Pre- and post-test mathematical modelling of the SPEAR building

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1 Pre- and post-test mathematical modelling of the SPEAR building P.Fajfar 1, M.Dolšek 1, D.Marušić 1, A.Stratan 2 1 Univ. of Ljubljana, Slovenia 2 Technical Univ. of Timisoara, Romania, formerly visiting researcher at the Univ. of Ljubljana ABSTRACT: Pre- and post-test analyses of structural response of the SPEAR building were performed, aimed at leading the test preparation and execution as well as studying the mathematical modelling. In the paper the results of initial parametric studies, of the blind pre-test predictions and of the post-test analysis are summarized. In all studies a simple mathematical model with one-component member models with concentrated plasticity was employed. The pre-test analyses were performed with the CANNY program. After the test results became available, the mathematical model was improved using an approach based on a displacement - controlled analysis. Basically, the same mathematical model as in pre-test analyses was used, only the values of some parameters have been changed. The OpenSees program was employed. A good agreement between test and numerical results was obtained. The results prove that relatively simple mathematical models are able to adequately simulate the detailed seismic response of reinforced concrete frame structures to a known ground motion, provided that the input parameters are properly determined. 1 INTRODUCTION Within the European research project SPEAR (Seismic performance assessment and rehabilitation of existing buildings), a real-size asymmetric three-storey reinforced concrete (RC) frame building was tested pseudo-dynamicaly in full-scale. The structure is referred in the following as the SPEAR building. Several partners in the project and other researchers have performed blind pre-test analyses and/or post-test analyses. A range of different modelling options and computer programs were explored. The wealth of experimental and numerical data related to the SPEAR building represents very valuable material for studying the mathematical modelling of existing RC buildings. In this paper only the results of analyses performed at the University of Ljubljana are discussed. Intentionally, only simple modelling options have been used, which can be readily performed with existing software in advanced design offices. In all studies, the mathematical model with one-component member models, composed of an elastic beam and two inelastic rotational springs, was employed. Only in initial parametric studies other models were also used. The pre-test and post-test analyses were performed with the CANNY (Li 2002) and OpenSees program (PEER 1999), respectively. In the paper the results of initial parametric studies, of the blind pre-test predictions and of the post-test analysis are summarized. 1

2 2 DESCRIPTION OF THE BUILDING Detailed description of the building is provided elsewhere in this monograph. Here only a 3D view and the plan with the denotation of columns are presented (Fig. 1). Figure 1. Overview and plan of the SPEAR building. Note that the positive sense of axes is different in the pre- and post-test models. 3 COMMON FEATURES OF MATHEMATICAL MODELS In all cases, the idealisation of the structure was based on elements placed at the middepths of members, and connected at the nodes. Rigid diaphragms were considered at the floor levels, due to monolithic RC slabs. Centreline dimension of elements vs. rigid offsets was considered as one of the modelling parameters in the initial parametric study. In the final pre-test and in the post-test analyses centreline dimensions were used with one exception: rigid elements were used for all models at the column C6 to account for the finite dimensions of this column. Using centreline dimensions, the storey heights were 2.75 and 3 m for the first und upper two stories, respectively. In the preliminary estimation of seismic response and in the final pre-test model (Chapters 4-6), the height of the first storey was artificially increased from 2.75 to 3.00 m in order to take into account potential bar slippage at the column-footing interface. This artificial increase was not applied in the post-test analyses. One element per member was used. Beam flexural behaviour was modelled by one-component lumped plasticity elements, composed of elastic beam and two inelastic rotational hinges (defined with the moment-rotation relationship). The plastic hinge was used for the major-axis bending only. The element formulation was based on the assumption of the inflexion point at the midpoint of the element. For columns flexural behaviour also a one-component lumped plasticity model was used, with two independent plastic hinges for bending about the two principal axes. In initial parametric studies, lumped and distributed plasticity fibre models were used in addition for columns. Elastic axial, shear and torsional response of the element was assumed, with the exception of axial behaviour of fibre columns. 2

