PSEUDO-ENERGY RESPONSE SPECTRA FOR THE EVALUATION OF THE SEISMIC RESPONSE FROM PUSHOVER ANALYSIS

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1 First uropean Conference on arthquake ngineering and Seismology (a joint event of the th C & 0 th General Assembly of the SC) Geneva, Switzerland, -8 September 00 Paper Number: 8 PSUDO-NRGY RSPONS SPCTRA FOR TH VALUATION OF TH SISMIC RSPONS FROM PUSHOVR ANALYSIS Marco MZZI, Fabrizio COMODINI, Matteo LUCARLLI, Alberto PARDUCCI and nrico TOMASSOLI SUMMARY The application of non linear static analysis method through the energy approach is based on the idea that the energy of the seismic input transferred to the structure is dissipated by the controlled damage of its members. The pushover curve is computed considering that, in each step, the work of the floor forces is equal to the structure internal work and is expressed in terms of energy capacity. It can be compared with energy response spectra representative of the seismic input to find the performance point defining the structural response to the design earthquake. The use of pseudo-energy spectra is proposed, alternative to the conventional reduced design spectra. Solutions are carried out for a case of study. The results are compared with those coming from non linear static analyses based on reduced spectra with controlled damping or ductility and from non linear dynamic analyses. The potential evolutions of the methodology are outlined.. INTRODUCTION The pushover analysis is a simplified method for the seismic design of structures taking into account their postelastic behavior: a nonlinear model of the building is subjected to monotonically increasing lateral forces up to the attainment of a target displacement at a predefined point of the structure. The global and local status should correspond to the maximum response obtained from a dynamic analysis under the design earthquake. The use of nonlinear static analysis procedures is continuously increasing in seismic design of structures and it is provided by the most advanced codes. It was introduced about ten years ago in ATC-0 [ATC, 99] and in FMA-7 [FMA, 997] and it is now included in the last revision of [urocode 8, 00] (C8) and in the new Italian guidelines [Ordinanza, 00] (O). Different methodologies have been developed and are reported in the various codes, generally based on the comparison of a capacity curve of the structure with a demand curve of the earthquake, both plotted in ADRS format. One of the most widespread method is the Capacity Spectrum Method, that can be applied following different procedures. The base shear, V base, vs. top displacement, d top, pushover curve of the whole structure is computed as response to increasing lateral floor loads. The capacity diagram of an equivalent SDOF system in ADRS format is computed by dividing the base shear by the effective modal mass M of the fundamental vibration mode and the top displacement by the factor Γ φ N, being Γ the modal participation factor and φ N the roof modal deformation for the fundamental mode. The curve can then be converted in an PP relationship throughout an energetic equivalence, so a conventional yielding value d y of the control displacement d top is defined. The performance point is then evaluated as intersection between the capacity diagram and a demand curve of the considered earthquake, consisting of a response spectrum reduced according to en equivalent damping or ductility value. The computed value is then converted into the structure target displacement at roof level. ATC-0 details three procedures to estimate the intersection of the capacity and demand diagram. While, Dipartimento di Ingegneria Civile ed Ambientale, Università di Perugia - Via G.Duranti 9-0 Perugia (Italia) mail : m.mezzi@unipg.it

2 according to C8 and O, a modified value of the displacement computed as the intersection of the capacity diagram with the elastic response spectrum of earthquake can be assumed. Reliability of pushover analysis depends on several parameters concerning the loading process: the choice of the control point, the distribution of loading forces, the conversion to SDOF response in ADRS format, the assumption of reduced spectra. A critical review of this aspects is reported in (Chopra, 999) and (FMA 0, 00). Other problematical questions are involved by the application of the pushover analysis: influence of higher modes in the response, influence of plasticization in the evolution of the load distribution, influence of torsion in D response. All these aspects have already been investigated and special methodology variants have been developed and suggested. They are not referred here because are out of the present interest. In any case it has to be reminded that the pushover analysis must not give an answer to all the questions because it is, and must be considered, only a "design" methodology and not a methods for evaluate the "true" dynamic response of a structure. Within this objective, this paper mainly focuses the assessment of the equivalent SDOF system and the performance point based on the use of energy concepts with the aim of simplifying the choosing procedures.. NRGY-BASD NON LINAR STATIC ANALYSIS Since the beginning of the modern earthquake engineering, the energy concept has guided the interpretation of the seismic resistant mechanisms of the structures [Housner, 9]. In the following development many researchers