Fiber Beam Analysis of Reinforced Concrete Members with Cyclic Constitutive and Material Laws

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1 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Internatonal Journal of Conrete Strutures and Materals ORIGINAL ARTICLE Open Aess Fber Beam Analyss of Renfored Conrete Members wth Cyl Consttutve and Materal Laws T. Rav Mullapud and Ashraf Ayoub * Abstrat Ths paper presents a non-lnear Tmoshenko beam element wth axal, bendng, and shear fore nteraton for nonlnear analyss of renfored onrete strutures. The strutural materal tangent stffness matrx, whh relates the nrements of load to orrespondng nrements of dsplaement, s properly formulated. Approprate smplfed yl unaxal onsttutve laws are developed for raked onrete n ompresson and tenson. The model also nludes the softenng effet of the onrete due to lateral tensle stran. To establsh the valdty of the proposed model, orrelaton studes wth expermentally-tested onrete spemens have been onduted. Keywords: ombned loadng, Tmoshenko beam, R/C beams, un-axal onsttutve relatons, tangent stffness Bakground The response of renfored onrete RC strutures s affeted by the ombned effet of bendng, shear, and axal loads. Aurate onsttutve model of RC elements for ombned loadngs s essental for relably predtng strutural behavor. In the past deades onsttutve models have markedly mproved, thereby mprovng the auray and effeny of modelng omplex RC strutures. Effent onsttutve models for onrete and renforng bars are typally establshed from large-sale panel testng, and assumng a smeared raked behavor. Belarb and Hsu 995 developed the Rotatng-Angle Softened-Truss Model RA-STM. They assumed that shear stresses exst along the rak dreton, and proposed a tenson stffenng funton to aount for ths effet. Ths model was further mproved by Pang and Hsu 996, who developed the Fxed-Angle Softened-Truss Model FA-STM. In ths model, raks were assumed to be orented at a fxed angle. Zhu et al. derved a proper shear modulus ompatble wth the FA-STM model and proposed a robust soluton algorthm for *Correspondene: Ashraf.Ayoub.@ty.a.uk Dept. of Cvl Engneerng, Cty Unversty of London, Northampton Square, London ECV OHB, UK Full lst of author nformaton s avalable at the end of the artle Journal nformaton: ISSN / eissn analyss of shear-rtal onrete elements. Later, the authors evaluated the Hsu/Zhu Posson s Rato Posson s rato for raked onrete Zhu and Hsu, whh resulted n the Softened Membrane Model SMM. The SMM proved to be able to smulate both the pre-peak and post-peak behavor of onrete elements. Aurate modelng of the omplex behavor of RC strutures s typally performed wth two-dmensonal membrane elements. However, these elements are omputatonally very expensve, whh renders the analyss tme-onsumng. Unlke membrane elements, fber beam elements proved to provde a good balane between auray and numeral effeny Belarb and Hsu 995. In fber-based beam elements, the spread of nelastty along the depth s evaluated through dsretzaton of the seton nto a large number of fbers wth approprate materal models. Typal fber elements aount for the axal-flexural nteraton effet owng to the assumpton that plane setons reman plane after deformaton. Lately, Mullapud and Ayoub Mullapud and Ayoub developed a fber-based beam element that aounts for shear-axal-flexure nteraton effets and further mproved the element to aount for full three-dmensonal effets, nludng the ombned bendng, axal, shear and torsonal nteraton Mullapud and Ayoub 3. The Authors 8. Ths artle s dstrbuted under the terms of the Creatve Commons Attrbuton 4. Internatonal Lense veo mmons.org/len ses/by/4./, whh permts unrestrted use, dstrbuton, and reproduton n any medum, provded you gve approprate redt to the orgnal authors and the soure, provde a lnk to the Creatve Commons lense, and ndate f hanges were made.

2 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page of 6 Researh Objetve The man objetve of ths paper s to derve an approprate materal tangent stffness matrx for fber beamolumn element formulaton of shear-rtal onrete members Fg.. The developed stffness matrx does not aount for materal nonlnearty. Smplfed yl unaxal onsttutve laws are developed for onrete n both ompresson and tenson. The formulaton of the proposed element s based on the flexblty method of analyss. Flexblty-based formulatons Mullapud and Ayoub 9; Labb et al. 3 are used to overome most of the lokng dffultes assoated wth the standard dsplaement model. Shear effets s smulated through a Tmoshenko-based approah Mullapud and Ayoub 9. The onrete onsttutve law s based on the aforementoned SMM model wth Hsu/Zhu ratos. The work also attempts to mprove the development of the onrete un-axal onsttutve relatons. The model s added to the lbrary of the fnte element program FEAPpv Taylor 5. 3 Conrete Consttutve Model The ACI 38 Ameran Conrete Insttute 8 buldng ode suggests that the shear strength of an RC member s the ombnaton of onrete strength V C and transverse renforement strength V S. The value of V C annot be alulated n the RA-STM model, beause the rak angle s assumed to be rotatng. However, the FA- STM and SMM theores are apable of aountng for the effet of V C ; beause the raks are assumed to be orented at a fxed angle and the proper onrete shear stress term τ s aounted for. To formulate the SMM model wth the nluson of FRP, three oordnate systems are defned as shown n Fg. : the frst x, y represents the loal oordnate of the fber; the seond, defnes the prnpal stresses; whle the thrd system r, d defnes the onrete prnpal oordnate system n whh the onrete shear stress τ =. In the fgure, α s the angle between the x- and -axes, and α r s the angle between the x- and r-axes Fg.. d σ y τ xy τ σ τ y σ σ σ τ xy r α r α r α σ x Fg. Loal oordnate x, y; prnpal stress dretons, ; and onrete prnpal oordnate system r, d of RC elements. x Y Renforng Steel f C Z σ = ξ f P C Conrete X P ξ = Y Z Fg. Fber model dsretzaton.