3 In all cases reported here, time-history analysis was performed. In the initial studies, 5% Rayleigh damping was used, for the first two modes of vibration. The instantaneous stiffness-proportional damping was applied. For the final pre-test analysis and for the post-test analyses, zero damping, as in the pseudo-dynamic tests, was assumed. All pre-test analyses were performed with the CANNY program. Due to the program limitation, second order (P-delta) effects were not considered in these analyses. However, they were considered in the post-test analyses, performed with OpenSees program. Masses and mass moments of inertia differed slightly in different analyses. 4 INITIAL MATHEMATICAL MODELLING PARAMETRIC STUDY First, a study on the influence of modelling parameters and assumptions was made. The structural performance was evaluated by non-linear dynamic analyses of several structural models under a set of seven bi-directional recorded ground motions. The following modelling issues were addressed: column flexural behaviour (using onecomponent model which neglects the M-M-N interaction and strength degradation, and the fibre elements, accounting for these effects), rigid offsets vs. centreline dimensions of elements, bilinear, tri-linear, and multi-linear (through fibre discretization) momentrotation element modelling, shape of hysteresis loops, post-yielding stiffness, beam effective width, expected vs. characteristic material strength, and evaluation of shear strength of members and joints. A modified version of Takeda hysteresis rules was used for flexural modelling of cyclic response of one-component models. The parametric study has demonstrated that the displacement demands were affected significantly when the global stiffness and/or strength of the structure changed. The seismic response of the analysed structure was most influenced by the bilinear ( yield stiffness) vs. tri-linear element modelling and rigid offsets vs. centreline element dimensions. On the other hand, equivalent bilinear model using effective ("cracked") stiffness showed a very good agreement with the tri-linear model. Note that bilinear and tri-linear element modelling corresponds to the part of the moment rotation envelope before the potential strength degradation starts. The effect of the M- M-N interaction and strength degradation for columns was moderate. Post-yielding stiffness, pinching, and beam effective width had little influence on the structural response of the investigated building. Evaluation of element and joint shear strength according to different procedures showed important variation. However, even considering the more conservative estimates for shear strength, the imposed demands did not exceed the shear capacities, suggesting that the brittle shear failure modes would probably not occur in the investigated building. Attention was paid to the dilemma if rigid offsets or centreline dimensions should be used. In structural modelling of RC moment resisting frames often the finite dimensions of beam-column joints are taken into account by considering rigid offsets for the interconnecting beam and column elements. On the other hand, joint deformations may be important, especially in the case of older existing frames, due to lack of joint transverse reinforcement and slippage of longitudinal reinforcement. Centreline modelling of elements may be used as a simple way to account for both the reduction of stiffness and strength due to additional deformations in the joint regions for nonlinear structural 3

4 assessment of RC frames. Both options have been used in parametric studies. Consideration of rigid offsets as an alternative to the centreline dimensions of elements led to important changes in global structural strength and stiffness, as well as change of relative storey shear strengths. Thus, beside higher displacement demands of the centreline model, there was a significant change in distribution of inter-storey drift demands along the height of the structure. Higher demands in brittle components (element shear forces) were another consequence of rigid offsets. Detailed results of preliminary parametric studies can be found in (Stratan and Fajfar 2002). 5 PRELIMINARY ESTIMATION OF SEISMIC RESPONSE In the next phase of the study, an estimation of the seismic response of the structure was made. Here only the basic data and the most important results are summarized. A detailed description is provided in (Stratan and Fajfar 2003). Based on the study of different parameters affecting the seismic response of the SPEAR structure summarized in the previous chapter, two 3D structural models, which were supposed to represent the "best-estimate" models, were considered in the next round of analyses. The first model employed the one-component lumped-plasticity model with tri-linear moment-rotations idealisations for both beams and columns, and the second employed the same model for beams and distributed plasticity fibre models for columns. Only the first model will be discussed here. Though the one-component model does not account for some important aspects such as strength degradation and M-M-N interaction for columns, it was chosen for its simplicity. Preparation of input data and interpretation of results are easy and suitable for design practice. Variants of one-component element models are well-known, and are readily available in some commercial computer programs. To authors' experience similar models showed adequate correlation with full-scale pseudo-dynamic tests in the past. The one-component model was based on expected material characteristics. Centreline dimensions were used for the elements to account for additional deformations not modelled directly (bar slippage and joint shear distortion). The first storey columns were considered the same length as the second and third story columns (3m), as the effect of strain penetration and bar slippage may equally occur at the column-footing interface. Rigid elements were used only at the column C6, to account for the finite dimension of this member. Beam effective width was computed according to the recommendations of Paulay and Priestley (1992). When assessing the beam flexural resistance under negative moments (top reinforcement in tension), only the top slab reinforcement effectively anchored was considered. Takeda hysteretic behaviour was used for the elements, without pinching. Expected material strengths were estimated according to Priestley (1997). Concrete was considered unconfined for establishing the stress-strain relationship. A standard moment-curvature analysis was carried out for each element. For columns, axial force corresponding to gravitational loading was considered. The cracking and yield curvatures and moments were determined by a numerical procedure. The ultimate curvature 4

