Global resistance methods for the design of fiber-reinforced concrete (FRC) beams with material nonlinear finite element analysis

Ci a ion: Shah bija i, K.B.; Ba os, J.A.O.; Valen e, I.B. Global Resis ance Me hods o he Design o Fibe -Rein o ced Conc e e (FRC) Beams wi h Ma e ial Nonlinea Fini e Elemen Analysis. Buildings 2023,13, 2848. h ps://doi.o g/10.3390/ buildings13112848 Academic Edi o s: Fede ico Acco ne o and Albe o Ca pin e i Recei ed: 28 Oc obe 2023 Re ised: 8 No embe 2023 Accep ed: 9 No embe 2023 Published: 14 No embe 2023 Copy igh : © 2023 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h ps:// c ea i ecommons.o g/licenses/by/ 4.0/). buildings A icle Global Resis ance Me hods o he Design o Fibe -Rein o ced Conc e e (FRC) Beams wi h Ma e ial Nonlinea Fini e Elemen Analysis Kamya B. Shah bija i * , Joaquim A. O. Ba os and Isabel B. Valen e ISISE, Depa men o Ci il Enginee ing, Uni e si y o Minho, 4800-058 Guima ães, Po ugal; [email p o ec ed] (J.A.O.B.); [email p o ec ed] (I.B.V.) *Co espondence: kamya [email p o ec ed] Abs ac : This a icle explo es he applica ion o he global esis ance me hods (GRMs) on he design o hyb id glass ibe - ein o ced polyme (GFRP) and s eel ibe - ein o ced conc e e (SFRC) beams. Add essing challenges posed by GFRP- ein o ced beams, his s udy aims o assess he impac o ma e ial unce ain ies on he beha io o such hyb id beams. The in es iga ion in ol es he expe imen al es ing o I-shaped SFRC beams, which a e used o de elop and alida e nonlinea ini e elemen analysis (NLFEA) models. These models inco po a e ma e ial non-linea i ies while minimizing unce ain ies ela ed o modeling assump ions. Th ough he applica ion o GRM, he s udy e alua es he global esis ance sa e y ac o , o e ing insigh s in o he s uc u al pe o mance o hyb id ein o cemen SFRC beams. Ul ima ely, his esea ch seeks o acili a e a ansi ion om adi ional localized app oaches o mo e accu a e and comp ehensi e analyses o he design o hyb id ein o cemen SFRC beams, con ibu ing o he ad ancemen o s uc u al enginee ing by p omo ing sa e , mo e esilien , and sus ainable cons uc ion sys ems. Keywo ds: hyb id lexu al ein o cemen ; ibe - ein o ced conc e e; global esis ance me hods; ma e ial nonlinea ini e elemen analysis; design o FRC beams 1. In oduc ion In ecen yea s, he cons uc ion indus y has wi nessed a su ge o in e es in ibe - ein o ced conc e e (FRC) as a sus ainable and eliable ma e ial, which lauded o i s enhanced mechanical and du abili y p ope ies [ 1 – 3 ]. S eel ibe - ein o ced conc e e (SFRC) is s ill he mos applied FRC, whe e s eel ibe s a e used o e ec i ely mi iga e b i le ailu e and con ol c ack de elopmen [ 4 , 5 ], posi ioning i sel as a p omising solu ion o mode n in as uc u e p ojec s. Employing enough s eel ibe s in ein o ced SFRC beams lacking s i ups can enhance hei shea capaci y, which migh po en ially esul in a ansi ion o he ailu e pa e n om being p edominan ly shea -d i en o being go e ned by lexu al mechanisms [6–9]. Fu he mo e, he u iliza ion o glass ibe - ein o ced polyme s (GFRPs) as a pa ial o o al eplacemen o adi ional s eel ba s has eme ged as a g owing end in s uc u al en- ginee ing [ 10 – 12 ], p ima ily d i en by he need o comba he ulne abili y o con en ional s eel ein o cemen o co osion [ 13 , 14 ]. The supe io co osion esis ance o GFRP ba s makes hem an a ac i e al e na i e; howe e , beams ein o ced solely wi h GFRP ba s ha e been ound o exhibi b i le ailu e modes [ 15 ]. Addi ionally, GFRP ba s demons a e a no able decline in hei mechanical p ope ies unde ela i ely high empe a u es [ 16 , 17 ], posing unique challenges in i e-p one en i onmen s. To enhance he duc ili y and o e all pe o mance o GFRP- ein o ced beams, he concep o hyb id ein o cemen , comp ising s eel and GFRP ba s, has been p oposed [ 18 – 22 ]. This inno a i e app oach capi alizes he inhe en s eng hs o bo h ma e ials, s iking a balance be ween co osion esis ance and duc ile beha io , hus ele a ing he eliabili y o s uc u al elemen s. Buildings 2023,13, 2848. h ps://doi.o g/10.3390/buildings13112848 h ps://www.mdpi.com/jou nal/buildings Buildings 2023,13, 2848 2 o 20 To p edic he beha io o FRC beams wi h hyb id ein o cemen unde bo h Se ice- abili y Limi S a e (SLS) and Ul ima e Limi S a e (ULS) condi ions, nonlinea ini e elemen analysis (NLFEA) has eme ged as a powe ul s a egy [ 23 – 25 ]. NLFEA e olu ionized he ield, enabling p ac i ione s and esea che s o explo e and eplica e he esponse o ein o ced conc e e (RC) membe s and sys ems, accoun ing o bo h ma e ial and geo- me ic non-linea i ies. The applica ion o NLFEA in s uc u al enginee ing, especially wi hin he con ex o RC s uc u es, has been he ocus o nume ous in es iga ions and guidelines [ 26 – 28 ]. Mo eo e , NLFEA has eme ged as a powe ul ool o assessing exis ing RC s uc u es and in as uc u es, conside ing he impac o de e io a ion p ocesses [ 29 , 30 ]. In he e e -e ol ing landscape o s uc u al enginee ing, he pu sui o sa e, esilien , and e icien designs has been an ongoing challenge o p o essionals wo ldwide [ 31 – 34 ]. As a chi ec s and enginee s encoun e inc easingly complex p ojec s, adi ional me hods o linea analysis and localized sa e y checks ha e p o en insu icien o add ess he in ica e beha io s o RC s uc u es [35]. His o ically, he analysis o conc e e s uc u es in ci il enginee ing p edominan ly elied on linea and pseudo-nonlinea app oaches [ 36 , 37 ]. Howe e , he inhe en nonlinea beha io o RC has spa ked cu iosi y and discussion among s uc u al designe s [ 38 ], p omp ing he need o mo e sophis ica ed me hods o add ess his complexi y. The ques o mo e accu a e and comp ehensi e design app oaches has led o he eme gence o global esis ance me hods (GRMs), a se o echniques ha ha ness he powe o NLFEA o explo e he holis ic pe o mance o s uc u es unde di e se loading condi ions [28,35,39]. GRMs acili a e he de ini ion o sa e y o ma s ha encompass bo h alea o y (me- chanical and geome ic) and epis emic (nume ical model) unce ain ies [ 40 – 42 ]. Using NLFEA, he global s uc u al esis ance can be be e es ima ed, accoun ing o ma e ial and geome ic unce ain ies ia app op ia e pa ial sa e y ac o s. This app oach p o ides a mo e e ined and comp ehensi e unde s anding o s uc u al eliabili y, allowing enginee s o compa e he design alues o ex e nal ac ions wi h he co esponding global s uc u al esis ances [42]. Comp ehensi e delibe a ions conce ning he global sa e y o ma s o he NLFEA o RC s uc u es a e a ailable wi hin he e e ences [ 40 , 43 , 44 ]. Cas aldo e al. [ 35 ] explo ed he impac o alea o y unce ain y and he sensi i i y o he nume ical model in accu a ely p edic ing ailu e modes and hei subsequen e ec s on NLFEA esul s in e ms o eliabil- i y. In es iga ions ha e also been conduc ed o include geome ic non-linea i ies [ 45 , 46 ] and nonlinea nume ical models o RC s uc u es [ 39 , 47 , 48 ]. Howe e , a comp ehensi e assessmen o he combined in luence o ma e ial and geome ic unce ain ies on hyb id GFRP-s eel