inelastic cyclic model for steel braces

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Inelastic Cyclic Model for Steel BracesJun Jin1 and Sherif El-Tawil, P.E., M.ASCE2

Abstract: A beam–column element that can accurately model the inelastic cyclic behavior of steel braces is presented. A bsurface plasticity model in stress-resultant space coupled with a backward Euler algorithm is used to keep track of spread ofthrough the cross section. Deterioration of cross-section stiffness due to local buckling is accounted for through a damage mproposed formulation has been implemented in a large deformation analysis program and is shown to be capable of predireasonable accuracy the experimentally observed inelastic behavior of a variety of members subjected to reversed cyclic loasubassemblage under simulated seismic conditions.

DOI: 10.1061/~ASCE!0733-9399~2003!129:5~548!

CE Database subject headings: Buckling; Postbuckling; Bounding surface; Plasticity; Cyclic loads; Bracing; Steel.

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IntroductionBraced steel frame structures are popular in regions of high semicity. The steel braces improve the lateral strength and stiffneof the structural system and contribute to seismic energy dissiption by deforming inelastically during an earthquake. Steel braccan be designed to resist only tensile forces, or to resist botensile and compressive axial forces. Tension-only braces are tstructural members that buckle early under compressive load, ahence their compressive capacity is ignored in design. Buildingthat include tension-only braces have performed rather poorduring strong earthquakes, and are unpopular in regions thatprone to strong seismic shaking~Bruneau et al. 1998!. Experi-ments have shown that, in general, tension–compression braprovide better performance during an earthquake, but their behaior under severe cyclic loading is complicated and not yet weunderstood.

The cyclic inelastic behavior of brace members is complex duto the influence of the following physical phenomena: yielding intension, buckling in compression, postbuckling deterioration ocompressive load capacity, deterioration of axial stiffness witcycling, low-cycle fatigue fractures at plastic hinge regions, anthe Bauschinger effect. These factors complicate the formulatioof efficient analytical models that are capable of accurately simlating the inelastic behavior of steel braces. Nevertheless, praccal and reliable analytical tools are essential for the transitiofrom current prescriptive seismic codes to performance-based dsign specifications, which require accurate predictions of inelaslimit states up to structural collapse.

1PhD, Assistant Professor, Dept. of Maritime Systems EngineerinTexas A&M Univ. at Galveston, P.O. Box 1675, Galveston, TX 775531675.

2PhD, Associate Professor, Dept. of Civil and Environmental Engneering, Univ. of Michigan, Ann Arbor, MI 48019-2125~correspondingauthor!. E-mail: [email protected]

Note. Associate Editor: Victor N. Kaliakin. Discussion open until Oc-tober 1, 2003. Separate discussions must be submitted for individual ppers. To extend the closing date by one month, a written request mustfiled with the ASCE Managing Editor. The manuscript for this paper wasubmitted for review and possible publication on December 11, 200approved on September 5, 2002. This paper is part of theJournal ofEngineering Mechanics, Vol. 129, No. 5, May 1, 2003. ©ASCE, ISSN0733-9399/2003/5-548–557/$18.00.

548 / JOURNAL OF ENGINEERING MECHANICS © ASCE / MAY 2003

J. Eng. Mech. 20

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This paper presents a formulation for a beam–column elemthat can be used to simulate the inelastic cyclic behavior of tular steel braces. The model is implemented in a computer anaprogram and is verified by comparing analytical calculationsexperimental data for individual steel braces and a three-sbraced steel frame.

Inelastic Brace Buckling Models

Inelastic frame analysis models can be broadly categorizeeither macro- or micromodels according to their resolutionmodeling the nonlinear behavior of beam columns. The emphin the former is on generalized stress versus generalized sbehavior~for example, bending moment versus curvature behior! as opposed to pointwise stress versus strain response ilatter. Macromodels are therefore more computationally efficthan micromodels and form the basis of most large-scale anaof two- and three-dimensional frames. Typical macromodelseither of the concentrated or the distributed type. Elements bon the concentrated approach lump all inelasticity at the endthe member, and thus deal with inelastic material behavior inapproximate yet computationally efficient manner. Distribumodels, on the other hand, are more rational than concentplasticity models in that cross-sectional behavior is monitoalong the length as opposed to only at the ends. However, thecomputationally more expensive than concentrated models.

