Response Sensitivity for Nonlinear Beam Column Elements

Transcription

1 Rspons Snsitivity for Nonlinar Bam Column Elmnts Michal H. Scott 1 ; Paolo Franchin 2 ; Grgory. Fnvs 3 ; and Filip C. Filippou 4 Abstract: Rspons snsitivity is ndd for simulation applications such as optimization, rliability, and systm idntification. h xact rspons snsitivity of matrial nonlinar bam column lmnts is drivd for th displacmnt- and forc-basd formulations. For displacmnt-basd bam column lmnts th rspons snsitivity is straightforward to comput bcaus th displacmnt fild is spcifid along th lmnt. A nw approach is prsntd for computing th rspons snsitivity of forc-basd bam column lmnts, in which th displacmnt fild is not spcifid. In this approach, th rspons snsitivity dpnds on th drivativ of unbalancd sction forcs bcaus th lmnt displacmnt fild changs with th lmnt stat. Exampl nonlinar static analyss of stl and rinforcd concrt structural systms vrify th xact rspons snsitivity for forc-basd lmnts using th nw approach. DOI: 1.161/ ASCE : CE Databas subjct hadings: Bam columns; Finit lmnts; Nonlinar analysis; Optimization; Paramtrs; Snsitivity analysis; Structural rliability. Introduction h computational simulation of structural systms has an intgral rol in arthquak nginring analysis and dsign. h ability to modl accuratly th rspons of a structural systm is crucial, particularly with th advnt of prformanc-basd arthquak nginring mthodologis. h simulatd rspons of a structur dpnds on modling assumptions, a majority of which ar basd on simplifid approximations. In addition, th dsign of a nw structural systm is a function of dsign paramtrs having uncrtain proprtis. h computation of rspons snsitivity provids guidanc to nginrs as to which paramtrs control th rspons of a structural systm. As a consqunc, th snsitivity of th systm rspons is just as important as th rspons itslf. Structural rliability, optimization, and systm idntification applications all rquir accurat and fficint rspons snsitivity computations for th convrgnc of itrativ sarch algorithms to an optimal solution point iu and Dr Kiurghian h snsitivity of a structural rspons quantity du to a chang in a paramtr can b computd in on of two ways. h first approach is th finit diffrnc mthod FDM, in which th simulation is rpatd with a prturbd valu of th paramtr. 1 Graduat Studnt Rsarchr, Dpt. of Civil and Environmntal Enginring, Univ. of California, Brkly, CA Rsarch Assistant, Dpt. of Structural and Gotchnical Enginring, Univ. of Rom, a Sapinza, Rom, Italy. 3 Profssor, Dpt. of Civil and Environmntal Enginring, Univ. of California, Brkly, CA Profssor, Dpt. of Civil and Environmntal Enginring, Univ. of California, Brkly, CA Not. Associat Editor: Marc I. Hoit. Discussion opn until Fbruary 1, 25. Sparat discussions must b submittd for individual paprs. o xtnd th closing dat by on month, a writtn rqust must b fild with th ASCE Managing Editor. h manuscript for this papr was submittd for rviw and possibl publication on Fbruary 25, 23; approvd on Sptmbr 1, 23. his papr is part of th Journal of Structural Enginring, Vol. 13, No. 9, Sptmbr 1, 24. ASCE, ISSN /24/ /$18.. his approach is tim consuming bcaus th analysis must b rpatd for ach paramtr that dfins th modl, and it is pron to numrical round-off rror for small paramtr prturbations. h scond mthod is th dirct diffrntiation mthod DDM, in which th govrning quations of structural quilibrium, compatibility, and constitution ar diffrntiatd xactly Klibr t al h DDM givs th rspons snsitivity as th analysis procds, rathr than by ranalysis with prturbd paramtrs, and it can b computd vry fficintly. In th nonlinar analysis of structural systms, thr ar two typs of formulations for bam column lmnts: displacmntbasd and forc-basd. Displacmnt-basd lmnts follow th standard