Development and Utilization of a Database of InfilledFrame Experiments for Numerical ModelingHonglan Huang, S.M.ASCE1; Henry V. Burton, M.ASCE2; and Siamak Sattar3
Masonry infill panels are often treated as nonstructural componentsin the design of reinforced concrete (RC) and steel frame buildings.However, observations from past earthquakes, as well as experi-mental and numerical investigations, have demonstrated the signifi-cant influence of masonry panels on the lateral strength, stiffness,and energy dissipation capacity of infilled frames (Asteris et al.2011; Liberatore et al. 2018; Mehrabi et al. 1996; Noh et al. 2017).Developing a reliable numerical model that is able to simulate thenonlinear response of a structural system from the onset of damagethrough collapse is a key step in implementing the PerformanceBased Earthquake Engineering (PBEE) framework (Haselton et al.2008; Koliou and Filiatrault 2017; Lignos and Krawinkler 2011).More specifically, reasonable assessments of the seismic perfor-mance of masonry-infilled frame structures require appropriatemodeling techniques as well as reliable predictive equations for es-timating key force-deformation parameters of the masonry infillpanels.
A variety of macro models and material laws for characterizingthe in-plane response of infilled frames have been proposed in thepast. The idea of using compression-only diagonal struts to re-present the infill panel was proposed in the 1960s (Holmes 1961;Mainstone and Weeks 1974; Polyakov 1960; Stafford Smith andCarter 1969), and has since been adopted by most of the recentmacro models. Among the more recently proposed macro models,there are two major developments: (1) the use of multiple struts in
each diagonal direction to incorporate more local interactionsbetween the frame and infill (Burton and Deierlein 2014;Chrysostomou et al. 2002; Crisafulli and Carr 2007; Crisafulli1997; El-Dakhakhni et al. 2003; Sattar and Liel 2016); and (2) uti-lizing shear springs to capture the sliding shear mechanism of theinfill panel (Crisafulli and Carr 2007; Crisafulli 1997). However,these improvements in the ability to capture local behavior withmore precision also increases the complexity in determining thegoverning parameters. On the other hand, several constitutive lawsthat define the hysteretic response of equivalent infill struts havealso been proposed (Cavaleri et al. 2005; Klingner and Bertero1978; Liberatore 2001). A more comprehensive summary regard-ing these strut models and material laws can be found in the state-of-the-art review by Noh et al. (2017).
Analytical models for estimating the stiffness and strength prop-erties of the equivalent diagonal struts and the frame-infill systemhave also been the subject of prior research. Stafford Smith andCarter (1969) proposed a nondimensional parameter to representthe relative stiffness between the frame and the infill, and developedanalytical methods that relate the contact length and effective strutwidth to the relative stiffness. The effective strut width, a keyparameter in estimating the stiffness and ultimate strength of theinfill strut, has been investigated in numerous studies (Decanini andFantin 1986; Holmes 1961; Mainstone 1971; Mainstone andWeeks1974; Saneinejad and Hobbs 1995; Stafford Smith and Carter1969), among which the most frequently used are the equationsproposed by Mainstone and Weeks (1974), which have also beenadopted in FEMA 306 (FEMA 2000). Several researchers proposedanalytical equations for the strength of the equivalent diagonalstruts (Decanini and Fantin 1986; Panagiotakos and Fardis 1996;Paulay and Priestley 2009; Priestley and Calvi 1991; Saneinejad andHobbs 1995; Stafford Smith and Carter 1969). The recently updatedASCE 41 provisions on masonry infills adopted a framework pro-posed by Stavridis et al. (2017) to estimate the backbone curveparameters used to model the lateral response of the infilled frame.
Although these macro models, constitutive laws, and analyticalequations cover important aspects of the seismic analysis of ma-sonry infilled frames, most of them are derived based on relativelysmall data sets from experimental and numerical studies. There-fore, their accuracy and generalizability need to be validated on
1Graduate Student, Dept. of Civil and Environmental Engineering,Univ. of California, Los Angeles, CA 90095 (corresponding author). Email:[email protected]
2Assistant Professor, Dept. of Civil and Environmental Engineering,Univ. of California, Los Angeles, CA 90095. ORCID: https://orcid.org/0000-0002-5368-0631
3Research Structural Engineer, National Institute of Standards andTechnology, 100 Bureau Dr., Gaithersburg, MD 20899.
more comprehensive infilled frame response data sets. In this re-gard, Liberatore et al. (2017) constructed a database consisting of162 specimens of masonry-infilled RC and steel frames and con-fined masonry structures to assess the performance of five analyti-cal models for strut strength and stiffness. They subsequentlyproposed a multilinear backbone curve for the infill panel lateralresponse and a framework to combine and modify previous ana-lytical equations to estimate the associated backbone curveparameters. De Risi et al. (2018) collected 219 specimens ofRC frames with unreinforced hollow clay brick panels, among
which 38 were used to evaluate the applicability of two constit-utive models for simulating the lateral response of the infill panel,and further modified the model proposed by Panagiotakos andFardis (1996).
