Engineering Structures 45 (2012) 104–116

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Engineering Structures

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A cast modular bracing system for steel special concentrically braced frames

K.M. Ward 1, R.B. Fleischman ⇑, G. Federico 1

Dept. of Civ. Engrg. and Engrg. Mec., School of Engrg., Univ. of Arizona, 1209 E. 2nd St., Tucson, AZ 85721, United States

a r t i c l e i n f o a b s t r a c t

Article history:Received 16 June 2011Revised 15 May 2012Accepted 16 May 2012Available online 20 July 2012

Keywords:Finite element analysisSeismic designCast steelConcentrically braced frames

0141-0296/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.engstruct.2012.05.025

⇑ Corresponding author. Tel.: +1 520 343 0223; faxE-mail addresses: kmward@email.arizona.edu (

arizona.edu (R.B. Fleischman), giovanni@email.arizon1 Tel.: +1 520 621 226.

A Cast Modular Ductile Bracing System (CMDB) is under development as an alternative to special concen-trically braced frames. The CMDB system introduces cast components at the ends and center of the bracein an attempt to produce a system with reliable strength, stiffness, and deformation capacity. A cruciformcross-section has been chosen for the cast component geometry, which is specially detailed to enhanceenergy dissipation and increase low cycle fatigue life thereby reducing the likelihood of fracture. In thispaper, capacity design parameters are established that describe the axial strength and flexural strength ofthe cast components relative to the main hollow structural section member. These parameters are variedin 2D finite element models to understand the nature of the system and identify the best performingdesigns. 3D FE models of the CMDB system and a typical special concentrically braced frame, in combi-nation with fracture indices, are used to compare the expected low cycle fatigue life of the two systems.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction section (HSS) on the bracing system behavior are examined. Design

Special Concentrically Braced Frames (SCBFs) are a common andeconomical lateral force resisting system in steel buildings. How-ever, recent research has indicated a number of performance issueswith respect to reliable performance in the post-buckling regimethat have called into question the seismic reliability of the system[1]. These include net section fracture at the gusset plates and low-cycle fatigue fracture at the brace member midspan [2–4]. Forthese reasons, improvements to SCBFs [3,5,6] and new innovativesystems have been proposed as alternatives to SCBFs [7–9].

This paper presents a new system under development, the CastModular Ductile Bracing system (CMDB), which seeks to eliminatecertain failure mechanisms present in SCBFs. The CMDB systemmakes use of cast components introduced at the ends and centerof a diagonal brace to produce a controlled energy dissipationmechanism. Cast components have been shown in previous re-search to be capable of producing excellent ductility and stable en-ergy dissipation due to the versatility in shape afforded by thecasting process [10]. In the CMDB system, the cast componentsare configured to increase buckling strength, improve low-cycle fa-tigue life, and eliminate nonductile failure modes.

This paper presents analytical research describing the develop-ment and evaluation of the CMDB system and compares its perfor-mance to a typical SCBF through nonlinear static and transientdynamic analyses. In particular, the effects of the relative strengthof the cast components with respect to the main hollow structural

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: +1 520 621 2550.K.M. Ward), rfleisch@email.a.edu (G. Federico).

guidelines for the effects of CMDB geometry on the critical buck-ling load and buckling direction control are developed in parallelresearch [11]. The results from this work will be used to design aprototype system for full-scale experimental work.

1.1. CMDB concept

The term CMDB component will be used in the paper to refer tothe castings, CMDB element will refer to the entire bracing element(castings and main HSS member), and CMDB system will refer tothe assemblages of columns, beams and bracing elements servingas the Lateral Force Resisting System (LFRS).

Fig. 1 shows a schematic of the CMDB system. The end and cen-ter cast ductile components, identified in Fig. 1 as EC and CCrespectively, are inserted into a bracing element composed of a(HSS) main member. The CMDB element is fabricated by shop (fil-let) welding the cast components to the HSS member and erectedby field bolting the CMDB element in place thus permitting areplaceable element [12]. The single diagonal brace configurationshown is used in this research for simplicity in evaluating theCMDB designs; however, the CMDB prototype design is anticipatedto be employed in a Chevron or X-shaped configuration.

The EC and CC are shown in detail in Fig. 2a and b. Each compo-nent is composed of: an interface region, of length Lint, where thecomponent is shop-welded to the brace (HSS) main member orbolted to the surrounding frame members; a special ductile regionpossessing a reduced cross-section, of length Lsd and depth d, inwhich energy dissipation is to concentrate; and a filleted radiustransition section in between, of length Ltr and initial depth dtr, thateliminates plastic strain concentrations at the interfaces. The lengthof the entire cast component is Lc. See Section 2.2 for a numerical

Notation

d depth of out-of-plane casting fin (cm)w depth of in-plane casting fin (cm)t width of out-of-plane casting fin (cm)b width of in-plane casting fin (cm)H height of bay (m)Ag cross-sectional area of casting (cm2)Fy steel material yield strength (MPa)Z plastic modulus (cm3)Mp plastic moment (MPa cm)Mp,m plastic moment of the HSS member (MPa cm)Mp,c plastic moment of the casting component (MPa cm)