finit lmnt procdur of spcifying an approximat displacmnt fild along th lmnt Zinkiwicz and aylor 2. In contrast, forc-basd lmnts intrpolat intrnal forcs, which is xact vn in th nonlinar rang of matrial bhavior Spacon t al Nunhofr and Filippou 1997 hav statd th advantags of forc-basd lmnts ovr displacmnt-basd lmnts, th most notabl bing th ability to us on lmnt to rprsnt th matrial nonlinar bhavior of a bam column mmbr, compard with svral displacmntbasd lmnts for a singl mmbr. h formulation of rspons snsitivity for displacmnt-basd lmnts is straightforward bcaus th lmnt displacmnt fild is spcifid. h drivation of rspons snsitivity for forc-basd lmnts, howvr, is mor difficult bcaus th displacmnt fild dpnds on th sction constitutiv rspons, which must b dtrmind such that quilibrium is maintaind with th lmnt forcs. h objctiv of this papr is to prsnt a uniform approach to th computation of rspons snsitivity for bam column lmnts, highlighting th similaritis and diffrncs btwn th displacmnt- and forc-basd formulations. h mthodology for computing th rspons snsitivity at th structural lvl is formulatd first. hn th rspons snsitivity of displacmntbasd lmnts is prsntd, followd by a nw drivation of th rspons snsitivity for forc-basd lmnts. Attntion thn turns to th computation of rspons snsitivity at th sction constitutiv lvl. h papr concluds with xampl applications showing th validity of th forc-basd lmnt rspons snsitivity. JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24 / 1281

2 Global Formulation for Snsitivity h global quilibrium quations for a structural systm whos rsisting forcs com from inlastic rat-indpndnt matrial modls hav th gnral form P r U,t, P f,t (1) whr P f is a vctor of th xtrnal forcs applid to th structur; and is a vctor of paramtrs for th structural modl. h xtrnal forcs ar a function of psudo-tim, t, as is th nodal displacmnt vctor U, which is obtaind by standard nonlinar solution procdurs such as th Nwton Raphson mthod. h rsisting forc vctor P r is assmbld from lmnt forcs, and it dpnds on th paramtrs xplicitly, as wll as implicitly via th nodal displacmnts. h focus of this papr is th drivation of rspons snsitivity for th lmnt rsisting forcs. As a rsult, Eq. 1 is limitd to static quilibrium ffcts bcaus th inclusion of inrtial and damping forcs for nonlinar dynamic analysis is accomplishd at th global lvl following wll stablishd procdurs, and is indpndnt of th lmnt formulation. Applying th chain rul to Eq. 1, th drivativ of th quilibrium quations with rspct to a paramtr, which blongs to th vctor, is P r U U P r P f (2) U whr P r / U is trmd th conditional drivativ of th rsisting forc vctor bcaus it givs th drivativ of th rsisting forcs with rspct to whil th displacmnts ar hld fixd. Physically, this drivativ rprsnts th chang in rsisting forcs rquird to kp th structur fixd at th currnt stat whil th paramtr changs. Using th dfinition of th tangnt stiffnss matrix, K P r / U, Eq. 2 givs a linar systm of quations for th nodal rspons snsitivity U/ U K P f P r (3) U h conditional drivativ of th rsisting forc vctor is assmbld from lmnt contributions in th sam mannr as th rsisting forc vctor. Rspons snsitivity analysis at th global lvl rquirs assmbly of th right-hand sid and solution of th factorizd linar systm of quations Eq. 3 for ach paramtr in th vctor. h drivativ of th xtrnal forc vctor, P f /, is nonzro only for paramtrs that dscrib th load applid to th structur. With th formulation at th global lvl, th lmnt contribution to th rspons snsitivity must b dtrmind. Elmnt Formulation for Snsitivity Bam column lmnts ar convnintly formulatd in a basic systm, fr of rigid body displacmnt mods. For this discussion, lmnt dformations ar assumd small. In th twodimnsional simply supportd basic systm, th lmnt dformation