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SEISMIC BEHAVIOUR OF EXISTING STRUCTURES

BY MEANS OF NON LINEAR STATIC ANALYSIS:

POLAR SPECTRUM & CAPACITY DOMAINS

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- 1. University of Salerno Department of Civil Engineering Contact Person: Luigi Petti, petti@unisa.it REPORT INNOVATIVE PROCEDURES TO ASSESS SEISMIC BEHAVIOUR OF EXISTING STRUCTURES BY MEANS OF NON LINEAR STATIC ANALYSIS: POLAR SPECTRUM & CAPACITY DOMAINS Luigi Petti, Ivana MarinoINTRODUCTIONAs known, the standard analysis for traditional seismic design is Linear DynamicProcedure (LDP) with assigned response spectrum (Response Spectrum Analysis). Morecomplex procedures, such as non-linear time history analysis (NDP), are seldom usedbecause of the difficulties in the definition of an accurate hysteretic model to describe thebehaviour of the materials under cyclic actions and the choice of a set of accelerogramsthat describes the real site conditions. Moreover, as it is well-known, NDP analyses presentconvergence difficulties and require major computational effort.Non-linear static analysis is instead becoming the key method to evaluate the seismicresponse of existing structures. This method, which is able to evaluate the collapsemechanism of structures, is suitable for the analysis of structures not explicitly designedfor seismic actions.However, non-linear static procedures can lead to unsuitable results when applied toirregular structures because of the difficulties in taking into account dynamic latero-torsional effects. As it is known, an asymmetric distribution of mass and stiffness, or ofstrength in plan, leads to high ductility demand for the elements near the soft or weak edge.In the last years, several proposals have been put forward to extend traditional pushoveranalysis, calibrated on plan systems, to the assessment of three-dimensional modelsbehaviour. At present no simplified procedure which can adequately account for thetorsional behaviour of asymmetric structures is implemented. The main questions concernthe combination of the forces in the two directions, and the best way to consider extraductility demand for the elements near the edges.This report presents and discusses the main results of the new methodology to assess theseismic response of irregular structures by means of capacity domains which allow us toevaluate the direction of least seismic capacity of the investigated buildings and polarspectrum which representing the seismic demand in plan. In particular, the capacitydomains, represented in terms of strength and displacement capacities, are evaluated usingtraditional pushover analyses by applying load distributions, triangular and uniform, atdifferent angles in plan to search for the least seismic - resistant direction. The polar 1
- 2. spectrum, instead, is a new spectral representation which are able to investigate the seismicdemand each in-plane direction. In particular, this tool is based on the spatial spectralsurface obtained by the spectral seismic response evaluated for different in-plan directionsand the in-plan projection is defined “Polar Spectrum”.Both polar spectrum and capacity domains allow researchers to investigate the non-linearseismic capacities of irregular structures. As is well-known, indeed, the collapse conditionsfor a three-dimensional system are also governed by the spatial features of the seismicevent. Therefore, the goal of the presented results is to investigate the seismic behaviour ofirregular structures by considering the least seismic-resistant directions of the investigatedstructure and the spatial features of the seismic event.DESCRIPTION OF THE BENCHMARK STRUCTURETo the scope of this report the benchmark structure proposed in the ReLUIS project (Retedei Laboratori Universitari di Ingegneria Sismica – Earthquake Engineering Test LabsNetwork) is considered.The structure, designed without seismic actions, is a five-storey L-shaped building with RCmoment frames in two orthogonal directions, as represented in Figure 1. Details of thebenchmark structure are contained in the ReLUIS project. Y e d c b a X 1 2 3 4 5 Fig. 1 – Benchmark Structure PlanA finite-element model has been implemented in Opensees software. The floors aremodelled by means of 50mm elastic shells and the elastic modulus is set to the value of30000MPa. Beams and columns are modelled by spread plasticity non-linear elements withfiber sections applied.The collapse condition for the generic section is defined in terms of section curvature,evaluated by imposing limit strain values for the concrete fibers (c=0,006 in compression)and for the reinforced steel fibers (s=0,03 in tension).The main dynamical properties of the model are reported in the Table 1. In particular, eachmode shape is described by the vibration period (T), the mass participation ratios (M%)and the sum of the participation ratios up to the considered modal shape (SUM). 2