PY,m axial yield load of the HSS (MPa)PY,c axial yield load of the casting (MPa)PY brace member axial yield load (MPa)Xb overstrength factor with respect to the plastic modulusXa overstrength factor with respect to the cross-sectional

areasL total length of the CMDB element (cm)Lsd length of the special ductile region (cm)K cast component length ratio�pl

eqv maximum equivalent plastic strain

K.M. Ward et al. / Engineering Structures 45 (2012) 104–116 105

description of all parameters used. A cruciform cross-section (seeFig. 2c) is used for the CMDB component due to its ease of castabil-ity, construction, and modulation of geometric parameters.

2. CMDB design parameters and evaluation frame

2.1. CMDB design parameters

The CMDB element is described by a number of design param-eters which determine the characteristics of CMDB system re-sponse. A unique aspect of the CMDB approach is the uncouplingof the cross-sectional and material properties of the cast compo-nents from the main bracing element, thereby partially uncouplingthe yield strength, buckling strength, and post-buckling strength.Thus the key parameters evaluated in this paper are capacity de-sign factors pertaining to the relative strength of the cast compo-nents to the main bracing members: a flexural overstrengthfactor, Xb , defined as

Xb ¼ Mp;m=Mp;c ð1Þ

where Mp,m and Mp,c are the plastic moment of the HSS member andcast component, respectively; and an axial overstrength factor, Xa,defined as

Xa ¼ PY;m=PY;c ð2Þ

where PY,m and PY,c are the axial yield load of the HSS and cast com-ponent, respectively. In both cases, the cast component strength isevaluated at its reduced section. The overstrength of the interface

Fig. 1. CMDB system schematic.

region is defined as Xint = Mp,int/Mp,c where Mp,int is the plastic mo-ment of the interface region.

Values of Xb and Xa greater than one indicate a weak castingrelative to the main HSS brace member, thus ‘‘overstrength’’ refersto the HSS. The flexural overstrength factor, Xb controls the extentto which the plastic region is contained within the cast componentand the post-buckling strength. The axial overstrength factor Xa

has an effect on the strength and controlling mechanisms in bothtension and compression.

Note that the plastic moment strength of the cast component willbe reduced due to axial effects, which can be significant for lowslenderness CMDB designs. The large area of the cruciform nearthe neutral axis (see Fig. 3a) creates a condition where the axial-flexural interaction is adequately approximated by (see Fig. 3b):

ðP=PyÞ3 þM=Mp ¼ 1 ð3Þ

Some comment on buckling direction is warranted. The CMDBgeometry can be biased to control the buckling direction [11]. Thuswhile the CMDB component possesses a different Xb value abouteach axis (y and z in Fig. 2c), the flexural strength only becomesimportant for a brace element in the post-buckling regime. There-fore, the Xb values used in this paper correspond to the y value, i.e.the in-plane primary buckling direction intended in design.

2.2. CMDB evaluation frame

To assess the effects of the different design parameters on thebehavior of the CMDB system, parametric studies were performed.These studies encompassed a broad range of designs in order toidentify values that provide optimal performance. The CMDBelements are evaluated through nonlinear static pushover analysesin a single-bay, single story diagonally-braced frame. The evaluationframe is based on a frame used in experimental testing of SCBFs[3,13]. The bay geometry is 3.66 m � 3.66 m, providing a bracelength considered typical of a chevron bracing element. The HSSsection is a 9 � 9 � 5/8 A500 Gr B. The surrounding frame membersare W16 � 45 beams and W12 � 72 columns of A992 material. Thein-plane depths of the three castings, d = 21.60 cm and dtr =33.02 cm, are held constant. The ductile region length isLsd = 22.86 cm, unless otherwise noted. This length was selectedfrom preliminary analyses that showing adequate tension ductility[14] for Xa > 1 designs, while still preventing local instabilities [11].This translates into a cast component length ratio of K = 4.4% where,

K ¼ Lsd=L ð4Þ

The effect of this parameter is examined in Section 4.4. Preli-minary analyses also showed that a transition length Ltr = 12.7 cmwith a radius of 20.32 cm is sufficient to contain plasticity in theductile region and eliminate plastic strain concentration. Xint is

(a) (b) (c)Fig. 2. Cast components: (a) end casting; (b) center casting; and (c) cross-section.

Fig. 3. Section axial-flexure behavior: (a) flexure reduction and (b) interaction curves.

0

100

200

300

400

500

600

700

0.00 0.10 0.20 0.30

Stre

ss [M

Pa]

Strain

CastingsBeams/ColumnsHSS

Fig. 4. Material stress–strain curves.

106 K.M. Ward et al. / Engineering Structures 45 (2012) 104–116

held constant at or near 0.5 and Lint = Lsd to maintain an elastictransition region and provide an adequate weld length [14]. Notethat due to fillet welds on all four faces of the HSS, shear lag effectsare minimized and thus a connection region of considerably short-er length than that of a typical SCBF [3] is achieved (see Section5.4).

