criteria for balanced design of diagonally braced moment resisting frames based on hierarchical...

Criteria for balanced design of diagonally braced moment resistingframes based on hierarchical yielding and failure sequencesand their application

M. Lotfollahi a,1, M.M. Alinia a,2, E. Taciroglu b,⇑,2

a Department of Civil Engineering, Amirkabir University of Technology, Tehran, Iranb Department of Civil and Environmental Engineering, University of California, Los Angeles, CA 90095, USA

a r t i c l e i n f o

Article history:Received 19 December 2013Revised 23 July 2014Accepted 8 December 2014Available online 14 February 2015

Keywords:Diagonally braced moment resisting frameGusset plateBalanced designEnergy dissipationDuctilityBucklingYield mechanismFailure modePlasticityFinite element

a b s t r a c t

Inelastic behavior of diagonally braced moment resisting frame (DBMRF) dual systems are investigated todetermine their yield mechanisms and failure modes, and to quantify the load sharing between the momentframe and the gusset–brace subsystems. An improved performance is sought through new balanced designcriteria that will increase a DBMRF system’s ductility, and permit yielding in multiple secondary stages atselected performance levels. The presented balanced design approach is based on a non-dimensional for-mulation, which addresses both tensile yielding and compressive buckling phases of DBMRF systemsand considers the participation of all system constituents. The satisfaction of the proposed balanced designcriteria is achieved through parametric studies carried out with high-fidelity three-dimensional finiteelement (FE) models that are globally and locally validated/verified against experimental data/numericalsimulations available in open literature. Using the validated FE models, the collapse behavior of a represen-tative set of DBMRF systems are examined, and the influences of the brace elements’ demand-to-capacityratios, as well as the gusset plate connection types/sizes on the yield mechanisms and the failure modes arescrutinized. Both pushover and cyclic analyses are carried out; and the overall system ductility and energydissipation values are investigated for different width-to-height ratios. The responses of system constitu-ents are evaluated using the specified, as well as the expected material properties. The worked examplesclearly demonstrate the utility of the proposed balanced design criteria in improving the DBMRF systems’ductility, and in avoiding premature failure modes.

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1. Introduction

The key aspect of a braced moment resisting frame (BMRF) sys-tem is its dual character, which is manifested through the behaviorand interaction of its two subsystems—viz., the moment frame andthe gusset–brace systems. The collapse mechanisms, energy dissi-pation, and ductility of a BMRF system can be controlled throughdesign; and thus, BMRF systems can be gainfully utilized to meetthe most severe seismic performance objectives. BMRFs come ina variety of different configurations and geometries, but typicallycomprise diagonal bracing members connected to a primarymoment frame system with gusset plate connections. Duringearthquake loading, the braces must be capable of sustaining

multiple cycles of inelastic tensile yielding as well as compressivebuckling without any significant deterioration in their stiffness orstrength. These primary mechanisms should be balanced withthe other (complementary) ductile mechanisms of the system, sothat the frame can tolerate inelastic deformations and dissipateenergy, while various undesirable failure modes (e.g., failure ofthe gusset-to-brace connections) are avoided.

Small inter-story drifts that occur during the initial stages of acyclic lateral loading scenario (e.g., minor earthquakes) can beaccommodated by the brace elements. Brace buckling and initialyielding occur as the lateral loads increase (e.g., moderate earth-quakes), and this behavior provides some energy dissipation. Addi-tional lateral loading can initiate plastic hinge formation in themiddle, and subsequently, at both ends of the brace element onthe gusset plates in the compression phase, and propagation ofyielding within the brace element during the tension phase of defor-mations. At this stage, the frame members become more active; butideally they will remain in the state of immediate-occupancy (IO)performance level. Further increases in lateral loads (e.g., severe

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⇑ Corresponding author. Tel: +1 310 267 4655; fax: +1 310 206 2222.E-mail address: [email protected] (E. Taciroglu).

1 Ph.D. Candidate. Also visiting Ph.D. Student at the University of California, LosAngeles (UCLA).

2 Professor.

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earthquakes) lead to the formation of plastic hinges in the panelzones and within the frame members. The aforementionedsequence of events represents the intended functions of the gus-set–brace and the moment frame systems for meeting life-safety(LS) and collapse-prevention (CP) performance objectives, andassures ductile response in the BMRF systems. Presently, the ANSISeismic Design Provisions [1] for Special Concentrically BracedFrames (SCBFs) and Special Moment Resisting Frames (SMRFs) stip-ulate the aforementioned performance objectives for such systems.

In a BMRF system, the overall performance can be significantlyinfluenced by the nonlinear behavior of its gusset–brace subsystem.In a general sense, the brace members are expected to exhibitductile behavior under earthquake-induced lateral motions. Numer-ous research efforts [2–14] have experimentally and analyticallyexplored the cyclic behavior of steel brace members and the find-ings are recognized in modern seismic design provisions. On theother hand, gusset-to-brace connections are required to exhibithigher capacity than the demands exceeding the capacity of thebrace members. Current provisions for gusset-to-brace connectionsare based on several experimental and numerical studies [15–24]wherein complete frame-action is excluded through inelasticpost-buckling deformations of the brace elements. Recent studies[25,26] showed that the existing provisions on gusset plate connec-tions may lead to unintended responses, and offered a modificationin the form of a ‘‘balanced design approach’’. This approach enablesthe connections to be properly designed such that undesirable fail-ure modes are suppressed, yielding through a secondary yieldmechanism in the gusset plate connections are assured, and onlythe desired failure modes in the gusset–brace system are observed.The behavior of gusset plate for buckling-restrained braces [27] andthe uses of low yield point and stainless steel gusset plate connec-tions [28,29] have also been the subject of other studies in this area.

The seismic design requirements of a BMRF system are mainlyaffected by the cyclic behavior of the complete frame system. Preli-minary studies had revealed the complex panorama of asymmetriccyclic behavior of braced frame systems, which is primarily due tothe alternating tensile yielding and compressive buckling responsesof their brace members [30–34]. In those studies, the braces’ width-to-thickness and the effective slenderness ratios were identified asthe main system parameters that control, respectively, the energydissipation capacity and resistance to local buckling. The post-buck-ling regime was reasonably bracketed, and then handled by applyinga buckling reduction factor to the brace element’s compressivestrength. In recent years, a number of studies have focused on theevaluation of the effects of different corner and mid-span gussetplate connection sizes and types on the overall performance of thebraced frame systems. These investigations have finally led to theproposition of a new elliptical clearance requirement in the designof gusset plate connections [35–37]. More recently, improved ana-lytical models for brace members have been presented in [38]. Whilethese aforementioned studies have improved the performance ofbraced frame systems, there is yet no work—to the best of theauthors’ knowledge—that comprehensively evaluates the overallnonlinear inelastic responses of BMRFs. The studies that exist haveconcentrated either on moment frames with tension-brace action[39], or those with buckling-restrained braces [40].

In the present study, we sought to evaluate multiple secondaryyield mechanisms and probable failure modes in diagonally bracedmoment resisting frame (DBMRF) systems under cyclic loads, aswell as within the tensile yielding and the compressive bucklingregimes. Since both the local and the global responses of a DBMRFsystem highly depend on the nonlinear behavior of its gusset–brace subsystem, accurate modeling of that behavior is critical.To that end, we developed highly detailed three-dimensional finiteelement models, and validated and verified them—both in terms ofglobal and local responses—using published data from both a

comprehensive experimental program involving 13 large-scalespecimens, which was carried out by Lehman et al. [25], and acompanion numerical study, which was executed by Yoo et al.[35]. We then utilized the validated models (and the analysis pro-cedures) in subsequent parametric sensitivity studies to identifyDBMRF systems’ collapse behavior by assessing their yield mecha-nisms and failure modes.

We are also proposing herein, a set of new and improved bal-anced design criteria for DBMRF systems, which prolong yieldingin the main frame system through multiple secondary yield mech-anisms, so that the system’s ductility is enhanced and its drift capacityis increased. The proposed criteria are based on a non-dimensionalformulation; address both tensile yielding and compressive buck-ling phases of DBMRF systems; and consider the participation ofall system constituents. Derivation of the said criteria involved bothcyclic and monotonic (pushover) parametric studies on DBMRFmodels with varying gusset-to-brace connection sizes and typesas well as different frame geometries and specifications. In theseanalyses, the DBMRFs’ ductilities and energy dissipation valuesare calculated, and the effects of expected yielding behavior in eachmodel are evaluated. Utilizing those results, we offer in this study:

(1) New hierarchical yielding/failure sequence criteria that sup-press the undesirable failure modes entirely and balance theprimary yield mechanism against a number of multiple sec-ondary yield mechanisms and desirable failure modes.

(2) A quantified understanding of the interplay between thedegree of brace-to-frame rigidity and the brace element’sdemand-to-capacity ratio (this issue is examined through theDBMRFs’ in-plane stiffness, and the brace-to-frame contributionshares in story shears at different stages of lateral loading).

2. Proposed balanced design criteria

To avoid premature (i.e., joint) failures in SCBF systems, Lehmanet al. [25] proposed the following expressions (cf., Eqs. (1) and (2)in [25]):Ryield;mean ¼ RyRyield 6 by1Ry1Ryield;1

6 by2Ry2Ryield;2 6 " " " 6 byiRyiRyield;i ð1Þ

Ryield;mean ¼ RyRyield 6 by1Ry1Ryield;1 6 bfail;1Rfail;1

6 bfail;2Rfail;2 6 " " " 6 bfail;iRfail;i ð2Þ

which separate and order the possible yield mechanisms and prob-able failure modes in the gusset-brace system. Here, Ry denotesratio of the expected yield stress to the minimum specified yieldstress, and Ryield,mean denotes the primary yield resistance. The nom-inal resistances for various secondary yield mechanisms (Ryield,i) anddifferent failure modes (Rfail,i) are separated by balancing factors (byi

and bfail,i) in order to control the resistance of possible secondaryyield mechanisms and to maintain a balanced state through theprobable failure modes. It is useful to note here that the b factorsin Eqs. (1) and (2) are essentially all equal to zero within the exist-ing Load and Resisting Factor Design (LRFD) approach [41].

The b factors considered in the present study are intended forductility evaluation and nonlinear displacement estimation for anentire DBMRF system. Hence, the above states are substituted withthe following expressions:

Dy 6 Dy1 6 Dy2 6 " " " 6 Dyði%1Þ 6 Dyi

& 1 6 ly1 6 ly2 6 " " " 6 lyði%1Þ 6 lyi ð3Þ

Dy 6 Dy1 6 Dy2 6 " " " 6 Dyði%1Þ 6 Dyi 6 Df 1 6 Df 2 6 " " " 6 Dfi

& 1 6 ly1 6 ly2 6 " " " 6 lyði%1Þ 6 lyi 6 lf 1 6 lf 2 6 " " " 6 lfi ð4Þ

M. Lotfollahi et al. / Engineering Structures 87 (2015) 198–219 199

which are unique to the present study. These inequalities imply thatthe primary yield mechanism displacement (Dy) is followed bymultiple secondary yield mechanisms—i.e., by a first secondarymechanism displacement (Dy1), which is followed by a second sec-ondary mechanism displacement (Dy2), and so on. All the multiplesecondary yield mechanism displacements (Dyi) are required tooccur in a particular (balanced) sequence—this point is elaboratedin detail later in the present study. Moreover, they need to be lessthan the potential failure mode displacements (Dfi) of theDBMRF—a performance goal that, incidentally, was also suggestedby Lehman et al. [25]. The proposed balanced design procedure pre-sented here additionally requires that the calculated ductilities ofall failure modes (lfi) exceed the ductility of the multiple secondaryyield mechanisms (lyi), and that less favorable failure modes withsmaller probabilities of occurrence to have greater separations(i.e., greater lf values) than the more favorable failure modes.