The primary objectives of the current study are to: (1) developan experimental database of masonry-infilled frames (steel andRC), (2) develop a modeling approach for RC-infilled frames usingthe pinching material (Lowes et al. 2004) (also denoted as theLowes-Mitra-Altoontash model) to simulate the nonlinear axial re-sponse of the infill equivalent diagonal struts, (3) calibrate the infillstrut force-deformation parameters based on the experimental re-sults, (4) develop empirical models for estimating the backbonecurve parameters of the equivalent diagonal struts, and (5) discussand provide recommendations for the associated hysteretic param-eters. It is important to note that while the calibration was per-formed using an implementation of the Lowes-Mitra-Altoontashpinching model in the Open System of Earthquake Engineering(OpenSees) (Mckenna 1999), the backbone curve parameters canalso be used for nonpinching materials to represent the nonlinearresponse of equivalent infill struts [e.g., the peak-oriented hyster-etic model formulated by Burton and Deierlein (2014)]. Moreover,the cyclic degradation and pinching parameters can be used withthe Lowes-Mitra-Altoontash model in non-OpenSees contexts.
Experimental Database
The entire experimental database consists of 257 one-bay one-storyand 7 multi-bay one-story masonry infilled frames, collected from49 journal publications, conference proceedings, and technical re-ports. The sources of the database are listed in Table 1. The frametypes include both reinforced concrete and steel. The infill panels inthe database cover a range of different types of masonry units, re-inforced and unreinforced masonry, panels with and without open-ings and with and without retrofit measures. A graphical summaryof the types of infilled frames in the database is provided in Fig. 1.Since the focus of the current study is on RC infilled frames, thedetailed characteristics of the infilled steel frames are omitted fromFig. 1. Table 2 presents the types of masonry units included in thedatabase and the corresponding number of specimens. Approxi-mately 60% of the specimens are made from clay brick, slightly
Table 1. Database sources
ReferenceNumber ofspecimens
Frametype Loading type
Abdul-Kadir (1974)a 12 Steel MonotonicAkhoundi et al. (2018)a 1 RC Quasi-static cyclicAl-Chaar et al. (2002)a 4 RC MonotonicAngel et al. (1994)a 7 RC Quasi-static cyclicAnil and Altin (2007) 7 RC Quasi-static cyclicBaran and Sevil (2010)a 3 RC Quasi-static cyclicBasha and Kaushik (2016)a 9 RC Quasi-static cyclicBergami and Nuti (2015)a 2 RC Quasi-static cyclicBillington et al. (2009)a 1 RC Quasi-static cyclicBlackard et al. (2009)a 4 RC Quasi-static cyclicBose and Rai (2014)a 1 RC Quasi-static cyclicCalvi and Bolognini (2008)a 4 RC Quasi-static cyclicCavaleri and Di Trapani (2014)a 12 RC Quasi-static cyclicChiou and Hwang (2015)a 4 RC Quasi-static cyclicColangelo (2005)a 11 RC Pseudo-dynamicCombescure et al. (1996)a 2 RC Quasi-static cyclicCrisafulli (1997)a 2 RC Quasi-static cyclicDa Porto et al. (2013) 6 RC Quasi-static cyclicDautaj et al. (2018)a 7 RC Quasi-static cyclicDawe and Seah (1989)a 28 Steel MonotonicFiorato et al. (1970)a 7 RC MonotonicFlanagan and Bennett (1999) 8 Steel Quasi-static cyclicGazic and Sigmund (2016)a 11 RC Quasi-static cyclicHaider (1995)a 4 RC Quasi-static cyclicKakaletsis andKarayannis (2008)
6 RC Quasi-static cyclic
Khoshnoud andMarsono (2016)a
2 RC Monotonic
Kumar et al. (2016)a 1 RC Quasi-static cyclicLeuchars and Scrivener (1976)a 2 RC Quasi-static cyclicLiu and Soon (2012) 10 Steel MonotonicMansouri et al. (2014)a 5 RC Quasi-static cyclicMarkulak et al. (2013) 6 Steel Quasi-static cyclicMehrabi et al. (1996)a 10 RC Monotonic/
Quasi-static cyclicMisir et al. (2016)a 5 RC Quasi-static cyclicMorandi et al. (2014)a 4 RC Quasi-static cyclicMosalam et al. (1997) 4 Steel Quasi-static cyclicPires et al. (1997)a 2 RC Quasi-static cyclicSchwarz et al. (2015) 3 RC Quasi-static cyclicSigmund and Penava (2013)a 9 RC Quasi-static cyclicStylianidis (2012)a 5 RC Quasi-static cyclicTasnimi and Mohebkhah (2011) 5 Steel Quasi-static cyclicTawfik Essa et al. (2014)a 3 RC Quasi-static cyclicTizapa (2009)a 3 RC Quasi-static cyclicVerderame et al. (2016)a 2 RC Quasi-static cyclicWaly (2000)a 2 RC Quasi-static cyclicYorulmaz and Sozen (1968)a 7 RC MonotonicYuksel and Teymur (2011)a 2 RC Quasi-static cyclicZarnic and Tomazevic (1985) 3 RC Quasi-static cyclicZhai et al. (2016)a 3 RC Quasi-static cyclicZovkic et al. (2013) 3 RC Quasi-static cyclicaExperiments used in calibration.
more than half of which are hollow. Concrete blocks, most of whichare solid, comprise approximately 30% of the specimens. Other lesscommon types of brick (e.g., calcarenite, vitrified ceramic, cementand pumice) are also included and there are nine autoclaved aeratedconcrete specimens.