- 3. Modo T(s) M%x M%y SumX SumY 1 1,35 10,9% 52,1% 10,9% 52,1% 2 1,22 56,9% 19,6% 67,8% 71,8% 3 0,99 10,8% 9,2% 78,6% 80,9% 4 0,42 0,9% 8,5% 79,4% 89,4% 5 0,38 9,4% 1,9% 88,8% 91,2% 6 0,31 1,5% 1,1% 90,3% 92,4% 7 0,23 0,1% 4,1% 90,4% 96,5% 8 0,20 4,2% 0,5% 94,6% 97,0% 9 0,17 0,9% 0,5% 95,5% 97,5% Table 1 - Modal properties for the Benchmark StructureTHE CONCEPT OF THE CAPACITY DOMAINSIn order to assess the non-linear static response of irregular structures it is possible toobtain capacity domains, in terms of control node displacement or base shear. In particular,such domains are obtained from the results of non-linear static procedures (pushover)carried out by varying the in-plan direction of the applied load distribution. The domains,represented in the following, depict the displacement of the control node or the base shearup to collapse.The analyses are carried out by applying two load distributions along the height of thestructure. The first one is a triangular distribution, taking into consideration the inertialmasses and the height above the ground. The second one considers only the floor masses.The forces are applied at beam-column joints, proportionally to the joints mass.For each applied load distribution, 24 capacity curves have been obtained by rotating thedirection of action in the x-y plane, 15° at each step. The control node is assumed to be thecentre of mass on the top floor in all the pushover analyses carried out.Using the described limit values for section curvature, the global collapse is assumed whenthe first primary element exceeds its limit. Each pushover curve is bi-linearized accordingto the FEMA 356 procedure.The capacity domains are compared to the non-linear dynamic response to recordedearthquake, selected among those proposed by ReLUIS. The records, considering both NSand EW components, are suitably scaled to study the structural response by varyingground acceleration as in the case of non linear incremental analysis (IDA).The main results are represented in the following. In particular, the figures 2 and 3refer tothe collapse of the analyzed benchmark structure by varying the direction of action in theplan in the case of triangular (TR) and uniform (UN) load distributions and represent thecapacity domains. These domains are plotted in terms of displacement of the control node(U) and shear base (Vb). As it is possible to observe for both domains a non-regularbehaviour of the structure above all in terms of limit displacements. In particular, theanalyses show that the benchmark structure presents lower displacement capacities in the150°- 330° direction. Otherwise, structural behaviour seems to be more regular in terms ofstrength. 3
- 4. U (cm) Vb (kN) 90 90 105 20 75 105 3000 75 120 60 120 60 2500 135 15 45 135 45 2000 150 10 30 150 1500 30 1000 165 15 165 15 5 500 180 0 0 180 0 0 195 345 195 345 210 330 210 330 225 315 225 315 240 300 UN 240 300 255 285 UN TR 255 285 270 270 TR Fig. 2 - Capacity Domains in terms of node control Fig. 3 - Capacity Domains in terms of base shear displacements at collapseIn order to evaluate the applicability of the capacity domains to accurately describe thecollapse state of a structure, incremental dynamic analyses have been performed. For boththe considered in-plan dispositions of the structures, 0° and 90°, the analyses have beencarried out considering the Iceland 2000 earthquake (code 06334, maximum PGA equal to7,07m/s2). The figures 4-7 show the dynamic response (NDP) for the analyzed cases. 0,2 3000 0,15 2000 0,1 1000 0,05 Uy (m) Vy (kN) 0 0 -0,2 -0,15 -0,1 -0,05 0 0,05 0,1 0,15 0,2 -3000 -2000 -1000 0 1000 2000 3000 -0,05 -1000 -0,1 -2000 -0,15 -0,2 -3000 Ux (m) Vx (kN) NSP uniform NSP Triangular NDP NSP uniform NSP Triangular NDP Fig. 4 – Comparition between capacity domains and Fig. 5 - Comparition between capacitys domain and NDP - Collapse amplification factor 0.40 - In plan NDP - Collapse amplification factor 0.40 - In plan orientation of benchmark structure 0° orientation of benchmark structure 0° 4
