respectively. These masses were adopted also in the pseudo-dynamic test on the full scale specimen. The design base shear versus the weight of the structure corresponded to about 16%, since the design base shear was 529 kn [Fardis (ed.), 1996]. The longitudinal and shear reinforcement of beams and columns is presented in Figures 2.1, 2.2 and 2.3. The reinforcement is presented only for the frames which are parallel to the direction of loading in the pseudo-dynamic test (Figure 2.1). Columns are reinforced with 8 to 12 bars with the total reinforcement ratio from 1% to 1.9% as presented in Figure 2.1. The diameters used for longitudinal reinforcement in columns vary from φ16 at upper stories and φ20 or φ25 in the bottom and in some cases in the first storey. Stirrups φ10/10 are usually used in the critical regions of columns (Figures 2.1 and 2.3). Table 2.1. The mean concrete strength and modulus of elasticity. Construction element Column Beam Storey Concrete strength f cm (MPa) Modulus of elasticity E cm (MPa)

6 Table 2.2. The mean yield strength of reinforcement bars. Bar diameter (mm) Yield strength (MPa) Ultimate strength (MPa) Ultimate deformation (%) Figure 2.2. The longitudinal and shear reinforcement in columns. Bars φ14 in combination with bar φ12 are used for the longitudinal reinforcement in beams. In all cases more than half longitudinal reinforcement in the beams is placed at the bottom of the beam (Figures 2.1 and 2.2). The detailed data of lap-splicing and anchorage of bars is not available Pseudo-dynamic tests on full scale specimen Different pseudo-dynamic tests were performed for the studied building at the European Laboratory for Structural Assessment (ELSA, Ispra) [Negro and Verzeletti, 1996, Negro et al., 1996]. The pseudodynamic tests were performed not only for the bare frame but also for different configurations of the masonry infills. For example, the uniformly infilled bare frame is presented in Figure

7 Figure 2.3. The longitudinal and shear reinforcement in beams. The so called low- and high level tests were performed on the same bare frame. The accelerogram used in the test is generated from the real accelerogram recorded during the 1976 Friuli Earthquake. The accelerogram and the corresponding spectrum are presented in Figure 2.5. It is shown that the acceleration spectrum shape approximately corresponds to the EC 8 shape of spectrum, normalized to peak ground acceleration of 0.3 g. The scale factors 0.4 (0.12 g) and 1.5 (0.45 g) for the acceleorgram were used for the low- and high- test, respectively, and the zero viscous damping was assumed in both tests. After the low-level test no visible damage were observed. It was assumed that structure become practically in the elastic region. During the high-level test crakes opened and closed in the critical regions of the beams of the first three stories and of most columns. Neither spalling of the concrete cover nor local buckling of reinforcement was observed. Besides the cracks at the end of beams and columns, which were considered as evidence of yielding in the rebars and of bond-slip in the joints, the specimen remained quite undamaged. However, the fundamental period of the building after the high level test was about 1.22, which is about two times higher than the period measured on the undamaged building (0.56 s). Quite uniform damage pattern was observed in both test, with exception of fourth storey, for which the drifts are significantly lower than these measured in other stories. The maximum top displacement in 5

8 Figure 2.4. The tested frame with masonry infills. Figure 2.5. The accelerogram used in the pseudo-dynamic test and the corresponding elastic acceleration spectrum compared with EC8 spectrum Mathematical modeling The mathematical model of the studied building is developed on basis of Eurocode 8 [CEN, 2004] requirements. According to Eurocode 8 a bilinear force deformation relationship may be used at the element level as the minimum representation of non-linear behaviour of structural elements. In ductile elements, expected to exhibit post-yield excursions during the response, the elastic stiffness of a bilinear relation should be the secant stiffness to the yield-point. The trilinear force-deformation relationships, which take into account pre-crack and post-crack stiffnesses, are allowed. Also, zero post-yield stiffness may be assumed. However, for elements in which the strength degradation is expected, the strength degradation has to be included in the force-deformation relationship. It is also recommended to use the mean values of the properties of the materials. For new structures, mean values of material properties may be estimated from the corresponding characteristic values. Gravity 6

9 loads has to be applied to appropriate elements of the mathematical models. Therefore the axial forces due to gravity loads should be taken into account when determining force-deformation relations for structural elements. The same basic principles of the modeling as presented by Fajfar et al. [2006] were employed also in this study. These principles are full compliance with Eurocode 8. Beam and column flexural behavior was modeled by one-component lumped plasticity elements, composed of an elastic beam and two inelastic rotational hinges (defined by the moment-rotation relationship). The element formulation was based on the assumption of an inflexion point at the midpoint of the element. For beams, the plastic hinge was used for major axis bending only. Additional plastic hinges in beams were modeled in series after the primary plastic hinge in order to simulate bar slip, which was observed in the experiment. For columns, two independent plastic hinges for bending about the two principal axes were used. All analyses were performed by OpenSees [PEER, 2007], which is an object-oriented software framework for simulation applications in earthquake engineering using finite element method, developed at the Pacific Earthquake Engineering Research Center. The tcl input files for the OpenSees were automatically generated with the OS modeler developed in Matlab [Dolšek, 2008]. Such approach has an advantage, since all the moment-rotation envelopes of plastic hinges can be automatically generated, which is not an option in OpenSees, when plastic hinges are represented by zero length elements. In the next Sections more precise description of mathematical modeling, especially, the procedure for determination of moment-rotation envelopes for plastic hinges, is presented Gravity loads Gravity load was modeled as a uniformly distributed load on beams and as point loads on columns. The uniformly distributed load on beams results from the self weight of slab and beams and also from the permanent load on slab. The point loads at top of the columns are used to model only the self weight of columns. The specific weight of the reinforced concrete 25 kn/m 3 was adopted. Since the gravity load slightly exceeds the weight calculated from the mass, which was assumed in the pseudo-dynamic experiment, the live load 2 kn/m 2, which was used for the design purpose, has been slightly decreased in order to obtain the same gravity load as result from the storey mass used in the pseudo-dynamic test. Values 7

