Creep and Shrinkage Analysis of Curved Composite Beams Including the Effects of Partial Interaction

Transcription

1 Paper 154 Civil-Comp Pre, 212 Proeeding of the Eleventh International Conferene on Computational Struture Tehnology, B.H.V. Topping, (Editor), Civil-Comp Pre, Stirlinghire, Sotland Creep and Shrinkage Analyi of Curved Compoite Beam Inluding the Effet of Partial Interation X. Liu 1, R.E. Erkmen 2 and M.A. Bradford 1 1 Centre for Infratruture Engineering and Safety Faulty of Engineering, The Univerity of New South Wale, Sydney, Autralia 2 Centre for Built-Infratruture Reearh, Shool of Civil and Environmental Engineering, Univerity of Tehnology, Sydney, Autralia Abtrat Thi paper invetigate the time-dependent reep and hrinkage behaviour of horizontally urved teel-onrete ompoite beam, inluding the effet of partial interation exiting between the onrete dek and teel girder. The flexibility of the hear onnetion in the radial diretion, a well a in the tangential diretion at the teel and onrete interfae, are taken into aount in the propoed formulation. The reep and hrinkage effet of the onrete dek inluding onrete age effet are onidered by uing the analyti age-adjuted effetive modulu method. Thi method i effiient in onverting the reep analyi into peudo-elati analyi. The auray and effiieny of the propoed approah i validated by omparing it reult with available experimental reult reported in the literature and thoe baed on a more ophitiated but omputationally le effiient ABAQUS hell finite element model. The effet of initial urvature and partial interation on the timedependent behaviour of urved ompoite beam under ervieability ondition are alo eluidated and the lip effet due to warping i examined. Keyword: ompoite beam, urvature, flexible, onnetion, reep, hrinkage, vioelatiity. 1 Introdution Compoite teel and onrete beam whih are urved in plan are ued widely in highway bridge, partiularly for motorway interhange where high peed require mooth hange in diretion. Depite thi, omparatively few analytial olution have been reported on ompoite urved beam. An analytial formulation for horizontally urved ompoite bridge beam ubjeted to utained load and impoed deformation wa developed by Giuani and Mola [1], in whih it wa aumed that there wa full interation between the teel girder and onrete dek. However, in ompoite beam, the flexibility of the hear onnetor that join the 1

2 dek and the girder aue partial hear interation whih may ignifiantly influene the deformation of ompoite beam [2,3]. Therefore, the well-known model of Newmark et al. [4] i ommonly adopted for the analyi of traight ompoite beam with partial interation in the longitudinal diretion. For urved ompoite beam however, partial interation not only exit in the tangential diretion but alo in the radial diretion ine radial defletion and twit of the beam our even under vertial loading. Erkmen and Bradford [5] developed a beam model that inorporate partial interation in the tangential diretion a well a in the radial diretion in the elati analyi of ompoite beam urved in plan and their model i adopted herein. On the other hand, in trutural engineering deign, atifying ervieability limit tate i a vital omponent for the deign of teel and onrete ompoite beam [6, 7]. In order to atify thee ervieability requirement, an aurate aement of the reep and hrinkage effet on the defletion i required. Muh work ha been reported on the time-dependent reep and hrinkage behaviour of traight ompoite beam with partial hear interation. One of the earliet tudie in thi field wa publihed by Tarantino and Dezi [8], in whih a method wa preented for the vioelati analyi of imply upported teel-onrete ompoite beam with flexible hear onnetor. Later, they extended their method to tudy the long-term behaviour of ompoite ontinuou beam [9, 1] and hear-lag effet in ompoite beam [11]. In their analyi, the olution i ahieved numerially by uing a tepby-tep proedure for the reep and hrinkage onideration. However, the numerial tep-by-tep proedure require retaining of the full hitory of the train and tree whih require exeive omputer torage [12]. Subtantial implifiation an be made to prevent the problem by uing o-alled algebrai method uh a the effetive modulu method (EMM), age-adjuted effetive modulu method (AEMM) and mean tre method (MSM), baed on whih reep and hrinkage effet an be determined in the urrent time tep diretly without integrating the full hitory. At the ame time and independent of the Tarantino and Dezi tudy [8], Bradford and Gilbert [13] utilized a boundary value modelling approah for the analyi of imply upported teel-onrete ompoite beam baed on the AEMM, and extended it to handling ontinuou beam [14]. Dezi et al. [15] performed a implified reep analyi of ontinuou ompoite beam with flexible hear onnetor. More reently, Ranzi and Bradford [16] preented a generi model for the time dependent analyi of teel-onrete ompoite beam with partial hear interation by uing the AEMM and MSM, while Amadio and Fragiaomo [17] evaluated reep and hrinkage effet in a ompoite beam with rigid a well a flexible onnetor baed on the AEMM. Erkmen and Bradford [18,19] onidered vioelati Maxwell hain model for the time-dependent analyi of urved teel and onrete beam. In thi paper the objet i to develop an aurate model for the reep and hrinkage analyi of teel-onrete ompoite beam with partial hear interation while onidering the effet of initial urvature. The work build ignifiantly on the kinemati model of Erkmen and Bradford [5] developed for urved ompoite 2