It is expedient to note here that the nonlinear performance andthe yield mechanisms of DBMRF systems are mostly affected bythe compression phase. The criteria presented by Lehman et al.[25], and later used by Roeder et al. [26], address only the tensileyielding phase with a single secondary yield mechanism, whichfocus solely on the gusset–brace system. In contrast, the approachpresented here addresses both tensile yielding and compressivebuckling phases, and evaluates the participation of all system con-stituents within the moment frame and the gusset–brace systemsto multiple secondary yield mechanisms under monotonic (push-over) as well as cyclic loads. Moreover, the proposed balanceddesign methodology is based on a non-dimensional formulationand makes use of integrated/non-dimensional response metrics—e.g., ductility (as opposed to resistance), energy dissipation ratio,and participation share of story shear. This format renders themethodology applicable to a broad range of DBMRF systems.

3. Finite element models and their validation and verification

The values of ductility, energy dissipation, and elastic/inelasticload shares of story shear for the separation required by the bal-anced design criteria proposed above can be obtained throughfinite element (FE) analyses. Models and procedures used in theseFE analyses should be globally and locally validated/verifiedagainst available experiments/simulations, and should be basedon first principles so that they can be confidently used as predictivetools. The ductility value for each possible secondary yield mecha-nism and probable failure mode can be calculated through numer-ical simulations, and the expected ductility can be increased ordecreased to reach the desirable outcomes by varying the designparameters. In addition, energy dissipation and participation share

of story shear are assessed at each DBMRF nonlinear performancelevel. In the present study, the FE analysis package ABAQUS [42]was used in all nonlinear monotonic (pushover) and cyclic as wellas linear modal analyses. The details of the modeling techniquesand method of analyses are omitted here for brevity and are pro-vided—along with their validation using experimental data by Leh-man et al. [25]—as Appendix A to this paper.

4. The model matrix

The results presented in Appendix A section indicate that it ispossible (and fairly straightforward) to extend and apply the vali-dated FE modeling and analysis techniques to DBMRF geometriesbeyond those that were tested by Lehman et al. [25]. With theaim of developing a procedure that yields balanced designs, multi-tudes of single-story single-bay DBMRF systems with differentwidth-to-height ratios (B/H = 0.66, 1.0, and 1.5, where H = 3.5 m)and hollow square section (HSS) brace elements are designed hereaccording to the AISC 341-10 provisions [1] and AISC 360-10 rules[41]. The DBMRFs should also comply with the requirements forlateral-seismic-force-resisting systems in the aforementioned AISCcodes. The considered loads and load combinations are obtainedfrom SEI/ASCE 7-10 [43]. More specifically, the gravity loads arebased on a typical story of an office building; and each DBMRF isdesigned to resist lateral loads in a highly seismic area—here, fora stiff soil site in Los Angeles, California.

Two different model matrices are considered (Fig. 1): Matrix I(Table 1) comprises models with a fixed width-to-height ratio ofB/H = 1.5 that comply with the current linear-clearance seismicprovisions for rectangular and tapered gusset plate connections;and Matrix II (Table 2) comprises models with different B/H ratios,including a number of specimens that feature the more compactrectangular gusset plate connections with elliptical-clearances thatare proposed by Lehman et al. [25]. The performances of modelsfrom these two sets are evaluated through numerical simulations,and are explored to attain balanced designs in the proposedDBMRF systems.

Gusset-brace systems of these DBMRFs are designed accordingto the ‘‘special concentrically braced frame requirements’’, whereinthe gusset-to-brace connections are designed according to theexpected yield strength of the brace members, and the gusset plateis assumed to accommodate excessive nonlinear deformations orrequired flexural strength due to the brace post-buckling effects.The brace should be also checked in order to meet slenderness,width-thickness, and required strength limitations. Beams and col-umns are designed according to the ‘‘special moment framerequirements’’ to allow for the extreme yielding of the braces in

Fig. 1. Typical DBMRF systems for different gusset-to-brace connections (G.P.: gusset plate).

200 M. Lotfollahi et al. / Engineering Structures 87 (2015) 198–219

tension phase, plastic hinge formation at the mid-sections of thebraces and within the gusset plates in compression phase, suchthat they can resist the corresponding forces from the gusset–bracesystem and would remain stable beyond the DBMRF systems’ ulti-mate capacity. Due to the ‘‘strong-column, weak-beam’’ design cri-terion, plastic hinges are only allowed to form at the beam-endsand at the lower ends of columns, so that the system can maintainstability even after full or partial mechanism formation within thegusset–brace system. As a result, the moment frames are strongenough to avoid excessive or premature yielding during seismicloading, but they are not too strong to adversely affect the systemperformance due to added damage to the brace or gusset plates.The systems almost remain elastic under gravity loads and duringthe initial stages of lateral loading.

The frames of all DBMRFs considered in the model matrices Iand II (Tables 1 and 2) are identical for each instance of B/H; butevery model has a different gusset plate—namely, a rectangularor tapered gusset plate designed according to current seismic pro-visions with 2tp linear-clearance requirements, or a rectangularplate with ntp elliptical-clearance requirement proposed by Leh-man et al. [25], where tp denotes the gusset plate thickness. Inorder to consider various brace-to-frame stiffness ratios, the braceelements are designed according to different demand-to-capacity(D/C) ratios; and the axial capacities of the gusset plate connec-tions are set exactly equal to the expected axial capacities of thebrace elements. Each model is tagged by the gusset plate shape—i.e., Rectangular (R), Tapered (T), or Elliptical (E)—, width-to-heightratio of the frame, gusset plate thickness (in mm), and value of thedemand-to-capacity ratio of the brace element, as presented inTables 1 and 2. The fixed-base boundary condition is simulatedby restraining the motion of the bottom nodes of both columns(flanges and web) in all directions. The out-of-plane displacementsof the top flanges of the beams are also restrained to mimic theconstraints imposed by the slabs of the story floors.

5. Material properties

For both model matrices, the ASTM A 500, ASTM A 572, andASTM A 992 conventional structural steel standards were adoptedfor brace elements, gusset plates, and frame members, respec-tively, as considered in previous research work by Yoo et al. [35].The uniaxial stress–strain diagrams for these three types of steel(all with E = 205 GPa and m = 0.3) are shown in Fig. 2. The stress–strain diagrams of all three cases are piecewise-linear. Data pointsof the stress–strain diagram for the frame members and gusset–brace systems through both the expected and the specified yieldstress values were obtained from a best-fit to the Ramberg–Osgoodformula [44], and the transition region from elastic to plasticbehavior was kept highly refined, which improved numerical con-vergence in the nonlinear FE simulations. Also, a 0.2% to 0.3% strainhardening was assumed through the plateau transition of the gus-set–brace system, while a higher strain hardening value of 0.5% isapplied for the plateau region of the frame members. The specified(expected) yield stress values for the brace, the gusset plates, andthe frame members were set at 345 (483) MPa, 415 (456.5) MPa,and 385 (423.5) MPa respectively.

An isotropic hardening rule was adopted for all materials inorder to evaluate strain hardening/softening and hardening fol-lowed by softening under monotonic (pushover) loads. For cyclicloads, a best-fit bilinear stress–strain behavior from the aforemen-tioned piecewise-linear material models is extracted with the con-straint(s) that the yield stress of (and the area under) the bilinearand the piecewise-linear plastic material models are identical. Thisresults in the calculation of different hardening functions for thesystem constituents that have 1% strain hardening as an upperbound, effective until the maximum stress of each material model,beyond which a plateau is specified. A kinematic hardening rule—the evolution of which follows the linear Ziegler law—is imple-mented to this bilinear model, in order to capture either the

Table 1DBMRF model matrix I.

Model tag (I) Brace section (HSS) Abrace (cm2) Ibrace (cm4) GP (mm) tp (mm) Clearance (mm) Lw (mm) Beam Column

R-1.5-15-0.45a 4.875 ' 4.875 ' 5.6/16 40.9 905.2 1099.9 ' 744.3 15 Linear, 50 283 W10 ' 26 W12 ' 50R-1.5-15-0.60 4.375 ' 4.375 ' 5.75/16 37.2 650.9 1006.1 ' 676.4 15 Linear, 50 251 W10 ' 26 W12 ' 50R-1.5-15-0.75 4.125 ' 4.125 ' 5.35/16 32.7 509.3 922.7 ' 619.4 15 Linear, 50 221 W10 ' 26 W12 ' 50R-1.5-15-0.85 4.125 ' 4.125 ' 4.45/16 27.6 441.6 834.9 ' 560.8 15 Linear, 50 189 W10 ' 26 W12 ' 50R-1.5-15-0.95 4.125 ' 4.125 ' 3.82/16 23.9 390.3 754.5 ' 507.3 15 Linear, 50 165 W10 ' 26 W12 ' 50R-1.5-15-1.25 3.875 ' 3.875 ' 3.6/16 21.2 304.7 723.1 ' 484.8 15 Linear, 50 152 W10 ' 26 W12 ' 50T-1.5-15-0.45 4.875 ' 4.875 ' 5.6/16 40.9 905.2 568.8 ' 456.7 15 Linear, 50 283 W10 ' 26 W12 ' 50T-1.5-15-0.60 4.375 ' 4.375 ' 5.75/16 37.2 650.9 519.5 ' 412.8 15 Linear, 50 251 W10 ' 26 W12 ' 50T-1.5-15-0.75 4.125 ' 4.125 ' 5.35/16 32.7 509.3 482.2 ' 379.8 15 Linear, 50 221 W10 ' 26 W12 ' 50T-1.5-15-0.85 4.125 ' 4.125 ' 4.45/16 27.6 441.6 448.9 ' 350.3 15 Linear, 50 189 W10 ' 26 W12 ' 50T-1.5-15-0.95 4.125 ' 4.125 ' 3.82/16 23.9 390.3 417.3 ' 322.4 15 Linear, 50 165 W10 ' 26 W12 ' 50T-1.5-15-1.25 3.875 ' 3.875 ' 3.6/16 21.2 304.7 401.4 ' 308.3 15 Linear, 50 152 W10 ' 26 W12 ' 50

a The gusset plate thickness has increased up to 15.8 mm in numerical analysis.

Table 2DBMRF model matrix II.

