On The Pushover Analysis As A Method For Evaluating The .

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Earthquake Resistant Engineering Structures V203On the pushover analysis as a method forevaluating the seismic response of RC buildingsP. P. Diotallevi & L. LandiDepartment of Civil Engineering DISTART, Bologna University, ItalyAbstractThe purpose of this work was to compare the non-linear pushover and dynamicmethods of analysis. Pushover analyses of a RC building were performedconsidering different load distributions and incremental dynamic analyses werecarried out considering a large number of earthquake motions. Then severalsimplified non-linear procedures based on the pushover analysis were applied inorder to assess their capability in the prediction of the seismic demand.Keywords: non-linear dynamic analysis, pushover analysis, RC buildings,Eurocode 8, simplified non-linear methods.1IntroductionIn the practical design applications the evaluation of seismic response is usuallybased on linear elastic structural behaviour. However this approach may be noteffective in limiting the damage levels of the buildings. To this purpose moreaccurate methods of analyses, which can predict the real behaviour under strongseismic actions, are required. The non-linear dynamic analysis is the mostrigorous method, but it is still too complex for design use. The non-linear staticpushover analysis seems to be a more rational method for estimating the lateralstrength and the distribution of inelastic deformations. In the last years severalsimplified non-linear procedures [1] were developed in order to predict theseismic demand by using the results of pushover analysis. These methods werealso implemented in recent guidelines [2, 3] based on the new performancebased engineering concepts.In the present research the pushover analysis was applied for studying theresponse of a RC building. Pushover analyses were performed consideringdifferent load distributions and incremental dynamic analyses were carried outWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)

204 Earthquake Resistant Engineering Structures Vconsidering a large number of earthquake motions. The purpose was to comparethe non-linear static and dynamic methods of analysis. Then well-knownsimplified procedures as the N2 [1] and the capacity spectrum method [4] wereused in the evaluation of seismic demand. The aim was to study the differencesbetween the various procedures and to compare them with the dynamic analyses.2Analysis method2.1 Structure under studyThe present research was carried out considering a RC frames with six storeysand three spans as structure under study (Fig. 1). The structure was designedaccording to the new Italian seismic code [5], very similar to the last version ofEurocode 8 [6].In the design a C25/30 concrete, with a cylinder strength equal to 25 MPa,and a reinforcing steel with a yield strength equal to 430 Mpa were considered.The gravity load amounted to 30 kN/m, and the live load to 12 kN/m. For allbeams a rectangular cross-section with a depth of 0.6 m and a width of 0.35 mwas assumed. For the columns square cross-sections with variable dimensionsfrom storey to storey were adopted. The dimensions of columns were determinedin order to limit the normalized axial force, to give adequate stiffness and to limitthe longitudinal reinforcement ratio to values not much larger then 0.01.The seismic design was performed through a modal analysis considering aresponse spectrum for medium soil condition and a peak ground acceleration(PGA) equal to 0.35 g. Assuming to develop the design according to the highductility class, a value of the behaviour factor q 5.85 was assumed. The designmoments in columns and the design shear forces in beams and columns wereevaluated according to capacity design criteria. In critical regions of beamsdifferent tension and compression reinforcement ratios were assumed. Thetransverse reinforcement in critical regions of beams and columns were definedaccording to detailing rules for local ductility. In most cases the diameter ofstirrups was set equal to 8 mm, and their spacing to 80 mm.3.20b 0.40b 0.353.20b 0.45b 0.353.20b 0.45b 0.353.20b 0.50b 0.403 .20b 0.50b 0.403.20b 0.55b 0.406.006.00Beamsh 0.60b 0.35Columnsb6.00Figure 1:bStructure layout.WIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V2052.2 Non-linear analysis modelNon-linear static and dynamic analyses of the designed structure were carried outusing a computer program based on a non-linear model developed by theauthors [7]. The finite element model is characterized by different sub-elementsconnected in series. They allow to determine the zones, with variable length,where the elastic limit is exceeded, and to account for the principal aspects of theresponse of RC elements, as the slip of reinforcing bars. The properties of theinelastic zones are derived from those of the control sections, located at theelement ends. The behaviour of the control sections is studied with a bilinearmoment-curvature relationship for monotonic loading, and with a degradinghysteretic model for cyclic loading. The slope of the elastic branch of themoment-curvature diagram was set equal to the secant stiffness at yield. Theanalytical model includes also a particular procedure for evaluating the effects ofchanging axial forces on the loading path of control sections and, as aconsequence, on the strength and stiffness of the structural elements.The deformation capacity of the structural elements was determined in termsof curvature considering the effect of concrete confinement due to stirrups. Theconcrete law proposed by Scott et al. [8] was adopted. The damage of thestructural elements was studied by calculating the ratio of required to availableductility of the control sections. This ratio was calculated at each instant duringthe loading history, accounting for the actual level of axial load. The ultimatecondition of the structure was defined through local and global criteria [9]. Thelocal one identifies the structural collapse with the occurrence of the first flexuralor shear collapse of an element. The global one associates the ultimate conditionwith the exceeding of a given value of the inter-storey drift (3% of the storeyhigh), usually corresponding to the formation of a collapse mechanism.2.3 Pushover analysesIn the pushover analysis a lateral force distribution representing the inertia forcesis applied statically to the structure with increasing intensity until the ultimatecondition is exceeded. The global response is represented by the base shear-topdisplacement curve, called also pushover or capacity curve.The choice of a proper load shape is a significant aspect because of itsinfluence on the structural response. There is not one only clear criteria to definethe load shape, and often one makes reference to literature, guidelines or codes.The Eurocode 8, similar to FEMA 273, requires to use at least two forcepatterns: one, termed uniform pattern, is based on forces proportional at eachstorey to the mass, the other, termed modal pattern, is based on forcesproportional at each storey to the mass multiplied by the corresponding modaldeformation. In general, assuming a load shape related to the displacementshape, the force at a level i can be expressed as:Fi miφiVbN m φj 1jjWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)(1)

