Gonc¸alo Carvalho - fenix.tecnico.ulisboa.pt

SEISMIC ANALYSIS OF REINFORCED CONCRETE BUILDINGSSTUDY OF ALTERNATIVES FOR NONLINEAR

MODELING AND ANALYSIS

Goncalo Carvalho∗

Department of Civil Engineering, Instituto Superior Tecnico,Technical University of Lisbon, Portugal

Abstract: Large material deformations generated by medium-high intensity earthquakeslead designed building structures to a nonlinear response. However, due to the seismic ac-tion uncertainty, the lack of nonlinear behavior knowledge, the complexity of mathematicalmodels, the computational effort required to perform nonlinear analysis and global time con-sumption, only simple linear methods are effectively used in design offices. Nevertheless,intensive research studies, increasing developments of computational skills and improvingcapabilities of existing software have recently raised nonlinear methods to a much morepromising level, especially for the analysis of complex irregular structures in which linearmethods have proven to be particularly inaccurate. In this study, a set of existing nonlin-ear finite element models are used in two advanced computer programs, SAP2000 [13] andSeismoStruct [15], to model an existing concrete building structure, over which nonlinearstatic and dynamic analyses were carried out. Consequent seismic response results, theamount of labor and time requirements were compared in order to evaluate the feasibilityand reliability of each model. Program capabilities and different analyses used were alsodiscussed.

Keywords: earthquakes; building structures; reinforced concrete sections; material non-linearity; lumped plasticity; distributed plasticity; nonlinear seismic analysis.

Introduction

The induced high levels of deformation of reinforcedconcrete buildings by a design seismic action lead itsstructural elements to their actual limits of resistance.Material nonlinear behavior arises as a generic crosssection reaches its maximum internal forces, and defor-mation can follow provided that necessary ductility isguaranteed. While the action proceeds, stress and defor-mation are redistributed throughout the structure, thusleading to a global nonlinear response.

When a linear behavior is considered, the internal forcesgenerated by a design seismic action become muchstronger than those which act on the structure during itsworking life. Designing a structure to resist these forcesis consequently neither economical nor practical, whichleads designers to ensure seismic resistance by ductilitydemands, rather than forces. Resisting forces are hencereduced, for the structure to hold minimum stiffness andto withstand the remaining design actions.

Nonlinear behavior should therefore be well expectedin the response of a building structure subjected to highintensity earthquakes. It is known however that regularstructures with an even stiffness and strength distribu-tion can reasonably be modeled with simple linear meth-

ods using an estimated behavior factor to easily accountfor nonlinearity. It is also known that this procedurecannot be applied accurately to complex irregular build-ings. In these cases, analysis of the structures seismic re-sponse under nonlinear regime is highly recommendedfor accurate damage control.

A wide number of nonlinear modeling alternatives,analyses and computer programs have been developedin the last two decades, and included in several researchworks, regarding their implementation and experimen-tation. Nevertheless, it is fact that true designers possessneither, in general, the required knowledge nor the timeto deal with structural nonlinearity problems nor time tospend with the complexity of the procedures involvedin their resolution. The increasing development of com-putational skills, however, has recently raised nonlinearanalysis to a much more promising level, meaning thatseismic engineering can now evolve in a faster way andexpand its applicability.

Objectives

It was intended with this work [2] to initially conducta survey of the existing nonlinear models and later touse some of these in different computer programs fora specified case study. In this paper, only the models

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SEISMIC ANALYSIS OF REINFORCED CONCRETE BUILDINGS April, 2011

used, featured in the computer programs SAP2000 andSeismoStruct, are presented. Three-dimensional modelswere built in these applications, and to evaluate the ac-curacy of the results, nonlinear static and dynamic anal-yses were performed.

Both the amount of work required to build each modelin the aforementioned software and the time consumedby each analysis and procedure prosecuted were takeninto account to conclude about the efficiency of each al-ternative. With the accomplished study, final advice wasgiven to support the future users on the choice betweenthe available possibilities of the seismic assessment ofreinforced concrete buildings.