According to Ikeda and Mahin~1986!, frame element modelswhich have been used to simulate the inelastic behavior ofbraces, can alternatively be classified as finite element, pheenological, and physical theory models. Finite element modelsessentially micromodels, whereas the phenomenologicalphysical theory models fall under the macromodel category.their very nature, finite element models of a brace member armost rigorous and rational. However, they are computationexpensive. Phenomenological models are based on simphysteretic rules that only mimic the observed axial force veraxial displacement response of a brace member. These mgenerally possess one local degree of freedom~axial deformation!and express the load–displacement hysteresis cycles using aber of piecewise linear segments. The use of phenomenolomodels requires specification of numerous empirical input pareters for each element analyzed. The empirical input data

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generally derived from experiments or the results of more refinanalysis models. Despite the difficulty of determining input datphenomenological models have been widely used for nonlinseismic analyses; for example in Nilforoushan~1973!; Singh~1977!; Jain et al.~1978!; Maison and Popov~1980!; Ikeda et al.~1984! and Fukuta et al.~1989!.

Physical theory models are more fundamental than phenoenological models and are formulated based on physical conerations that influence inelastic brace behavior. For exampphysical theory models account for the interaction between being and axial effects and are generally implemented as part olarge deformation analysis formulation. Unlike phenomenologicmodels, input parameters for physical theory models are basedmaterial properties such as steel yield strength and moduluselasticity as well as geometric properties including cross-sectioarea, moment of inertia, etc. The simplest and most commoused physical theory models of a brace member are concentrmacromodels. Typically, such models are comprised of elaselements connected by a plastic hinge at mid-span. The boundconditions are pinned–pinned, and when fixed–fixed conditioneed to be simulated, an ‘‘effective length’’ portion of the fixedfixed member is considered as pinned–pinned. Examples of smodels can be found in Nilforoushan~1973!; Nonaka ~1973,1977!; Gugerli and Goel~1982!; Shibata~1982!, and Remennikovand Walpole~1997b!.

Most of the published physical theory models suffer fromnumber of significant limitations, the most important being:~1!concentrated inelastic behavior is assumed in the plastic hinregion and therefore spread of plasticity along the member lenis not explicitly accounted for;~2! transition from elastic to plas-tic behavior is abrupt and does not account for the Bauschineffect; ~3! degradation of axial stiffness with cycling is not simulated; and~4! boundary conditions are pinned–pinned. Refinephysical theory models attempt to address one or more of thlimitations. For example, the models in Ikeda and Mahin~1984!and Remennikov and Walpole~1997a! account for the Bausch-inger effect and the degradation in axial brace stiffness with cclic loading.

This paper describes a beam–column model that can be uto represent inelastic brace buckling. The proposed modeldresses all of the limitations listed above, i.e., it accounts fgradual spread of plasticity along the length and cross sectisimulates the degradation of axial stiffness with cycling, and hno restrictions on the boundary conditions. The model is of tdistributed macrotype and makes use of a bounding surface pticity model applied at the cross section level to relate stresssultants to generalized cross section strains, i.e., centroidal astrain and curvatures. The proposed model is an extensionanother model previously developed by El-Tawil and Deierle~2001a,b! and is formulated taking into consideration the specichallenges associated with simulating the inelastic behaviorsteel brace members. The developed model is verified by comping analytical data to experimental results published by Popet al. ~1979!; Black et al.~1980!; and Ghanaat~1980!.

Stress-Resultant Plasticity

Stress-resultant plasticity models for beam columns involve aaptations of classical stress-space plasticity rules to model inetic behavior under combined axial forces and moments. For ccentrated hinge models, member forces are related to memberdeformations~axial shortening and rotations!, whereas in the dis-

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tributed model, proposed herein, cross-section forces are relato generalized cross-section strains~centroidal axial strain andcurvature! as shown in Fig. 1. The incremental section forcesdFand generalized strainsde are defined as

dF5^dP dMz dMy&T (1)

de5^de dfz dfy&T (2)

whereP5axial load;Mz andM y5major and minor bending mo-ments; e5axial centroidal cross-section strain; andfz andfy5major and minor axis bending curvatures. Incremental stracan further be separated into their elastic and plastic componeas follows:

de5dee1dep (3)