vctor, v, consists of thr componnts: on axial dformation and on rotation at ach nod, as shown in Fig. 1. hr-dimnsional lmnts hav six dformations. h corrsponding basic forcs, q,, ar a function of th lmnt dformations, as wll as th paramtr. At vry cross sction along th lmnt, thr ar sction dformations,, and th corrsponding sction forcs, s,. For both th Fig. 1. Simply supportd basic systm for two-dimnsional bam column lmnts displacmnt- and forc-basd lmnt formulations, th lmnt rspons is obtaind by intgration of th sction rspons, as prscribd by th quations of lmnt quilibrium and compatibility. o comput th lmnt rspons snsitivity, it is ncssary to diffrntiat th basic and sction forcs with rspct to th paramtr. By application of th chain rul, in a mannr similar to th global rsisting forcs, th drivativ of th basic forcs is k q (4) whr kq/ is th lmnt stiffnss matrix. h drivativ of th sction forcs is also obtaind by th chain rul k s (5) whr th sction stiffnss matrix is k s s/. h gomtric transformations of basic forcs q to global rsisting forcs P r and global displacmnts U to lmnt dformations v ar linar for small dformations and displacmnts. hs transformations ar carrid out by wll documntd structural analysis procdurs, and th rspons drivativs transform btwn th global and basic systms in th sam mannr as th rspons itslf. h computation of rspons snsitivity for path-dpndnt problms is a two-phas procss Zhang and Dr Kiurghian Phas on bgins with th assmbly of P r / U from th conditional drivativ of th basic forcs / v for ach lmnt. Solution for th nodal rspons snsitivity U/ by Eq. 3 concluds phas on. For path-dpndnt problms, th rspons snsitivity is also path-dpndnt. o track th path-dpndnt rspons snsitivity, snsitivity history variabls ar rquird. h computation of th drivativ of sction dformations / from th nodal rspons snsitivity prmits th updat of ths snsitivity history variabls during phas two. Which snsitivity history variabls must b stord and how thy ar updatd dpnds on th sction constitutiv modl. his procss is outlind by Zhang and Dr Kiurghian 1993 for th J 2 plasticity modl Simo and Hughs 1998 and in th Appndix of this papr for a simplifid uniaxial concrt modl. Du to th govrning quations of lmnt quilibrium and compatibility, th computation of th conditional drivativ of basic forcs and th drivativ of sction dformations is diffrnt for displacmnt- and forc-basd lmnts. h statmnts of quilibrium and compatibility, and th computational stps for th rspons snsitivity, ar prsntd in th following two sctions for ach lmnt formulation / JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24

3 Displacmnt-Basd Elmnts For displacmnt-basd bam column lmnts Zinkiwicz and aylor 2, compatibility along th lmnt is statd as a v (6) whr th matrix a a (x) contains intrpolation functions rlating sction dformations to lmnt dformations. h principl of virtual displacmnts lads to a wak form of quilibrium btwn basic forcs and sction forcs q a sdx (7) h lmnt stiffnss matrix is obtaind from linarization of Eq. 7 with rspct to th lmnt dformations k a k s a dx (8) ypically, th assumd displacmnt filds along th lmnt ar linar for th axial componnt and cubic Hrmitian for th transvrs componnt. hs displacmnt filds admit constant axial dformation and linar curvatur along th lmnt in Eq. 6, according to th Brnoulli bam thory. Du to this approximation of dformations, which is xact only for linar lastic, prismatic lmnts, it is ncssary to us svral displacmntbasd lmnts h rfinmnt to rprsnt th matrial nonlinar bhavior of a structural mmbr. It is possibl to us highr ordr intrpolation functions p rfinmnt, but th dformations along th lmnt rmain constraind to an approximat and gnrally inaccurat solution for matrial nonlinarity. h rspons snsitivity for displacmnt-basd lmnts is wll known