206 Earthquake Resistant Engineering Structures Vwhere mi and φi are the mass and the modal deformation of the same level i, Vb isthe base shear and N is the total number of storeys. If the deformations φi are setequal to the deformations of the fundamental mode shape φi1, then the modalpatterns is obtained. In the case of flexible or irregular buildings the anticipatedeffect of the higher modes can be evaluated considering load patterns derivedfrom the combination of different modes [4, 10]. When the elastic limit isexceeded the load shape should be continuously updated during the analysisdepending on the level of inelastic deformations. However the adaptive loadpatterns involve a significant increase of the computational cost.According to the provisions of EC8 the pushover analyses in the present workwere conducted applying both the modal and the uniform load shapes.2.4 Dynamic analysesThe dynamic analyses were carried out in order to verify the results of thepushover analyses. A set of twenty natural acceleration records, selected withinthe most frequently used records, was considered. These records, listedelsewhere [11], were obtained during the Imperial Valley (1940), the Irpinia(1980), the Northridge (1994), the Kobe (1995) and other relevant earthquakes.The average response spectrum of the selected records results in good agreementwith the code spectrum in the period range of interest [11]. All the accelerationrecords were scaled to the same PGA value. The dynamic analyses were repeatedconsidering for each record eighteen increasing values of PGA ranging from0.04 g to 1.2 g. These analyses had the purpose to evaluate the structuralresponse to earthquakes with an increasing intensity.3Comparison between dynamic and pushover analysesInitially the results of the static and dynamic analyses were compared in terms ofbase shear-top displacement curve. The response of each dynamic analysis wascharacterized by a point, whose coordinates are the maximum values, during theearthquake, of base shear and top displacement. Through the incrementaldynamic analyses it was possible to build for each earthquake record a "dynamicpushover curve" [9]. On the basis of the curves obtained with all records anaverage and two limit curves, corresponding to the upper and lower envelope,were determined. Figure 2 shows the results of the pushover analyses (POA) forthe two load shapes together with the average and the limit curves obtainedthrough the dynamic analyses (NDA).From Fig. 2 it is clear that the static pushover results are strongly affected bythe load shape. Moreover it is evident that the modal pattern provided a betteragreement with the average results of the dynamic analyses. However the curveobtained with the uniform pattern is quite close to the upper envelope of thedynamic analyses. Therefore it represents forces and deformations which incertain cases may occur in the structure.In the elastic range the modal pattern caused a response very close to theaverage dynamic analyses results, while the uniform pattern determined a largerWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V207stiffness. In the inelastic range the modal pattern gave an underestimation of baseshear, while the uniform pattern produced a significant overestimation. Howeverthe values of base shear obtained with the modal pattern are closer to the averagedynamic analyses results then the values derived with the uniform pattern, and inany case higher then the lower dynamic envelope. The underestimation of theinelastic response with the modal pattern is probably due to the higher modes.The differences observed between the results obtained with the modal anduniform pattern seems to be in agreement with the observations of other authors[9]. These differences are related with the position of the resultant of lateral load.The resultant of the uniform load distribution is located in a lower position thenthe resultant of the modal load distribution. To obtain a given displacement thebase shear must be larger with the uniform pattern then with the modal pattern.Therefore the uniform load shape tends to provide a conservative estimation ofthe required strength, particularly of the required shear strength in the elements.On the other hand, the modal load shape tends to provide a conservativeprediction of the deformation demand and of the maximum lateral strength.In Fig. 2 the points corresponding to the first local yield, to the local and tothe global collapse condition are marked. The Table 1 shows the values of topdisplacement, inter-storey drift and base shear associated with the yield andcollapse conditions. With the dynamic analyses the average value of base shearassociated to the first yield resulted very close to the design value. The averagevalue of PGA that caused the first yield is equal to 0.088 g, larger then the ratioof the design PGA to the design behaviour factor, equal to 0.06 g. With most ofthe earthquake (fourteen), and also with the pushover analyses, the first yieldoccurred in the external beam of the third storey.1000local collapse conditionfirst yielding800Base shear [kN]global collapse condition600PGA 0.35g400NDA - average200Design base shearNDA - upper & lower envelopePOA - modal shapePOA - uniform shape00Figure 2:100200300Top displacement [mm]400500Static pushover and dynamic analysis results.The points associated with the collapse condition in the average curve of thedynamic analyses and in the pushover curve obtained with the modal patternWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)