5 was determined at the attainment of ultimate strains in concrete or steel. The equivalent plastic hinge length was determined according to Paulay and Priestley (1992). The moment-rotation relationship was obtained by integrating the curvature distribution along the element length. The ultimate rotations, determined by the above approach for columns, amounted to from 13 (column C3 with the highest axial load) to 35 mrad (a corner column in the 3rd storey). For dynamic analyses, two sets of earthquake records were used. The first one was a suite of seven recorded bidirectional ground motions from Southern Europe, selected from the European strong motion databank. They were scaled (but not fitted) to approximately match the EC8 spectra for soil type C in the constant velocity range. A second suite of nine semi-artificial earthquake records (fitted to EC8 spectrum for soil class C) provided within the SPEAR project were added later to provide an easier comparison of results with other project tasks. Three intensities of ground motion were used, corresponding to 0.1, 0.2 and 0.3 g peak ground acceleration (a g ). The a g value includes the code soil coefficient, i.e. it represents the peak acceleration at the top of the soil layer. The main features of this initial model and of the final pre-test and post-test models (described in next chapters) are summarized in Tables 1 and 2. Although the results of numerical studies cannot be directly compared with the results of pseudo-dynamic tests due to differences in material characteristics, masses, ground motion time-histories, damping (5% versus zero damping), and initially damaged structure and sequence of tests in the case of test structure, they provide a qualitative estimate of the expected seismic response and a very rough quantitative estimate of the seismic capacity. Element model Table 1. Common features of all models (with one-component elements). Centreline dimensions Rigid offset at beam-column connection Additional deformations at element intersections Insufficient anchorage of bars M-M-N interaction Effective width Paulay and Priestley (1992) Effective height of the 2 nd and 3 rd storey columns 3.00 m One-component lumped plasticity elements based on moment-rotation relationship (plastic hinge defined for uni-axial bending). Takeda s hysteretic rules. Elastic axial, shear and torsional behaviour. Yes No, with the exception of connections to the large column Bar slippage and joint shear distortion considered using centreline dimensions without rigid segments at beam-column connections (with the exception of connections to the large column) Accounted for by reducing the yield strength of insufficiently anchored bars No 5

6 Table 2. Comparison of modelling assumptions. The mean results of dynamic analyses suggested that displacement demands were higher in the weaker X direction, and that they were different in the positive and negative Y direction. Average increase of top displacements at the flexible edge with respect to the centre of mass amounted to 30% and 16% for the X and Y direction, respectively. The corresponding decrease of top displacements at the stiff edge with respect to centre of mass amounted to 15% and 10% for X and Y directions, respectively. Inter-storey drift demands concentrated in the lower two storeys, but were more uniform in the Y direction. 6