- 5. 0,20 3000 0,15 2000 0,10 1000 0,05 Uy (m) Vy (kN) 0,00 0 -0,20 -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15 0,20 -3000 -2000 -1000 0 1000 2000 3000 -0,05 -1000 -0,10 -2000 -0,15 -0,20 -3000 Ux (m) Vx (kN) NSP uniform NSP Triangular NDP NSP uniform NSP Triangular NDP Fig. 6 – Comparition between capacity domains and Fig. 7 - Comparition between capacity domains and NDP - Collapse amplification factor 0.35 - In plan NDP - Collapse amplification factor 0.35- In plan orientation of benchmark structure 90° orientation of benchmark structure 90°As it is possible to observe, the collapse is attained in the direction of about 255°independently by plan orientation of benchmark structure. Moreover, the collapse occurs incorrespondence of the capacity domain for triangular load distribution, and the collapsedelement, column 15 at ground level, is the same for both dynamic and static analyses. Forboth analyses the collapse amplification factor in the case of NDP is represented. In thefigures, the capacity domains are indicated with NSP (Non-linear Static Procedure) for theuniform and triangular load distributions.As a further comparison between static and dynamic responses, figure 8 shows storeydisplacements at collapse for both analyses in the considered plan directions of benchmarkstructure (0° and 90°). 5 5 4 4 3 3 imp imp 2 2 1 1 0 0 0 0,1 0,2 0 0,1 0,2 U (m) U (m) NSP uniform NSP Triangular NDP Fig. 8 – Storey Displacement Comparison at collapse for in plan directions 0° and 90° 5
- 6. Results show that the collapse mechanism is a global one, being the displacement pathcarried out by NDP well described by the one obtained by NSP with triangular loaddistribution. Moreover, as it is possible to observe, the collapse as well as read with NDPprocedure intercepts always the capacity domains corresponding to triangular loaddistribution (outer domain in figures 4 and 6 and inner domain in figures 5 and 7).THE CONCEPT OF “POLAR SPECTRUM”As known, seismic events generally present a strong directivity effects due to site geo-morphological conditions, source mechanism and path from source to the site of theseismic waves.To take into account the directivity and site effects it is possible to refer to polar spectrum,a new tool, proposed by authors, which is able to investigate each in-plan direction. Inparticular, this tool is based on the spatial spectral surface obtained by the spectral seismicresponse evaluated for different in-plan directions and the in-plan projection of this surfaceis defined “Polar Spectrum”.Figure 9 shows the polar spectrum in terms of pseudo-acceleration for the seismic eventSouth Iceland aftershock (code 006334) with a damping factor =0,05 in the range ofperiods 0-2 seconds. In particular, the spectral surface has been obtained by consideringthe NS and EW components of recorded accelerograms projected along each direction inplan. In correspondence of each radius, the spectrum response evaluated in that direction isrepresented. In correspondence to each circumference, instead, the spectral demand for afixed period is plotted for each considered direction. The origin corresponds to 0 secperiod. As it is easy to observe, the polar spectrum is a polar-symmetric figure. Fig. 9 – Polar Spectrum in terms of pseudo-acceleration for South Iceland aftershock (code 006334)The analysis of polar spectrum shows that the considered seismic event presents a peakdemand along the direction (75°-255°) regardless of the vibration period value.The comparison between the polar spectrum of the considered seismic event and the resultsof Non-Linear Dynamic Procedure (NDP) shows clearly as the collapse of the benchmarkstructure are controlled by the directivity effects more than seismic capacity of theanalyzed structure.“CRITICAL” ALIGNMENTThe possibility of evaluating the worst structural response moving from observation of thepolar spectrum suggests performing dynamic analyses with the structure oriented so that its 6