SFRC beams, pa icula ly when subjec ed o ma e ials non-linea i ies wi hin he global esis ance me hods, is cu en ly lacking. This s udy assesses he applicabili y o global esis ance me hods in e ms o he maximum load and ailu e mode o hyb id GFRP-s eel ein o cemen SFRC beams. By u ilizing expe imen al es esul s on I-shaped SFRC beams, NLFEA models a e de eloped and alida ed. Ini ially, wo expe imen al es s selec ed h ough an expe imen al p og am conduc ed by Mazahe ipou e al. [ 49 ] on hyb id GFRP-s eel SFRC beams, each wi h a ying p es ess le els, we e simula ed wi h a mul i-di ec ional ixed smea ed c ack model (MDFSCM), implemen ed in o Femix V4.0, a so wa e based on he ini e elemen me hod (FEM) [ 50 ]. The global esis ance sa e y ac o is e alua ed, o e ing aluable insigh s in o he s uc u al pe o mance o hyb id ein o cemen SFRC beams and aiding in he applica ion o global esis ance sa e y o ma s, such as he Es ima ion o he Coe icien o Va ia ion (ECoV) me hod [42]. The pape begins by discussing he basic p inciples o GRM and hen p esen s a case s udy o illus a e he applica ion o GRM o he design o an FRC beam. Finally, a compa a i e analysis is conduc ed o assess he design ou comes achie ed wi h a ious sa e y o ma s in he con ex o FRC beam design. Buildings 2023,13, 2848 3 o 20 2. De ini ion and Cha ac e iza ion o Unce ain ies Acco ding o Sa e y Fo ma s o Nonlinea Fini e Elemen Analysis The eliabili y design me hod is an e ec i e app oach o assessing he p obabili y o s uc u al ailu e since i akes unce ain ies in o accoun in a quan i a i e manne . Howe e , i s implemen a ion p esen s challenges due o he conside able compu a ional complexi y, making i di icul o design enginee s o calcula e ailu e p obabili ies. Highe -le el eliabili y analysis me hods, such as Mon e Ca lo simula ion and i s -o de eliabili y me hods, a e also challenging o apply in complex s uc u al design scena ios [51,52]. The “local app oach” is widely emb aced in design codes o p ac ice and is highly e icien o p ac i ione s and designe s [ 53 , 54 ]. I in ol es s uc u al linea analyses o de e mine he esul an s esses in in ended c oss sec ions o he membe s o ming he s uc u e and he compa ison o he design alue o hese esul an s esses (E d ) wi h he co esponding design alue o he esis ing capaci y o he co esponding sec ions (R d ) h ough he pa ial ac o me hod [ 54 ]. Howe e , his local app oach can be oo o e - conse a i e, mainly in s uc u es o high edundan suppo s and/o when hey ha e a high s ess- edis ibu ion capaci y a e c ack ini ia ion [ 35 ], as is he case o FRC, due o he c ack-opening esis ing mechanisms o e ed by ibe s [55]. The “global esis ance app oach” allows o he es ablishmen o app op ia e sa e y c i e ia ha acili a e he compa ison be ween he design alues o ex e nal ac ions (F d ) and he co esponding design global esis ance (R d ) o he s uc u al elemen s o he en i e sys em [ 35 , 40 , 42 , 43 ]. The global s uc u al esis ance can be es ima ed h ough NLFEA, and i s design alue (R d ) can be calcula ed by conside ing he in luence o unce ain ac o s, such as ma e ial p ope ies, geome ic a ia ions, and nume ical model accu acy, by in oducing speci ic sa e y ac o s [ 42 ]. The pa ial sa e y ac o s can be assessed acco ding o p ede ined a ge le els o eliabili y, which di e en ia e be ween newly cons uc ed s uc u es and hose ha al eady exis [ 35 , 42 ]. NLFEA comp ehensi ely accoun o he global esponse o s uc u al elemen s o sys ems, conside ing he e olu ion o damage unde speci ic loading condi ions, he nonlinea beha io o conc e e and ein o cemen s, and e en local o global ins abili ies. Wi hin he global app oach, he design alue o ex e nal ac ions (F d ) is assessed ollowing he guidelines in Eu ocode [ 54 ] and hen compa ed o he design alue o global s uc u al esis ance (R d ), which can be e alua ed using NLFEA based on global esis ance me hods [42] exp essed as ollows: Fd≤Rd=RNLFEAx ep γRγRd (1) whe e RNLFEAx ep ep esen s he global esis ance o a s uc u e e alua ed by NLFEA by employing ep esen a i e alues ( x ep ) o bo h geome ical and ma e ial p ope ies in acco - dance wi h he chosen sa e y o ma . The le el o s uc u al eliabili y is add essed h ough he u iliza ion o wo dis inc global sa e y ac o s: γR , which is he global esis ance sa e y ac o ha accoun s o da a unce ain ies like hose associa ed wi h he ma e ial p ope ies and geome y [ 42 ], and γRd , which ep esen s he esis ance model’s unce ain y, depen- den on he p edic i e pe o mance o he nonlinea ma e ial model [ 28 , 47 , 56 ]. Besides he cha ac e is ics o he model, γRd also depends on he ype o s uc u al ailu e mode and is la ge in s uc u es p esen ing b i le ailu es (punching/shea ) and mixed ailu e modes, and i is smalle in s uc u es wi h duc ile bending ailu e modes (1.06 o 1.16) [42]. As pe ib Model Code 2010, MC2010 [ 57 ], he design esis ance (R d ) can be de e - mined h ough se e al app oaches ha inco po a e a ying deg ees o p obabilis ic heo y implemen a ion. These me hods include he pa ial sa e y ac o (PSF) me hod and he global esis ance me hod (GRM), each o e ing dis inc ways o accoun o unce ain ies and eliabili y in he design p ocess. The PSF me hod in ol es u ilizing he design alues d ( i.e, cd = ck/γc , c d = c k/γc, syd = syk/γs, . . .) as inpu pa ame e s o he NLFEA: Rd=RNLFEA( d)(2) Buildings 2023,13, 2848 4 o 20 The pa ial sa e y ac o s o he cha ac e is ic s eng h o s eel, conc e e, and GFRP du ing he ULS a e 1.15, 1.5, and 1.25, espec i ely [ 42 ]. Howe e , o he SLS, he pa ial sa e y ac o emains a 1. The global esis ance ac o (GRF) me hod is becoming inc easingly popula since i is he ecommended app oach in p ac ice codes [ 42 ] o he design and e alua ion o RC s uc u es using NLFEA. In his app oach, he design esis ance (R d ) is de e mined h ough he ollowing equa ion: Rd=RNLFEA cmd, sym, um γRγRd (3) whe e RNLFEA cmd, sym, um ep esen s he global esis ance o a s uc u e e alua ed by NLFEA using he mean alue o he yield s eng h ( sym = 1.1 syk ) o he s eel ein o cemen and a educed alue o conc e e s eng h ( cmd = 0.85 ck ) o accommoda e i s highe andom a iabili y. To calcula e he educed mean alues o o he ma e ial p ope ies o conc e e, including ensile s eng h, ac u e ene gy, and bond s eng h, he educ ion ac o o 0.85 should be applied. In he absence o a speci ic educed alue o GFRP ba s eng h in MC2010, he cha ac e is ic ensile s eng h ( um = uk ) is assumed o he mean ensile s eng h since i has a mo e b i le beha io han s eel bu no as b i le as conc e e. MC2010 ecommends a pa ial ac o o esis ance ( γR ) o 1.2 and a model unce ain y ac o ( γRd ) o 1.06. I is wo h emphasizing ha in ce ain building scena ios, γR can be calib a ed using he NLFEA, as demons a ed by se e al esea che s [35,41,44,48,58–60]. Acco ding o p obabilis ic s udies, he andom dis ibu ion o esis ance in RC mem- be s can be cha ac e ized by a wo-pa ame e o logno mal dis ibu ion wi h he lowe bound a he o igin [ 61 ]. Hence, he p obabilis ic me hod elies on his