Performance of Steel Moment Connections undera Column Removal Scenario. II: Analysis

Fahim Sadek, Ph.D., M.ASCE1; Joseph A. Main, Ph.D., A.M.ASCE2; H. S. Lew, Ph.D., P.E., F.ASCE3;and Sherif El-Tawil, Ph.D., P.E., F.ASCE4

Abstract: This paper presents a computational investigation of the response of steel beam-column assemblies with moment connectionsunder monotonic loading conditions simulating a column removal scenario. Two beam-column assemblies are analyzed, which incorpo-rate (1) welded unreinforced flange bolted web connections, and (2) reduced beam section connections. Detailed models of the assembliesare developed, which use highly refined solid and shell elements to represent nonlinear material behavior and fracture. Reduced modelsare also developed, which use a much smaller number of beam and spring elements and are intended for use in future studies to assess thevulnerability of complete structural systems to disproportionate collapse. The twomodeling approaches are described, and computationalresults are compared with the results of the full-scale tests described in the companion paper. Good agreement is observed, demonstratingthat both the detailed and reduced models are capable of capturing the predominant response characteristics and failure modes of the as-semblies, including the development of tensile forces associated with catenary action and the ultimate failure of the moment connectionsunder combined bending and axial stresses. DOI: 10.1061/(ASCE)ST.1943-541X.0000617. © 2013 American Society of CivilEngineers.

CE Database subject headings: Buildings; Connections; Finite element method; Nonlinear analysis; Progressive collapse; Seismicdesign; Steel structures; Columns.

Author keywords: Buildings; Connections; FEM; Nonlinear analysis; Progressive collapse; Seismic design; Steel structures.

Introduction

Disproportionate collapse is widespread failure of a structure thatresults from localized initial damage (e.g., failure of a column), inwhich the final extent of damage far exceeds the initial extent. Anaccurate characterization of the nonlinear, large-deformation be-havior associated with the redistribution of loads in such localdamage scenarios is critical in assessing the potential for dispro-portionate collapse. NIST is conducting research to develop reliablemethodologies for assessing the vulnerability of structures to dis-proportionate collapse and for quantifying the reserve capacity androbustness of structures at the system level. This research involvesthe development of three-dimensional models of various types ofstructural systems, which must be capable of representing thepredominant response characteristics and failure modes of each

structural system. This paper focuses on the analysis of steelmomentframes under a column removal scenario.

Two basic modeling approaches can be considered for collapseanalysis of structural systems: (1) detailed finite-element modeling,which uses highly refined solid and shell element meshes to repre-sent nonlinear material behavior and fracture, and (2) reduced finite-element modeling, which uses a much smaller number of beams andspring elements. Reduced models can be analyzed much morerapidly than detailed models, making them well suited for collapseanalysis of complete structural systems.

A number of previous studies have used detailed and reducedmodeling approaches for collapse analysis of steel framing systems.Khandelwal and El-Tawil (2007) used detailed models to study thebehavior of special moment frames (SMFs) with reduced beamsection (RBS) connections. Khandelwal et al. (2008) used reducedmodels to analyze SMFs with RBS connections, as well as in-termediate moment frames (IMFs) with welded unreinforced flangewelded web (WUF-W) connections. Sadek et al. (2008) andAlashker et al. (2010) used both detailed and reduced models tostudy single-plate shear connections, while Khandelwal et al. (2009)and Khandelwal and El-Tawil (2011) used reduced models to an-alyze steel braced frames. While these studies considered columnremoval scenarios, other studies focused on the development ofreduced models to represent the moment-rotation behavior ofconnections. Del Savio et al. (2009) present a reduced modelingapproach to represent the interaction of axial force and bending mo-ment for end-plate and web-cleat connections. Yim and Krauthammer(2010) used a reducedmodeling approach to represent themoment-rotation behavior of welded unreinforced flange bolted web(WUF-B) connections and compared the reduced model pre-dictions with results from detailed models. Alashker et al. (2011)investigated how various levels of approximations in collapsemodeling influence the modeled response.