Nonlinear models

Structural elements deformed by severe earthquakeground motions are expected to form plastic hinges atboth ends, where maximum internal forces are concen-trated. These element cross sections are thus subjectedto multiple excursions into the inelastic range and to acyclic degradation of its behavior. Flexural section be-havior is prevailed to shear mechanisms by its ductilecharacteristics, which is often modeled either with thedefinition of hysteretic rules or with a fiber discretiza-tion model.

Finite element models for the nonlinear material re-sponse of the structural elements have fallen into twocategories: concentrated plasticity and distributed plas-ticity. Both categories were used in this study.

Concentrated plasticity

Concentrated plasticity models are based on lumpingnonlinear phenomena to both end sections of the ele-ments. The most common formulation, initially pro-posed by Clough et al. [3], is the association in seriesof a nonlinear rotational spring at each end of a linearelastic element. For the inelastic behavior definition ofthe rotational springs, a plastic hinge length is generallyspecified for the section curvature integration.

This formulation is featured in SAP2000 by setting foreach section both moment and curvature at yielding andat ultimate limit, and by typing for each a plastic hingelength. Another formulation was also recently imple-mented in this program, that automatically determinesthe sectional moment-curvature behavior from a fiberdiscretization of each section, with the uniaxial behaviordefinition of each material.

In SeismoStruct, the hysteretic alternative is also fea-tured, while for the use of fiber models a slightly dif-ferent formulation proposed by Scott & Fenves [14] isimplemented, where a numerical quadrature is directlyapplied to the curve integration along the plastic hingelength.

Distributed plasticity

In a distributed plasticity model, more than two crosssections are modeled through the element to considerplasticity progression along its length. Although dis-tributed plasticity can be modeled by placing severalhinges along the element, as initially suggested byTakayanagi & Schnobrich [16], and by the SAP2000support system [12], more sophisticated formulationsare recently in use.

SeismoStruct features the fiber model approach, founde.g. in Hellesland & Scordelis [7], in which a finite num-ber of cross sections are directly used for numerical in-tegration of element curvatures, to compute its stiffnessor flexibility matrix.

Case study

The structure studied in this work consists of a five-story reinforced concrete building (see Figure 2) lo-cated in Turkey, asymmetric along the x-axis. Allfloors have the same geometry, same height (2.85 m),same element dimensions and same reinforcement. Theslabs are 0.10 and 0.12 m thick. Beam sections aremainly 0.20×0.50 m except for the 0.20×0.60 m lo-cated at the center of the building. Column sectionsrange from 0.25×0.50 m to 0.25×0.75 m and wallsfrom 0.20×1.00 m to 0.2×1.4 m. Confining stirrups arespaced at 20 cm in both beams and columns. For morestructural details, see Vuran [17].

Structural models

For this case study, six three-dimensional models weredeveloped, each composed by a different finite elementformulation:

a. elastic element coupled with two frame hinge el-ements in SAP2000, modeled with specified hys-teretic behavior (Figure 1a);

b. elastic element coupled with two frame hinge ele-ments in SAP2000, with fiber models of the crosssections (Figure 1b);

c. elastic element coupled with two nonlinear linkelements in SeismoStruct (Figure 1c);

d. plastic hinge elements in SeismoStruct (Fig-ure 1d);

e. distributed plasticity elements in SeismoStruct(Figure 1e);

f. elastic element coupled with two distributed plas-ticity elements in SeismoStruct (Figure 1f).

(a) (b) (c) (d) (e) (f)

Figure 1: Nonlinear finite element models used.

2 / Instituto Superior Tecnico, UTL, Lisbon, Portugal

Goncalo Carvalho

(a) plan view (b) lateral view

Figure 2: Existing five-story building structure (dimensions in m).

For models a., b., c., d., and e., a plastic hinge lengthdependent on a fixed ratio λ of the cross sections heightwas used. Parameter λ was considerer as being 0.25,0.50, 0.75, 1.00 and 1.25 for parametric study.

Rigid diaphragms were applied at each floor to modelconcrete slabs; mass was linearly distributed along thebeams; rotation at the base of the vertical elements wasfully restricted.

Materials and sections

Seismic nonlinearity in reinforced concrete structuresis mainly derived, as mentioned before, from materialnonlinear behavior. Therefore the correct modeling ofmaterial behavior is crucial for accurately assessing thestructural seismic response. For a clear definition of ma-terials uniaxial behavior, the Mander et al. [8] concreteconstant confinement model and the Menegotto & Pinto[9] steel model were used.

