materials

Article Theoretical Local Buckling Behavior of Thin-Walled UHPC Flanges Subjected to Pure Compressions

Jeonghwa Lee 1 , Seungjun Kim 2 , Keesei Lee 3 and Young Jong Kang 2,*

1 Future and Fusion Laboratory of Architectural, Civil and Environmental Engineering, Korea University, Seoul 02841, Korea; [email protected] 2 School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 02841, Korea; [email protected] 3 Department of Urban Infrastructure Research, Seoul Institute of Technology, Seoul 03909, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-02-3290-3317

Abstract: To enhance structural performance of concrete and reduce its self-weight, ultra-high- performance concrete (UHPC) with superior structural performance has been developed. As UHPC members with 180 MPa or above of the compressive strength can be designed, a rational assessment of thin-walled UHPC structural member may be required to prevent unexpected buckling failure that has not been considered while designing conventional concrete members. In this study, theoretical local buckling behavior of the thin-walled UHPC flanges was investigated using geometrical and material nonlinear analysis with imperfections (GMNIA). For the failure criteria of UHPC, a concrete



 damaged plasticity (CDP) model was applied to the analysis. Additionally, an elastic-perfectly plastic material model for steel materials was considered as a reference to establish differences in local Citation: Lee, J.; Kim, S.; Lee, K.; buckling behavior between the UHPC and steel flanges. Finite element approaches were compared Kang, Y.J. Theoretical Local Buckling and verified based on test data in the literature. Finally, this study offers several important findings Behavior of Thin-Walled UHPC Flanges Subjected to Pure on theoretical local buckling and local bending behavior of UHPC flanges. The inelastic local buckling Compressions. Materials 2021, 14, behavior of UHPC flanges was mainly affected by crack propagation due to its low tensile strength. 2130. https://doi.org/10.3390/ Based on this study, possibility of the local buckling of UHPC flanges was discussed. ma14092130 Keywords: ultra-high-performance concrete; UHPC; local buckling; stability; thin-walled flange; Academic Editors: Jean-Marc Tulliani, nonlinear analysis Patryk Rozylo, Hubert Debski and Katarzyna Falkowicz

Received: 18 March 2021 1. Introduction Accepted: 17 April 2021 Concrete has been widely used as a construction material for constructing infrastruc- Published: 22 April 2021 tures such as bridges, tunnels, buildings, pavements, etc. Although conventional concrete is weaker than steel, it has been among the most popular construction materials because of Publisher’s Note: MDPI stays neutral its high durability, low maintenance cost, excellent workability, and good fire resistance. with regard to jurisdictional claims in Conventionally, concrete members are reinforced using steel rebars for enhanced resistance published maps and institutional affil- iations. capacity to compensate for their low tensile strength. The failure mode of reinforced concrete (RC) structural members was mainly controlled by the yield of the reinforced rebars. The compressive failure or instability of concrete members was not a major concern in concrete structures before ultra-high-performance concrete (UHPC) with 180 MPa or above of compressive strength was developed. Copyright: © 2021 by the authors. Since ultra-high-performance concrete (UHPC) has been developed to enhance the Licensee MDPI, Basel, Switzerland. structural performance of conventional concrete structures and reduce their self-weight, it This article is an open access article has been widely used in various applications such as beams, columns, connections, decks, distributed under the terms and conditions of the Creative Commons and long-span bridge members in construction. The term UHPC was first introduced by Attribution (CC BY) license (https:// Larrard [1] in 1994; UHPC has been continuously developed in Europe, North America, creativecommons.org/licenses/by/ and Asia. In the 2000s, reactive powder concrete was developed by Bouygues of France [2], 4.0/).

Materials 2021, 14, 2130. https://doi.org/10.3390/ma14092130 https://www.mdpi.com/journal/materials Materials 2021, 14, 2130 2 of 25

Ductal® in North America [3], and CEMTEC [4] at LCPC in France. Furthermore, K- UHPC [5] was developed by the Korea Institute of Construction Technology in South Korea. Conventionally, UHPC is characterized by high compressive and tensile strengths. It features a compressive strength greater than or approximately equal to 180 MPa, which is achieved by reducing the water–cement ratio and curing at high temperatures. It is also characterized by a high tensile strength, which is approximately 10 MPa or above with ductility, owing to the mixing of steel fibers in the concrete mixture. Design standards applicable to UHPC have been introduced and applied in France, Japan, Australia, and South Korea [6–9]. Since the compressive strength of UHPC exceeds 180 MPa, which is similar to that of conventional structural steels, designing a thin-walled concrete member can be possible. It has been reported that the flange width-to-thickness ratios (λ f ) in the current design ranges of the bridge deck geometry range from 4 to 30 for outer UHPC bridge decks as indicated in Figure1. In general, the bridge deck with ordinary strength concrete (OSC) under 40 MPa of compressive strength has no possibility of buckling failure modes based on bridge data of OSC as shown in Figure1. However, for the UHPFRC bridge decks with 180 MPa of compressive strength, the possibility of buckling failure modes may drastically increase due to its high compressive strength, which is approximately the same level in compressive strength as steel flange’s (typical yield strength of steel flanges are ranged from 230 to 315 MPa). Furthermore, the slenderness ratio of practically designed outer decks can be increased up to 30, which may be considered as a highly slender flange section. Therefore, a rational assessment of the buckling stability of thin-walled UHPC structural members, which has not been considered in conventional concrete member designs, is now necessary for avoiding unexpected buckling failure and for careful design practice. Although realistic UHPFRC flanges are frequently stiffened for prestressing systems and designed based on effective-width approaches that produce conservative deck designs, it is important to verify the likelihood of local buckling phenomena in thin-walled UHPFRC members to secure more reasonable, safe, and efficient design practices for the application of the thin-walled UHPFRC members. The instability of concrete members has been rarely considered as a controlling failure mode in the previous studies regarding the lateral instability of concrete girders only. Pioneering studies concerning the lateral instability of concrete girders were conducted from 1950 to 1960. Marshall [10], Hansell and Winter [11], Sant and Bletzacker [12], and Massey [13] have demonstrated the phenomena of the lateral torsional buckling in concrete girders with rectangular sections and established the corresponding theoretical backgrounds through experimental studies. Since then, several studies have focused on the lateral torsional buckling of RC girders [14–18]. Additionally, ACI 318-14 [19], a design specification for concrete girders, stipulates that the slenderness limit of the girders should be considered to prevent their lateral torsional buckling. Recently, studies on the structural instability of UHPC compression and flexural members have been conducted. Illich et al. [20] conducted an experimental study on UHPC column specimens that were subjected to compressive forces to observe the global buckling of slender columns. Lee et al. [21] conducted an experimental study on slender UHPC girder specimens and established a design equation for elastic and inelastic LTB strength while considering nonlinear material properties and the effective moment of inertia. Lee [22] investigated the experimental tests and numerical evaluations of the local buckling in UHPC thin-walled I-girder flanges. Additionally, they presented moment capacity equations considering the local buckling phenomenon [23]. The buckling failures of the slender concrete members are illustrated in Figure2. Modern concrete designs such as long-span bridges employ thin-walled, and slender UHPC concrete structural members to reduce the self-weight of the entire structure. In general, the outer flange may be more likely to undergo instability, because a flange supported by one edge has a lower buckling coefficient (k) than those supported by two edges. Considering these perspectives, the buckling instability of thin-walled and Materials 2021, 14, x FOR PEER REVIEW 3 of 27

Materials 2021, 14, x FOR PEER REVIEW 3 of 27 Materials 2021, 14, 2130 3 of 25 edges. Considering these perspectives, the buckling instability of thin-walled and slender concrete members may possibly emerge as an important engineering issue and its signif- icanceedges. may Considering increase asthese the perspectives, strength of concretethe buckling increases. instability of thin-walled and slender concreteslender members concrete membersmay possibly may emerge possibly as emerge an important as an important engineering engineering issue and issue its signif- and its icancesignificance250 may increase may increase as the strength as the strength of concrete of concrete increases. increases. Practical UHPC, f'c = 180 MPa design range OSC, f'c = 40 MPa 250 Design data for OSC 200 Practical UHPC,Design f'c data = 180 for MPa UHPC design range OSC, f'c = 40 MPa Design data for OSC 200 Design data for UHPC 150

(MPa) 150

100(MPa) max f 100 max f 50 50 0 00 102030405060 0 102030405060 λf f Figure 1. A comparison between theoretical elastic local buckling strength (𝑓 ) with respect to the FigureFigure 1. 1.A Acomparison comparison between between theoretical theoretical elastic elastic local local buckling buckling strength strength (𝑓 ()f crwith) with respect respect to the to the slenderness ratio (𝜆𝜆) and practical design range of UHPC and ordinary strength concrete (OSC) slendernessslenderness ratio ratio ( (λ) fand) and practical practical design design range range of of UHPC UHPC and and ordinary ordinary strength strength concrete concrete (OSC) (OSC) for for forcompressioncompression compression bridge bridge decksdecks decks (data (data(data from fromfrom reference reference reference [23 [23]).]). [23]).

(a) (b) Figure 2. Examples of buckling(a) failure in concrete slender members: (a) local (bucklingb) of thin- walled ultra-high performance concrete (UHPC) flanges (a test specimen conducted from refer- FigureFigure 2. Examples 2. Examples of buckling of buckling failure failure in in concrete concrete slenderslender members: members: (a) ( locala) local buckling buckling of thin-walled of thin- ence [22]) and (b) lateral-torsional buckling (LTB) of slender UHPC I-girder (test specimen con- walledultra-high ultra-high performance performance concrete conc (UHPC)rete (UHPC) flanges flanges (a test (a specimen test specimen conducted conducted from reference from refer- [22]) ducted from reference [21]). ence and[22]) (b and) lateral-torsional (b) lateral-torsional buckling buckling (LTB) of (LTB) slender of UHPC slender I-girder UHPC (test I-girder specimen (test conducted specimen from con- ducted from reference [21]). referenceIn this [21 study,]). the local buckling strength of UHPC flanges was evaluated based on geometrical and material nonlinear finite element (FE) analysis with imperfections In this study, the local buckling strength of UHPC flanges was evaluated based (GMNIA)In this study,by considering the local two buckling different strength boundary of conditions. UHPC flanges For material was evaluated failure criteria, based on geometricalon geometrical and material and material nonlinear nonlinear finite finite el elementement (FE) analysisanalysis with with imperfections imperfections a (GMNIA)concrete damaged by considering plasticity two (CDP) different material boundary model conditions. with a tensile For material strength failure of approxi- criteria, a (GMNIA) by considering two different boundary conditions. For material failure criteria, matelyconcrete 5% damaged compared plasticity to its compressive (CDP) material stre modelngth was with applied a tensile to strength the UHPC of approximately flanges. An a concreteelastic-perfectly-plastic5% compared damaged to its plasticity compressive material (CDP) model strength material (plast wasic material model applied model)with to the a that UHPCtensile considers flanges. strength equal An of elastic-yield approxi- matelyinperfectly-plastic compressive 5% compared and material tensionto its compressive modelwas adopted (plastic streas material angth reference was model) materialapplied that considers modelto the toUHPC equalestablish flanges. yield the in An elastic-perfectly-plasticbasiccompressive differences and in tensionlocal material buckling was adopted model behavior (plast as abetween referenceic material the material UHPC model) and model that steel to considers flanges. establish The the equal pre- basic yield in sentedcompressivedifferences finite inelement and local tension buckling approaches was behavior foradopted analyzin between asg a thethe reference local UHPC buckling and material steel of flanges.UHPC model flanges The to presentedestablish were the basiccomparedfinite differences element and verified approachesin local basedbuckling for on analyzing test behavior data the in localbetween the literature buckling the ofUHPC[22]. UHPC This and flanges study steel wereoffers flanges. compared several The pre- important findings on the inelastic local buckling and local bending behavior of UHPC sentedand finite verified element based approaches on test data infor the analyzin literatureg the [22]. local This buckling study offers of severalUHPC importantflanges were flanges. It was concluded that the inelastic local buckling behavior of UHPC flanges was comparedfindings and on theverified inelastic based local on buckling test data and localin th bendinge literature behavior [22]. ofThis UHPC study flanges. offers It wasseveral mainlyconcluded affected that by the crack inelastic propagation local buckling because behavior of its low of UHPC tensile flanges strength. was Further, mainly affectedwhen important findings on the inelastic local buckling and local bending behavior of UHPC theby compressive crack propagation strength because and slenderness of its low tensile ratio are strength. equal, Further,the post-buckling when the compressivestrength of flanges.UHPCstrength It flanges was and concluded slendernesshas a marginal that ratio theeffect, are inelastic equal, as compared the local post-buckling buckling to that of strength be steelhavior flanges. of UHPCof UHPC Based flanges flangeson this has a was mainlystudy,marginal affected possibility effect, by of ascrack the compared local propagation buckling to that ofof because steelUHPC flanges. flanges of its Based lowwas ondiscussed.tensile this study,strength. possibility Further, of the when the compressivelocal buckling strength of UHPC and flanges slenderness was discussed. ratio are equal, the post-buckling strength of UHPC flanges has a marginal effect, as compared to that of steel flanges. Based on this study, possibility of the local buckling of UHPC flanges was discussed.

