nonlinear static methods vs. experimental shaking table test results
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Nonlinear Static Methods vs.Experimental Shaking Table Test ResultsDonatello Cardone aa Department of Structures, Geotechnics and Applied Geology ,University of Basilicata , Potenza, ItalyPublished online: 29 Nov 2007.
To cite this article: Donatello Cardone (2007) Nonlinear Static Methods vs. ExperimentalShaking Table Test Results, Journal of Earthquake Engineering, 11:6, 847-875, DOI:10.1080/13632460601173938
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Journal of Earthquake Engineering, 11:847–875, 2007Copyright © A.S. Elnashai & N.N. AmbraseysISSN: 1363-2469 print / 1559-808X onlineDOI: 10.1080/13632460601173938
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UEQE1363-24691559-808XJournal of Earthquake Engineering, Vol. 0, No. 0, May 2007: pp. 0–0Journal of Earthquake Engineering
Nonlinear Static Methods vs. Experimental Shaking Table Test Results
Nonlinear Static Methods vs. Experimental ResultsD. Cardone DONATELLO CARDONE
Department of Structures, Geotechnics and Applied Geology, University of Basilicata, Potenza, Italy
Three different Nonlinear Static Methods (NSM’s), based on pushover analysis, are applied to a 3-story,2-bay, RC frame. They are (i) the Capacity Spectrum Method (CSM), described in ATC-40, (ii) theDisplacement Coefficient Method (DCM), presented in FEMA-273 and further developed in FEMA356, and (iii) the N2 Method, implemented in the Eurocode 8. Pushover analyses are conducted withDRAIN-3DX by using four different lateral force distributions, according to the acceleration profileassumed along the height of the structure: uniform, triangular, modal-proportional, and multimodalfully adaptive. In the numerical model, RC members are modeled as fiber elements.
The numerical predictions of each method are compared to the experimental results of theshaking table tests carried out on two similar 1:3.3-scale structural models, with and withoutinfilled masonry panels, respectively. The comparison is made in terms of maximum story displace-ments, interstory drifts, and shear forces. All the NSM’s are found to predict with adequate accuracythe maximum seismic response of the structure, provided that the associated parameters are prop-erly estimated. The lateral load pattern, instead, is found to little affect the accuracy of the resultsfor the three-story model considered, even if collapse occurs with a soft story mechanism.
Keywords Pushover Analysis; Capacity Spectrum Method; Displacement Coefficient Method;Inelastic Demand Spectra; Shaking Table Tests
1. Introduction
In recent years, a new generation of simplified nonlinear methods for the design and theseismic assessment of buildings has been developed. All the methods combine the push-over analysis of a multi-degree-of-freedom (MDOF) model with the response spectrumanalysis of an equivalent single-degree-of-freedom (SDOF) system, to provide an estima-tion of the global displacement response of structures that exhibit nonlinear behaviorunder strong earthquakes. The main Nonlinear Static Methods (NSM’s) are: (i) the so-called Capacity Spectrum Method (CSM), originally proposed by Freeman [1978] andthen adopted by ATC-40 [1996]; (ii) the Displacement Coefficient Method (DCM),presented in FEMA-273 [1997] and then further developed in FEMA 356 [2000]; and(iii) the N2 Method [Fajfar, 2000], which has been recently implemented in the Eurocode8 [CEN, 2001].
The common feature of NSM’s is the use of a pushover analysis (POA) to character-ize the nonlinear behavior of a structure. The applicability of NSM’s is mainly limited bythe implicit assumptions in POA [Krawinkler and Seneviratna, 1998]. Thus, recentattempts to improve NSM’s basically consist in improvements of POA, to account for the
Received 15 May 2006; accepted 14 December 2006.Address correspondence to Dr. Donatello Cardone, DiSGG – University of Basilicata, Macchia Romana
Campus, 85100 Potenza, Italy; E-mail: [email protected]
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848 D. Cardone
contributions of higher modes [Chopra and Goel, 2002; Sasaki et al., 1998; Gupta andKunnath, 2000] and/or the redistribution of inertia forces due to possible story mecha-nisms through adaptive force distributions [Bracci et al., 1997; Satyarno et al., 1998;Elnashai, 2001; Requena and Ayala, 2000]. A multimodal fully adaptive pushover proce-dure has been proposed by Antoniou et al. [2002] and Antoniou and Pinho [2004a]. In thisprocedure, the lateral force distribution is continuously updated during the process,according to the modal shapes and participation factors derived by eigenvalue analysiscarried out at each analysis step. More recently, the same authors proposed an improvedversion of their adaptive pushover procedure, based on the use of a displacement ratherthan force loading vector [Antoniou and Pinho, 2004b; Pinho et al., 2006].
The most important difference among the Nonlinear Static Methods consists in thedefinition of the Demand Curve. Appropriately reduced and standard 5%-damped (for R/C structures) elastic response spectra are considered by CSM and by DCM, respectively,while the latest version of the N2 method makes use of inelastic spectra.
Several numerical studies [Albanesi et al., 2002; Zamfirescu and Fajfar, 2001; Bentoet al., 2004; Kim and D’Amore, 1999; Faella and Kilar, 1999; Lawson et al. 1994; Mwafyand Elnashai, 2000] proved that NSP’s lead to good estimates of seismic demands, when thestructural response is actually governed by the fundamental mode and the inelastic action isdistributed throughout the height of the structures (i.e., for symmetric low-rise and medium-rise buildings). The global quantities (i.e., top displacement and base shear) are generallypredicted with more accuracy than the local ones (e.g., plastic rotations at member ends).
Comparisons with experimental results have also been carried out, to assess thedegree of reliability of NSM’s. However, they were limited to single structural elements(e.g., R/C columns) [Lin et al., 2004] or to a few pseudodynamic tests on full-scale R/Cframe [Falcao and Bento, 2002].
In this article the results of an extensive program of shaking table tests on two similar1:3.3-scale R/C frames, with and without infilled masonry panels, are considered [Dolceet al., 2005] and compared to the predictions of the three above-mentioned NSM’s. Fivetests of increasing intensity (PGA from 0.08 g to about 0.6 g) were carried out on themodel without infills and six tests (with PGA from 0.08 g to about 0.9 g) on the modelwith infills. The comparison between numerical predictions and experimental results ismade in terms of maximum story displacements, interstory drifts, and story shears.
2. Overview of Nonlinear Static Methods
Nonlinear Static Methods (NSM’s) are simplified procedures in which the problem ofevaluating the maximum response of a building under strong earthquakes is converted intothat of estimating the maximum displacement of an equivalent SDOF system whichapproximate the MDOF behavior of the real structure. The term “Nonlinear” is used toindicate that various structural elements (or components) are described through a nonlin-ear mathematical model. The term “Static” is used to point out that the characteristiccapacity curve of the real structure is obtained through POA [Lawson et al., 1994; Fajfar,1996], in which a suitable distribution of lateral forces or displacements are statically (i.e.slowly) applied to the structure and the displacement of a specific point in the structure ismonitored and related to the force variation. Various lateral force patterns have been pro-posed and adopted in POA, ranging from the simple uniform and inverted triangular distri-butions to the more sophisticated modal or multi-modal distributions [ATC, 1996; FEMA,1997; FEMA, 2000; Chopra and Goel, 2002]. Fully adaptive force distributions thatattempt to follow more closely the time-variant distribution of inertia forces have beenlately proposed [Gupta and Kunnath, 2000; Elnashai, 2001; Antoniou et al., 2002].
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Nonlinear Static Methods vs. Experimental Results 849
In the NSM’s, the Demand Curve is described by either elastic or inelastic responsespectra, representative of the expected ground motions. The seismic performance of thestructure (typically expressed in terms of maximum top displacement and maximum baseshear) is evaluated by comparing, in a proper format, the Demand Curve with a suitableschematization of the Capacity Curve.