The CMDB brace for the evaluation frame has an effective slen-derness varying between 33 and 36. These values are obtainedusing an effective length factor, k, and an equivalent moment ofinertia, Ieq, which are based on the CMDB design parameters anddeveloped in the parallel research [11]. A low slenderness has beenrecommended for chevron braces [15] and is chosen here to clearlydemonstrate the effect of the casting strength on the system. TheCMDB performance for other effective slendernesses is presentedin Section 4.6.

The cast steel proposed for the CMDB components is a mildsteel developed for previous cast prototypes with a yield stressslightly lower than typical rolled material. Experimental materialproperties from coupon tests are used for the different elementsin the CMDB system (see Fig. 4), with yield stress for the castings,brace, and main members of 276 MPa [10], 317 MPa, and 345 MPa[3] respectively.

2.3. Analytical models

This analytical research was performed through finite elementanalysis, using the program ANSYS. A two-dimensional (2D) finiteelement (FE) model of the evaluation frame (See Fig. 5a) is usedand verified using three-dimensional (3D) models. The 2D modelallows for rapid evaluation of the design parameters. The 3D modelis also used for performance assessment in Section 5.

For the 2D model, the CMDB brace and members are modeledusing plane stress elements with thickness inputs, based on thetotal thickness of the cross-section (see Fig. 5b and c). The planestress elements possess both nonlinear geometry and nonlinearmaterial capabilities to allow for inelastic buckling and large strainformulations [16]. A mesh refinement study determined that nineelements across the width of the HSS in the 2D model were ade-quate for producing sufficient accuracy [14]. A finer mesh was usedin more critical areas. The element aspect ratio was biased oppositethe principal strain direction for accurate gauss point interpolationunder larger strain [16].

The 3D model allows for out-of-plane buckling and capturescomplex behaviors such as local buckling, shear lag, stress concen-trations, and tri-axial stress states. Four node shell elements pos-sessing nonlinear geometry and material capabilities were usedin these models. The simply supported boundary conditions usedin the 2D models were replaced with the boundary conditions

(a) (b) (c)Fig. 5. 2D FE model: (a) evaluation frame; (b) center casting; and (c) end casting.

Table 1Cast component geometry.

d (cm) w (cm) b (cm) t (cm) Xb Xa

1 21.60 2.54 5.59 5.33 1.76a 1.202 21.60 2.54 6.35 5.79 1.62a 1.103 21.60 2.54 7.80 6.25 1.50 1.004 21.60 2.54 14.58 6.15 1.50 0.905 21.60 2.54 23.06 6.02 1.50 0.806 21.60 2.54 10.95 4.62 2.00 1.207 21.60 2.54 15.57 4.55 2.00 1.109 21.60 2.54 21.11 4.47 2.00 1.00

10 21.60 2.54 27.89 4.37 2.00 0.9011 21.60 2.54 36.37 4.24 2.00 0.8012 21.60 2.54 24.26 2.84 3.00 1.2013 21.60 2.54 28.90 2.77 3.00 1.1014 21.60 2.54 34.42 2.69 3.00 1.0015 21.60 2.54 41.20 2.59 3.00 0.9016 21.60 2.54 24.28 2.49 3.00 0.80

a Lower limit of Xb for practical geometry.

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present in the test setup [3] in Section 5 for the experimentalcomparisons.

In all models, the brace is given an initial imperfection byupdating the brace geometry with the primary flexural bucklingmode shape imported from an eigenvalue analysis with a maxi-mum amplitude of L/1000 [11].

Multilinear kinematic hardening using von Mises plasticity [16]was used to model the material behavior [16]. For the large defor-mation analyses, the engineering stress–strain obtained in the cou-pon testing (Fig. 4) was converted to true stress–logarithmic strain.Residual stresses in the HSS and castings were included in patternsfor the HSS and castings based on external sources [17,18].

A Newton-Raphson procedure with a line search option andautomatic time stepping is used to solve the nonlinear FE equa-tions. Default convergence criteria are used for the nonlinear staticanalyses with a tolerance of .005 for force and moment. For the dy-namic analyses, the Newmark method is used with 3% Rayleighdamping.

3. CMDB design parameter study

The design parameter studies utilize nonlinear static monotonicpushover analyses of the 2D FE model. Lateral load is applied in adisplacement-controlled fashion at the top of the columns. A targetdrift of .02 radians [19] is selected for the performance evaluations.Certain selected designs are subsequently evaluated for seismicperformance through cyclic loading and nonlinear transient dy-namic analysis of simple structures under expected ground motionexcitation (see Section 5.5).

