9
Buckling control of cast modular ductile bracing system for seismic-resistant steel frames G. Federico, R.B. Fleischman , K.M. Ward Dept. of Civil Engineering and Engineering Mechanics, Univ. of Arizona, Tucson, AZ, United States abstract article info Article history: Received 16 June 2011 Accepted 21 November 2011 Available online 20 December 2011 Keywords: Steel castings Special concentric braced frames Steel lateral resisting system Earthquake resistant design Structural stability Modular construction The buckling behavior of a new ductile bracing concept for steel structures is examined. The system makes use of cast components introduced at the ends and the center of the brace to produce a special bracing detail with reliable strength, stiffness and deformation capacity. The system takes advantage of the versatility in ge- ometry offered by the casting process to create congurations that eliminate non-ductile failure modes in favor of stable inelastic deformation capacity. This paper presents analytical research performed to determine the buckling strength and buckling direction of the bracing element based on the geometries of the cast com- ponents. Limiting geometries are determined for the cast components to control the buckling direction. De- sign formulas for buckling strength are proposed. © 2011 Elsevier Ltd. All rights reserved. 1. Introduction A cast modular ductile bracing system is under development as an alternative to steel special concentric braced frames (SCBFs). The sys- tem, termed a cast modular ductile bracing system (CMDB), produces a controlled energy-dissipation mechanism through the use of spe- cially detailed cast steel components introduced at the ends and the center of the brace. These components are designed to produce con- trolled stable and ductile plastic hinge regions when the bracing ele- ment undergoes buckling and straightening cycles during seismic loading. The steel SCBF is a popular seismic resistant system. However re- cent research has indicated a number of seismic performance issues [1] related to: member low-cycle fatigue life; fracture at connections; induced distortion in the surrounding members, and unbalanced shear load in the beam [2]. Improvements to SCBFs have been pro- posed [24] and new innovative systems have been proposed as al- ternatives to SCBFs [58]. The CMDB system falls into the latter group. Ward et al. [9] have presented the CMDB component geometries that produce ductile mechanisms leading to greater low-cycle fatigue life and greater post-buckling strength. These designs were shown to have the potential for improved seismic response in comparison to a SBCF. A necessary further step in developing the CMDB prototype de- sign is the ability to: (1) reliably control the buckling direction to ensure the desired post-buckling mechanism; and, (2) adequately predict the CMDB critical load. This paper presents analytical research to establish the relationship between CMDB geometry and these behaviors. Design expressions for buckling strength and required geometry for buckling control are proposed. 2. Casting modular ductile brace concept Fig. 1 shows a schematic of the CMDB system in the evaluation frame used for this paper. The CMDB bracing element is constructed by insert- ing cast components at the ends and center of HSS (Hollow Structural Section) members. The components, termed the end and center (cast ductile) components, or EC and CC, are shown in detail in the next sec- tion. For simplicity, the CMDB design is developed using a single diago- nal conguration, as is commonly done in research. However, the system is anticipated to be used in chevron or X-brace conguration, which represents a more accommodating case in terms of tolerance. Ward et al. [9] demonstrated analytically that the CMDB system can develop a stable and ductile plastic mechanism in the post- buckling region. Fig. 2(a) shows the controlled plastic mechanism de- veloped in the CMDB analytical model, in which dark regions of the contour plot are elastic, while the lighter regions indicate the plastic hinge regions contained in the specially designed cast components (EC and CC). Ward et al. [9] compared the performance of the CMDB system to a SCBF of similar strength and bay geometry under cycling loading protocols. The accumulated plastic strain comparison is shown in Fig. 2(b), with the hatched area representing the predicted fracture range. The lower strain demand in the CMDB is due to the spreading of the inelastic demands in the special cast component, and the elim- ination of strain concentrations and local buckling in the HSS. Journal of Constructional Steel Research 71 (2012) 7482 Corresponding author. E-mail address: r[email protected] (R.B. Fleischman). 0143-974X/$ see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.11.010 Contents lists available at SciVerse ScienceDirect Journal of Constructional Steel Research

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Journal of Constructional Steel Research 71 (2012) 74–82

Contents lists available at SciVerse ScienceDirect

Journal of Constructional Steel Research

Buckling control of cast modular ductile bracing system forseismic-resistant steel frames

G. Federico, R.B. Fleischman ⁎, K.M. WardDept. of Civil Engineering and Engineering Mechanics, Univ. of Arizona, Tucson, AZ, United States

⁎ Corresponding author.E-mail address: [email protected] (R.B. Fleis

0143-974X/$ – see front matter © 2011 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2011.11.010

a b s t r a c t

a r t i c l e i n f o

Article history:Received 16 June 2011Accepted 21 November 2011Available online 20 December 2011

Keywords:Steel castingsSpecial concentric braced framesSteel lateral resisting systemEarthquake resistant designStructural stabilityModular construction

The buckling behavior of a new ductile bracing concept for steel structures is examined. The system makesuse of cast components introduced at the ends and the center of the brace to produce a special bracing detailwith reliable strength, stiffness and deformation capacity. The system takes advantage of the versatility in ge-ometry offered by the casting process to create configurations that eliminate non-ductile failure modes infavor of stable inelastic deformation capacity. This paper presents analytical research performed to determinethe buckling strength and buckling direction of the bracing element based on the geometries of the cast com-ponents. Limiting geometries are determined for the cast components to control the buckling direction. De-sign formulas for buckling strength are proposed.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

A cast modular ductile bracing system is under development as analternative to steel special concentric braced frames (SCBFs). The sys-tem, termed a cast modular ductile bracing system (CMDB), producesa controlled energy-dissipation mechanism through the use of spe-cially detailed cast steel components introduced at the ends and thecenter of the brace. These components are designed to produce con-trolled stable and ductile plastic hinge regions when the bracing ele-ment undergoes buckling and straightening cycles during seismicloading.