Three of the main NSM’s developed in the past have been applied in this study,namely: (i) the Capacity Spectrum Method (CSM) [ATC, 1996]; (ii) the DisplacementCoefficient Method (DCM) [FEMA, 2000]; and (iii) the N2 Method [Fajfar, 2000]. In thefollowing subparagraphs, the basic features of such procedures are briefly reviewed. Foreach method, the criteria followed in the selection of the optimal parameters are discussed.
2.1. Capacity Spectrum Method (CSM)
The CSM provides an estimate of the Performance Point (PP) of a structure, as defined bythe maximum roof displacement and maximum base shear experienced by the structureduring a given earthquake. It uses an iterative procedure, that considers a sequence ofequivalent linear SDOF systems. The capacity of the structure is directly compared to theseismic demand in the so-called Acceleration-Displacement Response Spectrum (ADRS)format, in which spectral acceleration vs. spectral displacements are plotted, with periodsrepresented by radial lines passing through the origin of the axes [ATC, 1996].
The capacity curve is converted to spectral coordinates by dividing the top displace-ment by the first modal participation factor at the top of the structure (i.e., ) and thebase shear by the effective modal mass of the fundamental vibration mode (i.e., ). Both
and are a function of the floor masses (mj) and of the first modal shape (F1). Thelatter can be obtained from modal analysis of an elastic MDOF model of the structure.
A bilinear representation of the Capacity Curve (CC) is then required, to easily esti-mate the effective damping of the equivalent SDOF system. The slope of the elasticbranch of the bilinear representation of CC is taken equal to the initial stiffness (Ki) of thestructure. The post-elastic branch must satisfy two conditions: (i) passing through theactual curve at the PP and (ii) having a slope such as the area below and above the curveare approximately the same. An iterative procedure is then needed, as the bilinear idealiza-tion of CC depends on the displacement demand.
The effective damping of the system (bef) is viewed as a combination of the viscousdamping inherent in the structure (bo, typically assumed equal to 5% for R/C structures)and the equivalent hysteretic damping (beq):
The equivalent hysteretic damping is calculated as [Chopra, 1995]:
where WD is the energy dissipated by the equivalent SDOF system in the cycle of maxi-mum amplitude and Ws is the associated maximum strain energy. The k-factor in Eq. (1)plays a crucial role in the CSM. It is a measure of how much the actual hysteretic behaviorof the building differs from the theoretical elasto-plastic behavior. The k-factor dependson the quality of the seismic resisting system and the duration of the ground motion. Forsimplicity, ATC-40 distinguishes three categories of structures, types A, B, C, for which
G1 1⋅φ N
M1*
G1 M1*
b b k bef o eq= + ⋅ . (1)
bpeq
D
s
W
W=
⋅ ⋅4(2)
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the k-factor assumes the values 1, 2/3, and 1/3, respectively, this latter corresponding topoor hysteretic behaviors, with strong pinching effects.
The seismic demand is represented by highly damped elastic response spectra(referred to as Demand Curves). They are obtained from the bo-damped elastic responsespectrum by means of appropriate reducing factors, which depend on bef. The intersectionbetween Capacity and Demand Curve provides an estimation of the Performance Point(PP). Obviously, an iterative procedure is needed to estimate PP, as capacity and demandare mutually related, through effective damping.
Finally, the coordinates of PP are re-converted to the maximum roof displacementand maximum base shear by multiplying them by and , respectively.
2.2. Displacement Coefficient Method (DCM)
Unlike CSM, DCM provides a direct numerical procedure to estimate the performancepoint of the structure, not requiring any conversion to the spectral format. Also in thismethod, however, the base shear vs. top displacement curve obtained from POA is ideal-ized by a bilinear relationship, with initial slope Ke and strain-hardening parameter a. Thepost-yield segment passes through the actual curve at the calculated target displacement,which requires an iterative procedure also for this method. Line segments on the idealizedforce-displacement relation are located using an iterative graphical procedure that approx-imately balances the areas above and below the two curves, like in CSM. The most impor-tant difference with respect to CSM is that the slope of the first segment of the bilinearcurve is taken as the secant stiffness Ke (instead of the initial stiffness Ki), calculated for abase shear equal to 60% of the effective yield strength of the structure [FEMA, 2000]. Aneffective elastic period of vibration ( ) is then defined and used to calcu-late the maximum displacement of the equivalent linear SDOF system (de), throughresponse spectrum analysis. The maximum top displacement expected for the real struc-ture (dt) is determined by adjusting the elastic displacement de with modification factors,according to the following equation:
where C0 relates the spectral displacement of the equivalent SDOF system to the topdisplacement of the real building, C1 relates the maximum inelastic displacement to theelastic one, C2 represents the effects of stiffness/strength degradation and pinching on themaximum displacement, and C3 accounts for P-Δ effects. Equations and numerical valuesfor these coefficients are specified in the FEMA 273/356 guidelines [FEMA 1997; FEMA,2000]. In Eq. (3), Sae represents the response spectrum acceleration for the effective elasticperiod of vibration (Te) and the viscous damping ratio of the elastic structure (typically 5%for R/C structures).
2.3. N2 Method
The N2 method can be considered a variant based on inelastic spectra [Fajfar, 1999] of theCSM. The inelastic demand spectra are derived from “smoothed” elastic response spectra,by applying a suitable reduction factor Rm, which depends on the hysteretic energy dissipa-tion capacity of the structure and can be expressed as a function of its ductility factor mand the elastic period of vibration T. Several formulations exist for the reduction factor Rm.
G f1 1⋅ N M1*
T T K Ke i i e= ⋅
dp
t e aeeC C C C d C C C C S
T= ⋅ ⋅ ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅
⋅0 1 2 3 0 1 2 3
2
24(3)
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Nonlinear Static Methods vs. Experimental Results 851
In this study, the approach of EC8 [CEN, 2001] has been followed, which makes use of abilinear relationship for Rm:
where Tc is the characteristic period of the ground motion, typically defined as thatcorresponding to the transition from the acceleration-sensitive to the velocity-sensitiveregion of the spectrum. Equations (4) and (5) are a slightly simplified form of the equa-tions proposed by Vidic et al. [1994].
The capacity curve obtained from POA is transformed into that of an equivalentSDOF system, similarly to CSM. The force-displacement relationship of the equivalentSDOF system is then idealized as elastic-perfectly plastic, according to the equal energyprinciple [Fajfar, 2000]. The elastic period of the idealized bilinear SDOF system is deter-mined as:
where represents the mass of the equivalent SDOF system, k* the associ-ated elastic stiffness, and F1 the normalized (F1top=1) first modal shape of the structure. T*
is used to derive Rm from Eqs. (4) and (5).A graphical procedure similar to CSM could be adopted to determine the PP. In this
case, a number of inelastic demand spectra, corresponding to different ductility values, arereported in the same graph with the idealized capacity curve. The yielding branch of thecapacity curve intersects the demand curve for different values of ductility. PP is deter-mined by the intersection where the ductility factor calculated from the capacity curvematches the ductility value associated with the demand curve. Actually, all the steps of theprocedure can be performed numerically without using any graph [Fajfar, 1999].
At the end of the calculations, the coordinates of PP are transformed back to maxi-mum roof displacement and maximum base shear.
3. Testing Model and Experimental Results
The accuracy of NSM’s in estimating the maximum global response of R/C framed struc-tures has been evaluated by comparing the numerical predictions of the three aforesaidmethods with the experimental results of a series of shaking table tests, carried out withinthe Brite-Euram MANSIDE project [Dolce et al., 2005].
The shaking table tests were conducted on seven similar 1:3.3-scale R/C planeframes, with and without masonry infills, under three different configurations: (i) fixed-base moment-resisting frame, (ii) base-isolated frame, and (iii) frame equipped with spe-cial braces. More details on the model configurations and passive control devices can befound in the final report on the MANSIDE project [NSS, 1999]. In this study, only the twofixed-base frames, with and without masonry infills, respectively, are considered. Theyhad the same geometrical characteristics and reinforcement detailing, with some minordifferences in the mechanical characteristics of the concrete material (e.g., compressive
RT
Tcm m= − ⋅ +( )1 1 T Tc<
Rm m= ≥T Tc
(4)
(5)
Tm*
*
*= ⋅2π
k(6)
m i* = ⋅∑mi f1
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strength of about 35 MPa, after 28 days of curing, for the model w/o infills while about 43MPa, after 90 days of curing, for the model with infills).