Kinematics of Bounding Surface Model

Kinematics of the plasticity formulation are governed by a bouning surface model. The loading and bounding surfaces of the pticity model are shown in Fig. 2@two-dimensional~2D versions ofthe surfaces shown for clarity!#. The model consists of two nestedsurfaces, where the inner~or loading! surface encloses a regionwhere cross-sectional behavior is assumed to be elastic. Thegion inbetween the two surfaces corresponds to partial yieldingthe cross section, and the bounding surface represents full plafication. Motion of the two surfaces is governed by a kinemahardening rule, where the surfaces translate but do not chashape or rotate. For computational convenience, the loadingface is assumed to be a scaled down version of the boundsurface so as to ensure that the loading surface stays withindoes not overlap the bounding surface.

Fig. 1. Stress resultants and corresponding generalized strain

Fig. 2. Bounding surface model for steel members

JOURNAL OF ENGINEERING MECHANICS © ASCE / MAY 2003 / 549

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Initial yielding ~plastic loading! of the cross section com-mences whenever the calculated stress resultant comes in cowith the loading surface, indicated by pointA in Fig. 2. Upon theinitiation of plastic loading, a conjugate point on the boundinsurface,A8, is determined, and with continued plastic loading, thloading surface translates along the lineu connecting pointA toA8. The bounding surface translates in the same direction asloading surface, but at a slower rate. As shown in Fig. 2, tconjugate pointA8 on the bounding surface is determined sucthat the normals,g, to the loading and bounding surfaces atA andA8 are parallel. The direction of the loading surface movementhe same as that implied by Mroz’s kinematic hardening ru~Mroz 1967!. To enforce the consistency condition, the magnituof surface motion is determined so as to maintain contact betwthe force point and the loading surface. Once the loading abounding surfaces touch, Mroz’s rule is no longer applicable bcauseu is undefined, and so surface motion is assumed togoverned by Prager’s rule Prager~1956!, where surface motionoccurs in the surface gradient direction, i.e., parallel tog in Fig. 2.Continued loading after both surfaces touch is handled usingbackward Euler scheme as discussed later on in the paper.

Generalized Force–Strain Relations

Using the loading and bounding surfaces, the plastic cross-secstiffness coefficients for the principal bending and axial loadidirections are given by the following expression and shownFig. 3:

Kp,i5Ke,iFk11k2S d

din2dD k3Gi

(4)

where Ke,i5elastic cross-section stiffness; andk1 , k2 , andk35modeling coefficients for each principal directioni. Referringto Fig. 3, thek1 parameter controls the residual hardening stiness, whereas thek2 andk3 parameters control the rate of softening after yielding initiates in the cross section. Thed/(din2d)term represents the proximity of the force point to the boundisurface. Whend5din the section is elastic and the plastic stiffnesmodulus is set to infinity. Whend50, the force point is at thebounding surface, implying full plastification with the plastistiffness equal tok1 . Between these two limits (d5din to 0!, theplastic stiffness varies as a function ofd.

Fig. 3. Force versus generalized plastic strain relationship


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Whereas in stress-based plasticity the distanced is usuallytaken as the distance between the force and conjugate pointsthe loading and bounding surfaces, this is inappropriate for streresultant plasticity, where the axial and bending moment capaties are often significantly different in magnitude. In the casemoment versus axial load, the capacities also differ in units. Tadopted approach separately evaluates the plastic stiffness mulii for each of the three principal directions using distinct distances,di , measured along each principal direction.

For each principal direction,din is the distanced at the initia-tion of a new plastic loading process. If for any reasond exceedsdin , din assumes the new value ofd. This may be brought aboutby either unloading, or a situation during which such a largchange of direction occurs so that, effectively, a new plastic prcess initiates~Dafalias and Popov 1975!. The initialdin values forall three loading directions are reset whend exceedsdin for anyone of the directions. Problems can arise due to overshootassociated with small unloading–reloading cycles, and strategfor dealing with this have been adopted based on work by Sfaanakis and Fardis~1991!.