The steel SCBF is a popular seismic resistant system. However re-cent research has indicated a number of seismic performance issues[1] related to: member low-cycle fatigue life; fracture at connections;induced distortion in the surrounding members, and unbalancedshear load in the beam [2]. Improvements to SCBFs have been pro-posed [2–4] and new innovative systems have been proposed as al-ternatives to SCBFs [5–8]. The CMDB system falls into the latter group.

Ward et al. [9] have presented the CMDB component geometriesthat produce ductile mechanisms leading to greater low-cycle fatiguelife and greater post-buckling strength. These designs were shown tohave the potential for improved seismic response in comparison to aSBCF. A necessary further step in developing the CMDB prototype de-sign is the ability to: (1) reliably control the buckling direction to ensurethe desired post-buckling mechanism; and, (2) adequately predict theCMDB critical load. This paper presents analytical research to establish

chman).

l rights reserved.

the relationship between CMDB geometry and these behaviors. Designexpressions for buckling strength and required geometry for bucklingcontrol are proposed.

2. Casting modular ductile brace concept

Fig. 1 shows a schematic of the CMDB system in the evaluation frameused for this paper. The CMDB bracing element is constructed by insert-ing cast components at the ends and center of HSS (Hollow StructuralSection) members. The components, termed the end and center (castductile) components, or EC and CC, are shown in detail in the next sec-tion. For simplicity, the CMDB design is developed using a single diago-nal configuration, as is commonly done in research. However, thesystem is anticipated to be used in chevron or X-brace configuration,which represents a more accommodating case in terms of tolerance.

Ward et al. [9] demonstrated analytically that the CMDB systemcan develop a stable and ductile plastic mechanism in the post-buckling region. Fig. 2(a) shows the controlled plastic mechanism de-veloped in the CMDB analytical model, in which dark regions of thecontour plot are elastic, while the lighter regions indicate the plastichinge regions contained in the specially designed cast components(EC and CC).

Ward et al. [9] compared the performance of the CMDB system toa SCBF of similar strength and bay geometry under cycling loadingprotocols. The accumulated plastic strain comparison is shown inFig. 2(b), with the hatched area representing the predicted fracturerange. The lower strain demand in the CMDB is due to the spreadingof the inelastic demands in the special cast component, and the elim-ination of strain concentrations and local buckling in the HSS.

EC (see Fig. (3a))

CC (see Fig. (3b))

bolted interfaces

Brace main member (HSS typ.)

7.62 m (L)

5.39 m

5.39 m

W14x48

W14x48

W14

x132

W14

x132

Fig. 1. CMDB element: evaluation frame schematic.

75G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

These analytical results suggest that the CMDB system may pro-duce a significantly improved low-cycle fatigue life. The CMDB designmay also permit relaxing of seismic local slenderness limits for theHSS since it remains essentially elastic. Likewise, the cruciformshaped interface decreases the reduction required for shear lag asso-ciated with net section fracture. Ward et al. [9] also showed that post-buckling strength can be improved in cases.

2.1. CMDB component

Each CMDB component is composed of (see Fig. 3): (1) an inter-face region of length, Lint, in which the CMDB element is connectedto the main member (typically shop-welded to the brace main mem-ber and bolted to surrounding main members, producing replaceablebrace elements); (2) a special ductile region of length, Lsd, possessinga reduced cross-section in order to control the inelastic mechanisms;and (3) a filleted radius transition section of length, Ltr.

A cruciform cross-section is chosen for the CMDB component dueto its ease of castability, facilitation of fabrication, reduction of shearlag effects, and straightforward modulation of in-plane and out-of-plane component properties [9]. Fig. 3(c) shows a schematic of theCMDB cross-section indicating the main geometric parameters inthe special ductile region, the in-plane (of the braced frame) depth,d; and corresponding thickness t; the out-of-plane width, b; and cor-responding thickness w. It is noted that while this study treats thecruciform as an assembly of rectangular elements, for castability thecruciform section will include fillet radii at each juncture. Section

Fig. 2. CMDB vs. SCBF comparison [9]: (a) plastic mechanism; (b) accumulated plasticstrain under cyclic load.

properties of the cast components are determined based on the re-duced cross-section geometry.

2.2. Design parameters

The CMDB element is described by a number of design parame-ters, typically geometry and/or material strength ratios. These designparameters control the characteristics of the response, as will beshown in this paper. The parameters include:

(1) a cast component stiffness ratio, γ, which is important in deter-mining the critical buckling direction:

γ ¼ Iy=Ix ð1Þ

where I is the moment of inertia of the CMDB component re-duced cross-section and the x–x and y–y axes are shown inFig. 3(c). A value of γ greater than unity indicates a propensityto buckle in-plane. Since the end castings and center castingneed not be identical, parameters γe and γc are defined forthese components, respectively. The term γm is used to describethis same property in the HSS section.