Figure 1 shows the general layout of the structural model under consideration. It rep-resents a 3-story, 2-bay, R/C frame prototype with usual dimensions (3.5 m interstoryheight, 5 m span length in the prototype scale).
The design of the full-scale prototype was made according to EC8 [CEN, 2001] andEC2 [CEN, 1992]. Main design parameters were: soil type-B response spectrum, 0.15 gpeak ground acceleration (PGA), “low” ductility class for detailing, 2.5 behavior factor q,C25/30 concrete, and S500 steel. The structural prototype was then scaled down by a 3.3factor, in order to fully exploit the dimensions (4 m × 4 m) and the payload capacity (about150 KN) of the seismic platform available at the Seismic Laboratory of the TechnicalUniversity of Athens, where tests were carried out.
Mass-similitude scaling required about 77.4 kN of added weight, made of steel blocksanchored to each floor slab by threaded steel bars. Total weight of the model (foundationbeam excluded) was 97 kN, without infill, and about 106 kN with infills.
All the columns of the model had constant cross section (150 mm height by 105 mmwidth) and the same steel reinforcement ((3+3) 4 mm diameter bars longitudinal rein-forcement and 4 mm diameter hoops at 50 mm spacing transverse reinforcement) over theheight of the structure. Story beams had T-shaped cross section (90+60) mm height by(50+105+50) mm width at all floors. The steel reinforcement of beams and columns wasdesigned with neither “capacity design” provisions nor special ductility detailing, accord-ing to the EC8 requirements for low ductility structures. Therefore a weak column/strongbeam collapse scheme was expected to occur.
Each experimental model was subjected to two alternate series of tests, namely (1)seismic tests and (2) characterization tests, always driving the shaking table only in thelongitudinal direction of the frame. The seismic tests were aimed at evaluating the struc-tural response under seismic motion of increasing intensity (PGA). Characterization testswere aimed at assessing the damage suffered by the structure during the previous seismictest, through the evaluation of its fundamental frequency of vibration.
FIGURE 1 General layout of the model [24]. Units in cm.
65.663
Shakingtable Shaking table
Load cells
Load cells
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Nonlinear Static Methods vs. Experimental Results 853
The input signal of the table for the seismic tests was a generated accelerogram, com-patible with the EC8 elastic response spectrum for soil type B [Dolce et al., 2005]. Its totalduration was 20 s, with 13 s. stationary signal, reached after a ramp of 2 s. The accelera-tion profile was scaled down in time by a (3.3)1/2 factor, for consistency with the scale ofthe model (see Fig. 2). PGA was progressively increased during the sequence of seismictests, up to the structural collapse or the operative limits of the table.
The input signal of the table for the characterization tests was a low intensity (0.07 g)white noise.
Twenty sensors were used to record the response of the testing models, namely: (i)four Celesco LVDT’s connected to a reference steel frame external to the table, measuringabsolute story displacements; (ii) three load-cells, measuring shear, moment, and axialforce at the base of the model; and (iii) eight Endevco accelerometers, with ± 2 g range,measuring horizontal and vertical floor accelerations.
The experimental seismic response of the fixed-base models is extensively describedin Dolce et al. [2005]. The attention is herein focused on some important response param-eters (story displacements, story shears, interstory drifts, and effective frequency of vibra-tion), which allow a direct comparison with the numerical results of NSM’s.
The top diagrams of Fig. 3 show the experimental values of maximum base shear vs.maximum top displacement (relative to the table), as recorded during the shaking tabletests on the model (a) without and (b) with infilled masonry panels. Each experimentalpoint is marked by the PGA of the corresponding test. For the tests at 0.48 g (on the modelw/o infills) and 0.9 g (on the model with infills), two different experimental points arereported, because the maximum base shear and the maximum top displacement occurredin two different instants of time. By joining together all the experimental points under con-sideration, the typical force-displacement behavior of a ductile R/C framed structure isobserved.
The bottom diagrams of Fig. 3 show the changes in the fundamental frequency ofvibration of the two models, due to seismic tests of increasing intensity (PGA). Two dif-ferent sets of data are reported. The “natural” frequencies are obtained from the transferfunctions of the signals recorded during the characterization tests. They represent the fun-damental frequency of vibration of the inelastic system vibrating within its linear elasticrange. The “effective” frequencies are obtained from the transfer functions of the signalsrecorded during the seismic tests. They represent the effective frequency of vibration ofthe inelastic system during the seismic excitations. Both natural and effective frequenciesreduce during the sequence of the tests, while structural and non structural damageprogress. The differences between natural and effective frequency are mostly related tothe amplitude of vibration [Dolce et al., 2005].
FIGURE 2 (a) Input acceleration profile used for shaking table tests, and (b) associated5%-damped response spectrum [24].
0
1
2
3
4
5
0 1 2 3
(sec)–1
–0.5
0
0.5
1
0 12(sec)
g/a
(a) (b)
acc/
PGA
Sa/P
GA
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Figure 4 shows the top displacement-time histories and base shear-time historiesrecorded during the tests at 0.19 g and 0.28 g, on the model w/o infills. The experimentalcurves are compared to those obtained from nonlinear direct-integration time-history anal-yses on the numerical model described in Sec. 4. In the numerical simulations, the tableacceleration-time histories recorded during the tests are used as ground accelerationrecords. No results are reported for the test at 0.48 g, as numerical and experimental time-histories diverge after the attainment of the peak strength of the structure. The accordancebetween experimental and numerical time-histories is excellent for the two tests underconsideration (at 0.19 g and 0.28 g). At 0.48 g, instead, only the maximum base shear (andthe corresponding top displacement) is captured with adequate precision.
The comparison between experimental and numerical time-histories shown in Fig. 4is important to substantiate the methodology followed in this study. Typically, indeed, theaccuracy of pushover-based methods is checked by directly comparing pushover againstIncremental Dynamic Analysis (IDA) results [Mwafy and Elnashai, 2000]. The examina-tion of Fig. 4 proves that the experimental results considered for comparison in this study
FIGURE 3 (Top) Maximum base shear vs. maximum roof displacement and (bottom)changes in the natural and effective frequency of vibration for the model (a) without and(b) with infilled masonry panels, while increasing the seismic intensity (PGA) of theexperimental test.
0
20
40
60
80
100
Top displacement (mm))
Nk(raehse sa
B
0
20
40
60
80
100
0
Top displacement (mm)
)Nk(raehs
esaB
0
2
4
6
8
10
0
PGA/g
(a) (b)
)zH(
ycneuqerF
0
2
4
6
8
10
PGA/g
)zH(
ycneuqer F
natural frequency
effective frequency
natural frequency
effective frequency
0.07g
0.14g
0.19g 0.28g
(0.48g)∗∗
(0.48g)∗
0.08g
0.16g
0.22g
0.34g
0.63g
(0.9g)∗∗
(0.9g)∗
0.15 0.3 0.45 0.6 0.75 0 0.15 0.3 0.45 0.6 0.75
30 60 90 120 0 30 60 90 120
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Nonlinear Static Methods vs. Experimental Results 855
are little affected by the loading history (i.e., previous dynamic tests) and can reliably beused to assess the accuracy of pushover-based methods, more realistically than IDA’s.
Figure 5 shows the experimental base shear vs. top displacement relationships rele-vant to the tests at 0.19 g and 0.28 g. The examination of Figs. 3 and 5, together with aseries of other considerations, relevant to the strain levels attained in the steel and concrete
FIGURE 4 Top displacement- and base shear-time histories recorded during the experi-mental tests at (a) 0.19 g and (b) 0.28 g. Comparison with the numerical curves obtainedfrom nonlinear direct-integration time-history analyses.