The plasticity parametersk1 , k2 , andk3 , used in this research,are calibrated using a fiber section model. The calibrationachieved by comparing the force versus generalized strainsponses calculated using Eq.~4! to fiber analysis data for aW3503147 (W14399 in English units! with elastic–perfectlyplastic steel properties. The reader is referred to El-Tawil aDeierlein ~2001b! for details about this process. This particulasection is chosen to represent typical medium weight rolledWshapes. Experience with Eq.~4! shows that cross-section responsis rather insensitive to the values of the chosen calibration paraeters, and that they work well for a range of section weights atypes including tubular members. The calibration parameterslisted in Table 1.

Plastic Flow

A function of the formf (F2a)5k describes the loading surfacein terms of the stress resultants,F, an offset vectora describingthe position of the surface in stress resultant space, and a harding parameterk. In the proposed formulation, this surface represents initial yielding of the cross section. Plastic loading occuwhen the incremental plastic~dissipated! energy is positive.Drucker ~1959! defines a work hardening~or stable! plastic ma-terial as one in which the work done during incremental loadingpositive and the work done in a loading–unloading cycle is nonegative~zero or positive!. This definition is generally known asDrucker’s postulate and, while originally developed for stressstrain behavior, may be extended to stress-resultant analysecross sections. This postulate leads to two important conquences for a work hardening cross section that is stableDrucker’s sense: convexity of the loading surface, and for ass

Table 1. Plasticity Calibration Parameters

Direction k1 k2 k3 zK zk2

Axial 0.005 6.0 1.0 0.1 1.0

Majorbending

0.005 0.7 1.2 2.0 3.0

Minorbending

0.005 0.7 0.85 2.0 3.0

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ciated flow, the normality condition. The normality condition im-plies that plastic deformations are normal to the loading~yield!surface, which is central to the derivation of the inelastic stiffnesmatrix and provides a convenient way to check for plastic loadinand unloading.

Enforcing the normality assumption, the incremental plastistrain vector is proportional to the normal,g, at the force point onthe yield surface

dep5dlg (5)

wheredl5plastic deformation parameter andg5gradient~nor-mal! to the cross-section loading surface, given by the followingequation:

g5] f ~F!

]F5 K ] f

]P

] f

]Mz

] f

]M yL T

(6)

The above-described flow rule results in inelastic coupling oaxial and bending terms through the plasticity relationships.

El-Tawil and Deierlein~1998! previously investigated the ac-curacy of stress-resultant plasticity assumptions through a serof analyses where plastic flow directions implied by the assumeloading surface and the normality condition are compared to platic flows determined using a fiber section analysis that is nosubject to these assumptions. They showed that although the nmality rule and surface translation according to Mroz is fairlyaccurate for partially plastified, bisymmetric sections subjected tbiaxial bending, the correlation is not good for partially plastifiedsections subjected to uniaxial bending and axial load. They alsshowed that there is consistent agreement between the numerand theoretical flow angles as the force point approaches the fuplastified condition at the bounding surface. As a result of thistudy, El-Tawil and Deierlein~1998! suggested that:~1! strictenforcement of the associated flow rule between axial and bening strains should be avoided for partially plastified sections—particularly when the loading surface is a scaled-down version othe fully plastified bounding surface and does not match the truinitial surface as assumed in this paper; and~2! the normalitycondition should be instated when the section is fully plastifiedThese recommendations imply that classical plasticity rules ano longer followed when the section is partially plastified, buthey were found to improve results particularly in the case ominor bending of steel wide flange sections, and are therefoadopted in this research.

Bounding Surface Equation

El-Tawil and Deierlein~2001b! proposed the following equationfor the bounding strength surface. Although originally calibratedto aW3503147 (W14399 in English units! it was found to givereasonable results for a wide range of wide flange members atubular members with rectangular and circular cross sections

S mz

12paD n

1S my

12pbD n

51 (7)

where p5(P2a1)/Pn5ratio of axial force to the nominalaxial strength; and mz5(Mz2a2)/Mzp , my5(M y2a3)/M yp5respectively, the ratio of bending moment to nominal plastic strength corresponding to strong and weak axis bending. Thvariablesa1 , a2 , and a35components of the offset vector de-scribing the location of the surface inP, Mz , andM y directions.The variablesa, b, andn5parameters defined according to sec-tion characteristics and are calibrated using a cross-section fib

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analysis such that:a51.35, b52.65, andn51.7(113p4). De-tails of the calibration process can be found in El-Tawil anDeierlein ~2001b!.