Model tag (II) Brace section (HSS) Abrace (cm2) Ibrace (cm4) GP (mm) tp (mm) Clearance (mm) Lw (mm) Beam Column

R-1.5-15-0.85 4.125 ' 4.125 ' 4.45/16 27.6 441.6 834.9 ' 560.8 15 Linear, 50 189 W10 ' 26 W12 ' 50T-1.5-15-0.85 4.125 ' 4.125 ' 4.45/16 27.6 441.6 448.9 ' 350.3 15 Linear, 50 189 W10 ' 26 W12 ' 50E-1.5-15-0.85 4.125 ' 4.125 ' 4.45/16 27.6 441.6 525.3 ' 354.4 15 Elliptical, 120 189 W10 ' 26 W12 ' 50R-1.0-14-0.85 3.25 ' 3.25 ' 4.2/16 20.2 195.7 654.4 ' 620.8 14 Linear, 46 156 W10 ' 22 W12 ' 45T-1.0-14-0.85 3.25 ' 3.25 ' 4.2/16 20.2 195.7 412.4 ' 382.4 14 Linear, 46 156 W10 ' 22 W12 ' 45E-1.0-14-0.85 3.25 ' 3.25 ' 4.2/16 20.2 195.7 524.4 ' 504.8 14 Elliptical, 84 156 W10 ' 22 W12 ' 45R-0.66-13-0.85 2.75 ' 2.75 ' 4.1/16 16.4 111.4 392.4 ' 490.8 13 Linear, 42 144 W10 ' 19 W12 ' 40T-0.66-13-0.85 2.75 ' 2.75 ' 4.1/16 16.4 111.4 312.4 ' 412.2 13 Linear, 42 144 W10 ' 19 W12 ' 40E-0.66-13-0.85 2.75 ' 2.75 ' 4.1/16 16.4 111.4 354.4 ' 464.8 13 Elliptical, 104 144 W10 ' 19 W12 ' 40


Bauschinger effect or the cumulative plastic strain under subse-quent cycles of lateral loading.

6. General behavior of DBMRFs

In the following section, the nonlinear responses of DBMRF mod-els are systematically evaluated to develop performance-based seis-mic design (PBSD) criteria for these systems. The procedure utilizesboth monotonic (pushover) and cyclic analyses to explore the sys-tem yield mechanisms and failure modes (cyclic loading protocolsare also used for examining nonlinear performance and probablefailure modes during unloading-reloading cycles).

6.1. The compression phase

The general behavior can be classified into four regimes asshown in Fig. 3(a). Attributes of behavior in each regime aredescribed as follows:

Part OA: The system is subjected to a relatively low lateral load.The brace element experience compressive elastic axial strainswhile participating in absorbing the story shear. By increasingthe lateral load—and due to the imposed initial imperfection—thebrace buckles and the system experiences a sudden loss of stiffness(point A in Fig. 3(a)). Meanwhile, the stress levels in the framemembers are very low.

Part AB: The system experiences temporary instability due tobrace buckling; and the brace experiences out-of-plane bendingdeformations due to the pre-calculated mode shapes imposed tothe brace members as initial imperfections. As the lateral displace-ment increases, the frame action becomes more effective and thetotal stiffness gradually increases. The system can carry more

additional loading; and post-buckling deformation of the brace ele-ments results in plastic hinge formation in the mid-section of thebrace elements (point B in Fig. 3(a)). Fig. 4(a) displays the vonMises stress distribution of a typical DBMRF at a load magnitudecorresponding to point B. The brace is now in its post-bucklingstate. It experiences plastic deformations; and its resistancedeteriorates.

Part BC: The brace response is dominated by kinematic andmaterial nonlinearities, and large post-buckling deformationsensue. Lateral loading increases almost at a constant rate with lat-eral displacement and yielding spreads out from both ends of thegusset plates. Further increase of the lateral load results in plastichinge formation in both gusset plates. Additional stiffness degrada-tion occurs as first yielding occurrences are seen in the framemembers (specifically, in the beams), which can lead to plastichinge formation (point C in Fig. 3(a)). Fig. 4(b) displays the vonMises stress distribution at point C. A partial mechanism developsacross the gusset–brace system (i.e., a triple plastic hinge forma-tion); and the plasticity spreads out within the beam.

Part CD: In the fourth stage, the moment frame response becomeshighly nonlinear, and the system dissipates significant energy. Thegusset–brace system has almost fully yielded, and partial or com-plete plastic hinges form in the frame members. The stiffnessremains nearly constant—but at a very low level—for a significantrange of increasing displacements, while fully plastic hinges formin the beams. The lateral stiffness of the system then graduallydecreases to zero, indicating the system ultimate state and the for-mation of plastic hinges in the columns (point D in Fig. 3(a)). Thevon Mises stress distribution at the ultimate state is shown inFig. 4(c). The gusset–brace system becomes completely inefficient;and excessive plasticity spreads throughout the frame members.

Fig. 2. Stress–strain backbone diagrams for steel of the considered model matrix.

Fig. 3. The general behavior of DBMRFs for lateral load vs. displacement and lateral stiffness vs. drift ratio curves, as well as the selected performance levels of DBMRFs indifferent regimes of the compression and tension phases (G.P.: gusset plate; P.Z.: panel zone; F.Y.: first yielding occurrence; P.H.: plastic hinge formation).


6.2. The tension phase

The general behavior can be classified into three regimes asshown in Fig. 3(b). Attributes of behavior in each regime aredescribed below:

Part OA: Under moderately low lateral loads, the brace memberundergoes elastic tensile axial strains while participating inabsorbing the story shear. Under increasing lateral loads, the braceelement yields (point A in Fig. 3(b)). The stiffness of the systemsgradually decreases due to local yielding within the brace ele-ments. Fig. 5(a) displays the von Mises stress distribution at pointA. The brace elements become fully plastic. Meanwhile, the stresslevels in frame members remain low.

Part AB: The system experiences a large loss of stiffness due tothe complete yielding of the brace element. With further increaseof lateral load, the first yielding occurrences arise within the framemembers (specifically, in the beams), which can cause further stiff-ness degradation of the overall system response. Subsequently,plastic hinges form within the beam elements (point B inFig. 3(b)); and the gusset–brace system becomes inefficient (notethat the gusset plates remain elastic even after the brace elementyields completely). Fig. 5(b) displays the von Mises stress distribu-tion at point B.

Part BC: In the final stage, the moment frame provides consider-able energy dissipation. The gusset–brace system is completelyineffective; and fully plastic hinges form in the beam elements.The lateral stiffness of the system gradually decreases as the plastichinge spreads out within the moment frame system. It eventuallyapproaches to zero, at which point the ultimate state is reacheddue to complete or partial plastic hinges formation within the col-umn elements (point C in Fig. 3(b)). Fig. 5(c) displays the von Misesstress at the ultimate state of the tension phase.

6.3. The cyclic response

In order to carefully evaluate the nonlinear behavior and toascertain the performance levels—which are indicated in Fig. 3—cyclic analyses are performed with increasing-amplitude driftcycles according to the ATC testing protocol [45]. The magnitudeof each cycle is a multiple of the different compressive bucklingand tensile yielding drifts from the preceding monotonic (push-over) analyses. The DBMRFs experience significant yielding andlocal buckling; however these responses contribute to the second-ary yield mechanisms rather than induce any of the probable fail-ure modes. The systems’ primary failure modes—i.e., tearing/fracturing of the welds or members—typically occur at locationswith stress/strain concentrations. The states of stress/strain atthese critical locations provide insight into the failure modes of(damage to) the DBMRF systems.

In numerical simulations, such occurrences may be estimatedby computing the equivalent plastic strain (epl

eqv) and by establish-ing threshold values using visual observations in experiments. Thecriteria adopted here are based on the approach presented by Yooet al. [35] who provided approximate ranges of epl

eqv that markedvarious states of the occurrences of weld cracking or brace fractur-ing within the cyclic loading experiments of the steel braced framesystem. To wit,

(i) epleqv in the range of 0.054–0.065 was recorded for crack at the

weld of the gusset plate to the frame members at the reen-trant corner of the column.

(ii) epleqv in the range of 0.033–0.055 was recorded for crack at the

weld of the gusset plate to the frame members at the reen-trant corner of the beam.

Fig. 4. Mises stress (MPa) distributions in different performance levels of the DBMRFs in compression phase.

Fig. 5. Mises stress distributions in different performance levels of the DBMRFs in tension phase.


(iii) epleqv in the range of 0.271–0.306 was recorded for fracture in

the center of the locally deformed area of the buckled bracesection.

The equivalent plastic strain above is computed through a user-defined subroutine that implements the general von Mises equa-tion, as follows:

epleqv ¼

1ffiffiffi2p

1þ ~mð Þepl

x % eply

" #2þ epl

y % eplz

" #2þ"epl

z % eplx

#2$

þ 23

cplxy

" #2þ cpl

yz

" #2þ"cpl

zx

#2% &'1=2

ð5Þ

where eplx ; epl

y ; cplxy, etc., are the plastic strain components, and ~m

denotes an ‘‘effective Poisson’s ratio of 0.3 (Yoo et al. [35])’’. Theequivalent plastic strain depends on the FE mesh refinement. There-fore, element sizes were kept constant among the FE models of theDBMRF model matrix in the critical locations—this fact is demon-strated later in Appendix A.

While the von Mises equivalent plastic strain is a reasonablyaccurate index for capturing failure modes at critical locations ofDBMRF systems, the aforementioned brackets are obtained viaanalyses involving shell finite elements, and they only implicitlyinclude/reflect the triaxial state of strain in the physical specimens.Explicit consideration of three-dimensional strain fields wouldlikely improve the range of validity of this metric—albeit with veryhigh computational cost. Moreover, consideration of hydrostaticstresses might be also necessary, as shown in recent studies (seefor example [46,47]).

It should also be noted that in the simulations presented here,the weld geometries were not explicitly modeled, and the epl

eqv val-ues were obtained using the shell element strains at the weld loca-tions and their brackets were established using experimentalobservations of weld fracture.

7. Nonlinear behavior and stiffness variation

The typical lateral load vs. displacement curves of the selectedDBMRF models are displayed in Figs. 6 and 7. As seen, the decreasein the brace’s D/C value (i.e., lower D/C ratio or higher brace-to-frame rigidity) results in a higher buckling load in the compressionphase, and a higher yield load in the tension phase. Also, a DBMRFwith a lower D/C ratio experiences more elastic loading before itsbrace buckles; but has a larger unstable displacement region, dueto the softening behavior of the system. The results also show that,for lower ratios of B/H, the tapered gusset plate connections havemore stable responses after brace buckling compared to the rect-angular gusset plates having higher ratios of B/H.

Tables 3 and 4 display the lateral load–displacement valuesassociated with the primary yield mechanisms and the ultimatestates of the DBMRFs in model matrix I. The primary yield mecha-nism displacements (Dyc = Dcr, Dyt = Dy) and loads (Pyc = Pcr, Pyt =Py) are related to the initial buckling and the complete yieldingof the gusset–brace subsystem in the compression and tensionphases of the DBMRF systems, respectively. The ultimate state dis-placement of the DBMRF systems in the compression and tension

phases Dcol:pl:

" #

com:¼ Dc;max; Dcol:

pl:

" #

ten:¼ Dt;max

" #are also separately

defined as the displacement at which full plastic hinges formwithin the DBMRF system, and is generally equal to the displace-ment reached when the system’s lateral stiffness becomes zero.In most cases, this ultimate state is beyond the well-known 2.5%drift ratio of the system, recommended by the current design codes(e.g., ASCE SEI 7-10 [43]).