Materials 2021, 14, 2130 4 of 25

Materials 2021, 14, x FOR PEER REVIEW 4 of 25

2. Material Properties of UHPC 2. MaterialThe UHPC Properties exhibited of UHPC a compressive strength of approximately 180 MPa and an elastic modulusThe UHPC of 45,000 exhibited MPa, a achieved compressive by reducing strength the of approximately water–cement 180 ratio MPa without and an a elastic coarse modulusaggregate. of In45,000 addition, MPa, high-temperature achieved by reducing steam the curing water–ce can bement employed ratio without to improve a coarse the aggregate.robustness In of addition, the UHPC high-temperature microstructure. The steam tensile curing strength can be and employed ductility ofto UHPCimprove can the be robustnessimproved byof addingthe UHPC approximately microstructure. 2% of The steel tensile fibers strength to the concrete and ductility admixture. of UHPC Examples can beof improved a conventional by adding UHPC approximately mix design are 2% listed of steel in Table fibers1 .to the concrete admixture. Exam- ples of a conventional UHPC mix design are listed in Table 1. Table 1. An ultra-high-performance concrete (UHPC) mix design example by weight. Table 1. An ultra-high-performance concrete (UHPC) mix design example by weight. Glass Water Steel Fiber W/B Cement Fine Sand Silica Fume Fine Silica Glass Powder Reducer Steel(Volume) Fiber W/B Cement Water Reducer 0.18 1Sand 0.92 Fume 0.25Powder 0.25 0.0108(Volume) 0.22–0.31 0.18 1 0.92 0.25 0.25 0.0108 0.22–0.31 Graybeal [3] presented the compressive strength and strain relationships of the ascend- ing branchGraybeal for [3] estimating presented UHPC the compressive compression materialstrength behavior.and strain In relationships this model, the of ultimate the as- cendingstrain of branch UHPC for can estimating be estimated UHPC as 0.0042 compressi underon a compressivematerial behavior. strength In ofthis180 model, MPa. Thethe ultimatecompressive strain strength of UHPC and can strain be estimated relationships as 0.0042 are presented under a ascompressive follows: strength of 180 MPa. The compressive strength and strain relationships are presented as follows: fc = εcE(1 − α) (1) 𝑓 =𝜀𝐸(1 − 𝛼) (1) εcE 0 α = ae b fc − a (2) 𝛼=𝑎𝑒 −𝑎 (2) where α is the percentage decrease in stress compared to the linear elastic predicted stress, where 𝛼 is the percentage decrease in stress compared to the linear elastic predicted εc is the compressive strain in concrete corresponding to fc, and a, b are constants related stress,to curing. 𝜀 is the compressive strain in concrete corresponding to 𝑓 , and 𝑎, 𝑏 are con- stantsThe related tensile to curing. strength of UHPC is considerably higher than that of ordinary concrete. Additionally,The tensile greater strength ductility of UHPC and tensile is considerably strength, attributed higher than to steelthat fibers,of ordinary can be concrete. observed Additionally, greater ductility and tensile strength, attributed to steel fibers, can be ob- after the first cracking. Therefore, the ultimate tensile strength ( ft) of UHPC can be defined served after the first cracking. Therefore, the ultimate tensile strength (𝑓 ) of UHPC can be as the post-peak strength ( fpeak) that develops after the crack strength ( fcr) induced by initial definedcracking. as The the conventionalpost-peak strength tensile behavior(𝑓) that of UHPCdevelops is illustrated after the incrack Figure strength3. Graybeal (𝑓) in- [ 3] ducedproposed by initial tensile cracking. strength The equations conventional for UHPC tensile while behavior considering of UHPC the compressive is illustrated strength, in Fig- ureas shown3. Graybeal in Equation [3] proposed (3). The tensile coefficients strength 7.8 equations and 8.3 in for Equation UHPC (3)while represent considering the lower the compressiveand upper bounds strength, of theas shown tensile in strength, Equation respectively, (3). The coefficients considering 7.8 steamand 8.3 curing. in Equation In the (3)case represent of untreated the lower specimens, and upper the coefficientbounds of wasthe tensile assumed strength, to be 6.7.respectively, considering steam curing. In the case of untreated specimens, the coefficient was assumed to be 6.7. p 0 p 0 ft = 7.8 fc or 8.3 fc in psi unit (3) 𝑓 =7.8𝑓′ or 8.3𝑓′ in psi unit (3) 0 where, ft is the tensile strength of UHPC and fc is the compressive strength of UHPC. where, 𝑓 is the tensile strength of UHPC and 𝑓′ is the compressive strength of UHPC.

Figure 3. Typical tensile behavior of UHPC. Figure 3. Typical tensile behavior of UHPC.

Materials 2021, 14, 2130 5 of 25

The relationship between the compressive strength and elastic modulus of concrete can be estimated using the following equation [3]:

p 0 Ec = 49, 000 fc in psi unit (4)

0 where Ec is the elastic modulus of the UHPC and fc is the compression strength of the UHPC. Ma et al. [24] presented an equation for the modulus of elasticity based on compressive strength, as follows:  f 0 1/3 E = 525, 000 c in psi unit (5) c 10

3. Theoretical Elastic Local Buckling Strength of Thin-Walled UHPC Flanges For thin-walled flange members with high yield strengths, it is important to determine the local buckling strength to avoid unexpected brittle failure before yielding. Although the stress acting on the cross-section of the flanges does not reach its yielding stress, a thin-walled flange member with a large slenderness ratio may be unstable with a large out-of-plane deformation, which can dominate the overall strength instead of the yielding strength. It implies that the geometric characteristic strength affected by the width to thickness ratio (slenderness ratio, λ f ), in which λ f can be calculated by half width divided by thickness (b f /2t f ) for the flange supported by one edge or width divided by thickness (b f /t f ) for the plate supported by two edges, becomes the governing strength for the thin-walled flange sections. This type of buckling failure frequently occurs in thin-walled flange members, including steel, and FRP flanges, which are characterized by high yielding or rupture strengths. Figure4 shows the typical equilibrium path and critical buckling strength for the perfect and imperfect flanges. In the critical local buckling state of the flange within the elastic ranges, the stiffness of the flange subjected to axial compression loads theoretically becomes zero. This is called bifurcation-type buckling modes (refer to Figure4). The elastic local buckling strength of the flange can be calculated using the theoretical buckling strength equation defined in Equation (6). The theoretical elastic local buckling strength is affected by the modulus of elasticity (E), width-thickness ratio (b f /2t f or b f /t f ), Poisson’s ratio (µ), and elastic buckling coefficient (k):

kπ2E f = (6) cr  2 2 12(1 − µ ) λ f

where fcr is the elastic local buckling strength, k is the elastic buckling coefficient that represents the support conditions, E is the modulus of elasticity, µ is the Poisson’s ratio, and λ f is the slenderness ratio of the flanges. The buckling of flanges is strongly influenced by the type of load and boundary conditions. In general, the design elastic buckling coefficient (kc) of a steel flange supported by one edge can be determined using web constraints, as follows [25]:

4 kc = (0.35 ≤ kc ≤ 0.76) (7) q H tw

where kc is the design elastic buckling coefficient, H is the web height, and tw is the thickness of web panels. Materials 2021, 14, x FOR PEER REVIEW 6 of 27

aggregates, sands, and cement, the modulus of elasticity and Poisson’s ratio are consider- Materials 2021, 14, 2130 6 of 25 ably lower than those of steels. These differences in material property could lead to a dif- ferent elastic buckling strength than that of steel flanges. loads Post-bucking

Compressive Compressive equilibrium path Post-buckling strength

Bifurcation type Theoretical critical buckling mode buckling load Approximate critical buckling load : Perfect flange : Imperfect flange with initial imperfection

Out-of-plane deflection Initial imperfection

FigureFigure 4. 4. TypicalTypical local local buckling buckling behavior behavior of of thin-walle thin-walledd members members in in perfect perfect and and imperfect imperfect flanges. flanges.

AccordingIn thin-walled to Lundquist UHPC flanges, and Stowell the UHPC[26], the is elastic a highly buckling complex strength quasi-brittle of a uniformly hetero- geneous material with different mechanical properties under tension and compression, compressed flange with simply supported edges (see Figure 5) is determined using the whereas steel is a homogeneous material with equal compressive and tensile strengths and boundary conditions, slenderness ratio, modulus of elasticity, and Poisson’s ratio. The high modulus of elasticity. As UHPC contains different types of materials, including aggre- elastic buckling coefficients (𝑘) can be calculated using Equation (8). gates, sands, and cement, the modulus of elasticity and Poisson’s ratio are considerably lower than those of steels. These differences1 πb inε materialc πb property could lead to a different ⎧ 1−𝜇+ + +c +𝜇c ⎫ 6 λ 2 2 λ elastic buckling strength⎪ than that of steel flanges. ⎪ According to Lundquist⎪ and Stowell [26], the elastic buckling strength⎪ of a uniformly compressed flange with⎪ simplyε c supportedπb edgesc (see Figure5) is determinedε ⎪ using the ⎪+ + +c −𝜇c + ⎪ boundary conditions, slenderness4 2 λ ratio, modulusπb of elasticity, andπb Poisson’s ratio. The ⎪ 2 2 ⎪ elastic buckling coefficients2 (k) can be calculatedλ using Equation (8).λ 𝑘= (8) π ⎨ 1 cε cε ⎬ +  +  ⎪  32 2a 4a 2 ⎪  1 − µ + 1 πb + ε c1 πb + c + µc  ⎪ 6 λ 2 2 λ 2 3 ⎪ ⎪   ⎪  2  2  ⎪ + ε c4 πb + c5 + c − µc + ε ⎪  4 2 λ πb 2 6 7 πb 2  2 ⎪ 2( ) 2( ) ⎪ k = λ λ 2 ⎩ 2 ⎭ (8) π  1 + c8ε + c9ε   3 2a3 4a2  where 𝑘 is the non-dimensional elastic buckling coefficient3 that depends on the condi-   tions of the edge restraint and shape of the flange, 𝜇 is Poisson’s ratio, 𝑏 is the width of   the flange, 𝑎 is the effective buckling length, 𝑚 is the number of buckling modes, 𝜆 is a parameter obtained by dividing 𝑎 with m (𝜆 = 𝑎/𝑚), 𝜀 is the restrain coefficient (= where k is the non-dimensional elastic buckling coefficient that depends on the conditions 4𝑆 𝑏/𝐷) 𝜀=0 𝜀=∞ 𝑆 of the edge ( restraint; Hinged and shapecondition, of the flange,; Fixedµ is Poisson’s condition), ratio, b is is thethe width stiffness of the per flange, unit lengtha is the of effective the elastic buckling restraining length, mediumm is theor the number moment of bucklingrequired modes,to rotateλ a isunit a parameter length of theobtained elastic bymedium dividing througha with one-fourthm (λ = a/ radian,m), ε is the𝐷 is restrain the flexural coefficient rigidity(= of4 Stheb /flangeD) (ε =per0 ; 0 unit length (𝐸𝑡 /(12(1 − 𝜇 ))), and 𝑐–𝑐 are the coefficients. Hinged condition, ε = ∞; Fixed condition), S0 is the stiffness per unit length of the elastic restraining medium or the moment required to rotate a unit length of the elastic medium through one-fourth radian, D is the flexural rigidity of the flange per unit length 3 2


Et / 12 1 − µ , and c1–c9 are the coefficients.

Materials 2021, 14, x FOR PEER REVIEW 7 of 27

f f

Restrained edge a 2a

: undeformed shape MaterialsMaterials 20212021, 14, ,14 x, FOR 2130 PEER REVIEW 7 of 25 7 of 27 : deformed shape b b

f Figure 5. Outstanding flange under edge compression. f By substituting a Poisson’s ratio (𝜇) of 0.2 and 0.3 for UHPC and steel, respectively, Restrained 𝑘 intoedge Equation (8), athe elastic local buckling2a coefficient ( ) of the UHPC and steel flanges according to the two boundary conditions and a/b can be obtained, as shown in Figure 6. Additionally, Table 2 shows the modulus: undeformed of elasticity shape (𝐸), Poisson’s ratio (𝜇), and min- : deformed shape imum elastic bucklingb coefficient (k) of the UHPC and steel flanges calculated using Equa- tion (8). By substituting theb values listed in Table 2 in Equation (6), the theoretical elastic

bucklingFigure 5. Outstandingstrength of flange UHPC under and edge steel compression. flanges can be obtained, as shown in Figure 7. In FigureFigure 5.7, Outstandingthe UHPC flange flange demonstratesunder edge compression a local buckling. strength that was approximately 1/4 timesBy substituting lower than a that Poisson’s of the ratio steel (µ )flange of 0.2 ands owing 0.3 for to UHPC its low and modulus steel, respectively, of elasticity (𝐸); however,intoBy Equation substituting it exhibited (8), the a elastica Poisson’shigher local elastic bucklingratio local (𝜇) coefficient ofbuckling 0.2 and coefficient ( k0.3) of for the UHPC UHPC (𝑘). Based andand steelsteel, on the flanges respectively, theoretical evaluation,intoaccording Equation to the the (8),local two the boundarybuckling elastic localphenomenon conditions buckling and was a/bcoefficient canmore be likely obtained, (𝑘) of to theoccur as UHPC shown in the inand UHPC Figure steel6 concrete. flanges Additionally, Table2 shows the modulus of elasticity ( E), Poisson’s ratio (µ), and minimum flangeaccording than to in the the twosteel boundary flange members conditions unde andr the a/b same can boundary be obtained, conditions as shown and in slender- Figure 6.elastic Additionally, buckling coefficient Table 2 shows (k) of the the UHPC modulus and steelof elasticity flanges calculated (𝐸), Poisson’s using Equationratio (𝜇), (8). and min- ness ratios (𝜆). imumBy substituting elastic buckling the values coefficient listed in Table (k) of2 thein Equation UHPC and (6), thesteel theoretical flanges calculated elastic buckling using Equa- strength of UHPC and steel flanges can be obtained, as shown in Figure7. In Figure7, the tion (8). By substituting the values listed in Table 2 in Equation (6), the theoretical elastic TableUHPC 2. flangeElastic demonstratesflange buckling a localcoefficient buckling (𝑘) according strengththat to the was Poisson’s approximately ratio (𝜇). 1/4 times buckling strength of UHPC and steel flanges can be obtained, as shown in Figure 7. In lower than that of the steel flanges owing to its low modulus of elasticity𝒌 (E); however, it FigureexhibitedCases 7, the a higher UHPC elastic 𝑬flange local demonstrates buckling𝝁 coefficient a local ( kbuckling). Based on strength the theoretical that was evaluation, approximately Hinge Support Fixed Support 1/4the times local bucklinglower than phenomenon that of the was steel more flange likelys owing to occur to in its the low UHPC modulus concrete of flangeelasticity (𝐸); however,thanSteel in the it steelexhibited flange205,000 a members higher el underastic 0.3 local the same buckling boundary coefficient 0.426 conditions (𝑘). Based and slenderness on the 1.289 theoretical evaluation,ratiosUHPC (λ f ). the local49,790 buckling phenomenon 0.2 was more 0.486 likely to occur in the UHPC 1.381 concrete flange than in the steel flange members under the same boundary conditions and slender- 3 ness ratios (𝜆). Fixed supports (UHPC) Hinge supports (UHPC) 2.5 Table 2. Elastic flange bucklingFixed coefficient suports (Steel) (𝑘) according to the Poisson’s ratio (𝜇). Hinge supports (Steel) 2 𝒌 Cases 𝑬 𝝁