3 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 3 of 6 To followng matrx Rθ s used to rotate the stress and stran vetors from one oordnate system to another: os θ sn θ os θ sn θ [Rθ] = sn θ os θ os θ sn θ os θ sn θ os θ sn θ os θ sn θ where θ s the angle between the two oordnate systems. The ompatblty equatons n the x y system are: where f sx and f sy are the renforng bar stresses n the x and y dretons respetvely, and ρ sx, ρ sy are the smeared steel ratos n the x and y dretons respetvely. The lateral stran ɛ y n fber s alulated from the seond of Equatons n 4, knowng that the value of σ y equals zero. In order to evaluate the value of the lateral stran ɛ y, an teratve proedure s needed owng to the nonlnear behavor of the materals used Mullapud and Ayoub. { x y.5γ xy } T = [R α ] {.5γ } T Smlar to the stran transformatons, stress transformaton equatons n the x y system are: { σx σ y τ xy } T = [R α ] { σ σ τ } T The transformaton equatons are graphally represented by Mohr s stress and stran rles n Fg. 3. For a fber Tmoshenko-type beam element formulaton, the state determnaton at the fber level uses the stran state {ɛ x, ɛ y, γ xy } to evaluate the fber stresses {σ x, σ y, τ xy }. In ths ase, the values of ɛ x, γ xy are gven, but the lateral stran ɛ y value s unknown and has to be alulated from the equlbrum equatons as desrbed next. 4 Proess to Evaluate Lateral Strans To evaluate the lateral stran, the equlbrum equatons between onrete and steel are gven below: σ x σ y τ xy = 3 os α sn α os α sn α sn α os α os α sn α os α sn α os α sn α os α sn α 5 Un Axal Consttutve Relatonshps of the Materals The baxal strans n the x y dreton { } T x y γ xy need to be onverted to equvalent unaxal strans n the prnpal dreton { } T γ n order to evaluate the onrete stresses as explaned n Mullapud and Ayoub. To evaluate the rotatng rak angle, α r orrespondng to a onrete shear stress τ = Fg. 3 the followng expresson s used: tan α r = x y The alulaton of the rotatng angle, α r s dependent on the stran state. The alulaton of the tral rotatng angle, αr from the Mohr rle Fg. 3 s based on the stran values σ σ τ + γ xy ρ sx f sx ρ sy f sy 5 4 a σ, + τ x xy τ b γ, +.5 γ x xy σ d τ σ, α r α α r α r σ, + τ σ σ r τ d,.5 γ α r α r * α α r α r, +.5 γ r γ σ y, τ xy σ σ y,.5 γ xy Fg. 3 Mohr Crle representaton of stresses and strans a Stress, b stran.

4 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 4 of 6 αr =.5 γ xy tan x y If the value of the dfferene between the axal and transverse strans ɛ x ɛ y =, then the value of the rotatng angle α r depends upon the value of the shear stran γ xy as follows. Rotatng angle α r = 45 when the value of γ xy >, and α r = 35 when the value of the γ xy <. If the value of the shear stran γ xy = then the rotatng angle depends upon the value of the ɛ x and ɛ y as follows. The rotatng angle α r = when the value of ɛ x > ɛ y and, α r = 9 when the value of ɛ x < ɛ y. If both the shear stran and the dfferene of the axal and transverse strans are non-zero numbers, then the followng laws wll be appled. If the value of ɛ x > ɛ y and the shear stran γ xy >, then the value of the rotatng angle α r beomes the value of the αr. If the value of ɛ x > ɛ y and the shear stran γ xy <, then the value of the rotatng angle α r = 8 αr. If the value of ɛ x < ɛ y and the shear stran γ xy >, then the value of the rotatng angle α r = 9 αr. If the value of ɛ x < ɛ y and the shear stran γ xy <, then the value of the rotatng angle α r = 9 + αr. After evaluatng theɛ y term that satsfes the equlbrum ondton Eq. 4, the prnpal angle α s evaluated as: 6 {.5γ } T = [Rα ] { x y.5γ xy } T The baxal strans n the x y dreton { x y γ xy } T need to be onverted to equvalent unaxal strans n the prnpal dreton { γ } T n order to evaluate the onrete stresses as explaned n Mullapud and Ayoub. Ths s done usng the Hsu/Zhu ratos μ, μ Zhu and Hsu. μ s the rato of the tensle stran nrement n dreton to the ompressve stran nrement n dreton, and μ s the rato of the ompressve stran nrement n dreton to the tensle stran nrement n dreton. Based on test data the followng expressons are proposed by Zhu and Hsu. µ = sf sf y, µ =.9 sf > y where ɛ sf s the stran n the steel bar that yelds frst and ɛ y s the yeld stran. After rakng, the value of the Hsu/Zhu rato μ s larger than maxmum value of.5 for Posson ratos of ontnuous materals. Before rakng, the Hsu/Zhu rato μ =. and, after rakng, μ =, meanng the tensle stran does not affet the ompressve stran. The Hsu/Zhu rato s used to elate the un-axal strans to the baxal prnpal strans: 8 9 { sx sy.5γ xy } T = [µ][r α ] {.5γ } T τ xy tan α = σ x σ y Smlar to Eq., the baxal prnpal strans are alulated as follow: 7 where [µ] = µ µ µ µ µ µ µ µ µ µ sx = sy = µ µ + µ µ + µ os α + µ µ µ sn α + µ µ The longtudnal and transverse renforement unaxal prnpal strans are then: sn α γ snα osα µ µ µ µ µ + µ µ µ + 3 os α + γ snα osα µ µ 4