The results of epleqv calculations for model matrix I are presented

in Fig. 8. These results are extracted from cyclic analyses usingboth the expected and the specified yield stress values, and typicaloutcomes are presented in Fig. 9. The results indicate that the cal-culated epl

eqv values are often less than the threshold values that

Fig. 6. Typical lateral load vs. displacement curves of the DBMRFs in compression phase (G.P.: gusset plate) (for the expected behavior plot, use the lower x-axis).

Fig. 7. Typical lateral load vs. displacement curves of the DBMRFs in tension phase (G.P.: gusset plate) (for the expected behavior plot, use the lower x-axis).


delineate the probable failure modes (i.e., brace fracture and gussetplate weld cracking). However, for the reentrant corners of the gus-set plate, there is some predicted failure before the system ulti-mate state. It appears that stiffer braces and stronger gussetplates have concentrated higher epl

eqv values at the reentrant cornerof the gusset plates, and thus, weld tearing may occur at theselocations before the presumed ultimate state of the system. Also,the likelihood of occurrence for brace fracture and gusset plateweld cracking will increase when the brace becomes steeper bythe frame geometry tending from B/H > 1 to B/H < 1.

It should be noted that the beam-to-column connections weredesigned according to the Welded Unreinforced Flange-WeldedWeb (WUF-WW) connection of FEMA 350 [48], which is classifiedas a ‘‘prequalified welded fully restrained connection’’. All codelimitations according to ANSI/AISC 358 [49] provisions wereconsidered in the design of these connections here, and the

connection components details and design parameters were care-fully modeled and implemented in the FE models. Therefore, it isreasonable to expect that a connection failure due to local fractureduring simulations would be a rare event. This assertion turned outto be true, as will be described later.

Lateral stiffness curves were utilized to evaluate the yieldmechanisms and to calculate the stiffness degradation due to theplastic hinge propagation within the DBMRF systems. The typicalstiffness curves during the compression and tension phases for var-ious demand-to-capacity ratios, different B/H and gusset plate con-nection types are shown in Figs. 10 and 11. These figures indicatethat the initial lateral stiffness is lost after brace buckling in thecompression phase, and brace yielding in the tension phase. It isalso observed that a lower D/C ratio results in a greater negativelateral DBMRF stiffness after brace buckling. Consequently, the sys-tem experiences a more sizable instability region within the push-

Table 3Primary yielding mechanisms and ultimate states for the DBMRF model matrix I with rectangular G.P. connections.

Model Peigen (kN) Pcr Dcr Pc,max Dc,max EDRmaxcom: ugs—br

com: ugs—brcom:

lc,max

Py Dy Pt,max Dt,max EDRmaxten: ugs—br

ten: ugs—brten:

(kN) (mm) (kN) (mm) MRF (%) + gs–br (%) elastic drift (%) 2.5% drift (%) lt,max

R-1.5-15-0.45 975.4 954.2 9.6 719.7 103.1, [98.1]b 33.8 + 37.2 95.9 30.8 10.81431.2 12.5 1789.3 112.9, [91.3]b 21.1 + 63.7 88.7 71.9 9.1

R-1.5-15-0.60 812.1 796.3 8.8 665.5 103.7 35.9 + 32.8 95.1 26.9 11.81321.8 12.4 1685.6 110.4, [96.1]b 21.7 + 60.6 87.8 68.7 8.9

R-1.5-15-0.60c 812.2 801.3 10.1 778.6 112.1 33.3 + 34.6 94.6 37.6 11.1(1823.4)a, 1643.4 (17.4)a, 14.8 2115.7 121.4, [106.1]b 20.9 + 60.7 89.4 77.6 6.9

R-1.5-15-0.75 640.2 630.8 7.8 630.1 104.2 37.9 + 28.8 94.2 23.7 13.31173.2 12.6 1526.1 107.5, [101.4]b 21.7 + 57.9 86.9 65.8 8.5

R-1.5-15-0.85 529.4 522.4 7.5 594.2 104.6 39.1 + 25.5 93.6 20.4 13.9998.7 13.1 1350.6 105.7 22.4 + 54.8 85.9 62.9 7.9

R-1.5-15-0.95 451.2 442.3 7.2 570.1 105.8 39.9 + 22.8 92.8 17.4 14.6871.9 13.4 1224.1 102.4 22.7 + 52.1 85.1 59.1 7.7

R-1.5-15-0.95c 451.2 445.3 9.4 646.3 115.4 37.7 + 24.1 92.9 19.6 12.31201.7 18.1 1530.6 116.1 20.9 + 54.6 85.8 67.4 6.4

R-1.5-15-1.25 363.1 360.1 6.9 552.1 106.9 41.3 + 19.4 91.7 14.2 15.6778.7 12.6 1140.9 101.7 23.2 + 49.2 84.1 55.7 8.1

G.P.: gusset plate; MRF: moment resisting frame system; gs–br: gusset–brace system; EDR (%): energy dissipation ratio; u (%): gusset–brace system share of story shear.a The value in the parentheses represents brace tension complete yielding after beam element first yielding.b The value in the brackets represents gusset–brace system failure occurrence before ultimate state displacement.c The bold font is used to denote the expected behavior results.

Table 4Primary yielding mechanisms and ultimate states for the DBMRF model matrix I with tapered G.P. connections.

Model Peigen (kN) Pcr Dcr Pc,max Dc,max EDRmaxcom: ugs—br

com: ugs—brcom:

lc,max

Py Dy Pt,max Dt,max EDRmaxten: ugs—br

ten: ugs—brten:

(kN) (mm) (kN) (mm) MRF (%) + gs–br (%) elastic drift (%) 2.5% drift (%) lt,max

T-1.5-15-0.45 885.5 877.6 10.1 659.6 106.4, [78.3]b 32.3 + 35.8 94.4 28.3 10.5(1390.7)a, 1227.6 (14.2)a, 11.6 1669.4 119.1, [79.4]b 22.1 + 64.6 90.1 74.9 8.3

T-1.5-15-0.60 664.8 655.5 8.6 611.2 107.6, [98.4]b 34.8 + 31.4 93.4 24.6 12.6(1287.7)a, 1095.6 (14.0)a, 11.2 1580.6 115.7, [85.8]b 22.9 + 61.6 89.1 71.1 8.3

T-1.5-15-0.60c 664.8 653.1 10.8 708.9 110.8, [101.3]b 31.7 + 33.9 93.5 28.4 10.2(1783.1)a, 1235.6 (20.8)a, 12.54 2019.8 125.1, [92.8]b 21.1 + 62.3 91.4 81.4 6.1

T-1.5-15-0.75 529.1 518.4 8.1 577.7 108.5 36.8 + 27.4 92.4 20.6 13.6(1145.7)a, 1089.9 (13.9)a, 12.4 1454.3 113.6, [96.4]b 23.4 + 59.6 88.2 67.4 8.2

T-1.5-15-0.85 449.8 442.1 7.7 554.5 109.6 38.4 + 23.7 91.4 17.6 14.2(977.8)a, 977.8 (14.6)a, 14.6 1288.4 110.8, [103.1]b 23.9 + 57.1 87.5 64.2 7.6

T-1.5-15-0.95 393.9 380.8 7.6 537.2 112.7 39.8 + 20.5 90.3 15.1 14.7855.6 14.3 1172.2 109.4 24.6 + 54.7 86.6 61.1 7.5

T-1.5-15-0.95c 393.9 389.4 10.5 610.2 117.3 37.6 + 23.1 90.7 18.9 11.2(1172.7)a, 960.4 (18.5)a, 15.1 1475.5 119.1 21.4 + 56.8 87.3 70.1 6.4

T-1.5-15-1.25 322.4 318.1 7.4 522.8 115.3 40.9 + 17.3 89.9 12.3 15.5764.3 13.7 1093.7 107.8 25.2 + 52.1 85.6 57.8 7.9

G.P.: gusset plate; MRF: moment resisting frame system; gs–br: gusset–brace system; EDR (%): energy dissipation ratio; u (%): gusset–brace system share of story shear.a The value in the parentheses represents brace tension complete yielding after beam element first yielding.b The value in the brackets represents gusset–brace system failure occurrence before ultimate state displacement.c The bold font is used to denote the expected behavior results.


Fig. 9. Typical cyclic analysis results of various DBMRFs in the model matrix I and II (for the expected behavior plot, use the lower x-axis).

Fig. 10. Typical lateral stiffness vs. drift ratio of the DBMRFs in compression phase (G.P.: gusset plate) (for the expected behavior plot, use the lower x-axis).

Fig. 11. Typical lateral stiffness vs. drift ratio of the DBMRFs in tension phase (G.P.: gusset plate) (for the expected behavior plot, use the lower x-axis).

Fig. 8. Calculated equivalent plastic strain as function of cyclic displacement range for the DBMRF model matrix I, X:R, rectangular G.P. and X:T, tapered G.P. (G.P.: gussetplate).


over curves. After the occurrence of plastic hinges in the middle ofbrace elements within the drift ratio ranges of 0.46–0.55% for B/H > 1, 0.48–0.53% for B/H = 1, and 0.49–0.56% for B/H < 1 in com-pression, or after the braces yielding within the drift ratio rangesof 0.36–0.45% for B/H > 1, 0.39–0.42% for B/H = 1, and 0.37–0.43%for B/H < 1 in tension, the gusset–brace system becomes less effec-tive. At that point, all curves tend to converge toward each other,and merge into the state of an open-frame system. After the driftratio ranges of 0.57–0.92% for B/H > 1, 0.43–0.56% for B/H = 1.0,and 0.59–0.98% for B/H < 1 in compression (when the gusset platesbecome fully plastic), or after the drift ratio ranges of 0.64–0.82%for B/H > 1, 0.69–0.77% for B/H = 1, and 0.67–0.79% for B/H < 1 intension (when the brace elements completely yield), the ‘‘trussaction’’ is nearly completely lost; and the ‘‘frame action’’ will besignificantly dominated. Accordingly, the added story shearbeyond the complete yielding of tension braces or the imposedstory shear due to the buckling of compression braces is resistedby the moment frame system. Note that the gusset plates do notundergo plastic deformations during the tension phase of theDBMRF systems.

By increasing the lateral load and with the spread of plasticity,the beam-ends become plastic within the drift ratio ranges of1.21–1.53% for B/H > 1, 1.38–1.55% for B/H = 1, and 1.18–1.51%for B/H < 1 in compression, or within the drift ratio ranges of1.05–1.37% for B/H > 1, 1.28–1.42% for B/H = 1, and 1.11–1.41%for B/H < 1 in tension. The columns, however, remain elastic dueto the ‘‘strong-column, weak-beam’’ design criterion. Therefore,an adequate displacement range is available to the DBMRF systemsto develop widespread plasticity before the formation of plastichinges in the frame members, which would allow them to meetthe capacity-design provisions of current seismic design codes.