10 for uniformly distributed gravity load as well as point load on columns are presented in Tables 2.3 and 2.4, respectively. Table 2.3. The uniformly distributed gravity load on beams. Beam Storey 1 g (kn/m) Storey 2 g (kn/m) Storey 3 g (kn/m) Storey 4 g (kn/m) Table 2.4. The point loads at the top of columns. Column Storey 1 G (kn) Storey 2 G (kn) Storey 3 G (kn) Storey 4 G (kn) Beam effective width Reinforced concrete beams and the slab act as a monolithic section since they are usually constructed at the same time. Therefore a contribution of slab to the stiffness and strength of beam has to be considered in the seismic assessment, although there are no straight rules how to define the beam 8

11 effective width. Different beam effective width can be used for modeling stiffness and strength especially in the design process. According to Paulay and Priestley [1992] the flange contribution to stiffness in T and L beams is typically less than the contribution to flexural strength. They suggested that the contribution of slab to the beam effective width for modeling of the beam stiffness should be half of that used for modeling the flexural strength of the beam if the flange is in compression. These values for beam effective width are presented in Table 2.5. However, different values are used for the effective tension reinforcement. Paulay and Pristeley [1992] recommended that in T and L beams, built integrally with floor slabs, the longitudinal slab reinforcement placed parallel with the beam, to be considered effective in participating as beam tension (top) reinforcement, in addition to bar placed within the web width of the beam. The tension reinforcement should include all bars within the effective width b eff, which may be assumed to be the smallest of the following one-fourth of the span of the beam under consideration (l b /4), extending each side from the center of the beam section, where a flange exists one-half of the span of a slab, transverse to the beam under consideration, extending each side from the center of the beam section where a flange exists one-forth of the span length of a transverse edge beam, extending each side of the center of the section of that beam which frames into an exterior column an is thus perpendicular to the edge of the floor. Table 2.5. The beam effective width according to Paulay and Pristley [1992]. Flexural strength (flange in compression) b eff Flexural stiffness b eff b w +16h s b w +8h s b w +(b eff,1 +b eff,2 )/2 b w +(b eff,1 +b eff,2 )/4 b b /4 b b /8 Figure 2.6. The beam effective width. 9

12 According to FEMA 356 [2000] the combined stiffness and strength for flexural and axial loading shall be calculated considering a width of effective flange on each side of the web equal to the smaller of the provided flange width eight times the flange thickness half the distance to the next web one-fifth of the span for beams. When the flange is in compression, both the concrete and reinforcement within the effective width shall be considered effective in resisting flexure and axial load. When the flange is in tension, longitudinal reinforcement within the effective width and that is developed beyond the critical section shall be considered fully effective for resisting flexural and axial loads. The Eurocode 8 [CEN, 2004] also prescribes that slab reinforcement parallel to the beam and within the effective flange width, should be assumed in the design process to contribute to the beam flexural capacities, if it is anchored beyond the beam section at the face of the joint. It is suggested that for primary seismic beams framing into exterior columns, the effective flange width b eff is taken, in the absence of a transverse beam, as being equal to the width of the column, or, if there is a transverse beam of similar depth, equal to this width increased by 2h s on each side of the beam. Also Eurocode 2 [CEN, 2004] suggests values of beam effective widths for all limit states (strength and stiffness) and are to be based on the distance l 0 between points of zero moments beff = beff, i + bw b (0.1) beff, i 0, 2 bi 0,1 l0 0, 2 l 0 = + (0.2) b eff, i b (0.3) i where b eff is the beam effective width, b w is the width of the beam, b i is the one half of the distance between the beams and l 0 is the distance between the points of zero moments. The distance l 0 should be taken as l b /2 in the seismic analysis. More extensive review regarding the beam effective width can be found in Stratan and Fajfar [2002]. In the mathematical model used in this case study the beam effective width was calculated according to the Eurocode 2. The results are presented in the Table 2.6. The beam effective width is the highest for the beams connected to the interior column and is in the range from 110 to 150 cm. For the exterior beams the beam effective width is in the range between 70 and 90 cm. 10

13 Table 2.6. The beam effective width considered in analysis. Beam b eff, EC2 (cm) B3, B5 70 B1, B2, B11, B12 80 B8, B10 90 B4 110 B6, B7 130 B Moment-rotation relationship for plastic hinges The only nonlinear behavior of the columns and beams was modeled by moment-rotation plastic hinges, which were considered at both ends of structural member. It was assumed that the momentrotation relationship is tri-linear with the material softening after the maximum moment (Figure 2.7). The three points at the increasing part of the moment-rotation envelope represent the points at the cracking of the concrete (CR), the point at the yielding of reinforcement (Y) and the point at maximum moment (M). After the maximum moment is attained the linear strength degradation is assumed and defined with the point at the near collapse (NC). Figure 2.7. Moment-rotation relationship for plastic hinges in beams and columns. Since the plastic hinges were modeled with the ZeroLengthSection elements, implemented in OpenSees [PEER, 2007], only the plastic rotation are used in the definition of the moment-rotation relationship. 11

14 Cracking (M cr ), yield (M y ) maximum (M m ) moment and moment at near collapse limit state (M nc ) The moment at the cracking of concrete is determined based on the elastic analysis of the cross-section according to the following equation P Mcr = W fct A (0.4) where W is section modulus, P is the axial force (compression is negative), A is cross-section area and the f ct is the mean tensile strength determined according to EC 2 f ctm = 0.3( f ) 23 (0.5) cm where f cm is the mean concrete compressive strength. The yield and the maximum moment were calculated based on the moment-curvature analysis of the cross-section. Axial forces due to the vertical loading were taken into account in columns, while in the beams zero axial force was assumed. For the concrete the stress-strain relationship prescribed by EC 2 for nonlinear analysis was adopted, while for the steel the elasto-plastic strain-stress relationship was assumed in the analysis. The yield moment M y is defined when the strain in the first reinforcing bar is equal to the yield strain of the steel f sy /E s, where f sy is the yield stress of the steel and E s the corresponding modulus of elasticity, which is assumed kn/cm 2. The maximum moment M m is directly determined from the moment-curvature analysis. The moment-curvature analysis was determined until the ultimate deformation of concrete ε cu =-3.5 or the ultimate deformation of reinforcing steel ε su =10 is reached. In some cases it may happened that the section collapse before the yield deformation is reached in the reinforcing bar. In this case the yield moment is determined based from the assumption of equal areas, which are determined from the moment-curvature envelope and the idealized M cr and M y points. The moment at the near collapse limit state M nc is defined according to EC 8 as the 0.8M m. Rotation at M cr and M y The hinge rotation corresponding to the M cr is calculated by assuming the linear curvature along the length of the element 12