) 1.5 Hinge Support Fixed Support k Steel( 205,000 0.3 0.426 1.289 1 UHPC 49,790 0.2 0.486 1.381 0.5 3 0 Fixed supports (UHPC) Elastic local buckling coefficient coefficient buckling local Elastic 0246810Hinge supports (UHPC) 2.5 Fixed suports (Steel) a/bHinge supports (Steel) 2 Figure 6. Comparison of elastic local buckling coefficient (k) for UHPC and steel. Figure 6. Comparison of elastic local buckling coefficient (k) for UHPC and steel. ) 1.5 k Table 2.( Elastic flange buckling coefficient (k) according to the Poisson’s ratio (µ). 1

0.5 k Cases µ 0 E Elastic local buckling coefficient coefficient buckling local Elastic 0246810 a/b Hinge Support Fixed Support Steel 205,000 0.3 0.426 1.289 FigureUHPC 6. Comparison of 49,790elastic local buckling 0.2 coefficient (k) for 0.486 UHPC and steel 1.381.

Materials 2021, 14, 2130 8 of 25 Materials 2021, 14, x FOR PEER REVIEW 8 of 25

Figure 7. Comparison of elastic local buckling strength of UHPC and steel flanges. Figure 7. Comparison of elastic local buckling strength of UHPC and steel flanges. 4. Inelastic Local Buckling of Thin-Walled UHPC Flanges Based on FEA 4.4.1. Inelastic Overview Local of FEA Buckling of Thin-Walled UHPC Flanges Based on FEA 4.1. OverviewAs mentioned of FEA in Section3, the theoretical elastic local buckling strength of the UHPC flangeAs maymentioned be 1/4 in times Section lower 3, the than theoretical that of the elastic steel flangeslocal buckling for the strength same boundary of the UHPC condi- flangetions andmay slenderness be 1/4 times ratios. lower This than implies that of that the thesteel elastic flanges local for buckling the same of boundary the UHPC condi- flange tionsis more and likely slenderness to occur. ratios. Additionally, This implies due tothat the the different elastic materiallocal buckling properties of the of UHPC UHPC flangeunder is compression more likelyand to occur. tension, Additionally considerably, due different to the inelasticdifferentlocal material buckling properties behavior of UHPCmay be under observed. compression and tension, considerably different inelastic local buckling be- haviorThe may inelastic be observed. local buckling of UHPC flanges was investigated based on geometrical andThe material inelastic nonlinear local buckling FE analysis of UHPC with initial flanges imperfections was investigated (GMNIA). based Here, on geometrical an inelastic andUHPC material material nonlinear model, FE whose analysis tensile with strength initial imperfections was 5% of its (GMNIA). compressive Here, strength, an inelastic based UHPCon Graybeal material [3]’s model, equation whose (Equation tensile (3))strength was employed.was 5% of Additionally,its compressive an strength, elastic-perfectly based onplastic Graybeal model, [3]’s which equation represents (Equation equal (3)) yield was strengths employed. under Additionally, compression an andelastic-perfectly tension, was plasticused to model, simulate which conventional represents steelequal flanges yield strengths as the reference under compression models for comparisons. and tension, was used toIn simulate general, conventional the local buckling steel flanges strength as ofthe flanges reference can models be classified for comparisons. into the critical bucklingIn general, strength the andlocal the buckling post-buckling strength strength. of flanges The cancritical be classified buckling into strength the critical is a bucklingtheoretical, strength characteristic and the post-buckling buckling strength strength obtained. The fromcritical the buckling elastic buckling strengthcoefficient is a theo- retical,(k), which characteristic is presented buckling in Table strength2. It is also obta estimatedined from based the elastic on the buckling compressive coefficient load and(k), whichout-of-deflection is presented relationships in Table 2. It is by also defining estimated strength, based where on the thecompressive stiffness ofload a structural and out- of-deflectionmember decreased relationships significantly by defining after strength, reaching where critical the buckling stiffness equilibrium of a structural conditions mem- ber(such decreased as the flange significantly with initial after imperfection reaching critical in Figure buckling4). In practice, equilibrium flange conditions members could(such asresist the flange additional with compression initial imperfection loading in induced Figure by4). theIn practice, redundancy flange of tensilemembers stress could until resist they additionalattain post-buckling compression strength. loading The induced post-buckling by the redundancy strength was of affected tensile bystress the until geometric they attainand material post-buckling nonlinearities strength. when The thepost-buck flange sectionling strength had a highwas affected slenderness by the ratio. geometric and materialIn UHPC nonlinearities flanges, UHPC when exhibited the flange a considerably section had a lower high slenderness tensile strength ratio. compared to itsIn compressive UHPC flanges, strength. UHPC These exhibited significant a consid differenceserably lower in the tensile tensile strength strength compared of UHPC flanges affected the inelastic buckling and post-buckling behaviors. Therefore, in this to its compressive strength. These significant differences in the tensile strength of UHPC study, the GMNIA analysis of the UHPC flange member was conducted to investigate the flanges affected the inelastic buckling and post-buckling behaviors. Therefore, in this differences in inelastic and post-local buckling behaviors by comparing it with the reference study, the GMNIA analysis of the UHPC flange member was conducted to investigate the steel flange models. The parameters used for the GMNIA analysis are listed in Table3. differences in inelastic and post-local buckling behaviors by comparing it with the reference steel flange models. The parameters used for the GMNIA analysis are listed in Table 3. Table 3. Parameters of the GMNIA analysis.

Table 3. ParametersE of the GMNIA analysis.f0 or f f or f b Cases µ c y t y f λ (MPa) (mm) f 𝑬 𝒇 or(MPa) 𝒇 𝒇 (MPa)or 𝒇 𝒃 𝝁 𝒄 𝒚 𝒕 𝒚 𝒇 𝝀 CasesSteel 205,000 0.3 180 2.5–65𝒇 (MPa) (MPa)180 (MPa) (mm)1000 UHPC 49,790 0.2 8.7 (26 cases at 2.5 intervals) Steel 205,000 0.3 180 180 1000 2.5–65

MaterialsMaterials 20212021, 14, 14x FOR, 2130 PEER REVIEW 9 of9 27 of 25

4.1.1. FEA Models 4.1.1. FEA Models The UHPC and steel flanges were modeled with 3D shell elements (S4R) using the The UHPC and steel flanges were modeled with 3D shell elements (S4R) using the ABAQUSABAQUS program program [27]. [27 The]. The models models featured featured a flange a flange width width of of1 m. 1 m.Flange Flange thicknesses thicknesses werewere determined determined by varying by varying the slenderness the slenderness ratio ratiofrom from2.5 to 2.565, toas 65,listed as in listed Table in 3. Table The 3. loadThe and load boundary and boundary conditions conditions are depicted are in depicted Figure 8. in The Figure flange8. models The flange were models supported were by supportedthe hinge and by thefixed hinge constraints and fixed along constraints the centerline along of the the centerline flange width, of the as flange shown width, in Figureas shown 8a,b. Uniform in Figure compressive8a,b. Uniform stresses compressive were applied stresses to both were ends applied, as shown to both in Figure ends,as 8c. shownVertical in displacementFigure8c. Vertical of the displacementmodels was prevented of the models using was equally prevented spaced using restraints equally alongspaced their restraints length, alongas shown their length,in Figure as shown8d. The in Figurelength8 d.affording The length minimum affording buckling minimum strengthbuckling was strength determined was determined using a convergence using a convergence analysis, and analysis, it was and assumed it was assumedto be 40 tom be and40 16 m m and for 16the m hinged for the and hinged fixed and support fixed supportconditions, conditions, respectively. respectively. The supported The supported spac- ingspacing along the along length the was length in wasconsideration in consideration of the buckling of the buckling modes modesof the flanges, of the flanges, and the and numberthe number of elements of elements along alongthe flange the flange width width and andlength length could could influence influence the theconvergence convergence of elasticof elastic local local buckling buckling strength. strength. Thus, Thus, the the supported supported spacing spacing and and number number of ofelements elements werewere determined determined using using convergence convergence analyses. analyses.

(a) (b) (c) (d)

FigureFigure 8. Compressive 8. Compressive load and load boundary and boundary conditions. conditions. (a) Hinge (supports:a) Hinge displacement supports: displacement constrains along constrains the Z-axis; along (b) the fixed supports: displacement constraints along the Z-axis and rotations along the Y-axis; (c) uniform loads at each end for Z-axis; (b) fixed supports: displacement constraints along the Z-axis and rotations along the Y-axis; (c) uniform loads the hinge and fixed condition; and (d) support along the Z-axis for the hinge and fixed conditions. at each end for the hinge and fixed condition; and (d) support along the Z-axis for the hinge and fixed conditions. 4.1.2. Material Properties for FEA 4.1.2. Material Properties for FEA With regard to the modulus of elasticity of UHPC, as proposed by Ma et al. [24], the lower valuesWith regarddetermined to the using modulus Equation of elasticity (5) were of UHPC,used for as conservative proposed by results. Ma et al. Tensile [24], the strengthlower was values calculated determined using using Equation Equation (3), as (5) presented were used by forGraybeal conservative [3]. results. Tensile strengthFor the wassimulation calculated of UHPC, using Equationascending (3), bran asches presented of the bycompressive Graybeal [ stress3]. and strain For the simulation of UHPC, ascending branches of the compressive stress and strain relationship was determined using Graybeal’s [3] models based on Equations (1) and (2). relationship was determined using Graybeal’s [3] models based on Equations (1) and (2). For the descending branch, Kappos and Konstantinidis’ [28] model, which was developed For the descending branch, Kappos and Konstantinidis’ [28] model, which was developed for high-strength concrete, was used to simulate the brittle failure of UHPC after peak for high-strength concrete, was used to simulate the brittle failure of UHPC after peak strength. Kappos and Konstantinidis’ [28] material constitutive models are shown below: strength. Kappos and Konstantinidis’ [28] material constitutive models are shown below: For the ascending branch, For the ascending branch, 𝜀 𝐸 𝑓 εc Ec fcc𝜀 𝐸 −𝐸 0<𝜀 ≤𝜀 ,𝜎= εccl Ec−Ecl < ≤ = (9) 0 εc εccl, σc Ec (9) 𝐸  𝜀 E−E Ec −−1+1 + εc c cl 𝐸Ec−𝐸−Ecl ε𝜀ccl For the descending branch, For the descending branch, 𝜀 −𝜀 " # 𝜀 >𝜀,𝜎 = 𝑓 1 − 0.5 − (10) 𝜀.εc −𝜀εccl εc > εccl, σc = fcc 1 − 0.5 (10) ε0.5 fcc − εccl where 𝜀 is the general concrete strain, 𝜀 is the axial concrete strain at the peak stress in confined concrete, 𝜎 is the general concrete stress, 𝑓 is the maximum compressive where εc is the general concrete strain, εccl is the axial concrete strain at the peak stress 𝐸 𝐸 strengthin confined of the concrete, σc is is thethe generaltangent concretemodulus stress,of elasticityfcc is of the the maximum concrete, compressive is the secant modulus of elasticity of the concrete, 𝜀 is the strain, at which the stress in plain strength of the concrete, Ec is the tangent modulus. of elasticity of the concrete, Ecl is the concrete drops to 0.5𝑓, and 𝜀 is the axial concrete strain at the peak stress.

Materials 2021, 14, 2130 10 of 25

Materials 2021, 14, x FOR PEER REVIEW 10 of 27

secant modulus of elasticity of the concrete, ε0.5 fcc is the strain, at which the stress in plain concreteThe crack drops displacement to 0.5 fcc, and forεccl tensileis the axial stress concrete was determined strain at the using peak the stress. “Design Guide- lines forThe K-UHPC crack displacement [8]”; detailed for material tensile stress proper wasties determined are listedusing in Table the “Design4. Considering Guidelines the materialfor K-UHPC properties [8]”; detailed for flanges, material two properties different aretypes listed of material in Table4 .mo Consideringdels, namely the concrete material damagedproperties plasticity for flanges, (CDP) two and different an elastic-perf types of materialectly plastic models, model namely (plastic concrete model), damaged were plasticity (CDP) and an elastic-perfectly plastic model (plastic model), were used in the used in the GMNIA. These two materials represent the material constitutive laws for con- GMNIA. These two materials represent the material constitutive laws for concrete and crete and steel, respectively (refer to Figure 9). steel, respectively (refer to Figure9). Table 4. Material properties for tensile behavior of UHPC applied to the CDP model. Table 4. Material properties for tensile behavior of UHPC applied to the CDP model.

