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Copyright 1996 Elsevier Science LtdPaper No. 1146. (quote when
citing this article)Eleventh World Conference on Earthquake
EngineeringISBN: 0 08 042822 3Iwens
EVALUATION OF THE RIC STRUCTURES SEISMIC RESPONSEBY MEANS OF
NONLINEAR STATIC PUSH-OVER ANALYSES
GIUSEPPE FAELLADipartimento di Analisi e Progettazione
Strutturale, Universita di Napoli Federico II, Italy
ABSTRACTThe prediction of the seismic behaviour of tlc framed
structures bymeans of nonlinear static push-over analysesis
compared with the results provided by nonlinear dynamic analyses.
The comparison is carried out byexamining the response of frames
characterized by a different number of stories and designed
according to theEurocode 8 regulations for high ductility
structures. The push-over analyses are conducted by assuming
anonlinear distribution of the lateral forces over the frame
height, which are amplified until a target topdisplacement,
computed byusing a capacity spectrum method, is achieved. The
dynamic analyses are performedby considering both simulated and
natural earthquake records, typical of stiff soil conditions. The
analysescarried out in the paper allow to identify the range of
applicability within which the nonlinear static push-overanalyses
can give relevant information on displacements, structural damage
and collapse mechanisms of thetlc building structures under
earthquake excitations.The results presented in this paper are
based on work developed with the contribution of Prof. Andrei
M.Reinhorn of the State University of New York at Buffalo, USA.
KEYWORDSPush-over analyses, tlc frames, prediction of seismic
behaviour, composite response spectra, damage indices.
INTRODUCTIONAn accurate and realistic prediction of the seismic
behaviour of reinforced concrete structures can be achievedby
nonlinear dynamic analyses. However, despite the increasing
efficiency of the available computational tools,at this time, the
step-by-step nonlinear dynamic analysis is unfeasible in most
practical applications. In fact,due to the need for implementing
complex behavioural models, such analysis requires a
significantcomputational effort, especially for large multistory
buildings; furthermore, besides the dependence on thestructure
modeling, the results depend on the assumed input ground motion.
For these reasons, all seismic codesallow to compute the pattern of
internal actions, needed todetermine the required strength ofbeams
and columns,through elastic analyses. Nevertheless, more relevant
information on the seismic behaviour of tic structurescan be gained
by performing nonlinear static push-over analyses. In such analyses
the structure is subjected toincremental lateral loads,
characterized by a selected distribution over the height of the
structure, and theresponse is computed as a selected deformation
parameter, usually the top displacement, increases up to a
targetvalue, which should be close to the one expected under the
design earthquake.The nonlinear static push-over analyses are
mainly a performance evaluation procedure which is usually
notintended as an alternative direct method for designing the
structural members but, rather, as a tool to obtainfurther insight
into the seismic behaviour of structures (Krawinkler 1994). In
fact, the push-over analyses cangive information on the structural
strength capacities and on the deformation demands as well as can
identifydiscontinuity in the strength distribution and the regions
potentially exposed to larger damage (Miranda andBertero 1991,
Lawson et al. 1994). In order to verify the range of applicability
within which the informationprovided by such analyses can be
considered satisfactory, a comparison with the results obtained
through
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step-by-step nonlinear dynamic analyses is carried out in this
paper. The comparison is performed by varyingtwo parameters which
are believed to be more significant, such as the number of stories
and the input groundmotion. The frames have been designed according
to the European seismic code provisions (Eurocode 8) forhigh
ductility structures and the dynamic analyses have been performed
byusing several- natural and spectrumcompatible generated -
accelerograms which are representative of stiff soil conditions.
