58 elnashai

8/3/2019 58 Elnashai

1/19


2/19


3/19

Advanced ine lasti c static (push over) analysis br earthquake applications

Fig. I Capacity-Spectrum method:and inelastic

T 0.5 s T = 1.0

increased (stiffness, strength andlor ductility), or the demand is reduced. Clearly, the demand canonly be reduced as a consequence of changes (not necessarily increase) in the responsecharacteristics of the structure. Changes in stiffness lead to period changes. hence a rotation aboutthe origin in Fig. I to perhaps lower requirements, whilst changes in strength andlor ductility wouldlead to a shift to a lower composite spectrum for higher ductility (equivalent damping).The capacity spectrum approach (in its original format, or as modified by Fajfar (1999) forinelastic spectra) is appealing, because it gives a visual representation of the supply-demandequation, suggests possible remedial action if the equation is not satisfied and easily incorporatesseveral limit states, expressed as stations on the load-displacement curve of the structure. It,howeve r, requires as a matter of necessity a realistic pushover curve for the structure that isrepresentative of its true dynamic behaviour under the design event (s). Since this approach is thepreferred design and assessment method (e.g. FEMA 1998), it follows that further development,refinement and verification of pushover analysis. as an acceptable alternative to inelastic dynamicanalysis, are worthwhile research objectives.

3. Review of previous work the state-of-analysisIt is instructive to review recent work on pushover analysis, as applied only in earthquake analysisof structures (the offshore industry also makes us of pushover analysis for stability and strengthassessment of fixed oil production platforms). This is undertaken below in chronological order.followed by a more detailed account of selected papers of more pertinence to the current work.The use of inelastic static analysis in earthquake engineering dates back to the work of Gulkanand Sozen (1974) or earlier, where a single degree of freedom system is derived to represent

equivalently the multi-degree of freedom structure. The load-displacement curve of this substitute tothe real structure is evaluated by either finite element analysis or hand calculation to obtain theinitial and postyield stiffness, the yield strength and the ultimate strength. Simplified inelasticanalysis procedures for multi-degree of freedom systems have also been proposed by Saudi and

T = 1.5 s.

T = 3,0 sc

0 10 20 30 40 50 60 70 80Spectral Displacement (cm)i)ema nd spectrum from Duzce (Kocaeli , Turkey 1999) earthquake, elastic


4/19


5/19


6/19


7/19

Advanced inelastic Static (pushover) analysis for earthquake apphcation.s 57

each load step if they are not held constant. These issues are dealt with below.

4.2 orce- ersu dIsp;auethht-coao1Considering that the objective is to simulate the dynamic response of the structure. the question

arises whether the displacements or the forces from a particular mode should he kept constant.notm2 that inelastic effects cause a deviation between the two options. It is interestin2 to visualise.as per Fig. 2. the possible effect of displacement- or forcedistribution on the response (Elnashai1997). Application of a constant displacement profile (a) lead to the concealment of the first storeysoft response apparent in the constant lbrce plot (b). Consequently. all storey drift-storey shearresults are affected. The next question is, which is closer to dynamic response proper. Conceptually,dynamic analysis is inertia force-driven hence the constant force seems more appropriate, but storeylbrces in a dynamic analysis, even when one mode is dominant, do not exhibit a constant multiplier.It is. however, clear that fixing the displacement distribution ma give seriously misleading results.Difficulties arise in conducting stable inelastic analysis under force control though. It is thereforenecessary to have the ability to:

a. prescribe a set of fOrces to be used as the basis for application of the storey forces in theincremental solution.

b. control one displacement, say the roof node,c. calculate the force corresponding to the constrained applied displacement at the roof node,d. impose forces at other storey levels in compliance with the fixed ratios specified in (a) above.This proceduie resolves an important problem. that of applying forces hut controlling

displacements. and it is the procedure utilised in the software package INDYAS (Elnashai el a!.2000) used for most of the analysis presented in subsequent secticns and the package ADAFFIC(Izzuddin and Elnashai l99). By adopting this approach. local effects may be identified in mostcases, provided that the force distribution used represents the dynamic response at its maximumdemand situation. The above procedure is the main reason why the conventionally-used force

story - storey2

storey3storey 4starey5storey6

(a) (h)Fig. 2 Store drift versus storev shear in pushover analysis of 6 story asymmetric frame under constant

displacement (a) and constant force (b> contributions


8/19

58 AS. Elnashaidistributions (codebased. uniform or mult imodal) yield less dispersed results than those reported in

!it ertur I e d!r! F 04.3 Contribution of higher modesWhereas the problem has been identified long ago, few studies have investigated the effect ofhigher modes in pushover analysis. Two of the references cited in Section 3 have discussed this.employing a single structure with tixed characteristics. In a recent study, twelve RC structuresdesigned for different ground acceleration and with different levels of ductile detailing (Carvalhoand Coelho 1998. where details of the structures are given) were studied, employing 8 earthquakeground motion records of ditTeren t frequency contents (Mwafy and Elnashai 2000). A sample of theresults is shown in Fig. 3 where a polynomial fit of 97 dynamic analyses is shown and Fig. 4 where