For the purpose of the current study, the subset (134) of speci-mens that include only RC frames with unreinforced masonry panelswithout openings or retrofit are first extracted. As one of the impor-tant steps in this study is to calibrate the backbone modeling param-eters of the axial response of the equivalent diagonal struts, explicitforce-displacement curves from the tests are needed. However, someof the investigations did not report the force-displacement curves forall tested specimens. After excluding those investigations that did notinclude this information, 113 specimens remained and were used tocalibrate and develop empirical equations for the backbone curveparameters of the infill struts.
Table 3 provides a statistical summary of the key structural var-iables in the specimen subset used for subsequent calibrations.These variables are selected based on a review of prior literatureon the behavior of infilled frames. hcol is the column height fromthe top of the base to the centerline of the beam, lbeam is the beamspan to the centerline of the column, ld is the length of the diagonalstrut (measured based on the centerline dimension of the frame, asis used in the structural models in subsequent sections), hw is theheight of the infill panel, lw is the length of the infill panel, hw=lw isthe aspect ratio of the infill panel, tw is the thickness of the infillpanel, and P=Agf 0
c is the column axial load ratio. In most experi-ments, the axial load was applied directly to the columns (to re-present gravity loads). However, it is worth noting that in realbuildings some of the gravity load can be transmitted from the framebeams to the infill panels. ρt;col is the column transverse reinforce-ment ratio, fm is the masonry prism strength, and Em is the masonrymodulus of elasticity. For consistency, the recorded values of fm andEm are based on prism compression tests with loading appliedperpendicular to the bed joint (some studies also performed the com-pression tests parallel to the bed-joints). In some cases where Em isnot available, an empirical relationship, Em ¼ 550fm, as recom-mended by Kaushik et al. (2007) and FEMA 306, has been adopted.
In addition, the relative stiffness parameter (λhcol) proposed byStafford Smith and Carter (1969) is also computed:
λhcol ¼�Emtw sinð2θÞ4EfIcolhw
�0.25
· hcol ð1Þ
where θ ¼ tan−1ðhw=lwÞ is the angle between the diagonal and hori-zontal of the panel,Ef = the modulus of elasticity of the frame, Icol =the moment of inertia of the column.
The infilled frame failure mode classification scheme proposedby Tempestti and Stavridis (2017) has been adopted for the currentstudy. In the original literature, the infilled frames are first classifiedinto four categories based on two quantitative metrics, then eachcategory is assigned a distinct failure mode. For the current study,because the failure observations in the experiments are available,we did not use the quantitative metrics for classification, but insteadassigned each specimen to a distinct failure mode based onthe available observations. Table 4 presents the adopted failuremode descriptions and the number of specimens correspondingto each one.
Numerical Modeling and Calibration of the InfillStrut Parameters
This section discusses the numerical modeling of infilled framesusing the Lowes-Mitra-Altoontash pinching model and details ofthe process used to calibrate the parameters of the equivalent diago-nal struts. The single-strut model (in each direction) used for thecalibration process is consistent with the modeling framework pro-posed by Bose et al. (2018) and Stavridis et al. (2017), and adoptedin ASCE 41 (ASCE 2017). As discussed earlier, while this model-ing approach may not be able to precisely capture any localizedfailure modes, the total response of the masonry infill and the framecan represent the global seismic response of the tested specimens.A schematic view of a one-bay one-story model is shown in Fig. 2.
Development of the Infilled Frame Model in OpenSees
The numerical model of the infilled frame is developed inOpenSees. However, it is worth noting that the overall approachis applicable to other platforms (e.g., LS DYNA) that have similarimplementations of the adopted material models. A concentratedplasticity model is used for the RC beams and columns, which con-sists of elastic beam-column elements with two zero-length hingesat the ends (Burton and Deierlein 2014; Haselton et al. 2008;Haselton et al. 2016; Noh et al. 2017), as shown in Fig. 2. The peakoriented hysteretic model developed by Ibarra et al. (2005) (alsodenoted as the Ibarra-Medina-Krawinkler model) is used to modelthe beam flexural hinges. As column failure is one of the primaryconcerns in the seismic performance of nonductile infilled frames,it is critical that this mechanism is incorporated in the simulation.Given this consideration, Burton and Deierlein (2014) used a flexu-ral spring with the Ibarra-Medina-Krawinkler model paired with ashear spring with rigid-softening material to incorporate flexuralhinging and column shear failure. Sattar (2014) used a flexuralspring with the Ibarra-Medina-Krawinkler model, a shear springwith the failure model developed by Elwood (2004), and an axialspring with the failure model also developed by Elwood (in series)to capture the complex flexure-shear-axial interactions. In thisstudy, flexural and shear springs are modeled in series at the col-umn ends (top and bottom), as shown in Fig. 2. The flexural springis modeled using the Ibarra-Medina-Krawinkler model and theshear spring is modeled with Elwood’s shear failure model. It isworth noting that this frame model can also capture column shearand flexural failure in the spans with no infill walls. Therefore, thesame column modeling approach can be used for all the columnswhen modeling a real building.