To model the elements cross sections of the present casestudy, three confining ratios kc were considered1, whichassume a medium-low confinement exploration. Neces-sary parameters were calibrated according to the avail-able information (see Figures 3 and 4).

εc0 fc0 [kPa] kc fcc [kPa] εu

(1) 0.002 16700 1.0 16700 0.0035(2) 0.002 16700 1.1 18370 0.0050(3) 0.002 16700 1.2 20040 0.0100

Figure 3: Concrete confining ratios.

Figure 4: Concrete capacity curves used.

Parameters for definition of the longitudinal reiforce-ment steel bars are listed in Figure 5. A cyclic responseto a given strain history is also presented.

fy [MPa] Es [MPa] b R(0) a1 a2 a3 a4 εsu

371.0 200.0 0.005 20.0 18.5 0.15 0.025 2.0 0.075

Figure 5: Steel capacity curves used.

1For this and other parameters of material definition refer to Carvalho [2].

Instituto Superior Tecnico, UTL, Lisbon, Portugal / 3


In order to define the hysteretic models, a previous studyof each hinge section had to be conducted. A Mat-lab application was therefore developed, in which mate-rial behavior was carefully modeled, and section prop-erties were introduced coupled with its medium axialforce, determined by linear analysis. The applicationthen computes the moment-curvature relation and ideal-izes it to a bilinear standard as supported by SAP2000and SeismoStruct. The extensive data is subsequentlyput into generated Ms.Excel sheets compatible with eachprogram for direct importation.

Rectangular beams were modeled as T-sections to ac-count for slabs flexural stiffness.

Seismic action definition

Regarding the seismic action definition, three realrecords from the PEER database [11] were consid-ered (see Table 1). The two horizontal componentswith higher peak ground acceleration were fitted to theEC8 [10] elastic response spectrum (Figure 6), using thesoftware RSPMatch2005 [6], to a 0.4 g ground peak ac-celeration. The components with lower peak ground ac-celeration was scaled so that each pair of componentshave the same peak ground acceleration proportion.

Table 1: Seismic records [11].

Reference Year Dist [km] Magnitude Site class.

Northridge-01 1994 37.19 6.69 Firm rockTabas, Iran 1978 13.94 7.35 Firm rock

Whittier Narrows-01 1987 40.61 5.99 Very firm soil

Figure 6: Spectrum.

Nonlinear analysis performed

To firstly assess the dynamic characteristics and the lin-ear response of the building structure, linear modal andresponse spectrum analysis were carried out. To vali-date each model, also linear static analyses were run tocheck equilibrium of vertical gravity loads.

Over all nonlinear models one pushover analysis in eachdirection was performed with the modal lateral load dis-tribution. With the distributed plasticity model, also theuniform load was applied to evaluate structural response

differences. To evaluate the influences of the plastichinge length on the results, pushover analyses were alsorun to all variants employed.

The N2 method prescribed by the EC8 [10], (see alsoFajfar [4, 5]) was applied to the capacity curves obtainedwith each pushover analysis, to evaluate the globalstructural response to the 0.4 g ground peak accelera-tion. Only the 0.75 value of λ was used to perform thisprocedure in the concentrated plasticity models.

Finally, nonlinear dynamic time-history analyses werecarried out for all models (again only with λ = 0.75)to the three intensities considered in this study (0.2, 0.3and 0.4 g) for the six semi-artificial accelerograms2.

It is noted here that not all models were able to oper-ate successfully, namely the two concentrated plasticitymodels in SeismoStruct (c. and d.). The problems weregradually reported to the support team who after severalattempts recognized that the model was too complex tobe computed by the program. Convergence difficultieswere also verified during the analyses on the models ofSAP2000, specially on the fiber models, which are stillunder development for implementation. In both of theconcentrated plasticity models used in SAP2000, con-vergence failure occurred at a given time step, whichrequired reduction of the seismic intensity.

Results

The most relevant results obtained with the linear andnonlinear analyses were summarized in this section.