PGA = 0.19g
–40
–20
0
20
40
(sec)
Top
dis
plac
emen
t (m
m)
Top
dis
plac
emen
t (m
m)
Numerical Experimental
PGA = 0.19g
–50
–25
0
25
50
(sec)
Bas
e sh
ear
(kN
)B
ase
shea
r (k
N)
Numerical Experimental
(a)
PGA = 0.28g
–40
–20
0
20
40
(sec)
Numerical Experimental
PGA = 0.28g
–50
–25
0
25
50
(sec)
Numerical Experimental
(b)
3 151311975
3 151311975 3 151311975
3 151311975
FIGURE 5 Experimental base-shear vs. top-displacement relationships relevant to thetests at (a) 0.19 g and (b) 0.28 g on the model w/o infills.
–50
–25
0
25
50
(mm)
)Nk(
–50
–25
0
25
50
(mm)
(kN
)
(a)
–40 40200–20 –40 40200–20
(b)
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fibers based on the numerical simulations, lead to the identification of the structural per-formance ranges of the two experimental models during the tests.
The structural response of the model without infills remains basically linear elastic upto 0.19 g. Actually, concrete cracking and some limited yielding occur at 0.19 g, but, gen-erally speaking, the structure responds within an Immediate Occupancy PerformanceLevel [ATC, 1996; FEMA, 2000] up to 0.19 g. The model exhibits a strong nonlinearbehavior at 0.28 g, attaining a maximum force of about 46 KN (corresponding to about0.474 the weight of the structure), and is severely damaged. Basically, the structureresponds within a Life Safety Performance Level [ATC, 1996; FEMA, 2000] at 0.28 g.Finally, the model reaches a condition of incipient collapse (Structural Stability [ATC,1996] or Collapse Prevention [FEMA, 2000] Performance Level) at 0.48 g, due to the for-mation of a soft story mechanism at the first story.
The structural response of the model with infills is essentially linear elastic up to 0.16 g.Some damage in masonry panels starts to develop at 0.22 g. Significant damages in R/Cmembers are observed starting from 0.63 g. During this test, a maximum strength of 95 kN,corresponding to about 90% of the model weight, is attained. Finally, a collapse conditionis reached during the 0.9 g test, due to the formation of a soft story mechanism at the firstfloor.
4. Numerical Model
The structure is modeled as a 3-D assemblage of nonlinear elements connected at nodesusing DRAIN-3DX [Prakash et al., 1994]. Three different analyses are carried out on thismodel, namely: static gravity, modal, and static load-to-collapse (“pushover”) analyses.
Figure 6(a) shows the finite element mesh of the R/C frame. Figure 6(b) shows theidealized constitutive laws assumed for concrete and steel, according to the results of theacceptance tests on steel and concrete. Steel bars are supposed to have the same behaviorunder tensile and compressive stresses.
The fiber element (element type 15) of DRAIN-3DX is used to model beams and col-umns. The cross section of each structural member is divided into concrete and steelfibers, so as to capture the effects of yielding and strain-hardening of steel, cracking,crushing, and post-crushing strength of concrete. Fourteen fibers of concrete and 2 fibersof steel are used for beams, 20 fibers of concrete and 2 fibers of steel are used for columns,as shown in Fig. 4(a). Concrete cover is simulated by 15 mm-thick concrete fibers.
Each structural member is divided into a number of elements (4 for columns and 6 forbeams, as shown in Fig. 6(a)), to take into account the actual arrangement of reinforce-ment bars. At the end of each structural member, rigid end zones are defined to simulatethe actual stiffness of beam-column joints. The length of the rigid end zones is assumedequal to 1/4 the cross section height of the orthogonal member. The load cells are modeledby elastic beam-column elements (element type 17), connected to the base beam throughrigid arms. At each floor, the total mass (structural+additional) is lumped at the nodes ofthe beams. The in-plane mechanical behavior of the infill panels is described by means ofequivalent diagonal compression-only struts (see Fig. 6(c)).
Reference to the model described in Panagiotakos and Fardis [1994] is made to simu-late the monotonic nonlinear behavior of masonry panels. Panagiotakos and Fardis assumea quadrilinear force-deformation law under monotonic loading (see Fig. 6(d)), whichreproduces the elastic stiffness Gw · Aw/H, the cracking strength a · Aw · tws, and the peakstrength b · Aw · tws of the infill panel, where Aw and H denote the horizontal cross-sectionarea of the panel and its clear height, Gw and tws are the shear modulus and shear strengthof masonry as measured on square wallette specimens in diagonal compression tests, a < 1
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Nonlinear Static Methods vs. Experimental Results 857
and b > 1 are two empirical modification factors which depend on the panel aspect ratio,masonry type, and its material characteristics [Biondi et al., 2000]. The secant stiffnessfrom cracking to peak strength is evaluated with the formula proposed by Klinger andBertero [1978], which provides the axial stiffness of the equivalent strut (i.e., Ew · As/ls asa function of its cross-section area (As) and length (ls) and of the masonry Young’s modu-lus (Ew), as resulting from diagonal compression tests on wallettes. The height of thecross-section of the equivalent strut is taken equal to the panel thickness. The width isdefined on the basis of the effective contact length between column and panel, accordingto Mainstone and Weeks [1970]. In the case under consideration, the strut width is esti-mated as 1/10 the diagonal length of the panel (ls).
The softening stiffness beyond the peak strength is taken equal to 0.5% of the initialone. Finally, a residual strength of about 10% of the peak strength is assumed.
FIGURE 6 Numerical modeling; (a) Finite element mesh; (b) Assumed constitutive lawof concrete and steel; (c) modeling of infill through equivalent compression-only strut;and (d) associated idealized force-displacement behavior.
δ
F
ε
σ
ε
σ
(a)
(b)
(c) (d)
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The mechanical properties of masonry (basically tws, a, and b) are calibrated by fit-ting the numerical floor displacement time-histories to the experimental ones [Dolce et al.,2005], through the least square method. The mechanical properties of masonry as obtainedfrom the calibration process were then critically reviewed, in order to assess their compat-ibility with typical experimental values [Biondi et al., 2000]. The choice of using an accu-rate numerical model was intentional, in order to eliminate possible conditionings in theevaluation of the effectiveness of the NSM’s, due to approximations in the numericalmodel.
Table 1 summarizes the first three modes of vibration of the models, with (w) andwithout (w/o) infills, in the test direction. For each mode, the period of vibration Ti, theeffective mass ratio Mi (expressed as percentage with respect to the total mass of the struc-ture), and the modal participation factor at the top ( ) are reported. As can be seen,more than 75% of the total mass participates in the fundamental mode of both models,which supports the assumption of a simplified lateral load distributions for the pushoveranalysis [ATC, 1996; FEMA, 2000].
The base shear vs. top displacement relationships obtained from the pushover analy-ses on the model w/o and with infills are plotted in Figs. 7(a) and 7(b), respectively. Fourdifferent vertical distributions of lateral forces are considered for the model w/o infills(see Fig. 7(a)), according to the acceleration distribution assumed along the height of thestructure: (a) an inverted triangular distribution; (b) a distribution consistent with theshape of the fundamental mode of the structure; (c) a multimodal adaptive distribution thatchanges as the structure is displaced beyond yielding; and (d) a uniform acceleration dis-tribution, in which the lateral forces at each level are simply proportional to the total massat that level.
The multimodal Adaptive Pushover Analysis (APOA) is carried out according to theprocedure described in Antoniou et al. [2002], considering 3 modes of vibration and 40steps of analysis. At the end of each load step of the APOA, an eigenvalue analysis is per-formed, to determine modal shapes, participation factors, and modal mass ratios of thestructure in its stressed state. The increment of lateral forces to be applied to the structurein the next step of the APOA is calculated based on the changed modal properties of thestructure, combined using the SRSS rule.