The structural responsehas been evaluated in terms of interstory
drifts and of structural damage; in particular, indices at story
leveland overall damage indices have been computed based on
deformation and energy "consumption".DESIGN OF FRAMES AND MODELING
ASSUMPTIONS
Three-bay 3-, 6- and 9-story tic symmetrical plane frames have
been analysed in order to consider differentrealistic cases. All
frames have bays 5000 mm wide and stories 3500mm high and are
subjected to gravity loadsin addition to the seismic forces; in
particular, a dead load DL of 30 Nlmm and a live load LL of 15Nlmm
havebeen considered. The design has been performed by using the
Eurocode 2 and 8 provisions (CEC 1990, CEN1994). Vertical and
earthquake loads have been applied in a combination ofDL+O.3LL+EL;
the earthquake loadEL has been computed by reducing the live loads
through a coefficient equal to 0.15 for all stories except thetop
one, where it has been fixed equal to 0.30. Therefore, each frame
has been designed for a seismic weightof 34.50 Nlmm at the roof and
of 32.25 Nlmm at the other floors; however, at each level the
relevant weight ofbeams and columns has been also considered. In
order to design the member strengths, the internal forces inbeams
and columns due to the seismic loading have been computed by a
modal response spectrum analysis.The soil profile A spectrum (stiff
soil) has been used and a value of 0.40 g has been chosen for the
design peakground acceleration. Eurocode 8 rules for high ductility
structures have been applied and a q-factor equal to 5has been used
to define the strength level of the frames. The column
cross-section depth has been spliced by50 m m every story; the
beams have been constrained to be identical at each floor and the
cross-section depthhas been spliced by 100mm every three stories.
The cross-section capacities have been computed by consideringa
characteristic cylinder strength of 20Nlmm: for concrete and a
characteristic yield strength of 440 Ntmm" forsteel. When designing
the longitudinal and the shear reinforcement of beams and columns,
the Eurocode 2 and8 provisions concerning the minimum amount of
reinforcement have been constantly taken into account Thebeams and
columns cross-section sizes are reported in Fig. 1, where the frame
fundamental periods Tf, basedon the member elastic properties, and
the periods To, computed including the contribution of the
reinforcement,are also indicated.Both the nonlinear static
push-over analyses and the step-by-step nonlinear dynamic analyses
have been carriedout by means of the ][DARC2D computer program
(version 3.2) where a detailed modeling of the tk:membersis
implemented (Kunnath and Reinhorn 1994). In particular, a member
bymember macro-modeling is adoptedfor beams and columns and a
distributed plasticity model is used to obtain the element
stiffness matrix. Thecyclic behaviour of the member cross-sections
ismodelled by a degrading moment-curvature relationship builton a
non-symmetric trilinear envelope curve. The main points of such
curve, corresponding to the cracking,the yielding and the ultimate
bending moments, have been computed by adopting constitutive
relationshipswhich account for the confinement in the concrete and
for the hardening in the steel; in such computation
thecharacteristic strengths have been used for concrete and
steel.Regarding to the hysteretic properties to be defined when
performing the dynamic analyses, the member cyclicbehaviour has
been represented with a hysteretic four parameter model (Kunnath
and Reinhorn 1994): inparticular, parameter ex,which the stiffness
degradation at unloading depends on, has been set equal to 2
whileparameters ~d and 1 3 . , which influence the ductility-based
strength decay and the energy-controlled strength
3-story frame: T, =0.38sec To=0.35sec6-story frame: T, = 0.61sec
To=0.54sec9-story frame: T, = 0.73sec To=0.64sec
30x603Ox50 3Ox60 3OXSO
3Ox55 3Ox65 3OXS5
3OXSO 3Ox70 3Ox60
3Ox65 3Ox75 3Ox65
3Ox70 3Ox80 3Ox70
3Ox75 3Ox85 3Ox75
3Ox80 3OX90 3Ox80
30x85 3Ox95 3OxS5
30x90 30x100 3Ox90
3Ox60
3Ox60
3Ox60 3Ox40 3OXSO 3Ox40
3Ox45 3Ox55 3Ox45
3OXSO 3Ox60 3OXSO
3Ox55 3OX65 3Ox55
3OXSO 3Ox70 3Ox60
3Ox65 3Ox75 3Ox65
3Ox703Ox60 3Ox70
30x60 3Ox70
3Ox603OX40 3OXSO 3Ox40
3Ox45 3OXS5 3Ox45
3Ox50 30x60 3Ox50
3Ox70 3OxSO
3Ox70 3Ox80Ox60
30x60 30x70 3Ox80
500 500 500 500 500 500 500 500 500
Fig. 1. Geometry of the examined frames.