- 7. least capacity direction (assessed by means of the capacity domains) is aligned with themaximum seismic demand (evaluated by means of the polar spectrum).Figure 10 depicts the results in this case. In particular, the structure is been rotated 105°counter-clockwise in order to match the least resistant direction (=150°- 330°) with themaximum spectral seismic demand direction (75°- 255°). 0,20 0,15 0,10 0,05 Uy (m) 0,00 -0,20 -0,15 -0,10 -0,05 0,00 0,05 0,10 0,15 0,20 -0,05 -0,10 -0,15 -0,20 Ux (m) Fig. 10 – Dynamic analyses results, in plan orientation of the benchmark structure 240°, NDP collapse amplification factor 0.30The obtained results show that when the least seismic resistance direction is aligned withthe maximum seismic demand, the minimum collapse amplification factor is reached.SOFTWARE IMPLEMENTATIONA first implementation of the capacity domains could be found in the softwareIPERSPACE which leads to evaluate these domains in terms of limit states according tothe Italian Design Code (NTC 2008). As an example, the following figures represent themain results obtained with Iperspace software for the analyzed benchmark structure: Fig.11- Capacity Domains in terms of node control displacements at collapse (on the left) and maximum base shear (on the right) obtained with Iperspace Software 7
- 8. The modest difference between the capacity domains obtained by Iperspace and byOpensees is due to the adopted collapse criteria. In the case of Iperspace, the NTC2008collapse criterion has been considered by controlling the ultimate plastic hinge rotation ofRC members.CONCLUDING REMARKSThe report shows new tools in order to assess the non-linear behaviour of a irregularbuilding and the spectral seismic demand in plan. In particular, the capacity domains areevaluated by means of pushover analyses by varying the load distribution in plan and leadto search for less seismic resistant directions. The polar spectra lead to a fully assessmentof the spatial features of the seismic demand and represent an opportunity to investigatethoroughly the seismic demand and the event characteristics, being able to clearly evaluatedirectivity and site effects.From the obtained results it is possible to observe that: The collapse condition, assessed by means of non-linear dynamic procedures, turns out to be coherent with the capacity domains based on the results of traditional non-linear static procedures carried out by varying the angle of the application of the load distribution; The non-linear dynamic analyses have shown that the directional components of the seismic event govern the collapse direction; The worst condition for a structure is obtained by aligning its least resistance capacity direction, evaluated on the basis of the capacity domains, with the maximum response demand direction identified by the polar spectrum.REFERENCESAyala A. G. and Tavera, E. A. [2002], “A new approach for the evaluation of the seismicperformance of asymmetric buildings”, Proc. of the 7th National Conference on EarthquakeEngineering, Boston.Chopra A. K. and Goel, R. K. [2004], “A modal pushover analysis procedure to estimate seismicdemands for unsymmetric-plan buildings”, Earthquake Engineering and Structural Dynamics33(8), pp. 903-927.Chopra A. K. [2008]. Application of modal pushover analysis to RC frames and IncrementalDynamic Analysis. Nonlinear Static Methods for Design/Assessment of 3D Structures, 5-6 May2008, Lisbon, Portugal.Faella G. and Kilar V. [1998], “Asymmetric multistory R/C frame structures: push-over versusnonlinear dynamic analysis”, Proc., 11th Eur. Conf. Earth. Engrg., A.A. Balkema, Rotterdam.Fajfar P. [2000], “A nonlinear analysis method for performance-based seismic design”, EarthquakeSpectra 16(3), pp. 573-592.Fajfar, P. [2002], “Structural analysis in earthquake engineering - a breakthrough of simplifiednonlinear methods”, Proc. of the 12th European Conference on Earthquake Engineering, London,UK, Keynote lecture.Fajfar P. and Gaspersic P. [1996], “The N2 method for the seismic damage analysis of RCbuildings”, Earthquake Engineering and Structural Dynamics 25, pp. 23-67. 8