logno mal dis- ibu ion assump ion, wi h wo key andom pa ame e s: R m (mean esis ance) and V R (coe icien o a ia ion o esis ance). This me hod is known as ECOV (Es ima ion o he Coe icien o Va ia ion). Using hese assump ions, he andom dis ibu ion o esis ance can be de e mined h ough he mean ( RNLFEA cm, c m, sym, um, . . . ) and cha ac e is- ic ( RNLFEA ck, c k, syk, uk, . . . ) alues o he ma e ial p ope ies used by he model in he NLFEA. Gi en a p obabili y o 0.05 o he cha ac e is ic alue, he coe icien o a ia ion o he dis ibu ion o he global s uc u al esis ance (VR) can be ob ained om VR=1 1.65 ln  RNLFEA cm, c m, sym, um, . . . RNLFEA ck, c k, syk, uk, . . . (4) The calcula ion o he global ac o o he mean esis ance ( γR ) is ela ed o he andom a ia ion in esis ance caused by basic ma e ial pa ame e s and ailu e modes and is de i ed om γR=eαR·β·VR(5) whe e αR is he i s -o de - eliabili y-me hod (FORM) sensi i i y ac o o he eliabili y o esis ance and is equal o 0.8 o a ailu e p obabili y o 1 ‰ [ 42 , 53 ], and he eliabili y index ( β ) is equal o 3.8 o a e e ence pe iod o 50 yea s [ 62 ]. Finally, he design esis ance (Rd) using he ECOV me hod is de e mined om Rd=RNLFEA cm, c m, sym, um, . . . γRγRd (6) The epis emic unce ain y in NLFEA a ises om “missing” knowledge, assump ions, and simpli ica ions conce ning cons i u i e laws, kinema ic compa ibili y, and o ce equi- lib ium [ 63 ]. I also includes unce ain ies ela ed o auxilia y non-physical a iables o indi idual choices made du ing he analysis. Buildings 2023,13, 2848 5 o 20 The model unce ain y ac o ( γRd ) o well- alida ed nume ical models is sugges ed as 1.06 [ 53 ]. Howe e , o models wi h lowe le els o alida ion, model unce ain y alues g ea e han 1.06 should be employed [ 42 ]. Consequen ly, a comp ehensi e in es iga ion in o he de e mina ion o he sa e y ac o s becomes impo an . 3. Expe imen al Tes s o he Hyb id GFRP-S eel Rein o cemen SFRC Beam Figu e 1illus a es he geome ic cha ac e is ics, ein o cemen con igu a ions, suppo , and loading condi ions o he beams analyzed in he expe imen al s udy conduc ed by Mazahe ipou e al. [ 49 ]. A 4 m long beam was chosen o enable a ho ough analysis o i s beha io while s aying wi hin he limi a ions o he a ailable esou ces and expe imen al se up. The adop ion o an I-shaped c oss-sec ional con igu a ion o his beam s uc u e is sugges ed, aiming o achie e enhanced lexu al pe o mance. This choice le e ages he supe io lexu al s i ness inhe en in he I-shaped sec ion, in con as o a ec angula c oss- sec ional beam wi h an equi alen olume o SFRC ma e ial. The igh suppo (Figu e 1) pe mi s ansla ional mo emen in he ho izon al di ec ion and o a ions a ound he e ical and la e al axes, while he le suppo allows only o a ion a ound he la e al axis. Fi e Linea Va iable Di e en ial T ansduce s (LVDT) we e ins alled a in e als spanning he beam’s leng h o moni o he e ical de lec ion. Figu e 1also depic s he a angemen o nine s ain gauges placed on he ma e ials o measu e hei s ain unde load, deno ed as “SG” o GFRP ba s, “SGs ” o s eel s ands, and “SGc” o SFRSCC. The selec ed expe imen al p og am comp ised wo I-shaped beams, namely IB5 and IB6, which bo h had he same ein o cemen bu di e en p es ess le els o he hyb id lexu al ein o cemen . Buildings 2023, 12, x FOR PEER REVIEW 6 o 23 Figu e 1. Geome y, ein o cemen de ails, and suppo and loading condi ions o he beams o he es p og am (dimensions in mm) (adap ed wi h pe mission om [49], Else ie , 2016). In he expe imen al p og am, each beam had lexu al ein o cemen consis ing o wo glass ibe - ein o ced polyme (GFRP) ba s, each wi h a diame e o 12 mm and a o al c oss-sec ional a ea o 269 mm2. Addi ionally, a single s eel s and wi h a diame e o 9 mm and a c oss-sec ional a ea o 51.6 mm2 was also used as longi udinal ein o cemen . The nominal mechanical p ope ies o hese ein o cemen s, ob ained om he manu ac u e s’ in o ma ion, a e p o ided in Table 1. Beam IB5 se ed as he e e ence beam and did no unde go any p es ess applica ion in ei he he GFRP ba s o he s eel s and. In beam IB6, he s eel s and ( p e s ) was p es essed a a le el o 800 MPa, while he GFRP ba s ( p e ) we e p es essed o a le el o 200 MPa, as speci ied in Table 2. The a e age p e-s ain losses a he ime o es ing we e app oxima ely 13.6% o he GFRP ba s and 9.8% o he s eel s and, compa ed o he p e-s ain measu ed on he day o p es essing hese ein o cemen s. Table 1. Nominal p ope ies o he GFRP eba s and s eel endons used in he p esen s udy. Type Diame e C oss Sec ion A ea Modulus o Elas ici y Yielding S ain Yielding S ess Ul ima e S ess Ul ima e S ain Weigh , s  , s AA , s EE εsy sy , u su ε ,ε u su (mm) (mm2) (GPa) (%) (MPa) (MPa) (%) (g/m) GFRP ba 2 ϕ 13.1 269.0 60.0 – – 1350 2.25 317 S eel s and ϕ9 51.6 187.5 ~0.8 ~1600 ~1900 >3.5 405 Table 2. Beam iden i ica ion, ein o cing a io o GFRP and s eel ba s, p es ess le el, maximum load, i s cen al de lec ion, and ailu e mode. Beam ID ρ ρs p e s P es ess Le el (S eel) p e P es ess Le el (GFRP) ,expu M max P De lec ion a max P *  * s  Mode o Failu e (%) (%) (MPa) (%) (MPa) (%) (MPa) (kN) (mm) (%) (%) IB5 0.29 0.07 0.0 0 0 0 148 185 47 1.05 - Shea IB6 0.29 0.07 800 50 200 15 186 232 64 1.09 - Flexo shea * Las alue o s ain eco ded du ing he mono onic es . SGc 150 3700 150 LVDT1 LVDT2 LVDT3 LVDT4 LVDT5 Sec. 1 Sec. 2 Sec. 3 Sec. 4 Sec. 5 1600 700 500 SG1 SG2 SG3 GFRP 1 S eel s and GFRP 2 SG4 SG5 SGs 700 Bending zone 200 Shea zone Shea zone 250 250 Figu e 1. Geome y, ein o cemen de ails, and suppo and loading condi ions o he beams o he es p og am (dimensions in mm) (adap ed wi h pe mission om [49], Else ie , 2016). In he expe imen al p og am, each beam had lexu al ein o cemen consis ing o wo glass ibe - ein o ced polyme (GFRP) ba s, each wi h a diame e o 12 mm and a o al c oss-sec ional a ea o 269 mm 2 . Addi ionally, a single s eel s and wi h a diame e o 9 mm and a c oss-sec ional a ea o 51.6 mm 2 was also used as longi udinal ein o cemen . The nominal mechanical p ope ies o hese ein o cemen s, ob ained om he manu ac u e s’ in o ma ion, a e p o ided in Table 1. Beam IB5 se ed as he e e ence beam and did no unde go any p es ess applica ion in ei he he GFRP ba s o he s eel s and. In beam IB6, he s eel s and ( p e s ) was p es essed a a le el o 800 MPa, while he GFRP ba s ( p e ) we e p es essed o a le el o 200 MPa, as speci ied in Table 2. The a e age p e-s ain losses a he ime o es ing we e app oxima ely 13.6% o he GFRP ba s and 9.8% o he s eel s and, compa ed o he p e-s ain measu ed on he day o p es essing hese ein o cemen s. Buildings 2023,13, 2848 6 o 20 Table 1. Nominal p ope ies o he GFRP eba s and s eel endons used in he p esen s udy. Type Diame e C oss Sec ion A ea Modulus o Elas ici y Yielding S ain Yielding S ess Ul ima e S ess Ul ima e S ain Weigh φ ,φsA ,AsE ,Esεsy sy u, su ε u,εsu (mm) (mm2)(GPa) (%) (MPa) (MPa) (%) (g/m) GFRP ba 2 φ13.1 269.0 60.0 – – 1350 2.25 317 S eel s and φ9 51.6 187.5 ~0.8 ~1600 ~1900 >3.5 405 Table 2. Beam iden i