5 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 5 of 6 f f Non Softened A ς, ςf C ς f.ςf O A The equvalent unaxal longtudnal and transverse steel stresses, f sx and f sy respetvely are evaluated from the orrespondng steel strans sx and sy through a proper steel onsttutve model. The equvalent unaxal strans and are also used to evaluate the onrete stresses σ and σ. 5. Conrete Model The unaxal onrete materal model adopted follows the well-establshed modfed Kent and Park model Park et al. 98. However, the model was modfed to aount for the followng effets: Frst, the softenng effet for both, the stresses and strans, s aounted for. Seond, the yl stffness degradaton for both, the unloadng and reloadng branhes, s ntrodued. Thrd, the tenson-stffenng effet s aounted for Belarb and Hsu 994. Aordng to Kent and Park Park et al. 98, the monoton stress stran envelope of onrete follows a parabol urve Fg. 4: [ f = f Softened f B ς o o o ς ς ].8ςf C 5 It was observed from expermental tests of onrete panels that the ompressve stress stran urve s redued due to the effet of perpendular tensle stresses. Ths effet s aounted for through a softenng oeffent ς. When the softened stress stran urve s developed, t s assumed that the lnes that onnet the orgn to the peak stress of the softened and non-softened urve have the same slope as shown n Fg. 4. Smlarly, the pre-peak and post-peak urves of the softened Fg. 4 Monoton softened onrete materal model. Stress, f E C F O H.5E r t, E r E -,-f r r R r member are assumed to follow a parabol shape. The desendng branh of the softened envelope s gently sloped untl the stress reahes a value that equals % of the maxmum stress ςf at a stran of ɛ. The resdual onrete ompressve strength s assumed to be % of the softened onrete ompressve strength ςf. The sse of ths value n the model s very ommon and has aurately predted the expermental results Mullapud and Ayoub 3. For the softened behavor, the followng relatonshps are adopted: [ Regon OA, ς, f = ςf D, f m m ] ς ς 6 Regon OA, ς, Tangent modulus E t = f ς 7 [ ] Regon AB, ς <, f = ςf ς.8 ς 8 Regon AB, ς <, Tangent modulus E t =.6ςf ς ς G Regon BC, >, f =.ςf B,. ς f f r Stran, Fg. 5 Cyl stress stran urve of softened onrete. Regon BC, >, Tangent modulus E t =. E C E C E C. ς f C C 9