Since buckling will cause bending moments to pick up aboonly one principal axis in members with bisymmetric sectionEq. ~7! will generally boil down to one of the following 2D ver-sions, the first of which is plotted in Fig. 2:

S mz

12paD n

51 or S my

12pbD n

51

Although the 2D form of these equations is particularly simpledeal with in the numerical calculations, it presents a special chlenge as described in the following section in integrating the plticity equations because of the top and bottom corners.

Incremental Elastoplastic Equations

Following standard practices of the theory of plastic flow, it cbe shown that the incremental forces are related to the incremtal displacements through the following relationship~El-Tawiland Deierlein 2001a!:

dF5@De2Deg@gT~De1Dp!g#21gTDe#de5Dtde (8)

where the term in brackets represents the inelastic cross-secstiffness matrixDt . The second term in the brackets is sometimreferred to as a ‘‘plastic reduction matrix’’ since it represents tdecrease in elastic stiffness due to yielding.De5diagonal matrixcontaining the sectional elastic stiffnesses

De5diag EA EIz EIy& (9)

Dp5matrix of plastic stiffness coefficients

Dp5diag Kp,p Kp,z Kp,y& (10)

The matrix operations involved in Eq.~8! are relatively simplesince the matrices are at most on the order of 3. Moreover,term that requires inversion,@gT(De1Dp)g#21, is a scalar andhence presents no difficulty.

Loading in Fully Plastified State

As a cross section is loaded with both bending moments and aforces, the loading surface will approach the bounding surfaand will eventually touch it—signaling that the section hareached the fully plastified condition. Continued loading will rsult in sliding between the surfaces as they move to satisfyimposed kinematic constraints. As a consequence of the surftouching, the Mroz rule governing surface kinematics in the ptially plastified state is no longer valid because the vectoru inFig. 2 is not defined. This will cause the force point to drift awafrom the touching surfaces, leading to numerical problems asanalysis proceeds. The following algorithm based on the Prarule ~1956! coupled with the backward Euler method is implemented to ensure that the force point lies on the touching sfaces.

The change in plastic strain as the load step is incremenfrom j to j 11 can be expressed as@contrast with Eq.~5!#

dep5dl b~12g!gj1ggj 11c (11)

The vectorgj gives the direction of plastic flow at the beginninof the increment, andgj 11 gives the direction of plastic flow atthe end of the increment, which is the direction of plastic flow


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the beginning of the next increment. The parameterg may begiven values that range from 0 to 1.

According to Stouffer and Dame~1996! wheng50, Eq. ~11!becomes the forward Euler algorithm

dep5dlgj (12)

Wheng51, Eq. ~11! becomes the backward Euler algorithm

dep5dlgj 11 (13)

At the beginning of the step, the stress resultant incrementdF isknown or estimated. The plastic multiplierdl, the plastic strainincrementdep , the yield surface at the end of the incrementf j 11 ,and the plastic flow vector at the end of the incrementgj 11 , areall unknown. The backward Euler algorithm has been frequenused in stress-space plasticity and is popular for its effectivenand simplicity, and is adopted herein for use in stress-resultspace. At the end of thejth increment the yield function may bewritten as

f j 115 f ~Fj 112aj 11!5k (14)

The first trial statef j 111 at the end of the increment is compute

by giving the force a trial incrementdF15dF, assuming that theentire strain increment is elastic. The superscript ‘‘1’’ is usedindicate that the affected quantities are associated with thetrial ~iteration!

f j 111 5 f ~Fj 11

1 2aj 111 !5k1 (15)

whereFj 111 5Fj1dF1 andaj 11

1 5aj1da1.The first increment of the offset vector is taken as zero, i.

da150, as well as the first increment of plastic straindep1. Using

the first term of a Taylor series expansion about the trial stproduces

f j 115 f j 111 1S ] f

]FUj 11

1 D T

dF1S ] f

]aUj 11

1 D T

da5k (16)

Subtracting Eq.~15! from Eq. ~16!