ica ion, ein o cing a io o GFRP and s eel ba s, p es ess le el, maximum load, i s cen al de lec ion, and ailu e mode. Beam ID ρ ρs p e s P es ess Le el (S eel) p e P es ess Le el (GFRP) Mu,exp Pmax De lec ion a Pmax ε* ε* s Mode o Failu e (%) (%) (MPa) (%) (MPa) (%) (MPa) (kN) (mm) (%) (%) IB5 0.29 0.07 0.0 0 0 0 148 185 47 1.05 - Shea IB6 0.29 0.07 800 50 200 15 186 232 64 1.09 - Flexo shea * Las alue o s ain eco ded du ing he mono onic es . The s eel ibe - ein o ced sel -compac ing conc e e (SFRSCC) con aining 90 kg/m 3 o s eel ibe s was p epa ed using a mixing me hod ou lined in a di e en sou ce [ 64 ]. This conc e e mix u e included hooked-end s eel ibe s wi h a leng h o 33 mm, an aspec a io o 65, and a yield s ess o 1100 MPa. To assess he comp essi e s eng h and Young’s modulus o he SFRSCC, a o al o 25 cylind ical specimens wi h a diame e o 150 mm and a heigh o 300 mm we e subjec ed o comp ession es s in acco dance wi h he ASTMC39 s anda ds [ 65 ]. The esul s showed an a e age comp essi e s eng h o 73 MPa wi h a coe icien o a ia ion (CoV) o 6%. Addi ionally, he a e age Young’s modulus was ound o be 35.4 GPa wi h a CoV o 3%. The a e age esidual lexu al ensile s eng h pa ame e s o he SFRSCC ( Ri , i=1 o 4 ) we e ob ained om he applied o ce (F) e sus he C ack Mou h Opening Displacemen (CMOD) diag ams [ 66 ] by execu ing h ee-poin no ched beam bending es s acco ding o he ecommenda ions o EN 1465 [ 67 ]. Two sepa a e se s o no ched beams we e in es i- ga ed, wi h each se con aining i e specimens. The i s se comp ised s anda d specimens measu ing 150 × 150 × 600 mm, and hese specimens had a no ch dep h o 35 mm. The second se consis ed o no ched beam specimens ob ained om in ac sec ions a bo h ends o he beams. These specimens had dimensions o 70 × 70 × 600 mm and a no ch dep h o 15 mm. The esul s o he i e no ched beam bending es s in ol ing hese specimens a e g aphically ep esen ed in Figu e 2, whe e he applied o ce has been no malized agains he CMOD. The de ailed esul s o he esidual lexu al ensile s eng hs R1 , R2 , R3, and R4 o a CMOD o 0.5, 1.5, 2.5 and 3.5 mm, espec i ely, ob ained om he expe imen al es s o bo h se s o specimens a e p o ided in Table 3. The pos -c acking beha io obse ed in ex ac ed membe s exhibi s a no ably high de- g ee o dispe sion o he da a wi h espec o he mean alues, which is p ima ily a ibu ed o he oo-small ac u ed su ace a ea o hese specimens ega ding he maximum dimen- sion o he agg ega es and leng h o ibe s adop ed o p oducing he SFRSCC. In his case, he numbe o ibe s c ossing he ac u e su ace is ela i ely small, as he pos -c acking ensile capaci y o hese small specimens is e y sensi i e o he dis ibu ion and o ien a ion o he ibe s, con ibu ing signi ican ly o he ob ained dispe sion o Ri . Since his ac u e is much lowe han wha is expec ed o occu in he ailu e o he beams, he Ri ob ained in he es s o he second se ies should no be ep esen a i e o he pos -c acking ensile beha io o he SFRSCC, bu speci ic esea ch is equi ed in his domain. This obse a ion highligh s he deba able na u e o assuming iso opic pos -c acking beha io when simu- Buildings 2023,13, 2848 7 o 20 la ing SFRSCC s uc u es; howe e , o he design in his s udy, he pos -c acking beha io esul s om specimens wi h s anda d size we e u ilized. Buildings2023,12,xFORPEERREVIEW7o 22  70×600mmandano chdep ho 15mm.The esul so  he i eno chedbeambending es sin ol ing hesespecimensa eg aphically ep esen edinFigu e2,whe e he applied o cehasbeenno malizedagains  heCMOD.Thede ailed esul so  he esidual lexu al ensiles eng hs R1, R2, R3,and R4 o aCMODo 0.5,1.5,2.5and3.5 mm, espec i ely,ob ained om heexpe imen al es s o bo hse so specimensa e p o idedinTable3.  (a)(b) Figu e2.Theno malizedapplied o ce e susCMODde i ed om heno chedbeam es s.(a) Fi s se ies(s anda d es ).(b)Secondse ies. Table3.Thea e age esidual lexu al ensiles eng hpa ame e so SFRSCC. ResidualFlexu alTensileS eng hPa ame e s 1 C M O D2 C M O D3 C M O D4 C M O D 1 F 1, Rm 2 F 2,m R 3 F 3,m R 4 F 4,m R  (kN)(MPa)(kN)(MPa)(kN)(MPa)(kN)(MPa) A e ageo  i s se ies(150 ×150×600mm) 30.65 [1.29] 11.59 [0.49] 30.81 [1.87] 11.65 [0.7] 28.25 [1.63] 10.68 [0.62] 25.26 [1.61] 9.55 [0.61] A e ageo secondse ies (70×70×600mm) 2.37 [0.96] 8.37 [3.37] 2.47 [1.13] 8.73 [4.01] 2.25 [1.09] 7.99 [3.86] 2.06 [1.00] 7.51 [3.80] The aluesenclosedwi hinb acke sdeno e hes anda dde ia ion. Thepos ‐c ackingbeha io obse edinex ac edmembe sexhibi sano ablyhigh deg eeo dispe siono  heda awi h espec  o hemean alues,whichisp ima ily a ibu ed o he oo‐small ac u edsu acea eao  hesespecimens ega ding he maximumdimensiono  heagg ega esandleng ho  ibe sadop ed o p oducing he SFRSCC.In hiscase, henumbe o  ibe sc ossing he ac u esu aceis ela i ely small,as hepos ‐c acking ensilecapaci yo  hesesmallspecimensis e ysensi i eo  hedis ibu ionando ien a iono  he ibe s,con ibu ingsigni ican ly o heob ained dispe siono  R i .Since his ac u eismuchlowe  hanwha isexpec ed ooccu in he ailu eo  hebeams, he R i ob ainedin he es so  hesecondse iesshouldno be ep esen a i eo  hepos ‐c acking ensilebeha io o  heSFRSCC,bu speci ic esea ch is equi edin hisdomain.Thisobse a ionhighligh s hedeba ablena u eo assuming iso opicpos ‐c ackingbeha io whensimula ingSFRSCCs uc u es;howe e , o  he designin hiss udy, hepos ‐c ackingbeha io  esul s omspecimenswi hs anda d sizewe eu ilized. Du ing heexpe imen alp og am, heI‐shapedbeamswe esubjec ed oa ou ‐ poin bending es con igu a ionusingase o‐hyd aulicac ua o un il ailu e.The es s Figu e 2. The no malized applied o ce e sus CMOD de i ed om he no ched beam es s. ( a ) Fi s se ies (s anda d es ). (b) Second se ies. Table 3. The a e age esidual lexu al ensile s eng h pa ame e s o SFRSCC. Residual Flexu al Tensile S eng h Pa ame e s CMOD1CMOD2CMOD3CMOD4 F1 R1,mF2 R2,m F3 R3,m F4 R4,m (kN) (MPa) (kN) (MPa) (kN) (MPa) (kN) (MPa) A e age o i s se ies (150 ×150 ×600 mm) 30.65 [1.29] 11.59 [0.49] 30.81 [1.87] 11.65 [0.7] 28.25 [1.63] 10.68 [0.62] 25.26 [1.61] 9.55 [0.61] A e age o second se ies (70 ×70 ×600 mm) 2.37 [0.96] 8.37 [3.37] 2.47 [1.13] 8.73 [4.01] 2.25 [1.09] 7.99 [3.86] 2.06 [1.00] 7.51 [3.80] The alues enclosed wi hin b acke s deno e he s anda d de ia ion. Du ing he expe imen al p og am, he I-shaped beams we e subjec ed o a ou -poin bending es con igu a ion using a se o-hyd aulic ac ua o un il ailu e. The es s we e conduc ed unde mono onic loading condi ions, wi h 500 mm be ween he applied loads, as illus a ed in Figu e 1. The ac ua o ’s pis on was displacemen -con olled a a speed o 0.01 mm/s. The load de lec ion esponse o es ed beams is illus a ed in Figu e 3. The expe imen al indings demons a e a clea end o inc easing he beam’s shea capaci y wi h he p es ess applied o he lexu al ein o cemen . This imp o emen is a ibu ed o he a o able mechanism o agg ega e in e lock and a la ge unc acked SFRC a ea. A compa ison among expe imen al esul s also indica es ha using he p es ess in ein o cemen esul s in a signi ican enhancemen in he load-ca ying capaci y. This end is also e iden in he de lec ion o he beams. In cases whe e he p es ess was applied, he load–de lec ion cu e exhibi ed a pla eau a e eaching a de lec ion o 35 mm, indica ing ha no u he inc ease in he load was obse ed. Mo eo e , as he p es ess le el inc eased, he GFRP ba s expe ienced highe s ains a he poin o ailu e. Buildings 2023,13, 2848 8 o 20 Buildings2023,12,xFORPEERREVIEW8o 22  we econduc edunde mono onicloadingcondi ions,wi h500mmbe ween heapplied loads,asillus a edinFigu e1.Theac ua o ’spis onwasdisplacemen ‐con olleda a speedo 0.01mm/s.Theloadde lec ion esponseo  es edbeamsisillus a edinFigu e 3.  