1Leader, StructuresGroup, Engineering Laboratory, National Institute ofStandards and Technology, 100 Bureau Dr., Stop 8611, Gaithersburg, MD20899-8611. E-mail: fahim.sadek@nist.gov

2Research Structural Engineer, Engineering Laboratory, NationalInstitute of Standards and Technology, 100 Bureau Dr., Stop 8611, Gai-thersburg, MD 20899-8611 (corresponding author). E-mail: joseph.main@nist.gov

3Senior Research Structural Engineer, Engineering Laboratory, NationalInstitute of Standards and Technology, 100 Bureau Dr., Stop 8611, Gaithers-burg, MD 20899-8611. E-mail: hsl@nist.gov

4Professor, Dept. of Civil and Environmental Engineering, Univ. of Mich-igan, Ann Arbor, MI 48109-2125. E-mail: eltawil@umich.edu

Note. This manuscript was submitted on August 5, 2011; approved onMarch 30, 2012; published online on April 3, 2012. Discussion period openuntil June 1, 2013; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Structural Engineering, Vol.139, No. 1, January 1, 2013. ©ASCE, ISSN 0733-9445/2013/1-108e119/$25.00.

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Because of the complexities of the nonlinear behavior thatconnections exhibit under collapse scenarios, physical tests areindispensible for establishing confidence in both detailed andreduced modeling approaches. A companion paper (Lew et al.2013) describes full-scale testing of two steel beam-column assem-blies, each comprising three columns and two beams. One specimenrepresents part of a seismically designed IMF, incorporating WUF-Bconnections, and the other represents part of an SMF, incorporatingRBS connections. In this paper, both detailed and reduced models aredeveloped of these two test specimens, and the computational resultsare comparedwith the experimental measurements. Thefinite-elementanalyses presented are conducted using explicit time integration inLS-DYNA (Hallquist 2007), a general-purpose finite-element softwarepackage. The analyses account for both geometrical and materialnonlinearities, including fracture. In all analyses, the center stubcolumn is pushed downward under displacement control using apredefined displacement time-history curve until failure occurs.Displacements are increased at a slow rate to minimize dynamiceffects, similar to the test conditions.

Description of Test Specimens

The full-scale test specimens described in Lew et al. (2013)consisted of two beam spans connected to three columns usingmoment connections. The span length of the beams (center-to-centerof columns) was 6.10 m. The beams selected for the first specimenwere W21 3 73 sections and were connected toW18 3 119 col-umns using WUF-B connections. The WUF-B connection includeda shear plate (shear tab) that was fillet welded to the column flangeand bolted to the beam web using three 25-mm diameter, high-strength bolts. The beam flanges were joined to the column flangeusing complete joint penetration (CJP) groove welds. Continuityplates were provided for both interior and exterior columns. Thebeams selected for the second specimen were W24 3 94 sectionsand were connected to W24 3 131 columns using RBS con-nections. The beam flanges and web were connected to the columnflange using CJP groove welds. The RBS connections utilizedcircular radius cuts in both the top and bottom flanges of the beam ata distance from the beam-column interface. Continuity plates wereprovided for both the center and end columns, while doubler plateswere required only for the center column. ASTM A992 structuralsteel (Fy 5 345 MPa) was used in all beams, columns, and doublerplates in the panel zone. ASTM A36 steel (Fy 5 248 MPa) wasused for the shear tabs and continuity plates at the connections,and ASTM A490 high-strength bolts were used for the boltedconnections.

For both specimens, the tops of the two end columns were re-strained by two diagonal braces for each column. The diagonalbraces simulated the bracing effect provided by the upper floors ina multistory building. A pair of columns straddling each beam at themidspan provided lateral bracing for the beams. The vertical loadwas applied to the top of the center stub column by a single hydraulicram acting through a load cell and a steel plate. A pair of steel plateson each side of the center stub column restrained out-of-planemotion at the lower end of the center stub column.

Modeling Approach

For each test specimen, both detailed and reduced models weredeveloped. This section describes the general approach used torepresent the various materials and components of the test speci-mens in the detailed and reduced models. Subsequent sectionsprovide further information on the models of each test specimen.

Steel Material Models

Various material models are available in LS-DYNA, which cancapture the nonlinear behavior and failure of the steel components.The primary material model used was a piecewise linear plasticitymodel (material Model 24 in LS-DYNA; Hallquist 2007). In thismodel, an effective stress versus effective plastic strain curve isspecified, along with a plastic strain to failure. Fracture is simulatedusing element erosion, in which elements are removed from themodel (i.e., their stiffness drops to zero) when the specified failurestrain is reached.

For the various types of steel used in the test specimens, thematerial model parameters were developed based on engineeringstress-strain curves obtained from coupon tensile tests for all steelsections and plates. The coupon tensile tests applied the ASTMA370-03a test standard (ASTM 2003). For both the WUF-B andRBS specimens, the coupon tests provided data for beam webs,beam flanges, column webs, column flanges, shear tabs, continuityplates, and any doubler plates.