In this study, the adaptive pushover curves have been employed within equally adap-tive NSM’s, in which the modal properties associated to each point of the MDOF adaptivepushover curve are exploited to derive the equivalent SDOF capacity curve. In Fig. 7(a),the capacity curves of the model w/o infills are compared to the experimental results, rep-resented by circular dots, whose coordinates correspond to the maximum top displace-ments and maximum base shears recorded during the seismic tests. As said before, for thetest at 0.48 g (model w/o infills) and 0.9 g (model with infills), the maximum base shear
TABLE 1 Periods of vibration (Ti), effective mass ratios (Mi), and participation factor at the top for the first three modes of vibration of the model with (w) and without (w/o) infills
Mode
Ti (sec) Mi (%) Gi Fi,top
w/o w w/o w w/o w
1 0.254 0.091 76.7 71.9 1.25 1.352 0.079 0.041 7.8 11.6 0.31 0.363 0.044 0.033 1.4 1.3 0.063 0.015
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Nonlinear Static Methods vs. Experimental Results 859
and maximum top displacement are not contemporary. As a consequence, two differentexperimental points are reported, whose coordinates represent (i) the maximum base shearand corresponding top displacement and (ii) the maximum top displacement and corre-sponding base shear, respectively. The examination of Fig. 7 points out that the numericalmodel reproduces with good accuracy the global behavior of the testing model, both in theelastic and post-elastic range. The only exception is represented by the experimental pointassociated to the maximum top displacement of the model w/o infills at 0.48 g, which issignificantly far from the numerical curves. Actually, the model w/o infills collapsedthrough a soft-story mechanism at the first story, triggered by the pull out of the reinforc-ing bars of the beam. Unfortunately, the pull out phenomenon was not considered in thenumerical model.
As it was expected, only minor differences are observed between the capacity curvesassociated to the different lateral load patterns, mostly limited to the phase of developmentof plastic hinges. This is because the structural model under consideration (a 3-story regu-lar frame) featured a 1st mode-dominated response, both in the elastic and post-elasticrange.
Similar considerations apply to the model with infills (see Fig. 7(b)), for which onlytwo lateral force distributions are considered. The numerical model describes very wellthe experimental global behavior of the structure both before and after the attainment ofthe peak strength.
Figure 7 also confirm that the tests which actually produced significant plastic defor-mations in the structure are the tests at 0.28 g and 0.48 g, for the model w/o infills, and thetests at 0.63 g and 0.9 g, for the model with infills. The NSM’s are then applied with reference
FIGURE 7 Base shear vs. top displacement relationships for the model (a) without and(b) with infills, as obtained from pushover analyses by considering (a) triangular, (b)modal, (c) adaptive, and (d) uniform distributions. Comparison with the experimentalresults.
(a)
0
10
20
30
40
50
60
70
80
90
100
(mm)
(kN
)
0
10
20
30
40
50
60
70
80
90
100
(mm)
(kN
)
(a) (b)
+ +
(c) (d)
(a) (b)
(c)
(d)
(b)
(d)
(b) (d)
0 12010080604020 0 12010080604020
(b)
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860 D. Cardone
to these tests. For the tests at 0.48 g and 0.9 g, the experimental points associated to themaximum base shear are considered.
5. Results and Comparisons
5.1. Capacity Spectrum Method (CSM)
The following assumptions have been made in applying CSM. The viscous dampinginherent in the structure (b0) has been taken equal to that exhibited by each model in theseismic test of the lowest intensity (about 0.08 g), i.e., about 2.5% for the model without(w/o) infills and about 8% for the model with (w) infills [Dolce et al., 2005]. The infillshave been considered partially effective at 0.63 g, while their contribution has beenneglected at 0.9 g, as they suffered severe damage during the test at 0.63 g and then col-lapsed at 0.9 g. The loss of effectiveness of the infills during the test at 0.63 g has beentaken into account by performing two different analyses, assuming b0=8% and b0=2.5%,respectively. b0=2.5% has been assumed at 0.9 g. The values of the k-factor have beendrawn from ATC-40 assuming structural behavior Type B (k = 0.67) and Type C (k = 0.33),for the model with and without infills, respectively.
Finally, a smoothed response spectrum, compatible with that obtained from the tableacceleration-time histories recorded during the tests, has been considered in the analysis,assuming the damping reduction factor recommended in [CEN, 2001] (i.e.,
) during the iterative procedure. The use of smoothed real response spectrawas aimed at avoiding that the results provided by the NSM’s were too much sensitiveto the local variations (peaks and troughs) of the spectra drawn from the recordedaccelerograms.
Figure 8(a) shows a graphical representation in the ADRS format of the CSM, consid-ering the modal distribution of lateral forces, for the test at 0.28 g on the model w/o infills.The capacity and demand spectra at the end of the iterative process and the PP of theequivalent SDOF system are shown in Fig. 8(a). The original pushover curve is convertedto the capacity spectrum by means of the first-mode spectral properties reported in Table 1.A bilinear representation of the capacity curve is then constructed and repeatedly updatedduring the following iterative process for the evaluation of PP. At the end of this process,the initial 2.5%-damped response spectrum is reduced to a 10.5%-damped response spectrum.The idealized bilinear capacity curve intersects the 10.5%-damped response spectrum inthe PP, which provide the maximum expected top displacement and base shear. The radialline passing for PP defines the effective period of vibration of the structure during theearthquake.
Similarly, Fig. 9(a) shows the graphical representation, in the ADRS format, of theCSM, for the test at 0.63 g on the model with infills, considering the modal distribution oflateral forces, b0=8% and k=0.33.
The numerical results provided by CSM are summarized in Table 2, for the model w/oinfills (model n. 1) and with infills (model n. 2). The numerical results obtained for theconsidered lateral load distributions are compared to the experimental outcomes. Thecomparison is made in terms of maximum top displacement (dtop), maximum base shear(Sbase), effective period of vibration (Tef), and effective damping ratio (bef). The naturalperiod of vibration of the structure (To) is provided in the “Experimental results” sectionand the values of viscous (b0) and hysteretic damping (beq) (see Eqs. (2) and (3)) are alsoreported in the “Model parameters” section. It is worth to emphasize that To and bo intendto represent the dynamic properties of the model at the beginning of each test, while Tefand bef represent equivalent dynamic characteristics exhibited by the model during the
h b= +10 5( )
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Nonlinear Static Methods vs. Experimental Results 861
FIGURE 8 Graphical representation of (a) CSM (b) DCM and (c) N2 method and evalu-ation of the Performance Point for the model without infills at 0.28 g.
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Sd (mm)
Sa
(g)
Sa
(g)
Sa
(g)
Bilinear idealization of CC
Effective period
Performance Point Capacity Curve (CC)
10.5%-damped ADRS
2.5%-damped ADRS
200 40 60 80 100 120
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
200 40 60 80 100 120
2.5%-damped elastic ADRS
Capacity curve (CC)
Bilinear idealization of CC Performance
Point
Effective period
Inelastic ADRS at μ ductility
μ
Rμ
Sd (mm)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Sd (mm)40 20 0 60 80 100 120
2.5%-damped ADRS
Capacity curve (CC)
Bilinear idealization of CC
Performance Point
Tef
C0 C1 C2 C3
ElasticResponse
Te
(a)
(b)
(c)
. . .
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862 D. Cardone
FIGURE 9 Graphical representation of (a) CSM (b) DCM and (c) N2 method andevaluation of the Performance Point for the model with infills at 0.63 g.
(a)
(b)
(c)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Sd (mm)
Sd (mm)
Sd (mm)
Sa
(g)
Sa
(g)
Sa
(g)
Bilinear idealization of CC
Performance Point
Capacity Curve (CC)
17.8%-damped ADRS
8%-damed ADRS
20 0 40 60 80 100 120 140 160 180
Effective period
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
8%-damped elastic ADRS
Bilinear idealization of CC
PerformancePoint
Effective period
Inelastic ADRS at μ ductility
μ Rμ
200 40 60 80 100 120 140 160 180
Capacity Curve (CC)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
8%-damped ADRS
Capacity Curve (CC)
Bilinear idealization of CC
PerformancePoint
Tef
Elastic response
Te
200 40 60 80 100 120 140 160 180
C0 C1 C2 C3
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Com
pari
son
betw
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expe
rim
enta
l res
ults
and
num
eric
al p
redi
ctio
ns p
rovi
ded
by C
SM f
or th
e m
odel
n. 1
(w
/o in
fill
s) a
nd f
or th
e m
odel
n.