In the analytical study, Xb (Eq. (1)) is varied from 1.5 to 3.5 andXa (Eq. (2)) is varied from 0.8 to 1.2. The Xb–Xa design space is

Fig. 6. Study matrix: EC and CC values.

shown in Fig. 6. A vertical array of diamonds represents a sequenceof runs in which Xb is held constant and Xa is varied. These samedesign points are evaluated in horizontal groups to isolate the ef-fect of Xb. Note that achieving simultaneous low values of Xb withhigh values of Xa is difficult for a cruciform section using practicalgeometries due to the need to minimize the CMDB component areawhile maximizing its plastic modulus. Thus, the Xb = 1.5 seriesdesign space shown in Fig. 6 is based on the minimum practicalXb rather than Xb � 1.5. Table 1 lists the corresponding parame-ters of the reduced cross-section cruciform geometry (refer toFig. 2c). It is noted that some geometries require a rectangularHSS with the strong axis out of plane to assure an in-plane bucklingmechanism [11].

4. Design parameter study analysis results

This section presents the results for the study described in Sec-tion 3. Loading places the CMDB element in compression unlessotherwise noted.

4.1. Global response

The global response is shown as brace axial force, Pbr, vs. driftangle with the axial force normalized by the nominal HSS axialyield strength, Py. These plots allow assessment of the stiffness,critical load (maximum strength) and sustained strength (post-buckled degraded strength) of the CMDB.

Fig. 7. Normalized brace force vs. drift: (a) Xa = 1.20; (b) Xa = 1.00; and (c) Xa = 0.80.

Fig. 8. Normalized brace force vs. drift: (a) Xb = 1.50; (b) Xb = 2.00; and (c) Xb = 3.00.

108 K.M. Ward et al. / Engineering Structures 45 (2012) 104–116

Each graph in Fig. 7 shows designs possessing a different con-stant Xa(1.2, 1.0, 0.8). In each plot, Xb is varied from 1.5 to 3. Sev-eral important trends can be identified from examining thesegraphs individually. First, the relative flexural strength of theCMDB component, Xb, is seen to have almost no effect on theCMDB element critical load. Secondly, as Xb increases, the post-buckling strength decreases more rapidly (steeper descendingbranch) and the residual strength at the target drift is lower. Ingeneral, values of Xb nearer to one possess higher and more stablepost-buckling regimes, characterized by a longer plateau for a loweffective brace slenderness. It is noted that at this effective slender-ness, the critical load is close to the brace axial yield strength, Py,for CMDB designs with Xa < 1. Thus, CMDB designs with anXa > 1 have a direct reducing effect on the critical load.

Fig. 8 shows the same data as Fig. 7, this time grouped for con-stant Xb and varying Xa in order to observe the effect of the latter.The first observation that can be made by examining each plot inFig. 8 separately is that Xa has a significant influence on the criticalload for this brace slenderness. As Xa decreases, the bucklingstrength of the CMDB element increases. A threshold is reachedfor Xa < 1.0 where the strength converges to a constant value forthis effective slenderness. Note also that the CMDB response forthe same Xb converge to similar values at the target drift(.02 rad) with slightly larger post-buckled strengths for lower Xa.

The secondary slope response is distinctive for different CMDBdesigns with low slenderness. For Xa < 1.0, the secondary responseinvolves a plateau response. As seen in Fig. 8, the plateau is partic-ularly long for Xa = 0.8 prior to initiating descending branchbehavior. This is due to HSS compressive deformation (see Section4.2). For designs with Xa P 1.0, the secondary slope involves animmediate descending branch response. Fig. 8 shows that thisslope descends more steeply as Xb increases.

The global response results indicate that the highest bucklingand post-buckling strength are obtained from a cast componentwith a minimized Xa and Xb. Producing a ductile mechanism cre-

ates a competing set of conditions on the cast component geome-try, which is considered in Section 4.3. First, however, CMDBcomponent deformation is examined.

4.2. CMDB component deformation

The brace element begins as an axial element, but is subjectedto both axial forces and flexure in the post-buckling region.Because of this transition in behavior, it is important to understandthe relative contribution of axial compressive deformation andplastic rotation for different CMDB designs in meeting the drift de-mands imposed by an earthquake.

Fig. 9 shows the component deformation as average compressiveaxial strain ð��Þ and curvature (/) of the CC and one HSS element as afunction of Xb with constant Xa = 1.0. This plot shows that highervalues of Xb lead to less axial strain in all components, more curva-ture in the castings, and less curvature in the HSS sections. Lowvalues of Xb lead to more axial strain and significantly more curva-ture in the HSS member, while reducing curvature in the castings.

Fig. 10 shows the same information as a function of Xa withconstant Xb = 2.0. An axially weak casting (Xa > 1) results in themajority of axial strain and curvature occurring in the castings,with little in the HSS. An axially strong casting (Xa < 1) reversesthese tendencies.

4.3. Plastic mechanism control

For preliminary design, an equivalent plastic strain limit [16] isselected as �pl

eqv ¼ 0:10 for the cast components and �pleqv ¼ 0:03 for

the HSS section. These limits are intended to approximately mapmonotonic demand at the target drift to adequate cyclic ductility[20]. The higher limit for the cast components is due to the specialdetailing of the reduced region that ensures stable ductile hyster-etic response. In Section 5.3, the low-cycle fatigue life of the CMDB

Fig. 9. Component deformation vs. lateral drift for varying Xb: (a) �� CC ; (b) / CC; (c)�� HSS; and (d) / HSS.