350350350350350350350350350
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Table 1. I np ut g ro und mo ti on s u se d in t he s tudy
.Earthquake Date Station ID Component Duration PGA[sec] [g]Friuli
6.5.76 Tolmezzo FT EW 36.44 0.313Montenegro 15.4.79 Petrovac MP NS
19.62 0.438Montenegro 15.4.79 Hercegnovj MH EW 25.00 0.230Parkfield
27.6.66 Parkfield PK N65E 43.78 0.495KemCounty 21.7.52 Taft 1F S69E
54.40 0.179
Simulated acce le rograms Al+A5 25.00 0.40deterioration, have
been set equal to 0.10 and to 0.15 respectively; a value of 0.25
has been selected at the beamend sections and at the
column-to-footing sections for the parameter y, which controls the
pinching, while avalue of 0.30 has been selected at the other
column end sections. The dynamic analyses have been
performedassuming a Rayleigh damping of 5%.In conclusion it has to
be underlined that the adopted structure modeling yields a
satisfactory simulation of theframe response if no significant
deterioration of the joint properties is expected under the cyclic
loading sincejoint behaviour is not explicitly represented by the
computer model.
INPUT GROUND MOTIONSFive simulated accelerograms and five
historical earthquakes, which can be considered representative of
stiffsoil conditions, have been used in the dynamic analyses in
order to be consistent with the adopted designspectrum. The
characteristics of the selected records are reported in Table 1,
but it has to be evidenced that inthe analyses all records have
been scaled at a peak ground acceleration of 0.4 g,which equals the
design groundacceleration. The simulated accelerograms have been
generated compatible with the elastic response spectrumdefined by
EC8 for stiff soils. The SIMQKE computer program (Gasparini 1976)
has been used to generate thesignals, by adopting an exponential
intensity envelope function, a peak ground acceleration equal to
0.4 g anda duration of 25 seconds.
STATIC LATERAL LOAD PATTERN AND COLLAPSE MECHANISMSIt is well
known that the nonlinear static push-over analyses involve applying
a settled lateral load pattern andpushing the structures under
these horizontal forces until a target deformation level is
achieved. The lateralload pattern should be representative of the
distribution of relative inertia forces which is acting on the
structureduring an earthquake. Obviously, relative values of forces
should change as yielding spreads throughout thestructural members
and, then, the assumption of a constant lateral load pattern
inevitably leads to anapproximation. Nevertheless, constant load
patterns are usually adopted and, frequently, the lateral
loaddistribution is selected by referring to the inertia forces
which activate in the elastic range of behaviour. In mostcases a
linear (inverted triangle) distribution, which is representative of
the forces associated with the firstvibration mode in low-rise
regular buildings, is used. As the number of stories increases, the
influence of highermodes of vibration is no longer negligible and
the fundamental vibration mode lies approximately between astraight
line and a parabola with a vertex at the base; therefore, a
nonlinear shape of the lateral load patternneeds to be adopted for
long period frames.In this paper, the distribution of story forces
has been defined as follows:
w.h~F. = J J Vb (1)I ~n hkL.j=l Wj jwhere Wi is the seismic
weight of the i-th floor, h i is the corresponding height from the
base, n is the numberof stories, Vb is the base shear and kis a
coefficient which can be assumed to be dependent on the period To
ofthe structure. The coefficient k can be settled equal to 1 for
structures having period lower than 0.5 sec, thusleading to the
inverted triangle distribution, and equal to 2 for T;>2.5 sec; a
linear variation between 1 and 2can be accepted to obtain a simple
transition between the two extreme values (FEMA 1991). Based on
valuesof periods To, the coefficient k has been set equal to 1.0,
1.06 and 1.1 for the 3-, 6- and 9-story frames respectively.The
obtained nondimensional base shear - top displacement curves are
plotted in Fig.2: in particular, the baseshear Vb has been
nondimensionalized with respect to the frame seismic weight W I
while the top displacementdlop isplotted as a percentage of the
frame total heightH. All curves are plotted up to a displacement
correspondingto the first achievement of the ultimate curvature in
a member cross-section. From Fig. 2 it can be seen that the
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50r----------------------------,V b [% )
40 Wt3-story frame30
6-story frame9-story frame
dtop/H [% )2.5 3.0.0 1.5 2.0
~VYrCV -rCEC y v VI I I IC C C vt-VV+VV+VCic c C yI I I IC C C
vI-v-v+VY +v-v ~E C C YI I I I. : r _ . . y . . Y
vrVCr-C_C"I""C_E.,
C y V VI I IY 0 0 vt-V_V+V_V +V_Ocoo VI I I E C C Yt-v-v+v-v
+y-v-tE C C VI I I IC C C VI-VV+VV+VYoE C C CI I I Iceo yt-v-y+y -y
+y-v;-; 9 9 9ceo y"VY+YY+VVIE E E CI I I IY V V Y
E-Elaetlo C-Cracked Y-Ylelded
~YYT"VYT"y-c.,C C C VI I I fo E E C~VY+VY+YV~C C v yI I I ICEO
0~Yv +y-v +y- v-iE 0 0 vI I I IE 0 0 Ct-v-v+ v-v +V -v ...E 0 0 0I
I I IE C 0 ct-Vv+v-V+VVe 0 C CI I I Io 0 0 Ct-YV+YV +v-v-lE E E CI
I I Icoo c"VV+ v-v +YYIE E E CI I I Iceo ot- y- V +v-v +- v- v to E
E E I I Io C V Y... Y - Y +v-y + Y - y-to c c CI I I IY Y V V- -F
ig . 2 . B as e sh ea r - to p d isp la ceme nt c urv es a nd sta
tic c olla pse m ec ha nism s f or th e e xam in ed f ram es .