The infill panel is modeled as two diagonal struts (one in eachdirection) using a truss element with the Lowes-Mitra-Altoontashpinching material model. Various material models have been usedby different researchers to simulate the nonlinear behavior of infillstruts. Some examples include an early version of the Ibarra-Medina-Krawinkler model (Burton and Deierlein 2014), the con-crete material model (Mohd-Yassin 1994; Bose and Stavridis2018), the Kent-Scott-Park model (Scott et al. 1982; Noh et al.2017), the Bouc-Wen hysteretic model (Bouc 1967; Noh et al.2017), and the Lowes-Mitra-Altoontash model (Furtado et al. 2015;Noh et al. 2017). In this study, the Lowes-Mitra-Altoontash modelis adopted because of its flexibility in controlling pinching effects
and cyclic degradations in the infill panel, and its effectiveness formodeling the infill struts has been illustrated by Noh et al. (2017).A schematic response of the Lowes-Mitra-Altoontash model andassociated parameters is shown in Fig. 3 (adapted from Loweset al. 2004). The parameters that are critical to the calibration pro-cess described in the next subsection are presented in bold font. Aresponse envelope, an unload-reload path and three damage rulesdefine the hysteretic response in the Lowes-Mitra-Altoontashmodel. More specifically, eight points (16 parameters) are usedto construct a multilinear response envelope (shown as solid linesin Fig. 3). The unload-reload path (shown as dashed lines in Fig. 3)is controlled by four points (six parameters). The unloading stiff-ness degradation, reloading stiffness degradation and strength deg-radation, are each controlled by five parameters, which predict thedamage index as functions of the displacement history and energyaccumulation. More details of the model can be found in Loweset al. (2004).
Calibration
The flexural hinge parameters of the reinforced concrete beams andcolumns are determined using the semi-empirical equations pro-posed by Haselton et al. (2016), whereas the column shear hingeparameters are obtained using the equations by Elwood (2004).Once the parameters of the reinforced concrete frame are deter-mined, those governing the axial response of the infill struts arecalibrated through an iterative procedure, where the lateral hyster-etic curve of the entire one-bay one-story infilled frame from thenumerical simulation is compared with the experimental hystereticcurve, and the parameters of the infill struts are adjusted until anacceptable visual match is achieved (Burton and Deierlein 2014),i.e., close visual resemblance in the response envelope, the pinch-ing effects, and the unloading-reloading slopes.
A good fit (compared to the experimental results) is achievedusing the Lowes-Mitra-Altoontash material model because of theflexibility to control a wide range of parameters. However, atrade-off between precision and complexity in parameter defini-tions must be considered when applying the model. For the pur-pose of this study, several simplifying assumptions are made todetermine the parameters for the Lowes-Mitra-Altoontash modelapplied to the equivalent infill struts, while seeking to maintain anadequate level of calibration accuracy. Some of the modeling
Fig. 2. Schematic view of the infilled frame model in OpenSees.
assumptions discussed by Noh et al. (2017) are also adopted. Thedetails of the calibration process are as follows:1. The points defining the multilinear backbone curve of the strut
axial response are calibrated such that there is a close matchbetween the simulated and experimental lateral response envel-ope of the infilled frame. In the Lowes-Mitra-Altoontash model,eight points define this envelope. However, as the infill panel ismodeled by a compression-only strut in each diagonal direction,only the negative (compression) branch [ðeNd1; eNf1Þ, ðeNd2;eNf2Þ, ðeNd3; eNf3Þ, ðeNd4; eNf4Þ] needs to be calibrated.The force level of the positive (tension) branch is taken to be1% of that in the compression branch (Noh et al. 2017), result-ing in a response envelope similar to the constitutive model de-veloped by Cavaleri and Di Trapani (2014). After calibrating acertain number of specimens, it was discovered that specifyinganother point between the yield and capping points does notsignificantly influence the calibration accuracy (as measuredby the dissipated hysteretic energy). Therefore, ðeNd2; eNf2Þis assigned by linear interpolation after obtaining ðeNd1; eNf1Þand ðeNd3; eNf3Þ. As a result, this step can be simplified tocalibrating three points: the yield point ðeNd1; eNf1Þ, the cap-ping point ðeNd3; eNf3Þ, and the point where the residualstrength is achieved ðeNd4; eNf4Þ.
2. The unload-reload path parameters are adjusted to match thepinching shape as exhibited in the hysteretic response measuredduring the experiment. The Lowes-Mitra-Altoontash model usessix parameters to control pinching effects (through the points de-fining the unload-reload path). In the current study, four para-meters are calibrated: uForceN which determines the strengthat the start of unloading from the tension branch, rDispN andrForce which define the point at which reloading begins, anduForceP, which is used to match the strength at which unloadingfrom the compression branch begins. As the tension branch isnegligible, rDispP and rForceP are not influential and valuesof rDispP ¼ 0.2 and rForceP ¼ 0.5 are assumed.
3. The Lowes-Mitra-Altoontash model uses 15 parameters to con-trol the strength, unloading stiffness and reloading stiffnessdegradation, which significantly adds to the complexity of the
calibration process. A simplified approach is adopted in thisstudy, which relates the strength and stiffness degradation tothe displacement history (Lowes et al. 2004). More specifically,gK1 and gK3 are used to define unloading stiffness degradation,with gKLim setting the limit for the damage index (Lowes et al.2004). Similarly, ðgD1; gD3; gDLimÞ and ðgF1; gF3; gFLimÞare used to control the reloading stiffness and strength degrada-tion, respectively. The other six parameters ðgK2; gK4; gD2;gD4; gF2; gF4Þ are set to zero.