For the solid and shell elements used in the detailed models,engineering stress-strain curves from coupon tensile tests wereconverted to true stress versus plastic strain curves, and the resultingtrue stress-strain curves were extrapolated linearly beyond the pointof necking onset. The postnecking tangent modulus and the failurestrain ɛf were adjusted to achieve quantitative agreement betweenmeasured and calculated engineering stress-strain curves in thesoftening region beyond the ultimate stress. Because of mesh-sizesensitivity in the modeling of softening behavior, finite-elementmodels of tensile coupons were developed using the same mesh sizeand type (shell or solid element) as those used in the various modelsof the test specimens for each steel type. This approach ensured thatthe measured nonlinear material behavior up to failure was accu-rately captured in the material model.

Because stress calculations for the beam elements used in thereduced models do not incorporate changes in the cross-sectionalarea as a result of axial strain, stress-strain curves for the beammaterial models were defined using engineering stress values, ratherthan true stress values. The corresponding plastic strain values wereadjusted, and therefore the engineering stress-strain curves obtainedcomputationally from uniaxial tension matched those obtained ex-perimentally. Values of ɛf for a particular beam element size weredetermined such that the extension at fracturewas consistent with theresults of a detailed solid or shell element model with a corre-sponding gauge length.

Fig. 1 shows examples of detailed finite-element models of a ten-sile coupon obtained from the flanges of the W213 73 beam sec-tion used in the WUF-B specimen. In modeling the beam flanges inthe WUF-B specimen, the detailed solid/shell element model used

Fig. 1. Sample finite-element models of tensile coupons: (a) solidelements; (b) shell elements

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two different mesh types: (1) a solid element mesh with an elementsize of approximately 4.6 mm, typical of that used in the vicinity ofthe connections, and (2) a shell element meshwith an element size ofapproximately 19 mm, typical of that used away from the con-nections. Fig. 2(a) shows the true stress-true strain curves used asinput for the beam flange steel for the solid, shell, and beam elementmodels. As is evident in the figure, coarser meshes entail the use ofsmaller values of ɛf to achieve consistent results. The true stressversus plastic strain curve used for the shell elements is extrapolatedhorizontally beyond the point of necking onset, while the curve usedfor the solid elements is extrapolated tangentially to provide the bestmatch to the tensile test data. Fig. 2(b) presents the measured en-gineering stress-strain curve from the coupon test samples alongwith the analysis results of the solid and shell element models of thecoupons. Strain values in the figure correspond to a gauge lengthof 51mm, and results from the beam elementmodel are not included,because they correspond to a longer gauge length. The comparisonbetween themeasured and calculated stress-strain behavior indicatesthat the material models capture the constitutive relationships for thesteel of the beam flange with reasonable accuracy. The processshown was repeated for all coupon tests to develop the constitutivematerial models for all steels used in the specimens corresponding tothe various element types and sizes considered in the finite-elementmodels. For the reduced models, in which wide flange sections wererepresented by beam elements, material models were based onstress-strain data from the flanges, rather than the webs.

High-Strength Bolts

The bolts connecting the shear tabs to the beam webs in the WUF-Bspecimen were 25-mm diameter ASTM A490 high-strength bolts.In the detailed model of the WUF-B specimen, the bolts were rep-resented using solid elements with a piecewise linear plasticitymaterial model (material Model 24 in LS-DYNA; Hallquist 2007),which was also used for the steel sections and plates. The stress-strain relationship for the bolt material was calibrated to matchexperimental data reported in Kulak et al. (1986) using a solid-element model of a double-shear bolt test. The finite-element meshused to model the bolt in the double-shear test was identical to thatused in the detailed model of the WUF-B specimen. Because ofsymmetry, only one-half of the double-shear test specimen wasconsidered in the analysis, with appropriate boundary conditions onthe plane of symmetry. Prestressing of the bolts was not considered,because test data (Kulak et al. 1986) and additional analyses showed

that prestressing slightly affects the initial response of the bolt inshear, but does not significantly affect the ultimate behavior orfracture of the bolt. In this model, contact was defined between thebolt and the plates to model the transfer of forces through the boltedconnection, including friction and bolt bearing. Fig. 3(a) shows thecalculated load-deformation curve along with the experimental datafrom Kulak et al. (1986), while Fig. 3(b) shows a section view of thefinite-element model at the ultimate load. Fig. 3(c) shows the solidelement mesh of the bolt after fracture. The load values reportedin Fig. 3(a) are for single shear. For the calculated shear load-deformation curve, the deformations reported are those for the boltonly (between the bolt head and shank), whereas the experimentaldata includes the test jig deformations. That may explain the dif-ference in elastic stiffness between the measured and calculatedresponses. Overall, the agreement between the measured andcalculated responses was good.