Figu e3.Applied o ce e susmid‐spande lec iono  he es edbeams. Theexpe imen al indingsdemons a eaclea  endo inc easing hebeam’sshea  capaci ywi h hep es essapplied o he lexu al ein o cemen .Thisimp o emen is a ibu ed o he a o ablemechanismo agg ega ein e lockandala ge unc acked SFRCa ea.Acompa isonamongexpe imen al esul salsoindica es ha using he p es essin ein o cemen  esul sinasigni ican enhancemen in heload‐ca ying capaci y.This endisalsoe iden in hede lec iono  hebeams. Incaseswhe e hep es esswasapplied, heload–de lec ioncu eexhibi eda pla eaua e  eachingade lec iono 35mm,indica ing ha no u he inc easein he loadwasobse ed.Mo eo e ,as hep es essle elinc eased, heGFRPba s expe iencedhighe s ainsa  hepoin o  ailu e. 4.Code‐BasedDesignP o isions o Hyb idGFRP–S eel‐Rein o cemen SFRC Beams:E alua ionandAnalysis In hissec ion,ab ie analysiso  hesimpli iedme hodologiesemployedby enginee sin he lexu alandshea designo SFRCbeamsispe o med. Tode e mine hedesign lexu alcapaci yo ac osssec ion, R d M , hes ain compa ibili yandcons i u i elawso  hein e enien ma e ials,and he o ces equilib iuma econside ed.Figu e4illus a es heexpe imen albeam’sc osssec ion alongwi h heassocia eds ainands essdis ibu ions.  Figu e 3. Applied o ce e sus mid-span de lec ion o he es ed beams. 4. Code-Based Design P o isions o Hyb id GFRP–S eel-Rein o cemen SFRC Beams: E alua ion and Analysis In his sec ion, a b ie analysis o he simpli ied me hodologies employed by enginee s in he lexu al and shea design o SFRC beams is pe o med. To de e mine he design lexu al capaci y o a c oss sec ion, MRd , he s ain com- pa ibili y and cons i u i e laws o he in e enien ma e ials, and he o ces equilib ium a e conside ed. Figu e 4illus a es he expe imen al beam’s c oss sec ion along wi h he associa ed s ain and s ess dis ibu ions. Buildings2023,12,xFORPEERREVIEW8o 22  we econduc edunde mono onicloadingcondi ions,wi h500mmbe ween heapplied loads,asillus a edinFigu e1.Theac ua o ’spis onwasdisplacemen ‐con olleda a speedo 0.01mm/s.Theloadde lec ion esponseo  es edbeamsisillus a edinFigu e 3.  Figu e3.Applied o ce e susmid‐spande lec iono  he es edbeams. Theexpe imen al indingsdemons a eaclea  endo inc easing hebeam’sshea  capaci ywi h hep es essapplied o he lexu al ein o cemen .Thisimp o emen is a ibu ed o he a o ablemechanismo agg ega ein e lockandala ge unc acked SFRCa ea.Acompa isonamongexpe imen al esul salsoindica es ha using he p es essin ein o cemen  esul sinasigni ican enhancemen in heload‐ca ying capaci y.This endisalsoe iden in hede lec iono  hebeams. Incaseswhe e hep es esswasapplied, heload–de lec ioncu eexhibi eda pla eaua e  eachingade lec iono 35mm,indica ing ha no u he inc easein he loadwasobse ed.Mo eo e ,as hep es essle elinc eased, heGFRPba s expe iencedhighe s ainsa  hepoin o  ailu e. 4.Code‐BasedDesignP o isions o Hyb idGFRP–S eel‐Rein o cemen SFRC Beams:E alua ionandAnalysis In hissec ion,ab ie analysiso  hesimpli iedme hodologiesemployedby enginee sin he lexu alandshea designo SFRCbeamsispe o med. Tode e mine hedesign lexu alcapaci yo ac osssec ion, R d M , hes ain compa ibili yandcons i u i elawso  hein e enien ma e ials,and he o ces equilib iuma econside ed.Figu e4illus a es heexpe imen albeam’sc osssec ion alongwi h heassocia eds ainands essdis ibu ions.  Figu e 4. S ain dis ibu ion and in e nal o ces o he I c oss sec ion o he es ed beams (adap ed wi h pe mission om [49], Else ie , 2016). The alue o MRd was ob ained wi h he o mula ion esumed in Table A1 o Appendix A, whe e design alues o he ma e ial p ope ies we e used. Mo e de ails on his o mula ion a e a ailable elsewhe e [49]. The design shea capaci y, VRd , o he es ed beams was assessed using he o mula ion ecommended by he MC2010, as ou lined in Table A2 o Appendix A. The de ailed desc ip ion can be ound in Ba os e al. [68]. By applying hese o mula ions, he lexu al and shea design esis ance o IB5 and IB6 we e de e mined and included in Table 4. Conside ing he beam is simply suppo ed (Figu e 1), he design ul ima e load, Rd, was ob ained, which is also included in his able. Buildings 2023,13, 2848 9 o 20 Table 4. Expe imen al load-ca ying capaci y and he design alues o he lexu al and shea capaci y acco ding o he adop ed o mula ions. Specimen Expe imen al Ul ima e Load (kN) MRd (kN.m) VRd (kN) Design Ul ima e Load Rd(kN) IB5 184.71 163.29 39.95 79.90 IB6 232.23 170.37 46.94 93.88 When conside ing he design alue o he IB5 and IB6 lexu al s eng h, hei load- ca ying capaci ies a e, espec i ely, 163.29/1.6 × 2 = 102 kN and 170.37/1.6 × 2 = 213 kN. Howe e , he load-ca ying capaci y o hese beams ( Rd ) is limi ed by hei shea s eng h, esul ing in he alues indica ed in Table 4. These alues a e abou 40% o he co esponding alues egis e ed expe imen ally, esul ing in a sa e y ac o o abou 2.35. The calcula ed design load mus be lowe han he design load ob ained a e applying he applicable code- p esc ibed pa ial sa e y ac o s o a ious load combina ions. This obse a ion highligh s he conse a ism o he ecommended design o mulas o hese beams acco ding o he applied code. 5. Modeling Hypo heses o he NLFEA o SFRC Beams and Compa ison wi h he Expe imen al Resul s The NLFEA o conc e e was ca ied ou using a c ack-shea -so ening law [ 50 ] wi h a mul i-di ec ional ixed smea ed c ack model (MDFSCM) a ailable in he FEMIX compu e p og am, whose de ails a e desc ibed elsewhe e [ 69 ]. Acco ding o he MDFSCM, a c ack (in eali y, c acks a e smea ed in he co esponding in eg a ion poin (IP)) is o med when he p incipal ensile s ess, σI , a ains he ensile s eng h o he ma e ial ( ) a less han an adop ed small ole ance. A new c ack is o med in an al eady-c acked IP when, besides he p e ious c i e ion, he angle o med be ween he new c ack and p e ious ac i e c acks (no comple ely closed) is highe han an adop ed h eshold angle (in gene al θ h ∈ [30–60] ◦ ). To a oid nume ical ins abili ies wi h he occu ence o se e al c ack s a us changes du ing he loading p ocess, a maximum numbe o c acks ( Nc max ) o h ee and ou is gene ally adop ed o , espec i ely, 2D- and 3D- ype MDFSCM. Fo simula ing he ac u e mode I (o hogonal o he c ack plane) and mode II (pa allel o c ack plane), he NLMM104 nonlinea ma e ial model o FEMIX was used. In an a emp o p ese e he esul s independen ly o he e inemen o he ini e elemen mesh, he c ack wid h and c ack sliding a e di ided