Linear dynamic analysis

The first three modes of vibration presented in Figure 7were obtained by a linear model in SeismoStruct. Thefirst mode is basically characterized by a global transla-tion motion of the stories along the x-axis, with a slightrotation about the z-axis, thus implying torsional sensi-tivity. The second mode, on the other hand, is describedas a pure translational motion, derived from the symme-try along the y-axis, composed by four shear walls ori-ented in that direction. In the third mode, almost puretorsional motion of the structure is verified.

To evaluate the influences of the concrete and steel stiff-nesses on the vibrating periods of the structure, a 50%reduction was considered to the elastic modulus of theconcrete and both half and zero stiffness was adoptedto the steel. Results (Figure 8) show a general similaritybetween SAP2000, a model were no reinforcement stiff-ness is taken into account, and SeismoStruct when thesteel elastic modulus is reduced to zero. It is also con-firmed that influences on the period are more sensitiveto variations in the concrete elastic modulus, which ismainly expected considering the ratio between the twomaterial quantities.

2Each of the three semi-artificial records has two orthogonal components which were applied in both directions and thus all forming a setof six semi-artificial records.


Goncalo Carvalho

Mode T [s] f [Hz] Meffdx [%] Meff

dy [%]1 0.615 1.63 76.7 0.0

SAP2000

SAP2000 v12.0.0 - File:Elástico - Deformed Shape (MODAL) - Mode 1 - Period 0.66984 - Kip, in, F Units

3-12-11 16:39:17

(a) first mode, plan view


dy [%]2 0.592 1.69 0.0 78.0

SAP2000


3-12-11 16:40:27

(b) second mode, plan view


dy [%]3 0.508 1.97 5.0 0.0SAP2000


3-12-11 16:42:57

(c) third mode, plan view

Figure 7: Last story modal displacement shapes.

Figure 8: Influences of concrete and steel stiffnesses onthe modal periods.

Nonlinear static analysis

The capacity curves obtained with the distributed plas-ticity model in SeismoStruct (e.) are shown in Figure 9,for modal and uniform lateral load distributions, andfor x and y directions. The structure presents a slightlygreater resistance along the y-axis, characterized by ahardening phase, whereas along the x-axis an evidencedsoftening behavior is demonstrated. Regarding the twodifferent load distributions, it is shown that the uni-form load is distinguished by higher resisting forces andstiffness. This fact is due to the presence of strongerforces in superior levels in the modal distribution, whichincreases the values of the total shear force of eachfloor for the same base shear, leading to higher defor-mations, affecting resistance itself. Since SeismoStructdoes not consider the loss of element strength when ul-timate strains occur in materials, a (red) point is shownon the curves when ten strain warnings3 are registered

(a) x-direction (b) y-direction

Figure 9: Capacity curves obtain with SeismoStruct distributed plasticity model (e.).

3Each warning indicates an ultimate strain reaching at some element, being computed by SeismoStruct throughout the analyses. Thedefinition of a limit number of warnings was done empirically.




Figure 10: Capacity curves obtain with SeismoStruct limited distributed plasticity model (f.).

in the vertical elements, after which the curve shall notbe taken as accurately representing the actual capacitycurve.

The capacity curves obtained with limited distributedplasticity model in SeismoStruct (f.) for different val-ues of the factor λ are presented in Figure 10. It can beseen that results were considerably close, even for verysmall values of λ. However as the plastic hinge lengthis decreased, more convergence difficulties were veri-fied and more computational effort was required, thusincreasing significantly the duration of the analyses. Infact, when inelastic progress is limited by the length ofthe distributed plasticity elements, greater values of cur-vature are concentrated in the element critical sections,generating higher section forces. An increase of struc-tural stiffness and a slight decrease of strength is there-fore observed.

Regarding the capacity curves obtained with the concen-trated plasticity model in SAP2000 using the fiber mod-els of the cross sections (b.), for different values of λ,see Figure 11. Firstly, a considerable reduced maxi-mum top displacement is verified, fact that is consis-tent with the red dots presented in Figure 9, as ultimate

strain values were directly described in the program ma-terials definition. This formulation leads to great con-vergence difficulties as element forces abruptly drop tozero and stresses are constantly redistributed throughoutthe structure. The same result in stiffness is observedwhen λ is modified.