Fig. 2. (a) True stress-strain curves for the solid, shell, and beam elements; (b) engineering stress-strain curves from coupon models and tests(beam flange steel from the welded unreinforced flange bolted web specimen)

Fig. 3. (a) Measured and calculated shear load-deformation curves fora 25-mm A490 bolt; (b) section view of finite-element model at theultimate load; (c) bolt mesh after fracture

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In the reduced model of the WUF-B specimen, spring elementswere used to represent the shear behavior of the bolts, along withbearing-induced deformations of the shear tab and beam web, usingthe piecewise linear shear load-deformation curve shown inFig. 4(a). This simplified piecewise-linear curve is based on theresults of a detailed solid-element model of the bolted lap joint il-lustrated in Fig. 4(b), and the shear load-deformation curve obtainedfrom this model is also shown in Fig. 4(a). The two plates in thismodel correspond to the shear tab and the beam web used in thedetailed model of the WUF-B specimen, which are about half asthick as the plates used in the double-shear test [Fig. 3(b)]. Thefailure displacement obtained from this single-shear model is morethan twice the failure displacement obtained from the double-shearmodel considered previously (Fig. 3), mainly because of moreextensive bearing-induced deformations of the thinner plates, whichalso permitted rotation of the bolt shank. These effects are evidentin Fig. 4(b).

Panel Zones

While shear deformations of the panel zones were modeled ex-plicitly in the detailed models using solid or shell elements, springelements were used to represent the shear behavior of the panelzones in the reduced models. Diagonal springs representing thepanel zones had an elastic-perfectly plastic load-deformation curvebased on the stiffness and strength of the panel zone (Khandelwalet al. 2008). The stiffness of the panel zone spring, kpz, was definedusing equilibrium relationships that equate the response of the panelzone spring model with the column web in the panel zone regiondeforming in pure shear as

kpz ¼ G�dc 2 tcf

�tpz�

db 2 tbf�cos2 u

ð1Þ

where dc 5 column depth, db 5 beam depth, tcf 5 column flangethickness, tbf 5 beam flange thickness, tpz 5 panel zone thickness,

u5 inclination angle of the diagonal spring from the horizontal, andG5 shear modulus of steel. Using similar equilibrium relationshipsand the panel zone strength equation in the AISC seismic provisions(AISC 2005a), the yield capacity of the panel zone spring is

fpz ¼ 0:6Fydctpzcos u

"1 þ 3bcf t2cf

dbdctpz

#ð2Þ

where bcf 5 column flange width. Because the panel zone region isvery ductile, it is unlikely that it will fail prior to other components inthe model. Therefore, a failure deformation limit on the panel zonespring was not enforced.

Diagonal Braces

In the experimental setup for both specimens (Lew et al. 2013), twodiagonal braces were rigidly attached to the top of each end columnto simulate the bracing effect provided by the upper-floor framingsystem in a multistory building. Because the details of the brace andits connections were not of primary importance to the computationaleffort, these diagonal braces were represented using an arrangementof spring elements in all models. The load-deformation character-istics of these springs, for each specimen, were estimated based onthe axial force and axial displacement (or shortening) of each bracedetermined from the experiment (Sadek et al. 2010). Fig. 5 shows theload-deformation curves used to represent the diagonal braces foreach specimen. Nonlinear behavior of the braces is evident, mostlikely associated with yielding and slippage where the braces wereconnected to the strong floor of the testing facility.

Finite-Element Models of a Welded UnreinforcedFlange Bolted Web Specimen

Two finite-element models of theWUF-B specimenwere developedto study the response characteristics of the specimen and to comparethe calculated response with that measured during the test. The firstwas a detailed model of the specimen with approximately 300,000solid and shell elements, while the second was a reducedmodel withabout 150 beam and spring elements. The subsequent sectionsprovide descriptions of the models and the analysis results.