(2) a bending stiffness factor, κ, which is an indicator of the mod-ification of flexural stiffness in the brace due to the presenceof the CMDB component:

κ ¼ Im=Ic ð2Þ

where Im and Ic are the moment of inertia of the HSS memberand the casting, respectively. Once buckling direction is deter-mined through the parameter γ, the κ value in the primarybuckling direction is used in the determination of bucklingstrength. Different values, κe, κc, can exist for the EC and CC.

(3) a length ratio, Λ, which affects the influence of the CMDB com-ponent properties on the brace element response:

Λ ¼ Lsd=L ð3Þ

where L and Lsd are defined in Figs. 1 and 3.

Two capacity design factors (casting relative to HSS), a flexuraloverstrength factor, Ωb, and an axial overstrength factor, Ωa, controlthe plastic mechanism as described in [9].

2.3. Controlled buckling response

In order to develop the intended ductile mechanism, a controlledbuckling direction is desired. Current SCBF systems essentially followthis approach since the gusset plate is significantly more flexible in theout-of-plane direction [10]. For controlled CMDB response, in-planebuckling is chosen because of less non-structural damage to partitions,or to glazing and other façade elements, for instance at the perimeterof structures, where bracing elements are currently discouraged. A“balanced” condition, in which the CMDB element buckles in a bi-axialflexural mode, is used to delimit the transition between buckling direc-tions for CMDB system.

2.4. Analytical modeling

The CMDB system is evaluated in this paper through three-dimensional nonlinear finite elementmodeling, using the general pur-pose commercial software ANSYS [11]. All models employ geometricnonlinearity. Depending on the study, models are created from two-node beam, eight-node solid, or four-node shell elements. Other thanthe initial elastic study, elements constitutive relationships possessma-terial inelasticity utilizing multi-linear kinematic hardening principles,the Von Mises yield function, and the Prandtl–Reuss flow equation[11]. Large deformation formulations are used with the true-stress

Fig. 3. Cast components: (a) end casting; (b) center casting; (c) cruciform dimensions (after [9]).

76 G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

logarithmic strain curves represented in detailed piecewise-linear fash-ion in order to accurately capture stress under high strain gradientswithin the plastic zones.

3. Buckling control: elastic response

The effect of CMDB component geometry on the elastic bucklingmodes is first examined. Eigenvalue solutions are performed on athree-dimensional finite element (3D-FE) elastic beam model of anisolated CMDB element (see Fig. 4). This simple model is well-suited to rapidly investigate a large parameter space.

Model accuracy was evaluated via a limited number of compari-sons to a 3D-FE shell model with fully-realized cross-sections usedsubsequently for inelastic buckling analysis (see Section 4). The com-parisons [12] indicated that: (1) the beam model approximated theshell model within 5%, and exhibited similar trends; and (2) the sim-pler boundary conditions of the isolated element caused a slight shift(approximately 5%) in the relative values of in-plane to out-of-planecritical loads relative to models with the surrounding frame (whichshow an slight increased propensity toward in-plane buckling dueto in-plane end-moment and shear from frame action). These differ-ences were deemed acceptable for the large parameter initial study.

Each segment of the CMDB brace element in Fig. 3 modeled with a3D elastic beam. Rigid offsets of length Lrgd are inserted at each end torepresent the projection distance from theworking point to the EC spe-cial detail region. The elements representing the special detail regionsinclude half the transition region, and thus are of length Lsd+Ltr, and in-corporate aweighted averagemoment of inertia based on Isd and Itr. Theinterface region, with length Lint, is assigned a moment of inertia basedon the sum of the HSS and cast component, Iint+Ihss.

Different analyses in a given sequence are performed by varying γ viaparametric representation ofmoments of inertia of the cast components,while all other dimensions are held constant: the length of the CMDB el-ement, L, is 7.62 m (25 ft); Lsd is 22.86 cm (9 in.) (Λc=Λe=0.03);Ltr=7.62 cm (3 in.); Lint=22.86 cm (9 in.); and, Lrgd=38.1 cm(15 in.). The brace main member section (initially a HSS 7×7×5/8;and later rectangular HSS members with values of γm greater thanunity) remains unchanged during a study sequence.

Fig. 4. 3D beam representation for the elastic buckling study.

For each sequence, γe and γc are varied independently: γe througha range of 1 to 10; γc through a range of 0.25 to 3. The selection of γe

values greater than 1.0, i.e. a larger out-of-plane moment of inertia,produces end-conditions promoting in-plane response. The extensionof the γc design space to values less than unity (center cast compo-nent with strong axis oriented in-plane) is motivated by the desireto develop in-plane buckling mechanisms with significant energy dis-sipation [9]. Since the CMDB component moment of inertia and plas-tic modulus are proportional, a γcb1.0 design will create higherenergy dissipation, provided the bracing element buckles in-plane.Thus, these analyses examine if the γe>1.0 end cast component canstill produce an in-plane buckling mode for center cast componentsoriented strong axis in-plane.