investigated this subject and a spread review of these studies is reported in [Anderson and Bertero, 00]. Nowadays, the energy concept returns to earn a great attention because of the use of the Performance Based Seismic Design (PBSD) in the assessment of earthquake resistant structures. One of the tools of the PBSD is the non linear static analysis based on the use of the pushover capacity curve of a simplified equivalent SDOF, giving a synthetic representation of the global seismic performance of the MDOF structural system. The capacity curve can be used in the PBSD for comparison with coherent acceleration demand spectra, reduced according to the actual dissipation capacity of the system. The comparison can be easily made if both the capacity and the demand are drawn in an ADRS representation. Since the PBSD fundamentally depends on energy concepts, methods for the evaluation of both the capacity and demand curve using only energy concepts are suitable. An application of non linear static analysis method through energy approach is based on the idea that the energy of seismic input transferred to the structure is dissipated by the controlled damage of its members. The energybased pushover analysis is founded on the definition of the equivalent SDOF system by means of an energy equivalence. The pushover curve, as shown in Figure, is defined considering that, in each load step, the work of the floor forces, Σ i (k), is equal to the structure internal work, (k). The equivalent displacement δ does not correspond to any actual point on the MDOF model, it is a virtual value equalizing the work done under the total shear force F=Σ(F i ). xpressing the shear in terms of acceleration f=f/m, the capacity curve can be drawn in the ADRS representation, and the method proceeds by comparing it with the energy demand spectra as usual [Parducci et al., 00]. Fi Fi(k+) Fi(k) Work done by the floor force Fi in the MDOF model Σ i(k) = di(k) di(k+) di F F(k+) F(k) Capacity curve of the equivalent SDOF system based on energy equivalence δ(k) m (k) δ(k+) δm δ m (k) Figure : nergy-based definition of the capacity curve nergy-based capacity curve δ(k+) δm δ

3 As a consequence of the construction procedure of the pushover curve above illustrated, it can be easily expressed in terms of energy capacity vs. displacement (Figure ) and could be compared with energy response spectra representative of the seismic input to find the performance point defining the structural response to the design earthquake, so outlining a seismic design procedure completely based on energy criteria. This approach is useful for a variety of aspects: the arbitrary choice of displacement control point is eliminated; the equal-energy capacity curve has physical meaning since its integral is the actual elastic-plastic energy dissipated by the structure; the ductility coefficient computed through the capacity curve is the global ductility of the actual system; the evaluation of the performance point does not require iterative procedure for equalizing the ductility or damping of the reduced spectra conventionally derived from the elastic one. Figure shows a sample application of the methodology [Parducci et al., 00]. In this case the spectrum of the input energy, I, was derived from the pseudo-velocity spectrum, S v, and then from the pseudo-acceleration spectrum, S a, defined by the code I = ms v = m Sa T π () Finally, the input energy was expressed as a function of displacement on an nergy Displacement Response Spectra (DRS) plane. The solution can be found, starting from the elastic capacity curve in terms of "equal energy" or "equal displacement", because the above defined spectrum is elastic and has not information on the ductility demand. elastic capacity demand PPD PP nonlinear capacity δd="qual-displacement" displ. δ ="qual-nergy" displ. PD="qual-Displacement" PP P="qual-nergy" PP δd δ δ Figure : Performance point identification on the DRS plane The energy-based capacity curve can be computed directly [Parducci et al., 00], but the assumption of seismic response spectra in terms of energy needs some considerations. The evaluation of energy spectra in terms of pseudo-velocity can lead to results that are not in good agreement with the actual input energy computed from direct evaluations using the ground motions. The argument requires more investigations because the PBSD of high dissipating systems founds its validity on energy-based correlations, keeping away from any analytical step based on elastic concepts used in traditional design methods, and evaluations derived from the direct comparison of the energy capacity and demand could have more significance than those based on the use of elastic pseudoacceleration spectra empirically reduced for taking into account the energy dissipation. Moreover, the capacity curves of the ductile structures, extended in the field of the plastic deformations, can be easily modelled with a bi-linear representation on which a significant value of ductility can be read.. NRGY SPCTRA The energy demand of an earthquake is an easy intuitive concept, but it is a practical hard one. The input energy, I, can be expressed as a sum of contributions defined by the energy balance equation [Uang and Bertero, 988]: I = s + k + h () + s is the recoverable elastic strain energy, k is the kinetic energy, h is the irrecoverable plastic hysteretic energy, and ξ is the irrecoverable damping energy.