15 M L 3EI cr 0 Θ cr = (0.6) where the M cr is the cracking moment, L 0 is the length between the end of the element and the zero moment point, in our case is for beams and columns assumed half of the element length, and the EI is the product of the modulus of elasticity an moment of inertia. Since only the plastic rotation is needed to define in the plastic hinge, the rotation according to Eq. (0.6) is used to define the elastic rotation at the plastic hinge. Of course, the elastic rotation increases or decreases if the moment increases or decreases. The yield rotation of plastic hinge is determined according to Fischinger [1989] 2 L 0 M cr M cr M cr Θ y = 1+ φcr + 2 φ y (0.7) 6 M y M y M y where L 0 is the length between the end of the element and the zero moment point, in our case is for beams and columns assumed half of the element length, M cr and M y are cracking and yield moment, respectively, φ cr = M cr EI is the curvature corresponding to M cr and φ y is the curvature, which corresponds to M y, and is directly determined from moment-curvature analysis of the cross-section. Since the Eq. (0.7) represents an elastic and plastic part of the rotation the plastic part of the rotation was determined according to following expression M L y 0 Θ y, p=θy. (0.8) 3EI Rotation at maximum moment and near collapse limit state for columns Recently a non-parametric empirical approach, called the conditional average estimator (CAE) method [Grabec and Sachse, 1997], has been implemented for the estimation of the flexural deformation capacity of reinforced concrete rectangular columns [Peruš, Fajfar, Poljanšek, 2006]. The CAE method enables relatively simple empirical modeling of different physical phenomena. The method has been adapted by the authors and has already been used for predicting the capacity of RC walls in terms of shear strength, ductility, ultimate drift and failure type [Peruš, Fajfar and Grabec, 1994], and for the modeling of attenuation relationships [Fajfar and Peruš, 1997]. In the CAE method the experimental database is need for prediction of the parameter, which is the subject of the prediction. According to Peruš, Fajfar and Poljanšek [2006], the PEER Structural Performance Database complied at the University of Washington, with 156 test specimens, was selected. 13

16 Four different parameters were selected in order to determine the rotation at maximum moment and the rotation at near collapse limit state Θ nc. These parameters are: axial load index ( P ) shear span index ( L ) concrete compressive strength (f cm ) confinement index multiplied by confinement effectiveness factor αρ s = αρs fsy fcm. The axial load index is defined as the ratio between the axial load P (positive for compression) and the Po = bh fcm, where b represents the width of the compression zone, h the depth of rectangular column in the direction of loading and f cm concrete compressive strength. The shear span index is defined as L v /h, where L v =M/V the length from the location of plastic hinge and the zero moment, and h the depth of rectangular column in the direction of loading. The confinement effectiveness factor is defined according to Eurocode 8 [CEN, 2004] 2 s h s h b i α = bo 2h o 6hobo (0.9) where b o and h o represents the width and the depth of the confined core (measured between the centrelines), respectively, s h is the spacing of stirrups and b i is the centreline spacing of the longitudinal bars (indexed by i) laterally restrained by a stirrup corner or a cross-tie along the perimeter of the cross-section. The parameter ρ s = Asx bsh is the ratio between the transverse steel parallel to the direction of loading and the area defined as the product of the width of the column and the spacing of stirrups s h. Θ m The input parameters b l for determination of rotation at maximum moment collapse limit state Θ nc, which are P [ 0 0.6], [ 2 6] αρ s [ ] were normalized bl bl,min bl = b b l,max l,min Θ m and rotation at near L, [ ] f and cm (0.10) where b l,min and b l,max are the selected lower and upper bounds of the input parameters presented above in the brackets. The rotation Θ, which can be equation Θ m or Θ = Θ nc, is then calculated according to following N AnΘn (0.11) n= 1 14

17 where N is the number of experiments in the database, Θ n is the rotation, e.g. the rotation at the maximum moment or the rotation at near collapse limit state, of the n-th experiment from the database and A n = a n N i= 1 a i (0.12) where a n 1 = exp ( 2 ) D ( b ) 2 l bnl 2w D 2 2 π wn 1 wnd l= 1 nl. (0.13) The parameter b l is the l-th input parameter from among P, L, f cm and αρ s, b nl is the l-th parameter of the n-th experiment from the database and D is the number of the input parameters. The w nl is the so called smoothing parameter. It determines how fast the influence of data in the sample space decreases with increasing distance from the point whose coordinates are determined by the input parameters. The selection of the proper value for w nl is important. According to Peruš, Poljanšek and Fajfar [2006] the parameter w nl is not constant since the non-constant value of the parameter w nl improves the prediction of the rotations. It has trapezoidal shape as presented in Figure 2.8. Figure 2.8. Rule for a non-constant smoothness parameter w nl. The CAE method employed for the prediction of the rotation at maximum moment and at the limit state of near collapse has proven to be an efficient method if an appropriate database is available. More details about the CAE method implemented for the estimation of characteristic rotation for rectangular columns can be found in Peruš, Poljanšek and Fajfar [2006]. Rotation at maximum moment and at near collapse limit state for beams The rotation at near collapse limit state for beams was determined according to Eurocode 8-3 [CEN, 2005] 15