The upper limit of γe=10.0 represents a practical limit of asym-metry in the cruciform section: for higher values of γe it will becomemore difficult to realize an actual casting cruciform section (due tolocal slenderness, i.e. b/t requirements [13] and casting integrity thin-ness limitations [14]). Values of γc>3.0 are not considered since theyproduce too low of a CMDB element buckling strength.

Two cases are considered for varying γ: (1) κ≥1.0, i.e. the CMDBcomponent strong axis moment of the inertia (larger of Ix and Iy) isheld equal to Im, and, (2) κ≤1.0, i.e. the CMDB component weakaxis moment of inertia is held equal to Im. As an illustration, Table 1shows each case for the design point (γe=5, γc=0.5).

3.1. Elastic buckling results

The results of the elastic buckling study for the square HSS areshown in Fig. 5 (black lines are Case 1; gray lines are Case 2). Fig. 5(a) plots the ratio of the in-plane to out-of-plane CMDB elastic criticalloads, ρcr ¼ Pcr;y

Pcr;x; Fig. 5(b) plots the ratio of the CMDB fundamental

critical load normalized by the elastic buckling load correspondingto the analogous SCBF, based on the accepted effective length factorof 1.0.

Focusing first on Case 1 κ≥1.0 (black lines), an examination of theρcr plot in Fig. 5(a) indicates that as γc or γe increases, ρcr decreases.As expected, the combination of high γc and γe produce ρcr belowunity, denoting in-plane buckling response; and low γc and γe pro-duce ρcr above unity, denoting out-of-plane buckling response. How-ever, note that combinations involving γc≤1.0 can produce ρcr valuesbelow unity provided γe is sufficiently large. This finding is useful be-cause it implies that the center casting can be biased toward high in-plane energy dissipation and post-buckling strength (large in-planeplastic modulus associated with a low γc [9]) with the desired in-

Table 1Elastic buckling study: example γ values for castings.

Case 1: κ≥1.0 Case 2: κ≤1.0

EC CC EC CCγ 5.0 0.5 5.0 0.5Ix/Im 0.2 1.0 1.0 2.0Iy/Im 1.0 0.5 5.0 1.0

0.4

4.0

3.0

2.0

1.0

5.0

0.6

0.8

1.0

1.2

1 2 3 4 5 6 7 8 9

1 2 3 4 5 6 7 8 9

10

10

r cr

P C

MD

B/P

SCB

F

γe

γe

c = 0.25

c = 0.5 c = 1.0

c = 2.0

c = 3.0

a

b

c = 0.25 c = 0.5 c = 1.0 c = 2.0 c = 3.0

Pcr x controls c = 0.25

Fig. 5. Elastic buckling results: (a) ρcr vs. γe; (b) PCMDB/PSCBF vs. γe.

a

b

Fig. 6. Elastic buckling results, γm varied: (a) ρcr vs. γe; (b) PCMDB/PSCBF vs. γe.

Fig. 7. 3D shell model of CMDB element: (a) end view; (b) side view.

77G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

plane buckling mode still obtained via the out-of-plane bias of theend cast components.

Designs with ρcr values near unity may not reliably produce thedesired direction control due to the presence of initial imperfections,secondary forces, etc. Fig. 5(a) indicates that sufficiently high valuesof γe produce ρcr well below unity such that reliable in-plane bucklingresponse can be obtained. However, it is also seen in Fig. 5(b) that thecontrolling buckling load will decrease as ρcr decreases. Note that dueto the fixity provided by the EC, these elastic buckling strengths arestill well above the value associated with typical SCBFs. Thus, a designpoint can be obtained that maximizes buckling strength, minimizesγc for good post-buckling energy dissipation, and provides a reliablevalue of ρcr to assure in-plane buckling. One such design point residesat γe=2.5, γc=0.5 (black dot), which provides ρcr=0.83 (Fig. 5(a)),and a PCMDB only about 20% reduced from the maximum value (Fig. 5(b)), thus still significantly larger than that of the SCBF.

The light gray lines in the background of Fig. 5 represent Case 2(κ≤1.0). As seen, these curves are more tightly spaced and rapidlyconverge to a stable solution (ρcr=0.80 to 0.90 depending on γc)once γe reaches approximately 3.0, and produce a fundamental in-plane critical load (PCMDB) that remains relatively constant. Thus itis seen biasing of cast components to flexural rigidities greater thanthe main HSS brace members has a muted effect on buckling controlsince the main member itself dominates the buckling response.Therefore, Case 1 κ≥1.0 is only considered further for directioncontrol.

In Fig. 6, the same response indices as Fig. 5 are presented for aCMDB system with rectangular HSS sections. The HSS sections are bi-ased toward in-plane response (γm=3.0, γm=2.25 and γm=1.5 re-spectively), as shown in the inset of Fig. 6(a). For Fig. 6(b) PSCBF isbased on the square HSS (γm=1.0). The HSS out-of-plane dimensionis selected within the typical maximum width allowed architecturally.