7 Based on the results of the study, damage to the structure was dictated by weak columns and by significantly lower moment capacity under positive bending for beams. Pullout of bottom beam reinforcement and extensive concrete spalling were dominant failure modes. Though there was a lack of proper transverse reinforcement both in elements and in beam-column joints, shear capacity checks indicated that neither elements nor joints were susceptible to shear failures. Column demand to capacity ratios (DCR) were higher for columns at the flexible edges (higher demands) and for elements with higher axial loads (lower rotation capacities). The critical column was the first storey central column (C3). Another three columns from the first storey (C1, C2, and C4) followed closely with slightly lower DCRs. Extensive cover concrete spalling for the C1-C4 columns was predicted at the 0.2g intensity level. Initiation of significant global structural damage was evaluated to 0.24g. Though a similar response was obtained for both suites of earthquake records, torsional response was significantly lower in the case of the semi-artificial records. The results indicated that at the a g = 0.3 g level the concrete in the most critical columns started to crush. Maximum displacements were in the region where the pushover curve started to soften. However, at this ground motion level, the structure was still stable. 6 FINAL PRE-TEST ANALYSIS ( PRE-TEST MODEL ) Initial analyses of the test building were based on assumed material properties, since the test building has not been built yet and the actual material properties were not known. Moreover, the time-history of ground motion to be used in the tests has not been determined yet. After a series of intermediate analyses, aimed at providing information needed for the decision about the ground motion to be simulated during the experiment, and after the test building was built, the final pre-test analyses were performed using basically the same model as in the preliminary analyses. The ground motion chosen for the test is represented by two orthogonal semiartificial ground motions, based on a record obtained in Hercegnovi during the 1979 Montenegro earthquake and fitted to the EC8 spectrum (type I, Soil C). Both horizontal components and the spectrum are shown e.g. in (Negro et al. 2004). The ground motion will be denoted as HN-1. The components were applied in the -X and -Y direction of the pre-test model (see Fig. 1). It is important to note that the selected ground motion produced much larger torsional rotations than an average record in the suite of candidate ground motion. In Fig. 2 maximum top displacements normalised by maximum displacement in the centre of masses at the top are presented for a g = 0.3 g and 5% structural damping. In addition to the response due to HN-1 ground motion, the mean results and standard deviation are presented for 56 ground motions and for 8 variants (the same ground motion applied in different combination of X- and Y- directions with + and - sense) of HN ground motion. In the case the HN-1 ground motion, the displacements at the flexible edge are comparable to the mean plus sigma values. The behaviour is typical for torsionally stiff structures: the response is governed by a single mode, displacements increase due to torsion at the flexible side and decrease at the stiff side. Extreme response is predominantly in one, fundamental mode. The results in Fig. 2 are shown for ground motion scaled to a g = 0.3 g. 7

8 X-direction 1.4 Y-direction u/ucm HN-1 HN-mean Mean +σ Stiff CM Flex. Stiff CM Flex. Figure 2. Normalized displacements u/u CM in the horizontal plane at the top of the building (pre-test model, a g = 0.3 g, ζ = 5%). From the intermediate analyses, only the influence of damping is presented here. It was studied in order to obtain a quantitative estimate of the difference between 5% damping, used in parametric studies described in previous chapters, and zero damping, used in pseudo-dynamic. In Fig. 3 results for 0% and 5% damping are compared. Time-histories of storey drift in the 1st storey are plotted. Ground motion was HN-1, scaled to 0.3 g. As expected, maximum seismic demand in the case of zero damping is substantially larger (up to 40%) than in the case of 5% damping. The influence of the sequence of three tests with increasing intensity (0.04 g, 0.12 g and 0.20 g) was also studied. The results suggested that the response during the highest ground motion was not substantially influenced by the previous ground motions. The final pre-test model was basically the zero damped model, employed in the pretest estimation of seismic response (Chapter 6), adapted to the new (measured) data on the material properties. The same properties of the concrete (Table 2) were assumed for the whole structure. For steel, two different characteristics were used (Table 2). Compared to the previous models, the reinforcement was stronger and more ductile whereas the strength of concrete was lower. Due to better steel properties the ultimate values of the deformation capacity of members of the new model were generally somewhat larger (ultimate rotations in columns amounted to from 14 mrad for column C3 to 40 mrad for a corner column in the 3 rd storey, and the overall lateral strength was up to 8% higher comparing to the previous model. Note, that the final pre-test model was still not fully compatible with the test structure. There was a substantial difference in mass moments of inertia. The initial damage of the test structure was not taken into account. The main features of the final pre-test model are summarized in Table 2). Based on the analysis of the pre-test model, inter alia, the following conclusions were made: In the case of HN-1 ground motion a g = 0.20 g should be an appropriate intensity to be used for testing. The analyses indicate that such ground motion would lead to damage serious enough to apply desired retrofitting techniques on the test building in the second stage of the experiment, yet it would not cause the collapse of any member. (Fajfar et al. 2003). HN-1 ground motion normalised to a g = 0.2 g causes substantial damage to critical columns. At a g = 0.25 g local failure (crushing of concrete) of critical columns occurs. Critical columns are columns C1, C2 and C4, 8