Zhang and Dr Kiurghian 1993, but it is drivd in this papr by an approach that lnds insight into th drivation of rspons snsitivity for forc-basd lmnts. o dtrmin th rspons snsitivity, th quilibrium rlationship, Eq. 7, is diffrntiatd with rspct to a dx (9) Aftr insrtion of th basic and sction forc drivativs, from Eqs. 4 and 5, Eq. 9 xpands to k q a k s dx (1) h solution for th conditional drivativ of basic forcs givs a dx a k s dx k (11) h drivativ of th lmnt compatibility rlationship, Eq. 6 a (12) is combind with Eq. 11 to giv th following xprssion: a dx a k s a dx k (13) From th dfinition of th lmnt stiffnss matrix, Eq. 8, th last two trms on th right-hand sid of Eq. 13 ar qual, and th conditional drivativ of th basic forcs for th displacmntbasd formulation rducs to a dx (14) In th implmntation of displacmnt-basd bam column lmnts, th basic forcs and thir conditional drivativ, from Eqs. 7 and 14, rspctivly, ar valuatd by th Gauss gndr numrical intgration rul. For th assumption of a cubic Hrmitian transvrs displacmnt fild along th lmnt, two Gauss gndr intgration points is sufficint. h rsult in Eq. 14 could hav bn obtaind dirctly from Eq. 9 bcaus th condition of fixd nodal displacmnts lads to fixd lmnt and sction dformations in th displacmntbasd lmnt formulation. Howvr, diffrntiation of th lmnt quilibrium and compatibility rlationships and subsqunt combination of ths drivativs is ncssary in th drivation of rspons snsitivity for forc-basd lmnts. Forc-Basd Elmnts In th forc-basd lmnt formulation Spacon t al. 1996, th quilibrium rlationship is statd in strong form as s bq (15) whr th matrix b b(x) contains quilibrium intrpolation functions that xprss sction forcs in trms of basic forcs. In th absnc of lmnt loads, th axial forc is constant, whil th bnding momnt varis linarly along th lmnt. From th principl of virtual forcs, th compatibility rlationship btwn sction dformations and lmnt dformations is v b dx (16) inarization of Eq. 16 with rspct to basic forcs givs th lmnt flxibility matrix f b f s bdx (17) whr f s k 1 s is th sction flxibility matrix. Invrsion of th lmnt flxibility matrix givs th lmnt stiffnss matrix, k f 1. h drivation of rspons snsitivity for forc-basd lmnts is not as straightforward as that for displacmnt-basd lmnts. h difficulty ariss from th structur of th lmnt quilibrium and compatibility rlationships in th forc-basd formulation. o dmonstrat this difficulty, diffrntiation of th quilibrium rlationship, Eq. 15, with rspct to givs b (18) Aftr substitution of th drivativs of basic and sction forcs from Eqs. 4 and 5, Eq. 18 xpands to k s b k q (19) Howvr, th conditional drivativ of basic forcs cannot b dtrmind from Eq. 19 bcaus th intrpolation matrix b is not invrtibl. Adding to th difficulty, diffrntiation of th lmnt compatibility rlationship, Eq. 16, givs b dx (2) JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24 / 1283

4 abl 1. Stps for Computation of Bam Column Elmnt Rspons Snsitivity Phas Stp Forc-basd Displacmnt-basd I Sction flxibility 1 fs k s Elmnt flxibility f b f s bdx Elmnt stiffnss Sction forc conditional drivativ Basic forc conditional drivativ k f 1 b f s dx k a dx Assmbly and solution for nodal rspons snsitivity K U P f P r U II Elmnt dformation drivativ Sction dformation drivativ f sbk s f b a in which th drivativ of th sction dformations appars in th intgrand on th right-hand sid of Eq. 2. From th form of Eqs. 19 and 2, it is apparnt that furthr manipulation is rquird to driv th conditional drivativ of th basic forcs. h following drivation stablishs th rspons snsitivity for forc-basd lmnts. As a first stp, th drivativ of th sction dformations can b obtaind from Eq. 19 by solving for / f sbk s f b (21) h quantity in parnthss on th right-hand sid of Eq. 21 rprsnts an unbalancd sction