In Fig. 6(a), each region indicated represents the rangewith an upperlimit of γc=0.25 and a lower limit of γc=1.0. In Fig. 6(b), these regions“collapse” into lines since the CMDB in-plane elastic critical load, Pcr,y, isnot influenced by γcb1.0 (for Case 1). As seen, the introduction of rect-angular HSS members provides an extended range of in-plane bucklingfor low γc values and lower γe values. For instance, selecting ρcr=0.85as a reliable limit for in-plane response, an acceptable design point isnow γm=2.25, γe=1.5 and γc=0.25 (black square), which producesa ρcr=0.8 and a PCMDB/PSCBF ratio=3.75 (see Fig. 6(b)). This design

provides higher energy dissipation than for the square HSS design caseof Fig. 5, since both the EC and CC have lower γ, and hence larger in-plane plastic modulus [9].

4. Buckling control: inelastic response

The controlling limit state of the CMDB element in most cases willbe inelastic flexural buckling, particularly for low-slenderness bracesas has been promoted for SCBFs [4]. Further, the post-buckling re-sponse of the CMDB element during seismic demand, like mostSCBF systems, will involve inelastic response regardless of the initialnature of the buckling. Accordingly, the analytical evaluation mustbe extended to inelastic models. In these models, the material valuesare based on coupon tests, with yield stress for the castings, brace,and main members of 276 MPa [15], 317 MPa, and 345 MPa [2] re-spectively. True-stress logarithmic strain constitutive relationships[9] are used for the nonlinear material large deformation analyses[11].

The nonlinear 3D FE model (see Fig. 7) employs inelastic shell el-ements for the CMDB element, as shown in Fig. 7 inset. The model al-lows the appropriate 3D plastic zone to form in the CMDB and HSScomponents, thereby capturing the inelastic buckling response ineach direction. The 3D model produces more accurate stress fieldtransitions between the different elements, including shear lag andconcentrations. Rigidly-connected 3D inelastic beam elements areused for the framing beams and columns that provide the back-upframe action.

2

1

00 0.25 0.5 0.75

0 0.25 0.5 0.75

2

1

0

a

b

Fig. 8. Transverse brace displacement vs. Δ: (a) δy, δx (b) δx/δy.

78 G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

The onset and eventual controlling direction of inelastic bucklingis examined for different combinations of γc and γe. The parameterγe is varied discretely from 0.25 (biased toward out-of-plane buck-ling) to 4.0 (biased toward in-plane buckling). The parameter γc isvaried from 0.5 to 1.0 so as to evaluate designs with high in-planeenergy-dissipation capabilities.

4.1. Study matrix

Table 2 shows the CMDB component cross-section dimensionsand design parameters examined. A square HSS section (γm=1.0)is used initially so that the influence of the cast component geometryon buckling direction can be isolated. This section, a HSS 7×7×5/8,possesses moment of inertia, Im, of 3887.6 cm4, cross-sectional areaA=90.3 cm2, plastic modulus Z=542.4 cm3, and polar moment ofinertia J=6576.5 cm4. Note in Table 2 that the larger moment ofinertia of the cast component is set nominally to Im. This practiceresults in κ≈γ for the study (refer to Eqs. (1) and (2)). For thissection in the evaluation frame (see Fig. 1), the effective slendernessof an SCBF would be 108, while for the CMDB designs in Table 2, therange is 57 to 67 based on a lower CMDB effective length factor (seeSection 5). Once the relationship for the square HSS is established,CMDB designs using the rectangular HSS sections (γm>1.0) shownin Fig. 6(a) inset are evaluated.

4.2. Inelastic buckling results

Consider first the case of varying EC stiffness ratio γe for a sym-metric center casting design, γc=1.0 and square HSS (γm=1.0). Re-sults for this case are shown in Fig. 8. In Fig. 8(a), the midspantransverse buckling displacement is shown for selected cases1.0≤γe≤4.0. The buckling displacement is decomposed into its in-plane δy (solid lines) and out-of-plane δx (dashed lines) components,as indicated in Fig. 7. Transverse displacement, δ, is expressed as per-centage of CMDB brace length, L. Frame drift, Δ, is expressed as

Table 2CMDB component dimensions and section properties.

γ d(cm)

t(cm)

d/2t b(cm)

w(cm)

b/2w Ix(cm4)

Iy(cm4)