18 θ nc ( ω ') ( ω) sy γ max 0.01, s d L αρ ν v fcm = 0.16 ( 0.3 ) fcm 25 ( 1.25 γel max 0.01, h f 100ρd ) (0.14) where γ d is the parameter considering the seismic detailing, which can be assumed instead of 1 in the members without detailing for seismic resistance. In our example it was assumed 0.85, which is a suggested value in one of the former Eurocode 8-3. The parameter γ el takes into account the importance of the structural member. In the case of primary seismic element γ el = 1.5, otherwise 1.0. The normalized axial force ν is defined like the axial load index P in the previous Section, and is assumed 0 for beams. Similarly α, ρ s, L v and h have the same meaning as defined in previous Section. The mechanical reinforcement ratio of the tension ω and the compression reinforcement ω are defined as A f A f = = (0.15) st sy sc sy ω, ω Ac fcm Ac fcm where A st and A sc are areas of longitudinal reinforcement in tension and compression, in our case determined at the maximum moment resulted from moment-curvature analysis. A c is the gross area of the cross-section and the f sy and f cm are the yield and the compressive strength of steel and concrete, respectively. Lastly, the parameter ρ d is the steel ratio of diagonal reinforcement in each diagonal direction. In order to determine the rotation at maximum moment the ratio between the rotation at total collapse and the rotation at maximum moment was assumed to be constant for all beams. A value of 3.5 was adopted. From rotation at near collapse limit state (Eq.(0.14)) and from the assumed ratio between the rotation at total collapse and the rotation at maximum moment the rotation at maximum moment can be calculated. Some other researchers [Haselton, 2006] proposed the empirically based formulas for prediction of the rotation at maximum moment. However the observed dispersion between the empirical results and results predicted based on empirical formula is very high. Bond-slip model Many authors have studied the bond-slip problems by performing experiments or by developing the model for simulating the bond-slip in the reinforced concrete members [Park and Pauly, 1975, Filippou et al, 1992, Saatcioglu et al., 1992, Ayoub and Filippou, 1999, Sezen and Moehle, 2003]. These studies have shown that elongation and slip of the tensile reinforcement, especially at the beamcolumn interface, could result in significant fixed-end rotations, which are not included in the usual 16

19 flexural analysis. This phenomenon has been also observed during the pseudo-dynamic test of the studied structure [Negro et al., 1996] and also considered in this study. In this study the simple model proposed by Park and Paulay [1975] and used also by Filippou et al. [1992] was adopted. The model is based on the assumption of uniform bond stress μ along the development length l d (Figure 2.9). Therefore the stress in the reinforcing bar σ s uniformly decreasing in the region of the length l d (Figure 2.9). It is also assumed that the anchorage length of the bar is sufficient. It means that the yield stress of the reinforcing bar can be developed (σ s =f sy ). The force in the reinforcing bar can be written as 2 π db F = σs = μ πdbl d (0.16) 4 where d b is the average bar diameter. The required development length can be simply expressed from Eq. (0.16) l d σ sdb =. (0.17) 4μ The slip or the elongation of the reinforcing bar, assuming the Hook law σ s =E s ε s, can be determined according to following equation ld ld σs σsld ε s El s d Es. (0.18) s = dl = l dl = The slip s can be now expressed if the development length l b in Eq. (0.18) is substituted with an Eq. (0.17) 2 2 σ s db s =. (0.19) 8E μ s According to Park and Paulay [1975] the uniform bond stress is approximated as 1.35 f cm, where f cm is the mean concrete strength. The rotation Θ s due to the bar slip can be simply calculated as the ratio between the bar slip and the length between the reinforcement layers (Figure 2.9). Based on adopted bond-slip model, the additional plastic hinge was modeled at both ends of beam hinges. The bilinear moment rotation was assumed. The yield moment and the hardening for the plastic hinge were assumed as calculated from the cross-section analysis of the beam section. Therefore this model increase the deformation due to the bar slip but does not reduce the overall strength of structure, since the sufficient anchorage length was assumed. 17

20 Figure 2.9. Determination of the bar slip Eigen value analysis The periods and the first three mode shapes are presented in Table 2.7 and 2.8. The first and the second periods is practical the same as the period, which was measured on the full scale specimen. The first mode shape is predominantly translational in X direction, second mode is translational in Y direction and third mode and the third mode is predominantly torsional. The first and the second mode shape will be used for determination of the horizontal force shape for pushover analysis. Table 2.7. Periods for the first six modes and the measured period on the full scale specimen. T 1,exp (s) T 1 (s) T 2 (s) T 3 (s) T 4 (s) T 5 (s) T 6 (s) Pushover analysis The force pattern for pushover analysis is calculated by multiplying mode shape (Table 2.8) and the storey masses (Section 2.1). The values for force pattern are practically the same for the analysis in X and Y direction, and amount to 0.28, 0.60, 0.87 and 1, respectively from first to top storey. The pushover curves are presented in Figure The shape of the pushover curves is almost the same for both directions and senses of loading. The base shear versus weight ratio amounts to about 0.32 for all pushover curves presented in Figure The maximum base shear is obtained at the top displacement of about 30 cm. After this displacement the structure degrades. The 20% reduction of the maximum base shear is obtained at the top displacement of about 80 and 70 cm, respectively, for analysis in X direction and Y direction. 18

21 The drift angles, which correspond to the maximum base shear and to the 80% of the maximum base shear measured in the degrading part of pushover curves, are presented in Tables 2.9 and Quite uniform distribution of the drifts along the stories is observed. The only exception is the drift at the top storey, which is smaller in comparison to the drifts obtained in other stories. The maximum drift is about 0.03 and 0.07, which correspond to the maximum base shear and 80% of the maximum base shear in the degrading part, respectively. Table 2.8. Mode shape 1, 2 and 3. Mode 1 Storey UX UY RZ Mode 2 Storey UX UY RZ Mode 3 Storey UX UY RZ The yielding of reinforcement, the maximum moment and the rotation at the near collapse limit state at the hinge level of beams and columns is indicated on the pushover curves (Figures 2.11 and 2.12). These results can be used for different purpose, for example, to obtain the damage on the structure for calculated target displacement, or to obtain the α u /α 1 ratio, used for determination of the behavior factor according to Eurocode 8 [CEN, 2005]. In our case the α u /α 1 ratio is equal to 1084/751=1.44 and 19