ultimate base shear coefficient VJ W t for the 3-story frame is
larger than the values obtained for the 6- and9-story frames, which
are similar. Besides the different overstrengths deriving from the
design, such result isalso due to the difference in the EC8 design
spectrum values at the frame periods, which are equal to 0.20
g,0.151 g and0.134 g for the 3-,6- and 9-story frame respectively:
in fact, the variation in the base shear coefficientsis rather
proportional to the variation in the corresponding spectral
values.Figure 2 also presents the collapse mechanisms provided by
the push-over analyses for the three examinedframes. Such
mechanisms are quite similar to those obtained through the
nonlinear dynamic analyses byincreasing the PGA (not shown for
brevity): it follows that the push-over analyses can provide a good
estimateof yielding distribution and of deformation demands in
beams and columns and, therefore, they can clearlyidentify the
critical member cross-sections. In the examined cases, it is
particularly evident that EC8 provisionssuccess to enforce the
intended weak-beam strong-column dissipation mechanism: in fact,
almost all of thebeam end sections yield while most of the columns
remain elastic or crack.
EVALUA nON OF THE TOP DISPLACEMENTThe performance evaluation
derived from the nonlinear static push-over analyses should be
performed at adeformation level, usually controlled by the top
displacement, similar to the one experienced by the structureunder
the considered earthquake. A faithful evaluation of such target top
displacement can be obtained by usingthe capacity spectrum method
(Freeman 1994): such procedure requires the computation of a
spectrumrepresentative of the inelastic strength and displacement
demands and the computation of a capacity curve ofthe structure.
Ifone considers MDOF systems, the demand curve may berepresented by
anelastic roof spectrum,computed for an equivalent viscous damping.
Alternatively, for the sake of simplicity, the demand curve canbe
obtained by computing the elastic spectrum of a SDOF system and
modifying the spectral values, in orderto define an approximate
inelastic spectrum. In both cases, the demand spectrum has to be
represented in theAcceleration Displacement Response Spectrum
(ADRS) format, that is by plotting the spectral accelerationsSa
against the spectral displacements Sd with the frame fundamental
period represented by radial lines. Infact,the intersection of such
curve with the capacity curve, which can be identified with the
base shear - topdisplacement curve, gives a prediction of the top
displacement experienced by the structure under the
selectedearthquake. In this paper, the demand curve has been
computed by using both the aforementioned procedures,in order to
verify their reliability in the examined cases (rIc high ductility
frames subjected to stiff soilearthquakes).The composite highly
damped spectra can be computed by assuming that the ratio of period
1j of the j-thvibration mode to the fundamental period To is
constant for any mode in respect to the first, independently ofthe
value of To' Under such hypothesis, the floor displacements d , are
given by (Reinhorn et al. 1995):
while the base shear Vb is given by:v ,W (T,v) =
t
(2)
(3)in which 'l'ij is the mass normalized modal shape j, g j is
the modal participation factor and Vj is the damping
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60 60Sa/g [% ] 3-story Sa/g [%]50 50- To lmezzo .. . .. .. - -.