𝑬 (MPa) 𝒇𝒄 0(MPa) 𝜺𝒖ε u 𝒇𝒕 (MPa) 𝒘𝒖 (mm) 𝒘𝒍𝒊𝒎 (mm) E (MPa) fc (MPa) ft (MPa) wu (mm) wlim (mm) 49,79049,790 180 180 0.0042 0.0042 8.67 8.67 0.3 0.3 5.3 5.3

300 200 10 200 8 , MPa) ,

c 150 f , MPa) ,

100 t f

, MPa) 6 f 0 100 -100 4 Stress ( Stress 50 -200 2 -300 ( stress Tensile Compressive stress ( stress Compressive 0 0 -0.02 -0.01 0 0.01 0.02 0 0.002 0.004 0.006 0.008 0246 Strain (ε) Strain (εc) CMOD (w, mm)

(a) (b) (c)

FigureFigure 9. 9. TensileTensile material material behavior behavior obtained obtained via via finite finite element element analysis: analysis: ( (aa)) stress–strain stress–strain relationship relationship of of the the plastic plastic material material modelmodel for for steel steel flanges, flanges, ( (bb)) compressive compressive stress–strain stress–strain relationship relationship of of the the co concretencrete damaged damaged plasticity plasticity (CDP) (CDP) material material model model for UHPC flanges, and (c) relationship between tensile strength and crack mouth opening displacement of the CDP model. for UHPC flanges, and (c) relationship between tensile strength and crack mouth opening displacement of the CDP model. For the CDP model, 𝜀, 𝜎 /𝜎 , 𝐾 , and 𝜓 were assumed to be 0.1, 1.16, 2/3, and 36, For the CDP model, ε, σb0/σc0, Kc, and ψ were assumed to be 0.1, 1.16, 2/3, and 36, respectively,respectively, where where 𝜀ε is is the the eccentricity,eccentricity, which which isis calculatedcalculated asas the ratio of of tensile tensile strength strength 𝜎 /𝜎 toto the the compressive compressive strength; strength; σb0/σc0 is is the the ratio ratio of of the the strength strength in in the the biaxial biaxial state state to to that that in inthe the uniaxial uniaxial state, state,Kc 𝐾is the is the parameter parameter related related to the to failurethe failure surface surface of concrete, of concrete, and ψandis the𝜓 isdilation the dilation angle. angle.

4.1.3.4.1.3. Initial Initial Imperfections Imperfections InIn general, general, local local buckling strengthstrength isis sensitive sensitive to to initial initial imperfections, imperfections, which which represent repre- sentthe initialthe initial defects defects in the in flanges the flanges that occurredthat occurred during during the production the production process. process. In the caseIn the of casetypical of typical steel flanges, steel flanges, the initial the imperfectionsinitial imperfections are mainly are mainly caused caused by welding by welding heat during heat duringthe fabrication the fabrication process. process. The initial The initial imperfections imperfections in UHPC in UHPC flanges flanges can be can induced be induced from fromboth both welding welding heat inheat the in steel the steel forms forms andthe and initial the initial casting casting stage. stage. Furthermore, Furthermore, methods meth- for odsestimating for estimating the initial the imperfections initial imperfections in the UHPC in the flangesUHPC offlanges actual of structures actual structures have not have been notwell been established. well established. According According to an experimental to an experimental studyby study Lee by [22 Lee], initial [22], imperfectionsinitial imper- fectionsin UHPC in flangesUHPC flanges are less are than less the than recommended the recommended value for value steel for flange. steel flange. Therefore, Therefore, in this instudy, this study, the conventional the conventional initialimperfections initial imperfec oftions the steel of the girder steel flanges girder specifiedflanges specified in the AWS in theBridge AWS Welding Bridge Welding Code [29 Code] were [29] used. were For used conservative. For conservative estimation, estimation, the provision the provision of initial ofimperfection initial imperfection for a flange for a supported flange supported by one edge by one is defined edge is asdefined follows: as follows:   b𝑏f 0.30.3𝑎a δ𝛿=𝑚𝑖𝑛= min ,, (11)(11) 150 150150

where 𝛿 is the initial imperfection, 𝑏 is the width of the flange, and 𝑎 is the unbraced length or the length of the half-sine buckling mode of the flange.

Materials 2021, 14, 2130 11 of 25

MaterialsMaterials 2021 2021, 14,, 14x FOR, x FOR PEER PEER REVIEW REVIEW 11 of11 27 of 27

where δ is the initial imperfection, b f is the width of the flange, and a is the unbraced length or the length of the half-sine buckling mode of the flange. Figure 10 shows the local buckling mode shapes obtained from the elastic buckling FigureFigure 10 10 shows shows the the local local buckling buckling mode mode shapes shapes obtained obtained from from the the elastic elastic buckling buckling analysis for the hinge and fixed boundary conditions. The buckling mode shape for the analysisanalysis for for the the hinge hinge and and fixed fixed boundary boundary conditions. conditions. The buckling The buckling mode mode shape shape for the for GMNIAGMNIAthe GMNIAwas was determined determined was determined based based on based onthe the nodal on nodal the displacements nodaldisplacements displacements of ofthe the 1st of1st mode the mode 1st shapes, modeshapes, ob- shapes, ob- tainedtainedobtained from from the from the elastic theelastic elasticbuckling buckling buckling analysis analysis analysis results. results. results. Figure Figure Figure 11 11shows 11shows shows the the elastic the elastic elastic buckling buckling buckling strengthstrengthstrength corresponding corresponding corresponding to tothe tothe magnitude the magnitude magnitude of ofinitial ofinitial initial imperfections, imperfections, imperfections, obtained obtained obtained using using using the the the GMNIA.GMNIA.GMNIA. Based Based Based on onthe onthe convergence theconvergence convergence analysis, analysis, analysis, the the maximum maximum the maximum initial initial imperfections initial imperfections imperfections were were de- de- were termined to be 6.67 mm (𝑏/150) for the hinged condition and 1.625 mm (0.3𝑎/150) for termineddetermined to be to 6.67 be mm 6.67 ( mm𝑏/150 (b f)/ for150 the) for hinged the hinged condition condition and 1.625 and 1.625mm (0.3𝑎/150 mm (0.3a) /for150 ) thethe fixedfor fixed the condition, fixedcondition, condition, based based on based onthe the onAWS AWS the Bridge AWS Bridge Bridge Welding Welding Welding Code Code Code[29], [29], [as29 asthe], asthe convergence the convergence convergence analysisanalysisanalysis for for the for the two the two boundary two boundary boundary conditio conditio conditionsns nsyields yields yields appropriate appropriate appropriate estimations. estimations. estimations.

(a) (a) (b)( b) FigureFigureFigure 10. 10.Local 10. LocalLocal buckling buckling buckling mode mode mode shapes shapes shapes based based based on onelastic on elastic elastic buckling buckling buckling analysis: analysis: (a) (hingea) hingehinge support supportsupport condi- condition condi- tiontion andand and ( (bb) )( fixedb fixed) fixed support support support condition condition. condition. .

1.41.4 HingeHinge support support 1.21.2 FixedFixed support support 1 1

cr,theory 0.8 cr,theory f 0.8 / f / 0.60.6 cr,GMNIA cr,GMNIA f 0.4f 0.4 ; for; forhinge hinge support support 0.20.2 ; for; forfixed fixed support support 0 0 00 50 50 100 100 150 150 200 200 250 250 300 300 n n

Figure 11. Critical buckling strength corresponding to the magnitude of imperfection (δ) when the Figure 11. Critical buckling strength corresponding to the magnitude of imperfection (δ) when the Figureslenderness 11. Critical ratio buckling (λ = b strength/2t ) is 20.corresponding to the magnitude of imperfection (δ) when the slenderness ratio (𝜆 =𝑏f/2𝑡 )f is 20f. slenderness ratio (𝜆 =𝑏 /2𝑡 ) is 20. 4.1.4. Definition of Critical and Post-Buckling Strength 4.1.4. Definition of Critical and Post-Buckling Strength 4.1.4. DefinitionThe critical of localCritical buckling and Post-Buckling of flanges occurs Strength within the elastic range during the initial stageTheThe critical of critical the local compression local buckling buckling of load; flangesof flanges out-of-plane occurs occurs within deflectionwithin the the elastic startselastic range to range be during developed during the the initial after initial the stagestagecompressive of theof the compression compression stress reaches load; load; out-of-plane the out-of-plane critical buckling def deflectionlection strength. starts starts to With beto bedeveloped respect developed to after the afterpost-buckling the the com- com- pressivepressivebehavior, stress stress the reaches appliedreaches the loadingthe critical critical can buckling buckling be gradually strength. strength. increased With With respect until respect it to reaches tothe the post-buckling ultimate post-buckling strength behavior,behavior,with largethe the applied out-of-planeapplied loading loading deformation. can can be begradually gradually The increased ultimate increased until strength until it reaches it canreaches be ultimate defined ultimate strength as strength the post- withwith bucklinglarge large out-of-plane out-of-plane strength. deformation. Several deformation. studies The The have ulti ultimate reportedmate strength strength methods can can be to bedefined determine defined as asthe critical the post- post- local 2 bucklingbucklingbuckling strength. strength. strengths Several Several [30 ].studies Instudies this have study, have reported freported− w methodswas methods used to to determineto approximate determine critical critical the lowerlocal local buck- boundary buck- lingling strengthscritical strengths buckling [30]. [30]. In strength Inthis this study, ofstudy, UHPC 𝑓−𝑤 𝑓−𝑤 flanges. was was Forused used a conventionalto toapproximate approximatef − thew thecurve, lower lowerthe boundary boundary equilibrium criticalcriticalpath buckling hasbuckling a nonlinear strength strength relationship,of ofUH UHPCPC flanges. asflanges. shown For For ina conventional Figurea conventional 12a. If the𝑓−𝑤 𝑓−𝑤f − curve,w curve curve, the is the equilib- transformed equilib- 2 riumriumto path the path hasf − has wa nonlinearacurve, nonlinear the relationsh equilibrium relationship,ip, pathas asshown exhibitedshown in inFigure an Figure approximately 12a. 12a. If theIf the 𝑓−𝑤 linear 𝑓−𝑤 curve relationship, curve is is transformedtransformedas shown to bytheto thethe 𝑓−𝑤 𝑓−𝑤 dotted curve, line curve, inthe the Figure equilibrium equilibrium 12b.Thereafter, path path ex hibitedexhibited the intersectionan anapproximately approximately points linear between linear relationship,relationship,the straight as asshown line shown extending by bythe the dotted fromdotted line the line equilibriumin Fiingure Figure 12b. 12b. path Thereafter, Thereafter, and the ftheaxis the intersection canintersection be defined points points as the betweenbetween the the straight straight line line extending extending from from the the equilibrium equilibrium path path and and the the f axisf axis can can be bede- de- finedfined as asthe the approximate approximate critical critical buckling buckling strength. strength. The The ultimate ultimate buckling buckling strength strength after after

Materials 2021, 14, x FOR PEER REVIEW 12 of 27 Materials 2021, 14, 2130 12 of 25

the critical buckling strength can be defined as the post local buckling strength of these flanges.approximate critical buckling strength. The ultimate buckling strength after the critical buckling strength can be defined as the post local buckling strength of these flanges. ) ) f f stress ( stress stress ( stress Compressive Compressive Compressive Compressive

Post buckling Post buckling strength strength

: GNIA analysis Critical buckling : GNIA analysis strength Critical buckling : GMNIA analysis strength : GMNIA analysis

Out-of-plane deflection (w) w2

(a) (b)

FigureFigure 12. 12. DefinitionDefinition of of critical andand post-bucklingpost-buckling strengthsstrengths based based on on FEA FEA (GNIA (GNIA means means geometric geometric nonlinear nonlinear analysis analysis with . withimperfections, imperfections, and and GMNIA GMNIA means means geometric geometric and and material material nonlinear nonlinear analysis analys withis with imperfections). imperfections) (a) f(a−) fw − relationship,w relation- ship, and (b) f − w2 relationship. and (b) f − w2 relationship.