2 (
wit
h in
fill
s)
Exp
erim
enta
l res
ults
Mod
el p
aram
eter
sN
umer
ical
res
ults
Mod
el n
.PG
A (
g)d t
op
(mm
)S
base
(KN
)T
o (s
ec)
Tef
(s
ec)b e
f (%
)L
oad
patt
ern
b 0
(%)
k-fa
ctor
b eq
(%)
b ef
(%)
d top
(m
m)
Sba
se
(KN
)T
ef
(sec
)Δd
(%
)ΔS
(%
)
10.
2835
.45
45.8
0.34
00.
4910
.1tr
iang
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2.5
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24.4
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50.
3324
.810
.526
.19
44.1
70.
41−2
6.1
−3.6
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tive
2.5
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24.5
10.4
26.0
144
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0.43
8−2
6.6
−3.1
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26.7
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4−0
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70.9
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2–
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1.8
−0.0
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same test. In the last two columns of Table 2, the percent differences between experimen-tal and numerical values of top displacement (Δd) and base shear (ΔS) are reported.
The following observations can be made from Table 2. Firstly, the numerical resultsare little sensitive to the shape of the lateral load distribution, as expected by consideringFig. 7 and related comments. Secondly, CSM apparently underestimates the maximum topdisplacements recorded during the experimental tests. Thirdly, a greater accuracy in theevaluation of the maximum base shear is observed.
Actually, for the model w/o infills (model n. 1 in Table 2), the percentage differencesin terms of top displacement range from about 25–35%, for both the tests. On the contrary,the differences in terms of maximum base shear are limited to a few percents. The under-estimation of the maximum displacement at 0.28 g is partly due to a certain overestimationof the effective stiffness of the structural model during the seismic tests. This is substanti-ated by the comparison between the experimental values of the effective period (column 6of Table 2) and the corresponding numerical estimation obtained at the end of the iterativeprocess (column 15 of Table 2). On the other hand, the effective damping exhibited by thestructural model is reproduced very well (compare columns 7 and 12 of Table 2), at leastfor the test at 0.28 g, for which experimental results are fully available. Another reason forthe underestimation of the maximum displacement can be found in the use of smoothedresponse spectra. As can be noted in Fig. 8(a), indeed, the radial line passing for the PP(whose slope corresponds to the effective period of vibration of the structure during theseismic test) intercepts the real response spectrum near to a peak, which implied highertop displacements and base shears.
A significant improvement in the precision of the numerical predictions can bereached by deriving the k-factor from the experimental outcomes. This can be done bycomparing experimental and theoretical force-displacement cycles at the same displace-ment amplitudes, as shown in Fig. 10(a). A value of the k-factor equal to 0.21 is thenfound at 0.28 g, which halves the percent differences on the maximum top displacementgiven in Table 2, passing from 25–30% to 10–15%.
For the model with infills (model n. 2 in Table 2), an excellent accuracy is reached at0.63 g, by using the modal load pattern and with the following combinations of numerical
FIGURE 10 Comparison between theoretical and experimental base-shear vs. top-dis-placement cycles for the model (a) without infills at 0.28 g and (b) with infills at 0.63 g.
(a)
–100
0
100
–35(mm)
(KN
)
Pushovercurve Experimental
cycle
Theoreticalcycle
0
(mm)
(kN
)
Pushovercurve
Experimentalcycle
Theoreticalcycle
–35 35
–100
100
0 350
(b)
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parameters: b0=8% and k=0.33 (masonry effective with significantly pinched hysteresisloops) or, alternatively, b0=2.5% and k=0.67 (R/C frame lacking of infills with stablehysteresis loops). It is worth to observe that the experimental value of the k-factor at 0.63gresults to be 0.44 (see Fig. 10(b)), i.e., practically in-between the values suggested byATC-40 for structural behavior types B and C. In this case, however, an intermediatevalue of b0 should also be assumed, leading to results similar to the already satisfactoryones obtained with the above mentioned values of the couple (b0, k).
No numerical results are reported for the test at 0.9g, as the procedure does not converge.
5.2. Displacement Coefficient Method (DCM)
Figure 8(b) shows a graphical representation (in the ADRS format) of the DCM, consider-ing the modal distribution of lateral forces, for the test at 0.28 g on the model withoutinfills. The bilinear idealization of the capacity curve (CC) at the end of the iterative pro-cess is reported. The demand curve is represented by a smoothed 2.5%-damped responsespectrum, consistent with that obtained from the table acceleration-time history recordedduring the test under consideration. The intersection between the projection of the firstsegment of the bilinear curve (whose slope corresponds to the effective elastic period Te)with the response spectrum defines the maximum elastic displacement. The expectedmaximum inelastic top displacement is calculated by applying the modification factors C0,C1, C2, and C3, as previously defined. The radial line passing for the PP provides the effec-tive period of vibration (Tef) of the model during the seismic test.
The graphical representation for the test at 0.63 g on the model with infills is reportedin Fig. 9(b). In this case, the demand curve is represented by a smoothed 8%-dampedresponse spectrum, consistent with that derived from the table acceleration time historyrecorded during the test.
The assumptions made in the application of DCM are reported in the “Model Parame-ters” section of Table 3. The modification factor C0 has been assumed equal to the firstmodal participation factor at the top of the structure. As can be deduced from Table 3, forthe model w/o infills (model n. 1 in Table 3), C0 progressively reduces while increasingthe lateral forces applied. It is equal to 1.25 under gravity loads only (triangular, modal,and uniform load pattern), then decreasing to 1.13 when a top displacement of about 25 mmis reached (adaptive load pattern). Finally, it tends to 1 when a soft story mechanismoccurs. C1 and C3 have been calculated using the equation suggested by FEMA 356, as afunction of the effective fundamental elastic period of vibration of the structure (Te). It isworth to note that C3 is always equal to 1 for the model w/o infills (capacity curve withpositive post-yield stiffness) while it assumes values greater than 1 for the model withinfills (capacity curve with negative post-yield stiffness). Finally, C2 has been varied dur-ing the analyses, according to the seismic intensity of the tests. More precisely, referencehas been made to the values recommended by FEMA 356 (1.0 < C2 < 1.5) [FEMA, 2000],which depend on the Structural Performance Level (SPL) of the building and its funda-mental elastic period. In this study, the SPL has been correlated to the state of the experi-mental model at the end of each test. Thus, the model w/o infills (model n. 1 in Table 3)has been deemed to comply with the “Safety life” SPL at 0.28 g and the “Collapse Preven-tion” SPL at 0.48 g [FEMA, 2000]. The model with infills (model n. 2 in Table. 3) hasbeen considered to respond within a “Damage Control” structural performance range[FEMA, 2000] at 0.63 g and within the “Collapse Prevention” SPL at 0.9 g. As far as thefundamental elastic period is concerned, reference has been made to the period of vibra-tion of the first mode, as obtained from modal analysis (see Table 1). The values of C2adopted in each analysis case are listed in Table 3.
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pari
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betw
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ults
and
num
eric
al p
redi
ctio
ns p
rovi
ded
by D
CM
for
the
mod
el n
. 1 (
w/o
infi
lls)
and
the
mod
el n
. 2
(with
infi
lls)
Exp
erim
enta
l res
ults
Mod
el p
aram
eter
sN
umer
ical
res
ults
Mod
el n
.PG
A
(g)
d top
(m
m)
Sba
se
(KN
)T
o (s
ec)
Tef
(s
ec)
Loa
d pa
tter
nb 0
(%
)T
e (s
ec)
Co
C1
C2
C3
d top
(m
m)
S bas
e (K
N)
Tef
(s
ec)
Δd
(%)
ΔS
(%)
10.