Fig. 10. Component deformation vs. lateral drift for varying Xa: (a) �� CC ; (b) / CC; (c) �� HSS; and (d) / HSS.

K.M. Ward et al. / Engineering Structures 45 (2012) 104–116 109

preliminary designs and a typical SCBF are compared and evalu-ated under a cyclic loading protocol.

Capacity design plays an important role in the CMDB perfor-mance as the relative strength of the cast components and HSS sec-tions will determine the pattern, location, and magnitude of plastic

strain demands. Fig. 11 shows plastic mechanisms that result fromdifferent combinations of Xb and Xa. These plots show equivalentplastic strain contours at the target drift. The dark regions in thecontour plot indicate portions of the brace remaining elastic; thelighter regions represent plastic zones.

Fig. 11. Equivalent plastic strain patterns for different values of Xb and Xa.

Fig. 12. Brace diagrams: (a) axial force; (b) moment; and (c) interaction (Eq. (3)).

110 K.M. Ward et al. / Engineering Structures 45 (2012) 104–116

First consider Fig. 11a showing the mechanism for a Xa > 1.0and a high Xb(2.0) design. In this case, the Xa and Xb values aresufficiently large to contain the plastic zone entirely within thecast components. Next, Fig. 11b shows that an Xa < 1.0 in conjunc-tion with a high Xb (2.0) will contain the hinge better thanXa < 1.0, Xb < 1.5 and have a lower �pl

eqv than Xa > 1.0, Xb < 2.0. Fi-nally, Fig. 11c shows a design with Xa < 1.0 and a low Xb(1.5). Inthis case, the buckling mechanism begins in the main memberbefore the flexural response becomes large enough to induce plas-tic bending in the cast components.

The behavior in Fig. 11c can be explained by examining theinteraction between axial force and moment in the CMDB ele-ment. Fig. 12a,b show time progressions of the force and momentdemands along the brace for an Xa = 0.80 and Xb = 1.50 CMDBdesign just prior to the brace reaching its critical compressiveload (at a drift of .005 rad), soon after it reaches its critical com-pressive load (.014 rad), and at the target drift (.02 rad). Prior toreaching its critical load, the axial force dominates over the flex-ural causing the HSS to reach its limit first (see Fig. 12c, Eq. (3)).

Fig. 13. Plastic strain progression (a) Xa =

These yielded sections trigger the plastic mechanism in the HSS,even though the CC is weaker flexurally. As the buckling displace-ment grows, the large secondary moment at the brace midspanelevates the interaction value in the cast components to a greatervalue than the HSS, shifting the plastic hinging into the castcomponents.

The plastic strain initiation described above (and hence theeventual plastic mechanism and demands) is strongly affected byXa. Fig. 13 shows the progression of plastic strain in the differentcomponents for the same Xb and different values of Xa. The com-ponent plastic strain limits are also indicated on the plots forreference. In Fig. 13a, Xa = 0.8, the plastic strain initiates in theHSS, then spreads to the EC, and finally to the CC as the flexuralresponse begins to dominate. After a certain point (.012 rad), addi-tional plastic strain demand occurs entirely in the cast componentswhile the plastic strain in the HSS no longer increases. However,the HSS strain has already exceeded its target strain limit. Nextconsider the Xa > 1 CMDB design shown in Fig. 13c. This designcauses the plastic strain to initiate in the cast components. The

0.80; (b) Xa = 0.90; and (c) Xa = 1.05.

Fig. 14. Equivalent plastic strain at the target drift verses Xa: solid lines representthe center castings, dashed the end castings, and dotted the HSS with lines ofconstant Xb.

K.M. Ward et al. / Engineering Structures 45 (2012) 104–116 111

HSS sees almost no plasticity and as a result the plastic strain in theCC exceeds its target limit. Finally, a CMDB design with a larger Xa

that is still less than one (Fig. 13b), hastens the shift of the plasticzone to the cast components, thereby keeping the HSS plasticstrain demands below the target limit.

Fig. 14 shows a parametric plot relating plastic strain demand atthe target drift to Xb and Xa. The plastic strain limits are indicatedon the plot for reference. Fig. 14 indicates the HSS component canbe kept below the target strain limit for a design with Xa > 1.0 andany design with Xb > 3.0. Xa = 0.9 and Xb = 1.5 represents a bound-ary for acceptable HSS behavior. The CC has acceptable plasticstrain for Xb = 1.5 and Xa = 0.8 or Xa = 0.9 and also for Xb = 2.0and Xa = 0.8 at this component length. The plastic strain in theEC is acceptable for all designs.

4.4. Effect of component length

The designs shown thus far have equal length for the CC and EC.However, Fig. 14 indicates that the CC typically experiences more

Fig. 15. Effect of K on �pleqv in CC.