4 A first problem is that of defining the type of input energy, absolute or relative, to be considered for the design evaluations. [Uang and Bertero, 988] suggest to assume the absolute input energy, other authors [Faifar et al., 99] use the relative energy formulation. In this paper the relative energy is used, considering that significant differences between the absolute and relative demand can be found only for extreme values of the system natural period, out of the interval s, that lie outside the objectives of the applications. A second problematic aspect concerns the correlation between the energy absorbed by the structure and its response in terms of maximum displacements and forces, which are the parameters interesting the design. The energy-based pushover curve should allow the correspondence. The influence of the external factors - magnitude, focal mechanism, distance from the source, local soil characteristics - on the input energy, can be significant and is discussed in [Decanini and Mollaioli, 998]. In the following, the energy considerations have been derived making reference to a seismic input consisting of synthetic spectrum-fitting accelerograms. Nine accelerograms were generated using the code SIMQK [Gasparini and Vanmarcke, 97]. The spectrum type S provided by C8 for subsoil class A (stiff soil) was assumed as target elastic response spectra in the generation with spectrum-fitting rules according to those provided by C8. Accelerograms are characterised by a peak acceleration of 0. g, by a duration of s with a growing range of. s and an intense phase of s. The average response spectra of the nine generated accelerograms have been calculated for the single energy components I,, K, H, D. Constant ductility spectra of relative input energy have been computed, for six values of displacement Ductility Ratio (DR), going from to in step of, assuming an elastic-perfectly plastic non linear behaviour. The input energy spectra are reported in Figure a and show that the total energy transmitted by the earthquake to the structure strongly depends on the ductility ratio, that is on the design factor R, in the range of periods greater than T B. The most variable factors are the hysteretic and damping contributes. Also the kinetic and elastic contributes show strong variation. As a consequence the "equal displacement" criterion, among nonlinear solutions differing in plastic threshold, appears more adequate than that of the "equal energy" in the range of periods greater than T B. This is confirmed by the observation of the constant ductility response spectra in terms of displacement reported in Figure b..0.0 (a).0.0 (b) Relative Input nergy (m /s ) DR = DR = DR = DR = DR = DR = Relative Input nergy (m /s ) DR = DR = DR = DR = DR = DR = Period (s) Displacement (m) Figure : Constant ductility spectra of relative input energy vs. periods (a) and vs. displacements (b). PSUDO-NRGY SPCTRA The greatest problem in using energy demand spectra in a static non linear analysis consists of the fact that the most representative value of the energy transferred by the earthquake to the structure is the input energy, but this value is not related to the maximum responses of the structure and, deriving from a cyclic behaviour, cannot be correlated to a monotonically growing pushover curve. At the same time the kinetic energy cannot be correlated with the ductility value and, using these spectra, a response can be computed only under the hypothesis of "equal displacement" or "equal energy" for different ductility ratios, as shown previously. In this research a proposal has been advanced of using a new parameter for representing the energy demand of an earthquake, that derives from both the maximum force and displacement of the response and that can be compared with the energy associated to a monotonic pushover curve. This parameter has been defined "pseudo-