Fig. 4. (a) Shear load-deformation curves for bolted lap joint;(b) section view of finite-element model showing deformations at theultimate load

Fig. 5. Axial load-displacement curve for diagonal braces for thewelded unreinforced flange bolted web and reduced beam sectionspecimens

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Detailed Model

An overview of the detailed model used in the analysis of theWUF-B specimen is shown in Fig. 6(a). Because of symmetry,only one-half of the beam-column assembly with appropriateboundary conditions was modeled. The detailed model consisted offinely meshed solid (brick) elements representing the beams,columns, continuity plates, shear tabs, bolts, and welds in the vi-cinity of the connection [Fig. 6(b)]. Four layers of solid elementswere used for all beam and shear tab plates andwelds, which resultedin an element size in the range of 2. 824. 8 mm. Two to 4 layers ofsolid elements were used for all column and continuity plates, whichresulted in an element size in the range of 5. 62 4. 8 mm. Solidelements with a minimum dimension of 1.8 mm were used for thebolts. Contact was defined between the bolts, shear tabs, and beamwebs to model the transfer of forces through the bolted connections,including friction and bolt bearing. Hand calculations showedthat the stresses in the fillet weld connecting the shear tab to thecolumn flange were low compared with their strength. As a result,the shear tab wasmodeled as rigidly connected to the column flange.In addition, no weld-induced residual stresses were considered forthe CJP groovewelds between the beam flanges and column flanges.Such residual stresses could potentially lead to earlier yielding of

the beam flange, but the ultimate axial and flexural capacities ofthe section would not be altered (Bruneau et al. 2011).

Away from the connection zones, the beam and columns weremodeled with shell elements (Fig. 6). Appropriate constraints wereimposed at the interface between the shell and solid elements toensure that nodes on the solid elements remained in the same di-rection as the fiber of the shell elements. Spring elements were usedto model the diagonal braces at the top of the end columns, using thenonlinear load-displacement curve shown in Fig. 5. All nodes werefixed at the bases of the end columns. Similar to the test configu-ration, out-of-plane lateral displacements of the beam flanges wereconstrained at the midspan.

The deflected shape of the WUF-B specimen is shown inFig. 7(a). The beam remained essentially elastic between con-nections, as evidenced by the nearly linear displacement profile ofthe beam span in Fig. 7(a). Plastic deformations, including axialextension and flexural deformation, were concentrated in the con-nection regions, where significant yielding was observed, as illus-trated by the contours of the plastic strain shown in Fig. 7(b) for theconnection to the center column. The failuremode of the connection,shown in Fig. 7(b), was very similar to that observed in the ex-periment (Lew et al. 2013). The lowermost bolt connecting the beamweb to the shear tab at the center column failed in shear at a vertical

Fig. 6. (Color) Detailed model of the welded unreinforced flange bolted web specimen: (a) overview; (b) connection to center column

Fig. 7. (Color) Detailed model results for the welded unreinforced flange bolted web specimen: (a) deflected shape at a center column displacementof 512 mm; (b) failure mode at the connection to the center column

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displacement of the center column of about 445 mm, which wasimmediately followed by failure of the middle bolt. The bottom flangeof the beam near the weld access hole began to fracture at a centercolumn vertical displacement of approximately 483 mm. The fractureinitiated at the root of the access hole (center of flange) and propagatedoutward until the bottom flange completely fractured.

Reduced Model

The reduced model of the WUF-B specimen, shown in Fig. 8, usedHughes-Liu beam elements (Hallquist 2007) to represent the beamsand columns, as well as the shear tabs and beam flanges in theconnection regions. The beam elements used cross section in-tegration with the proper cross-sectional geometry defined for eachcomponent. A piecewise linear plasticity model was used to rep-resent the steel materials, with stress-strain curves based on tensiletest data and fracture modeled using element erosion, as previouslydiscussed. An arrangement of beam and spring elements connectedwith rigid links was used to model the WUF-B connections, asshown in the figure. Zero-length spring elements (shown with finitelength in Fig. 8 for clarity) were used to model the single shearbehavior of the bolts, including bearing-induced deformations of theshear tab and beamweb, using the load-displacement curve shown inFig. 4(a). These elements are capable of representing bolt shear inany direction in the vertical plane, allowing modeling of combinedvertical and horizontal shear. Spring elements were also used tomodel the diagonal braces (Fig. 5) and the shear behavior of the panelzone [Eqs. (1) and (2)].

Two analyses were conducted in which the bases of the endcolumns were modeled as either fixed or pinned, representing lim-iting cases for column fixity. The deflected shape of the WUF-Bspecimen based on the reduced model with fixed column bases isshown in Fig. 9(a), while the corresponding failure sequence of thecenter column connection is indicated in Fig. 9(b). The lowermostbolt connecting the beamweb to the shear tab failedfirst, followed bythe nearly simultaneous failures of the middle bolt and the bottomflange of the beam. This failure sequence is very similar to what wasobserved in both the experiment and the detailed model.