400 600 800 1000Top Disp. (mm)Fig .3 Analys is of mutl i-stroey RC frame. Ninty seven inelas tic dynamIc analysis results using 8 earthquakerecords at different intensit ies are fitted with a polynomial for comparison with static push over res ults(Mwafy and Hnashai 2000)

Top Disp. (mm)Fig . 4 Comparison between dynam ic bes t-fit and pLishover curves for a number of fo rce distributions (samestruc ture as in Fig. 3)

z

20000

15000

10000

5000

0

RF.H030 Rz = 0.6935(97 runs)

Art-red Art-rcc2 Artrec3 Art-rec4A Kobe liii Loma Prieta (HJ Kobe IH+VJ Loma Prieta IH+VjPoly. (Ideal envelop)

7.0 200

=

200 400 600 800 1000


9/19


10/19

Cb=0.050[bCb = 0075eCb =0.100*-Cb=0.125

= 0.150fCb 0.175

= 0200a Cb = 0.225

Force, kNFig. 5 Step-wise force distribution for a ten storey RC frame with updating the forces using increments ofnew force distribution added to exisiting forces (cb is the load factor by which the resultant appliedforce is scaled)

Force, kNFig. 6 Stepwise force distribution for the same structure as in Figure 5 but calculated fr om th e total force ateach step (cb as above)

Fig. 7 Snap-shot of force distribution during dynamic analysis of the same structure as in Fig.5. Legend as inFigs. 6 and 7 (different lines for values of cb, th e load factor)

AS, E1,uiclu,iI0

V

00 50 100 150 200

I09876

0

3

0 50 100 150 200

I..cj

0

A X

0

xAX

50 100Force, kN

150 200


11/19

Advanced inelastic static (pushover) ana/vsi.c for earthquake applications 610.300

0.11)0

0,050

0.000

Fig. 8 Comparison of pushover curves for different lorce stepping methods with dynamic(using IDARC. no adaptive solution employed)analysis results

The two figures can therefore be viewed as incremental and total updating of the force distribution.Two issues are reflected in these plots, as follows:

The force distribution near collapse is completely different to that at the start of the analysis. It istherefore emphasised that the fixed force distribution ft)rmat is wholly inadequate to describe thebehaviour of this structure.The two distributions are very different. and would yield different results. especially on the locallevel. Whereas the total approach is probably more accurate, it is doubtful that generalpurposeanalysis packages are capable of accommodating such severe and abrupt changes in the appliedactions.

It is not obvious at this stage which of the two is more realistic, but it is clear that the distributionin Fig. 5 is superior to a fixed format and is likely to enable a stable solution to progress up tostructural collapse. To assess the accuracy. as opposed to the stability, of the two extreme cases, thelbrce distribution during dynamic analysis using IDARC. shown in Fig. 7 is examined (Lefort2000). Also, the pushover curves without using the internal adaptive facility of IDARC hut ratherby including in the input data series of force distributions representative of the incremental-Fig. 5and the total-Fig. 6 cases. respectively, are studied. These are shown in Fig. 8. Comparison of thethree force distributions confirm that. qLlalitativelv, the total force stepping option gives a betterrepresentation of the dynamic force distribution, with the exception of the top levels at largeamplitudes, where a whipping response is observed in the dynamic case. This isa function of thecharacteristics of the input motion (El Centro), a full account of which is beyond the scope of thispaper.Several important observations emanate from Fig. 8. Firstly, up to a drift of about 2% the total

force stepping (as per Fig. 6) gives results very close to the dynamic pushover curve (obtained fromsuccessively scaling the earthquake input and running inelastic dynamic analysis. Scaling in thiscase is simply by acceleration, whereas scaling of the records for the analysis results shown in Fig. 3is by velocity spectrum intensity: Mwaf and Elnashai 2000). Above 2% drift, the results of themanually-adaptive IDARC analysis is unreliable, because the scaling procedure is inaccurate in this