22 1102/715=1.54 for the analysis in X and Y direction, respectively. Note that the base shear force for analysis in X and Y direction (positive sign) amounts, respectively, to 1084 kn and 1102 kn, and that the base shear force, which corresponds to first yielding in beam is 751 kn for the pushover analysis in X and 715 for pushover analysis in Y direction. Figure Pushover curves for positive and negative X and Y direction. Table 2.9. Maximum drift angle determined at maximum base shear. Drift angle at maximum base shear Storey 1 Storey 2 Storey 3 Storey 4 + X X Y Y Table Maximum drift angle determined at 80% base shear measured in the degrading part of pushover curve. Drift angle at 80% base shear in the degrading part Storey 1 Storey 2 Storey 3 Storey 4 + X X Y Y

23 Figure The relationship between the damage of plastic hinges and pushover curve for analysis in X direction. The damage of hinges is indicated by yielding of reinforcement, maximum moment and rotation at NC limit state for columns and beams. 21

24 Figure The relationship between the damage in plastic hinges and pushover curve for analysis in Y direction. The damage in hinges is indicated by yielding of reinforcement, maximum moment and rotation at NC limit state for columns and beams. 22

25 2.6. Nonlinear dynamic analysis and comparison with pseudo-dynamic test The nonlinear dynamic analysis was performed for the same ground motion record, which was used in the pseudo-dynamic test (Figure 2.5). This enables comparison between experimental and calculated results. Since two pseudo-dynamic tests were performed at ELSA Laboratory also two nonlinear dynamic analyses were performed in a series. First analysis corresponds to the so called low (L) test, for which the peak ground acceleration is about 0.12 g. The peak ground acceleration of the 0.45 g, was used for the second analysis, so called high (H) test. Zero viscous damping was assumed in pseudo-dynamic test and consequently also in the nonlinear dynamic analysis. Storey displacements for L and H test are presented in Figures 2.13 and Despite very simple mathematical model very good comparison between measured and calculated storey displacement can be observed. The storey shear time histories are presented in Figures 2.15 and A close relation between calculated and measured storey shear forces is observed. Since peak value of the calculated base shear force is a bit underestimated, if compared with peak value of measured base shear force (Figure 2.16), it can be concluded that the strength of the structure is slightly underestimated Incremental dynamic analysis The Incremental Dynamic Analysis (IDA) was developed by Vamvatsikos and Cornell [2002, 2004] and it is a powerful tool for estimation of seismic demand and capacity for multiple levels of intensity and can be used for different applications (e.g. [Dhakal et al, 2006], [Fragiadakis et al, 2006]). However, it requires a large number of inelastic time-history analyses. The IDA analysis was performed for the same ground motion record as used in the pseudo-dynamic test (Figure 2.5). The IDA curves, presented in terms of peak ground acceleration versus maximum top displacement and peak ground acceleration versus maximum storey drift angle, are shown in Figure The peak ground acceleration, which causes the dynamic instability of the structure, determined with the precision of 2%, is about 1.5 g. This is rather high value. However, structure stars degrading if peak ground acceleration is about 1 g. This can be concluded if capacity diagram is compared with the IDA curve (Figure 2.16). 23

26 Figure Measured and calculated displacement for the L test. 24

27 Figure Measured and calculated storey displacement for the H test. 25

28 Figure Measured and calculated storey shear forces for the L test. 26

29 Figure Measured and calculated storey shear forces for the H test. 27

30 Figure The IDA curve and the capacity diagram presented for peak ground acceleration versus maximum top displacement and maximum storey drift angle. More details regarding the relation between the damage on the structure and the peak ground acceleration is presented in Figure The damage at the hinge level is indicated with yielding of reinforcement, maximum moment and the rotation at near collapse limit state, which is defined at the 85% of the maximum moment measured in the degrading part of the moment-rotation relationship. It can be observed that yielding of the reinforcement starts at peak ground acceleration of about 0.2 g, maximum moment in columns in first storey is reached at peak ground acceleration of 0.6 g and the near collapse limit state in beam hinges starts at about 1.0 g Determination of the target displacement by N2 method The N2 method is a simplified nonlinear method for seismic assessment of structures [Fajfar, 2000], which combines pushover analysis of a multi degree-of-freedom (MDOF) model with the response spectrum analysis of an equivalent single-degree-of-freedom (SDOF) model. The N2 method has been implemented in Eurocode 8 (EC8) [CEN, 2005], and extended to infilled frames [Dolšek and Fajfar 2004, 2005] and to probabilistic seismic assessment [Dolšek and Fajfar, 2007]. In this Section the target displacement is calculated according to Eurocode 8 [CEN, 2005] for the structure subjected to an earthquake in Y direction only. The seismic demand is determined for two levels of peak ground acceleration, firstly for design peak ground acceleration 0.3 g and also for the peak ground acceleration 0.45 g, which was used in pseudo-dynamic test (Section 2.2). The spectrum shape has the shape of the elastic response spectrum according to EC 8. However, the parameters of 28

31 Figure The relationship between the damage in plastic hinges and IDA curve for analysis in Y direction. The damage in hinges is indicated by yielding of reinforcement, maximum moment and rotation at NC limit state for columns and beams. 29

32 The following input data, parameters and/or assumption were used in the determination of target displacement the horizontal force shape f is calculated from the most important mode shape Φ for analysis in Y direction (Section 2.4) and assumed storey masses M used in pseudo-dynamic test (Section 2.2) f = MΦ= = (0.20) The defined horizontal force shape pattern was also used in the pushover analysis in Y direction (Section 2.5). Further on only the results of the pushover analysis in positive Y direction will be employed. the mass of equivalent SDOF system is m m i i t 2 = Φ = = (0.21) the transformation to an equivalent SDOF model is made by dividing the base shear and top displacement of the MDOF model with a transformation factor Γ, which is defined as T T * Φ M1 m Γ= = = = T * T Φ M Φ L (0.22) the pushover curve, for analysis in positive Y direction, was idealized by elasto-plastic relationship. The yield displacement d y was determined in a such way that the areas under the actual and the idealized force-displacement curves are equal. The displacement d m was assumed at the maximum force, and at the yield force F y of the idealized system was assumed equal to the maximum force. The yield displacement can be then calculated according to EC8 30