forc drivativ bcaus it is th diffrnc btwn th intrpolation of th conditional drivativ of th basic forcs and th conditional drivativ of th sction forcs. h conditional drivativ of th basic forcs must b computd, and th ky stp is th substitution of Eq. 21 into Eq. 2 b f s bdx k b f s b dx (22) From th dfinition of th lmnt flxibility matrix in Eq. 17 and th idntity fk I, th trm on th lft-hand sid and th first trm on th right-hand sid of Eq. 22 ar qual. Eq. 22 thn rducs to b f s b dx (23) Eq. 23 is important bcaus it rquirs th unbalancd sction forc drivativs to b zro in an avrag sns along th lmnt. Aftr invrsion of th lmnt flxibility matrix in Eq. 23, th final xprssion for th conditional drivativ of th basic forcs in th forc-basd formulation is k b f s dx (24) It is worth noting th similarity btwn Eqs. 14 and 24 for th two bam column formulations sinc it can b shown that th intrpolation matrix rlating th incrmnt in sction dformations to an incrmnt in lmnt dformations is a f s bk for forc-basd lmnts. h intrpolation of sction dformations from lmnt dformations dpnds on th currnt lmnt stat bcaus th lmnt stiffnss changs du to nonlinarity in th sction constitutiv rspons. As a rsult, th unbalancd sction forc drivativ must b includd in th drivativ of th sction dformations, as sn in Eq. 21. For displacmnt-basd lmnts, th drivativ of th sction dformations is computd dirctly from th drivativ of th lmnt dformations in Eq. 12 according to th spcifid lmnt displacmnt fild. In th numrical implmntation of forc-basd bam column lmnts, th compatibility rlationship, Eq. 16, must b solvd by an itrativ mthod,.g., Nwton Raphson, or by a nonitrativ approach Nunhofr and Filippou h intrmdiat stps in finding lmnt compatibility do not affct th rspons snsitivity for ths lmnts bcaus th snsitivity is only computd at a convrgd stat in which compatibility is satisfid. o assur th computation of th xact rspons snsitivity for forcbasd lmnts, th consistnt sction flxibility and lmnt stiffnss matrics from th compatibl stat must b usd in Eqs. 21 and 24. Furthrmor, th Gauss obatto numrical intgration rul is applid to Eqs. 16 and 24 bcaus it placs intgration points at th lmnt nds, whr th bnding momnts ar largst in th absnc of lmnt loads. ypically, thr to fiv Gauss obatto intgration points along th lmnt accuratly rprsnt th matrial nonlinar bhavior. h two-phas procss for th computation of rspons snsitivity for bam column lmnts is summarizd in abl 1. h conditional drivativ of th basic forcs is computd by ithr Eq. 14 or 24. hn, aftr assmbly and solution for th nodal rspons snsitivity in Eq. 3, th drivativ of th sction dformations is dtrmind by Eq. 12 or 21. h computational stps outlind in abl 1 hav bn implmntd in th OpnSs finit lmnt analysis framwork McKnna t al / JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24

5 Fig. 2. oad displacmnt rspons of stl cantilvr bam for forc- and displacmnt-basd bam column lmnts Sction Formulation for Snsitivity Having formulatd th rspons snsitivity at th global and lmnt lvls, attntion turns to th formulation at th sction lvl. hr ar two mthods to spcify th sction constitutiv rspons. h first mthod is th spcification of th sction forcs as a function of th sction dformations by a rsultant plasticity modl. Dirct diffrntiation of th sction rspons givs th conditional drivativ of th sction forcs, /. h scond mthod is intgration of th matrial strss rspons ovr th sction ara s A a s da (25) h compatibility matrix a s givs th matrial strain as a function of th sction dformations a s (26) By a drivation idntical to that for displacmnt-basd lmnts, th conditional drivativ of th sction forcs for this mthod is a s da (27) A Fig. 3. Stl cantilvr bam rspons snsitivity through on sinusoidal load cycl with rspct to yild strss computd by dirct