3 beam. However, in ontrat to Erkmen and Bradford [18,19], the reep and hrinkage effet are inluded in thi tudy by the AEMM whih i omputationally more effiient. In addition, we preent a more detailed tudy on the warping indued lip. The propoed finite element formulation i validated by omparing the reult with available experimental reult reported in the literature and with thoe baed on a ophitiated but omputationally le effiient ABAQUS hell element model. The effet of initial urvature and partial interation on the time-dependent behaviour of ompoite beam urved in plan are illutrated and the lip effet due to warping i larified. 2 Kinemati relationhip 2.1 Bai aumption Figure 1 how a ompoite beam urved in plan for whih the following aumption are made. The teel girder i a doubly ymmetri I-beam that i urved in plan. The dek ha a retangular ro-etion and ha the ame initial urvature a the girder in the undeformed onfiguration. There i no uplifting between the girder and the dek. Radiu of urvature i ontant along the beam. The ro-etion remain rigid and maintain it hape throughout the deformation, o that there i no ditortion and no loal bukling. The hear onnetion between the girder and the dek i flexible in both tangential and radial diretion. Rotation, defletion and train are mall. The hear train due to bending and warping are negligible, o that the hear train on the ro-etion i indued by uniform torion only. 2.2 Axial ytem and finite train A body attahed urvilinear axi ytem i ued to deribe the geometry of the urved beam a hown in Figure 1. In the undeformed onfiguration, the axi ytem i in the poition oxy. The axi o i oriented along the axial diretion of the urved beam while the axe ox and oy are in the plane of the ro-etion. The total deformation are onidered to reult from rigid tranlation due to diplaement u(), v(), and w() of the entroid along the tangential diretion of the axe ox, oy and o, repetively, rotation of the ro etion through an angle φ () about the axi o, uperpoed warping diplaement of the whole ro-etion due to nonuniform torion, and diplaement funtion Ω x ( ), Ω ( ) and Ω κ ( ) due to lip ation in the radial and tangential diretion in the horizontal plane and lip ation indued by warping of the ro-etion, repetively. 3

4 r Px,y, ( ) i i o y x u 1/ v u p w p P w o 1 x 1 1 y 1 Figure 1: Coordinate ytem and deformation of the urved ompoite beam. By adopting the Green-Lagrange train and omitting the eond and higher order mall term, the non-zero normal train at an arbitrary point P on the ro etion of horizontal urved ompoite beam an be obtained [5] a ( ) ( ) ( ) ε = +Ω Ω κ + κ +Ω κ κ φ ω φ + κ +Ω, (1) w x x u w y v v κ in whih ()' d() d, u = u + wκ and w = w uκ. In Equation (1), κ i the initial urvature of the beam about vertial axi and ω i the normalied etion warping funtion. The funtion Ω x, Ω and Ω κ are aumed poitive for the girder and negative for the dek. The total lip diplaement between the teel girder and onrete dek at the interfae an be written a for the radial diretion and u = 2Ω (2) lip x w lip = 2Ω 2ωΩ (3) κ for the tangential diretion. The non-zero hear train due to the uniform torion an be obtained [5] a ( ) γ = φ + κ +Ω, (4) 2r v κ where r i the perpendiular ditane from the mid-urfae of a plate egment to the point P on the ro-etion. 4

5 3 Variational formulation of the equilibrium equation The equilibrium equation an be obtained by uing the priniple of virtual work, whih an be ated a T T T T T δπ= δ da d + δ da d + δ lip dd x δ Q δ q d= LA LA Lb L εσ εσ d q u Q u q, (5) in whih the firt two integral are the internal virtual work due to the deformation of the teel girder and the onrete dek where A and A are the ro-etional area of the teel and onrete omponent repetively. Alo in Equation (5), L i the total length of the beam, and ε, σ, ε and σ are the vetor of train and tre omponent for the teel girder and onrete dek repetively. The third integral in Equation (5) i the internal virtual work due to the lip at the interfae between the girder and the dek, in whih d lip and q are the vetor of the relative lip diplaement and hear tree between the dek and the girder repetively. The interfae hear fore applied by the hear onnetor are aumed to be ontinuouly ditributed, and b i the width of the effetive interfae urfae. The lat two term in Equation (5) are the virtual work done by the external onentrated fore Q on the aoiated onjugate diplaement δu Q and the virtual work done by ditributed member fore q on the aoiated onjugate diplaement δu q. 3.1 Variation of train From Equation (1) and (4), the firt variation of the normal and hear train for both the teel girder and the onrete dek an be written a δε = SB δθ, (6) k in whih index k an be hanged to for the teel girder and for the onrete dek. The matrie S and B in Equation (6) are k k and 1 x y ω S = 2rα (7) B k κ 1 akκ ak 2 κ 1 2κ akκ = 1 κ, (8) κ 1 ak κ 1 ak 5