5 energy" and has computed, from the cyclic response of the P oscillator, as the area of the curve consisting of the positive or negative envelope, whichever includes the maximum displacement, of the cyclic behaviour, and then formed by the first elastic branch and by the plastic branch until the maximum inelastic displacement, as illustrated by the hatched area in Figure. Y U O Figure : Pseudo-energy definition for an P response Spectra of the above defined pseudo-energy have been calculated for the nine accelerograms: the average spectra for the six considered ductility ratio are reported in Figure. The pseudo-energy is a consistent parameter for the comparison with the energy values of the capacity curve derived by a pushover analysis. It is an energy parameter allowing for accounting at the same time for the displacement, ductility, and resistance parameters of the response. The use of the pseudo-energy spectra, together with the energy capacity spectrum of the structure, for the evaluation of the performance point is illustrated in the following chapter Pseudo-nergy (m /s ) DR = DR = DR = DR = DR = DR = Displacement (m) Figure : Constant ductility pseudo-energy spectra vs. displacement. SOLVING PROCDURS The first step of the solution procedure consists of the evaluation of the performance point, represented by a reliable, consistent, compatible intersection between a demand and capacity curve. The energy consistency of the intersection is automatically assured, and must not to be searched for through iterative procedures like in other methods, thanks to the representation of demand and capacity in true terms of energy. Two procedures have been pointed out for directly finding the performance point. The first solution provides for a ductility consistent solution that can be directly found if the energy capacity curve is expressed as a function of the ductility, instead of the maximum displacement, after a transformation of the continuous force-displacement capacity curve in a bilateral curve as shown in Figure (on the left). The bilinearization can be carried out under the hypothesis of conserving the same dissipated energy, or, what is the same, maintaining the same integral of the force-displacement curve. Therefore, an elastic limit displacement, δ y, can be computed

6 δ y ( δ / f ) = m Π m m () as a function of the maximum displacement, δ m, and the corresponding force, f m, and energy, Π m. The ductility corresponding to the displacement δ m can then be computed, in terms of energies, as µ + Π ( ) = / m / Π e () where Π e = / δ y f m, is the elastic contribute of energy, that is the energy corresponding to the elastic branch. The only ductility-consistent intersection of the capacity curve with a constant ductility demand spectrum, is the performance point for which the ductility of the reduced spectrum corresponds to the abscissa of the intersection point, as shown by Figure. F/m equal-energy elastic-plastic capacity curve DR= elastic-plastic pushover curve fy DR= capacity curve or the structure DR= DR= PP δy δm DR= DR= DR= DR= δ,dr Figure : lastic-plastic force-displacement capacity curve (left) and first procedure for detecting the performance point on the DRS plane (right) The second procedure does not require the modification of the capacity curve and avoids the arbitrary choice of the equivalent elastic-perfectly-plastic assumption connected with the bi-linearization. It requires the evaluation of both a linear and non linear energy capacity curve, therefore both a linear and non linear pushover analysis must be performed preliminarily. The graphical representation of the solution procedure is shown in Figure 7. The intersection between the elastic capacity curve (curve C) and the constant ductility pseudo-energy demand spectrum corresponding to a ductility ratio (curve D), represents the elastic solution of the structure, that is the response of the structure performing elastically and characterized by its fundamental period. Assuming that the non linear performing structure is characterized by the same period, in its elastic performance range, the actual performance point, represented by an intersection of the non linear capacity curve with a constant ductility demand spectrum, must result to be associated to this period. All the possible solutions characterized by the same period are represented by the constant period energy-displacement curve starting from the elastic performance point (curve DT). The intersection between the constant period curve (curve DT) and the non linear capacity curve (curve CN) represents the performance point which individuate the solution of the current problem. The ductility associated to the solution can be estimated from the constant ductility demand spectrum fixed out by the performance point. C CN D D D D PP DT Legenda: D, D, D, D Constant ductility demand spectra for DR =,,, C = lastic capacity curve CN = Nonlinear capacity curve DT = Constant period demand curve δ Figure 7: Second procedure for detecting the performance point on the DRS plane