4. The dmgType parameter, which is set to “Cycle,” relates thedamage indices to the displacement history. gE is taken as10 for all specimens (Noh et al. 2017).The calibration process follows the sequence of steps 1, 2, and 3
discussed previously, i.e., first the response envelope is adjusted,then the pinching shape is addressed, and finally the cyclic degra-dation parameters are determined. Table 5 lists the parameters in theLowes-Mitra-Altoontash model that are used in the calibration, in-cluding a description of how each parameter is used. Also note thatfor the calibration, the strut area and other infill parameters are notpre-determined using previously developed equations. Instead, aunit cross section area is used for the strut such that the calibratedstress parameters (for example, eNf1) are equal to the actual strutforce values.
A calibration example for specimen TA2 of the experimentalinvestigation by Morandi et al. (2014) is illustrated in Fig. 4.The experimental hysteretic curve of the infilled frame is shownwith the solid line in Fig. 4(a), while the numerical simulation resultis represented by the dashed line. A reasonable match between thesimulated and experimental hysteretic curve in terms of the overallresponse envelope, pinching effect and cyclic degradation, is ob-served. A further comparison is made between the cumulative en-ergy dissipation of the experimental and simulated response, asillustrated in Fig. 4(b), where the horizontal axis is the displace-ment level of the test and the vertical axis is the cumulative energy.A reasonable visual match is achieved between the experimentaland simulated results.
Fig. 5 further presents four calibration results based on experi-ments by Cavaleri and Di Trapani (2014). Specimen S1B-2 is used
Fig. 3. Schematic response of the Lowes-Mitra-Altoontash model. (Adapted from Lowes et al. 2004.)
as an example to illustrate the process for determining the transitionpoint between the negative post-peak stiffness and the residualstrength ðeNd4; eNf4Þ [the circled point in Fig. 5(a)]. This pointis estimated by observing the start of the stabilizing portion of thehysteretic response, where the load remains approximately constantwith increased displacement. Once the residual strength transitionpoint is identified, the corresponding deformation is converted tothe diagonal direction and used as eNd4 (needs to be divided by ldto obtain strain) in the Lowes-Mitra-Altoontash model. eNf4,which specifies the strut residual strength, is then adjusted until thesimulated hysteretic response of the entire infilled frame reasonablymatches the experimental response, as shown in the calibration re-sult in Fig. 5(a).
The calibration process is performed on the data set of113 specimens until an adequate visual match between experimen-tal and simulated response is achieved for each experiment.
For experiments with available digital force-deformation data, acumulative energy dissipation comparison is also conducted forfurther validation.
Empirical Model for the Backbone Curve Parametersof the Equivalent Infill Struts
This section presents the statistical analysis results for the cali-brated model parameters that define the axial force-displacementrelationship of the infill struts. The main focus of this paper iscalibrating and predicting the force-displacement parameters thatdefine the backbone curve, which can be used to model the infillstruts using pinching materials such as the one by Lowes-Mitra-Altoontash and others (e.g., peak-oriented models like the Ibarra-Medina-Krawinkler model). These parameters are presented here as
Table 5. Calibrated parameters in the Lowes-Mitra-Altoontash model
Parameter type Parameter name Description
Envelope eNd1 Strain in the infill strut at the yield pointeNf1 Stress (force) in the infill strut at the yield pointeNd3 Strain in the infill strut at the capping pointeNf3 Stress (force) in the infill strut at the capping pointeNd4 Strain in the infill strut at the point where the residual strength initiateseNf4 Stress (force) in the infill strut at the point where the residual strength initates
Unloading-reloadingand pinching effects
uForceN Stress (force) developed upon unloading from the tension branchrDispN Strain at the start of reloadingrForceN Stress (force) at the start of reloadinguForceP Stress (force) developed upon unloading from the compression branch
the initial stiffness (Ke), the yield strength (Fy), the capping strength(Fc), the deformation corresponding to the capping strength (dc),the post-capping stiffness (Kpc), and the residual strength (Fres),as shown in Fig. 6. Table 6 provides the relationship between
these backbone parameters and those that define the Lowes-Mitra-Altoontash model. An empirical equation is established to relate eachbackbone parameter to several geometric and material properties ofthe infilled frame. No regression analysis is performed for the pinch-ing and cyclic degradation parameters. However, recommendationsfor the appropriate range of the parameter values are presented laterin this section.
Fig. 5. Calibration results for specimens: (a) S1B-2; (b) S1A-1; (c) S1C-1; and (d) S1C-3 of the Cavaleri and Di Trapani (2014) experiments.
Fig. 6. Axial force-deformation response of the infill strut model.
Table 6. Relationship between the infill strut backbone parameters andthose of the Lowes-Mitra-Altoontash model
Backbone parameterCalibrated Lowes-Mitra-
Altoontash model parameters
KeeNf1
eNd1 · ldFy eNf1Fc eNf3dc eNd3 · ld
KpceNf4 − eNf3
ðeNd4 − eNd3ÞldFres eNf4
Note: Unit strut area is used so the calibrated stress is the same as theassociated force.
The empirical model is developed using multivariate regressionanalysis. The general functional form is an additive model in thelogarithmic scale, which is given by
ϵ ∼ Nð0; σ2Þ ð3Þwhere Y = the strut model parameter (response variable), fXjjj ¼1; 2; : : :pg = the predictor variables (each represents a structuralproperty), fβjjj ¼ 0; 1; : : :pg = the regression coefficients, ϵ =a random error term, and σ2 = an unknown constant variance ofϵ. This functional form has been adopted in empirical modelsfor the response parameters for steel components (Lignos andKrawinkler 2011), reinforced concrete beam-column elements(Haselton et al. 2008, 2016), and reinforced concrete beam-columnjoints (Jeon et al. 2014). After obtaining the regression coefficients,the resulting equation can be equivalently expressed in the originalspace using a multiplicative functional form
Y ¼ eβ0ðX1Þβ1ðX2Þβ2 : : : ðXpÞβp ð4Þ
which is more comparable to the form of the mechanistic analyti-cal relationships between the response parameters and structuralproperties.