33 d y E m = 2 dm = = m F y 1102 (0.23) where yield force F y, displacement at maximum force d m and the deformation energy E h, which was calculated up to the displacement d m, were determined from the pushover curve. The pushover curve and the idealized force-displacement relationship are presented in Figure The transformation of the characteristic force-displacement points of the idealized MDOF system to the characteristic force-displacement points of the SDOF system can be simply obtained by dividing the forces and displacement of the idealized MDOF system by factor Γ the elastic period of idealized system is * mdy T = 2π = 2π = 0.87 s (0.24) F 1102 y the relation between the (spectral) acceleration and the yield force F y of the MDOF system is defined as S ay Fy 1102 = = = = Γ m m s g. (0.25) Figure The pushover curve for analysis in positive Y direction and the idealized forcedisplacement relationship. 31

34 The target displacement for the SDOF system can be obtained through relationships between the reduction factor R, the ductility μ, and the period T (the R-μ-T relations). In our case, where we determine the target displacement for a given seismic loading the unknown parameter is the ductility demand, which can be according to EC 8 determined as TC ( R 1) + 1 T < T μ = T R T T C C (0.26) where the reduction factor R due to energy dissipation capacity is defined as the ratio of the acceleration demand S ae in terms of the elastic spectral acceleration for the period T, to the acceleration capacity S ay (i.e. spectral acceleration corresponding to the yield force, Eq.(0.25)) R S S ae = (0.27) ay The acceleration demand S ae in terms of the elastic spectral acceleration for the period T can be obtained from the defined seismic loading in terms of the elastic spectral acceleration. For two levels of peak ground acceleration these values are S ae =0.517 g and S ae =0.775 g (Figure 2.19). The corresponding reduction factors calculated according to Eqs. (0.27) and (0.25) are R=1.341 and R=2.011, respectively, for a peak ground acceleration 0.3 and 0.45 g. Since the period of the idealized system (Eq. (0.24)) exceeds the corner period at the upper limit of the constant acceleration region of the elastic spectrum (T C ) the ductility demand μ is equal to the reduction factor R. The target top displacement d t is then obtained as the product of the ductility demand μ and the yield displacement of the idealized system d y and is d t = m= 12.5 cm for the peak ground acceleration 0.3 g and d t = m= 18.7 cm. The determination of the target displacement can be also graphically presented in accelerationdisplacement (AD) format. In this case all variables are related to the equivalent SDOF system. In Figure 2.20 the elastic spectrum, inelastic spectrum and capacity diagram are presented for the peak ground acceleration 0.3 g and 0.45 g. The displacement demand for SDOF system is determined by the intersection between the capacity diagram and the inelastic spectrum. Note that the inelastic spectrum (inelastic displacement S d versus acceleration at yielding of SDOF system S a ) is defined as 32

35 μ Sae Sd = Sde, Sa = (0.28) R R where S ae is the elastic acceleration spectrum and the elastic displacement spectrum S de determined from the elastic acceleration spectrum S de 2 T = S 2 ae. (0.29) 4π Figure The acceleration spectra used for determination of target displacement. Figure Seismic demand and capacity in AD format for a) peak ground acceleration of 0.3 g and b) for peak ground acceleration 0.45 g. 33

36 2.9. Determination of the target displacement by SDOF-IDA approach The SDOF-IDA approach for determination of the target displacement is based on the same philosophy as the N2 method. The difference is appears in the determination of the displacement demand of the SDOF system, which is in this case determined by nonlinear dynamic analysis of the defined SDOF system and not by employing the R-μ-T relations. This in general enables any shape of idealized force-displacement relationship and not only ideal elasto-plastic relationship, which was used in the case of N2 method (Section 2.8). In addition, the displacement demand can be determined for single ground motion record as well as for a set of ground motion records. The top displacement is determined for the same ground motion record as used in the dynamic and IDA analysis (Sections 2.6 and 2.7). Zero damping was assumed in the analysis in order to obtain comparable results with the results of IDA analysis for the MDOF model (Section 2.7). The pushover curve was idealized with the four-linear force-displacement relationship as presented in Figure The transformation factor Γ=1.279 as calculated in Section 2.8 was employed also in this case for transformation of MDOF to SDOF quantities. Similarly the equivalent mass of the SDOF system is t (Section 2.8). However, period of the equivalent system T=0.52 s, which in general is not needed for the prediction of the SDOF-IDA curve, differs from that determined for ideal elasto-plastic system (T=0.87 s). Based on the presented quantities of the SDOF system the SDOF-IDA curve was calculated and it is presented in Figure Very good correlation between the top SDOF-IDA and IDA curves is observed, especially for the peak ground acceleration which is less then 1.2 g. Figure The pushover curve for analysis in positive Y direction and the idealized forcedisplacement relationship employed for SDOF-IDA analysis. 34

37 Figure The peak ground acceleration versus top displacement for IDA and SDOF-IDA analysis in positive Y direction Influence of some modeling uncertainty on the response of structure In the seismic performance assessment of structures the determination of the seismic hazard and the selection of the ground motion records probably represent main sources of uncertainty. However, in addition to the uncertainty related to random nature of earthquakes, which is treated as the aleatory uncertainty, the epistemic uncertainty can also has an important influence, especially in the case of the seismic response, which is a subject of many modeling uncertainties [Dolšek, 2007], and since the data for the seismic response analysis are usually insufficient. Herein only the influence of some modeling parameters on the seismic response through nonlinear dynamic and pushover analysis is studied. The results are compared with the results of the original model (Model 0) or with the experimental results. Three additional models were created (Table 2.11) in order to study the influence of the bar slip in beams, which was modeled in the original model, the influence of the beam effective width and the influence of the material strength. In model 1 the bar slip was neglected. In model 2, beams were modeled as rectangular sections and consequently zero reinforcement from the slab was assumed. In the last example (model 3), the characteristic strength of concrete and steel was adopted. Therefore the concrete strength was assumed 25 MPa and the yield strength was assumed 500 MPa. These values are less than the mean values used in the original model (model 0) and are presented in Tables 2.1 and 2.2. The fundamental periods and the most important mode shapes are presented in Tables 2.12 and Periods are compared with the measured period. It can be observed that the period for Model 1 is reduced in comparison with the period of the original model. This is expected results, since no slip is 35