Fig. 16. Tension response: (a) brace force v

plastic strain than the EC. This outcome is consistent with rigidplastic mechanism theory where the CC will be expected to under-go a rotation twice that of the EC. As the CC strain can be the lim-iting factor for the CMDB design, lengthening the CC is nowexamined. As seen in Fig. 15, this modification lowers the plasticstrain. Designs that exceeded the plastic strain limit for a cast com-ponent length ratio of K = 4.4%(Eq. (4)), meet the target limit atK = 7.4%.

4.5. CMDB tension behavior

The tensile behavior of the system is now considered. Fig. 16ashows the brace axial tension response for different Xa. Whenthe cast component is weaker than the HSS member (Xa > 1), theCMDB yield strength is lower than the HSS nominal axial yieldstrength. At large drifts, however, strain hardening in the cast com-ponents permits the CMDB element to eventually reach thisstrength.

Fig. 16b shows the distribution of equivalent plastic strain inthe HSS main member and cast components during tensionresponse. An Xa > 1 design will result in the plastic strain concen-trating in the cast components, incurring a plastic strain of approx-imately 5.5% at the target drift (for K = 4.4%). While not exceedingthe limit, this strain may be sufficient large to lower the CMDB lowcycle fatigue life. A design with Xa = 1 shares the plastic strain be-tween the cast components and the HSS member, thereby loweringthe demands on the cast components. Designs with Xa < 1 containthe tensile plastic strain entirely within the HSS member, thoughstill at low values due to the HSS length.

4.6. CMDB element slenderness

The evaluation frame geometry used in previous sections re-sults in a low effective brace slenderness. As the CMDB systemmay be employed with a variety of frame geometries, analyseswere performed for slenderness kL

r varying from 34 to 78. The dif-ferent slendernesses were achieved by varying the brace length(bay dimensions). The CMDB effective length factor, k, and equiva-lent radius of gyration, r, are based on design equations from [11].

Fig. 17a shows that as slenderness increases, the effects of Xa onbrace strength are reduced. At the highest slenderness, still in theintermediate (inelastic buckling) range, Xa is seen to have virtuallyno effect on CMDB strength. Fig. 17b demonstrates that the effectof Xb on post-buckling strength at the target drift is diminishedas the effective slenderness increases.

Fig. 17c,d show the maximum plastic strain in the HSS memberand CC, respectively, at the target drift. The effect of Xa on maxi-mum �pl

eqv in both the HSS and CC are reduced considerably as effec-

s. drift and (b) plastic strain demand.

Fig. 17. Effect of brace slenderness at target drift: (a) PBR/PY; (b) PBR/PY; (c) �pleqv HSS; and (d) �pl

eqv CC.

Fig. 18. Post-buckling strength vs. �pleqv: (a) CC and (b) HSS.

112 K.M. Ward et al. / Engineering Structures 45 (2012) 104–116

tive slenderness increases. These plots indicate that lower Xa de-signs can be considered at higher effective slenderness.

4.7. CMDB design space

Fig. 18 shows the relationship between post-buckling strengthand maximum equivalent plastic strain in the (a) CC and (b) HSSas a function of Xb and Xa (low slenderness designs). The diagramcontains lines of constant Xb, which are approximately horizontal,and lines of constant Xa, those approximately vertical. The quad-rant of desired performance is in the upper-left corner of theseplots where post-buckling strength is high and plastic strain islow. This region is seen to contain low Xb and Xa designs. The lightgray region shows the area of the Xb-Xa design space which over-laps between the two plots (note the Xa = 1.2 contour is on the farright in the Fig. 18a and on the far left in Fig. 18b), representing

potential CMDB designs. Higher slenderness tend to make thesecurves collapse down and to the left.

5. CMDB cyclic performance assessment

The CMDB system’s cyclic performance is evaluated and directlycompared to that of an SCBF.

5.1. CMDB cyclic behavior

The CMDB cyclic behavior is assessed using two cyclic protocols(see Fig. 19): (a) a standard increasing amplitude [13] and (b) thelateral drift record of a CMDB system subjected to a Berkeleydesign basis earthquake [14]. The latter possesses an initial largeexcursion. CMDB designs are analyzed (Xb = 1.5) for a cast

Fig. 19. Cyclic response: (a) increasing amplitude and (b) seismic history.

(a) (b)

(d)(c)Fig. 20. Cumulative plastic strain: (a) inc. amp., K = 4.4%; (b) inc. amp., K = 7.4%; (c) seismic, K = 4.4%; and (d) seismic, K = 7.4%.

K.M. Ward et al. / Engineering Structures 45 (2012) 104–116 113

component length ratio of K = 4.4% (see Section 2.2) and K = 7.4%and Xa = 0.80,1.05.

Fig. 19 shows a hysteretic force comparison for K = 4.4%. Theforce reponse for K = 7.4% is similar. Fig. 20 plots the cumulativeplastic strain vs. loading step. These plots show that the increasingamplitude protocol exhibits the same trends noted in Fig. 13 regard-ing delayed yield initiation in the cast components, while this doesnot occur for the seismic history with an early large excursion.Fig. 20c and d shows that increasing the CC length to 7% has a signif-icant effect on evening demands among the components.