P eb 'ovac \40 \40 ...... H e rc eg no vj II!.,30 .. ., 30 ,.'" '
.. . .. .20 20 ,I'-,10 10
dtop/H [%]0 00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0 0.2
0.4F ig . 3 . C omp os ite h ig hly d amp ed s pe ctra v s c ap ac
ity o f f ram es .
9-story- Tolmezzo- - _. P etro vac. .- -- - He rc e gn o vj
dtop/H [%]0.6 0.8 1.0
ratio. In order to account for the hysteretic energy
dissipation, it is possible to compute an equivalent viscousdamping
Veq:4 ) '( l l m a x - 2)v =V+dV=V+--------eq 1 t l l m a x [ 2 +X
( l l m a x - 2)]
which derives from an energy equivalence over a typical
hysteretic loop (Reinhom et al. 1995). In Equation(4), < X and
'Y are coefficients which account for the pinching which
characterizes the behaviour of r ic members,while I l m a x is the
maximum deformation ductility. With reference to this, it has to be
evidenced that, particularlyfor highly inelastic structures, there
are many uncertainties about the existence of a stable relationship
betweenthe hysteretic energy dissipation and the equivalent viscous
damping, and especially about the values ofparameters which have to
be introduced. However, referring to Equation (4), if coefficients
< X and 'Y cover arange ofvalues typical forrlc structures, itis
easy toverify that the increment indamping av is slightly
influencedby the variation in the coefficient < X while Av is
more sensitive to the variation in y, furthermore, values of zivare
rather independent on the ductility for values of I l m a x larger
than about 5. Inthis paper, coefficients < X and'Y have been
assumed equal to 0.05 and to 0.50 respectively, i.e. equal to the
intermediate values among thoserecommended for ric structures.
Therefore, assuming I l m a x = 5 , Equation (4) gives a viscous
equivalent dampingVeq equal to0.23. Figure 3 shows, as anexample,
the evaluation of the top displacement for the 3- and the
9-storyframes subjected to the Tolmezzo, Petrovac and Hercegnovj
records.
(4)
By adopting the aforementioned second procedure, the inelastic
spectra have been determined by a simplemethod derived from the one
proposed by Vidic et al. (1994): in particular, the elastic
spectral accelerationshave been reduced by a coefficient k,assumed
to vary linearly between 1 and I ld (being I l d the ductility
factor)in the short period region, while a constant reduction
through k a = J . l d has been considered in the medium andlong
period range, within which the equal-displacement principle is
usually accepted; the displacements havebeen modified through the
ratio I l j k Q ' The transition period which separates the above
two regions has beenassumed equal to the predominant period of the
input ground motion. The evaluation of the target topdisplacement
by means of such procedure is shown in Fig. 4 for the same frames
and records as those of Fig.3.Table 2 presents the maximum top
displacements obtained by means of dynamic inelastic analysis for
the threeexamined frames subjected to the selected input ground
motions, aswell as the corresponding values computedbyusing the
modal (MS ) and the inelastic (IS) spectra; the displacements
relative to the simulated accelerogramsare the mean of the five
computed values. Table 2 shows that, on the average, the estimate
through the inelastic
60 60Sa/kag [% ] 3-story Sa/kag [ 0 / 0 ] 9-story50 50
- To lmezzo - Tolmezzo- - _ . P etro vac . . . . . . .1 --_ . P
etrov ac40 40 ~ I. .- -- . He rc e gn o vj ~ , .. . -. . He rc e gn
o vj,.Ao,,/ II30 30 c.:I,_ . . . . . . . .-'20 2010 10dtop/H [%]
dtop/H [%]0 - --:-- --- ---0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
2.0 0.2 0.4 0.6 0.8 1.0
F ig . 4 . C omp os ite i ne la sti c s pe ctra v s c ap ac ity
o f f ram es .
