ThePushover Analysisn ItsSimplicity Rahul Lesli e, Assistant Direct or Buildings Design DRIQ Board Keral a PWD Trivandrum eismic esign One of the emerging fields in seismic design of structures is the Performance ased Design The subject is still in the realm of re se ar ch an d ac ade mi cs and is only s lowl y e mer gi ng out into the r ac ti tio ner s a re na. Sei smi c des ign is t ran sf orm ing f rom a stage wh ere li ne ar el as ti c ana ly si s f or a st ruc tu re wa s s uf fic ien t f or b ot h it s e la st ic an d duct il e d es ig n t o a s ta ge w he re a s pe ci al ly d ed ic at ed on-linear procedure is to be done w hi ch f in al ly i nf lu en ce s the seismic design as a whole. The basis for the linear approach lies in the concept of the Response Reduction f actor R. When a structure is designed for a R es po ns e R ed uc ti o f ac t or of say R= 5 it m ea ns t ha t 1 15 th o f th e s ei smi c f or ce i s t a ke n b y t he L im it S ta te c ap ac it y. F ur th er d ef l ec t io n The needfora simp lemet hod to predictthenon -lin ear behaviour of a structure underseismic loadssawlight in what is nowpopularly knownas the Pushover Analysis PA . It can helpdemonstrate howprogressiveailu rein building s reallyoccurs, and identify the modeo f final failure. is in it ductile behaviour and is t aken by the ductile capacit y of the structure. In Reinforced Concrete RC) structures, the members beams and columns) are detailed such as to make sure that the s tr uc tur e can take the full impact w it hou t c ol la ps e bey on d i ts L im it St at e ca pac it y up t o it s d uc ti le c apa cit y. I n f ac t w e n ev er ana ly se f or the duct ile pa rt , but only follow the reinforcement detailing guidel ines for the same. The drawback is that the response beyond the limit state is neither a simple extrapolation, nor a perfectly ductile b eh av io ur w it h pr e- d et er mi na bl e d ef orma ti on c ap ac it y. This is due to various reasons: the change in stiffness of members due to cr cking and yielding, P-delta eff ects, change in the final seismic force estimated, to list a few - putting vaguely. A though elastic a na ~ si s gives a good indication of elas ti c capa ci ty of st r uc tu re s a nd shows whe e yielding will first occur, it cannot account for r ed is tr i bu t io n of fo rce s d ur in g t he p ro gr es si ve y ie ld in g t ha t fo ll o ws a nd p re di ct its fa il u re m ec ha n ism s, o r d et ec t p os si bi li ty a nd loc at io n of any premat ur e failure. A non- li near st at ic an alysiscanpredict these more accurately since it considers the inelast ic behaviour of the s tr uc tu re . It can help id e nt if y c ri ti c al m em be rs l ik el y t o r ea ch c ri ti ca l states during an earthquake for whic attention should be given duri ng design and detail ing. T he n ee d f or a s i mp le m et ho d t o pr e di c t th e n on -l i ne ar b eh av io ur of astr u ct ur e u nd er s ei sm ic l oa d s sa w l ig ht i n w h, at i s n ow p op ul ar ly k no wn as t he Pu sh ov er Ana ly s is PA) . I t ca n hel p dem ons tr at e h ow pr og re s si ve f ai lu r e i n b ui ld i ng s r ea ll y o cc ur s, a nd i de nt if y th e m od e o f f in al f ai lu re . P ut ti ng si m pl y, PAis a n on -l in ea r a na ly s is p ro ce du re t o estimat e the strength capacity of a structure beyond its elastic limit meaning Limit State) up to its ultimate strength in the post- elasti c range. Inthe process,the method al so pred icts potential weak a re as in t he s tru ct ur e, by k ee pi ng t rac k of t he s eque nc e o f da mag es o f e ac h a nd ev er y m emb er in t he s tr uc tu re by us e o f wh at ar e c all ed hinges the y hol d). PushoverVersus onventional nalysis I n order to underst and PA, he best approach would be to first see t he si mil ar it ie s be twe en PA and the c onv en ti ona l s ei sm ic ana ly si s SA) , bo th Se is mic C oe ff ic ie nt and Re sp ons e Sp ec tr um me tho ds described in IS:1893, which most of the readers are familiar wit h, and then s ee ho w t hey a re di ff er ent . Bot h S A a nd PA a pp ly l at era l l oad of a p red ef ine d pat ter n on t he structure. InSA, the lateral load isdistri buted either parabo li call y in Seismic Coefficient method) or proportional to the modal combination in Response Spectrum method). In PA, the distribution is proportional to height raised to the power of k , where k can be equal to 0 uniform di st ri bu ti on) , 1 t he i nv er ted t ri angl e di st ri but io n), 2 pa rab ol ic d is tr ibut ion as in the s eis mic coeff icient method) or any value between 1 and 2, the value of k be in g ba se d on c er tai n c ri ter ia li st ed i n t he FEMA 3 56 F ed er al E me rg en cy M an ag em en t A ge n cy , U SA ) c od e. Th e d is tr ib u ti on can also be proportional to either the first mode shape, or a c om b in a ti on of modes. In both SA and PA, the maximum lateral load estimated for the s tr uc tur e i s c al cul at ed based on the fund mental t ime pe ri od of the str ucture. And the last point above is precisely where t he difference starts. Whi le i n S A the in it ia l t im e per io d i s tak en t o be a co ns tan t, i n PA th is is continuously re-calculated as the analysis progresses. The d if fe re nc es b et we e n t he p ro ce du re s a re a s f ol l ows: 118 CE CRJUNE 2 12
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ThePushoverAnalysisn ItsSimplicity
Rahul Leslie, Assistant Director Buildings Design DRIQ Board Kerala PWD Trivandrum
eismic esign
One of the emerging fields in seismic design of structures is the
Performance ased Design The subject is still in the realm of
research and academics and is only slowly emerging out into the
practitioner s arena. Seismic design is transforming from a stage
where a linear elastic analysisfor a structure was sufficient for both
its elastic and ductile design to a stage where a specially dedicated
non-linear procedure is to be done
which finally influences
the
seismic design as a whole.