3.97 21.6 1.1 9.44 23.9 3.7 3.24 1053 41793.84 22.9 1.0 11.25 26.7 2.5 5.25 1045 40173.01 20.3 1.9 5.33 25.4 3.0 4.17 1386 41712.89 22.7 1.3 8.95 25.5 2.7 4.79 1282 37002.08 26.5 1.1 11.61 26.5 2.4 5.50 1811 37631.95 25.4 1.5 8.33 25.4 3.0 4.17 2139 41711.49 26.7 1.5 8.75 26.7 2.3 5.83 2435 36211.47 26.7 1.8 7.50 27.9 2.3 6.11 2839 41661.25 28.2 1.5 9.25 28.2 1.9 7.40 2864 35671.23 26.7 1.9 7.00 26.7 2.4 5.65 3038 37501.18 26.7 1.8 7.50 28.2 1.8 7.93 2822 33341.18 29.2 1.9 7.88 29.3 2.2 6.79 3875 45581.00 26.7 2.4 5.65 26.7 2.4 5.65 3763 37631.00 29.5 1.7 8.92 29.5 1.7 8.92 3530 35300.95 29.2 2.0 7.19 29.2 1.9 7.57 4237 40290.95 27.9 1.8 7.86 26.8 1.9 7.05 3247 30840.90 29.3 2.0 7.22 29.2 1.9 7.88 4291 38710.90 28.2 1.8 7.93 26.7 1.9 7.05 3334 30050.85 29.3 2.2 6.79 29.2 1.9 7.88 4558 38750.85 28.2 1.8 7.93 26.7 1.8 7.50 3334 28220.76 25.9 2.5 5.10 25.9 1.9 6.80 3696 27930.74 28.2 1.9 7.40 26.7 1.7 8.08 3567 26260.68 27.9 2.3 6.11 26.7 1.8 7.50 4166 28390.67 26.7 2.3 5.83 26.7 1.5 8.75 3621 24350.51 25.4 3.0 4.17 25.4 1.5 8.33 4171 21390.49 26.2 2.5 5.15 26.7 1.1 11.67 3792 18400.26 26.7 2.5 5.25 22.9 1.0 11.25 4017 10450.25 23.9 3.7 3.24 21.6 1.1 9.44 4179 1053

percentage of the frame height, h (refer to Fig. 7). It is of interest tonote in Fig. 8(a), that for controlled buckling in one direction, a com-ponent of buckling deformation exists in the other direction. As the γe

value approaches unity, the magnitude of the non-controlling dis-placement component increases. Note that for these cases, the post-buckling displacement gradient of the non-critical direction may orig-inally be quite steep, but at a certain point, the increase in this dis-placement abates and the non-critical direction displacementplateaus and then returns slightly back toward the original position.The designs with γeb1.0 are omitted for clarity since their resultsare essentially the same as their inverse (e.g., γe of 2.0 and 0.5),with in-plane (δy) and out-of-plane (δx) displacements reversed.

Fig. 8(b) shows the same information instead as plots of displace-ment ratio, δx/δy, and for all the design combinations. As seen in Fig. 8(b), the inelastic buckling response moves from fully in-plane, δx/δysmaller than unity (e.g., γe=4.0) to essentially out-of-plane, δx/δygreater than unity (e.g., γe=0.25) response as the end stiffnessratio is varied, as expected. The design combination with γe=0.95shows the most balanced behavior (δy≈δx). The reason that γe isslightly offset from unity is due to added end-moment and shear inthe plane of the brace due to frame action.

Fig. 9 shows the effect of γc on the control of the buckling direc-tion for different γe, again for a square HSS (γm=1.0). In the plot,the displacement ratios (δx/δy) are taken at a frame drift Δ/h=0.5%,i.e., beyond the point of buckling. The effect of γc, though not asgreat as the effect of γe, is still seen to be significant. Balanced condi-tions range from γe from 0.85 to 2.6 as γc changes from 1.0 to 0.5. Alower γc leads to a shift in the “balanced” condition toward greatervalues of γe, as expected. As was implied in the elastic analyses, theCMDB system is able to force in-plane buckling (δx/δyb1.0) for de-signs with γcb1.0 by specifying a sufficiently high value of γe.

c = 0.50

c = 0.75

c = 1.00

Balanced

/ h = 0.5%

e

0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 0

2

4

6

8

Fig. 9. Effect of γc (Δ/h=0.5%).

Balanced

max 0 x

max 0 y

"perfect"

0.0

1.0

2.0

3.0

4.0

0.25 1 1.75 2.5

e

/ h = 0.5%

Fig. 11. Imperfection sensitivity: γc=γm=1.0, Δ/h=0.5%.

4 a

79G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

4.3. Sensitivity to imperfection

The contours in Fig. 9 are from analyses of initially straight ele-ments. The sensitivity of CMDB buckling control due to mill, fabrica-tion and erection tolerances is now examined.

The initial imperfection is introduced into the model by applyingan initial displacement field with a maximum amplitude δo, in eitherorthogonal direction as shown in Fig. 10. The imported displacementfield is the first (primary) buckling mode shape, determined throughan eigenvalue analysis [11] of the FE model in its elastic state, andscaled to the allowable tolerance of L/480 as specified in AISC [16],equal to 1.6 cm for the evaluation frame.

Consider first a CMDB design with γc=γm=1.0 1.0 and γe variedfrom 0.25 to 3.0. The response is shown in Fig. 11 as displacement ra-tios (δx/δy), at a frame drift Δ/h of 0.5%. Results are for the maximumallowable in-plane (δoy=L/480) and out-of-plane (δox=L/480) im-perfection. The initially perfect case is shown for reference.

As seen, the range of possible initial imperfections shifts the δx/δycurves significantly. In this case, where a balanced condition is real-ized for an initially straight member for γe≈0.95, an increase in γe

to nearly 1.75 is required to preserve in plane buckling response forδox=L/480.

The influence of γc and γm is now included in order to determinethe limiting values for these parameters. Fig. 12 shows contours of(γc, γe) pairs required to produce the balanced condition. In Fig. 12(a) the contours of balanced response are shown for a square HSS(γm=1.0). A range of acceptable (γe, γc) designs, assuming out-of-plane imperfection at the tolerance limit, are seen to exist from (2.0,1.0) to (4.0, 0.66).