by he c ack bandwid h, lb , which is a ce ain leng h associa ed wi h he geome y o he FE. In he p esen e sion, he same lb was adop ed o he c ack-opening and -sliding p ocess, which was equal o he squa e oo o he a ea o he in eg a ion poin o he c acked ini e elemen ( √AIP ). In he NLMM104 model, i is assumed ha he conc e e in comp ession exhibi s linea elas ic beha io , which is an accep able assump ion in he es ed beams since he maximum comp essi e s ain le el is expec ed o be much smalle han he SFRC c ushing s ain. The sys em o nonlinea equa ions is sol ed using he s anda d New on–Raphson i e a i e me hod employing an ene gy con e gence c i e ion. Simul aneously, he displacemen a he poin o load applica ion was con olled using he a c-leng h me hod. The s ess–c ack-opening ela ionship o he SFRSCC in each se ies o no ched beam bending es s was ob ained h ough in e se analysis (IA) [ 70 ] using he o ce–CMOD esponse by i ing he load e sus he CMOD egis e ed expe imen ally. Fo nume ical simula ions, wo models we e employed. In he i s model, he p ope ies o he op and bo om langes we e assigned o he i s se ies o specimens (150 × 150 × 600 mm), while he p ope ies o he web we e associa ed wi h he second se ies (70 × 70 × 600 mm). In he second model, he p ope ies o all SFRSCC elemen s we e assigned based on he s anda d size ecommended by Eu ocode 2, which co esponds o he i s se ies. Failu e c i e ia in all simula ions a e de e mined acco ding o wo dis inc scena ios. The i s occu s when he maximum load-ca ying capaci y is achie ed and is obse able in he load-de lec ion esponse o he beam h ough ully open c acks in ini e elemen s, Buildings 2023,13, 2848 16 o 20 Table 9. Resul s ob ained o he design ul ima e load, compu ed based on a ious sa e y o ma s. Specimen Expe imen al Ul ima e Load (kN) Sa e y Fo ma Global Resis ance Fac o γR Design Ul ima e Load Rd(kN) IB5 184.71 PSF - 108.22 GRM 1.2 138.03 ECOV 1.54 121.73 IB6 232.23 PSF - 109.39 GRM 1.2 174.21 ECOV 1.34 170.55 The ecommended global esis ance ac o o 1.2 o he GFR me hod, as ou lined in Table 9, is in acco dance wi h MC2010. The global esis ance ac o o ECOV is de e mined h ough he applica ion o Equa ions (4) and (5). The design ul ima e load alues, as p esen ed in Table 9 o PSF, GRM, and ECOV, we e de e mined based on he maximum load ob ained om NLFEA, and in acco dance wi h Equa ions (2), (3), and (6), espec i ely. The esul s demons a e ha inco po a ing design pa ame e s and accoun ing o sa e y ac o s makes he inal design mo e han 40% conse a i e, making i he mos conse a i e design among he a ious sa e y o ma s employed. Ne e heless, i emains mo e economical han he esul s ob ained using he MC2010 ecommended alues. Addi ionally, when design alues o ma e ial p ope ies a e applied, he ensile capaci y o he SFRSCC is subjec o mo e subs an ial penaliza ion compa ed o he lexu al ein o cemen , pa icula ly he s eel componen . This dispa i y can be a ibu ed o he ac ha he beha io o he beam is no ably in luenced by he p ope ies o he lexu al ein o cemen . Employing he GRM sa e y o ma wi h a ecommended global esis ance ac o o 1.2 as pe MC2010 leads o a mo e economical design compa ed o o he me hods used. Speci ically, he e was a 25% conse a ism in he ul ima e load be ween he expe imen al and GRM o IB5 and IB6. On he o he hand, u ilizing ECOV sa e y o ma esul ed a global esis ance ac o ha exceeded he ecommended alue p o ided by MC2010. This di e gence is p ima ily a ibu ed o dispa i ies in he ul ima e load ob ained om NLFEA when using he mean and cha ac e is ic ma e ial p ope ies. The conside able s anda d de ia ion in ac u e p ope ies o SFRC ma e ials u he ampli ies he con as be ween cha ac e is ic and mean ul ima e loads, consequen ly leading o a highe global sa e y ac o . 7. Conclusions The objec i e o his s udy was o e alua e and con as a ious sa e y app oaches in he con ex o es ima ing he global design s eng h o hyb id s eel-GFRP SFRC beams. To achie e his, NLFE models we e accu a ely de ined o eplica e he expe imen al es s and subsequen ly used o conduc nume ous NLFEA, aligning wi h he dis inc sa e y app oaches, o each SFRC beam. Upon e iewing he esul s, we can conclude ha all sa e y o ma s yield he design o ul ima e loads ha a e lowe han he obse ed expe imen al ou come. Addi ionally, conside ing he non- o al iso opy o SFRC ma e ials, i is ad isable o conside a global sa e y ac o highe han he ecommended alue o 1.2 speci ied in MC2010. Despi e his, in beams wi hou s i ups, whe e shea is he go e ning ailu e mode, NLFEA is an icipa ed o p o ide a mo e dependable design as opposed o conse a i e simpli ied equa ions, which a e ecommended by MC2010. Employing he GRM sa e y o ma wi h a global esis ance ac o o 1.2 as pe MC2010 guidelines eme ges as he mos cos -e ec i e app oach. Howe e , he ECOV sa e y o ma exhibi s a highe global esis ance ac o o 1.5 due o dispa i ies be ween he cha ac e is ic and mean ul ima e loads de i ed om NLFEA. These di e ences a e accen ua ed by he signi ican s anda d de ia ion in ac u e p ope ies o SFRC ma e ials. As a esul , ECOV p o es o be a mo e conse a i e design in he case o in es iga ed SFRC beams, su passing he ecommended MC2010 alues. Buildings 2023,13, 2848 17 o 20 Au ho Con ibu ions: K.B.S.: Concep ualiza ion, Me hodology, Fo mal analysis, In es iga ion, Visu- aliza ion, W i ing—o iginal d a . J.A.O.B.: Concep ualiza ion, Me hodology, Valida ion, Supe ision, W i ing— e iew and edi ing. I.B.V.: Valida ion, Supe ision, W i ing— e iew and edi ing. All au ho s ha e ead and ag eed o he published e sion o he manusc ip . Funding: The i s au ho g a e ully acknowledges he inancial suppo o “Fundação pa a a Ciência e Tecnologia” (FCT-Po ugal), h ough he PhD g an SFRH/BD/09253/2020. Da a A ailabili y S a emen : Da a on which his pape is based a e a ailable om he au ho s upon easonable eques . Acknowledgmen s: The au ho s acknowledge he suppo p o ided by FCT h ough he p ojec FemWebAI, e e ence PTDC/ECI-EST/6300/2020, and PID2021-125553NB-I00 (MCI/AEI/FEDER, UE). This wo k was pa ly inanced by FCT / MCTES h ough na ional unds (PIDDAC) unde he R&D Uni Ins i u e o Sus ainabili y and Inno a ion in S uc u al Enginee ing (ISISE), unde e e - ence UIDB/04029/2020, and unde he Associa e Labo a o y Ad anced P oduc ion and In elligen Sys ems ARISE unde e e ence LA/P/0112/2020. Con lic s o In e es : The au ho s decla e no con lic o in e es . Appendix A Table A1. Flexu al design esis ance analysis o he beams: bending o mula compila ion. Analy ical Momen Fo mula Mn=   Fsyds−β1c 2+F ud −β1c 2+σc c ,b(Ag−cβ2b)d −β1c 2ρ ≤ρhb F d −β1c 2+Fsyds−β1c 2+σc c (Ag−cβ2b)d −β1c 2ρ >ρhb (A1) ρhb =β2α1β1 cm u +m εcu εcu +ε u −εp e −msρs−m ρc(A2) ρc=Ag bd ,ρs=As bds,ρ =A bd (A3) ε =d c−1εcu +εp e (A4) Fsy =As sy,F u =A u (A5) d =1 2(h+c)(A6) c=cb+γ(ρ −ρhb)(A7) γ=m2 u β2(m1 cm −σc c ,b)d (A8) m1=(1+ρ /ρhb)α1β1ρ ⩽ρhb α1β1ρ >ρhb ,m2=1ρ ⩽ρhb ρhb/ρ ρ >ρhb (A9) α1=   1 2Ecεc cm 1 β1εc⩽εcp 1−1 2εcp εc1 β1εcp <εc⩽εcu ,β1=(2 3εc⩽εcp 1−εcp εc+1 3εcp εc2εcp <εc⩽εcu A ul ima e comp essi e s ain o conc e e (εc=εcu):α1=0.949 and β1=0.752 (A10) m =σc c ,b u (A11) σc c ,b=G ,u ε u (A12) β2=(1cb⩽h1 h1 cb−1−h1 cb1−cb−h1 2bh2(b−bw)h1<cb⩽h1+h2 (A13) cb=εcu εcu +ε u −εp e d (A14) ms=ds d sy u (A15) Buildings 2023,13, 2848 18 o 20 Table A2. MC2010 app oaches o p edic ing shea esis ance o SFRC beams wi hou s i ups. Analy ical Shea Fo mula VRd =0.18 γckh100ρs,eq1+7.5 F uk c k  cki1/3 +0.15σcpbwds,eq (A16) σcp =Nsd Ac <0.2 ck γc(A17) k=1+q200 ds,eq ≤2.0 (A18) ds,eq =Asds+(EGFRP/Es)AGFRPdGFRP As+(EGFRP/Es)AGFRP (A19) ρs,eq =As bwds+EGFRP Es AGFRP bwdGFRP (A20) F uk =0.45 Rk,1 −0.6(0.65 Rk,1 −0.5 Rk,3)⩾0 (A21) Re e ences 1. Lee, H.; Choi, M.K.; Kim, B.