9 which are located in the first storey at the flexible edges and carry a relatively large axial load, and column C3 with the largest axial stress. (Marušić and Fajfar 2003). Figure 3. Storey drift time-history for the 1st storey (average of all columns at the floor, ξ = 0% and 5%; HN-1, a g = 0.3 g. 7 SUMMARY OF TEST RESULTS Tests and test results are described in details elsewhere (Negro et al. 2004, Jeong and Elnashai 2005). Here only the main results, important for the comparisons with analyses are summarized. In the first phase of pseudo-dynamic testing, three tests were performed in sequence. Horizontal loading was applied in both horizontal directions simultaneously. The same acceleration time-histories were used in all three tests, but they were differently scaled. Before the tests started, the structure was damaged during transportation to the final position in the lab. The analyses made by the partners (see the detailed study performed by Jeong and Elnashai 2005) suggested that the initial structural damage only slightly influences the structural response at higher ground motion intensities. It is clear, however, that the initial conditions have been substantially changed compared to»uncracked«structure and that the results of the first small intensity test (a g = 0.02 g), intended to simulate the elastic response, cannot be compared with the results of elastic analyses of initially undamaged structure. Moreover, the initial damage has most probably influenced the detailed behaviour. The intensity, expected to cause significant damage, without going beyond the reparability stage, was set to a g = 0.15 g, as a compromise between different predictions of different project partners. However, only minor damage occured at this intensity level. Minor cracks were concentrated mainly at the top of the columns and in the beams intersecting into column C6. No damage was visible at the bottom of columns The maximum top displacements in CM were about 70 and 50 mm in the X and Y direction, respectively, and the maximum rotation at the top was about 12 mrad. The most affected storey was the 2nd storey (Negro et al. 2004). Due to small damage, it was decided to continue with the test sequence and perform a test with a g = 0.20 g. The maximum top displacements in CM were about 100 mm in both X and Y direction, and the maximum rotation at the top was about 20 mrad. Again, the 2nd storey was the most affected storey. Maximum interstorey drifts in the 9