38 modeled in the beams. On the other hand the period of Model 2 exceeds the period of the original model for about 14% since the structure is much more flexible if beams are modeled with rectangular instead of the T cross sections. In the last case (Model 3), a slight difference for period is observed if compared with the period of the original structural model (Model 0). The difference is the consequence of the model for bond slip, which depends on the material strength. Table Description of the models created for studying the influence of modeling uncertainties. Model No. Description 0 The original model described in Section Bar slip was neglected Beams were modeled rectangular, consequently 2 reinforcement from the slab was neglected Characteristic material strength was used (f 3 c =25 MPa, f sy =500 MPa) Table Fundamental periods for all models compared to the measured period. Experiment Model 0 Model 1 Model 2 Model Table The mode shape 1 and 2 for all models. Mode shape 1 (predominant in X direction) Storey Model 0 Model 1 Model 2 Model Mode shape 2 (predominant in Y direction) Storey Model 0 Model 1 Model 2 Model

39 The pushover analysis was performed for all three models to see the difference in strength and deformation capacity. The force patterns for pushover analyses were calculated as discussed in Section 2.5. Since the difference in modes shapes are not significant also the force patterns are practically the same for all pushover analyses performed for different models. The pushover curves are presented in Figure Quite big difference can be observed. The deformation capacity in terms of top displacement is significantly reduced if the slippage of reinforcement is neglected in beams (Model 1). The highest deformation capacity is observed if beams are modeled with rectangular sections (Model 2). However, in this case the strength is significantly reduced if compared to the strength of the original model. Similarly also the strength is reduced if the characteristic material properties are used (Model 3). Figure Pushover curves for different structural models based on analysis in positive X and Y direction. 37

40 The storey drifts angles, which were determined at the 80% of maximum base shear measured in the degrading part of pushover curves, are for different structural models presented in Table The most uniform drifts are observed for Model 2, while for other models the drift angle at the top storey is significantly reduced in comparison to the drift angles at other stories. Quite big scatter is also observed if the top displacement and base shear time histories are compared for different structural models (Figure 2.24). The results are compared also with the experimental results for the low level (0.12 g) and the high level test (0.45 g) discussed in Section 2.2. The most deviation from the test results can be observed for models 2 and 3. However in all cases, the maximum strength is underestimated as already discussed in Section 2.6). Table Maximum drift angles determined at 80% of maximum base shear measured in the degrading part of pushover curves. Pushover in positive X direction Storey 1 Storey 2 Storey 3 Storey 4 Model Model Model Model Pushover in positive Y direction Storey 1 Storey 2 Storey 3 Storey 4 Model Model Model Model Table Comparison between maximum top displacement and maximum base shear force. Low level test (0.12 g) High level test (0.45 g) U (cm) U (cm) F (kn) F (kn) Experiment Model Model Model Model

41 Figure Measured and calculated base shear force and top displacement time histories for different models and for both low level and high level tests. 39

42 3. ICONS frame 3.1. Description of the structure The test structure is a four-storey plane reinforced concrete RC frame. The building had been designed to reproduce the design practice in European countries about forty to fifty years ago [Carvalho and Coelho (Eds.), 2001]. However, it may also be typical of buildings built more recently, but without the application of capacity design principles (especially the strong column - weak beam concept), and without up-to-date detailing. In such buildings a soft first story effect may appear in bare frame or even in uniformly infilled frame [Dolšek and Fajfar, 2001]. The elevation, the plan and the typical reinforcement in columns are presented in Figure 3.1. All beams in the direction of loading are 0.25 m wide and 0.50 m deep. The slab is 0.15 m thick. The reinforcement for columns in the first and second storey is also presented in Figure 3.1. The reinforcement is reduced in the top two stories in columns B and C. In column B only three φ12 mm bars are disposed on each side of the long side of the column. Such configuration results to 6φ12 mm bars in the cross-section of column B. For column C in the top two stories only 4φ16 mm bars are used in the corners of cross-section, while the same reinforcement (2φ12 mm bars) is used in the middle of the long side of the cross-section. The bottom longitudinal reinforcement in beams consists of 2φ12 mm bars. Three φ12 mm bars are used for the top longitudinal reinforcement at the connection to the column A (Figure 3.1). The top reinforcement in the beams connected to the column B and to the column D amounts to 2φ12+2φ16 mm bars. Beams, connected from both sides to column C, have the strongest reinforcement at the top of the beam (2φ12+5φ16 mm bars). The slab reinforcement is φ8/10 cm. The design base shear coefficient amounted to In the design, concrete of quality C16/20 and smooth steel bars of class Fe B22k (according to Italian standards) were adopted [Carvalho and Coelho (Eds.), 2001]. The mean strength of concrete amounts to 16 MPa and the mean yield strength of steel amounts to MPa Pseudo-dynamic tests The studied bare frame was tested in full scale at ELSA Laboratory (Figure 3.2). Two tests, B475 and B975, with the same ground motion record (Figure 3.3) were performed in a series. In order to obtain the peak ground acceleration of the acceleration time history, which was used in the pseudo-dynamic tests, the moderate to high seismic hazard scenario with the return periods of 475 and 975 years were 40

43 defined. The peak ground acceleration is 0.22 g and 0.29 g, respectively, for the first (B475) and second (B975) test. The results of the experiments can be found in ECOEST2-ICONS Report No.2 [Carvalho and Coelho (Eds.), 2001]. The concentration of the damage was observed in the third storey after the second test (B975). The maximum drift angle after the second test (B975) in the third storey was 2.41% and the maximum measured base shear force was 217 kn. Assuming that the mass of the structure is the same as used in the test (178 t) and that the design base shear coefficient amounted to 0.08 it can be concluded that the maximum measured base shear force exceeds the design base shear force for about 55%. The ratio between the base shear and the weight of structure amounts Figure 3.1. The view, the plan and the typical reinforcement in columns of the test structure. Figure 3.2. The tested specimen at ELSA Laboratory. 41