The basis for the linear approach lies in the concept of the
Response Reduction factor R. When a structure is designed for a
ResponseReduction factor of say R = 5 it means that 115thof the
seismic force istaken by the Limit State capacity. Further deflection
Theneedfora simplemethodto predictthenon-linear
behaviourofa structureunderseismicloadssawlightin
whatisnowpopularlyknownasthe PushoverAnalysis PA .
It canhelpdemonstratehowprogressiveailurein buildings
reallyoccurs,andidentifythe modeof finalfailure.
is in its ductile behaviour and istaken by the ductile capacity of the
structure. In Reinforced Concrete RC) structures, the members
beams and columns) are detailed such as to make sure that the
structure can take the full impact without collapse beyond its Limit
State capacity up to its ducti le capacity. Infact we never analyse for
the ductile part, but only follow the reinforcement detailing guidelines
for the same. The drawback is that the response beyond the limit
state is neither a simple extrapolation, nor a perfectly ductile
behaviour with pre-determinable deformation capacity. This is due
to various reasons: the change in stiffness of members due to
cracking and yielding, P-delta effects, change in the final seismic
force estimated, to list a few - putting vaguely. Although elastic
ana~sis gives a good indication of elastic capacity of structures and
shows where yielding will first occur, it cannot account for
redistribution of forces during the progressive yielding that follows
and predict its failure mechanisms, or detect possibility and location
of any premature failure. A non-linear static analysiscanpredict these
more accurately since it considers the inelast ic behaviour of the
structure. It can help identify critical members likely to reach critica
states during an earthquake for which attention should be give
during design and detailing.
The need for asimple method to predict the non-linear behaviou
of astructure under seismic loads saw light in wh,at is now popularl
known as the Pushover Analysis PA). It can help demonstrate how
progressive failure in buildings really occurs, and identify the mod
of final failure. Putting simply, PAis a non-linear analysis procedur
to estimate the strength capacity of a structure beyond its elasti
limit meaning Limit State) up to its ultimate strength in the post
elastic range. Inthe process, the method also predicts potential wea
areas in the structure, by keeping track of the sequence of damage
of eachand every member in the structure by useof what arecalle
hinges they hold).
Pushover Versus onventional nalysis
In order to understand PA, the best approach would be to first se
the similarities between PA and the conventional seismic analys
SA), both Seismic Coefficient and Response Spectrum method
described in IS:1893, which most of the readers are familiar with
and then see how they are different.
Both SAand PAapply lateral load of a predefined pattern on th
structure. InSA, the lateral load isdistributed either parabolicall
in Seismic Coefficient method) or proportional to the mod
combination in Response Spectrum method). In PA, the
distribution is proportional to height raised to the power of k
where k can be equal to 0 uniform distribution), 1 the inverted
triangle distribution), 2 parabolic distribution as in the seism
coeff icient method) or any value between 1 and 2, the value o
k being based on certain criteria listed inthe FEMA 356 Federa
Emergency Management Agency, USA) code. The distribution
can also be proportional to either the first mode shape, or
combination of modes.
In both SA and PA, the maximum lateral load estimated for th
structure iscalculated based on the fundamental time period o
the structure.
And the last point above is precisely where the difference starts
While inSAthe initial time period istaken to be a constant, in PAthi
is continuously re-calculated as the analysis progresses. Th
differences between the procedures are as follows:
118
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SA uses an elastic model, while PAuses a non-linear model. In
the latter this is incorporated in the form of non-linear hinges
inserted into an otherwise linear elastic model which one
generates using a common analysis-design software package,
but of course, using one that has facilities for PA.
The Hinges
inges are points on a structure where one expects cracking and
eldingto occur inrelatively higher intensity so that they show high
exural (or shear) displacement, as it approaches its ultimate
rength. These are locations where one expects to see cross diagonal
acks in an actual building structure after a seismic mayhem and
y would be at either end of beams and columns, the cross being
a smalldistance from the joint - that is where one is expected to
ert the hinges inthe corresponding computer model. Hinges are
various types - namely, flexural hinges, shear hinges and axial
nges. The first two are inserted into the ends of beams and columns.
cethe presenceof masonry
infill hassignificantinfluence
onthe
ismic behaviour of the structure, modelling them using equivalent
agonal struts is common inPA,unlike inthe conventional analysis,
re its inclusionisa rarity.Theaxialhingesare insertedat either
d of the diagonal struts thus modelled, to simulate cracking of
llduring analysis.
Basicallya hinge represents localised force-displacement relation
a member through its elastic and inelastic phases under seismic
ads. For example, a flexural hinge represents the moment-rotation
lation of a beam of which a typical one is as represented in Fig.1.
represents the linearrange from unloaded state (A)to its effective
eld (B), followed by an inelastic but linear response of reduced
stiffness
fromBtoC.CDshowsa suddenreductioninload
sistance, followed by a reduced resistance from Dto E,and finally
total lossof resistance from Eto F.Hingesare inserted ina framed