-J. S uc u al and Func ional P ope ies o Fibe Rein o ced Conc e e Composi es o Cons uc ion Applica ions. J. Ind. Eng. Chem. 2023,125, 38–49. [C ossRe ] 2. Rossi, P. Nume ical Designing o Fibe Rein o ced Conc e e Eco-Cons uc ions. Ma e ials 2023,16, 2576. [C ossRe ] [PubMed] 3. Gha ehbaghi, K.; Rahmani, F.; Pa e no, D.; Gha ehbaghi, S. Pe o mance o FRC and GPC o High-Rise Cons uc ion: Case S udies. IOP Con . Se . Ma e . Sci. Eng. 2020,829, 012002. [C ossRe ] 4. Ahmed, T.; Chidamba am, R.S. Shea s eng h o s eel ibe ein o ced conc e e beam—A e iew. Ma e . Today P oc. 2022 ,64, 1087–1093. [C ossRe ] 5. Liao, L.; Zhao, J.; Zhang, F.; Li, S.; Wang, Z. Expe imen al s udy on comp essi e p ope ies o SFRC unde high s ain a e wi h di e en ibe con en and aspec a io. Cons . Build. Ma e . 2020,261, 119906. [C ossRe ] 6. Zhao, M.; Li, J.; Xie, Y.M.; Shen, J. Semi-empi ical syne ge ic analysis o he shea capaci y o s eel ib e ein o ced conc e e slende beams o ec angula -sec ions wi hou s i ups. Eng. S uc . 2023,285, 116035. [C ossRe ] 7. Yu, Y.; Zhao, X.Y.; Xu, J.J.; Wang, S.C.; Xie, T.Y. E alua ion o shea capaci y o s eel ibe ein o ced conc e e beams wi hou s i ups using a i icial in elligence models. Ma e ials 2022,15, 2407. [C ossRe ] 8. Yang, J.-M.; Kim, J.-K.; Yoo, D.-Y. Flexu al and shea beha iou o high-s eng h SFRC beams wi hou s i ups. Mag. Conc . Res. 2019,71, 503–518. [C ossRe ] 9. Almasabha, G.; Mu ad, Y.; Alghossoon, A.; Saleh, E.; Ta awneh, A. Sus ainabili y o Using S eel Fibe s in Rein o ced Conc e e Deep Beams wi hou S i ups. Sus ainabili y 2023,15, 4721. [C ossRe ] 10. Adam, M.A.; E an, A.M.; Habib, F.A.; El-Sayed, T.A. S uc u al beha io o high-s eng h conc e e slabs ein o ced wi h GFRP ba s. Polyme s 2021,13, 2997. [C ossRe ] 11. Xiao, S.-H.; Lin, J.X.; Li, L.J.; Guo, Y.C.; Zeng, J.J.; Xie, Z.H.; Wei, F.F.; Li, M. Expe imen al s udy on lexu al beha io o conc e e beam ein o ced wi h GFRP and s eel- ibe composi e ba s. J. Build. Eng. 2021,43, 103087. [C ossRe ] 12. Si imon ee, S.; Keawsawas ong, S.; Thongchom, C. Flexu al beha io o conc e e beam ein o ced wi h GFRP ba s compa ed o conc e e beam ein o ced wi h con en ional s eel ein o cemen s. J. Appl. Sci. Eng. 2021,24, 883–890. 13. Han, S.; Fan, C.; Zhou, A.; Ou, J. Shea beha io o conc e e beams ein o ced wi h co osion- esis an and duc ile longi udinal s eel-FRP composi e ba s and FRP s i ups. Eng. S uc . 2023,278, 115520. [C ossRe ] 14. Nguyen, T.H.; Nguyen, V.T.; Phan, M.T. Expe imen al s udy on he lexu al beha iou o co oded conc e e beams ein o ced wi h hyb id s eel/GFRP ba s. S uc . In as uc . Eng. 2022, 1–12. [C ossRe ] 15. Al udaini, T.M.S. Op imiza ion o conc e e beams ein o ced wi h GFRP ba s. J. So Compu . Ci . Eng. 2022,6, 18–38. 16. Zhou, C.; Pan, J.; Zhang, Z.; Zhu, Y. Compa a i e s udy on he ensile mechanical beha io o GFRP ba s unde and a e high empe a u e exposu e. Case S ud. Cons . Ma e . 2022,16, e00905. [C ossRe ] 17. Zhao, J.; Pan, H.; Wang, Z.; Li, G. Expe imen al and Theo e ical S udy on Flexu al Beha io o GFRP-and CFRP-Rein o ced Conc e e Beams a e High-Tempe a u e Exposu e. Polyme s 2022,14, 4002. [C ossRe ] 18. Xingyu, G.; Yiqing, D.; Jiwang, J. Flexu al beha io in es iga ion o s eel-GFRP hyb id- ein o ced conc e e beams based on expe imen al and nume ical me hods. Eng. S uc . 2020,206, 110117. [C ossRe ] 19. Moolaei, S.; Sha ba da , M.K.; Khey oddin, A. Expe imen al e alua ion o lexu al beha io o HPFRCC beams ein o ced wi h hyb id s eel and GFRP ba s. Compos. S uc . 2021,275, 114503. [C ossRe ] 20. Abbas, H.; Abadel, A.; Almusallam, T.; Al-Salloum, Y. Expe imen al and analy ical s udy o lexu al pe o mance o conc e e beams ein o ced wi h hyb id o GFRP and s eel eba s. Eng. Fail. Anal. 2022,138, 106397. [C ossRe ] 21. De a aj, R.; Olo injana, A.; Ge be , C. Making a Case o Hyb id GFRP-S eel Rein o cemen Sys em in Conc e e Beams: An O e iew. Appl. Sci. 2023,13, 1463. [C ossRe ] 22. Xu, J.; Zhu, P.; Ma, Z.J.; Qu, W. Fa igue lexu al analysis o conc e e beams ein o ced wi h hyb id GFRP and s eel ba s. Eng. S uc . 2019,199, 109635. [C ossRe ] 23. Elwakkad, N.Y.; Heiza, K.M.; Mansou , W. Expe imen al S udy and Fini e Elemen Modelling o he To sional Beha io o Sel -Compac ing Rein o ced Conc e e (SCRC) Beams S eng hened by GFRP. Case S ud. Cons . Ma e . 2023 ,18, e02123. [C ossRe ] 24. Mus a a, S.A.; Hassan, H.A. Beha io o conc e e beams ein o ced wi h hyb id s eel and FRP composi es. HBRC J. 2018 ,14, 300–308. [C ossRe ] Buildings 2023,13, 2848 19 o 20 25. Sol anzadeh, F.; Edala -Behbahani, A.; Ba os, J.A.; Mazahe ipou , H. E ec o ibe dosage and p es ess le el on shea beha io o hyb id GFRP-s eel ein o ced conc e e I-shape beams wi hou s i ups. Compos. Pa B Eng. 2016,102, 57–77. [C ossRe ] 26. Díaz, R.A.S.; No a, S.J.S.; da Sil a, M.C.T.; T au wein, L.M.; de Almeida, L.C. Reliabili y analysis o shea s eng h o ein o ced conc e e deep beams using NLFEA. Eng. S uc . 2020,203, 109760. [C ossRe ] 27. Luu, C.; Mo, Y.; Hsu, T.T. De elopmen o CSMM-based shell elemen o ein o ced conc e e s uc u es. Eng. S uc . 2017 ,132, 778–790. [C ossRe ] 28. Cas aldo, P.; Gino, D.; Be agnoli, G.; Mancini, G. Resis ance model unce ain y in non-linea ini e elemen analyses o cyclically loaded ein o ced conc e e sys ems. Eng. S uc . 2020,211, 110496. [C ossRe ] 29. Ha i i-A debili, M.A.; Sanchez, L.; Rezakhani, R. Aging o conc e e s uc u es and in as uc u es: Causes, consequences, and cu es (C3). Ad . Ma e . Sci. Eng. 2020,2020, 9370591. [C ossRe ] 30. Vecchi, F.; Belle i, B. Capaci y assessmen o exis ing RC columns. Buildings 2021,11, 161. [C ossRe ] 31. Fu ado, A.; Rod igues, H.; Va um, H.; Cos a, A. E alua ion o di e en s eng hening echniques’ e iciency o a so s o ey building. Eu . J. En i on. Ci . Eng. 2017,21, 371–388. [C ossRe ] 32. Melche s, R.E.; Beck, A.T. S uc u al Reliabili y Analysis and P edic ion; John Wiley & Sons: Hoboken, NJ, USA, 2018. 33. Bazan , Z.P.; Le, J.-L. P obabilis ic Mechanics o Quasib i le S uc u es: S eng h, Li e ime, and Size E ec ; Camb idge Uni e si y P ess: Camb idge, UK, 2017. 34. Zou, Y.; Ki iniemi, A.; Jones, S.W. A e iew o isk managemen h ough BIM and BIM- ela ed echnologies. Sa . Sci. 2017 ,97, 88–98. [C ossRe ] 35. Cas aldo, P.; Gino, D.; Mancini, G. Sa e y o ma s o non-linea ini e elemen analysis o ein o ced conc e e s uc u es: Discussion, compa ison and p oposals. Eng. S uc . 