When a hysteretic rule was imposed in SAP2000 tomodel the behavior of each section (a.), different resultswere obtained with the capacity curves (see Figure 12).In both directions, a lower value of the maximum shearforce is reached, and a considerably higher ductility fac-tor is demonstrated. As seen with the other models, thegreater the plastic hinge length is defined, more defor-mation is observed as well as higher resistance is ex-pected. In fact, when the plastic hinge length is in-creased, more rotation capacity is given to the hystereticmodels and thus more deformation capacity is acquiredby the structure. Occasionally, e.g. the y-axis in thiscase, as the structural displacements grow higher, resist-ing forces on the linear elastic elements are increased,thus experiencing a greater value of Vb. Note that untilyielding is reached, structural response is kept the samedue to the rigid behavior of plastic hinges.


Figure 11: Capacity curves obtain with SAP2000 concentrated plasticity model with fiber models (b.).


Goncalo Carvalho


Figure 12: Capacity curves obtain with SAP2000 concentrated plasticity model with hysteretical models (a.).

The target displacements determined with the N2method are listed in Table 2. It is shown that withthe concentrated plasticity model with fiber hinges inSAP2000 (b.) it was not possible to obtain a target dis-placement, as the model appears not to have necessaryductility to resist to the deformation imposed by theseismic action defined. For the remaining models, thetarget displacements were very similar, even in both di-rections.

Table 2: Roof target displacements by the N2 method.

Models dxu [m] dy

u [m]

a. histeretic conc.plast. (SAP2000) 0.118 0.123b. fiber conc..plast. (SAP2000) – –e. distributed plast. (SeismoStruct) 0.126 0.126f. limited distr.plast. (SeismoStruct) 0.122 0.125

In Figure 13, the interstory drifts obtained with this pro-cedure show that models a., e. and f. conducted to con-siderably consistent results. Compared to the results ob-tained with the modal response spectrum analysis, it isseen that nonlinear models led to a greater deformation,concentrated in the first three stories in the x-directionand uniformly distributed through the stories in the y-direction. In fact, prevailing wall systems (y-direction)tend to form plastic hinges at the base of the walls, ho-mogenizing the spread of story drifting in height, due tothe greater stiffness of the walls compared to beams. Onthe contrary in a frame system (x-direction) exceedingdrift of the base floor caused by plastic excursion doesnot affect the upper floors. Torsional effects are alsocompared in Figure 14, where the top displacements ofthe corner frames P1 and P23 (Figure 2) are normalizedwith the roof center of mass displacement in each direc-tion. While in the y-direction, due to symmetry, no rota-tion is observed, in the x-axis slight displacements of thecorner frames are verified to the center of mass, which isconsiderably smaller compared to the ones experiencedin the response spectrum analysis. These results are ac-cording to what was expected as the torsional effects arereduced in the a structure when nonlinear behavior ex-cursions occur.


Figure 13: Interstory drifts obtained with the N2 methodand with the response spectrum analysis.


Figure 14: Normalized roof displacements obtainedwith the N2 method, and with the response spectrumanalysis.

Nonlinear dynamic analysis

The roof displacements obtained with the distributedplasticity model in SeismoStruct (e.), for the three valuesof the peak ground acceleration in the first combinationof the Northridge record, are shown in Figure 15. Abovethe displacement was placed a chart where each bar in-dicates two times the duration between maximums and



Figure 15: Top displacements in the x-direction obtain with the distributed plasticity model (e.) in the first combi-nation of the Northridge record, for different ground peak accelerations.

minimums, i.e., the response periods. It is seen that in-creasing the seismic intensity, maximum roof displace-ments are near-proportionally increased, as well as theresponse periods, the later being caused by the greaternonlinear excursions and consequent loss of stiffness.

When compared to the equivalent linear behavior (seeFigure 16), the difference between periods of vibrationis more significant. It is also shown that maximum dis-placements are very similar, fact that was confirmed inall remaining records and intensities.

Regarding the limited distributed plasticity model builtin SeismoStruct (f.), it is observed in Figure 17 that nomajor difference was detected, as expected by the capac-ity curves shown in Figure 10. A very small increase inperiods of vibration and amplitudes is verified betweenthe two curves.