6 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 6 of 6 The yl model Fg. 5 onsders the onrete damage and aounts for the rak openng and losng. The envelope for the yl stress stran urves of onrete adheres to the monoton stress stran urve. The unloadng and reloadng path of the ompressve sde s smplfed, as all the loadng paths start from a ommon pont R, whh determnes the degradaton stffness.e. the rato between the slope of any gven loadng path and that of the monoton envelope at the orgn, whh lmt s provded by the slope of the path RB. The unloadng modulus E at pont B of the monoton envelope urve s.ςf f r r. E and must be determned expermentally. The stress and stran at the nterseton of pont R and the orgn are gven by the followng expressons: r =.ςf E E E f r = E r 3 n whh E s the ntal tangent modulus at orgn n ompresson; n the urrent model, t s assumed to equal f. The unloadng stress fm and stran m values at pont D on the ompressve monoton envelope are used to alulate the reloadng modulus and stran t at zero stress pont H from the followng expressons: E r = f m f r m r t = m f m E r 4 5 From any unloadng pont D, the stress wll reah the zero stress axs at pont H after ompletng two smaller yles that are defned by these expressons: Maxmum stress lne HD fmax = fm + E r m, t m Mnmum stress lne HE fmn =.5Er t, t m. 6 7 The loadng and unloadng yles are arred out wth the assumpton that the model follows a lnear behavor wth modulus E. The tral stress f T and tangent modulus E t are based on a lnear elast behavor wth ntal tangent modulus E ; later ths assumpton s orreted to fall under the lne HD and lne HE. f T = f P + E 8 f s f p f y f y f Steel stress Here, f P s the prevous stress and Δɛ s the stran nrement. The atual stress f and tangent modulus E t are alulated based on the tral stress state f mn f T f T E s Bare steel bar E p Steel stran f max then f = f T E p y Fg. 6 Smeared mld steel stress stran urve. Smeared steel bar and E t = E < f mn then f = f mn and E t =.5E r 9 3 f T > fmax then f = fmax and E t = Er. 3 When the unloadng begns from ponts D to E, the reloadng wll follow the same path bak to D. When the unloadng reahes pont F, then reloadng wll result n the loop DEFGD. If unloadng reahes pont H, then reloadng wll follow loop DEHD. The reloadng path wll always rejon the ompresson envelope at the pont of ntal unloadng, D. If unloadng ontnues below pont H, then reloadng begns n tenson. After the start of the reloadng n ompresson, the model wll re-enter the ompresson branh at pont H. Subsequent loadng n the tenson branh wll not affet the behavor one the model returns bak to the ompresson branh. 5. Steel Model The smeared stress stran relatonshp of steel embedded n onrete under un-axal loadng has been developed by Belarb and Hsu 994, 995. The steel stran at raked setons typally nreases rapdly ompared to adjaent regons beause part of the stress s ressted by the onrete. Steel stresses are averaged along the renforng bar rossng several raks, and the resultng smeared steel stress at frst yeld s redued ompared to the loal yeld stress of a bare bar. The smeared E s p s

7 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 7 of 6 stress stran relatonshp of embedded steel bars sold urves, as well as that of bare bar dotted urve are plotted n Fg. 6. The dfferene between the bare bar yeld stress f y and smeared steel bar yeld stress f y depends on the parameter B defned by Belarb and Hsu 994, 995. The parameter Bs derved to be a funton of three varables suh as perentage of steel ρ, rakng strength of onrete f r, and yeld stress of the bare bar f y. The parameter B depends on the rak wdth, rak spang and nterfae bond slp behavor of steel bar and onrete. It an be seen that when the steel rato ρ s dereased or when f r s nreased, then the smeared yeld stress f y dereases and the smeared stress stran plot moves downward. As shown n Fg. 6, the shape of both, the bare bar and the smeared steel bar, urves of mld steel follow two straght lnes. These two straght lnes have slopes of E s before yeldng and E p after yeldng for smeared steel, and slope of E p after yeldng for bare steel bar as llustrated n Fg. 6. The slope of the stran-hardenng branh of the bare steel bar s assumed to equal.5 E s. The stress value at whh the two straght lnes of smeared steel nterset s the smeared yeld stress f y and the orrespondng stran s the smeared yeld stran y. The equatons of the pre yeld and post yeld lnes are gven as: f s = E s s when f s f y f s = f o + E p s when f s > f y. The vertal nterept of the post yeld lne f s evaluated o as: f o = E s E p f y E s Belarb and Hsu 994, 995 defned the parameter Bas: B = ρ fr f y.5, where f r =.3 f MPa and ρ.5%. The smeared yeld stress of steel bar s alulated as: f y =.93 Bf y The smeared steel bar yeld stran s alulated as: y = f y E s. The slope of the stran hardenng branh of the smeared steel bar s alulated as E p The smeared steel stress before yeldng an be alulated as: 4 When the steel stress f s reahes a value of f p and begns to unload, then the unloadng branh follows a straght lne wth a slope of E s. The unloadng stress s expressed as: f s = f p E s p s where s < p 4 The yl response of the smeared steel bar s formulated aordng to the Flppou et al. 983 model whh aounts for sotrop stran hardenng and Baushnger effet. 6 Tangent Materal Consttutve Relatons The onrete onsttutve law s smplfed as an orthotrop materal wth two perpendular planes of elast symmetry. Dretons one and two are the loal prnpal materal axes that are normal to the planes of symmetry. Wth the equvalent un-axal strans, the stffness values Ē and Ē are determned from a materal unaxal stress stran relatonshp. The materal behavor s expressed as: { } σ = [Dlo ] { In ths equaton, { } 4 σ} s the onrete stress vetor, {ɛ } s the prnpal stran vetor, and [D lo ] s the prnpal loal un-axal onrete materal tangent stffness matrx. The tangent stffness matrx of an RC element s defned as: =. + B. f s = E s when s y f s =.9 Bf y BE s s when s > y. d [ ] +s Dgl = d σ x σ y τ xy x y γ xy, 43