Comparison with Experimental Measurements

Fig. 10(a) shows the applied vertical load, Fig. 10(b) shows the beamaxial force, Fig. 10(c) shows the horizontal (inward) displacement ofthe end columns at beam midheight, and Fig. 10(d) shows the endcolumn axial force, all plotted against the vertical displacement ofthe center column. These plots compare the results of the detailed andreduced finite-element models, with experimental measurements

described in the companion paper (Lew et al. 2013) showing goodagreement between the experimental and computational results.While the reduced model uses far fewer elements than the detailedmodel, the results show that it is capable of capturing the primaryresponse characteristics of the test specimen.

Consistent with the experimental results, the results of the de-tailed and reduced models shown in Fig. 10(a) indicate that theassembly remained in the elastic range up to a vertical displacementof the center column of about 50 mm. The analyses indicate that inthe early stages of the response the behavior was dominated byflexure, as indicated by the axial compression in the beams shown inFig. 10(b). With increased vertical displacement, axial tensiondeveloped in the beams, and the behaviorwas dominated by catenaryaction. The axial tensile force in the beams increased with increaseddownward displacement of the center column, as Fig. 10(b) indi-cates, until the connection could no longer carry the combinedaxial and flexural stresses, resulting in the failure of the assembly.The development of axial tension in the beams was associated withincreased inward displacement of the end columns at beam mid-height, as shown in Fig. 10(c). Similar to the experimental results,the results of the detailed and reduced models shown in Fig. 10(d)indicate that in the initial stages of the response, the end columnswere in compression, but with increased vertical displacement, axialtension developed in the end columns, as a result of the largecompressive axial loads carried by the diagonal braces.

Fig. 8. Reduced model of the welded unreinforced flange bolted web specimen: (a) overview; (b) connections to the center column

Fig. 9. (Color) Results from the reduced model of the welded un-reinforced flange bolted web specimen (fixed columns): (a) deflectedshape at a center columndisplacement of 511mm; (b) failure sequence atthe connection to the center column

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Fig. 11 shows the bending moment-axial force interaction diagramfor the WUF-B connection based on the results of the detailed andreduced models. For the reduced model, only the results with fixedcolumn bases are presented for clarity, because the results with pinnedcolumn bases are almost indistinguishable. The bending moment andaxial force were calculated at the face of the center column, wherefailure of the connection was observed. Superimposed on the con-nection response are the limiting bending moment-axial tension in-teraction diagrams for the gross beam cross section (W213 73),calculated based on the actual yield and ultimate strengths of the beammaterial. The limiting interaction diagrams for the beam cross sectionwere calculated based on Chapter H of ANSI/AISC 360-05 (AISC2005b). The figure shows that the capacity of the WUF-B connectionexceeds the yield capacity of the beam gross cross section undercombined axial and flexural loads, but is somewhat less than the ul-timate strength of the beam.

Finite-Element Models of Reduced Beam SectionSpecimen

Similar to the WUF-B specimen, two finite-element models of theRBS specimen were developed to study the response characteristicsof the connections and to compare the calculated response with thatmeasured during the experiment. The first was a detailed model ofthe assembly with approximately 235,000 shell elements, while thesecond was a reduced model with about 130 beam and spring ele-ments. The following sections provide descriptions of the modelsand the analysis results.

Detailed Model

An overview of the detailed model of the RBS specimen is shown inFig. 12. The model consisted of shell elements representing thecolumns, beams, continuity and doubler plates, and welds. In thevicinity of the reduced section, the beams and columns weremodeled using a fine shell element mesh with an element size ofabout 6.4 mm. Away from the connection zones, the beams and

Fig. 10. (a) Applied vertical load; (b) beam axial force; (c) horizontal displacement of the end column at beam midheight; (d) end column axial forceversus center column displacement for the welded unreinforced flange bolted web specimen (estimated coefficient of variation in measurements: 1%)

Fig. 11. Axial load versus bending moment for the welded un-reinforced flange bolted web connection to the center column

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columns were modeled using a coarser mesh with an element size of25 mm. Spring elements were used to model the diagonal braces atthe top of the end columns, using the nonlinear load-displacementcurve shown in Fig. 5. All nodes were fixed at the bases of theend columns. Similar to the test configuration, out-of-plane lat-eral displacements of the beam flanges were constrained at themidspan.