208 Earthquake Resistant Engineering Structures Vresulted quite close. This shows again that with the modal pattern it is possible tohave a good prediction of the seismic response. The average value of PGA thatcaused the collapse condition is equal to 0.6 g in the case of local collapse, andto 0.65 g in the case of global collapse condition. Both values seemed to be muchlarger then the design PGA, showing a high safety level of the structure.Table 1:Top displ.[mm]Inter-storeydrift [mm]Base shear[kN]Response quantities at yielding and collapse conditions.First yieldingLocal collapse mGlobal collapse 6.9With the uniform pattern the displacement at both local and global collapsecondition resulted significantly lower then with the dynamic analyses. Moreoverthe local collapse occurred much earlier then the global collapse, and the singlestructural elements collapsed before the formation of a mechanism. This can beseen in Fig. 3, which illustrates the distribution of plastic zones obtained withboth load patterns at local collapse. In the same figure the collapsed zone ismarked. With most of the earthquake (fifteen), and also with the pushoveranalyses, the local collapse occurred in a base section of the columns.(a)Figure 3:(b)Plastic zones obtained with modal (a) and uniform load shape (b).The ratio of the base shear at the collapse condition to the base shear at thefirst yield, called αu/α1 in Eurocode 8, corresponds to an overstrength ratiorelated to the structural redundancy. In the EC8 this ratio, called αu/α1, is used inthe definition of the behaviour factor, and is set equal to 1.3 for multi-storey andmulti-bay frames. Higher values are allowed, but the values assumed cannot beWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)