8 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 8 of 6 σ x σ y τ xy y y s = [R α ] steel σ σ τ + where ρ s s the renforement rato n the th dreton; and [R α ] and [R α s ] are the transformaton matres from the loal oordnate and the x s y s renforement oordnate system, to the x y oordnate system, respetvely Fg. 7. Substtutng Eq. 44 nto Eq. 43 gves: [R α ] [ ] +s Dgl = α s σ σ τ Equaton 45 s splt nto a onrete stffness [D gl ] and a renforement stffness [D gl ] s as: σ [R α ] σ [ ] τ Dgl =, 46 x y γ xy Thus the total stffness beomes: [ ] +s [ ] [ ] s, Dgl = Dgl + Dgl + [R α s] x y γ xy ρ s f s [R α s ], and ρ s f s [R α s] [ ] s Dgl =. x y γ xy x s α α α Fg. 7 Coordnate system for renfored onrete element. s x ρ s f s where [D gl ] s the tangent materal onsttutve matrx of onrete; and [D gl ] s s the tangent materal onsttutve matrx of the steel bars. The equvalent un-axal strans an be alulated from the global strans usng: γ = [µ] [Rα ] where [μ] s the Hsu/Zhu rato matrx as shown n Eq.. After substtutng Eq. 49 nto Eq. 46: σ σ σ γ [ ] σ σ σ Dgl = [R α ] γ [µ] [Rα ]. From Eq. 5, the onrete loal tangental stffness an be wrtten as: [D lo ] = σ σ σ σ τ τ After substtutng Eq. 5 nto Eq. 5: [ ] Dgl = [R α ] [D lo ] [µ] [Rα ]. The dagonal terms n Eq. 64 matrx an be found dretly from the un-axal stresses and strans n the respetve dretons. The frst dagonal term σ = Ē s the tangental un-axal modulus of onrete n the -dreton, the seond dagonal term σ = Ē s the tangental un-axal modulus of onrete n the -dreton, and the thrd dagonal term τ = σ σ γ = G s the shear modulus. The off-dagonal terms σ and σ are obtaned usng the un-axal stresses and the unaxal strans n the orthogonal dreton. These off-dagonal terms are not zero beause the stress and strans of the onrete n ompresson s softened by the perpendular tensle strans. Therefore, [D lo ] an be wrtten as: [D lo ] = Ē τ τ σ γ σ γ τ γ σ σ Ē G. x y γ xy, τ γ

9 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 9 of 6 The off-dagonal terms an be determned wth three ases as desrbed below. 6. Equvalent un axal strans > and > When > and > then the un-axal onrete stresses σ and σ are alulated only from un-axal strans and, respetvely. Therefore, σ C = and σ C =. 6. Equvalent un axal strans > and < When the un-axal stran s > then the un-axal ompressve stress σ s alulated dretly from the, and σ s not a funton of the orthogonal onrete stran σ. Therefore, C =. To obtan σc, the onsttutve law of the onrete strut n ompresson s needed. In order to obtan σc, the onsttutve law of onrete n ompresson s needed. [ ] σ C = ζ f C, ζ ζ σ C = ζ f C / ζ 4 /, ζ ζ, where the softenng oeffent ζ Hsu and Zhu ζ >, α ζ =.9 r f MPa In Eq. 57, α r s n degrees. If the value of the α r s n radans then the value of the 4 should be onverted nto radans. From panel tests up to MPa of onrete ompressve strength, the devaton angle α r s equal to or less than 4 Wang 6 After makng the requred hanges, Eq. 57 beomes 5.8 ζ =.9 5 α r, f MPa + 4 π 58 α r =.5 tan γ and 59 = µ and = µ, 6 where ɛ and ɛ are the baxal strans n the oordnate system and and are the equvalent un-axal strans n the oordnate system. From Eqs. 59 and 6: αr =.5 γ tan, 6 + µ + µ where μ and μ are the Hsu/Zhu ratos Zhu and Hsu σ C an be wrtten as σc = σc. 6 an be derved after substtutng Eq. 6 nto Eq. 58: 5.8 f = MPa + 4 4π 5 tan γ + µ + µ tan γ = +µ +µ f MPa π γ + µ γ +µ +µ + γ +µ +µ 63 64

10 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page of 6 where σc an be derved as follow when ζ : when ζ > : Equaton 67 an be wrtten as: where Let σ C σ C ζ f C [ ] ζ ζ = = f C ζ = f C = f C = f C ζ ζ ζ = / ζ f C ζ 4 / ζ = f C / ζ ζ 4 /. ζ Let / ζ ζ 4 / = A, ζ σ C = f C A, / / ζ A = 4 / ζ + ζ ζ 4 / ζ / ζ 7 4 /. ζ B = / ζ 4 /, ζ 7 Equaton 7 an be wrtten as: / / ζ A = 4 / ζ + ζ ζ 4 / B, ζ 73 where B = / ζ 4 / ζ = / 4 4 ζ. 74 Substtutng Eq. 74 nto Eq. 73 gves: A = / ζ / 3 4 ζ / ζ + 4 / ζ +, >. ζ ζ 75 Substtutng Eq. 75 nto Eq. 7 gves: σ C = f C / ζ / 3 4 ζ / ζ + 4 / ζ ζ Substtutng Eqs. 66 and 64 nto Eq. 6 gves:when, σc = f C ζ. ζ 77 Substtutng Eqs. 76 and 64 nto Eq. 6 gves:when >, σc ζ = f C / ζ / 3 4 ζ / ζ + 4 / ζ +. ζ 78 When the equvalent un-axal strans < and >, then the same proedure should be followed. 6.3 Equvalent un axal strans < and < When < and < then onrete wll not soften; nstead t nreases ts ompressve strength n one dreton dependng on the onfnng stress n the orthogonal dreton. Beause of ths reason, the value