10 2nd storey in CM amounted to about 55 and 45 mm in X and Y direction, respectively. Due to torsional influence the drifts at the flexible edges increased to about 70 cm (i.e. about 2.8% of the clear storey height) in both directions (columns C1, C2 and C5 in X direction and columns C4 and C7 in Y direction). The damage was concentrated mainly in the columns, more at the top than at the bottom and more in the second than in the first storey. The column C3 (with the highest axial load), suffered the most for extended spalling at the top. The columns at the flexible edges exhibited a significant level of spalling and cracking at the top. Some damage was detected in the beams and floor slabs connected to the strong column C6. (Negro et al. 2004). 8 EVALUATION OF THE PREDICTIONS BY THE PRE-TEST MODEL A comparison of results obtained by the pre-test model and in two tests is shown in Figs 4 and 5. The results of the pre-test model for a g = 0.2 g do not take into account the sequence of the loading. For a g = 0.15 g, a good correlation of top displacement histories can be observed. The correlation of top rotations is poor, also due to substantial underestimation of the mass moments of inertia in the pre-test model. Envelopes (Fig. 5) of storey drifts show that the predicted drifts in the first storey were overestimated in both directions, while in the second storey they were underestimated in Y- direction. This disagreement might be to a large extent attributed to the artificially increased height of the first storey in the pre-test model. The correlation for a g = 0.2 g is poor (Fig. 4). The tests have not provided pushover curves, therefore the pre-test base shear top displacement relations are compared with the results of the more accurate post-test model (Fig. 6). The distribution of lateral loading was based on the fundamental load shape in the direction under consideration. The main difference between the pre- and post-test results is in strength, which is largely overestimated in the pre-test model. On the other hand, the prediction of the global stiffness was fair. Based on the pushover curves in Fig. 6 and considering the maximum top displacements measured during the two tests, fair and poor correlations for 0.15 g and 0.20 g tests, respectively, can be easily explained. Since only limited plastification occurred during the 0.15g test, the response was controlled mainly by stiffness. On the other hand, difference in strength was decisive for the disagreements in the case of the 0.20 g test. The fact, that the strength degradation and the P- effect were not included in the pre-test model, did not substantially influence the response. Large torsional amplifications were predicted and this prediction was confirmed in tests. For the assessment of the structural behaviour it is necessary to compare demand and capacity. The predicted capacity in terms of drift was based on semi-empirical approach for determination of the ultimate rotation capacity with plastic hinge according to Paulay and Priestley (1992). Unconfined concrete was assumed. The test results have demonstrated that the predicted capacity was underestimated. It is interesting that the suggested test intensity (0.20 g) was proved to be correct. This might be interpreted as a good fortune, because both demand and capacity were underestimated. However, it may be explained also by the robustness of the estimates of the global seismic structural response. 10

11 In the case of the initial small intensity (0.02 g) test it is not possible to compare the test and analytical results due initial damage of the test structure, which very substantially influences the response in the case of a week ground motion. Figure 4: Top displacements and rotations at CM obtained by the pre-test model and test results for a g = 0.15 g and a g = 0.2 g. Figure 5: Envelopes of storey drifts at CM obtained by the pre-test model and test results for a g = 0.15 g. 11

12 Figure 6. Pushover curves for pre- and post-test models. Maximum values measured in tests at CM are indicated. Positive sense of axes corresponds to the post-test model (Fig. 1). 9 POST-TEST MODEL The comparison of the pre-test predictions and test results suggests that the pre-test model is not capable to simulate the detailed seismic behaviour of the test structure properly. An improved mathematical model was prepared by employing the test results. A simple trial and error approach, based on blind changing of structural parameters is very time consuming and, because of a large number of parameters involved, does not necessarily lead to an appropriate model. In our study, we employed the approach based on displacement-controlled analysis, developed by Dolšek and Fajfar (2002). Using this approach, it was possible to detect that the global strength and the energy dissipation capacity of the pre-test model were substantially overestimated. Furthermore, based on the observation that the second storey of the test structure was more damaged than the first one, it could be concluded that the artificial increase of the length of the columns at the bottom of the first storey was not realistic. Test results also suggested that the damage was larger at the top than at the bottom of the columns. A similar observation was made and explained by Calvi et al. (2002). The above conclusions represented the guidelines for the modification of parameters employed in the pre-test model. Since a displacement-controlled analysis option has not been implemented in the CANNY program, we choose the OpenSees program for the post-test analyses. The post-test model consists of zero-length section elements with hysteretic uniaxial material and elastic beam-column elements. The former elements represent plastic hinges at both ends of beams and columns, while the later elements model the elastic part in between of plastic hinges. As an additional advantage of changing the program it was possible to include the strength degradation in the envelope of the element moment - rotation relation and the P- effect at the global level. Although these features are substantial for analyses at larger intensities (as a part of other investigations), they did not exhibit an important influence on the response at the 12