To evaluate nonlinear excursions to which the structure

is subjected in the different seismic intensities consid-ered, the total base shear force was compared with andwithout considering inelastic behavior. Figure 18 showsthe values of base shear in the x-direction obtained withthe distributed plasticity model in SeismoStruct (e.), andthe correspondent values obtained with a linear dynamictime-history analysis. For this comparison, only the firstcombination of the Tabas record is represented. Eachpair of the shear forces are very close during the first3s, evidencing a linear elastic response of the struc-ture. From that instant, the three curves representingthe distributed plasticity models nearly follow the samecourse while linear elastic curves experience high peakskeeping the same proportion. This indicates, that eventhough the seismic intensity is reduced by half, a strongnonlinear behavior still affects the structure.

Such conclusion was very important to the analysis ofthe remaining concentrated plasticity models (a. and b.)

Figure 16: Top displacements in the x-direction obtain with the distributed plasticity model (e.) in the first combi-nation of the Northridge record, and with a linear dynamic time-history analysis.


Goncalo Carvalho

Figure 17: Top displacements in the x-direction obtain with the limited distributed plasticity model (f.) and withthe distributed plasticity model (e.) in the first combination of the Northridge record.

in SAP2000, which had great convergence difficulties,as mentioned before. By knowing this, the peak groundacceleration could be reduced and it could still be possi-ble to compare models in the inelastic range, which wasthe initial purpose of this work.

Figure 19 shows the top displacement obtained withthese two concentrated plasticity models, comparedwith the distributed plasticity model (e.) and that ob-tained with a linear dynamic time-history analysis. Itis seen that the results obtained with the concentratedplasticity model with hysteretic behavior modeling (a.)are reasonably accurate in the first 10s of the analysis,when compared to the reference model (e.), however

presenting lower vibrating periods. The flexibility ex-hibited with the fiber concentrated plasticity model inSAP2000 (b.) is considerably higher compared to theother models, which lead to higher values of deforma-tion. It was not possible to detect a fair reason for thisproblem since it is not consistent with the high stiffnesspresented in the capacity curves shown in Figure 11. Infact, however, capacity curves have shown weak ductil-ity characteristics for this model, which can actually beresponsible for the observed convergence failures.

Figure 18: Base shear force in the x-direction, obtained with the distributed plasticity model (e.) in the firstcombination of the Tabas record, and with a linear dynamic time-history analysis.



(a) ground peak acceleration of 0.4 g

(b) ground peak acceleration of 0.3 g

(c) ground peak acceleration of 0.2 g

Figure 19: Top displacements in the x-direction obtained with different models (a., b. and e.), in the first combina-tion of the Tabas record, and with a linear dynamic time-history analysis.

Conclusions

Throughout the entire work of building each model andperforming each analysis, it was noticed that both pro-grams still have several issues in terms of being able tobe used in the design activity of real structures.

When modeling a building structure, each program hasdifferent limitations: In SeismoStruct, although consid-

ering nonlinear behavior is almost automatic due to thesection fiber models, building the complex model itselfbecomes heavy. In SAP2000 however, despite the easeof defining the structure geometry by using its intuitivegraphical interface, when it comes to model nonlinearbehavior, the user has no other way but to resort to ex-ternal applications to compute the large amount of hys-teretic relations.


Goncalo Carvalho

As concentrated plasticity models in SeismoStruct (c.and d.) were not able to work in any type of analysis,its use should be cautious and is not advisable for build-ing structures of equivalent or higher complexity, at themoment. The limited distributed plasticity model (f.)is also not recommended as its results were so similarto those obtained with the distributed plasticity models,with the additional convergence difficulties and increaseof duration both nonlinear analyses performed. Theconcentrated plasticity model in SAP2000, with fibermodels (b.), has also proven not to be currently advis-able (in this version of SAP2000), as both in the capacitycurves it presented a very low ductility behavior and inthe nonlinear dynamic analysis it showed convergencefailure at early time steps, with extreme high displace-ments.

Only the results obtained with the concentrated plastic-ity model in SAP2000, with defined hysteretic models

for the plastic hinges (a.), could be compared to the dis-tributed plasticity model in SeismoStruct (e.). In spiteof being the base shear resisting force underestimated byits capacity curves, the N2 method, when compared withthe results obtained with the nonlinear dynamic analy-ses, led to considerably precised values of deformation.However, for high intensities of the seismic action, con-vergence failure occured at early time steps.