8. Collapse assessment via ductility evaluation and energycalculation

The system ductilities for each yield mechanism hierarchy, fail-ure mode sequence, and ultimate state within the DBMRF modelmatrix are calculated as lyic = Dyic/Dyc, lf1c = Df1c/Dyc, and lc,max =Dc,max/Dyc in the compression phase, and as lyit = Dyit/Dyt, lf1t =Df1t/Dyt, and lt,max = Dt,max/Dyt in the tension phase, separately.The yield point in the compression phase (Pyc, Dyc) is marked bythe initial buckling of the gusset–brace system, which is obtainedby recording the out-of-plane displacement at the mid-point ofthe buckled brace element, and by extracting the bifurcation pointof the gusset–brace system. The yield point in the tension phase(Pyt, Dyt) is marked by the complete yielding of the gusset–bracesystem, which is obtained by considering the von Mises stress dis-tribution, and by identifying the plasticity propagation throughoutthe gusset–brace system. The compression and tension phasedisplacements—Dyic, Df1c, and Dc,max as well as Dyit, Df1t, and Dt,-

max—are separately defined in Table 5. The presented hierarchicalyielding and failure sequences in Table 5 are the optimal/desiredsequences, which will cause yielding throughout the gusset–bracesystem first, and then within the moment frame system, and sub-sequently permit the occurrence of desirable failure modesthroughout the whole system. These hierarchies ensure that super-added damage in system constituents are prevented, undesirablefailure modes are suppressed, and thus, maximum system ductilityis attained.

On the other hand, the system energies for each DBMRF modelin each increment of lateral loading are evaluated as EI = EP + ES +EA, where EI, EP, ES, and EA are the accumulated input energy, plasticdissipation energy, strain energy, and artificial strain energy,respectively. Finally, the Energy Dissipation Ratio (EDR) for eachconstituent of DBMRF system is calculated as EDR = EP/EI.

Fig. 12 displays the aforementioned ductility evaluations formodel matrix I—individually, for each D/C ratio considered—inthe compression and tension phases. Fig. 13 presents the ductilityevaluation of model matrix II. Also, Fig. 14 displays the EDR valuesfor various DBMRFs of model matrices I and II at the performancelevels presented in Fig. 3. Finally, the maximum ductility and theEDR values for each D/C ratio and separate tension and compres-sion phases of model matrix I are shown in Tables 3 and 4. Theresults are summarized and discussed in the following subsections.

Remark: It should be noted that the considered panel-zonethicknesses, and beam-to-column connection details are basedon the current codes and seismic provisions wherein capacity isset equal to demand. The yield and buckling strengths of the gussetplates are calculated using the ‘‘Whitmore width’’ and ‘‘modifiedThornton design expressions’’. These values are compared to thetensile and compressive strengths of the brace elements, respec-tively. The gusset plate buckling is avoided, and the Whitmorewidth is defined by a 30" projected angle from the start to theend of the welded joint (see AISC 2010 [50]). Also, the gusset plategeometry is evaluated so that the resultant force of the brace ele-ment passes through the gusset plate centroid and the work pointof the beam-to-column connection. The calculated design value ofthe gusset plate thickness is equal to 11.3, 10.5, and 9.4 mm forDBMRFs with B/H > 1.0, B/H = 1.0, and B/H < 1.0, respectively; buthere they have been recalculated and assumed to be 14.2 mm,13.1 mm, and 12.3 mm to prevent all undesirable failure modesof the gusset-to-brace connections in the form of brace net-sectionfailure, brace-to-gusset interface failure, block shear, gusset platebuckling and Whitmore fracture failure.

8.1. The compression phase

In the case of linear-clearance rectangular gusset plate connec-tions with D/C P 0.95, a desirable sequence of multiple secondaryyield mechanisms are obtained. Thus, the system meets therequirement of a ‘‘balanced design’’. For D/C = 0.85, the first yieldingof the beam element occurs prior to the gusset plate plasticity. ForD/C 6 0.75, plasticity of the gusset plate is recorded after the firstyielding of the beam and full yielding of the panel-zone, whichresults in a lower DBMRF ductility (a thinner gusset plate herewould yield a DBMRF system with higher ductility). In the taperedgusset plate connections, for D/C 6 0.85, a desirable hierarchy of

Table 5The critical displacements for ductility evaluation of DBMRF systems.

Symbol Event

Compression Phase

Dy1c P.H. in the middle of the brace elementDy2c Full plasticity of the G.P.Dy3c F.Y. in the frame members (beams)Dy4c 50% yielding of the P.Z.Dy5c Complete yielding of the P.Z.Dy6c P.H. in the frame members (beams)Df1c Primary (desirable) failure mode

Dcol:pl:

" #

com:¼ Dc;max

P.H. in the frame members (columns)/ultimate state ofthe system

Tension Phase

Dy1t F.Y. in the frame members (beams)Dy2t Complete yielding of the P.Z.Dy3t P.H. in the frame members (beams)Df1t Primary (desirable) failure mode

Dcol:pl:

" #

ten:¼ Dt;max

P.H. in the frame members (columns)/ultimate state ofthe system

G.P.: gusset plate; P.Z.: panel zone; F.Y.: first yielding occurrence; P.H.: plastic hingeformation.


multiple secondary yield mechanisms is obtained; and the DBMRFacts as an almost balanced system. However, for D/C P 0.95, thegusset plate plasticity occurs prior to the plastic hinge formationof the mid-section of brace element. As such, a thicker gusset plateis required to attain a higher DBMRF ductility. Also, using taperedor elliptical-clearance rectangular gusset plates can provide moresuitable sequences of multiple secondary yield mechanisms for dif-

ferent ratios of B/H. However, the ductilities obtained from taperedgusset plates are higher than those from rectangular ones withelliptical-clearances. The use of linear-clearance rectangular gussetplates, on the other hand, delays the gusset plate plasticity, andresults in an unsuitable secondary yield mechanism. Moreover, itmay only provide mediocre ductility for the panel zones and theframe members. In all ratios of B/H, the use of gusset–brace

Fig. 12. Calculated ductilities of Y.Ms. and F.Ms. for the DBMRF model matrix I (Y.Ms.: yield mechanisms; F.Ms.: failure modes). Note: the void bars refer to the primary failuremode occurrence after ultimate state of the DBMRFs.

Fig. 13. Calculated ductilities of Y.Ms. and F.Ms. for the DBMRF model matrix II (Y.Ms.: yield mechanisms; F.Ms.: failure modes). Note: the void bars refer to the primary failuremode occurrence after ultimate state of the DBMRFs.

Fig. 14. Calculated EDR for different performance levels of various DBMRFs in the model matrix I and II (G.P.: gusset plate) (for the expected behavior plot, use the lower x-axis).


systems with D/C 6 0.85, featuring tapered or elliptical-clearancerectangular gusset plates, is recommended. This would provide ahigher ductility and ensure a balanced design.

Moreover, the energy dissipation at different DBMRF perfor-mance levels increases gradually with D/C in both rectangularand tapered gusset plate connections. To wit, from the conven-tional D/C = 0.95 to the balanced D/C = 0.75 (0.60) for the linear-clearance rectangular (tapered) gusset plate in the B, C, and Dperformance levels (see Fig. 3), the EDR values increase by 19.8%(37.7%), 14.4% (22.4%), and 6.4% (9.8%), respectively. Interestingly,the participation share of energy dissipation by the gusset–bracesystem is higher than that for the frame members for the proposedbalanced D/C (see Tables 3 and 4). This is a more favorable collapsepropagation arrangement for DBMRFs. For the ultimate state of thesystem, from the conventional D/C = 0.95 to the balanced D/C = 0.75 (0.60) for the linear-clearance rectangular (tapered) gussetplate, the EDR in the gusset–brace system increases by 26.3%(53.1%). This coincides with a decrease in EDR in the momentframe system by 5.7% (14.3%).

8.2. The tension phase

In the linear-clearance rectangular gusset plate connections, thedesirable sequence for multiple secondary yield mechanismsoccurs for D/C 6 0.60 and the system exhibits the benefits of a ‘‘bal-anced design’’. For D/C = 0.75, the beam plastic hinge and full yield-ing of the panel zone occur simultaneously. For D/C P 0.85, thecomplete panel zone yielding is recorded after the full plasticityof beam elements, which is due to the ‘‘strong-column, weak-beam’’ design criterion. For this case, a more slender panel zoneor a heftier beam section is recommended. On the other hand,for the tapered gusset plate connections, first yielding occurs inthe beam elements prior to the brace element yielding (i.e., the pre-ferred primary yield mechanism) for D/C 6 0.75; and in all cases,full plasticity of the beam element occurs prior to the full yieldingof the panel zones. Thus, brace elements with D/C 6 0.60 are rec-ommended for those cases so that a dissipative balanced perfor-mance of the system is assured. The use of linear-clearancerectangular gusset plates produces greater ductilities from theframe members. However, tapered and elliptical-clearance rectan-gular gusset plates provide greater overall ductilities, and result inmore suitable secondary yield mechanisms. As such, for all ratios ofB/H, the use of gusset-brace systems with preferably D/C 6 0.60,featuring a tapered or elliptical-clearance rectangular gusset plate,is recommended to attain a desirable sequence of multiple second-ary yield mechanisms.

Likewise, the D/C recommended above results in increasedenergy dissipation values. To wit, using the balanced D/C = 0.75(0.60) instead of the conventional D/C = 0.95 increases the EDR inthe A, B, and C performance levels (see Fig. 3) by 32.4% (51.2%),8.7% (10.1%), and 6.3% (6.8%) for the linear-clearance rectangular(tapered) gusset plate, respectively. Notably, using the lower ratioof D/C results in a higher participation share of energy dissipationthrough the gusset–brace system than the frame members, andyields a more suitable collapse propagation sequence for DBMRFs(see Tables 3 and 4). For the ultimate state of the system, from theconventional D/C = 0.95 to the balanced D/C = 0.75 (0.60) for the lin-ear-clearance rectangular (tapered) gusset plate, the EDR in the gus-set–brace system increases by 11.4% (14.3%). This coincides with adecrease in EDR in the moment frame system by 4.8% (7.6%).

9. Frame width-to-height ratio and gusset plate connectiongeometry

In each considered B/H ratio, DBMRF models with tapered andelliptical-clearance rectangular gusset plates have similar stiffness-

es, and the displacements corresponding to the buckling and yield-ing of the brace elements are almost the same. However, for thelinear clearance rectangular gusset plates, the initial buckling andyielding of brace elements occur earlier than they do for thetapered and elliptical-clearance rectangular gusset plates—viz.,6.1% for B/H > 1, 3.1% for B/H = 1, and 5.8% for B/H < 1 in bucklingdisplacement, and 13.7% for B/H > 1, 4.4% for B/H = 1, and 12.8%for B/H < 1 in yielding displacement. Plastic hinge formation inthe middle of brace element with linear-clearance rectangular gus-set plates occurs at a lower lateral displacement—viz., 11.5% for B/H > 1, 6.4% for B/H = 1, and 7.8% for B/H < 1. Also for different ratiosof B/H, the gusset plate plasticity displacement is delayed in thecase of the linear-clearance rectangular plate—viz., 56.7% for B/H > 1, 18.4% for B/H = 1, and 46.8% for B/H < 1.