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Table 2. Dynamic, IS and MS predictions of the frame top
displacement (inpercent of 11).3-story frame 6-story frame 9-story
frame
Earthquake Dyn MS IS Dyn MS IS Dyn MS ISTolmezzo 0.71 0.66 0.72
0.52 0.56 0.54 0.35 0.35 0.36Petrovac 1.01 0.90 1.00 0.53 0.50 0.54
0.35 0.41 0.34
Hercegnovj 0.39 0.36 0.38 0.27 0.30 0.25 021 0.20 0.18Parkfield
0.88 0.95 0.94 0.75 0.90 0.65 0.66 0.80 0.50
Taft 0.61 0.78 0.70 0.55 0.70 0.45 0.42 0.60 0.38Simulated 0.48
0.46 0.48 0.34 0.41 0.34 0.31 0.38 028
spectra is better than the one obtained by using the modal
spectra; however, the error is nearly always lowerthan 10%, except
for some values relative to the Parkfield and the Taft
records.EVALUATION OF DYNAMIC RESPONSE VERSUS PUSH-OVER RESULTS
As it has been evidenced in the introduction, the nonlinear
static push-over analyses are mainly a tool whichcan be used to
obtain further insight into the seismic behaviour of structures.
Inorder to evaluate the ability ofsuch analyses to give information
on the structural damage, dynamic and static results in terms of
interstorydrifts and in terms ofdamage indices are compared in the
following. Based on the results of the previous section,the
response provided by the push-over analyses has been evaluated as
top displacements equal to the onespredicted by using the composite
inelastic spectra have been achieved.Interstory driftThe interstory
drift is a widely used parameter to get information on the seismic
performance of multistorybuildings: in fact, it can provide
relevant information on deformation demand and, in many cases, it
adequatelyreflects the potential structural or nonstructural
damage. Since maximum interstory drifts at each level
areexperienced at different instants during the cyclic response,
the maximum dynamic top displacement is smallerthan the sum of the
maximum dynamic interstory drifts. Therefore, if the top
displacement is used to relate theresults of the push-over analyses
to the ones of the dynamic analyses, is advisable to compute the
static interstorydrifts for a maximum roof displacement higher than
the dynamic one. Obviously, the increment in the targettop
displacement strongly depends on the characteristics of the input
ground motion: moreover, an approximateevaluation can be performed
independently of the seismic input but by increasing the target top
displacementthrough a coefficient which increases with the number
of stories.In this paper, an increment of 10%,20% and 30% has been
adopted for the 3-, 6- and 9-story frames respectivelyand the
results are reported in Fig. 5, where the maximum interstory drifts
d/h obtained by the push-over andby the dynamic analyses are
compared as the input ground motion varies. Figure 5 shows that
values of d/hprovided by the push-over analyses satisfactorily
match the dynamic results, except for the 9-story framesubjected to
the Parkfield and the Petrovac records. In the first case, the
difference is due to the wrong predictionof d r o P ' while in the
second case the maximum dynamic story drift is achieved at the top
story because of thefrequency content of the input and of the
higher mode effects: in these conditions the push-over analysis is
notable to predict the story deformations. However, based on the
above results, it can be concluded that when thepush-over analyses
are used toevaluate the behaviour at the story level, the adoption
of a target roof displacementlarger than the dynamic one is really
needed. The increment used herein in most cases leads to a
satisfactoryprediction of d / h,even if the estimate of d r o p is
not particularly adequate (seeresults obtained for the Taft
record).Damage indexThe following modified version of the Park and
Ang damage index (1985) has been used to determine damageat each
member cross-section (Kunnath and Reinhom 1994):
8m-8r l 3 e EhD= +- (5)8u -8T u,where e m is the maximum
rotation attained during the load history, e u is the ultimate
rotation capacity of thesection, 8r is the recoverable rotation at
unloading, J j e is the strength degrading parameter, My is the
yield momentof the cross-section, Eh is the dissipated hysteretic
energy. Damage computed through the dynamic analyses isrelative to
the entire duration ofearthquakes while the corresponding damage
provided by the push-over analysesis representative of damage at
the achievement of the target top displacement; in the last case,
the energy Ehhas been computed as the area beneath the base shear -
top displacement curve. The comparison between staticand dynamic
results has been carried out both at story level and at overall
frame level: in order to compute the
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1.50 1.50d/h [% ) d/h [%)1.25 ~ 1.25. . . . . . . . . . . .1.00
1.00
0.75
- D ynam ic P u sh -o v er
~ """"'" . Taft........ ...............:::: .c:;.. t: : . : : :
: : . : : : : . . . . . . . . : : a
Simulated
. . . . . . . . . . . . . . . . . . . . . . . . . . . . ~= = . .