5.2. SCBF model calibration

A 3D FE SCBF model is created and calibrated for comparison toa 3D FE model of the CMDB. The 3D models are essential to capturekey behaviors such as out-of-plane buckling and local buckling.

Full-scale experimental results [13] are used to validate the 3DFE SCBF model. The experimental results come from a 1-bay fullscale SCBF tested cyclically to assess effect of gusset plate design

on system performance. The brace tested was an HSS 5 � 5 � 3/8in a 3.66 m � 3.66 m bay.

The analytical model was compared to the global and localexperimental data [14] measured in the testing. The test data in-clude base shear, frame lateral drift, brace force, brace out-of-planebuckling deflection, fracture initiation and location, visual observa-tion of local buckling modes, and strain gage readings. It is notedthat the model development and calibration made use of recom-mendations provided in the testing program related to needed fea-tures in SCBF models [5].

Fig. 21a shows a comparison of the FE SCBF model to thephysical test results under the test cyclic loading protocol. Analyt-ical backbone curves in compression and tension are also shown.As seen the model adequately captures key global responses: yield-ing in tension, buckling in compression, the post buckling regime,and the hysteretic characteristic under cyclic loading. Fig. 21bshows that the model accurately captures local buckling modes.Fig. 21c shows a comparison of the out of plane buckling undermonotonic loading, which also shows the analytical results matchthe experimental.

Fig. 21. Comparison of experimental results to SCBF 3D FE model, adapted from [13].

Table 2Calibration of fracture indices.

Ref. Fracture index Index value Fracture cycle 3D FE model

�Cumpl �pl

eqv

[3] Actual test/calibrated FE model NA 33 1.79 0.281 [21]

Df;pred ¼ 2L FY;meas2E þ 0:046 bd

t2

� ��0:1KLr

� �0:3� �2

( )4.06 32 1.16 0.24

2 [21] Df;pred ¼FY;meas

E Lð2:4þ 8:3kÞ 2.57 33 1.80 0.29

3 [22] �f ¼ 2HHþB

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðDmaxþjDmin jÞ

LB

q0.22 32 1.14 0.22

114 K.M. Ward et al. / Engineering Structures 45 (2012) 104–116

As the FE model does not capture fracture behavior directly, it isnecessary to introduce numerical indices to predict fracture, cali-brated using experimental fracture results. Table 2 lists the resultsof three fracture indices [21–23] along with experimental datafrom the SCBF tests [3] and corresponding FE models. Includedare the value of the fracture index (col. 4) and the predicted cycleof fracture (col. 5), which may be compared to the actual fracturecycle in the experiment (row 1). The three indices all predict bracefracture within one cycle of the experimentally recorded fracture.The last two columns of Table 2 show the instant and cumulativeplastic strain from the FE model at the cycle of fracture. These mea-sures are used to estimate fracture in subsequent SCBF FE models.

5.3. CMDB low cycle fatigue life assessment

A seismic design [24] for Los Angeles [25] is chosen for the SCBFperformance comparison. A CMDB system (Xb = 2.00, Xa = 1.00)was designed to the same nominal strength and geometry using[14]. The brace used is an HSS 8 � 8 � 3/8 in a 3.66 m � 3.96 mbay. Fig. 22a shows the SCBF base shear vs. drift of the systemsubjected to the same cyclic loading protocol of Fig. 21. Table 3shows the range of fracture predictions from the various fractureindices along with the plastic strains recorded at HSS midspan atthe predicted time of fracture.

The large cumulative plastic strains in Table 3 are a result oflocal buckling in the plastic hinge region at the HSS midspan.Reversal of loading as in an earthquake can cause this regionto fracture. Fig. 22b compares the accumulated plastic strainsin the SCBF and CMDB during the cyclic loading. The range offracture initiation, taken from Table 3, is indicated on the plot.Based on these analytical results, high plastic strains in the

HSS of the SCBF indicate probable fracture in the SCBF. Theseanalytical tests indicate improved low cycle fatigue behaviorfor the CMDB, but these results must be confirmed throughexperimental tests.

5.4. Net section fracture

Net section fracture is now considered for the CMDB. Unlike theSCBF, the CMDB cruciform section connects to all four sides of theHSS section, which should lead to improved effective section effi-ciency related to shear lag. However, each of the four connectionsis shorter and the net section of the HSS is smaller in the CMDBsystem.

The potential for fracture in the connection region in the elasticand inelastic state is examined here using two measures: (1) ashear lag factor,

U ¼ Ae=An ½24� ð5Þ

which is a measure of effective net section efficiency, where U isanalytically measured based on the ratio of average to maximumaxial stress across the net section (Table 4) and (2) a ductile fractureindex,

W ¼ �pleqv

�f½26� ð6Þ

where

�f ¼ cðe�1:5T1 Þ ð7Þ

and the triaxiality index T1 is taken as [27],

T1 ¼rm

reqvð8Þ

Table 3Fracture prediction for HSS 8 � 8 � 3/8.