~k12k!1~gj 111 !TdF2~gj 11

1 !Tda50 (17)

where

gj 111 5

] f

]FUj 11

1

5] f

]aUj 11

1

is the derivative evaluated at the trial force state.During the iteration the total strain is held fixed, i.e., Eq.~3! is

equated to zero. Using Eq.~13!, the force increment is thereforegiven by

dF5Dedee52Dedep52dl Degj 111 (18)

Since the correction iterations will be required when the loadiand bounding surfaces are touching, and surface motion is gerned by the Prager rule~1956!, surface motion can be expresseas

da5dFn5Dp dep5dl Dpgj 111 (19)

Substituting Eqs.~18! and ~19! into Eq. ~17! and solving for theplastic multiplierdl

dl5k12k

h(20)

whereh is given by

h5~gj 111 !TDegj 11

1 1~gj 111 !TDpgj 11

1 (21)


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The plastic strain, offset, and force increments may be computas

dep25dlgj 11

1 1dep1 (22)

da25dl Dpgj 111 1da1 (23)

dF252dl Degj 111 1dF1 (24)

After updating the plastic strain, offset, and force increments,new value of the trial yield function is computed, if the value issufficiently close to the target value, the iteration is terminateOtherwise,dF1 is replaced withdF2 andda1 is replaced withda2

and additional iterations are performed.Since the loading and bounding surfaces adopted in this mod

have corners, it is important to develop a strategy that deals wthis situation. Yield functions with corners can be approximateusing a smooth local rounding~see for example, Orbison 1982!.This, however, can lead to increased iterations around trounded region because of its extreme localized curvature. Tbackward Euler algorithm discussed above can be extendeddescribed in Crisfield~1997! and Jin~2002! to explicitly treat thecase of a corner without resorting to corner rounding. The retuis made to the corner region at which the yield functions intersec

Stiffness Degradation

Global buckling of a steel member is generally accompanied blocal buckling at one or more critical sections. The local bucklincauses degradation in cross-sectional strength and stiffness prerties, which increases with the amplitude of the global bucklinand with the number of applied cycles. In this work, strength anstiffness degradation at a cross section is assumed to be direrelated to the accumulated plastic energy per unit length. Teffect of degradation on the behavior of the model is achieved bmodifying some of the attributes of the formulation as the analysis proceeds.

The accumulated plastic work per unit lengthWp accumulatedover n load increments is,

Wp5(1

n

FTdep (25)

where FsecT 5total member forces at the cross section an

dep5incremental plastic strains. The energy density given by E~25! is normalized to allow a calibration that is independent of thsection properties. This is done by dividing the dissipated enerby an arbitrary normalizing elastic strain energy densityWnorm,which is calculated as

Wnorm5WnormP 1Wnorm

Mz 1WnormMy (26)

where WnormP , Wnorm

Mz , and WnormMy 5elastic strain energy densities

associated with the axial, major axis bending, and minor axbending capacities of the cross section, respectively. Thesecalculated as follows:Wnorm

P 5Pn2/2.E.A, Wnorm

Mz 5Mzp2 /2.E.I , and

WnormMy 5M yp

2 /2.E.I , whereMzp and M yp are the plastic momentcapacities in thez andy directions, respectively. The normalizeddissipated plastic work indexVp is defined as

Vp5Wp

Wnorm(27)

As discussed in El-Tawil and Deierlein~2001b!, the dissipatedplastic work index is used both as a damage index and a histo

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variable to model stiffness degradation of the element. Degration is applied to the unloading elastic stiffness and thek2 hard-ening parameter

r Ki 50.110.9•1.12jK

i•Vp (28)

r k2

i 51.012jk2

i•Vp (29)

Using Eqs.~28! and ~29!, the instantaneous unloading elaststiffnessKe,i and k2,i parameters for a specific cross section agiven, respectively, as

Ke,i5r Ki•Ke,i

initial (30)

k2,i5r k2

i•k2,i

initial (31)

The z parameters in Eqs.~28! and~29! are calibrated to test datafor steel tubes in Black~1980! and Popov et al.~1979! and aregiven in Table 1.