The adequacy of the predictive model is evaluated using the co-efficient of determination (R2) and the residual standard error (σ),which is an estimate of the standard deviation (σ) of the randomerror term (ε) that quantifies the uncertainty in the model predic-tion. Note that the R2 and σ values are evaluated based on the re-gression model developed using the functional form in Eqs. (2) and(3), not the transformed multiplicative form in Eq. (4).
As a first step toward developing the empirical equations, thetrends between each model parameter and several geometric andmaterial properties are investigated to gain a general understandingof the relationships and to select the necessary variables to be usedin the regression analysis. The trends are illustrated using a set ofscatterplots. A linear trend line is shown in each plot to illustrate therelationship between the model parameter and the predictor varia-bles. This line only serves as an illustration and does not representthe final equation because the actual regression model takes on themultivariate nonlinear form shown in Eq. (2).
Multivariate regression analysis is performed for each backboneparameter, incorporating the predictor variables deemed influentialbased on a prior understanding of the basic mechanics that governinfilled frame behavior (after reviewing the prior literature) andfrom the trend analysis. A stepwise selection strategy based onthe F test is implemented to exclude predictors that are not statisti-cally significant at the 95% level (Haselton et al. 2008; Hocking1976; Lignos and Krawinkler 2011). Once an initial model is ob-tained, an outlier detection strategy based on plotting jackknifedresiduals (Cook and Weisberg 1982) against predicted values isperformed to determine whether some data points need to be re-moved to obtain a final model.
Empirical Equation for Initial Stiffness Ke
Fig. 7 shows the trends between the initial stiffness Ke and sixstructural parameters. An increase in Em, fm, tw, ld, and λhcol areassociated with an increase inKe, while an increase in hw=lw resultsin a decrease in Ke.
Table 7 presents a summary of the regression results includingall six predictors discussed earlier. Based on the individual t-tests,
lnðfmÞ, lnðλhcolÞ, lnðldÞ and the intercept term are not significant atthe 95% level. Tables 8–10 present, in sequence, the results fromperforming backward stepwise elimination based on an F-test(Hocking 1976). At each round, the variable with the smallest in-fluence on the prediction is eliminated, as indicated with “*” inTables 8–10. After performing three rounds of backward elimina-tion, the remaining variables are all significant at the 95% levelbased on the F-test.
Fig. 8 shows the residual plot after fitting the reduced model[with lnðEmÞ, lnðtwÞ, and lnðhw=lwÞ as predictors]. The jackknifedresidual (t) is plotted against the predicted lnðKeÞ value. The resid-uals are generally evenly scattered and no curved pattern is ob-served, which indicates reasonable model performance. The solidlines added to the plot represent a boundary of jtj ¼ 2. Based on arule of thumb (Cook and Weisberg 1982), seven data points withjtj > 2 are detected as outliers and subsequently removed. Note thatthe intercept term is statistically insignificant based on the t-test.However, it is included in the final model because excluding it in-troduces significant bias when removing the outliers and refittingthe regression model.
Eq. (5) presents the empirical equation for Ke obtained from theanalysis discussed previously
Ke ¼ 0.0143E0.618m t0.694w
�hwlw
�−1.096
R2 ¼ 0.54; σ ¼ 0.56 ð5Þ
where Ke, Em, and tw are in units of kN=mm, MPa, and mm, re-spectively. An alternative way to estimate Ke incorporating theresults from the statistical analysis is by using the following ana-lytical formulation:
Ke ¼EmAe
ld¼ Emwetw
ldð6Þ
where Ae = the equivalent diagonal strut area, we = the effectivestrut width. The empirical formulation Eq. (5) incorporates thepositive effects of Em and tw on Ke, while the effect of we=ldis implicitly accounted for by hw=lw. To use Eq. (6), the effectivewidth we needs to be determined. A widely used empirical equa-tion for estimating the effective width is given by (Mainstone1971)
we ¼ ðλhcolÞ−0.175ld ð7Þ
Based on the data set used in this study, another empirical equa-tion for effective width can be provided as follows:1. From Eq. (6), the ratio between the effective width we and strut
length ld can be derived when Ke, Em, tw, and ld are given
we
ld¼ Ke
Emtwð8Þ
Thus, we=ld can be determined for all the specimens in thecurrent data set.
2. Subsequently, a regression analysis is performed on we=ld,resulting in the following equation:
we
ld¼ 0.318ðλhcolÞ−0.661
�hwlw
�−0.871
R2 ¼ 0.25; σ ¼ 0.59 ð9Þ
Eq. (9) is consistent with Eq. (7) because it does not introduce anew variable [see Eq. (1), the formula for λhcol], so it is an adjusted
version of Eq. (7) based on the current data set. The much lower R2
value obtained for we=ld (0.25) compared to Ke (0.54) [based onEq. (5)] is worth noting and may influence the decision regardingwhich of the two formulations [Eq. (5) or (6)] to use.