In Fig. 12(b), the balanced contour for maximum allowable out-of-plane imperfection (δ0x) is shown for different rectangular HSS sec-tion oriented strong-axis out-of-plane (γm>1.0). As seen in Fig. 12(b), the balanced condition is shifted to lower values of γe with in-creased γm. Thus an extended range for in-plane response is pro-duced through the use of a rectangular HSS section. The possibilityof CMDB designs with low γ EC and CC components is advantageousin terms of system post-buckling strength and in-plane energy dissi-pation [9]. Thus, the shaded area represents more desirable designs.Valid (γe, γc, γm) design sets for in-plane buckling in this region in-clude: (1.0, 1.0, 3.0), (1.25, 0.75, 3.0), and (1.5, 0.75, 2.25).

4.4. Local and torsional buckling

The use of a cruciform shape for the cast components requires theelimination of local and torsional buckling modes. Inelastic localbuckling is measured here using the ratio of the out-of-plane dis-placement of the cruciform element tip δxtip, see inset Fig. 10(b), to

a b

Fig. 10. Initial imperfection: (a) in-plane, δ0y; (b) out-of-plane, δ0x.

the length of the ductile region, Lsd. In Fig. 13, this ratio is plotted ver-sus local slenderness of the pertinent cruciform cross-sectional ele-ment. The local slenderness is the cast component element width-to-thickness ratio (maximum of d/2t and b/2w). Different ductile re-gion lengths were considered, as indicated by Λ. The local slendernesslimit (λps ¼ 0:3

ffiffiffiffiEFy

q) for seismically compact sections as specified by

the AISC Seismic Provisions [13] is referenced in the figure. The mea-surement is taken at a drift Δ/h=0.75%.

An increased ductile region length lowers the plastic curvature [9],and hence the propensity for local buckling for a given slendernessratio (see Fig. 13). Increased local buckling is observed for higherlocal slenderness, as expected. An allowable local slenderness canbe selected based on the amplification of Equivalent Plastic Strain(EPS) [11], measured as the ratio of EPS demand at the cruciformtip in an analysis free to local buckle to one where the local bucklingis inhibited. Amplified EPS is plotted versus δxtip/Lsd, in Fig. 14(a), andlocal slenderness in Fig. 14(b).

Using an amplified EPS demand limit of 20%, a local slendernesslimit slightly stricter than the AISC Seismic provision [13] is proposed:

d/2t, b/2w=7.5, or λ ¼ 0:28ffiffiffiffiEFY

q(see Fig. 14b).

Inelastic torsional modes did not control any of the analyses in thestudy. Such modes were only observed for cases where the EC wasgiven a polar moment of inertia less than 1.5% of the HSS memberand the cast component axial strength less than 30% of the HSS mem-ber, values that are not used for typical CMDB designs [12].

5. CMDB design buckling strength

Design expressions are developed for the CMDB. It is noted thatthe EC, CC and the HSS member can possess different axial yield

max o x"perfect"

max o y

/ h = 0.5 %

m = 1.0 m =2.25

m =3.0

/ h = 0.5 %

0

1

2

3

0.50 0.75 1.00

c

0.50 0.75 1.00

c

e

4

0

1

2

3

e

b

Fig. 12. Contours of balanced response: (a) γm=1.0; (b) γm>1.0.

d/2t or b/2w

= 2%

= 3%

= 5%"

0.3

0.2

0.1

05 6 7 8 9 10 11 12

ps

Fig. 13. Inelastic local buckling: Λ varying.

a

b

Fig. 15. Expression calibration: (a) effective length; (b) equivalent moment of inertia.

80 G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

strengths. The relative axial strength of these elements has an effecton the CMDB post-buckled plastic mechanism [9]. The minimumvalue of the three is the controlling axial strength, termed as theCMDB yield strength, PY, and is used to normalize the results to follow.

5.1. Effective slenderness and equivalent moment of inertia

The influence of the EC and CC components on the compressionresponse of the CMDB elements are summarized as follows: (1) theEC component has a direct influence on the CMDB element end rota-tional restraint; (2) the CC component has a non-negligible influenceon the effective flexural rigidity of the CMDB element, since it is locat-ed at midspan where maximum secondary bending moment occurs.Accordingly the EC properties are incorporated in an effective lengthfactor, k; while the CC properties are incorporated in an equivalentmoment of inertia for the CMDB element, Ieq. Both effects are calibrat-ed from the elastic response obtained in Section 3.

The effective length factor, k, is then expressed using a curve fit ofSection 3 results as (see Fig. 15(a)):

k ¼ β 0:53þ 0:045þ Λe

3

� �κe−1ð Þ

� �ð4Þ

a)

b)

Fig. 14. Amplification on EPS versus: (a) δxtip/Lsd; (b) local slenderness.

The coefficient β represents the flexibility due to the bolted inter-face, estimated as 1.07 based on nonlinear FE pushover analyses ofthe CMDB system including the bolted interface [17].