Earthquake Resistant Engineering Structures V209665544StoreyStoreylarger then 1.5, and they have to be confirmed through non-linear analysis. Inthis research a value of αu/α1 1.75 was obtained with the modal pattern, and avalue of αu/α1 1.5 was obtained with the uniform pattern. Therefore in this casethe code provisions seemed to be conservative.The results of the static and dynamic analyses were compared not only interms of global response parameters, but also in terms of displacement andductility demand at the various levels. The comparison was performedconsidering the results of pushover analyses in correspondence with a topdisplacement equal to the average value obtained from the dynamic analyseswith the earthquake records scaled to the design PGA.32NDA - averageNDA - average st.dev.21POA - modal shapePOA - uniform shape1000100200300Displacement [mm]Figure 4:6040020406080Interstorey drift [mm]100Distribution of displacement and inter-storey drift.6NDA - averageNDA - average st.dev.55POA - modal shapePOA - uniform shape4Storey4Storey333221100.20.40.6 0.81( mr / ma) columnsFigure 5:1.200.20.4 0.6 0.8(mr / ma) beams11.2Distribution of damage in beams and columns.Figure 4 shows the diagrams of displacement and inter-storey drift along theheight of the building, while the Figure 5 displays the diagram of damage inbeams and columns. The displacement diagram obtained with the modal shapeWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)

210 Earthquake Resistant Engineering Structures Vand with the dynamic analyses resulted very similar. The modal shape causedvalues slightly larger then the dynamic analyses at the fourth and fifth floors.The pushover analysis with uniform load shape produced a differentdisplacement diagram, with much larger values then the other methods ofanalysis at the first three floors. The inter-storey drift diagram reflects whatobserved about the different position of the resultant of the two loaddistributions. Moving from the modal to the uniform load pattern the drop of theposition of the resultant caused a significant increment of inter-storey drift at thefirst two storeys, and a reduction at the higher storeys. On the contrary the modalshape provided values very similar to those of the dynamic analyses at the firstthree storeys. Moreover the modal shape was capable to predict with enoughaccuracy the maximum value, and the storey where this value was attained.In Figure 5 the maximum of the values calculated in beams and columns ateach storey of the ratio of required to available ductility is reported. As observedabout the inter-storey drift diagram, the uniform shape caused larger values at thefirst two storeys and lower values at the higher storeys then the other methods ofanalysis. The diagram of damage in columns obtained with the modal patternseems to be similar to that of the dynamic analyses. As a consequence ofcapacity design criteria the damage of columns was concentrated at the base,while the damage of beams was more distributed along the height.4Determination of the seismic demand through simplifiedmethods4.1 Capacity spectrumWithin the simplified procedures the seismic demand is usually evaluated byusing the response spectra or the time-history analysis with reference to a SingleDegree-of-Freedom (SDOF) system. Therefore the structure under study needsto be transformed into an equivalent SDOF system. The following descriptionrefers to the procedure used by Fajfar [1], which has the peculiarity of adopting adisplacement shape related to the load shape through the equation (1). The forcedisplacement curve of the SDOF system is obtained by means of the equations:*Dt D*Γ(2)Vb F *Γ(3)*where D and F are the force and displacement of the SDOF system, and Dt isthe top displacement of the MDOF structure. The constant Γ is the modalparticipation factor of the assumed displacement shape, and is defined as:NΓ mφi 1Ni i mφi 12i iWIT Transactions on The Built Environment, Vol 81, 2005 WIT Presswww.witpress.com, ISSN 1743-3509 (on-line)(4)

Earthquake Resistant Engineering Structures V211where mi is the storey mass, and φi is storey deformation normalized in order tohave unity value at the top. The value in the numerator is the equivalent mass m*:Nm* miφi(5)i 1The initial stiffness of the equivalent SDOF system is equal to that of the MDOFsystem since forces and displacement are transformed in the same way. The useof response spectra for the seismic analysis of the SDOF system requires anadditional step, i.e. the determination of a bilinear idealization for the capacitycurve. This is a significant as

pushover analysis seems to be a more rational method for estimating the lateral strength and the distribution of inelastic deformations. In the last years several simplified non-linear procedures [1] were developed in order to predict the seismic demand by using the results of pushover analysis. These methods were also implemented in recent guidelines [2, 3] based on the new performance-based .

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