4.2.4.2. Estimation Estimation of of Inelastic Inelastic Local Local Bu Bucklingckling Strength Strength of of Thin-Walled Thin-Walled UHPC UHPC Flanges Flanges TheThe characteristics characteristics of of inelastic inelastic local local buckling buckling behavior behavior and and the the strength strength of of UHPC UHPC flangesflanges were were investigated investigated based based on on the the GM GMNIANIA analysis. analysis. The The CDP CDP material material model model was was usedused to to simulate simulate the nonlinearnonlinear materialmaterial model model of of UHPC. UHPC. Additionally, Additionally, the the elastic-perfectly elastic-per- fectlyplastic plastic material material model model was appliedwas applied to the to referencethe reference flange flange models, models, which which represent represent steel steelflanges flanges with with equal equal yield yield strength strength under under compression compression and and tension. tension. TheThe flange flange models models analyzed analyzed in in this this study study were were supported supported by by a ahinge hinge and and fixed fixed boundaryboundary conditions, conditions, including including web web constraints. constraints. If the If web the webis thick, is thick, it is similar it is similar to a fixed to a supportfixed support condition, condition, which does which not does allow not displacement allow displacement or rotation. or If rotation. the web Ifis theconsider- web is ablyconsiderably thin, it allows thin, itan allows almost an free almost rotation, free rotation, similar similar to the tohinge the hinge boundary boundary condition. condition. In practicalIn practical scenarios, scenarios, steel steel I-girder I-girder flanges flanges supported supported by one by web one webare generally are generally located located be- tweenbetween the thehinge hinge andand fixed fixed boundary boundary conditions, conditions, as indicated as indicated in Equation in Equation (7). However, (7). However, in thein case the caseof UHPC of UHPC flanges, flanges, the web the webconstraint constraintss can be can considered be considered as fixed as boundary fixed boundary con- ditions,conditions, because because thick thick UHPC UHPC webs webs with with shear shear rebars rebars are are preferred preferred to to secure secure sufficient sufficient shearshear resistance resistance capacity. capacity. In In this this study, study, th thee two two boundary boundary conditions conditions were were compared compared for for thethe purpose purpose of of theoretical theoretical investigations. investigations. The The analysis analysis results results were were evaluated evaluated in in terms terms of of thethe critical critical buckling buckling strength strength and and post-buckling post-buckling behavior behavior of of the the UHPC UHPC flanges. flanges.

4.2.1.4.2.1. Axial Axial Compressive Compressive Strength Strength and and Ou Out-of-Planet-of-Plane Displacement Displacement Relationships Relationships • • Elastic-PerfectlyElastic-Perfectly Plastic Plastic Model: Model: Steel Steel Flanges Flanges TheThe elastic-perfectly elastic-perfectly plastic plastic material modelmodel (hereinafter,(hereinafter, plastic plastic model) model) was was considered consid- eredas a as conventional a conventional material material model model to simulate to simulate the localthe local buckling buckling phenomenon phenomenon for thefor steelthe steelflanges. flanges. The The plastic plastic material material models models have have equal equal yielding yielding strengths strengths under under compression compres- and siontension; and tension; these models these models further characterizefurther characterize the excellent the excellent ductility ductility induced induced by the plateauby the plateauwith a with high a strain high strain rate after rate yielding.after yieldi Here,ng. Here, the plastic the plastic material material model model was was used used as a asreference a reference model model to demonstrate to demonstrate the conventionalthe conventional local local buckling buckling phenomenon phenomenon of a flange of a flangewith thewith properties the properties of steel. of steel. In the In following the following section, section, the local the bucklinglocal buckling behaviors behaviors of the ofUHPC the UHPC flanges flanges and theand steel the flangessteel flanges are compared. are compared. FigureFigure 13 13 shows shows the the compressive stressstress andand out-of-plane out-of-plane deflection deflection relationships relationships of theof theflanges, flanges, as simulatedas simulated by the by plasticthe plastic model. model. As plotted As plotted in Figure in Figure13, typical 13, criticaltypical bucklingcritical bucklingand post-buckling and post-buckling behavior behavior that demonstrated that demonstrated the same the trend same (Figure trend 12 (Figurea) were 12a) observed. were The critical and post-buckling strengths increase as the slenderness ratio decreases. The

Materials 2021, 14, x FOR PEER REVIEW 13 of 27

Materials 2021, 14, 2130 13 of 25

observed. The critical and post-buckling strengths increase as the slenderness ratio de- creases. The post-buckling strength of the plastic model was considerably higher than the post-buckling strength of the plastic model was considerably higher than the measured measured approximate critical buckling strength when the slenderness ratio exceeded ap- approximate critical buckling strength when the slenderness ratio exceeded approximately proximately 12.5 and 22.5 for the hinge and fixed boundary conditions, respectively. If the 12.5 and 22.5 for the hinge and fixed boundary conditions, respectively. If the slenderness slenderness ratio is below these values, the material nonlinearity becomes a controlling ratio is below these values, the material nonlinearity becomes a controlling factor instead factor instead of geometrical nonlinearity. In these cases, as it is difficult to distinguish of geometrical nonlinearity. In these cases, as it is difficult to distinguish between critical bucklingbetween andcritical post-buckling buckling and states, post-buckling the maximum states, strength the maximum can be considered strength ascan the be ultimate consid- strengthered as the of theultimate flange. strength of the flange.

200 λf = 2.5 200 λf = 2.5 λf = 5 λf = 5 180 λf = 7.5 180 λf = 7.5 λf = 10 λf = 10 λf = 12.5 λf = 12.5 160 λf = 15 160 λf = 15 λf = 17.5 λf = 17.5 λf = 20 140 λf = 20 140 λf = 22.5 λf = 22.5 λf = 25 λf = 25 120 λf = 27.5 120 λf = 27.5 λf = 30 λf = 30 λf = 32.5 λf = 32.5 100 λf = 35 100 λf = 35 λf = 37.5 λf = 37.5 80 λf = 40 80 λf = 40 λf = 42.5 λf = 42.5 λf = 45 λf = 45 60 λf = 47.5 60 λf = 47.5 λf = 50 λf = 50 λf = 52.5 λf = 52.5

Compressive stress (MPa) stress Compressive 40 40 λf = 55 (MPa) stress Compressive λf = 55 λf = 57.5 λf = 57.5 20 λf = 60 20 λf = 60 λf = 62.5 λf = 62.5 0 λf = 65 0 λf = 65 0 100 200 300 400 500 600 0 20406080100 Out-of-plane deflection (mm) Out-of-plane deflection (mm)

(a) (b)

FigureFigure 13. Relationship 13. Relationship between between compressive compressive stress and stress out-of-plane and out-of-plane deflection deflectionfor the steel for flanges the steel modeled flanges using modeled the elastic- us- perfectly plastic model according to slenderness ratio (𝜆): (a) hinge support condition and (b) fixed support condition. ing the elastic-perfectly plastic model according to slenderness ratio (λ f ): (a) hinge support condition and (b) fixed support condition. • CDP Model: UHPC Flanges • CDPThe local Model: buckling UHPC strength Flanges of UHPC flanges was investigated based on the CDP ma- terialThe model. local In buckling the analysis, strength CDP of material UHPC models flanges with was investigateddifferent failure based strengths on the under CDP materialcompression model. and In tension the analysis, were CDP considered; material the models compressive with different and failuretensile strengthsstrengths under were compressionconsidered as and 180 tensionMPa and were 8.7 MPa, considered; respecti thevely. compressive The nonlinear and compression tensile strengths and tension were consideredconstitutive as relations 180 MPa were and consid 8.7 MPa,ered respectively. as listed in TheTable. nonlinear 4 and plotted compression in Figure and 9. tension Figure constitutive14 shows the relations compressive were consideredstress and asout-of-pla listed inne Table deflection4 and plotted relationships. in Figure Post-buckling9. Figure 14 showsbehavior, the which compressive demonstrates stress anda trend out-of-plane similar to deflectionthat shown relationships. in Figure 12a, Post-bucklingwas observed behavior,in the FE analysis which demonstrates results. However, a trend considerable similar to that reductions shown in in Figure the post-buckling 12a, was observed strength in thewere FE observed. analysis results. However, considerable reductions in the post-buckling strength were observed. The post-local buckling strength of the flanges was measured, except in the case of low slenderness ratio (approximately less than 25 and 50 for the hinge and fixed support conditions, respectively). The post-buckling strength of the CDP model, as shown in Figure 14, decreased drastically compared to that of the elastic-perfectly plastic model (Figure 13), regardless of the boundary conditions. This implies that the post-buckling behavior of UHPC flanges had a significant effect on material yielding, particularly the tensile strength, as compared with fully plastic material models, such as those for steel flanges. This is because the tensile strengths of UHPC materials were lower than their compressive strengths, leading to crack behavior during the post-buckling behavior of UHPC flanges.

Materials 2021, 14, x FOR PEER REVIEW 14 of 27

200 λf = 2.5 200 λf = 2.5 λf = 5 λf = 5 180 λf = 7.5 180 λf = 7.5 λf = 10 λf = 10 λf = 12.5 λf = 12.5 160 λf = 15 160 λf = 15 λf = 17.5 λf = 17.5 140 λf = 20 140 λf = 20 λf = 22.5 λf = 22.5 λf = 25 λf = 25 120 λf = 27.5 120 λf = 27.5 λf = 30 λf = 30 100 λf = 32.5 100 λf = 32.5 λf = 35 λf = 35 λf = 37.5 λf = 37.5 80 80 λf = 40 Materials 2021,, 1414,, 2130x FOR PEER REVIEW λf = 40 λf = 42.5 14 of 2527 λf = 42.5 λf = 45 60 λf = 45 60 λf = 47.5 λf = 47.5 λf = 50 40 λf = 50 λf = 52.5 λf = 52.5 (MPa) stress Compressive 40 Compressive stress (MPa) λf = 55 λf = 55 λf = 57.5 20 λf = 57.5 20 λf = 60 λf = 60 200 λf = 62.5 200 λfλf = =62.5 2.5 λf = 2.5 0 λf = 5 0 λfλf = 65= 5 180 λf = 7.5 180 λf = 7.5 0 100 200 300λf = 400 10 0 102030405060λf = 10 λf = 12.5 λf = 12.5 160 Out-of-plane deflection (mm) λf = 15 160 Out-of-plane deflection (mm) λf = 15 λf = 17.5 λf = 17.5 140 λf = 20 140 λf = 20 λf = 22.5 λf = 22.5 (a) λf = 25 (b) λf = 25 120 λf = 27.5 120 λf = 27.5 λf = 30 λf = 30 100 λf = 32.5 100 λf = 32.5 Figure 14. Relationship between compressive stress andλf = out-of-pla 35 ne deflection for UHPC flanges modeled usingλf =the 35 CDP λf = 37.5 λf = 37.5 material model80 according to slenderness ratio (𝜆): (a) hinge support80 condition and (b) fixed support condition.λf = 40 λf = 40 λf = 42.5 λf = 42.5 λf = 45 60 λf = 45 60 λf = 47.5 λf = 47.5 λf = 50 The post-local bucklingλf = 50 strength of the flanges was measured, except in the case of 40 λf = 52.5 λf = 52.5 (MPa) stress Compressive 40 Compressive stress (MPa) λf = 55 low slenderness ratio (approximatelyλf = 55 less than 25 and 50 for the hinge and λffixed = 57.5 support 20 λf = 57.5 20 λf = 60 conditions, respectively).λf = The 60 post-buckling strength of the CDP model, as λfshown = 62.5 in Fig- 0 λf = 62.5 λf = 65 ure 14, decreased drastically compared0 to that of the elastic-perfectly plastic model (Figure 0 100 200 300 400 0 102030405060 Out-of-plane13), regardless deflection of (mm) the boundary conditions. ThisOut-of-plane implies that deflection the post-buckling (mm) behavior of UHPC flanges had a significant effect on material yielding, particularly the tensile strength,(a) as compared with fully plastic material models,(b) such as those for steel flanges. This is because the tensile strengths of UHPC materials were lower than their compressive Figure 14. Relationship between strengths, compressive leading stress to crack and out-of-planeout-of-plabehaviorne during deflectiondeflection the forpost-buckling UHPC flangesflanges behavior modeled of using UHPC the CDPflanges. 𝜆 . material model according toto slendernessslenderness ratioratio ((λf ): ( a) hinge support condition and ( b) fixed fixed support condition.condition 4.2.2. Inelastic Local Buckling Strength of UHPC Flanges 4.2.2.The Inelastic post-local Local buckling Buckling strength Strength of of the UHPC flanges Flanges was measured, except in the case of Figure 15 shows a comparison of the theoretical local buckling strength calculated low slenderness ratio (approximately less than 25 and 50 for the hinge and fixed support using FigureEquations 15 shows (6) and a (8) comparison with the critical of the an theoreticald inelastic localbuckling buckling strengths strength obtained calculated from conditions, respectively). The post-buckling strength of the CDP model, as shown in Fig- theusing GMNIA Equations under (6)two and support (8) with conditions the critical (Figure and 15). inelastic The critical buckling and strengthsinelastic buckling obtained ure 14, decreased drastically compared to that of the elastic-perfectly plastic model (Figure strengthsfrom the from GMNIA the GMNIA under two were support obtained conditions based on (Figure the compressive 15). The critical stress andand inelasticout-of- 13), regardless of the boundary conditions. This implies that the post-buckling behavior planebuckling deflection strengths relationships from the GMNIAin Figures were 13 and obtained 14. The based critical on thebuckling compressive strength stress and andin- of UHPC flanges had a significant effect on material yielding, particularly the tensile elasticout-of-plane buckling deflection strength relationshipscan be distinguished in Figures based 13 and on the14. intersection The critical bucklingbetween theoret- strength strength, as compared with fully plastic material models, such as those for steel flanges. icaland buckling inelastic strength buckling curves strength and canthe beFE distinguishedanalysis data points, based onas shown the intersection in Figure between15. Thistheoretical is because buckling the tensile strength strengths curves of and UHPC the FE materials analysis were data points,lower than as shown their compressive in Figure 15. strengths, leading to crack behavior during the post-buckling behavior of UHPC flanges. 1.2 1.2 Theory (k = 0.486) Theory (k = 0.486) 4.2.2. Inelastic Local Buckling Strength of UHPC Flanges Theory (k = 1.381) Theory (k = 1.381) 1 1 Figure 15 shows a comparisonHinge & CDP modelof the theoretical local buckling strengthFixed & CDP calculated model ' ' c using Equations (6) and (8)Hinge with & thePlastic critical model and inelasticc buckling strengthsFixed &obtained Plastic model from