The deflected shape of the RBS specimen based on the detailedanalysis is shown in Fig. 13(a). Only half of the model is shown inFig. 13(a) because of the symmetry of the deflected shape. Thebeams remained essentially elastic between the reduced sections, asevidenced by the nearly linear displacement profile of the beam spanin Fig. 13(a). Plastic deformations, including axial extension andflexural deformation, were concentrated in the reduced sections,where significant yielding was observed, as illustrated by thecontours of the plastic strain shown in Fig. 13(b) for the reduced

section near the center column. The failure mode from the analysis,shown in Fig. 13(b), was very similar to that observed in the ex-periment (Lew et al. 2013). The failure was characterized by thefracture of the bottom flange in the reduced section near the centercolumn, which displaced about 838 mm. The fracture immediatelypropagated through the web until the vertical load-carrying capacityof the specimen was lost.

Reduced Model

The reduced model of the RBS specimen, shown in Fig. 14, con-sisted of Hughes-Liu beam elements representing the beams andcolumns in the specimen. In addition, each RBS was modeled usingfive beam elements with varying section properties. The beam ele-ments used cross section integration with the proper cross-sectional

Fig. 12. (Color)Detailedmodel of the reduced beam section specimen: (a) overview; (b) reduced beam section near the connection to the center column

Fig. 13. (Color) Detailed model results for the reduced beam section specimen: (a) deflected shape at a center column displacement of 875 mm; (b)failure mode in the reduced section near the center column

Fig. 14. Reduced model of the reduced beam section specimen: (a) overview; (b) connection to the center column showing the reduced beam section

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geometry defined for each type of section. A piecewise linearplasticitymodelwas used to represent the steelmaterials, with stress-strain curves based on tensile test data and fracture represented usingelement erosion, as previously discussed. Spring elements were usedto model the diagonal braces (Fig. 5) and the shear behavior of thepanel zone [Eqs. (1) and (2)]. Both fixed and pinned bases wereconsidered for the end columns.

In general, the results from the reduced model were consistentwith those from the detailed model. The deflected shape of the RBSspecimen based on the reduced model with fixed column bases isshown in Fig. 15. The failure mode of the specimen based on thereduced model (for both fixed and pinned end columns) was con-sistent with that observed in the experiment and the detailed model.The beam element at the center of the reduced section experienced

large strains, associated with combined flexural and axial tensilestresses. Once a critical strain was reached, the element was eroded,resulting in the failure of the assembly.

Comparison with Experimental Measurements

Figs. 16 and 17 compare the results from the detailed and reducedfinite-element models with experimental measurements described inthe companion paper (Lew et al. 2013). Fig. 16(a) shows the appliedvertical load, Fig. 16(b) shows the beamaxial force, Fig. 16(c) showsthe end column axial force, and Fig. 16(d) shows the axial com-pression in each diagonal brace, all plotted against the verticaldisplacement of the center column. Fig. 17(a) shows the horizontal(inward) displacement of the end columns at beam midheight, andFig. 17(b) shows the horizontal displacement at the top of the endcolumns, also plotted against the vertical displacement of the centercolumn. Good agreement is observed between the experimental andcomputational results, and the results show that the reduced model,with far fewer elements than the detailedmodel, is able to capture theprimary response characteristics of the test specimen. The reducedmodel results with fixed and pinned column bases generally bracketthe experimental results.

Consistent with the experimental results, the results of the de-tailed and reduced models shown in Fig. 16(a) indicate that theassembly remained in the elastic range up to a vertical displacementof the center column of about 50 mm. The analyses indicate that inthe early stages of the response, the behavior was dominated byflexure, as indicated by the slight compressive axial forces in the

Fig. 15. (Color) Deflected shape from the reduced model of the re-duced beam section specimen (fixed columns) at a center column dis-placement of 867 mm

Fig. 16. (a) Applied vertical load; (b) beam axial force; (c) column axial force; and (d) axial compression in the brace versus center columndisplacement for the reduced beam section specimen (estimated coefficient of variation in measurements: 1%)

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beams shown in Fig. 16(b). With increased vertical displacement,tensile axial forces developed in the beams, and the behavior wasdominated by catenary action. As Fig. 16(b) indicates, the axialtensile force in the beams increased with increased downwarddisplacement of the center column until the connection could nolonger carry the combined axial and flexural stresses, resulting infracture of the bottom flange at the center of the reduced section.Similar to the experimental results, the results of the detailed andreduced models shown in Fig. 16(c) indicate that in the initial stagesof the response (when the response was dominated by flexure), theend columns were in compression. With increased vertical dis-placement, axial tension developed in the end columns as a result ofthe large compressive axial loads carried by the diagonal braces, asshown in Fig. 16(d). The development of axial tension in the beamswas associated with increased inward displacement of the endcolumns at beam midheight, as shown in Fig. 17.