11 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page of 6 of ζ should be greater than or equal to. The urrent researh uses the Veho s Veho 99 smplfed verson of Kupfer et al. 969 baxal ompresson strength envelope. These equatons are strength-based; stran-based equatons are not avalable n the lterature and need to be nvestgated n future. Aordngly: = σ C = σc σ C = σc and = = = The global tangental onsttutve matrx of steel [D gl ] s an be derved From Fg. 7 as: x y γ xy = [Rα ] [µ] [Rα s α ] s s γ s 8 where s s the equvalent un-axal longtudnal stran of the renforement, s s the equvalent un-axal transverse or dowel stran of the renforement and γ s s the equvalent un-axal shear stran of the renforement. Substtutng Eq. 8 nto Eq. 47 gves: ρ s f s [R α s] [ ] s Dgl =, s [Rα ] [µ] [Rα s α ] s = = [R α s] ρ s f s s f s s f s γ s [Rα ] [µ] [Rα s α ], [R α s ] ρ s f s s [Rα s α ] [µ] [Rα ], f s s f s γ s γ s Let [D lo ] s = ρ s f s s f s s f s γ s where f s s = Ē s, whh s the equvalent un-axal tangental modulus n the longtudnal dreton of the renforement. The dowel aton of the renforement s negleted, thus f s =, and also the shear deformaton s n the renforng bar s negleted, thus f s γ s =. 86 Wth these smplfatons Eq. 86 an be wrtten as: [D lo ] s = ρ s Ē s. 87 Equaton 85 an be wrtten as: [ ] s Dgl = [R α s ] [D lo ] s [Rα s α ] [µ] [Rα ]. After substtutng Eqs. 88 and 5 nto Eq. 48 the global tangent materal onsttutve matrx [D gl ] +s an be evaluated as: The total seton stffness s evaluated from the sum of onrete and steel stffness: The total seton fore s evaluated from the sum of onrete and steel fores n ther respetve dretons: where n s the number of onrete and steel fbers n a seton. The element stffness and fores are alulated wth numeral ntegraton of seton stffness and seton fores at dfferent seton ponts along the length of the element., k fber = [ D gl ] +s = [R α ] [D lo ] [µ] [K Seton ] = {F Seton } = [Rα ] + [T α s ] [D lo ] s [Rα s α ] [µ] [Rα ] n k fber, n F fber

12 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page of 6 7 Numeral Correlatons wth Experments The smplfed un-axal materal and onsttutve laws are mplemented n the developed fber beam element and analyzed the strutural members under yl loadng. Model predted the expermental results throughout the loadng hstory and ould be used to smulate the behavor of RC strutures under sesm loadng. 7. Comparson of Conrete Model wth the expermental D Cyl Stress Stran Curves The un-axal materal models developed n ths paper are ompared wth the expermental results of Mansour. Mansour tested three panels of the CVE3 seres. The steel grds n these panels are set parallel to the appled prnpal stresses n horzontal and vertal dretons. The three panels of ths seres CVE3-, CVE3- and CVE3-3 are subjeted to -D yl loadng n the horzontal dreton, whle mantanng a onstant lateral tensle stran t of.44,. and.3. Eah panel has the followng dmensons: length 397 mm, heght 397 mm, and thkness 78 mm. The panels are renfored n eah dreton wth No. 6 bars at 67 mm spang. The onrete ompressve strengths of CVE3-, CVE3-, and CVE3-3 are 48, 4, and 43 MPa. The yeld stress of longtudnal and transverse steel of panels CVE3-, CVE3-, and CVE3-3 are 45.4 MPa respetvely. The analytal result of the three panels CVE3-, CVE3-, and CVE3-3 wth the urrent model and orrespondng expermental results are presented n Fgs. 8, 9, and. In these fgures, the horzontal axes represent the smeared onrete stran n the longtudnal dreton, and the vertal axes represent the smeared onrete stress n the longtudnal dreton. Conrete stress n longt. dreton MPa Fber Beam Element Experment t = Conrete stran n longt. dreton Fg. 9 Panel CVE3- onrete stress stran behavor omparson wth fber beam element, and experment Mansour. Comparson of the urrent model results for the three panels n the CVE3 seres showed an nrease n lateral tensle stran; the ultmate value of the horzontal ompresson stress dereases beause of the softenng behavor of the onrete. Fgures 8, 9,,,, 3 show that the urrent model predts farly well the expermental behavor at both the ompresson and tenson regons. 7. Smulaton of Columns Three hollow, retangular prototype brdge pers PI, PI, and PS are tested under reverse ylal loadng at the Natonal Center for Researh on Earthquake Engneerng n Tawan Yeh and Mo 999. These prototypes are analyzed usng the fber beam element. The spemens dmensons along wth the materal propertes of onrete and renforement as gven n Table. Fgure shows the detals of the spemens ross seton dmensons and renforement detals. The olumns are tested Fber Beam Element -. Experment -5. t = Conrete stran n longt. dreton Fg. 8 Panel CVE3- onrete stress stran behavor omparson wth fber beam element, and experment Mansour. Conrete stress n longt. dreton MPa Conrete stress n longt. dreton MPa Conrete stran n longt. dreton Fber Beam Element Experment t =.3 Fg. Panel CVE3-3 onrete stress stran behavor omparson wth fber beam element, and experment Mansour.