/ f 0.8

/ f 0.8 cr cr f the GMNIA under two support conditions (Figuref 15). The critical and inelastic buckling 0.6 0.6 or or strengths from the GMNIA were obtained based or on the compressive stress and out-of- y y

plane/ f deflection relationships in Figures 13 and/ f 14. The critical buckling strength and in- 0.4 0.4 cr cr cr f elasticf buckling strength can be distinguished based on the intersection between theoret- ical0.2 buckling strength curves and the FE analysis0.2 data points,Practical as shown in Figure 15. design ranges 0 0 1.20 5 10 15 20 25 30 35 40 45 50 55 60 65 1.20 5 10 15 20 25 30 35 40 45 50 55 60 65 Theory (k = 0.486) Theory (k = 0.486) λ (b /2tTheory) (k = 1.381) λ (b /2tTheory) (k = 1.381) 1 f f f 1 f f f Hinge & CDP model Fixed & CDP model ' ' c (a) Hinge & Plastic model c (b) Fixed & Plastic model

/ f 0.8

/ f 0.8 cr cr f Figure 15. Comparisons between theoretical local bucklingf strength and flange local buckling 0.6 0.6 or or or or y strength based on GMNIA according to slenderness ratioy and boundary conditions: (a) hinge support

conditions/ f 0.4 and (b) fixed support conditions. / f 0.4 cr cr cr f f

0.2Considering the hinge support condition, the0.2 approximatePractical critical buckling strength of the steel flanges, obtained using the plastic model, was similardesign ranges to the theoretical buckling strength,0 as compared in Figure 15a. However, the0 approximate critical bucking strength 0 5 10 15 20 25 30 35 40 45 50 55 60 65 0 5 10 15 20 25 30 35 40 45 50 55 60 65 of the UHPC flanges was lower than the theoretical value, when the slenderness ratio was λ (b /2t ) λ (b /2t ) less than 25. f f f f f f Similarly, for the fixed condition, the inelastic local buckling strength of the steel (a) (b) flange reduced when the slenderness ratio was between 15 and 22.5, whereas that of

Materials 2021, 14, 2130 15 of 25

the UHPC flanges, as determined using the CDP material model, exhibited a nonlinear relationship, with a significant reduction in strength for a slenderness ratio of 2.5–50. Thus, the intersection point distinguishing the critical and inelastic buckling strengths of UHPC flanges could attain a higher slenderness ratio than that of steel flanges. It was noted that, under the fixed condition, the effects of material nonlinearity were more critical than those under the hinged condition. Additionally, considering practical deck geometric conditions supported by a thick web, realistic deck constraints can be approximately assumed as the fixed support condi- tions with a slenderness ratio of 5–30 for the flanges, as reported in the literature [22]. Here, realistic buckling modes may be assumed as inelastic local buckling rather than elastic local buckling within the practical design ranges, as shown in Figure 15b. Figure 15b does not confirm that the practical UHPC bridges could fail with local buckling. It is because the practical bridge deck is usually designed based on effective width-based design approaches that may provide conservative deck designs. Additionally, conventional UHPC concrete decks are stiffened to achieve adequately prestressed systems in which the stiffened bridge decks possess a higher local buckling coefficient than unstiffened flange sections. However, it should be noted that the possibility of the local buckling phenomenon could exist not only for steel structures but also for the UHPC bridges within the design ranges. Thus, research on the local buckling phenomena could be necessary to achieve more reasonable, safe, and careful design practices for the application of thin-walled UHPC bridge systems.

5. Evaluation of FEA Approaches Based on Test Data 5.1. Summary of the Experimental Test in the Literature Lee [23] investigated the local buckling behavior of UHPC flanges subjected to pure compression based on experimental tests (Figure 16a). The girder specimens with a three- different thickness of thin-walled flanges were made and four-point bending loading was applied using spreader beams to introduce pure compression of the flanges on the test zone (Figure 16b). The specimens have 22–27.5 mm flange thickness with 730 mm and 740 mm width to induce flange local buckling (Figure 16c). UHPC concrete used for producing the specimens has 162 MPa of compressive strength, and 47,166 MPa of young’s modulus, which was determined by axial compression material tests. For tensile behavior, three-point bending tests were conducted to measure crack width (w). Using the measured crack displacement, crack width at tensile strength (wu) and crack width at zero strength beyond the tensile strength (wlim) were determined. Tensile strength was determined by p 0 7.8 fc, which is a lower boundary strength suggested by Graybeal [3]. The summary of the material test results was listed in Table5. To earn critical strain rates when the flanges of test specimens buckle, longitudinal strain values of the top and bottom side of flanges were measured during the tests (Figure 16d). Three-dimensional finite element analysis models for evaluating the test specimens considering the local buckling behavior were established in their research. Four-point loadings and simply supported boundary conditions with lateral supports to prevent overturning of the specimens used in the test were applied in the presented FEA models to simulate similar structural behavior of test specimens as possible as shown in Figure 16e.

Table 5. Material properties based on test data from the literature [23].

0 εu E (MPa) fc (MPa) ft (MPa) wu (mm) wlim (mm) 47,166 162 0.00396 8.24 0.22 6.43 Materials 2021, 14, x FOR PEER REVIEW 17 of 27

Materials 2021, 14, 2130 16 of 25

(a) (b)

730-740 22-27.5 30

70-80 475-480 662-667.5

160 490-500

(c)

Strain gauges Strain gauges L/3 L/3 L/3 at top surface at bottom surface

⑩ ⑳ Rear ⑧ ⑱ ⑥ ⑯ ④ ⑭

Front ② ⑫ ① ⑪ : Longitudinal strain gauges

(d)

(e)

Figure 16. Test specimens and test set-up in literature. (a) Illustration of test setup, (b) test zone of the specimens, (c) dimensions of cross-section of flanges at test zone, (d) longitudinal strain gauge

locations of flanges in the test zone, and (e) an example of the established three-dimensional FEA model (test specimens conducted from reference [23]). Materials 2021, 14, x FOR PEER REVIEW 18 of 27

Figure 16. Test specimens and test set-up in literature. (a) Illustration of test setup, (b) test zone of the specimens, (c) dimensions of cross-section of flanges at test zone, (d) longitudinal strain gauge locations of flanges in the test zone, and (e) an example of the established three-dimensional FEA model (test specimens conducted from reference [23]).

Table 5. Material properties based on test data from the literature [23].

𝑬 (MPa) 𝒇𝒄 (MPa) 𝜺𝒖 𝒇𝒕 (MPa) 𝒘𝒖 (mm) 𝒘𝒍𝒊𝒎 (mm) Materials 2021, 14, 2130 17 of 25 47,166 162 0.00396 8.24 0.22 6.43

In order to determine the maximum strength of the compression flange sections, strain valuesIn along order the to determineflange width the maximumwere measured strength as of describe the compression the location flange of sections,strain strain gauges invalues Figure along 16d. theThe flange strain width gauges were were measured installed ason describe the top and the locationbottom surfaces of strain of gauges in the flangeFigure section 16d. as The described strain gauges in Figure were 16d. installed Figureon 17 theshows top andstrain bottom values surfaces of both of top the flange and bottomsection surfaces as described of the flanges in Figure along 16d. the Figure flange 17 shows width strainwere valuesmeasured of both at ultimate top and bottom strengthsurfaces state during of the the flanges tests. alongBased theon flangethe strain width values, were the measured average at maximum ultimate strengthcom- state pressiveduring strain was the determined. tests. Based The on themaximum strain values, average the compressive average maximum strain (𝜀 compressive) induced strain by compressivewas determined. stress was The determined maximum as average listed in compressive Table 6. Additionally, strain (εavg) induceddesign moment by compressive M strengthstress (𝑀 was) and determined test strength as listed (𝑀 in) Tabledue to6. the Additionally, local buckling design were moment also provided strength in ( design) Table 6. andFrom test the strength established (Mtest 3D) due FEA to models the local corresponding buckling were to also the provided test specimens, in Table nor-6. From the malized establishedcompressive 3D stresses FEA models (𝑓 /𝑓 corresponding′) considering to fiber the test orientations specimens, factor normalized (𝐾 ) due compressive to 0 materialstresses uncertainty ( fFEA during/ fc) considering concrete casting fiber we orientationsre also presented factor (inK fTable) due 6. to More material detailed uncertainty informationduring of design, concrete test, casting and wereFEA alsovalues presented were presented in Table6 .in More the literature detailed information [23]. of design, test, and FEA values were presented in the literature [23].

-4000 -4000 -4000 Top surface Top surface Top surface Bottom surface Bottom surface Bottom surface Avg. Avg. -3000 Avg. -3000 -3000 ) ) ) -6 -6 -6

-2000 -2000 -2000 Strain (×10 Strain (×10 Strain (×10 Strain -1000 -1000 -1000

0 0 0 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 Distance from web center (mm) Distance from web center (mm) Distance from web center (mm)

Gauge numbers ① to ⑩

-4000 -4000 -4000 Top surface Top surface Top surface Bottom surface Bottom surface Bottom surface Avg. Avg. -3000 Avg. -3000 -3000 ) ) -6 -6

-2000 -2000 -2000 Strain (×10 Strain (×10 Strain (×10-6) Strain -1000 -1000 -1000

0 0 0 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 -600 -400 -200 0 200 400 600 Distance from web center (mm) Distance from web center (mm) Distance from web center (mm)

Gauge numbers ⑪ to ⑳ (a) (b) (c)

Figure 17.Figure Compressive 17. Compressive strain distributions strain distributions in the longitudinal in the longitudinal direction directionof UHPFRC of UHPFRC flanges: ( flanges:a) specimen (a) specimen #1, (b) specimen #1, (b) specimen #2, and (c#2,) specimen and (c) specimen #3 (data from #3 (data reference from reference [23]). [23]).

Table 6. Comparisons between average strain (𝜀 ) and maximum material strain (𝜀 ) (data from reference [23]). Table 6. Comparisons between average strain (εavg) and maximum material strain (εcu) (data from reference [23]). 𝒃 𝒕 𝒇 𝒇 Test TestFEA FEA Specimens 𝝀𝒇 𝑴,𝒅𝒆𝒔𝒊𝒈𝒏 (mm) (mm) 𝑴𝒕𝒆𝒔𝒕 𝜺𝒖 𝜺𝒂𝒗𝒈 𝜺𝒂𝒗𝒈/𝜺𝒖 𝒇𝑭𝑬𝑨/𝒇𝒄′ bf tf εavg εavg/εu 0 Specimens λ M M (mm) (mm) f ,design test fFEA/fc Mtest M εu design Gauges Gauges Gauges Gauges (Kf = 1.7) 1 to 10 1 to 10 1 to 10 1 to 10 Specimen #1 730 27.5 12 2221 1103 0.497 0.00396 0.001766 0.00176 0.446 0.444 0.401 Specimen #2 730 25 13.2 2056 1005 0.489 0.00396 0.001516 0.001346 0.383 0.34 0.426 Specimen #3 740 22 15 1984 1111 0.56 0.00396 0.001387 0.001779 0.35 0.45 0.416 Materials 2021, 14, x FOR PEER REVIEW 19 of 27

𝑴𝒕𝒆𝒔𝒕 Gauges Gauges Gauges Gauges (𝑲 =𝟏.𝟕) 𝒇 𝑴𝒅𝒆𝒔𝒊𝒈𝒏 ① to ⑩ ① to ⑩ ① to ⑩ ① to ⑩ Specimen #1 730 27.5 12 2221 1103 0.497 0.00396 0.001766 0.00176 0.446 0.444 0.401 MaterialsSpecimen2021 #2, 14, 730 2130 25 13.2 2056 1005 0.489 0.00396 0.001516 0.001346 0.383 0.34 0.42618 of 25 Specimen #3 740 22 15 1984 1111 0.56 0.00396 0.001387 0.001779 0.35 0.45 0.416

5.2.5.2. Comparisions Comparisions between between FEA FEA and and Test Test Data Data FigureFigure 18 18 shows shows a acomparison comparison of of maximum maximum average average strain strain values values from from the the test test data data andand maximum maximum stress stress values values based based on on flange flange models establishedestablished inin thisthis study. study. Additionally, Addition- ally,the the maximum maximum stress stress values values based based on 3Don FEA3D FEA models models corresponding corresponding to the to testthe test specimens spec- imenspresented presented in the in literature the literature [23] also [23] compared.also compared. The finiteThe finite element element analysis analysis was conductedwas con- ductedbased based on the on material the material test data test in data Table in5 Table. Maximum 5. Maximum stress values stress accordingvalues according to the slender- to the slendernessness ratio wereratio analyzedwere analyzed based based on the on analysis the analysis results results simulated simulated with the with same the analysis same analysisapproaches approaches presented presented in the previous in the previous section. section. Based on Based measured on measured data from data the from literature, the literature,3 mm of 3 initial mm of imperfections initial imperfections were applied. were applied. As shown As shown in Figure in Figure 18, the 18, test the data test shows data showssimilar similar trends trends compared compared with finite with element finite el analysisement analysis results. results. The test The data test were data located were at locatedlower partsat lower of theparts graph of the than graph the analysisthan the resultsanalysis of results fixed and of fixed hinge and boundary hinge boundary conditions. conditions.All test data All shows test data better shows correlations better correla withtions hinged with boundary hinged boun conditions.dary conditions. Considering Con- the sideringuncertainty the uncertainty of the material of the properties material proper of concreteties of concrete and realistic and realistic boundary boundary conditions, condi- the tions,test resultsthe test mayresults show may good show correlations good correlations with the with presented the presented finite elementfinite element analysis. analysis.