The thicker and larger gusset plates (i.e., the linear-clearancerectangular gusset plates in DBMRFs with B/H > 1 and B/H < 1) havea critical effect on the panel zone and beam elements, and cause ear-lier plastic hinge formation displacement in these members com-pared to the thinner and smaller gusset plates (tapered andelliptical-clearance rectangular plates in DBMRF with B/H = 1)—viz., 17.2% (29.2%) for B/H > 1, and 14.4% (10.4%) for B/H < 1 in com-plete compression (tension) yielding of panel zone, and 11.2% (6.5%)for B/H > 1, and 16.3% (7.4%) for B/H < 1 in compression (tension)plastic hinging of the beam. Also, higher system ductilities are gen-erally obtained with tapered gusset plate connections that are usedwith different ratios of B/H, as compared to those with the rectangu-lar gusset plate connections, as shown in Fig. 13. In the case of linear-clearance rectangular gusset plates, heftier beam elements andthicker panel zones than those suggested in the current seismicdesign guidelines for DBMRF systems are recommended. Taperedand elliptical-clearance rectangular plates have generally similareffects on the collapse behavior of the DBMRFs for different B/Hratios. On the other hand, tapered gusset plate connections usedwith lower ratios of B/H results in smaller instability regions in theDBMRF response due to the buckling of the brace element, as com-pared to rectangular gusset plate connections in higher ratios of B/H, as indicated in Figs. 6 and 10. The use of the tapered gusset plateconnections with different ratios of B/H can delay the formation ofplastic hinges in the frame members; and consequently it providesframe members’ higher ductility in compression than the linear-clearance rectangular gusset plate connections—viz., 8.3% for B/H > 1, 5.2% for B/H = 1, and 9.8% for B/H < 1—but the frame membersmay suffer lower ductility in tension—viz., 18.6% for B/H > 1, 6.7% forB/H = 1, and 12.8% for B/H < 1.

Moreover, in view of the constant D/C, the participation sharesof energy dissipation between the gusset–brace and the momentframe systems are affected moderately by the frame geometry.The mean values of this parameter in the compression (tension)phase of the gusset–brace system are 39.4% (71.7%) for B/H > 1,32.7% (65.4%) for B/H = 1, and 28.1% (60.4%) for B/H < 1,respectively.

In summary, the obtained results suggest that the tapered andelliptical-clearance rectangular gusset plates can increase the duc-tility and improve the nonlinear behavior of DBMRF systems fordifferent ratios of B/H. However, stress concentrations increase atthe reentrant corners of the gusset plates where cracks may initi-ate. This is because the tapering of the plate reduces its effectivearea. As a result, weld crack initiation would be expected at smallerstory drifts for the tapered plate, as indicated in Table 6.

10. Discussion of results for the expected and the specified yieldstress values

As one might anticipate, using the expected yield stress valuesresults in the same lateral stiffness, but greater ultimate loads as


Table 6Calculated equivalent plastic strain at different locations for the DBMRF model matrix II.

Model Compression Phase Tension Phase

Middle of brace Reentrant corner of column Reentrant corner of beam Middle of brace Reentrant corner of column Reentrant corner of beam

epleqv

" #

maxDc,max epl

eqv

" #

maxDc,max epl

eqv

" #

maxDc,max epl

eqv

" #

maxDt,max epl

eqv

" #

maxDt,max epl

eqv

" #

maxDt,max

R-1.5-15-0.85 0.096 104.6 0.024 104.6 0.023 104.6 0.044 105.7 0.021 105.7 0.027 105.7R-1.5-15-0.85b 0.156 112.7 0.027 112.7 0.025 112.7 0.042 116.9 0.019 116.9 0.024 116.9T-1.5-15-0.85 0.084 109.6 0.035 109.6 0.031 109.6 0.031 110.8 0.036 110.8 0.038 110.8

(103.1)a

E-1.5-15-0.85 0.087 108.5 0.031 108.5 0.027 108.5 0.037 112.2 0.026 112.2 0.035 112.2(104.8)a

R-1.0-14-0.85 0.143 108.4 0.049 108.4 0.029 108.4 0.084 112.5 0.041 112.5 0.036 112.5(104.3)a

T-1.0-14-0.85 0.102 109.8 0.064 109.8 0.041 109.8 0.068 115.1 0.061 115.1 0.052 115.1(99.1)a (102.4)a (107.4)a (96.4)a

E-1.0-14-0.85 0.127 110.2 0.059 110.2 0.037 110.2 0.073 115.8 0.057 115.8 0.046 115.8(104.3)a (105.3)a (111.2)a (100.6)a

E-1.0-14-0.85b 0.176 116.2 0.062 116.2 0.041 116.2 0.071 123.7 0.051 123.7 0.043 123.7(109.2)a (111.7)a (108.2)

R-0.66-13-0.85 0.278 103.2 0.052 103.2 0.044 103.2 0.115 108.1 0.051 108.1 0.047 108.1(101.7)a (97.1)a (99.8)a

T-0.66-13-0.85 0.207 105.5 0.078 105.5 0.055 105.5 0.083 111.2 0.069 111.2 0.069 111.2(87.3)a (94.7)a (98.8)a (86.9)a

T-0.66-13-0.85b 0.252 110.7 0.084 110.7 0.058 110.7 0.079 117.6 0.061 117.6 0.065 117.6(91.6)a (99.4)a (104.4)a (92.2)a

E-0.66-13-0.85 0.217 105.9 0.067 105.9 0.048 105.9 0.094 113.5 0.063 113.5 0.058 113.5(95.2)a (99.7)a (104.7)a (94.6)a

a The value in the parentheses represents gusset–brace system failure occurrence before ultimate state displacement.b The bold font is used to denote the expected behavior results.

210M

.Lotfollahietal./Engineering

Structures87

(2015)198–219

well as reduced instability after brace buckling for the DBMRFsstudied (see Figs. 6 and 7). However, the compressive buckling loadis not affected by the use of the expected yield stress; and a greatertensile yielding load is reached than that obtained with the speci-fied yield stress, as presented in Tables 3 and 4.

Also, greater lateral displacements are obtained with theexpected yield stress for primary and multiple secondary yieldmechanisms, and for failure modes. However, the final calculatedductilities do not significantly vary between the expected vs. thespecified yield stress values—neither for different ratios of D/C(see Fig. 12), nor for different gusset plate connection types andframe geometries (see Fig. 13). More importantly, the ductilityvariations for all primary/secondary yield mechanisms and proba-ble failure modes are reasonably continuous; and thus, the pro-posed balanced design criteria are attainable in principle.

The mean increases of the primary yield mechanism displace-ment due to the expected yield stress in the compression (tension)phase are 27.4% (29.6%) for B/H > 1, 34.1% (23.6%) for B/H = 1, and21.4% (34.3%) for B/H < 1. Also, the mean increases for the multiplesecondary yield mechanisms displacement on the performancelevels B and C in compression (B in tension) of Fig. 3 are 11.3%and 13.8% (7.1%) for B/H > 1, 10.3% and 14.3% (9.6%) for B/H = 1,and 9.1% and 8.2% (14.3%) for B/H < 1. Finally, the mean increaseon the ultimate state displacement of the DBMRF systems in thecompression (tension) phase due to the expected yield stress are6.7% (9.3%) for B/H > 1, 5.4% (8.1%) for B/H = 1, and 4.6% (6.2%) forB/H < 1, respectively.

Likewise, the expected and the specified yield stress valuesresult in slight differences between the EDR values at different per-formance levels of Fig. 3. Finally, the energy dissipation values upto the ultimate state of the DBMRF systems are almost the same;but they correspond to different ultimate displacement values.The mean differences for the B, C, and D performance levels in

the compression phase are 10.1%, 5.1%, and 1.9% for B/H > 1,11.4%, 6.3%, and 2.1% for B/H = 1, and 15.3%, 7.8%, and 3.2% for B/H < 1, respectively. The mean differences for the A, B, and C perfor-mance levels in the tension phase are 45.8%, 8.3%, and 2.8% for B/H > 1, 36.7%, 7.4%, and 2.1% for B/H = 1, and 28.1%, 6.2%, and 2.4%for B/H < 1, respectively.

Furthermore, the elastic share of story shear between the gus-set–brace and the moment frame systems for the expected andthe specified yield stress values are almost the same (seeFig. 16). The mean increases for the gusset–brace share of storyshear at 2.5% drift ratio for the compression (tension) phase is18.9% (13.4%) for B/H > 1, 27.4% (10.3%) for B/H = 1, and 41.3%(8.2%) for B/H < 1. This corresponds to a mean decrease of 5.1%(28.6%) for B/H > 1, 5.8% (14.8%) for B/H = 1, and 6.7% (9.8%) for B/H < 1 in the moment frame share of story shear.

Finally, the equivalent plastic strain variations under cyclicloading depend on the expected and the specified yield stress val-ues. In the ultimate state of the system, the mean decrease at theweld of the gusset plate to the frame members at the reentrant cor-ner of the beam (column) due to the expected yield stress are 12.2%(10.7%) for B/H > 1, 6.3% (12.8%) for B/H = 1, and 3.1% (15.2%) for B/H < 1, respectively. This corresponds to a mean increase of 57.3%for B/H > 1, 36.4% for B/H = 1, and 22.6% for B/H < 1 in the middleof the brace element. Accordingly, the mean increase of the failuremode displacement in the compression (tension) phase are 8.4%(9.8%) for B/H > 1, 5.6% (6.9%) for B/H = 1, and 4.7% (5.6%) for B/H < 1, respectively (see Fig. 8).

11. Load participation shares of story shear

In this section, the portions of story shear taken up by the gus-set–brace and the moment frame systems are examined to evalu-ate their relative effectiveness. Fig. 15(a) displays the said shares

Fig. 15. Results of absorbed shear force, and percentage share of story shear by the gusset-brace and moment frame systems for the DBMRF model matrix I (G.P.: gusset plate;fr. members: frame members; F.Y.: first yielding occurrence; P.H.: plastic hinge formation; Com.: compression phase; Ten.: tension phase; PCT: percentage).