diffrntiation and finit diffrnc mthods: a forc basd, on lmnt and b displacmnt basd, fiv lmnts Stl Cantilvr In th first xampl, a cantilvr stl bam is usd to compar th displacmnt- and forc-basd formulations. h rspons of th W21 5 sction is intgratd by th midpoint rul with 24 layrs from th strss strain bhavior rprsntd by a uniaxial vrsion of th J 2 plasticity modl. h lastic modulus is E MPa, th yild strss is y 25 MPa, and 2% kinmatic strain hardning is assumd. h bam is loadd at its fr nd through on sinusoidal cycl of pak intnsity P 134 kn. h load displacmnt rspons of th stl cantilvr is shown in Fig. 2. A msh of fiv displacmnt-basd lmnts capturs th matrial nonlinar bhavior of th bam whr only on forc-basd lmnt is rquird. h rspons snsitivity is computd with rspct to th stl yild strss for both lmnt formulations, and is shown in Fig. 3. h rsults obtaind by th FDM convrg to thos obtaind by th DDM for both th displacmnt- and forc-basd lmnt formulations. Discrt jumps ar sn in th snsitivity as th stl matrial switchs from lastic to plastic stats Cont t al Diffrntiation of th statmnt of sction compatibility, Eq. 26, givs th rlationship btwn th drivativs of matrial strain and sction dformations a s (28) his rlationship is ncssary to account for path-dpndnt bhavior in th computation of rspons snsitivity at th matrial lvl. Exampls o show th validity of th nw DDM for th forc-basd lmnt formulation, th rspons snsitivity is compard with that obtaind from th FDM for small paramtr prturbations,. For dcrasing paramtr prturbations, th snsitivity obtaind by th FDM should convrg to th DDM snsitivity lim U U U (29) h convrgnc of th FDM to th DDM indicatd in Eq. 29 is dmonstratd in th following nonlinar static analyss of stl and rinforcd concrt structural systms. Fig. 4. Stl cantilvr bam local rspons at th pak load for forc- and displacmnt-basd lmnt formulations: a curvatur distribution and b snsitivity of curvatur distribution with rspct to yild strss, computd by dirct diffrntiation mthod JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24 / 1285

6 Fig. 7. Stl fram rspons snsitivity through on sinusoidal load cycl with rspct to yild momnt computd by dirct diffrntiation and finit diffrnc mthods: a forc basd, on lmnt pr mmbr and b displacmnt basd, fiv lmnts pr mmbr Fig. 5. Stl momnt-rsisting fram modl for snsitivity xampl An application of rspons snsitivity to structural dsign is to dtrmin th chang in displacmnt rsulting from a chang in th slctd paramtr. Plottd in Fig. 3 is th quantity ( U/ y ) y, th snsitivity of th tip displacmnt with rspct to yild strss, scald by th yild strss. Multiplication of this quantity by a prcnt chang in yild strss givs th rsulting chang in tip displacmnt. For xampl, at th pak load of 134 kn, th tip displacmnt is 43.4 mm for th forc formulation, and th scald rspons snsitivity computd by th DDM is mm. A 5% incras in yild strss givs a 1.7 mm 25% rduction in th tip displacmnt. Although th global rspons of th stl bam agrs vry wll for th displacmnt- and forc-basd lmnt formulations, th rspons snsitivity is quit diffrnt, as sn in Fig. 3. o assss this discrpancy in rspons snsitivity, th local rspons of th bam is xamind. hr is a noticabl diffrnc in th distribution of curvatur in th plastic hing zon at th pak load of 134 kn, as shown in Fig. 4 a. h scald rspons snsitivity of th curvatur distribution at th pak load is shown in Fig. 4 b. At th pak load, th maximum curvatur prdictd by th forc formulation is /mm, and th associatd scald rspons snsitivity is /mm. A rduction of /mm 39% in maximum curvatur is stimatd from a 5% incras in th stl yild strss. Stl Momnt-Rsisting Fram h scond xampl is a fiv-story, on-bay stl momntrsisting fram, as shown in Fig. 5. All mmbrs ar W21 5 with E MPa and y 25 MPa. h flxural bhavior is rprsntd by a