Fig. 7. Trends between Ke and (a) Em; (b) fm; (c) hw=lw; (d) tw; (e) ld; and (f) λhcol.
Table 8. Stepwise elimination based on the partial F-test (round 1)
Fig. 9 shows the trends for the calibrated Fc. Strong associationsbetween Fc and both tw and ld can be observed. It is worth notingthat the lines in Figs. 9(a and b) do not represent the univariate
effects of Em and fm on Fc. The actual effects need to be examinedbased on a fundamental understanding of the mechanics governingthe behavior of infilled frames and multivariate statistics.
Eq. (10) presents the equation obtained from the regressionanalysis. It is observed that fm and tw are positively correlated withFc, which is consistent with our understanding of the basic me-chanics governing the response of infilled frames
Fc ¼ 0.003766f0.196m t0.867w l0.792d
R2 ¼ 0.80; σ ¼ 0.38 ð10Þ
where Fc, fm, tw, and ld are in units of kN, MPa, and mm, respec-tively. From the standpoint of basic mechanics, the cappingstrength Fc is known to be positively correlated with the compres-sive strength of the infill fm, the thickness of infill and the effec-tive strut width (or contact length between the frame and infill) atcrushing, which is proportional to the strut length ld [Eq. (7)].Moreover, the physical validity of Eq. (10) is consistent with sev-eral previously developed empirical equations (Mainstone andWeeks 1974; Paulay and Priestley 2009), but with the λhcolparameter excluded.
Fig. 9. Trends between Fc and (a) Em; (b) fm; (c) hw=lw; (d) tw; (e) ld; and (f) λhcol.
Fig. 10 shows a scatterplot of Fy against Fc. A strong associationbetween Fy and Fc is observed. Therefore, Fy can be simply com-puted as a proportion of Fc after the latter is determined. Afterregressing Fy against Fc, a simple equation is obtained
Fy ¼ 0.72Fc R2 ¼ 0.98; σ ¼ 0.13 ð11ÞEq. (11) provides a simplified approach to estimating the yield
point in the backbone curve using Fc, which is consistent with pre-viously developed models (e.g., Stavridis et al. 2017; Dolšek andFajfar 2008).
Empirical Equation for the Residual Strength F res
The analysis for Fres is based on 75 experimental observationswhere the residual strength is reached as indicated by the load-displacement curves. Fig. 11 shows a strong trend between the cali-brated Fres and Fc. Therefore, similar to Fy, Eq. (12) provides asimplified approach to estimating Fres using Fc.
The obtained empirical equation is given by
Fres ¼ 0.4Fc R2 ¼ 0.81; σ ¼ 0.41 ð12Þ
Empirical Equation for Deformation atPeak Strength dc
This subsection presents the analysis of the axial deformation ofthe infill strut at peak strength (dc) normalized by the length of the
strut (ld). Fig. 12 shows the relationships between dc=ld and fivepotential predictors. An increase in hw=lw and tw leads to a largerdc=ld value, while a higher Em decreases dc=ld. A significant as-sociation between dc=ld and λhcol is not observed in the scatterplot.
The obtained equation is presented in Eq. (13)
dcld
¼ 0.0154E−0.197m
�hwlw
�0.978
R2 ¼ 0.34; σ ¼ 0.47 ð13Þ
As Em increases, the infill panel becomes stiffer and the cappingstrength is reached at smaller displacement. As the aspect ratio hw
lwincreases, the entire frame tends to have a more ductile response,and the infill panel often crushes because of compression at largedrifts instead of sliding shear, which tends to occur at smaller driftdemands. As a result, the displacement at the point where thestrength starts to degrade tends to be larger.
Empirical Equation for the Ratio between thePost-Capping Stiffness and Initial Stiffness Kpc=Ke
Fig. 13 presents the trends between the ratio of the post-cappingto initial stiffness (Kpc=Ke) and six parameters. Increases in Em,fm, tw, and λhcol are associated with an increase in Kpc=Ke(a decrease in magnitude because Kpc is negative). The aspect ratioof the infill (hw=lw) does not appear to have a significant influenceon Kpc=Ke.
The empirical equation for Kpc=Ke is given by
Kpc
Ke¼ −1.278f−0.357m tw−0.517
R2 ¼ 0.32; σ ¼ 0.46 ð14Þ
where fm and tw are in units of MPa and mm, respectively.It is worth noting that the R2 values for Eqs. (13) and (14)
are relatively low. This means that the variance of the response var-iable is not fully captured by the adopted predictors. Nevertheless,the statistical significance indicates that the predictors providevaluable information about the response variable. More impor-tantly, the computed coefficients can reasonably capture the rela-tionships between the response variable and the adopted predictors.
Fig. 14 presents comparisons between the parameter values pre-dicted using the developed empirical equations and the originalcalibrated parameters after removing the outlier datapoints. Themean and median of the ratios of the predicted to calibrated param-eter values are listed in Table 11. The ranges of the mean andmedian ratios are 0.95–1.14 and 0.84–1.08, respectively, indicatingthat the empirical models predict the backbone parameters withreasonable accuracy.
Cyclic Degradation and Pinching Parameters
This section discusses the calibration results for the parameters thatcontrol the cyclic degradation and pinching effects which are notincluded in the regression analysis. Although the material modelsprovide well-defined rules to simulate the response incorporatingthe degradation effects, the influence of the structural characteris-tics on the degradation parameters could not be extracted from theexperimental data and is still not well understood. More in-depthstudies are necessary in this respect. Nevertheless, the statisticalvalues of these degradation parameters are used to recommendappropriate ranges for use in numerical modeling.