2019,193, 136–153. [C ossRe ] 36. Suzuki, T.; Elwood, K.; Pu anam, A.; Lee, H.J.; Hsiao, F.P.; Hwang, S.J. D i demand es ima es o inelas ic o sional esponse o hal -scale se en-s o ey RC specimens. In P oceedings o he 2021 New Zealand Socie y o Ea hquake Enginee ing Annual Technical Con e ence, Ch is chu ch, New Zealand, 14–16 Ap il 2021. 37. Oukaili, N.; Pee a, I. Beha io al nonlinea modeling o p es essed conc e e lexu al membe s wi h in e nally unbonded s eel s ands. Resul s Eng. 2022,14, 100411. [C ossRe ] 38. Sucha da, O.; Ma ecko a, P.; Bilek, V. Non-linea analysis o an RC beam wi hou shea ein o cemen wi h a sensi i i y s udy o he ma e ial p ope ies o conc e e. Slo ak J. Ci . Eng. 2020,28, 33–43. [C ossRe ] 39. Cas aldo, P.; Gino, D.; Ma ano, G.C.; Mancini, G. Alea o y unce ain ies wi h global esis ance sa e y ac o s o non-linea analyses o slende ein o ced conc e e columns. Eng. S uc . 2022,255, 113920. [C ossRe ] 40. Ce enka, V. Reliabili y-based non-linea analysis acco ding o ib Model Code 2010. S uc . Conc . 2013,14, 19–28. [C ossRe ] 41. Allaix, D.L.; Ca bone, V.I.; Mancini, G. Global sa e y o ma o non-linea analysis o ein o ced conc e e s uc u es. S uc . Conc . 2013,14, 29–42. [C ossRe ] 42. Wal a en, J.C.; Bigaj- an Vlie , A. The 2010 ib Model Code o S uc u al Conc e e: A new app oach o s uc u al enginee ing. S uc . Conc . 2011,12, 139–147. [C ossRe ] 43. Ce enka, V. Global sa e y o ma s in ib Model Code 2010 o design o conc e e s uc u es. In P oceedings o he 11 h In e na ional P obabilis ic Wo kshop, B no, Czech Republic, 6–8 No embe 2013. 44. Blom o s, M.; Engen, M.; Plos, M. E alua ion o sa e y o ma s o non-linea ini e elemen analyses o s a ically inde e mina e conc e e s uc u es subjec ed o di e en load pa hs. S uc . Conc . 2016,17, 44–51. [C ossRe ] 45. Dash i, F.; Dhakal, R.P.; Pampanin, S. A pa ame ic in es iga ion on applicabili y o he cu ed shell ini e elemen model o nonlinea esponse p edic ion o plana RC walls. Bull. Ea hq. Eng. 2019,17, 6515–6546. [C ossRe ] 46. Mucede o, G.; Pe one, D.; Mon ei o, R. Epis emic unce ain y in poo ly de ailed exis ing ames accoun ing o mason y in ill a iabili y and RC shea ailu e. Ea hq. Eng. S uc . Dyn. 2022,51, 3755–3778. [C ossRe ] 47. Cas aldo, P.; Gino, D.; Be agnoli, G.; Mancini, G. Pa ial sa e y ac o o esis ance model unce ain ies in 2D non-linea ini e elemen analysis o ein o ced conc e e s uc u es. Eng. S uc . 2018,176, 746–762. [C ossRe ] 48. A uda, M.; Dei idas, M.; Vadimas, K. S a e o he a on s uc u al ein o ced conc e e design guidelines wi h non-linea analyses. Mech. Ad . Ma e . S uc . 2023, 1–15. [C ossRe ] 49. Mazahe ipou , H.; Ba os, J.; Sol anzadeh, F.; Sena-C uz, J. De lec ion and c acking beha io o SFRSCC beams ein o ced wi h hyb id p es essed GFRP and s eel ein o cemen s. Eng. S uc . 2016,125, 546–565. [C ossRe ] 50. Ba os, J.A.; Baghi, H.; Ven u a-Gou eia, A. Assessing he applicabili y o a smea ed c ack app oach o simula ing he beha iou o conc e e beams lexu ally ein o ced wi h GFRP ba s and ailing in shea . Eng. S uc . 2021,227, 111391. [C ossRe ] 51. Chen, W.; Gong, C.; Wang, Z.; F angopol, D.M. Applica ion o i s -o de eliabili y me hod wi h o hogonal plane sampling o high-dimensional se ies sys em eliabili y analysis. Eng. S uc . 2023,282, 115778. [C ossRe ] 52. Dudzik, A.; Po zeszcz-Su , B. Hyb id app oach o he i s o de eliabili y me hod in he eliabili y analysis o a spa ial s uc u e. Appl. Sci. 2021,11, 648. [C ossRe ] 53. EN 1992-1-1 (2004); Eu ocode 2: Design o Conc e e S uc u es—Pa 1-1: Gene al Rules and Rules o Buildings. B i ish S anda d Ins i u ion: London, UK, 2004. 54. EN 1990 (2002); Eu ocode—Basis o S uc u al Design. Eu opean Commi ee o S anda diza ion: B ussels, Belgium, 2002. 55. Neu, G.E. Compu a ional App oaches o a Robus Segmen al Lining Design in Mechanized Tunneling; Depa men o Ci il and En i onmen al Enginee ing, Ruh Uni e si y: Bochum, Ge many, 2023. Buildings 2023,13, 2848 20 o 20 56. Gino, D.; Cas aldo, P.; Gio dano, L.; Mancini, G. Model unce ain y in non-linea nume ical analyses o slende ein o ced conc e e membe s. S uc . Conc . 2021,22, 845–870. [C ossRe ] 57. Wal a en, J.C. Model Code 2010-Final D a : Volume 1; ib Fédé a ion In e na ionale du Bé on: Lausanne, Swi ze land, 2012; ib Bulle in No. 65. [C ossRe ] 58. Schlune, H.; Plos, M.; Gyll o , K. Sa e y o ma s o non-linea analysis o conc e e s uc u es. Mag. Conc . Res. 2012 ,64, 563–574. [C ossRe ] 59. Ce enka, V. Global sa e y o ma o nonlinea calcula ion o ein o ced conc e e. Be on S ahlbe onbau 2008 ,103 (Suppl. S1), 37–42. [C ossRe ] 60. Pimen el, M.; B ühwile , E.; Figuei as, J. Sa e y examina ion o exis ing conc e e s uc u es using he global esis ance sa e y ac o concep . Eng. S uc . 2014,70, 130–143. [C ossRe ] 61. Pacheco, J.; de B i o, J.; Chas e, C.; E angelis a, L. Unce ain y models o ein o ced conc e e beams in bending: Code compa ison and ecycled agg ega e inco po a ion. J. S uc . Eng. 2019,145, 04019013. [C ossRe ] 62. Haso e , A.M.; Lind, N.C. Exac and in a ian second-momen code o ma . J. Eng. Mech. Di . 1974,100, 111–121. [C ossRe ] 63. Nannapaneni, S.; Mahade an, S. Reliabili y analysis unde epis emic unce ain y. Reliab. Eng. Sys . Sa . 2016 ,155, 9–20. [C ossRe ] 64. Sol anzadeh, F.; Ba os, J.A.; San os, R. High pe o mance ibe ein o ced conc e e o he shea ein o cemen : Expe imen al and nume ical esea ch. Cons . Build. Ma e . 2015,77, 94–109. [C ossRe ] 65. ASTM C469/C469M-14; S anda d Tes Me hod o S a ic Modulus o Elas ici y and Poisson’s Ra io o Conc e e in Comp ession. ASTM: Wes Conshohocken, PA, USA, 2010; p. 469. 66. Ma os, L.M.; Ba os, J.A.; Ven u a-Gou eia, A.; Calçada, R.A. A new in e se analysis app oach o p edic ing he ac u e mode I pa ame e s o ib e ein o ced conc e e. Eng. F ac . Mech. 2021,246, 107613. [C ossRe ] 67. EN 1465; Tes Me hod o Me allic Fib e Conc e e—Measu ing he Flexu al Tensile S eng h (Limi o P opo ionali y (LOP), Residual). CEN: B ussels, Belgium, 2005; p. 17. 68. Ba os, J.A.; Fos e , S.J. An in eg a ed app oach o p edic ing he shea capaci y o ib e ein o ced conc e e beams. Eng. S uc . 2018,174, 346–357. [C ossRe ] 69. Ven u a-Gou eia, A. Cons i u i e Models o he Ma e ial Nonlinea Analysis o Conc e e S uc u es Including Time Dependen E ec s. Ph.D. Thesis, Depa men o Ci il Enginee ing, Uni e sidade do Minho, B aga, Po ugal, 2011. 70. Mazahe ipou , H. S uc u al Beha io o Hyb id GFRP and S eel Rein o ced FRC P es essed Beams. Ph.D. Thesis, Uni e sidade do Minho, B aga, Po ugal, 2015. 71. Lee, G.G.; Fos e , S.J. Beha iou o S eel Fib e Rein o ced Mo a in Shea 1: Di ec Shea Tes ing; Uni esi y o New Sou h Wales: Sydney, Aus alia, 2006; Repo No.: R-444. 72. Lee, G.G.; Fos e , S.J. Beha iou o S eel Fib e Rein o ced Mo a in Shea 2: Gamma Ray Imaging; Uni esi y o New Sou h Wales: Sydney, Aus alia, 2006; Repo No.: R-445. 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