1.2 Theory (k = 1.381) Theory (k = 0.486) 1 Test data FEA (Test models)

u FEA (Fixed) ε / 0.8 FEA (Hinged) avg ε 0.6 or or ' c /f

cr 0.4 f

0.2

0 0 5 10 15 20 25 30 35 40 45 50

λf ( bf / 2tf )

FigureFigure 18. 18. ComparisonsComparisons between between presented presented FEA FEA results results an andd test test data data in in the the literature literature (test data from fromreference reference [23]). [23]).

6.6. Characteristic Characteristic of of Local Local Buckling Buckling Behavior Behavior of of Thin-Walled Thin-Walled UHPC UHPC Flanges Flanges 6.1. Inelastic Local Buckling Behavior of UHPC Flanges 6.1. Inelastic Local Buckling Behavior of UHPC Flanges The buckling mode of the fixed support condition simultaneously demonstrated a The buckling mode of the fixed support condition simultaneously demonstrated a conventional sine curve along the longitudinal direction (B-B line) with a half-sine curve conventional sine curve along the longitudinal direction (B-B line) with a half-sine curve along the transverse direction (A-A line) (refer to Figure 19a). However, in the case of along the transverse direction (A-A line) (refer to Figure 19a). However, in the case of hinged support conditions, a conventional sine curve along the longitudinal direction hinged support conditions, a conventional sine curve along the longitudinal direction (B- (B-B line) was observed, along with rigid body rotation along the transverse direction B line) was observed, along with rigid body rotation along the transverse direction (A-A (A-A line) (refer to Figure 19b). These differences between the buckling modes along the line) (refer to Figure 19b). These differences between the buckling modes along the trans- transverse directions may result in considerable local bending under high bending stresses, verseparticularly directions in themay case result of the in fixedconsiderable support local condition. bending Due under to the high local bending bending stresses, behavior particularlywith higher in curvatures, the case of thethe materialfixed support nonlinearity condition. effects Due under to the fixedlocal supportbending conditionsbehavior withhave higher a greater curvatures, influence the on material the inelastic nonlinea localrity buckling effects strength under fixed of UHPC support flanges. conditions Further havedetailed a greater evaluations influence of on these the aspectsinelastic are local presented buckling in strength the following of UHPC sections. flanges. Further detailed evaluations of these aspects are presented in the following sections.

Materials 2021, 14, x FOR PEER REVIEW 20 of 27 Materials 2021, 14, x FOR PEER REVIEW 20 of 27 Materials 2021, 14, 2130 19 of 25

B B B B AA AA AA AA 2a 2a 2a 2a

: undeformed shape b b : undeformeddeformed shape shape b B b B : deformed shape B B AA AA AA AA Half sine curve Half sine curve Rigid body rotation BBBBRigid body rotation BBBB Full sine curve Full sine curve Full sine curve Full sine curve

(a) (b) (a) (b) Figure 19. Comparison of buckling mode shapes between (a) fixed support condition, and (b) hingeFigureFigure support 19. 19. ComparisonComparison condition. of of buckling buckling mode mode shapes shapes between between (a (a) )fixed fixed support support condition, condition, and and ( (bb)) hinge hingesupport support condition. condition. 6.2. Estimation of Local Buckling Behavior of UHPC Flanges 6.2.6.2. Estimation Estimation of of Local Local Buckling Buckling Behavior Behavior of of UHPC UHPC Flanges Flanges In this section, local buckling behavior is discussed in terms of the local bending In this section, local buckling behavior is discussed in terms of the local bending stressesIn thisand section,strains along local withbuckling the longitudinal behavior is anddiscussed transverse in terms directions of the for local the twobending dif- stresses and strains along with the longitudinal and transverse directions for the two ferentstresses models, and strains namely along the withCDP the and longitudinal plastic material and transversemodels. The directions CDP model for the represents two dif- different models, namely the CDP and plastic material models. The CDP model represents theferent behavior models, of namelyUHPC flanges,the CDP while and plastic the plas materialtic material models model. The represents CDP model conventional represents the behavior of UHPC flanges, while the plastic material model represents conventional steelthe behavior flange sections of UHPC and flanges, serves whileas the thereference plastic model. material This model comparison represents is conductedconventional to steel flange sections and serves as the reference model. This comparison is conducted investigatesteel flange the sections effects and of the serves nonlinear as the material reference models model. and, This CDP comparison model on is the conducted local bend- to to investigate the effects of the nonlinear material models and, CDP model on the local inginvestigate and buckling the effects behavior of the ofnonlinear UHPC flanges.material Figure models 20 and, shows CDP the model relationship on the local between bend- bending and buckling behavior of UHPC flanges. Figure 20 shows the relationship between theing normalizedand buckling compressive behavior ofstress UHPC and flanges. out-of-p Figurelane displacement 20 shows the of therelationship flanges according between the normalized compressive stress and out-of-plane displacement of the flanges according tothe the normalized two different compressive material models. stress and The out-of-p flange lanewith displacement a slenderness of ratio the of flanges 55 was according selected toto the the two two different different material material models. models. The The flange flange with with a aslenderness slenderness ratio ratio of of 55 55 was was selected selected andand fixed fixed boundary boundary conditions conditions were were adopted. adopted. Out-of-plane Out-of-plane displacement displacement was was measured measured atand the fixed node boundary where maximum conditions displacement were adopted. occurred, Out-of-plane and the displacement compressive was stresses measured were atat the the node node where where maximum maximum displacement displacement occurred, occurred, and and the the compressive compressive stresses stresses were were calculatedcalculated by by dividing dividing the the compressive compressive loads loads acting acting on on each each end end of of the the flange flange with with the the cross-sectionalcalculated by dividing areas. the compressive loads acting on each end of the flange with the cross-sectionalcross-sectional areas. areas. 3 3 Plastic model CDPPlastic model model 2.5 CDP model 2.5 2 2 cr,theory f /

cr,theory 1.5 f

/ 1.5 GMNIA GMNIA

f 1 GMNIA GMNIA

f 1 0.5 0.5 0 0 0 20406080100120 0 20406080100120w (mm) w (mm) Figure 20. Relationship between normalized local buckling strength and slenderness ratio under the Figure 20. Relationship between normalized local buckling strength and slenderness ratio under Figurefixed support20. Relationship condition between when λ normalizedf = 55. local buckling strength and slenderness ratio under the fixed support condition when 𝜆 = 55. the fixed support condition when 𝜆 = 55. The plastic material model exhibited conventional post-buckling behavior, where The plastic material model exhibited conventional post-buckling behavior, where bucklingThe plastic strengths material gradually model increased exhibited with conv largeentional out-of-plane post-buckling deformation behavior, after the where critical bucklingbuckling strengths strength. gradually However, increased in the case with of the large CDP out-of-plane material model, deformation the initial after stiffness the crit- and icalbuckling buckling strengths strength. gradually However, increased in the case with of la therge CDP out-of-plane material deformationmodel, the initial after stiffness the crit- icalultimate buckling strength strength. of the However, flangewere in the significantly case of the CDP lower material than those model, of the plasticinitial stiffness material model. To further understand such local buckling behaviors, the local bending stresses

Materials 2021, 14, 2130 20 of 25

in the flange section were evaluated using strain and stress diagrams along the thickness direction for each loading phase determined by the material yielding states. To estimate the local bending behavior of UHPC and steel flanges, the stress and strain states at each loading phase were classified into four different phases, as shown in Figures 21 and 22. The notations #1–#4 in Figures 21 and 22 denote (1) the first crack or yielding initiation phase in the longitudinal axis near the centerline, (2) the second crack or yielding initiation phase in the transverse axis near the flange tips, (3) the ultimate buckling strength phase after the crack or yielding propagation, and (4) the final phase after the analysis, respectively. In the case of the CDP material model, the first crack occurred due to the transverse stress and strain along the web line (refer to phase #1 in Figure 21d). After the initiation of the crack, the stiffness of the compressive stress and out-of-plane displacement curve exhibited a decline (refer to phase #1 in Figure 21a), because the effective compressive area started decreasing due to crack propagation. Second, the strain along the longitudinal direction resulted in a crack at the flange tips (refer to phase #2 in Figure 21c); these cracks propagated via the transverse stress along the web line, leading to a significant reduction in stiffness (refer to phase #2 in Figure 21d). Third, crack propagation increased along with the longitudinal and transverse directions until the ultimate strength state was reached (refer to phase #3 in Figure 21a–d). A further increase in strength was not observed owing to the decrease in the effective compressive area, which was caused by changes in the neutral axis of the flanges due to crack propagation (refer to phase #4 in Figure 21a–d). The stiffness and ultimate strength of the reference steel flange simulated using the plastic material model under fixed boundary conditions were significantly different from those of the UHPC flanges, which were modeled using the CDP material model. For the plastic material model, the first yield along the web line, induced by the transverse stress and strain components, occurred immediately before the ultimate strength state (refer to phase #1 in Figure 22d). After the initiation of yield, stiffness declined marginally, which is reflected by the compressive stress and out-of-plane displacement curve (refer to phase #1 in Figure 22a). Subsequently, the compressive strain along the longitudinal direction at flange tips reached the yield strain (refer to phase #2 in Figure 22c). Yielding in the compression and tension sides of the flange progress; thereafter, transverse stress occurred along the web line owing to the decline of stiffness (refer to phase #2 in Figure 22d). Furthermore, after the propagation of the yield along the longitudinal direction at the flange tips, the ultimate was attained (refer to phase #3 in Figure 22a,c). Compressive stress decreased as the yield propagates along the longitudinal direction (refer to phase #4 in Figure 22a–d). The FE analysis results indicated that there were significant differences in the local buckling and local bending behaviors of the CDP material and plastic material model within inelastic ranges. These differences were attributed to the considerably lower tensile strength of UHPC, as compared to that of conventional steel flanges. In the UHPC flanges, initial cracks developed near the centerline; a reduction in stiffness, caused by the loss of the effective cross-section (Ae f f ) and the neutral axis changes, was observed due to crack propagation, as illustrated in Figure 23. For the plastic material model, although yielding was initiated near the centerline of the flange, the stiffness and strength reduction were not significant, because the gross cross- sectional area remains effective for the local bending behavior. This may further account for the tensile capacity of UHPC, which was approximately 20 times lower than that of the plastic models. Additionally, the plastic models did not exhibit strength reduction due to their ductility. Materials 2021 14 Materials, 2021, 2130, 14, x FOR PEER REVIEW 22 of 27 21 of 25

1.2 1 #3 #4 0.8 #2 cr,theory

/f 0.6 #1 0.4 GMNIA f 0.2 0 0 10203040 w (mm)

(a) (b)

ft = 8.69 MPa ft = 8.69 MPa ft = 8.69 MPa ft = 8.69 MPa 12 12 12 12 10 10 10 10 8 8 8 8 6 6 6 6 Crack propagation Layer Layer Layer Crack propagation Layer 4 4 4 4 Crack initiation 2 Elastic range 2 2 2 0 0 0 0 -200 0 200 400 600 -200 0 200 400 600 -200 0 200 400 600 -200 0 200 400 600 Stress (MPa) Stress (MPa) Stress (MPa) Stress (MPa)

-3 -3 -3 ε = 1.745 10-3 εt = 1.745 10 εt = 1.745 10 εt = 1.745 10 t 12 12 12 12 10 10 10 10 8 8 8 8 6 6 6 Crack propagation 6 Layer Layer Layer Layer 4 4 4 4 Crack initiation Crack propagation 2 2 Elastic range 2 2 0 0 0 0 -0.004 0 0.004 0.008 0.012 -0.004 0 0.004 0.008 0.012 -0.004 0 0.004 0.008 0.012 -0.004 0 0.004 0.008 0.012 Strain Strain Strain Strain

#1: initial phase #2: Crack initiation #3: Ultimate strength phase #4: Final phase : Crack propagation

(c)

(d)

Figure 21. Comparisons of compressive stress and strain for UHPC flanges, modeled using the CDP material model, according to loading phases: (a) definition of loading sequence, (b) definition of local bending location and element

layers, (c) normal stress distribution of local bending along the X-axis, and (d) normal stress distribution of local bending along the Y-axis. Materials 2021, 14, x FOR PEER REVIEW 23 of 27

Figure 21. Comparisons of compressive stress and strain for UHPC flanges, modeled using the CDP material model, ac- cording to loading phases: (a) definition of loading sequence, (b) definition of local bending location and element layers, Materials(c) normal2021 ,stress14, 2130 distribution of local bending along the X-axis, and (d) normal stress distribution of local bending along 22 of 25 the Y-axis.