Fig. 18 shows the bending moment-axial force interaction dia-gram for the RBS connection based on the detailed and reducedmodels. For the reduced model, only the results with fixed columnbases are presented for clarity, because the results with pinnedcolumn bases are very similar. The bending moment and axial forcewere calculated at the center of the reduced section near the centercolumn, where failure of the connection was observed. Super-imposed on the connection response are the limiting bendingmoment-axial tension interaction diagrams for the gross beamcross section (W243 94) and for the RBS calculated based on theactual ultimate strength of the beam material. The limiting in-teraction diagrams were calculated based on Chapter H of ANSI/AISC 360-05 (AISC 2005b) using yield and ultimate stress valuesobtained from coupon tensile tests. Fig. 18 shows that the capacity ofthe connection slightly exceeds the capacity of the reduced sectionunder combined axial and flexural loads, as calculated based onANSI/AISC 360-05 (AISC 2005b), but is less than the calculatedultimate capacity of the gross beam cross section.

Influence of Column Restraint Conditions

In the test setup for both assemblies, illustrated in Fig. 19(a), di-agonal braces were used to provide lateral restraint to the columntops to enable the development of catenary action. These braceswereintended to represent the lateral restraint provided in a configurationsimilar to that illustrated in Fig. 19(b). In this configuration, the endcolumns extend 1 story (4.19 m) above and below the beam level,

and the tops and bottoms of the end columns are fixed to representthe restraint provided by the upper- and lower-floor framingsystems. To assess how closely the test setup in Fig. 19(a) rep-resents the configuration in Fig. 19(b), both the WUF-B and RBSassemblies were analyzed in the configuration of Fig. 19(b) usingthe reduced models developed in this study. Fig. 20 shows thevertical load versus the vertical displacement of the center columnfor the WUF-B assembly [Fig. 20(a)] and the RBS assembly[Fig. 20(b)] in the two configurations shown in Fig. 19. For both theWUF-B and the RBS assemblies, good agreement is observedbetween the results computed for the two configurations, con-firming that the column restraint conditions in the two config-urations yield consistent results.

Summary and Conclusions

This paper presented a computational assessment of the perfor-mance of beam-column assemblies with two types of momentconnections under vertical column displacement. The connectionsconsidered include a WUF-B connection and an RBS connection.The study considered two levels of modeling complexity: (1) de-tailed models with a large number of elements, primarily solid and

Fig. 17. Horizontal displacement of an end column at (a) beam midheight and (b) the column top versus center column displacement for the reducedbeam section specimen (estimated coefficient of variation in measurements: 1%)

Fig. 18.Beam axial load versus bending moment for the reduced beamsection near the center column

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shell elements; and (2) reduced models with a limited numberof elements, primarily beam and spring elements. The analysesconducted using thesemodels provided insight into the behavior andfailure modes of the connections, including their capacity to sustaintensile forces that developed in the beams.

For both the WUF-B and RBS assemblies, the analyses showedan initial elastic response dominated by flexural behavior. Withincreased vertical displacement, yielding occurred at the beam-to-column connections, and axial tension developed in the beams.The analyses confirmed that the moment connections were able tosustain significant axial forces after the formation of plastic hinges,and thus were able to resist the applied vertical loads throughcatenary action. The ultimate fracture of the connections in both

assemblies was because of combined axial and flexural stressesassociated with increasing axial tension in the beams. Comparisonof the computed bending moment and axial force values with theinteraction equation for the beam cross sections showed that themoment connections were capable of developing a significantfraction of the cross-sectional capacity of the beams under combinedaxial and flexural loads.

Comparison of the computational results with experimentalmeasurements described in the companion paper (Lew et al. 2013)showed good agreement. Both the detailed and reducedmodels werecapable of capturing the primary response characteristics and failuremodes, providing validation of the modeling approaches. The re-ducedmodels developedwill be valuable in the analysis of complete

Fig. 19. Alternate configurations for the restraint of the end columns: (a) diagonal braces; (b) restraint at the upper-floor level

Fig. 20. Influence of column restraint conditions on vertical load versus center column displacement: (a) welded unreinforced flange bolted webspecimen; (b) reduced beam section specimen

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structural systems for assessing the reserve capacity and robustnessof building structures.

Disclaimer

Certain commercial entities, equipment, products, or materials areidentified in this document to describe a procedure or conceptadequately. Such identification is not intended to imply recommen-dation, endorsement, or implication that the entities, products, mate-rials, or equipment are necessarily the best available for the purpose.

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