Member Stiffness Matrix and Geometric Nonlinearity

The stress-resultant bounding surface plasticity model descriabove provides a convenient way to model inelastic behaviorsteel beam–column cross sections. Following is a descriptionhow the inelastic stiffness matrix is calculated using the proposformulation. The cross-sectional forces at the integration poiare computed using the member-end forces calculated duringprevious analysis step. The distancesd anddin from the previousincrement are used to compute the cross-sectional plastic snesses using Eq.~4! at each integration point. The cross-sectionelastic@Eq. ~9!# and plastic@Eq. ~10!# stiffness matrices are cal-culated accounting for the stiffness degradation described by E~25!–~31!. The cross-sectional inelastic stiffness matrix at eaintegration point is then computed using Eq.~8!. The cross-sectional inelastic stiffness matrix is inverted to give the crossectional flexibility matrix, which is integrated along the memblength to yield the member flexibility matrix. The member flexibility matrix is inverted yielding the member stiffness matrixand is then transformed and expanded to account for rigid bomodes and torsional stiffness terms. A member geometric sness matrix is then added to account for geometric nonlinearitThe flexibility approach is utilized because it is formulatedterms of force interpolation functions which are unaffectedspread of inelasticity. The reader is referred to El-Tawil aDeierlein ~2001b! for a critique of the flexibility method and ad-ditional details of the formulation.

Incremental Solution of Nonlinear Problem

After assembly of the global stiffness matrix, the nonlinear prolem is solved using an Euler incremental process with an endstep unbalanced force correction. Experience has shown thata simple solution method is acceptable because the timetaken in nonlinear dynamic analysis of building structures sujected to seismic loading is usually small ranging from 0.00250.01 s. The incremental member end deformations are extrafrom the incremental global structure deformations and the incmental member end forces are recovered by multiplying the meber stiffness matrix by the incremental member end deformatioThe incremental cross-sectional forcesdF at the integrationpoints are computed from the member-end forces. When the loing surface touches the bounding surface, kinematics of the p

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ticity model are handled by the backward Euler method@Eqs.~23!and ~24!#. The incremental cross-sectional strainsde anddee arecalculated and the incremental plastic strain vector is calculfrom dep5de2dee and is used in the stiffness degradation cculations of the next increment.

Computer Implementation

The above-described formulation has been implementedgraphically interactive computer program namedDYNAMIX ~dy-namic analysis of mixed systems!. The program has been deveoped through this and previous research, with the capabilitieperform inelastic static and dynamic analysis of 3D steelmixed steel–concrete frames~El-Tawil and Deierlein 2001a,b!.The program accounts for full geometric nonlinearities (P-D andP-d effects! as well as semirigid connection behavior. A panzone model is available in the program for representing the fisize of the panel zone and the shear distortions that occur inregion. The formulations implemented inDYNAMIX have beenverified extensively by comparisons with test data and thegram has been used successfully to analyze steel frame builsubjected to severe seismic loading~Chi et al. 1998!. Followingare some of the verification studies conducted, which supporaccuracy of the brace member formulations presented inpaper. All the analyses were conducted with the same set ofbration parameters.

Popov et al. (1979)

Popov et al.~1979! studied the cyclic inelastic buckling of tubulabraces in a Southern California offshore platform. Tests wereducted on six tubular struts, which were subjected to cyclic aloading. The test specimens are shown in Fig. 4. Test resultfour of the tested struts are presented here. The diameter ofwas 101.6 mm~4 in.!, and the wall thickness ranged from 2.1 m~0.083 in.! to 3.05 mm~0.12 in.!. The beam–column models useare shown in Fig. 5. The model for the pinned–pinned specimconsists of four members. The inner two members representube, while the outer two members represent the more rigid cland end plate connections. The measured initial camber is apat the intersection of the two middle members. The model forfixed–fixed specimens consists of only two members as showFig. 5, with an initial central camber equal to that measured in

Fig. 4. Specimens used in tests by Popov et al.~1979!


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test. Steel properties were taken as reported in the reference.analytical versus test results for axial force–displacementsponse are plotted in Figs. 6 and 7. The figures show thatmodel reasonably reproduces the inelastic behavior of the stru

Black et al. (1980)

Black et al.~1980! conducted cyclic axial loading experiments on24 structural steel struts with a wide range of cross-section geo

Fig. 5. Beam–column models representing specimens testedPopov et al.~1979!


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etries. Among the tested specimens, strut 15 was rather slen(Kl /r 580) and was chosen to serve as a rigorous test fordeveloped methodology. The strut was 101.6 mm~4 in.! in diam-eter and was tested in a manner similar to that in Popov et~1979!. The strut was modeled as described in the previous stion, and measured geometric and material properties were uin the analysis. Fig. 8 shows a comparison between the analytand experimental results. It is clear that the analytical methpredicts well the complex inelastic cyclic behavior.