- 9. Fajfar P. and Kilar V. [1997], “Simple push-over analysis of asymmetric buildings”, EarthquakeEngineering and Structural Dynamics, Vol. 26, pp. 233-249.Fajfar P., Kilar V., Marusic D., Perus I. and Magliulo G. [2002], “The extension of the N2 methodto asymmetric buildings”, Proc. of the fourth forum on Implications of recent earthquakes onseismic risk, Technical report TIT/EERG, 02/1, Tokyo Institute of Technology, Tokyo, pp. 291-308.Fajfar P., Marusic D. and Perus I. [2004], “Influence of ground motion intensity on the inelastictorsional response of asymmetric buildings”, Proc. of the 13th World Conference on EarthquakeEngineering, Vancouver, Canada, Paper No. 3496.Fajfar P., Marusic D. and Perus I. [2005], “Torsional effects in the pushover-based seismic analysisof buildings”, Journal of Earthquake Engineering, Vol. 9, No. 6, pp. 831-854.Fajfar P., Marusic D., Perus I., Kreslin M. [2008]. “The N2 method for asymmetric buildings”,Nonlinear Static Methods for Design/Assessment of 3D Structures, 5-6 May 2008, Lisbon,Portugal.Fujii K., Nakano Y. and Sanada Y. [2004], “Simplified nonlinear analysis procedure forasymmetric buildings”, Proc. of the 13th World Conference on Earthquake Engineering,Vancouver, Canada, Paper No. 149.Goel R. K. [2008] Generalized pushover curves for nonlinear static analysis of three dimensionalstructures. Nonlinear Static Methods for Design/Assessment of 3D Structures, 5-6 May 2008,Lisbon, Portugal.Hejal R. and Chopra A. [1987], “Earthquake response of torsionally-coupled buildings”, ReportUCB/EERC-87/20.Kan C. L. and Chopra A. K. [1979], “Linear and nonlinear earthquake responses of simpletorsionally coupled systems”, Report UCB/EERC-79/03.Kilar V. and Fajfar P. [2001], “On the applicability of pushover analysis to the seismicperformance evaluation of asymmetric buildings”, European Earthquake Engineering 15, pp. 20-31.Marino I., Petti L. [2011], “Seismic Assessment of Asymmetric Structures Behaviour by usingStatic Non-Linear Analysis”, in corso di stampa sulla rivista Journal of Civil Engineering andArchitecture ISSN 1934-7359.Moghadam A. S. and Tso K. [2000], “Pushover analysis for asymmetric and set-back multistorybuildings”, Proc. of the 12th World Conference on Earthquake Engineering, Auckland, NewZealand, Paper No. 1093.Open System for Earthquake Engineering Simulation (OpenSEES), Pacific EarthquakeEngineering Research Center, University of California, Berkley (USA);Petti L., Marino I., Cuoco L., Palazzo B. [2008], “Assessment of Seismic Response of Plan-Asymmetric Structures through simplified Static Nonlinear Analyses”, Proc.the 14th WorldConference on Earthquake Engineeering, Beijing, China.Pettinga J. D., Priestley M. J. N., Pampanin S., Christopulos C. [2007], “The Role of InelasticTorsion in the Determination of Residual Deformations”, Journal of Earthquake Engineering, vo.l11 2007, supplement 1, Proceedings of the Sixth ROSE Seminar.ReLUIS project - Rete dei Laboratori Universitari di Ingegneria Sismica – Earthquake EngineeringTest Labs Network, www.reluis.unina.it.Reyes C. Juan and Anil K. Chopra [2011], “Evaluation of three-dimensional modal pushoveranalysis for unsymmetric-plan buildings subjected to two components of ground motion”,Earthquake Engineering and Structural Dynamics; 40:1475 -1494. 9
- 10. Rutenberg A. [2002], EAEE Task Group (TG) 8: “Behaviour of irregular and complex structuresasymmetric structures - progress since 1998”, Proc. of the 12th European Conference onEarthquake Engineering, London, UK, Paper No. 832.Rutenberg A., Levy R. and Magliulo G. [2002], “Seismic response of asymmetric perimeter framesteel buildings”, Proc. of the 12th European Conference on Earthquake Engineering, London, UK,Paper No. 107.Soft.lab, Software Partnership http://www.soft.lab.it, Via Borgo II° - 82030 - PONTE (BN) -IT, IperSpace MaxTso W. K. and Moghadam A. S. [1997], “Seismic response of asymmetrical buildings usingpushover analysis”, Proc. of the International Workshop on Seismic Design Methodologies for theNext Generation of Codes, Bled, Slovenia, eds. P. Fajfar and H. Krawinkler (A.A. Balkema,Roterdam, Brookfield), pp. 311-321.Tso W. K. and Myslimaj B. [2002], “Effect of strength distribution on the inelastic torsionalresponse of asymmetric structural systems”, Proc. of the 12th European Conference on EarthquakeEngineering, London, UK, Paper No. 081.Yu Q.S.K., Pugliesi R., Allen M. and Bischoff C. [2004], “Assessment of modal pushover analysisprocedure and its application to seismic evaluation of existing building”, Proc. of the 13th WorldConference on Earthquake Engineering, Vancouver, Canada, Paper No. 1104.Zarate A. and Ayala A. G. [2004], “Validation of single storey models for the evaluation of seismicperformance of multi-storey asymmetric buildings”, Proc. of the 13th World Conference onEarthquake Engineering, Vancouver, Canada, Paper No. 2213. 10

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