7 . SAMPL APPLICATION A six story reinforced concrete plane frame, shown in Figure 8, is considered as sample structure. The span of the bays is.80 m for the lateral ones, and.0 m for the central ones. The story height is.0 m. The columns have dimensions 00x700 mm for the first three storeys and 00x00mm in the successive storeys. All the beams have section 00x00 mm. The frame is designed according to the specification of C8 for high ductility class (DCH). The design is performed by a modal response spectrum analysis with reference to the importance category I=.0, assuming a peak ground acceleration equal to 0. g and the parameters shaping a soil profile A spectrum. Dead loads of. kn/m (due to self weight, finishes and permanent partitions) and live loads of.0 kn/m are assumed at the intermediate floors. At the top floor, dead and live loads are,0 kn/m and. kn/m respectively. A concrete type C/0 and a steel with yield strength of 0 MPa are considered. For both columns and beams the required different strength levels are obtained by varying the amount of reinforcement. The reinforcement percentage is. % for the columns 00x700,.% for the columns 00x00 and varies from.% and.9% for the beams. T = 0.7 s and T = 0. s are the periods of the first two modes. Their mass participating factors are 0.79 and 0. respectively. 00x00 00x00 00x00,0 m 00x700 00x700 00x00 00x00 00x00 00x700 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x00 00x700 00x700 00x00 00x00 00x00 00x700 00x700 00x00 00x00 00x00 00x700 00x700 00x700 00x700 00x700 00x00 00x00 00x00,0 m,0,0,0,0,0,80,0,80 m Figure 8: Sample r/c frame The seismic response of the frame is computed with both nonlinear static and dynamic analyses. The previously defined nine spectrum-fitted accelerograms are used as input in the dynamic analyses. All the analyses - both linear and non linear, both static and dynamic - were carried out by the code SAP000 [SAP000, 00] which is able to reproduce the inelastic behaviours of the structure through the definitions, for the beams and columns, of the moment-rotation relationships of the potential plastic hinges at the elements ends, derived assuming the usual nonlinear relationship for the stress-strain curves of concrete and steel defined in [urocode, 99]. The nonlinear static analyses have been carried out by means of the energy-based method previously illustrated applying the second procedure based on the use of constant period curves. Two distributions of the lateral loads have been assumed for carrying out the pushover analyses: in the first one the force at each floor is proportional to the floor mass multiplied by the floor lateral displacement in the first mode, in the second one the forces are proportional to the masses. The average pseudo-energy response spectra of the generated accelerograms, reported in Figure, are used as constant ductility demand curves. A comparison is made with the solution resulting from the application of the N method provided by C8 and O. Figure 9 reports, for the two load distributions, the lateral story displacements computed using the energy-based method (green bars), the code method (blue bars), the red bars represent the "exact" solution, that is the average of the solutions from the nine dynamic analyses under the nine generated accelerograms. It is evident that the energy-based method gives values in good accord with the code method, both the methods well approximate the "exact" dynamic solution. Also the story drift ratios, ratios between the inter-story displacements and the story heights, are quite similar for the two analysis methods, as results in Figure 0, where the values from the energy-based method (green bars) and the code method (blue bars) are reported. 7

8 Dynamic nergy push Code push Dynamic nergy push Code push Lateral displacement (m) Lateral displacement (m) Figure 9: Floor displacements for modal (left) and mass proportional (right) force distribution nergy push Code push nergy push Code push Drift ratio Drift ratio Figure 0: drift ratios from pushover analysis for modal (left) and mass proportional (right) force distribution The agreement of the computed drift ratios with the exact dynamic solution is assured only at the lower stories, as shown by the Figure, where the values from the energy-based method (green bars) and the dynamic analysis (red bars) are reported. The values at the higher stories do not cover the dynamic results, but this fact depends on some limitations of the static pushover analysis and on the characteristics of the sample structure examined. The