44 Figure 3.3. The acceleration spectrum and accelerogram for test B475 (a g,max = 0.22g) Mathematical modeling The mathematical model is based on the bilinear moment-rotation relationship assuming the linear moment degradation after the point which defines the maximum moment. The moment-rotation envelopes were calculated by using the OS Modeler function CBeamEnvelopeHystereticSlip and CColumnEnvelopeHystereticSlip. The yield and the maximum moment in columns were calculated taking into account the axial forces due to the vertical loading on the frame, which amounts to 9.1 and 8.0 kn/m 2 for the bottom three stories, and for the top storey, respectively. These uniform loads were calculated from defined masses for the pseudo-dynamic test, which amount 46 t and 40 t for the bottom three stories, and for the top storey, respectively. The potential reduction in strength (moment) due to insufficient anchorage length of the reinforcing bars was not considered when determining the moment-rotation envelopes. An effective slab width of 75 cm and 125 cm [CEN, 2004a] was considered for the short and long beams, respectively. The characteristic rotations, which describe the moment-rotation envelope of a plastic hinge used in the model, were determined according to the procedure described by [Fajfar et al. 2006]. The zero moment point was assumed to be at the mid-span of the columns and beams. The ultimate rotation Θu in the columns at the near collapse (NC) limit state, which corresponds to a 20% reduction in the maximum moment, was estimated by means of the CAE method [Peruš et al. 2006]. The values of ultimate rotations Θu estimated for weak columns A, B and D (Figure 3.1) (28 to 30 mrad) are substantially lower than that estimated for the strong column C (41 to 48 mrad). For the beams, the EC8-3 [CEN 2005] formulas were used for determination of ultimate rotations. Due to the absence of seismic detailing, the ultimate rotations were multiplied by a factor of Low values were adopted for the confinement effectiveness factor, α=0.5, and for the ratio of the transverse reinforcement, ρsx= The parameter γel was assumed equal to

45 3.4. Pushover analysis for model H and model T The pushover analysis was performed in positive and negative X direction. Three different force shape pattern were used for pushover analysis. They were determined by multiplying the diagonal storey mass matrix by assumed storey deformation vectors, which were defined to have triangular shape, first mode shape and uniform shape. The last two are also prescribed for the pushover analysis in Eurocode 8 [CEN, 2004b]. The pushover curves are presented in Figure 3.4. The shape of the pushover curves are almost the same for both senses of loading. However, the pushover curves based on the uniform deformation pattern have higher strength and deformation capacity if compared to other pushover curves. The base shear versus weight ratio amounts to about 0.15, quite higher value as observed in the pseudo-dynamic test (Section 3.2). More realistic seismic performance assessment can be determined based on the other pushover curves for which the base shear versus weight ratio is equal and amounts to and 0.122, respectively, for pushover analysis in positive and negative X direction. These values are in very good agreement with the experimental results (0.124) presented in Section 3.2. The maximum base shear is obtained at the top displacement of about 8 cm. After this displacement the structure degrades. The 20% reduction of the maximum base shear is obtained at the top displacement of about 13 cm. The drift angles, which correspond to the maximum base shear and to the 80% of the maximum base shear in measured in the degrading part of pushover curves, are presented in Tables 3.1 and 3.2. The results are presented only for the analysis with assumed triangular and uniform deformation shape. For the analysis based on the assumption of triangular deformation shape the largest drifts are observed in the third storey, not only if they correspond to the 80% of the maximum base shear measured in the degrading part, but also if they corresponds to maximum base shear. The maximum drift, calculated at the 80% of the maximum base shear in the degrading part of pushover curve, is slightly less than This indicates that the soft storey mechanism was formed in the third storey, like also observed in the pseudo-dynamic test (Section 3.2). Different collapse mechanism is observed for the pushover analysis performed by assumed uniform deformation shape pattern. In this case the deformations are concentrated in bottom two stories (Table 3.2), while in the third storey the deformation is only about half of these observed in bottom two stories. In addition the drift angles determined in the degrading part of pushover curve at 80% strength are less than the drift angle observed in the case of pushover analysis with assumed triangular force deformation patter. This indicates that the beams, which in our case have lower deformation capacity 43

46 in comparison to the columns, are important source of strength degradation. The opposite can was observed for the pushover analysis based on assumed triangular deformation shape. In this case only source of deformation capacity are columns of the third storey (Figure 3.5). Figure 3.4. Pushover curves for model three different force shape pattern. Analysis is performed in positive and negative X direction. Table 3.1. Maximum storey drift angles determined at the maximum base shear. Results are presented for pushover analysis based on assumed triangular and uniform deformation shape pattern Triangular Uniform Drift angle at maximum base shear Storey 1 Storey 2 Storey 3 Storey 4 + X X X X Table 3.2. Maximum storey drift angles determined at the 80% of maximum base shear measured in the degrading part of the pushover curve. Results are presented for pushover analysis based on assumed triangular and uniform deformation shape pattern. Triangular Uniform Drift angle at 80% base shear in the degrading part Storey 1 Storey 2 Storey 3 Storey 4 + X X X X

47 Figure 3.5. The relationship between the damage in plastic hinges and pushover curve for assumed triangular deformation shape. The damage in hinges is indicated by yielding of reinforcement, maximum moment and rotation at NC limit state for columns and beams. 45

48 3.5. Nonlinear dynamic analysis and comparison with pseudo-dynamic test The nonlinear dynamic analysis was performed with intention to simulate pseudo-dynamic test. Therefore two analyses were performed in a series by subjecting the structure with the acceleration time history presented in Figure 3.3. The peak ground accelerations amount to 0.22 g and 0.29 g, respectively, for first (B475) and second (B975) test. The time history results are presented in Figure from 3.6 to 3.8. Figure 3.6. The comparison between measured and calculated storey displacement time histories for the B475 and B975 test (Section 3.2). 46

49 In Figures 3.6 and 3.7 the time histories for storey displacement and storey drifts are presented. Despite very simple mathematical model very good relation between measured and calculated results can be observed. Slightly more difference can be observed for the storey shear time histories. Figure 3.7. The comparison between measured and calculated storey drift time histories for the B475 and B975 test (Section 3.2). 47

50 Figure 3.8. The comparison between measured and calculated storey shear force time histories for the B475 and B975 test (Section 3.2). 48