0.200

-. 0.1,0

0

Stepping

Dynamic Envelope - Emeryvifl Dynamic Envelope - El Centro 2 RecordA Dynamic Envelope - Hoilister Record Dynamic Envelope Kobe Record

0.00 0.50 1.00 1.50Drift % 2.002.50 3.00


12/19

400 600 800 1000Top Disp. (mm)

Fig. 9 Comparison of pushover curves for eight s to rey RC frame with and without spectrum scaling anddifferent force distribution br spectrum-compatible artificial record

1ig. It) Comparison of pushover curves for eight storev RC fram e w ith an d without spectrum scaling anddiffei ent force distribution for Kohc ( Kobe University I record

s f.S. I.fnos/uii

i,ii LulL Iii IliIIILillliL5 il ol)ainiiiL iLnsLliili(ns fLr li i lt\Ll5 of fuhal stillness. lllcrem_nL(Ipoint of application of the iestiltant trans\ else lorce. The results from the El (enlio aual sR sho.\that features of the strong--motion record hereh a single large pulse (socalled 11mg. or nearsunrcL effect arc capable of defeating the static pusho\ er anal sis, lhis issue is he ond the scopeof this oi k but is urth of ftirther discussion. Ii is reasonable to s ta le herein that t he t ot al forcesti,-ppiiig opuon esults in more realistic and accurate solutions. Hoeer. it the anaRsis package isilL apalile of accoininodiiting such abrupt changes in applied forces, the incremental option is stilltar SU i 0 a li sed ft rce I )pri iac Ii.5.2 The role of spectral shapesThe expectation that one single inelastic analysis will provide design or assessment information

180001500012000900060003000

0

(a) usingelastic periodsand designspectrum Design

spectrum

0 200

Dynamic analysis (Art-red)

z1)

1800015000t2000900060003000

0a Dynamic analysis (Kube)

0 200 400 600 800 1000Top Disp. (mm)


13/19

Top Disp. (mm)Fig. II Comparison of pushover curves for ten storey RC frame with and without spectrum scaling anddifferent force distribution fo r Loma Prieta (Sylmar Hospital) record

for a structure under all earthquake input records regardless of their frequency content is ratherunrealistic. The idea of including in the incremental fbrce updating process a measure of spectralshape is therefore a natural extension worthy of consideration. This approach can be employed inprograms that do not necessarily have a specific facility for spectral scaling. Preliminary trials usingthe analysis package ADAPTIC (izzuddin and Elnashai 1992) have given promising results, asshown in Figs. 9, 10 and II. The frame referred to is one of the structures described by Carvalhoand Coelho (1998).

The plots have been obtained by deriving a fixed load distribution using a scaling procedure forthe first two modes utilising the spectral ordinates corresponding to the three elastic periods (as incurves marked a) or cracked member periods (as in curves marked h). The latter was estimatedfrom Fourier analysis of the dynamic response. In case (a). the design spectrum was used to scalethe modal contributions, whilst in case (b) the actual eai-thquake spectrum (Artificial. Kobe. LornaPrieta) was employed. The process is depicted graphically in Fig. 12. noting that the artificial record

0 0.5 I 1.5

;ldVa!lced inelastic static (pushover) analysis /r earthquake applications 6318000

za

a

12000900060003000

0

(b)

Dynamic Analysis (Loma Prieta)

0 200 400 600 800 1000

1.41.2

000,

EC8 elastic spectrum- .Kobe (KBU)

l.oma Prieta (SARITt,T2 dT3: .Ipriok

0.80.60.4

T T.

0.20 TI

Period (sec)2 2.5 3

Fig. 12 Scaling of force distrubtion using spectral ordinates for three modes


14/19

64 A. S. Elnasiueiwas compatjble with the EC$ spectrum.and dynamic inelastic analysis results. [his improvement varies between marginal (e.g. Fig. I U) an dsignificant (e.g. Fig. 9). The two curves for spectrumscaled forces a re much closer to the dynamicresponse points than either of the two others. Moreover. for the range of displacement responsecorresponding to moderate-to-heavy damage, use of inelastic periods gives better estimates of thedynamic response of the structure. It is reiterated that the inelastic periods are calculated fromdiscrete Fourier amplitude spectra of the inelastic dynamic response. The uniform distribution offorces leads to. as expected, higher resistance due to the lower point at which the resultant force isapplied. In Fig. 9, for example, there is no difference between the design and record spectra. sincethe record is spectrum-compatible. Hence the difference in results is due to difference in spectrumordinates corresponding to elastic and inelastic periods. This indicates that the simple extension topushover analysis using existing soflware packages (IDARC. DRAIN etc) by scaling forcesspectrally using estimates of cracked stiffness would improve the results in comparison withdynamic analysis.Whereas this simple idea leads to improved results, it is still open for interpretation, in terms of

the procedure of scaling the force vector. In Section 4.3. two distinct force distributions duringpushover analysis were given, as employed in IDARC analysis (but using the internal adaptiveprocedure). These were derived from the expressions:

C (2)

for force distribution in Fig. 5 (3)F1CV for force distribution in Fig. 6 (4 )Where

storey numberj : increment numberiii : number of modesii : number of stories

mode shape vectorF : mass participation factorSd spectral acceleration ordi nateF,, : base shear

increment of base sheark. / : counters

On the other hand. the results shown in Figs. 9. 11) and 11 use direct scaling of the modal forcevectors by the corresponding spectral ordinate. Other alternatives exist and this is still an area offurther investigation (Papanikolaou 2000, Antoniou 2002).


15/19

Adraiue,/ i,alasti static (pushover) analvsi.v for earthquake appIicatiot.c

0 40 80 120 160 200 240 280 320 360 400 440 480Top Displacement (mm)

65

Fig. 1 3 INDYAS results for a 6 DOE idealised inelastic regular structure. Shown comparison betweenconventional and fully adaptive analysis with and without spectrum scaling under the Loma Prieta(Corralitos, Santa Cruz) earthquake of 17 October 1989

1207

1007-a

80:6,60

20

Top Displacement (mm)Fig. 14 INDYAS results for an idealised inelastic 6 DOF structure with a soft s to rey (2nd floor). Shown arecomparisons between conventional and fully adaptive analysis under the Loma Prieta (Emeryville,

California) earthquake of 17 October 1989

5.3 Current status and further developmentThe outcome of the above discussion and assessment of results lead to the development of a fully

adaptive pushover facility in the analysis program IIDYAS (Elnashai ci a!. 2000. featuring thefollowing:

Fibre modelling of all sections for RC an d steel analysis. utilising advanced formulations forsteel plasticity and concrete inelasticity including the effect of confinement and cyclicaldegradation.

Geometric nonlinearity taken into account. in order that flexible elastic structures can heanalysed effectively.

1207

10080ci,

56 6040

20 Dynamic 44alysis

160

140

40

0 40 80 120 160 200 240 280 320 360 400 440 480

Dynamic analysis


16/19

6

5

4

3

S. Elnashai

0 120 240 360 480Top Displacement (mm)

Fig. 15 Stores drift dist ribut ion for three types of analysis (regular structure)

At pre-assigned. or at all, deformation levels, during pushover analysis. the instantaneousnonlinear and inelastic stiffness matrix and the mass matrix are used to calculate neweigensolutions.

The new eigenvectors are used to update the applied load vector according to user-definedoptions such as absolute sum, square root of sum of squares or complete quadratic combinationof modal contnbutions.

User defined options to scale the individual contributions above by a given spectrum. with theknowledge of the new eigenvalues. There is also the possibility of feeding in a time-history andINDYAS would evaluate the spectrum and use it for scaling purposes.

The analysis is kicked-off with an assumed force distribution and force control, with anautomatic transition to displacement control to cross hmit points.

In Fig. 13 an idealised yielding structure is used to test the new developments. The yielding ischaracterised on the stress-strain level and tile structure represents a six storey shear frame subjectedto the Corralitos (Santa Cruz. California) and Enleryville (Oakland. California) records from theLoma Prieta earthquake of 17 October 1989. Fig. 14 gives results for the same structure but with asoft second storev. The comparisons show the accuracy of the fully adaptive procedure. compared tofixed mode pushover analysis. Both adaptive solutions, with and without spectrum scaling. are closeto the dynamic analysis results. It is also evident that other fixed force distributions perform betterfor regular structures. The results collectively also underplay the role of spectrum scaling. This isbecause the idealised structures are all of shortto-medium periods, hence tile important effect ofhigher niode amplification (especially under the Emervville record) is not significant. Thedifferences between fixed and adaptive analysis is even more striking on tile stores drift level. Thisis indicated in Fig. 15 where the structure profile is shown at different levels of base shear.Considerable errors in deformation response would result from using a fixed triangular distributionof forces, in contrast to the adpative solution in INDYAS. Finally, it is noteworthy that whereas thenonlinear dynamic analysis failed to converge, even with the re-taking of non-convergent steps asimplemented in INDYAS. whilst the static analysis continued without problems.

The analyses required 500 re-evaluations of the inelastic-nonlinear eigenvalues and vectors. The


17/19


18/19


19/19

58 elnashai

Documents