Analysis of 0.6 Scale Concentrically Braced FrameModel

Ghanaat~1980! conducted earthquake simulator testing of a 0scale three-story X-braced steel frame. The frame was 1.8333.66m ~6312 ft! in plan, 5.28 m~17 ft–4 in.! high, and was fabricatedfrom A36 wide flange sections. Three bracing system were tes1/2 in. diameter rods with turnbuckles, 3/4 in. diameter pipe, a13131/8 in. double angle. Each pipe or double angle brace uwas welded together at the center and to connections at the e

The beam–column model of the frame with the pipe bracesshown in Fig. 9. The analytical model accounted for shake tabehavior through two vertical springs and was constructedcording to the information contained in the report. The base aceleration was measured from a plot of the recorded shaking tamotion for test EC-400 with peak acceleration 0.283g and wasused as input in the analysis. All braces were represented byequal length elements that intersect in such a way that a smcamber (L/500) is created in the middle of each brace. The anlytical and test results of the roof drift time history are plotte

by

Fig. 6. Comparison between analysis and Popov et al.~1979! test results for pinned-pinned specimens

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Fig. 7. Comparison between analysis and Popov et al.~1979! test results for fixed–fixed specimens

together in Fig. 10. It is clear that the model is capable of reasoably reproducing the experimentally recorded roof drift up toabout 4 s, beyond which the accuracy of the model deterioratePossible reasons for the discrepancy include:~1! errors in mea-suring the input signal~especially high frequency components!from the original report; and~2! the actual strength of some of theframe members was not reported, and nominal strength was usinstead.

General Comments

Success of the proposed stress resultant plasticity model depeon careful calibration of several parameters. These parametersinto three categories pertaining to:~1! cross-section strength:a,b, and n in Eqs. ~7!; ~2! cross-section force versus generalizedstrain behavior:k1 , k2 , andk3 in Eq. ~4!; and~3! degradation ofcross-section properties with cycling:zK

i andzk2i in Eqs.~30! and

~31!. Experience with these parameters has shown that the cabrated values work well for a wide variety of situations representative of those likely to be encountered in practice. However, it iimportant to keep in mind that the model will not necessarily baccurate outside the general conditions it has been calibratedFor example, the behavior of steel braces that have cross sectithat are substantially different than those considered in the papmay not be captured correctly. Nevertheless, recalibration is

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Fig. 8. Comparison between analysis and Black et al.~1980! testresults for slender strut


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straightforward process. As previously discussed, the fiber sectimethod can be conveniently used to compute the calibration prameters in the first two categories, i.e., those related to crossection strength and force–strain behavior. Degradation paraeters in the third category~related to degradation! can becalibrated to test data or to the results of more detailed analystechniques such as continuum finite element models.

Conclusions

A refined model for simulating the inelastic cyclic behavior ofsteel braces was presented. The proposed model employs stre

Fig. 9. Analytical model for three story braced steel frame

Fig. 10. Comparison between calculated and measured roof drift fobraced steel frame


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resultant plasticity concepts that focus on forces and correspoing generalized strains and is computationally efficient. Thmodel has no restrictions on brace boundary conditions~i.e., thebrace can be pinned–pinned, fixed–fixed, or pinned–fixed! andaccounts for gradual spread of plasticity through the cross sectand along the member length as well as degradation of axstrength and stiffness with cycling. A flexibility formulation isused to obtain the member stiffness matrix and geometric nonearity is captured through the geometric stiffness matrix aproach. The proposed model is shown to be capable of predictwith reasonable accuracy the experimentally observed inelabehavior of a variety of members subjected to reversed cycloading and a subassemblage under simulated seismic conditiThe developed model is well suited for conducting analyseslarge multistory braced steel frames subjected to seismic loadiThe writers are currently conducting parametric studies of susystems with the aim of developing performance-based seismcriteria for braced steel frames.

Acknowledgments

Financial support for this research was provided in part by tU.S. National Science Foundation~Grant No. CMS 9870927!, theNippon Steel Corporation, and the Univ. of Central Florida. Thopinions stated here are those of the writers and do not necesily reflect those of the sponsors.

References

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