As discussed earlier, this study assumes gK2, gK4, gD2, gD4,gF2, and gF4 to be zero. Three sets of parameters: ðgK1; gK3;gKLimÞ, ðgD1; gD3; gDLimÞ, and ðgF1; gF3; gFLimÞ, are usedto relate cyclic unloading stiffness degradation, reloading stiffnessdegradation, and strength degradation to the displacement history.Table 12 presents the statistical summary of the calibrated cyclicdegradation parameters. In general, using values close to the meanin Table 12 provides an approximate match between the simulatedand experimental hysteretic curves. A more precise match requiresfurther adjustments.
Table 13 presents the statistical summary for the parameters thatcontrol the pinching effect. rDispP and rForceP are assumed tobe 0.2 and 0.5, respectively, as discussed earlier. The pinchingparameters vary significantly across different specimens. Therefore,a generalized set of values might not provide a reliable approxima-tion. However, the ranges of values presented in Table 13 can providesome suggestions for setting these parameters. uForceP is usuallyassigned a negative value. The larger magnitude of this parameter isbecause the actual pinching point is adjusted by uForceP · ePf3,while ePf3 is a tension branch parameter that is set to be small,
as discussed in the calibration process. The ranges of rDispN,rForceN, and uForceN are 0.02–0.3, 0.02–0.4, and 0–0.3,respectively.
Conclusion
The main purpose of this study is to characterize the nonlinearforce-deformation parameters of infill panels modeled using equiv-alent diagonal struts, which is essential to performing reliable sim-ulations of the nonlinear response of infilled frame systems subjectto seismic loads and performance-based assessments.
An experimental database of infilled frames was assembled aspart of the study. The entire database consists of 264 infilled framespecimens subjected to monotonic, quasi-static, and pseudo-dynamic loading, among which 191 specimens have reinforcedconcrete frames and 73 specimens have steel frames. To achieveconsistency in the generalized modeling assumptions, 113 rein-forced concrete infilled frame specimens are used to calibrateand develop empirical equations for the infill strut backbone curveparameters.
Fig. 12. Trends between dc=ld and (a) Em; (b) fm; (c) hw=lw; (d) tw; and (e) λhcol.
The calibration of the strut model parameters is based on thepinching material model proposed by Lowes et al. (2004). An iter-ative procedure is used such that the simulated lateral hysteretic re-sponse curve of the entire one-bay one-story infilled frame achieves agood visual match with the values reported in the experiment.
Based on the calibration results, multivariate regression analysesare performed to develop empirical predictive equations for thebackbone curve parameters of equivalent diagonal struts, includingthe initial stiffness (Ke), the yield strength (Fy), the cappingstrength (Fc), the deformation at capping strength normalized bythe strut length (dc=ld), the post-capping stiffness (Kpc), and theresidual strength (Fres). The predictor variables include severalgeometric and material properties of the infill panels, which areeasily obtained from design or experimental documentation. Theprediction errors are quantified to inform the uncertainties associ-ated with using the empirical equations. Statistical information forthe parameters controlling pinching effects and cyclic degradationis also presented and discussed. Despite using a pinching materialas the basis of the calibration, the backbone curve parameters ob-tained from the empirical equations can be used for modeling infillstruts with non-pinching materials (e.g., the peak-oriented hystereticmaterial model). Subsequently, the strut model can be incorporated
in the nonlinear analysis models of multi-bay multi-story masonryinfilled reinforced concrete frame systems for limit state (e.g., col-lapse, demolition) assessments.
There are several unavoidable limitations associated with thecurrent study. Although the collected database covers a varietyof masonry-infilled frame tests, the primary scope of this studyis on reinforced concrete frames with unreinforced solid infill pan-els, which limits the number of specimens that can be utilized in thecalibration and the applicability of the empirical equations. Furtherresearch efforts incorporating specimens with other characteristics,such as masonry panels with openings and masonry infilled steelframes, are needed to support the development of more generalizedand comprehensive empirical models. In addition, because of thedifferences in the objectives and details provided among the pub-lications used to assemble the database, several structural propertieshaving potential influence on the nonlinear behavior of the infillpanels could not be collected for many of the test specimens, whichreduces the range of available predictors. The calibration process isbased on a simplified numerical model that is able to capture theglobal seismic response of the infilled frame. The accuracy of thecalibration could be further improved by using more complexnumerical models for the frame and infill. Also, the developed
Fig. 13. Trends between Kpc=Ke and (a) Em; (b) fm; (c) hw=lw; (d) tw (e) ld; and (f) λhcol.
empirical equations are for the backbone curve parameters of theinfill struts. The parameters associated with pinching effects andcyclic degradation have not been examined in detail. More in-depthstudies are needed to improve the characterization of these param-eters. Finally, because of the small number (relatively speaking) ofavailable experiments, the predictive capability of the empirical
models was evaluated using the training data (i.e., the same dataused to develop the model). Ideally, the model performance shouldbe evaluated using a data set that is different from the one used in itsdevelopment (i.e., testing data).
Fig. 14. Calibrated versus predicted response parameters: (a) Ke; (b) Fc; (c) Fy; (d) Fres; (e) dc=ld; and (f) Kpc=Ke.
Table 11. Mean and median of the ratio of the predicted to the calibratedparameters
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