3 #1 #2 #3 2.5 2 #4 cr,theory

/f 1.5 1 GMNIA f 0.5 0 0 50 100 150 w (mm)

(a) (b)

fy = 180 MPa fy = 180 MPa fy = 180 MPa fy = 180 MPa 12 12 12 12 10 10 10 First 10 Yielding 8 8 8 8 Yielding 6 6 6 6 Yielding propagation Layer Layer Layer First Yielding Layer propagation 4 Elastic 4 Yielding 4 4 propagation 2 range 2 2 2 0 0 0 0 -700 -350 0 350 700 -700 -350 0 350 700 -700 -350 0 350 700 -700 -350 0 350 700 Stress (MPa) Stress (MPa) Stress (MPa) Stress (MPa)

ε = 0.0036 ε = 0.0036 εy = 0.0036 y y εy = 0.0036 12 12 12 12 10 10 10 First 10 Yielding 8 8 8 8 Yielding 6 6 6 6 Yielding propagation Layer Layer First Layer 4 4 4 Yielding Layer propagation Elastic Yielding propagation 4 2 range 2 2 2 0 0 0 0 -0.014 -0.007 0 0.007 0.014 -0.014 -0.007 0 0.007 0.014 -0.014 -0.007 0 0.007 0.014 -0.014 -0.007 0 0.007 0.014 Strain Strain Strain Strain #1: Elastic phase #2: Yielding initiation #3: Ultimate strength phase #4: Final phase : Yielding propagation

(c)

fy = 180 MPa f = 180 MPa f = 180 MPa f = 180 MPa 12 12 y 12 y 12 y 10 10 10 10 Yielding Yielding propagation 8 8 propagation 8 Yielding 8 6 6 6 propagation 6 Yielding Layer Layer Yielding Layer Layer 4 4 4 propagation 4 Yielding First propagation 2 Yielding 2 2 2 propagation 0 0 0 0 -700 -350 0 350 700 -700 -350 0 350 700 -700 -350 0 350 700 -700 -350 0 350 700 Stress (MPa) Stress (MPa) Stress (MPa) Stress (MPa)

ε = 0.0036 ε = 0.0036 ε = 0.0036 ε = 0.0036 12 y 12 y 12 y 12 y 10 10 10 10 Yielding Yielding propagation 8 8 propagation 8 Yielding 8 6 6 6 propagation 6

Layer Layer Layer Yielding Layer Yielding 4 4 4 propagation 4 First propagation Yielding 2 Yielding 2 2 2 propagation 0 0 0 0 -0.014 -0.007 0 0.007 0.014 -0.014 -0.007 0 0.007 0.014 -0.014 -0.007 0 0.007 0.014 -0.014 -0.007 0 0.007 0.014 Strain Strain Strain Strain #1: Yielding initiation #2: Yielding propagation #3: Ultimate strength phase #4: Final phase : Yielding propagation

(d)

Figure 22. Comparisons of compressive stress and strain for ordinary steel flanges, modeled using the plastic material model, according to loading phases: (a) definition of loading sequence, (b) definition of local bending location and element layers, (c) normal stress distribution of local bending along the X-axis, and (d) normal stress distribution of local bending along the Y-axis. Materials 2021, 14, x FOR PEER REVIEW 24 of 27

Figure 22. Comparisons of compressive stress and strain for ordinary steel flanges, modeled using the plastic material model, according to loading phases: (a) definition of loading sequence, (b) definition of local bending location and element layers, (c) normal stress distribution of local bending along the X-axis, and (d) normal stress distribution of local bending along the Y-axis.

In the case of the CDP material model, the first crack occurred due to the transverse stress and strain along the web line (refer to phase #1 in Figure 21d). After the initiation of the crack, the stiffness of the compressive stress and out-of-plane displacement curve exhibited a decline (refer to phase #1 in Figure 21a), because the effective compressive area started decreasing due to crack propagation. Second, the strain along the longitudinal di- rection resulted in a crack at the flange tips (refer to phase #2 in Figure 21c); these cracks propagated via the transverse stress along the web line, leading to a significant reduction in stiffness (refer to phase #2 in Figure 21d). Third, crack propagation increased along with the longitudinal and transverse directions until the ultimate strength state was reached (refer to phase #3 in Figure 21a–d). A further increase in strength was not observed owing to the decrease in the effective compressive area, which was caused by changes in the neutral axis of the flanges due to crack propagation (refer to phase #4 in Figure 21a–d). The stiffness and ultimate strength of the reference steel flange simulated using the plastic material model under fixed boundary conditions were significantly different from those of the UHPC flanges, which were modeled using the CDP material model. For the plastic material model, the first yield along the web line, induced by the transverse stress and strain components, occurred immediately before the ultimate strength state (refer to phase #1 in Figure 22d). After the initiation of yield, stiffness declined marginally, which is reflected by the compressive stress and out-of-plane displacement curve (refer to phase #1 in Figure 22a). Subsequently, the compressive strain along the longitudinal direction at flange tips reached the yield strain (refer to phase #2 in Figure 22c). Yielding in the com- pression and tension sides of the flange progress; thereafter, transverse stress occurred along the web line owing to the decline of stiffness (refer to phase #2 in Figure 22d). Fur- thermore, after the propagation of the yield along the longitudinal direction at the flange tips, the ultimate was attained (refer to phase #3 in Figure 22a,c). Compressive stress de- creased as the yield propagates along the longitudinal direction (refer to phase #4 in Fig- ure 22a–d). The FE analysis results indicated that there were significant differences in the local buckling and local bending behaviors of the CDP material and plastic material model within inelastic ranges. These differences were attributed to the considerably lower tensile strength of UHPC, as compared to that of conventional steel flanges. In the UHPC flanges, initial cracks developed near the centerline; a reduction in stiffness, caused by the loss of Materials 2021, 14, 2130 23 of 25 the effective cross-section (𝐴) and the neutral axis changes, was observed due to crack propagation, as illustrated in Figure 23.

1.2 l

1 #3 tf #4 #2 0.8

cr,theory : Cracked Area

/f 0.6 #1 : Effective Area (Aeff) Aeff ≈ 0.2Ag Ag : Gross Area GMNIA 0.4 Aeff ≈ 0.4 Ag Aeff ≈ 0.1Ag f l : Unit length

0.2 tf : Flange thickness Aeff = Ag 0 0 10203040 w (mm)

Figure 23. VariationVariation of effective area during each loading sequences.

The second reason is the different failure mechanisms between UHPC and steel flanges. For steel flanges, which are represented by the plastic material model, the main strength reduction under inelastic local buckling behavior was induced by compression yielding

at the flange tips (refer to Figure 22c,d). Thus, compression steel flanges did not exhibit strength reduction until they reached compression yielding. In contrast, the controlling failure modes in UHPC flanges were mainly tensile fractures caused by local bending and not compression-induced failure modes. Therefore, the post-buckling strength of UHPC flanges was particularly susceptible to local failure due to the cracks in the tension side of the flanges. Alternatively, the local failure of the plastic material model was mainly caused by the initiation of compression yielding. Thus, the tensile strength of UHPC can be considered as a major design parameter on the inelastic local buckling strength of UHPC flanges.

7. Conclusions In this study, theoretical elastic and inelastic local buckling strengths, and behaviors of UHPC flanges were evaluated using the GMNIA. Parameters affecting the inelastic local buckling strength of UHPC flanges, including the tensile strength, and boundary conditions, were evaluated and compared with those of plastic material models with equal yield strengths under compression and tension. The important characteristics of UHPC flanges thus identified are presented as follows: • The elastic buckling strength of UHPC flanges was affected by boundary conditions and Poisson’s ratio. Although the UHPC flanges exhibited a higher elastic buckling coefficient than the steel flanges, the buckling strength of UHPC flanges possessed 1 4 times lower values than that of the steel flanges. • Nonlinear finite element analysis strategies to simulate the local buckling behavior of UHPC flanges were established based on geometric and material nonlinear analy- sis with imperfections (GMNIA). It was verified based on test data conducted in the literature. The finite element analysis results and test data show good correlations in ac- cordance with maximum average strain values along with the longitudinal directions. • The post-buckling strength of UHPC flanges was found to be considerably lower than that of the reference steel flanges, under both hinged and fixed support condi- tions. Particularly, UHPC flanges with fixed support conditions were susceptible to a considerable reduction in post-buckling strength because these flanges underwent local bending and tensile cracks caused by simultaneous longitudinal and transverse stress components. • Considering practical deck designs, local buckling of the UHPC flange deck could potentially exist within the design range with severe cases. In these cases, realistic buckling modes may be considered as the inelastic local buckling instead of elastic local buckling. Materials 2021, 14, 2130 24 of 25

Author Contributions: Supervision, Y.J.K.; Writing—original draft, J.L., S.K. and K.L.; Writing— review & editing, Y.J.K., J.L., S.K. and K.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2019R1I1A1A01059684) and supported by a National Research Foundation of Korea (NRF) grant funded by the Korean Government (MIST) (Grant No. 2020R1A2C201445012). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are available on request from the corresponding author. Conflicts of Interest: The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

References 1. Larrard, F.; Sedran, T. Optimization of Ultra-High-Performance Concrete by the Use of a Packing Model. Cem. Concr. Res. 1994, 24, 997–1008. [CrossRef] 2. Richard, P.; Cheyrezy, M. Composition of Reactive Powder Concrete. Cem. Concr. Res. 1995, 25, 1501–1511. [CrossRef] 3. Graybeal, B.A. Material Property Characterization of Ultra-High Performance Concrete; FHWA: Washington, WA, USA, 2006. 4. Rossi, P.; Arca, A.; Parant, E.; Fakhri, P. Bending and Compressive Behaviours of a new cement composite. Cem. Concr. Res. 2005, 35, 27. [CrossRef] 5. KICT. SUPER Bridge 200/Development of Economic and Durable Hybrid Cable Stayed Bridges; Korea Istitute of Construction Technology: Goyang-si, Korea, 2012. 6. SETRA-AFGC. Ultra High Performance Fibre-Reinforced Concretes, interim Recommendations; AFGC Groupe de Travail BUP: Paris, France, 2002. 7. JSCE. Recommendations for Design and Consruction of Ultra High Strength Fiber Reinforced Concrete Structures (Draft); Japan Society of Civil Engineers: Tokyo, Japan, 2004. 8. KCI. Design Guidelines for K-UHPC; Korea Concrete Institute: Seoul, Korea, 2014. 9. SAA. Australian Standard: Concrete Structures, 2nd ed.; Vols. AS3600-94; SAA: Sydney, Australia, 1996. 10. Marshall, W.T. The Lateral Stability of Reinforced Concrete Beams. J. Inst. Civ. Eng. 1948, 30, 194–196. [CrossRef] 11. Hansel, W.; Winter, G. Lateral Stability of Reinforced Concrete Beams. Am. Concr. Inst. J. 1959, 56, 193–214. 12. Sant, J.K.; Bletzacker, R.W. Experimental Study of Lateral Stability of Reinforced Concrete Beams. Am. Concr. Inst. J. 1961, 58, 713–736. 13. Massey, C. Lateral Instability of Reinforced Concrete Beams under Uniform Bending Moments. Am. Concr. Inst. J. 1967, 64, 164–172. 14. Revathi, P.; Menon, D. Estimation of Critical Buckling Moments in Slender Reinforced Concrete Beams. Am. Concr. Inst. J. 2006, 103, 296–303. 15. Girija, K.; Menon, D. Reduction in Flexural Strength in Rectangular RC Beams due to Slenderness. Eng. Struct. 2011, 33, 2398–2406. [CrossRef] 16. Hurff, J.B.; Kahn, L.F. Lateral-Torsional Buckling of Structural Concrete Beams: Experimental and Analytical Study. J. Struct. Eng. 2012, 138, 1138–1148. [CrossRef] 17. Kalkan, I. Lateral Torsional Buckling of Rectangular Reinforced Concrete Beams with Initial Imperfections. Aci Struct. J. 2014, 111, 71–81. 18. Kalkan, I.; Bocek, M.; Aykac, S. Lateral Stability of Reinforced-Concrete Beams with Initial Imperfections. Proc. Inst. Civ. Eng. Strut. Build. 2016, 169, 727–740. [CrossRef] 19. ACI. ACI 318-14: Building Code Requirements for Structural Concrete and Commentary; American Concrete Institution: Indianapolis, IN, USA, 2014. 20. Illich, G.; Tue, N.V.; Freytag, B. Schlanke vorgespannte Stützen aus UHPC-Experimentelle Untersuchung und Nachrechnung”. Beton Stahlbetonbau 2014, 109, 534–543. [CrossRef] 21. Lee, K.; Andrawes, B.; Lee, J.; Kang, Y.J. Lateral Torsional Buckling of Ultra-High-Performance-Fibrereinforced Concrete Girders. Mag. Concr. Res. 2020, 72, 820–836. [CrossRef] 22. Lee, J. Inelastic Flange Local Buckling Strength of Ultra High Performence Concrete I-Girders; Korea University: Seoul, Korea, 2018. 23. Lee, J.; Byun, N.J.; Lee, Y.W.; Kang, S.Y.; Kang, Y.J. Experimental and Numerical Evaluations of Local Buckling in Thin-walled UHPFRC Flanges. Thin-Walled Struct. 2021, 163, 107662. [CrossRef] Materials 2021, 14, 2130 25 of 25

24. Ma, J.; Dehn, F.; Tue, N.V.; Orgass, M.; Schmidt, D. Comparative Investigations on Ultra-high Performance Concrete with and without Coarse Aggregates. In Proceedings of International Symposium on Ultra High Performance Concrete (UHPC); Kassel University Press: Kassel, Germany, 2004. 25. AASHTO. Aashto Lrfd Bridge Design Specifications, 8th ed.; American Association of State Highway and Transportation Officials: Washington, DC, USA, 2020. 26. Lundquist, E.E.; Stowell, E.Z. Critical Compressive Stress for Outstanding Flanges; NACA Technical Report, Report No. 734; US Government Printing Office: Washington, DC, USA, 1 April 1941. 27. SIMULIA. ABAQUS Manual 6.18; SIMULIA: Johnston, RI, USA, 2018. 28. Kappos, A.J.; Konstantinidis, D. Statistical Analysis of Confined High Strength Concrete. Mater. Struct. 1999, 32, 734. [CrossRef] 29. AWS. Bridge Welding Code, ANSI/AASHTO/AWS L1.5-95; Joint Publication of American Association of State and High-Way Transportation Officials and American Welding Society; American Welding Society: Miami, FL, USA, 1995. 30. Roorda, J.; Venkataramaiah, K.R. Analysis of Local Plate Buckling Experimental Data. In Proceedings of International Specialty Conference on Cold-formed Steel Structures; Missouri S&T: Rolla, MO, USA, 1982.