The equivalent moment of inertia, Ieq is expressed using a curve fitof the Section 3 results as (see Fig. 15(b)):

IeqIm

¼ 2−κ1:25Λ0:7c

c ð5Þ

Note that once in-plane buckling response is enforced through theγe–γc–γm values, the in-plane section properties need only be consid-ered. As such, γm will not influence κ and Ieq, and κ=γ for γ>1.0,κ=1 for γb1.0, provided Case 1 is used (as will occur in typicalCMDB design).

5.2. Inelastic buckling strength

Most CMDB designs will be controlled by inelastic bucklingstrength. For inelastic buckling, the influence of residual stressesmust be considered. The residual stress patterns introduced in themodel as initial stress states are shown in Fig. 16 for the (a) HSSand (b) for the cast component, based on [18,19].

Fig. 17 shows the critical load for analyses of CMDB designs withγm=3.0, and γe varied between 1.0 and 2.0 and γc=1.0 (γc≠1.0 isnot shown for clarity but produces similar response). The brace ele-ment length is varied to change CMDB slenderness within typicalbrace ranges [10]. The slenderness parameter, kL/r, is calculatedbased on the section properties of the controlling section, where theminimum yield strength PY occurs. In addition to the residual stress

Fig. 16. Residual stresses patterns: (a) HSS; (b) cruciform.

Fig. 17. Compressive strength for varying CMDB slenderness: γc=1.

81G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

pattern in Fig. 16, an initial in-plane imperfection, δ0y=L/1000 [2] isintroduced.

The results in Fig. 17 are normalized by the CMDB yield strengthPY. The bounding elastic/plastic cases and the code nominal compres-sive strength curve [16] are shown for reference.

As seen the CMDB inelastic strength is similar to but not exactly fitby the code expressions [16] and at low slenderness is governed bythe controlling axial strength. The curves on different γe are seen tobe close, indicating that the effect of γe is reasonably captured by k.

5.3. Expression to predict CMDB critical load

An expression to predict the CMDB element critical buckling loadis proposed based on the results shown in Fig. 17. This expression isa modified version of the code compressive strength curve [16]. Thenominal compressive strength, PN-CMDB is:

PN�CMDB ¼ FcrA0

g ð6Þ

where the area, Ag′, is modified to account for the axial strength re-duction in the castings and set equal to:

A0

g ¼ Ag

Ωa≤Ag ð7Þ

where Ωa [9] is equal to:

Ωa ¼PY�HSS

PY�castð8Þ

The radius of gyration, r, is calculated as:

r ¼ffiffiffiffiffiffiIeqA

0g

sð9Þ

The flexural critical stress is determined using code procedures forcompression members. In Fig. 18, the ratio between the predictedstrength, PN-CMDB, and the strength from FE results, PN, is plottedwith respect to the CMDB element slenderness, kL/r, showing goodagreement and slightly conservative behavior for most cases.

Fig. 18. CMDB strength prediction verification: γc=1.

6. Conclusions

The buckling behavior of a new bracing concept for seismic-resistantsteel frames has been presented. The system, which is intended to pro-vide ductile response in an earthquake, requires control of the bucklingdirection to produce the desired mechanism, and an estimate of itsbuckling strength for design. The following conclusions are made:

1. The effect of design parameters, γe, γc and γm on the control ofbuckling direction has been shown, including the effects of initialimperfection.

2. γe,γc,γmdesign sets producing a controlled in-planemechanismwithassumed maximum out of plane imperfection are given in Fig. 12(b).

3. A local slenderness (d/2t, b/2w) of 7.5 is recommended to mitigatethe effect of inelastic local buckling.

4. A design expression for the CMDB buckling load is developed usingthe current code compression member strength procedures [16] inconjunction with a CMDB effective length factor (Eq. (4)), andequivalent moment of inertia (Eq. (5)).

In combination with design recommendations for developing aductile plastic mechanism [9], the results from this paper are beingused to develop a physical CMDB prototype for full-scale experimen-tal work. A family of modular bracing elements will be proposed thatcovers the range of strengths needed in design.

Notation

The following symbols are used in this paper

E Elastic modulus;EPS Equivalent Plastic Strain;FY yield stress;I moment of inertia;J polar moment of inertia;k equivalent effective length factor;L length of the brace between working point (member

centerlines);Mp plastic moment;PY axial yield load;Pcr Euler critical load;PCMDB, PSCBF Elastic critical load of the CMDB element, equivalent

SCBF;PN strength of the CMDB element;PN-CMDB predicted strength of the CMDB element;Z plastic modulus;β bolted interface coefficient for equivalent effective length

factor;δ displacement of the brace midspan;δΟ initial out-of-straightness of the brace midspan;Δ frame drift;γ bending stiffness ratio;κ bending stiffness factor;λps local slenderness limit;Λ length ratio;ρcr Euler critical loads ratio;Ωb bending overstrength factor;Ωa axial overstrength factor;

Acknowledgements

This research was supported by NSF Award Grant No. CMS-0324664.Supplemental funds were provided by the American Institute of SteelConstruction (AISC) and the Steel Founder's Society of America (SFSA).The writers are grateful for this support. Any opinions, findings, and

82 G. Federico et al. / Journal of Constructional Steel Research 71 (2012) 74–82

conclusions or recommendations expressed in this material are those ofthe writers and do not necessarily reflect the views of the National Sci-ence Foundation.

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