2835
.445
.80.
340.
49tr
iang
ular
2.5
0.31
1.25
1.04
41.
151.
029
.144
.40.
45−1
8−3
mod
al2.
50.
261.
251.
150
1.15
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.8
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Nonlinear Static Methods vs. Experimental Results 867
The numerical predictions provided by DCM are summarized in Table 3 for each dis-tribution of lateral forces considered in the POA. The comparison with the experimentalresults is made in terms of maximum top displacement (dtop), maximum base shear (Sbase)and effective period of vibration (Tef) at the performance point.
For the model w/o infills (model n. 1 in Table 3), no significant differences withrespect to the results provided by CSM are found. Top displacements are significantlyunderestimated (from about 18 to 34% at 0.28 g and from just 6 to 28% at 0.48 g), whilebase shear forces are captured with great accuracy. This can be partly ascribed to a certainoverestimation of the lateral elastic stiffness of the structure, which can be noted by com-paring the values of the effective elastic period of vibration used in the numerical analyses(column 9 of Table 3) with the corresponding experimental values recorded during thecharacterization tests (column 5 of Table 3). On the other hand, it is worth to observe thatthe effective elastic period of vibration (column 9 of Table 3) results to be very close tothe period associated with the transition from the constant acceleration segment to theconstant velocity segment of the demand curve (equal to 0.335 s in the case under consid-eration), beyond which the equal displacement rule applies (i.e., C1=1). This may suggesta re-appraisal of C2 (whose evaluation in the DCM is somewhat questionable, beinglargely based on individual judgment) towards higher values.
For the model with infills (model n. 2 in Table 3) the results appear considerablyimproved, compared to those provided by the CSM, not only because the predictions aresomewhat more accurate and conservative, but especially because the procedure con-verged at 0.9 g, providing base shears in good accordance with the experimental ones,though top displacements are appreciably lower (by about 20%), due to the more pro-nounced slope of the softening segment of the pushover curve (see Fig. 7(b)). The greateraccuracy of the method for the model with infills could be ascribed to a better accordancebetween experimental and numerical values of the effective elastic period of vibration,especially for the test at 0.63 g (compare column 5 and 9 of Table 3). In addition, the val-ues of C2 suggested by FEMA result more acceptable, due to the lower decay suffered bythe structural model during the experimental tests. For the test at 0.9 g, a fundamental roleis played by the modification factor C3 (> 1), which account for increased displacementsdue to dynamic P-Δ effects, as the softening effect occurs.
5.3. N2 Method
Figures 8(c) and 9(c) show a graphical representation (in the ADRS format) of the N2method, considering the modal distribution of lateral forces, and the evaluation of the PPfor the model without infills at 0.28 g and for the model with infills at 0.63 g, respectively.
The N2 method has been applied under the same assumptions as the two previousmethods, in particular for what concerns the elastic response spectrum and the inherentviscous damping of the experimental models. In the “Numerical results” section of Table 4,the numerical values of maximum top displacement (dtop), maximum base shear (Sbase)and effective period of vibration at the performance point (Tef), are summarised. In the“Model parameters” section of Table 4, the corresponding values of the ductility factor m,elastic period T*, and reduction factor Rm of the idealized bilinear SDOF system (see Eqs.(4)–(6)) are reported.
Looking at the test at 0.28 g on the model w/o infills (model n. 1 in Table 4), theresults provided by the N2 method appear to be more accurate than those obtained before,with percent differences in terms of maximum top displacements less than 15% (exceptfor the uniform load pattern). To this regard, it is worth to emphasize the excellent agree-ment between experimental and numerical values of the effective elastic period of vibration
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TA
BL
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Com
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Nonlinear Static Methods vs. Experimental Results 869
(compare columns 5 and 10 of Table 4). The examination of the numerical predictions rel-evant to the test at 0.48 g confirms the inadequacy of the NSM’s in capturing, with goodprecision, the displacement response of the model w/o infills near structural collapse.
For the model with infills (model n. 2 in Table 4) the correctness of the solutionstrongly depends on the inherent viscous damping considered in the calculations. By con-sidering the energy dissipation capacity of the infills (b0=8%), the numerical predictionsat 0.63 g turn out to be excellent. The contrary holds at 0.9 g, where the assumption ofb0=2.5% is fundamental to achieve suitable results.
The examination of the reduction factors (Rm) used in the N2 method to pass fromelastic to inelastic spectra, points out the different ductility demands (m) of each experi-mental model, while increasing the seismic intensity of the tests. For the model w/o infills(model n. 1), the ductility demand increases from about 1.7 to about 2.9, while PGAincreases from 0.28–0.48 g (see Table 4). For the model with infills (model n. 2), the duc-tility demand increases from about 1.5 to about 4, while PGA increases from 0.63–0.9g(see Table 4). Correspondently, the reduction factor Rm increases from about 1.6÷1.3 toabout 2.7. The aforesaid values of Rm have to be compared with the behavior factor q0=2.5assumed in the design, divided by the overstrength ratio au/a1=1.3 [CEN, 2001]. Thus, adesign value of Rm equal to about 1.9 is obtained, which is within the experimental rangepreviously defined. The associated PGA value (0.15 g), however, is much lower thanthose reached during the tests. This must be ascribed to the increase of the actual strengthof the structure, compared with that expected from design, due to the actual mechanicalproperties of concrete and steel, the real amount of reinforcement, the role of the infilledmasonry panels (disregarded in the design), etc.
5.4. Comparison Between Methods and Procedures
Figure 11 compares the maximum story displacements, story shears and interstory driftsrecorded during the test at 0.28 g on the model w/o infills to those predicted by the threeNSM’s. Each diagram refers to a different distribution of lateral forces used in the POA,namely: (a) triangular, (b) modal, (c) force-based multimodal adaptive, and (d) uniform.As can be seen, the deformed shapes of the model appear quite regular, with a slight con-centration of damage at the first story, well pointed out by the comparison between thedistribution of drifts over the height of the structure. All the NSM’s provide a suitable esti-mate of the maximum seismic response, with the exception of the first interstory drift,which is slightly underestimated. The N2 method provides the best results, regardless thelateral load pattern considered in the POA. Only a slight improvement in the accuracy ofthe results is observed by deriving the capacity curve from multimodal adaptive POA. Thelatter conclusion was already drawn in the past by other researchers [Antoniou and Pinho,2004a; Papanikolau et al., 2006], who related this “underperformance” of the force-basedadaptive pushover algorithm to the use of a single constant response spectrum and of theSRSS (or CQC) rule to combine modal forces [Antoniou and Pinho, 2004a].
Similar diagrams are reported in Fig. 12 for the test at 0.48 g, which determined acondition of collapse for the structural model w/o infills. It should be noted that the exper-imental values of Fig. 12 represent the story displacements, shear forces, and interstorydrifts simultaneous to the maximum base shear. Higher story displacements and interstorydrifts are reached after the attainment of the peak strength of the structure, which are notconsidered in this study, due to the inability of the pushover curves to follow the experi-mental behavior of the structural model up to collapse (see Fig. 7).
The accordance between numerical and experimental profiles of Fig. 12 appears to beacceptable, considering that the numerical methods do not take into account the material
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decay suffered by the structural model in the experimental test at 0.28 g. The DCMmethod provides the best results, regardless the lateral load pattern considered in the POA.
6. Summary and Conclusions
Three different nonlinear static methods (NSM’s) for the evaluation of the structuralresponse of buildings to strong earthquakes are considered in this study, namely: theCapacity Spectrum Method (CSM), the Displacement Coefficient Method (DCM), and theN2 Method. Common characteristic of all the methods is the use of Pushover Analysisfor the description of the nonlinear behavior of the structure. With the aim of evaluat-ing the level of accuracy of each method in estimating the maximum seismic responseof R/C framed structures, the experimental results of shaking table tests on two similar
FIGURE 11 Maximum story displacements, story shear forces, and interstory drifts ofthe model w/o infills at 0.28 g, as predicted by CSM, DCM, and N2 method for differentlateral load patterns, i.e., (a) triangular, (b) modal, (c) adaptive and, (d) uniform. Compar-ison with the corresponding maximum experimental values.