Fig. 16. Results of shear force percentage taken up by the gusset–brace and moment frame systems for the DBMRF model matrix II ( : R-X-Y-0.85;: T-X-Y-0.85; : E-X-Y-0.85) ( : R-X-Y-0.85, Expc.; : T-X-Y-0.85, Expc.; : E-X-Y-0.85, Expc.) (Com.:

compression phase; Ten.: tension phase).


for DBMRFs having brace elements with various demand-to-capac-ity ratios (D/C) under tension and compression, and with rectangu-lar or tapered gusset plate connections. The percent-contributionshares of the corresponding gusset–brace systems are shown inFig. 15(b). The absorbed shear forces in the gusset–brace systemsare calculated by integrating the axial stresses at the mid-sectionof the brace elements. The results indicate high contributions bythe gusset–brace systems prior to their buckling or yielding. Themean drift ratio at the compressive buckling of the brace elementsis 0.22% for B/H > 1, 0.20% for B/H = 1, and 0.24% for B/H < 1; and atthe tensile yielding, it is 0.40% for B/H > 1, 0.42% for B/H = 1, and0.39% for B/H < 1. After these stages, the gusset–brace contributionshares in the compressive phase decrease rapidly, up to the pointwhere plastic hinges form in the middle of braces at a mean driftratio of 0.49% for B/H > 1, 0.51% for B/H = 1, and 0.53% for B/H < 1.The same stage occurs in the tension phase at a mean drift ratioof 0.53% for B/H > 1, 0.55% for B/H = 1, and 0.51% for B/H < 1, anddisplays a broader spreading of plasticity within the brace ele-ments. Note that the aforementioned stages generally correspondto the first yielding of the frame members.

The curves tend to flatten after the formation of plastic hingesin the gusset plates, which occur at a mean drift ratio of 0.69%for B/H > 1, 0.49% for B/H = 1, and 0.77% for B/H < 1 in compression,and—with complete brace yielding—at a mean drift ratio of 0.73%for B/H > 1, 0.75% for B/H = 1, and 0.71% for B/H < 1 in tension. Uponthe formation of first plastic hinges in the beam elements—at amean drift ratio of 1.39% for B/H > 1, 1.47% for B/H = 1, and 1.35%for B/H < 1 in compression, and 1.21% for B/H > 1, 1.35% for B/H = 1, and 1.27% for B/H < 1 in tension—the gusset–brace systemsbecomes ineffective. Consequently, the lateral load is primarilyresisted by the moment frame systems. It is expedient to note that,due to the high rigidity of the gusset–brace system during the ini-tial linear stages, the critical and the yield loads of the DBMRF sys-tems are very close to those of the corresponding gusset–bracesystems. However, at large drift ratios (i.e., beyond the drift ratioof 1.75%), the ultimate strength of the system is considerablygreater than the load-bearing capacity of the gusset–brace sys-tems. This is due to the more significant contribution of themoment frame system than that of the gusset–brace systembeyond the said drift limit.

Fig. 16 displays the contribution shares of the gusset–brace andthe moment frame systems throughout the lateral loading historyof DBMRFs with gusset plate connections (frames) having differentsizes and types (geometries and specifications). Not surprisingly,the gusset–brace and the moment frame systems display differentcontributions in the compression and the tension phases.

In the compression phase, the gusset–brace systems absorb ahigh percentage of the story-shear during the early stages of thelateral loading. After brace-buckling and the formation of plastichinges at the middle of the braces (also upon the first yielding inthe frame members), the gusset–brace system gradually loses itseffectiveness. After the mean drift ratio of approximately 0.83%(1.13%) for B/H > 1, 0.63% (0.76%) for B/H = 1, and 0.51% (0.64%)for B/H < 1 for the specified (expected) yield stress, both themoment frame and the gusset–brace systems absorb equal shares,and then the moment frame becomes more effective. The drift atwhich plastic hinges form and first yielding occurs in the framemembers depends on the brace-to-frame rigidity ratio, and therigidities of the brace-ends.

In the tension phase, the gusset–brace systems absorb highershares of the story shear throughout the lateral loading history,depending on the system’s brace-to-frame rigidity ratio. Afterbrace yielding, the percent-share of the gusset–brace systemdecreases considerably, and consequently, the moment frame sys-tem becomes more effective, as seen in Fig. 16. However, during all

stages of lateral loading, the gusset–brace contribution is alwayshigher than 50% of the total load.

From the PBSD point of view, minor earthquake loads—i.e.,small inter-story drifts that less than 0.22% and 0.41% in the com-pression and tension phases, respectively—are dissipated by thegusset–brace systems. In moderate earthquakes, the buckled braceelements (compressive phase) form plastic hinges in their middle(0.51% mean inter-story drift) or at their both ends, within the gus-set plates (0.65% mean inter-story drift). Also, when the brace ele-ments yield (tensile phase), the plasticity will continue to spreadwithin the brace and the gusset plates (0.73% mean inter-storydrift), while the frame members are either elastic, or on the vergeof yielding. This latter state appears to be an ideal choice, becausethe moment frame systems retain their IO performance levelswhile the gusset–brace systems—which are easily repairable—arein their LS performance level (or in further stages, they even reachthe CP performance level). Under severe (e.g., strong earthquake)loads, the structure dissipates noticeable energy, and widespreadplasticity occurs within the frame members (the first plastic hingeformation in the beams occur at 1.41% and 1.28% mean inter-storydrift levels in the compression and the tension phases, respec-tively). Nevertheless, the structure does not collapse due to the highductility of the moment frame system and the ‘‘strong-column,weak-beam’’ design provisions. In short, if the system is calibratedto display a properly pre-defined yield mechanism, its overall duc-tility and performance will increase significantly—that is,unwanted failure modes will be suppressed; and irrecoverabledamages (single and multiple failure modes) and system collapsewill be significantly delayed.

12. Summary and conclusions

A comprehensive set of DBMRF systems with different demand-to-capacity ratios—viz., D/C = {0.45, 0.60, 0.70, 0.85, 0.95, 1.25}—and different frame geometries/specifications were designed andanalyzed. The outcomes were utilized (i) to examine different gus-set plate connection sizes and types on the nonlinear behavior andcollapse assessment, (ii) to evaluate the effects of brace-to-framerigidity ratios on the multiple secondary yield mechanisms andfailure modes, (iii) to calculate the values of ductility/ratios ofenergy dissipation at different stages of lateral loading (levels ofsystem performance), and (iv) to investigate the load participationshares of story shear between the moment frame and gusset–bracesystems in elastic and inelastic response regimes. The resultsobtained in this study may be summarized as follows:

) If DBMRF systems are designed according to the AISC provi-sions, then they provide adequate ductility; but they do not nec-essarily result in a suitable/optimal yielding sequence. Designswith significantly better ductility can be achieved. Specifically,plastic hinges may form within the frame members before thecomplete yielding of the gusset–brace systems. This is in con-tradiction with capacity-design principles that require ‘‘systemductility shall be primarily provided by [the] plastic deforma-tion of the gusset–brace system’’ (AISC 341-10 [1]).) The aforementioned weakness may be remedied through the

application of balanced design criteria that were proposed inthe present study for dual DBMRF systems. Previous designimprovement proposals have focused on the calibration of thegusset plate connection only [25,26]. The methodology pre-sented here improves the overall system performance throughductility assessment of all DBMRF constituents.) The gusset–brace systems are highly effective only during the

initial stages of lateral loading. In practical DBMRF systems(i.e., those with linear-clearance gusset plate connections, B/


H = 1.5, and D/C = 0.95 to 0.6), once either a plastic hinge formsat the mid-section of the brace element or the brace elementyields, the gusset–brace gradually loses its effectiveness, whichoccurs after the drift ratio ranges of 0.47% to 0.51% (0.51% to0.55%) for rectangular (tapered) gusset plates in compression,or of 0.46% to 0.49% (0.51% to 0.54%) for rectangular (tapered)gusset plates in tension. Most notably, after the drift ratioranges of 0.78% to 1.12% (0.54% to 0.75%) for rectangular(tapered) gusset plates in compression, and of 0.63% to 0.69%(0.71% to 0.75%) for rectangular (tapered) gusset plates in ten-sion, the gusset–brace system becomes nearly completely inef-fective, and the excess (imposed) story shear—i.e., beyond thecomplete yielding of the tensile braces (due to the buckling ofthe compressive braces)—is resisted by the frame action.) In compression braces, the preferred hierarchy for the second-

ary yield mechanisms is attained with D/C P 0.75 for linear-clearance rectangular gusset plate connections, and D/C 6 0.85for tapered and elliptical-clearance rectangular gusset plateconnections. In tension braces, the preferred hierarchy of thesecondary yield mechanisms is attained with D/C 6 0.75 for lin-ear-clearance rectangular gusset plate connections. For taperedand elliptical-clearance rectangular gusset plate connections,only brace elements having D/C 6 0.60 are recommended. Assuch, the appropriate value of D/C in engineering practice wouldbe D/C * 0.75 for linear-clearance rectangular gusset plate con-nections, and D/C * 0.60 for tapered and elliptical-clearancerectangular gusset plate connections.) The elastic load shares of gusset–brace/moment frame systems

for B/H > 1 are 93.8%/6.2%, and 86.1%/13.9% in the compressionand tension phases, respectively. The same shares are 86.4%/13.6%, and 83.2%/16.8% for B/H = 1, as well as 79.3%/20.7%, and77.3%/22.7% for B/H < 1. The maximum difference of the abovevalues between different gusset plate connections are 3.3%, and2.8% for B/H > 1, 2.6%, and 1.9% for B/H = 1, as well as 4.1%, and3.2% for B/H < 1 in the compression and tension phases,respectively.) The 2.5% drift ratio inelastic load shares of moment frame/gus-

set–brace systems for B/H > 1 are 79.8%/20.2%, and 37.6%/62.4%in the compression and tension phases, respectively. The sameshares are 84.4%/15.6%, and 43.2%/56.8% for B/H = 1, as well as89.3%/10.7%, and 52.7%/47.3% for B/H < 1. The maximum differ-ence of the above values between different gusset plate connec-tions is almost the same as the difference in the elastic load share.) The use of tapered and elliptical-clearance rectangular gusset

plate connections provide almost the same results with regardto yield mechanisms, propagation of plastic hinges, and failure

modes within DBMRF systems. However, tapered gusset plateconnections provide more ductility compared to rectangularplates for all ratios of B/H in both the tension and the compres-sion phases.) In the case of linear-clearance rectangular, the use of gusset

plates with lower thicknesses is recommended, provided thatgusset-plate buckling is avoided. Also, the use of heftier beamswith rectangular gusset plates (than currently provisioned) isrecommended in order to attain a more suitable collapse mech-anism for the overall DBMRF system.

The proposed balanced design criteria can be used to calibrateDBMRF systems to achieve optimal hierarchies of yielding, anddesirable sequences of failure. They can be applied to a wide rangeof geometries, gusset plate connection types/sizes, and brace ele-ment demand-to-capacity ratios. The design consequences areindependent of the use of the expected vs. the specified yield stressvalues. Moreover, the proposed design methodology is based onnon-dimensional design parameters and robust (integrated/non-dimensional) response metrics—e.g., ductility (as opposed to resis-tance), energy dissipation ratio, and participation share of storyshear. Thus, it should be applicable to any DBMRF of a multi-storymulti-bay structural system; however, additional parametric stud-ies are needed to delineate the effects of frame-to-frameinteractions.

Acknowledgments

The first author gratefully acknowledges the support from theDepartment of Civil and Environmental Engineering at UCLA. Theauthors would like to acknowledge Dorian Krausz who digitizedthe experimental data, which were procured from open literature,and the anonymous reviewers whose comments helped improvethe manuscript.