Fracture index Index value Fracture cycle �Cumpl �pl

eqv

1 3.01 30 1.59 0.292 4.41 33 3.45 0.103 0.22 31 2.19 0.22

Table 4Shear lag factor (U) (Eq. (5)).

SCBF (code value) [24] .83SCBF (analytically calculated value) .72CMDB CC (analytically calculated value) .81CMDB EC (analytically calculated value) .79CMDB (implied code value) .92

Fig. 23. Triaxiality index for the SCBF and CMDB vs. drift.

Fig. 24. Earthquake response for different CMDB designs and SCBF.

Fig. 22. (a) Force comparison of SCBF and CMDB of similar strength and geometry. (b) Plastic strain comparison.

K.M. Ward et al. / Engineering Structures 45 (2012) 104–116 115

where rm = (r1 + r2 + r3)/3, the hydrostatic stress, and reqv is theVon Mises equivalent stress,

reqv ¼ffiffiffi12

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr1 � r2Þ2 þ ðr2 � r3Þ2 þ ðr3 � r1Þ2

qð9Þ

c is a material constant which may be solved for consideringT1 = 0.33 and �f = 0.20 for a uniaxial state of tension for the HSSsteel. A value of W greater than one indicates a high likelihood offracture. The CMDB system in Fig. 22 was examined using this indexunder the same cyclic loading protocol. Its value of W did not ex-ceed 1.0 for these loading conditions.

Fig. 23 compares the ductile fracture index for the critical CMDBand SCBF net sections under tension. As seen, the ductile fractureindex in the CMDB is lower than in the SCBF, and everywhere be-low unity. The CMDB system was also examined under the cyclic

loading protocol shown in Fig. 22 and the value of PSI did not ex-ceed unity at any time.

5.5. Earthquake simulations

The previous results indicate the potential for higher low cyclefatigue life for the CMDB relative to a SCBF for similar load proto-cols. However, the EC rotational stiffness and enhanced energy dis-sipation should also tend to lower seismic demand. Accordingly,earthquake simulations are conducted on the CMDB system andSCBF to compare the seismic demands of different CMDB designs.

Fig. 24 shows the global seismic response (shown as base shearvs. roof drift) for three different Xb and Xa CMDB designs and aSCBF. Each is designed for and subjected to a SDC level D [28] de-sign basis earthquake (DBE) [14] of Fig. 19b using 3% Rayleighdamping [14].

As seen: (1) the Xa = 1.2, Xb = 3.0 design is too weak relative tothe SCBF and incurs larger drifts; (2) the Xa = 1.2, Xb = 1.5 designwith improved post-buckling strength and energy dissipation hassimilar response; and (3) the Xa = 0.8, Xb = 1.5 design with im-proved strength, post-buckling strength and energy dissipationhas lower seismic demands. Thus, not only do the CMDB designshave the potential for improved deformation capacity to an SCBF,they also have the potential to lower seismic demands.

6. Conclusions

A Cast Modular Ductile Bracing system is under developmentthrough analytical research. The CMDB serves as an alternative toSpecial Concentrically Braced Frames (SCBFs). The CMDB designis described by two main strength parameters, a flexural ((Xb))and an axial (Xa) overstrength factor. The following conclusionsare made:

116 K.M. Ward et al. / Engineering Structures 45 (2012) 104–116

1. A low Xa maximizes buckling strength at low brace slender-ness. Strength is rather insensitive to this parameter at highslenderness.

2. A low (Xb) will maximize post-buckling strength but has littleeffect on the critical load.

3. A sufficient length is required to limit tension deformationdemands and extend low-cycle fatigue life. An end cast compo-nent length ratio of 4.4% and a center cast component lengthratio of 7.4% seem sufficient for typical brace lengths.

4. A CMDB design with Xa > 1.0, (Xb > 1.0) will concentrate theplastic zone within the cast components. An Xa < 1.0 will initi-ate plastic strain in the HSS member. The most efficient designin terms of strain demand relative to capacity is an Xa slightlyless than 1.0 and an (Xb) slightly greater than 1.0.

5. Potential designs are identified in Fig. 18 that maximize buck-ling and post-buckling strength within expected plastic defor-mation limits for the components.

6. Analyses indicate the CMDB net section may have similardemands to an SCBF.

7. Analytical results indicate that CMDB designs in cyclic analysesmay possess a higher low cycle fatigue life than an SCBF, and inearthquake simulations, have the potential for lower seismicdemands. These results must be confirmed through futureexperimental testing.

Design strength expressions and geometric limits for bucklingcontrol are provided in [11]. The results of these two papers arebeing used to create a physical prototype for laboratory testing.

Acknowledgments

This research was supported by NSF Grant No. CMS-0324664.Supplemental support was provided by the American Institute ofSteel Construction (AISC) and the Steel Founder’s Society ofAmerica (SFSA). The writers are grateful for this support. Anyopinions, findings, and conclusions or recommendations expressedin this material are those of the writers and do not necessarilyreflect the views of the National Science Foundation.

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