++
Experimental CSM N2 DCME
0
1
2
3
0
(mm)0
1
2
3
(mm)0
1
2
3
(mm)0
1
2
3
(mm)
1-0
2-1
3-2
(kN)1-0
2-1
3-2
(kN)1-0
2-1
3-2
(kN)1-0
2-1
3-2
(kN)
0
1-0
2-1
3-2
(%)1-0
2-1
3-2
(%)1-0
2-1
3-2
(%)1-0
2-1
3-2
(%)
20 40 0 20 40 0 20 40 0 20 40
000 20 6040 20 6040 20 6040 0 20 6040
1
(a) (b) (c) (d)2 0 1 2 0 1 2 0 1 2
stor
yin
ters
tory
inte
rsto
ry
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Nonlinear Static Methods vs. Experimental Results 871
1:3.3-scale R/C plane frames, with and without infilled masonry panels, have been con-sidered and compared to the numerical predictions of each method.
Modal and Pushover Analyses have been performed by means of an accurate numeri-cal model, using fiber elements. A refined mesh has been adopted for both cross sectionsand elements. Moreover, the constitutive laws of steel and concrete have been derivedfrom the results of experimental tests on the materials. This has been intentionally done, toavoid possible conditioning in the evaluation of the effectiveness of the NSM’s, due toinaccuracies of the structural modeling.
Four different distributions of lateral forces have been considered in the POA, accord-ing to the distribution of acceleration along the height, namely: (i) a “triangular” distribu-tion, in which accelerations are proportional to story heights; (ii) a “modal” distribution, inwhich the lateral forces are consistent with the shape of the fundamental mode of thestructure; (iii) a “multimodal adaptive” load pattern, in which the lateral force distribution
FIGURE 12 Maximum story displacements, story shear forces, and interstory drifts ofthe model w/o infills at 0.48 g, as predicted by CSM, DCM, and N2 method for differentlateral load patterns, i.e., (a) triangular, (b) modal, (c) adaptive and, (d) uniform. Compar-ison with the experimental values corresponding to the maximum base shear.
+ +
Experimental CSM N2 DCME
0
1
2
3
(mm)0
1
2
3
(mm)0
1
2
3
(mm)0
1
2
3
(mm)
(kN)1-0
2-1
3-2
(kN) (kN) (kN)
1-0
2-1
3-2
(%)1-0
2-1
3-2
(%)1-0
2-1
3-2
(%)1-0
2-1
3-2
(%)
1-0
2-1
3-2
0 20 6040
inte
rsto
ryst
ory
inte
rsto
ry
0 20 6040
1-0
2-1
3-2
0 20 6040
1-0
2-1
3-2
0 20 6040
0 60 120 0 60 1200 60 120 0 60 120
0 3
(a) (b) (c) (d)6 0 3 6 0 3 6 0 3 6
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changes as the structure is displaced beyond yielding; and (iv) a “uniform” distribution, inwhich the lateral forces are proportional to the masses at each floor level.
The comparison between numerical predictions and experimental results has beenmade in terms of capacity curves and in terms of maximum story displacements, interstorydrifts, and story shears, with reference to four different seismic intensities (PGA’s), equal to0.28 g and 0.48 g for the model without (w/o) infills, 0.63 g and 0.9 g for the model withinfills. The tests at 0.28 g and 0.63 g produced significant structural and non-structuraldamage in the experimental models w/o and with infills, respectively. Those at 0.48 g and0.9 g determined collapse conditions.
The results of this study clearly point out the importance of developing a refinedstructural model, in order to achieve a good estimation of the actual seismic response ofthe building. As long as the pushover curve turns out to be consistent with the experimen-tal behavior of the structure, indeed, all the NSM’s seem to be able to predict the maxi-mum seismic response of the building with adequate accuracy, provided that theassociated numerical parameters are properly estimated. The lateral load pattern used inthe POA, instead, has been found to little affect the accuracy of the results, for the struc-tural model considered in this study (a 3-story regular frame), characterized by a 1stmode-dominated response, both in the elastic and post-elastic range.
A number of critical numerical parameters, whose selection play a fundamental rolein the accuracy of the results, have been identified for each method. The key parameter inthe CSM is the k-factor, which is a measure of how much the actual hysteretic behavior ofthe building differs from the theoretical elasto-plastic behavior. The key parameter in theDCM is the modification factor C2, which takes into account the increase of response dueto stiffness/strength degradation and pinching of hysteresis loops. The critical point is thatthe evaluation of the aforesaid parameters is based on individual judgment. A set of valuesfor both k and C2 are suggested in ATC-40 and FEMA-356, respectively. They appear rea-sonably accurate when the building responds within a “Safety Life” structural perfor-mance level, while they seem somewhat imprecise and underestimating when the buildingresponds within a “Collapse Prevention” (i.e., “Structural Stability”) structural perfor-mance level.
The key parameters in the N2 method are the effective elastic period of vibration (Te)and the reduction factor (Rm). The effective elastic period of vibration corresponds to theinitial period of the equivalent SDOF system. It depends on the approach used in the bilin-ear idealization of the actual pushover curve, which goes through individual judgment. Inthis study, reference has been made to the approach of Eurocode 8, which is based on theequal energy principle. The reduction factor Rm depends on the hysteretic energy dissipa-tion capacity of the structure and can be expressed as a function of the ductility factor mand effective elastic period Te. Several formulations exist for the reduction factor Rm. Inthis study, the relationships adopted in Eurocode 8 have been used. Under these assump-tions, the N2 method provides numerical results similar to those predicted by DCM. Forstructures with negative post-yield stiffness subjected to very strong earthquakes, DCMappears more accurate than the N2 method, due to the presence of the C3 modificationfactor. Perhaps, the introduction of an analogous factor could improve the N2 method.
The inherent viscous damping (b0) plays an important role for all the methods and itshould be selected with great care. On the contrary, the lateral load pattern seems to affectnegligibly the accuracy of the results. Obviously, this conclusion should be limited to low-rise buildings (such as the experimental models considered in this study), in which thestructural response is dominated by the first mode of vibration, while the same conclusioncould not hold for high-rise buildings, for which multi-modal pushover analysis proce-dures have been found to greatly improve the prediction [Chopra and Goel, 2002].
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Finally, it should be mentioned that CSM, DCM, and N2 method also differ in sim-plicity, reliability, and theoretical backgrounds. The CSM is formulated in the ADRS for-mat and it requires an iterative graphical procedure to determine the performance point, asstructural capacity and seismic demand are mutually related. Another important limitationof CSM is the lack of convergence under strong earthquakes (see test at 0.9 g on the modelwith infills), which deform the structure into the region of negative post-yield stiffness,with significant decay in the lateral strength. Unlike the CSM, DCM provides a directnumerical procedure to define the displacement demand of the structure, which does notrequire any conversion in the ADRS format. The bilinear idealization of the capacitycurve, however, depends on the displacement demand and the procedure becomes itera-tive. In the N2 method all the steps can be performed numerically. Moreover, if a simpleconservative assumption is made (i.e., To=Tc, being To the characteristic period of theground motion and Tc the transition period form the constant-acceleration to the constant-velocity segment of the response spectrum), no iterations are needed.
Acknowledgments
The experimental tests have been carried for the MANSIDE project, funded by the Euro-pean Commission, D.G. XII, within the BRITE-EURAM program of the IV framework(EC Project BE95-2168). The re-evaluation of the experimental tests and the numericalwork has been carried out within the framework of the Reluis – Line 4 research program(Reluis – Italian Network of University Laboratories of Earthquake Engineering).
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