13 ground motion intensities used in the tests. In order to allow the strength degradation, a bilinear relationship was used in the first part of the moment rotation relationship (before the maximum moment is reached). (The hysteretic material in OpenSees is limited to tri-linear moment rotation envelope). An effective initial stiffness was employed, as suggested by initial parametric studies (Chapter 4). In spite of this simplification, we were able to simulate the structural response well, as demonstrated in Chapter 10. However, it was difficult to fit the parameters in such a way that a good simulation was obtained for both tests (0.15 g and 0.20 g). It is obvious that a tri-linear envelope, taking into account a change in stiffness after the cracking of the concrete, is better suited for detailed simulations than a bilinear with an effective initial stiffness. Based on the above observations and restrains, the following modifications of the pre-test model were made: 1. The height of the first storey columns was reduced from 3.00 to The overall strength was reduced by assuming that the anchorage of reinforcement bars is insufficient not only in beams (as assumed in the pre-test model), but also at the top of the columns. Insufficient anchorage of bars was taken into account by reducing the yield strength of insufficiently anchored bars according to the procedure suggested in FEMA 356 (2000). 3. New moment - rotation envelopes (Fig. 7) were determined for elements. The yield moment M y and the maximum moment M max were determined from the crosssection analysis, like in the case of the pre-test models. The effective yield rotation was calculated by the formula θ y = (M y L)/(6EI ef ), where L is the length/height of a beam/column, E is the modulus of elasticity, and I ef is the effective moment of inertia of the element. The values of I ef /I, where I is the moment of inertia of the uncracked section, were determined based on observations made by comparing analysis with low intensity (a g = 0.02 g) test and a trial and error procedure. The values for all beams amounted to 0.5, whereas the values for columns decreased from the bottom to the top and amounted to about 0.47, 0.44 and 0.41 for the 1 st, 2 nd and 3 rd storey, respectively. The rotations θ max and θ u, (the later corresponds to 20% drop of the strength and is considered representative for the Near Collapse limit state), were estimated by the CAE method (Peruš et al. 2005), using experimental data bases for columns. The θ u values for the critical columns, e.g. for column C2 amounted from 35 (first storey) to 47 mrad (top storey) and for column C3 from 27 to 39 mrad for first and top storey, respectively. Note that these values are larger than the capacities used in the pre-test analyses. 4. The energy dissipation capacity was reduced by a further decrease (compared to the pre-test model) of the unloading stiffness in the element hysteretic rules. 5. The strength of the concrete, masses and mass moments of inertia were adjusted to the measured values and values used in the test. Only the change of mass moments of inertia was substantial. 6. For the column C6, the yield rotation was increased by a factor of 15 at the outer side at the bottom of the 1 st storey and at the inner side at the top of the 2 nd storey because of substantial initial cracks at these two sections. 13

14 Note that fine tuning of some parameters was made for changes 3, 4, and 6, using a trial and error procedure, combined with the displacement-controlled analysis. An overview of the features of the post-test model is provided in Table 2. Figure 7. The moment - rotation envelopes for the post-test model. 10 RESULTS OF POST-TEST ANALYSES AND COMPARISON WITH TEST RESULTS Selected results of analyses performed with the post-test model and comparison with test results are shown in Figs The analyses were performed in a sequence. First, ground motion was scaled to a g = 0.15 g and than to a g = 0.2 g. The results include time-histories of top displacements and rotations in CM (Fig. 8), and envelopes of storey drifts in CM (Fig. 9) and in the column at the flexible corner C2 (Fig. 10). All results are shown for both ground motion intensities. More data, including input file and detailed results for the described model and some other slightly different models, are provided in post-test report by Dolšek and Fajfar (2005). A fair correlation of all computed and measured results can be observed. The aim of our post-test analysis was to discover the main inaccuracies in the pre-test model and to develop an improved model which can properly simulate the seismic response of the test structure. An additional aim was to prove that simple mathematical models are able to describe the main features of non-linear structural response provided that proper values of input parameters are employed. By further fine tuning of parameters it would be possible to obtain even a better correlation, but we see no sense in such efforts. A relatively simple improvement of the element model, which would facilitate the modelling and improve some results, represents a tri-linear initial part of the moment rotation relation, taking into account pre- and post-crack stiffness. 11 CONCLUSIONS Mathematical modelling of existing RC frame structures, not designed for earthquake loading, proved to be a difficult task. Slippage of reinforcement bars due to insufficient anchorage and joint shear distortion will probably occur in the case of a severe ground motion and these phenomena should be properly taken into account in a mathematical model. In the case of the investigated building, the blind pre-test predictions were only partially successful in spite of known ground motion, which in the real world repre- 14