bilinar momnt curvatur rlationship with 2% kinmatic hardning. h axial bhavior of th mmbrs is assumd to b linar lastic and uncoupld from th flxural bhavior. h fram is loadd latrally through on sinusoidal cycl by an invrtd triangular distribution. h maximum forc applid at th roof lvl is P 15 kn, which givs a pak bas shar of 45 kn for th givn latral load distribution. h rspons is computd with on lmnt pr mmbr for th forc-basd formulation fiv Gauss obatto intgration points, and fiv lmnts pr mmbr for th displacmnt-basd formulation. h bas shar is plottd against th roof displacmnt in Fig. 6 for both lmnt formulations. h snsitivity of th roof displacmnt with rspct to th yild momnt, M y,of th W21 5 sction, is shown in Fig. 7. h snsitivity obtaind by th FDM convrgs to th DDM snsitivity and discrt jumps appar in th snsitivity as plastic hings form throughout th structur. his xampl shows th validity of th rspons snsitivity computation for lmnts in which a rsultant plasticity modl rprsnts th sction bhavior, rathr than th intgration of matrial strss, as in th first xampl. Rinforcd Concrt Cantilvr Bam h final xampl is of a rinforcd concrt cantilvr bam. h sction is rctangular, with two layrs of fiv No. 9 stl bars Fig. 6. oad displacmnt rspons of stl momnt-rsisting fram for forc- and displacmnt-basd bam column lmnts Fig. 8. Strss strain rlationship for uniaxial concrt modl 1286 / JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24

7 stl yild strss and th concrt comprssiv strngth for a singl forc-basd lmnt with fiv Gauss obatto intgration points. As shown in Fig. 1, th bam displacmnt is much mor snsitiv to th stl yild strss than to th concrt comprssiv strngth. Conclusions Fig. 9. oad displacmnt rspons of rinforcd concrt cantilvr bam for on forc-basd lmnt and 5 mm covr. h concrt is rprsntd by th phnomnological strss strain rlationship shown in Fig. 8, in which monotonic bhavior in comprssion is rprsntd by a parabolic function up to th pak comprssiv strngth of f c at a strain of c, followd by a linar dscnding branch to a crushing strain of u, whr th concrt loss all strngth. For simplicity, th concrt modl unloads and rloads along th sam branch, which passs through th origin, and th tnsil strngth is zro. h drivativs of th strss strain rspons for this concrt matrial modl ar prsntd in th Appndix. For this xampl, th concrt has a comprssiv strngth of f c 28. MPa, crushing strain c.2, and ultimat strain u.6. h stl is bilinar with 2% kinmatic hardning, lastic modulus E MPa, and yild strss y 42 MPa. A constant axial load qual to 1% of th gross sction capacity and a transvrs sinusoidal load of pak intnsity P 26 kn ar applid to th bam. h intgration of th stl and concrt matrial strss rspons ovr th sction ara accounts for axial momnt intraction, and th midpoint intgration rul is usd with 2 concrt layrs. h load displacmnt rspons for th bam is shown in Fig. 9. h rspons snsitivity is computd with rspct to both th h xact rspons snsitivity of forc-basd bam column lmnts has bn dvlopd from a consistnt dfinition of th drivativs of th lmnt quilibrium, compatibility, and sction forc dformation rlationships. Dirct diffrntiation of th govrning statmnts of lmnt quilibrium and compatibility provids a uniform approach to computing th rspons snsitivity for both displacmnt- and forc-basd lmnts. his approach ovrcoms th difficulty that ariss in th drivation of rspons snsitivity for forc-basd lmnts bcaus th displacmnt fild along th lmnt is not spcifid, but changs according to th lmnt stat. h rspons snsitivity computd by th finit diffrnc mthod convrgs to th xact snsitivity computd by th dirct diffrntiation mthod, as dmonstratd by th nonlinar analysis of stl and rinforcd concrt structurs. h ability to comput th rspons snsitivity accuratly and fficintly for forc-basd