drift at the higher stories is exalted, in dynamic analyses, by the contribution given to the lateral deformation by the higher modes. The maximum values of drifts do not occur when the lateral displacements are maximum and therefore cannot be correctly reproduced by a static pushover load, nor proportional to masses, neither to the first modal shape. The inaccuracy in the evaluation of the drifts is not so relevant for design aims because the deformations at the higher stories do not involve plastic deformations of the structural elements Drift ratio Drift ratio Figure : drift from dynamic (red bars) and pushover analysis (green bars) for modal (left) and mass proportional (right) force distribution 8

9 7. CONCLUSIONS An energy-based method for static nonlinear analysis is outlined using both capacity and demand curves expressed in terms of energy. Constant ductility pseudo-energy spectra are proposed as demand design spectra alternative to the conventional reduced spectra for the direct evaluation of the performance point by means of two application procedures, both leading to the estimation of the global ductility demand. Sample solutions are carried out for a r/c frame. The results are in good agreement with those coming from non linear static analyses based on reduced spectra with controlled damping or ductility and from non linear dynamic analyses. The potential application of the outlined method can be extended, providing for the use of standard pseudoenergy demand spectra derived by the ordinary pseudo-acceleration design spectra, instead that using the energy demand spectra derived by the actual dynamic response of accelerograms. These standard constant ductility pseudo-energy spectra could be derived by means of transformation rules, already developed and reported in literature, of plastic thresholds and maximum displacement as a function of the ductility ratio. 8. RFRNCS Anderson, J.C., Bertero, V.V. (00), Use of nergy Concepts in arthquake ngineering: A Historical Review, Proceedings of the 8 th National Conf. on arthquake ngineering, Paper 908, San Francisco, California. ATC-0 (99) Seismic valuation and Retrofit of Concrete Buildings, Applied Technology Council, Report No. ATC 0. Chopra, A.K., and R.K. Goel (999) Capacity-demand-diagram methods for estimating seismic deformation of inelastic structures: SDF systems, Report No. PR-999/0, Pacific arthquake ngineering Research Center, University of California, Berkeley, USA. Decanini, L.D., Mollaioli, F. (998) Formulation of lastic arthquake Input nergy Spectra, arthquake ngineering Structural Dynamics, 7, 0-. urocode (99) NV 99-- Design of concrete structure. Part. - General rules and rules for buildings, CN uropean Committee for Standardisation, Bruxelles. urocode 8 (00) FINAL DRAFT prn Design of structures for earthquake resistance. CN uropean Committee for Standardisation, Bruxelles. Fajfar, P., Vidic, T., Fishinger, M. (99), On nergy Demand and Supply in SDOF System, in Nonlinear Seismic Analysis and Design of RC Buildings, lsevier Applied Science. FMA-7 (997) NRP Guidelines for the Seismic Rehabilitation of Buildings, Federal mergency Management Agency, Washington, D.C. FMA-0 (00) Improvement of Nonlinear Static Seismic Analysis Procedures (ATC-), Federal mergency Management Agency, Washington, D.C. Gasparini, D.A, Vanmarcke,.H. (97) Simulated earthquake motions compatible with prescribed response spectra. R7-, Dept. of Civil ngineering, MIT, Cambridge, Massachusetts. Housner, G.W. (9), Limit Design of Structures to Resist arthquakes, Proceedings of the st World Conference on arthquake ngineering, Berkeley, California. Ordinanza PCM /0 (00) Ulteriori modifiche ed integrazioni alla Ordinanza PCM 7 Primi lementi in Materia di Criteri Generali per la Classificazione Sismica del Territorio Nazionale e di Normative Tecniche per le Costruzioni in Zona Sismica (in Italian). Parducci, A., Comodini, F., Mezzi, M. (00) Approccio nergetico per le Analisi Pushover, Congr.Naz. "L'Ingegneria Sismica in Italia", Genova, Italia (in Italian). Parducci, A., Comodini, F., Lucarelli, M., Mezzi, M., Tomassoli. (00) nergy-based Non Linear Static Analysis, st uropean Conf. on arthquake ngineering and Seismology, Paper 78, Geneve, Switzerland. SAP000 NonLinear v.0.0. (00). Computers and Structures, Inc. Berkeley, CA. Uang, C.M., Bertero, V.V. (988) Use of nergy as a Design Criterion in arthquake-resistant Design, Report No. UCB/RC-88/8, arthquake ngineering Research Center, University of California, Berkeley. 9