Finally, the distributed plasticity model in SeismoStruct(e.), taken in this study as the reference model, consti-tutes one of the most trusted nonlinear model and its useis widely recommended. It has been tested in a signif-icantly great amount of research studies, and some re-sults compared with experimental (Bento et al. [1]). Infact, time consumption is worth noticing in this model,and even though with the concentrated plasticity modelof SAP2000 (a.) it is reduced by half, nonlinear model-ing in the former is straightforward.

References[1] BENTO, R., PINHO, R. & BHATT, C. (2008). Nonlinear Static Procedures for the Seismic Assessment of the

3D Irregular Spear Building.

[2] CARVALHO, G. (2011). Analise Sısmica de Edifıcios de Betao Armado – Estudo de Alternativas deModelacao e Analise Nao-Linear. Master’s thesis, Istituto Superior Tecnico, Universidade Tecnica de Lis-boa, Portugal.

[3] CLOUGH, R., BENUSKA, K. & WILSON, E. (1965). Inelastic earthquake response of tall buildings. Pro-ceeding of Third World Conference on Earthquake Engineering, New Zealand 11.

[4] FAJFAR, P. (2000). A nonlinear analysis method for performance based seismic design. Earthquake Spectra16(3), 573–592.

[5] FAJFAR, P., MARUSIC, D. & PERUS, I. (2005). The Extension of the N2 Method to Asymmetric Build-ings. Proc. of the 4th European Workshop on the Seismic Behaviour of Irregular and Complex Structures,Thessaloniki, Greece.

[6] HANCOCK, J., WATSON-LAMPREY, J., ABRAHAMSON, N. A., BOMMER, J. J., MARKATIS, A., MCCOY,E. & MENDIS, R. (2006). An improved method of matching response spectra of recorded earthquake groundmotion using wavelets. Journal of Earthquake Engineering 10(1), 67–89.

[7] HELLESLAND, J. & SCORDELIS, A. (1981). Analysis of RC Bridge Columns Under Imposed Deformations.IABSE Colloquium, Delft, Netherlands.

[8] MANDER, J., PRIESTLEY, M. & PARK, R. (1988). Theoretical stress-strain model for confined concrete.Journal of Structural Engineering 114(8), 1804–1826.

[9] MENEGOTTO, M. & PINTO, P. (1973). Method of analysis for cyclically loaded r.c. plane frames includ-ing changes in geometry and non-elastic behaviour of elements under combined normal force and bending.Symposium on the Resistance and Ultimate Deformability of Structures Acted on by Well Defined RepeatedLoads, International Association for Bridge and Structural Engineering, Zurich, Switzerland , 15–22.

[10] NP EN 1998-1 (2010). Eurocodigo 8 – Projecto de estruturas para resistencia aos sismos, Parte 1: Regrasgerais, accoes sısmicas e regras para edifıcios. CEN, IPQ.

[11] PEER (2010). (Pacific Earthquake Engineering Research Center) – Strong Ground Motion Database.http://peer.berkeley.edu.

[12] SAP2000 (1995). Analysis Reference Manual. For SAP2000 R©, ETABS R© and SAFETM. CSI.

[13] SAP2000 (2008). v12.0.0 Advanced. Computers and Structures, Inc.


http://peer.berkeley.edu/peer_ground_motion_database/


[14] SCOTT, M. H. & FENVES, G. L. (2006). Plastic hinge integration methods for force-based beam-columnelements. Journal of Structural Engineering ASCE 132(2), 244–252.

[15] SeismoStruct (2010). v5.0.5. A computer program for static and dynamic nonlinear analysis of framedstructures. SeismoSoft, Ltd. Available from: www.seismosoft.com.

[16] TAKAYANAGI, T. & SCHNOBRICH, W. (1979). Non linear analysis of coupled wall systems. EarthquakeEngineering and Structural Dynamics 7(1), 1–22.

[17] VURAN, E. (2007). Comparison of Nonlinear Static and Dynamic Analysis Results for 3D Dual Structures.Master’s thesis, Universit? degli Studi di Pavia.


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