lmnts broadns th application of ths lmnts in th filds of structural analysis, dsign, rliability, and optimization. Appndix. Uniaxial Concrt Modl h rspons drivativs for th uniaxial concrt matrial modl shown in Fig. 8 ar prsntd in this Appndix. hr ar two history variabls to track path-dpndnt bhavior: min, th largst comprssiv strain, and min, th strss on th backbon corrsponding to min. h only paramtr considrd for diffrntiation in this xampl is th comprssiv strngth f c. h rspons drivativs for ach branch labld in Fig. 8 ar shown blow. Parabolic Ascnding Branch ( Ë min and Ì c ) Strss rspons f c 2 2, c (3) Drivativ of strss f c f c 1 c f c Conditional drivativ of strss (31) 2 2 (32) f c Fig. 1. Rinforcd concrt cantilvr bam rspons snsitivity through on sinusoidal load cycl with rspct to: a Stl yild strss and b concrt comprssiv strngth; computd by dirct diffrntiation and finit diffrnc mthods for on forc-basd lmnt inar Dscnding Branch Ë min and Ë c Strss rspons f u c (33) u c Drivativ of strss JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24 / 1287

8 f c u u c Conditional drivativ of strss f c u c f c (34) u (35) f c u c Unloading ÕRloading Branch Ì min Strss rspons Drivativ of strss min min f c Conditional drivativ of strss f c min min (36) f c min f c min min 2 min min min min f c min f c min f c (37) (38) 2 min Givn a strain and th history variabls min and min, th conditional drivativ of th strss is computd from ithr Eq. 32, 35, or 38, dpnding on th activ branch of th strss strain rlationship. his conditional drivativ of strss contributs to th conditional drivativ of th sction forcs in Eq. 27 during phas on of snsitivity computations. h drivativs min / f c and min / f c ar rquird to comput th conditional drivativ of strss on branch 3, as sn in Eq. 38. As a rsult, ths drivativs must b stord as snsitivity history variabls. Whn on branchs 1 and 2, min / f c is st to / f c, as givn by Eq. 28 during phas two of snsitivity computations. h drivativ min / f c is thn computd by ithr Eq. 31 or 34. Whn on branch 3, min dos not chang, so th snsitivity history variabls min / f c and min / f c ar not updatd during unloading and rloading. Notation h following symbols ar usd in this papr: a sction dformation intrpolation matrix; a s sction compatibility matrix; b sction forc intrpolation matrix; sction dformation vctor; f lmnt flxibility matrix; f s sction flxibility matrix; K global tangnt stiffnss matrix; k lmnt stiffnss matrix; k s sction stiffnss matrix; P f global xtrnal forc vctor; P r global rsisting forc vctor; q lmnt basic forc vctor; s sction forc vctor; U global displacmnt vctor; v lmnt dformation vctor; matrial strain; paramtr prturbation; paramtr; paramtr vctor; and matrial strss. Rfrncs Cont, J. P., Vijalapura, P. K., and Mghlla, M Consistnt finit lmnt snsitivitis in sismic rliability analysis. Proc., 13th ASCE Enginring Mchanics Division Conf., Rston, Va. Klibr, M., Antunz, H., Hin,. D., and Kowalczyk, P Paramtr snsitivity in nonlinar mchanics, Wily, Nw York. iu, P.., and Dr Kiurghian, A Optimization algorithms for structural rliability. Struct. Safty, 9, McKnna, F., Fnvs, G.., Jrmic, B., and Scott, M. H. 2. Opn systm for arthquak nginring simulation. Nunhofr, A., and Filippou, F. C Evaluation of nonlinar fram finit-lmnt modls. J. Struct. Eng., 123 7, Simo, J. C., and Hughs,. J. R Computational inlasticity, Springr, Nw York. Spacon, E., Ciampi, V., and Filippou, F. C Mixd formulation of nonlinar bam finit lmnt. Comput. Struct., 58, Zhang, Y., and Dr Kiurghian, A Dynamic rspons snsitivity of inlastic structurs. Comput. Mthods Appl. Mch. Eng., 18, Zinkiwicz, O. C., and aylor, R.. 2. h finit lmnt mthod: Volum 1, th basis, 5th Ed., Buttrworth Hinman, Stonham, Mass / JOURNA OF SRUCURA ENGINEERING ASCE / SEPEMBER 24