13 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 3 of 6 a 5 a b j 64-# a #4@8 5 b 7 5 j 5 b a b j 64-# #3@ 7 5 a 5 b 7 5 j a b j 64-#7 5 a b #3@ 5 j Fg. Dmensons of ross setons and renforement detals of spemens a PS, b PI and PI Dmensons are n mm Yeh and Mo 999.

14 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 4 of 6 Fg. Comparson of load dsplaement behavor of spemen PS wth fber beam element, and experment Yeh and Mo 999. renforements. Conrete enveloped by the strrups s modeled as onfned onrete, whle the remanng onrete mostly n the over s onsdered as unonfned. The horzontal fores are nreased based on the yl dsplaement ontrol sheme. The analytal shear fore versus dsplaement relatonshps of the spemens are predted wth the -D fber beam element as shown n Fgs., 3, 4; the results are then ompared to the expermental data. The moment arms for spemens PS, PI, and PI are 6.5, 4.5, and 3.5 m respetvely. The renforement of the olumns s desgned suh that olumn PS s domnated by a flexural falure, olumn PI s domnated by a flexure-shear falure, and olumn PI s domnated by shear falure. The expermental results Yeh and Mo 999 showed that spemens PS and PI faled n a flexure mode wth the formaton of plast hnges at the bottom of the olumn and spemen PI faled under shear falure mode wthout rupturng the longtudnal renforement. The falure modes and dutlty levels are refleted n the shape of the load dsplaement plots. The rebar yelded sgnfantly pror to rushng of the onrete n spemens PS and PI Fgs. and 3, whh resulted n a long yeld plateau and hgher energy dsspaton. Spemen PI Fg. 4 showed a muh shorter yeld plateau and pnhng wth less energy dsspaton Fg. 3 Comparson of load dsplaement behavor of spemen PI wth fber beam element, and experment Yeh and Mo 999. under dsplaement ontrol wth ylally-reversed horzontal load. The spemens are modeled wth only one fore based element and fve Gauss Lobatto ntegraton ponts. The ross seton s dvded nto 8 fber setons. The boundary ondton at the bottom s assumed to be fxed whle the lateral load s appled to the top of the olumn. The longtudnal and transverse steel ratos are alulated based on the dmensons and spang of the Fg. 4 Comparson of load dsplaement behavor of spemen PI wth fber beam element, and experment Yeh and Mo 999. Table Propertes of brdge pers Yeh and Mo 999. Spemen name f MPa Length mm Longtudnal renforement Transverse renforement Da. mm f y MPa f su MPa Da. mm f y MPa Spang mm PS PI PI f, Conrete ompressve strength; f y, Steel yeldng strength; f su, Steel ultmate strength; Da., Dameter of steel bar.