15 sents the major source of uncertainty. Knowing the structural response during the tests, it was possible to obtain a good simulation of the test results by adapting some parameters of the simple pre-test model, based on one-component member models with concentrated plasticity. The simple model, appropriate for practical applications, proved to be robust and efficient. Engineering judgement, needed in mathematical modelling of complex existing RC structures, can be easily exercised. The typical computing time for one time-history analysis amounted to about 2 minutes on a PC with Intel Pentium 4 processor (3.0 GHz, 512MB RAM). Figure 8: Top displacements and rotations at CM obtained by the post-test model and test results for a g = 0.15 g and a g = 0.2 g. Figure 9: Envelopes of storey drifts at CM obtained by the post-test model and test results for a g = 0.15 g and a g = 0.2 g. 15

16 Figure 10: Envelopes of storey drifts at C2 obtained by the post-test model and test results for a g = 0.15 g and a g = 0.2 g. 12 REFERENCES Calvi, G.M., Magenes, G., Pampanin, S., Relevance of beam-column joint damage and collapse in RC frame assessment. Journal of Earthquake Engineering 6(1): Dolšek, M., Fajfar, P Mathematical modelling of an infilled RC frame structure based on the results of pseudo-dynamic tests. Earthquake Eng. Struct. Dyn.31: Dolšek, M., Fajfar, P Post-test analyses of the SPEAR test building, University of Ljubljana. Fajfar, P. Dolšek, M. Marušić, D. Stratan, A. Peruš, I. Lapajne, J SPEAR Annual Report for the period September 2002 August 2003, University of Ljubljana. FEMA 356, Prestandard and commentary for the seismic rehabilitation of buildings, Federal Emergency Management Agency, Washington (DC). Jeong, S.-H., Elnashai, A Analytical assessment of the seismic performance of an irregular RC frame for full-scale 3D pseudo-dynamic testing, Part I: analytical model verification. Journal of Earthquake Engineering 9(1): Jeong, S.-H., Elnashai, A Analytical assessment of the seismic performance of an irregular RC frame for full-scale 3D pseudo-dynamic testing, Part II: Condition assessment and test development. Journal of Earthquake Engineering 9(2): Li, K.N., Dimensional nonlinear static and dynamic structural analysis computer program. CANNY Consultants Pte Ltd., Singapore. Marušić, D., Fajfar, P., Final pre-test analysis of the SPEAR test building, University of Ljubljana. Negro, P., Mola, E., Molina, F.J., Magonette, G.E., Full-scale testing of a torsionally unbalanced three-storey non-seismic RC frame. Proceedings 13WCEE, paper 968. Paulay, T., Priestley, M.J.N., Seismic Design of Reinforced Concrete and Masonry Buildings, John Wiley & Sons, Inc., New York. PEER Open System for Earthquake Engineering Simulation (OpenSees), Pacific Earthquake Eng. Research Center, Univ. of California, Berkeley, Peruš, I., Poljanšek K., Fajfar P Flexural deformation capacity of rectangular RC columns determined by the CAE method. Paper in preparation. Priestley, M.J.N., Displacement-Based Seismic Assessment of Reinforced Concrete Buildings, Journal of Earthquake Engineering 1(1): Stratan, A., Fajfar, P Influence of modelling assumptions and analysis procedure on the seismic evaluation of reinforced concrete GLD frames, University of Ljubljana. Stratan, A., Fajfar, P Seismic assessment of the SPEAR test structure, University of Ljubljana. 16

17 Filename: SPEARpaper2 Directory: D:\Clanek2005\SPEAR Template: E:\WINWORD6\SJABLOON\BALWORA4.DOT Title: Title Subject: Author: A.A. Balkema Uitgevers B.V. Keywords: Comments: Creation Date: :57:00 Change Number: 13 Last Saved On: :35:00 Last Saved By: MDOLSEK Total Editing Time: 295 Minutes Last Printed On: :01:00 As of Last Complete Printing Number of Pages: 16 Number of Words: 6,215 (approx.) Number of Characters: 33,502 (approx.)