~ . ~ . . . . = = =. . .. ~. . . o : : : : : . : : : : : : . . . .
. _To lmezzo 0.75
Parkfield...1
0.50 0.50 : " Hergegnovj..3....................~n n.25
L----...JL- ---L- -'-----..J 0.25 L-----L -L- L---...J
3 6 9 3 6 9F ig .5 . Dynam ic v er su s s ta ti c i nt er sto ry
d ri ft
corresponding indices, values ofthe indexD obtained through
Equation (5) have been combined using weightingfactors based on the
dissipated hysteretic energy at the component and at the story
levels respectively.Figure 6 displays damage in beams (D J and
columns (Dc) at each story level for the examined frames
subjectedto the Tolmezzo record. It has to be evidenced that under
such ground motion the procedure based on the useof the composite
inelastic spectra provides values of the top displacements
practically equal to the dynamicones, but the results shown in Fig.
6 are representative also of those obtained by using the other
selected inputground motions. Representation of Fig.6 allows to
easily identify the location and the severity of damage aswell as
to compare the results from the push-over analyses with those from
the dynamic analyses. As it can beseen, the beams experience more
damage than the columns, thus confmning the conclusion which has
beendrawn by examining the collapse mechanisms, that is the success
ofEe8 provisions to enforce the weak-beamsstrong-columns behaviour.
The comparison between dynamic and static damage distribution,
instead, showsthat the push-over analyses well predict damage in
the beams while underestimate damage in the columns:obviously, the
excellent fitting of the beams damage is also due to the good
prediction of the target topdisplacement. In any case, results of
Fig. 6 confirm those obtained in terms of interstory drifts and,
therefore,it seems advisable that also the prediction ofdamage in
the columns through the push-over analyses isperformedconsidering
an increased target top displacement (larger than the dynamic
one).The comparison in terms of overall structural damage indices
DJis presented in Fig. 7 for all examined frames.In this case,
values of damage indices provided by the push-over analyses have
been computed for topdisplacements equal to those evaluated by the
dynamic analyses, in order to verify the capacity of such
analysesto predict the global damage, independently of errors inthe
top displacement evaluation. Values ofDfconfmnthat the push-over
analyses are an efficient tool to predict the structural damage,
notwithstanding a poorcorrelation has been found in terms of
dissipated energy. In the examined cases, the maximum error in
Dfislower than 12% (except two cases) and it has to be underlined
that such result has been obtained for frames
3-story 6 ... 6-story\5 "'-,",4
-
8/3/2019 Fa Ella 1996
8/8
0.30 Df0.250.200.150.100.05
A1-A5 TF MH
3 - s t o r y 6 - s t o r y 9 - s t o r y
PK MP FT A1-AS TF MH PK MP FT A1-A5 TF MH PK MP FTFig. 7.
Dynamic versus static overall structural damage index.
which experience different inelastic actions; in fact, Dj ranges
between 0.07 and 0.23, being the value 0.40considered as the
threshold beyond which the structure IS not repairable. Finally, it
has to be evidenced that thegood estimate obtained in the examined
cases derives also from the use of input ground motions for which
mostof damage is due to the kinematic component.
CONCLUSIONThe comparison of the response prediction obtained by
means of nonlinear static push-over analyses with resultsfrom
nonlinear dynamic time history analyses has allowed to show the
capacity of the push-over analyses toassess the seismic response of
ric building structures, especially for the initial design steps.
The analyses carriedout in the paper have evidenced that such
analyses can identify the collapse mechanisms and the critical
regions,where larger deformations have to be expected, as well as
may give relevant information on interstory driftsand structural
damage of ric framed structures. It has been evidenced that the
main issue is the prediction ofthe target top displacement up to
which the response provided by the push-over analysis has to be
evaluated;furthermore, it has been also shown that the interstory
drifts and the columns damage have to be computed ata top
displacement larger than the maximum one expected under the
considered earthquake. Based on the resultsshown in the paper, a
prediction of the target displacement by using a capacity spectrum
method seems to bea promising procedure, but further analyses need
in order to have an efficient tool as the input ground
motioncharacteristics vary. Finally, the suitability of the
push-over analysis for structures subjected to input groundmotions
typical of soft soil conditions still need an assessment.
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