15 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 5 of 6 Table Comparson of strength values for spemens PS, PI, and PI Yeh and Mo 999. Spemen Strength n postve yles expermental Strength n postve yles analytal Strength n negatve yles expermental Strength n negatve yles analytal PS PI 3 5 PI than spemens PS and PI. Spemen PS developed a dsplaement dutlty of.8, spemen PI developed a dsplaement dutlty of 7.8, and spemen PI developed a dsplaement dutlty of only 3.7. The analytal load dsplaement relatonshps wth the fber beam element aurately aptured the dfferent behavors of eah of the three spemens. The fber beam element predted the ntal stffness, yeld pont, ultmate strength, and dutlty of spemens PS and PI very well. The predted yl load dsplaement urve of symmetr spemen PI Fg. 3 showed less ultmate strength n the negatve yles ompared to expermental results. The fber beam element also predted the behavor of spemen PI Fg. 4 to a degree n the postve dreton, nludng the ultmate strength and strength degradaton n the desendng branh. The predted ultmate strength of spemen PI n the negatve dreton s slghtly hgher than the test result, whle the ultmate strength s slghtly less wth the plane stress element analyss. The predted hysteress loops of spemen PI wth fber beam element showed muh less energy dsspaton and predted the expermental results. Table summarzes the spemens strength values for both the expermental results and analytal predtons. 8 Conlusons Ths paper represents a new element for yl analyss of renfored onrete strutures. A fber-based beam element s developed to analyze renfored onrete strutures wth the norporaton of mehansms of shear deformaton and strength. Smplfed yl un-axal onsttutve relatons are developed and heked wth the -D yl test panels of Mansour. The tangent stffness s formulated wth the nluson of the softenng and dlataton effets. The reverse yl analyses of dfferent olumns wth retangular ross-setons are analyzed wth the -D fber beam element. The yl analyss of olumns tested by Yeh and Mo 999 are analyzed wth the -D fber beam element. The olumns wth dfferent aspet rato exhbt dfferent behavors. The numeral results onernng the olumns agree wth the expermental data throughout the entre loadng hstory. The fnte element analyss usng the generalzed Softened Membrane Model predted the expermental results throughout the loadng hstory, nludng the ntal stffness, yeld pont, ultmate strength, and falure mode; and ould therefore be used to smulate the behavor of RC strutures under sesm loadng. Authors ontrbutons TRM and AA developed the model presented. TRM onduted the omputer odng and drafted the frst verson of the manusrpt. Both authors read and approved the fnal manusrpt. Author detals CTLGroup, 7834 Burhfeld Grove Ln, Katy, TX 77494, USA. Dept. of Cvl Engneerng, Cty Unversty of London, Northampton Square, London ECV OHB, UK. Competng nterests The authors delare that they have no ompetng nterests. Publsher s Note Sprnger Nature remans neutral wth regard to jursdtonal lams n publshed maps and nsttutonal afflatons. Reeved: 3 November 7 Aepted: 6 June 8 Referenes Ameran Conrete Insttute. 8. ACI 38, Buldng ode requrements for strutural onrete and ommentary. Farmngton Hlls, MI, USA: Ameran Conrete Insttute. Belarb, A., & Hsu, T. T. C Consttutve laws of onrete n tenson and renforng bars stffened by onrete. Strutural Journal, Ameran Conrete Insttute, 9, Belarb, A., & Hsu, T. T. C Consttutve laws of softened onrete n baxal tenson-ompresson. Strutural Journal of the Ameran Conrete Insttute, 95, Flppou, F. C., Popov, E. P., Bertero, V. V Effets of bond deteroraton on hysteret behavor of renfored onrete jonts. Report SESM 77-, Dvson of Strutural Engneerng and Strutural Mehans. Berkeley: Unversty of Calforna. Hsu, T. T. C., & Zhu, R. R. H.. Softened membrane model for renfored onrete elements n shear. Strutural Journal of the Ameran Conrete Insttute, 994, Kupfer, H. B., Hldorf, H. K., & Rush, H Behavor of onrete under baxal stresses. Strutural Journal, Ameran Conrete Insttute, 668, Labb, M., Mullapud, T. R. S., & Ayoub, A. S. 3. Analyss of RC strutures subjeted to mult-dretonal shear loads. Journal of Advaned Conrete Tehnology,, 34. Mansour, M.. Behavor of renfored onrete membrane elements under yl shear experments to theory. Ph.D. dssertaton. Department of Cvl and Envronmental Engneerng, Unversty of Houston. Mullapud, T. R. S., & Ayoub, A. S. 9. Fber beam element formulaton usng the softened membrane model. Ameran onrete nsttute speal publaton pp Farmngton Hlls: SP-65, ACI.

16 Rav Mullapud and Ayoub Int J Conr Strut Mater 8 :6 Page 6 of 6 Mullapud, T. R. S., & Ayoub, A. S.. Modelng of the sesm behavor of shear-rtal renfored onrete olumns. Journal of Engneerng Strutures, 3, Mullapud, T. R. S., & Ayoub, A. S. 3. Analyss of renfored onrete olumns subjeted to ombned axal, flexure, shear and torsonal loads. Journal of Strutural Engneerng ASCE, 394, Pang, X. B., & Hsu, T. T. C Fxed-Angle softened-truss model for renfored onrete. Strutural Journal of the Ameran Conrete Insttute, 93, Park, R., Prestley, M. J. N., & Gll, W. D. 98. Dutlty of square onfned onrete olumns. Journal of Strutural Davson, ASCE, 84, Taylor, R. L. 5. FEAP User Manual v.. Berkeley: Department of Cvl and Envronmental Engneerng, Unversty of Calforna. ley.edu/~rlt/feap/. Veho, F. J. 99. Fnte element modelng of onrete expanson and onfnement. Journal of Strutural Engneerng, ASCE, 89, Wang, J. 6. Consttutve relatonshps of prestressed onrete membrane elements. Ph.D. dssertaton. Houston: Unversty of Houston. Yeh, Y. K., Mo, Y. L Full sale tests on dutlty, shear strength and retroft of renfored onrete hollow olumns I. Report, No. NCREE Tape: Natonal Center for Researh on Earthquake Engneerng. n Chnese. Zhu, R. H., & Hsu, T. T. C.. Posson effet of renfored onrete membrane elements. Strutural Journal, Ameran Conrete Insttute, 995, Zhu, R. H., Hsu, T. T. C., & Lee, J. Y.. Ratonal shear modulus for smeared rak analyss of renfored onrete. Strutural Journal, Ameran Conrete Insttute, 984,