Appendix A. Finite element models and their validation andverification

The DBMRF constituents including frame members, connectiondetails, gusset plates and brace element were all modeled usingABAQUS [42] by the four-node reduced integration shell element‘‘S4R’’ with hourglass control that has six degrees of freedom ateach node. This is a general-purpose conventional shell elementthat allows transverse shear and provides finite element mem-brane strain and large deformation formulations. The reduced inte-gration option was used, because it provides adequately accurate

Fig. A1. Test setup and typical test specimen of the experimental program (Lehman et al. [25]), as well as the proposed mesh distribution plan of HSS1, HSS2, and HSS5models considered in the present study (db, dc, and L are respectively beam, column cross-sectional height, and brace length; all dimensions are in mm).


results while significantly reducing the run-times, which was amajor concern for the detailed three-dimensional models used inthe present study. Both geometric and material nonlinearities wereconsidered in all monotonic (pushover) and cyclic analyses. Mate-rial nonlinearities were incorporated through the von Mises mate-rial model with associated flow rule using both isotropic andkinematic strain hardening models.

Two supplementary techniques for predicting the bucklingmode shapes and buckling loads of the DBMRF systems wereemployed. First, an eigen-buckling analysis was conducted toobtain the relevant buckling mode shapes. These mode shapeswere then scaled with properly selected imperfection amplitudesto generate the initial brace geometries. Subsequently, nonlinearFE analyses were carried out using the initially imperfect geome-tries as the preliminary configurations to accurately capture thepost-buckling responses. The excessive magnitude of the initialimperfections may alter the actual governing equations of the sys-tem leading to the formation of soft-story mechanisms. Thus, theminimum gusset–brace out-of-plane configurations are separatelyaccounted for each DBMRF system so that the original stiffnessmatrix does not change—a veritable demonstration of this fact ispresented in [51].

A.1. Global responses

Data from experimental studies by Lehman et al. [25] were uti-lized to validate the FE analysis procedures, the mesh sizes, and theboundary conditions. The experimental test set up and specimens

are shown in Fig. A1a and b. Different specimens from their stud-ies—labeled ‘‘HSS1’’, ‘‘HSS2’’, and ‘‘HSS5’’—were modeled and ana-lyzed (see Fig. A2a–c). The geometries of the related gusset–bracesystems are provided in Table A1, and the material parameters andmembers’ thicknesses are given in Table A2.

Different mesh sizes and beam-to-column connection typeswere considered in successive numerical simulations to determinethe optimal model attributes. Various mesh arrangements wereconsidered while using brace imperfection magnitudes of L/750 = 4.54 mm in the HSS1 model and of L/1000 = 4.03 mm inthe HSS2 model. In each arrangement, the mesh size was variedindependently within the gusset–brace systems and the beam/col-umn cross-sections. For the gusset–brace systems, the maximummesh sizes (MMS) of 12.5 ' 12.5 mm, 25 ' 25 mm,37.5 ' 37.5 mm, and 50 ' 50 mm were used, along with corre-sponding MMS of 20 ' 20 mm, 40 ' 40 mm, 60 ' 60 mm,80 ' 80 mm for the beam/column cross-sections (Fig. A1c). Theresults for these four mesh arrangements are shown in Fig. A3.

The percent-error values of the simulated buckling loads for theHSS1 and HSS2 specimens were computed for different levels ofmesh refinement (Fig. A4). The maximum FE sizes of 25 ' 25 mmfor the gusset–brace systems and 40 ' 40 mm for the beam/col-umn cross-sections were found to yield adequately accurateresults (i.e., less than +1%), and thus, were selected for subsequentanalyses. Analyses with the selected mesh arrangements enabledthe fine-tuning of the brace imperfection amplitudes.

Fig. A5 displays the lateral load vs. displacement curves of theHSS1 and HSS2 specimens obtained using FE models with different

Fig. A2. Proposed FE models of the HSS1, HSS2, and HSS5 specimens at the compression ultimate state.

Table A1Geometry of the gusset–brace systems for HSS models (adopted from Lehman et al. [25]).

Specimen Gusset plate size (mm) Gusset plate thickness (mm) Clearance requirement Brace-to-gusset length (mm) Brace length (mm)

HSS1 863.6 ' 767.0 12.8 2t = 25.6, linear 374.6 3478HSS2 635.0 ' 533.4 12.4 6t = 74.4, elliptic 374.6 4030HSS5 635.0 ' 533.4 9.5 8t = 76.0, elliptic 374.6 4030


imperfection values. These and additional trial-and-error simula-tions indicated that the imperfection magnitudes of L/910 = 3.82 mm and L/1070 = 3.76 mm resulted in the bucklingloads of 880 kN and 785 kN in the HSS1 and HSS2 models, respec-tively; and these were deemed to be in very good agreement withthe experimental records (i.e., 885 kN, and 792 kN).

Additionally, two different beam-to-column connection typeswere considered in the analyses. The first connection type con-sisted of ‘‘rigid’’ connections, which were employed at both endsof the beam. These connections were modeled by merging thenodes between the beam (flanges and web) and their correspond-ing nodes on the column flange, with the aim of mimicking com-plete joint penetration welds for the beam flanges and web, andsupplemental fillet-weld between the beam web and the sheartab. The second connection type comprised a rigid beam-to-columnconnection employed at the beam’s end with the gusset plate con-

Table A2Material properties for HSS1 (HSS2) [HSS5] FE models (adapted from Lehman et al. [25]).

HSS1 (HSS2) [HSS5] Beam Column Gusset Brace

Thickness (mm) Web: 8.4 Web: 11.2 12.8 (12.4) [9.5] 9.5 (9.5) [9.5]Flange: 14.3 Flange: 17.0

Yield strength (MPa) 410 (405) [395] 414 (388) [409] 820 (454) [447] 482 (482) [505]Tensile strength (MPa) 516 (507) [501] 505 (516) [522] 875 (552) [602] 529 (529) [549]Tangent stiffness (MPa) 1831 (1771) [1849] 1581 (2208) [1961] 986 (1695) [2768] 809 (809) [782]Material standard ASTM A992 ASTM A992 ASTM A572 ASTM A500Material model Bilinear & R-O Multilinear Bilinear & R-O Multilinear Bilinear & R-O Multilinear Bilinear & R-O Multilinear

Fig. A3. Mesh refinement study for the HSS1 and HSS2 models.

Fig. A4. Percent errors of the simulated experimental buckling load vs. meshrefinement for the HSS1 and HSS2 models.

Fig. A5. Lateral load vs. displacement results for different levels of imperfection amplitude of the HSS1 and HSS2 models.


nection, and a ‘‘simple’’ connection (i.e., bolted shear-tab connec-tion) used at the other beam-end without the gusset plate connec-tion. The simple beam-to-column connection was modeled byusing a series of nonlinear spring element (‘‘basic connector’’) tomodel the bolts that connect the beam web and shear tab at theseconnections. The nonlinear force–deflection curves for the springswere based on the experimental work by Liu and Astaneh-Asl [52],while the translation constraint and/or rigid rotational restraintswere also imposed. This approach had also been considered in pre-vious research by Yoo et al. [35] to model the inelastic elongation

of bolt-holes. The computed lateral load vs. displacement relation-ships for the abovementioned beam-to-column connection typesare shown in Fig. A6. These results display a maximum of 6.7%and 5.3% difference for the HSS1 and HSS2 models, respectively,between the two connection types; and they have a very goodagreement with previous results by Yoo et al. [35]. The second con-nection type was adopted for the experimental validation in thesubsequent steps of Appendix A.

Results of the aforementioned parametric studies wereemployed to carry out pushover simulations for the HSS1 and

Fig. A6. Results of different beam-to-column connection types for the HSS1 and HSS2 models.

Fig. A7. Comparison between experimental results and proposed FE model for the HSS1 and HSS2 specimens (pushover analysis).

Fig. A8. Comparison between experimental results and proposed FE model for the HSS1 and HSS2 specimens (cyclic analysis).


HSS2 specimens. An axial compressive load of 1557.5 kN wasapplied to each column to simulate gravity loads from the upperstories; and the lateral forces were transmitted gradually to theframe by a load-beam to replicate the test. Fig. A7 displays theresults obtained from these simulations along with the experimen-tal data by Lehman et al. [25], and a prior FE simulation (for brace

under tension only) by Yoo et al. [35]. As this figure indicates, theagreement between the present and the earlier (experimental andsimulated) responses is very good—with maximum differencesbeing 3.6% (4.1%) and 2.2% (3.1%) in tension and compressionregions of the HSS1 (HSS2) model, respectively. Fig. A8 presentsthe typical results of cyclic analysis for the HSS1 and HSS2 models.

Fig. A9. Experimentally observed (Lehman et al. [25]) and simulated (this study) yield mechanisms in different locations of the HSS5 specimen (the photographs of HHS5specimen are reproduced from [25]).

Fig. A10. Comparison of equivalent plastic strain values computed in the present study with those obtained by Yoo et al. [35] for the HSS1 and HSS2 specimens.


The results show accurate comparison between subsequent cyclesof lateral loading between the proposed FEM and experimentalresults.

A.2. Local responses

A.2.1. Evaluation of yield mechanismsThe HSS5 model—which is similar to the HSS2, but has a differ-

ent gusset plate thickness (Table A1)—is considered here. The vonMises stresses were used as an indicator to identify the propaga-tion of yielding within members. Fig. A9a displays the observedand the simulated yield patterns in the gusset-to-brace connectionof the HSS5 model. The FE analysis accurately predicts the locationas well as the extent of material yielding. Fig. A9b indicates a real-istic comparison between the observed and simulated out-of-planedeformations of the brace in the HSS5 model at the same story driftof the frame. The results show that localization of plastic deforma-tions in the middle of the brace due to the large out-of-plane buck-ling of this member is accurately captured by the proposed FEmodel (Fig. A9c). Moreover, the yielding of the inner flange of thecolumn, the sustained residual rotations of the gusset plate as aresult of out-of-plane brace buckling (Fig. A9d), the flexural yield-ing of the inner flange of the bottom beam adjacent to the gussetplate as a result of out-of-plane brace deformation (Fig. A9e), andthe flexural yielding and local buckling of the left column adjacentto the gusset plate (Fig. A9f) are all well captured by the proposedFE model. These results are in very good agreement with experi-mental results, as well as earlier FE simulations carried out byYoo et al. [35]. In summary, the onset of plastic deformationsand their propagation within the specimens are accurately simu-lated with the current FE models.

A.2.2. Evaluation of the failure modesThe equivalent plastic strain (epl

eqv ) is a reasonable scalar metricof local strain concentrations, which occur in the regions withpotential fracture and weld tearing. The value of epl

eqv is sensitiveto the FE mesh sizes, and herein a constant element size of25 mm and 40 mm is utilized at the critical locations of the gus-set–brace systems and the beam/column cross-sections in everymodel. Fig. A10 displays these strains against the lateral displace-ment ranges of HSS1 and HSS2 models in the middle of the bracesand at the welded regions of the gusset plates. Again, the resultsobtained here are in very good agreement with those obtainedby Yoo et al. [35].

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