6 where a =.5 for the teel girder, and a =.5 for the onrete dek.. In Equation (6), θ i the diplaement vetor of the eleted origin whih an be written a θ = u u u v v v w w φ φ φ 2Ωx 2Ω x 2Ω 2Ω 2Ωκ 2Ω κ T. (9) 3.2 Stree It i aumed that the material behaviour of the teel girder i linear elati under ervie load. Thu from elementary elatiity theory, the tree of the teel girder an be written in term of train a σ σ E ε = = τ = G γ E ε, (1) where E i the matrix of material propertie in whih E and G are the Young modulu and hear modulu of teel repetively. On the other hand, onrete dek i onidered a an aging linear vioelati material. The vioelati repone of the onrete i idential in both ompreion and tenion a reommended in Gilbert [2], and Bazant and Oh [21] for tre level in ompreion le than about one half of the ompreive trength of the onrete and in tenion le than about one half of the tenile trength of the onrete. The tre tate onidered in thi paper are aumed to remain in thi tre range. By applying the uperpoition priniple [22], the well-known integral ontitutive relationhip of time-dependent onrete behaviour an be written by uing the reep funtion a () () () ( ) ( ) ( ) ( ) t ε 1 1 t εh t σ t d σ t = J( t, t ) + J( t, t ) γ 2( 1 ) 2 ( 1 t + ν τ t + ν + ) dτ t t, (11) where σ and τ are the time dependent onrete normal and hear tree repetively, ε h i the tre independent hrinkage train, t i the time at the ating of the onrete, t i the time at initial loading in day, t + indiate that the lower limit of the integration i taken after the initiation of loading, and J(t, t ) i the reep funtion defined a the train at time t due to a ontant unit tre ating from time t to time t. The Poion ratio of onrete i taken a ν =.2 and i aumed to be ontant [23]. 3.3 Variation of lip diplaement and hear flow fore at the interfae The firt variation of lip diplaement an be written a 6

7 T δdlip = δulip δwlip = SΩBΩδθ, (12) in whih The element of matrix 1 S Ω = 1 ω. (13) B Ω an be written a B Ω 1 = 1. (14) 1 The behaviour of the tud hear onnetor i aumed to be linear elati; thu the hear flow fore at the interfae an be obtained in the term of lip diplaement a q T ρu ulip = qu qw = lip ρ = w ρd, (15) w lip where ρ i the matrix of the hear onnetion tiffnee in both the radial and tangential diretion. In Equation (15), ρ u and ρ w are the tiffnee of the hear onnetion (with unit of fore/length 3 ) whih an be defined a the hear tree in the radial and tangential diretion for a unit lip diplaement, repetively. 3.4 External loading The external onentrated load vetor Q an be written a where Q x and { Q } T x Qy Q Q =, (16) Q y are the onentrated radial and vertial fore in the x and y diretion, and Q i the external onentrated axial fore in the diretion. Similarly, the external ditributed loading vetor ating on the member an be written a q = q q q, (17) where q x, q y and { } T x y q are the ounterpart ation to Q x, Q y and Q whih are ditributed in the diretion along the member. The external load are aumed to at along the beam axi; aordingly the diplaement u Q and u q at the point at 7

8 whih load Q and q at an be written in the term of the diplaement of the beam axi. 4 Creep and hrinkage analyi 4.1 Age-adjuted effetive modulu method (AEMM) Sine time-dependent tre-train relation baed on the integral in Equation (11) require a tep-by-tep numerial integration [12], imple and aurate algebrai method uh a the effetive modulu method (EMM), mean tre method (MSM) and age-adjuted effetive modulu method (AEMM) are often preferred [2], whih replae the integral equation with imple algebrai equation. Intead of the tepby-tep proedure, a ingle tep analyi provide the tre-train relation for the urrent time. Among thee method, the mot general one i the AEMM developed by Bazant [24], while the other two impler method an be derived a the peial ae of the AEMM. By uing the AEMM, the onrete tree at time t an be rewritten a [24, 25] σ ( ) () ( ) ( ) () ( ) ( ) σ 1 t ε t εh t σ t = = Ee( t, t ) ϕ τ 1/2( 1 t ν +, (18) + ) γ t τ t τ are the normal and hear tree repetively whih an be obtained from the intantaneou elati analyi of ompoite beam urved in plan at time t. In Equation (18), Ee ( t, t ) i the age-adjuted effetive modulu and ϕ i the reep effet fator, whih an be written a in whih, σ ( t ) and ( t ) E e ( t, t ) E ( t ) ( tt) φ ( tt) = 1 + χ,, (19) and ( tt, ) χ( tt, ) 1 + χ( tt) φ ( tt) φ ϕ = 1,,, (2) where ( ) E t i the elati modulu of the onrete at time t, ( tt, ) φ i the reep oeffiient defined a the ratio between the reep train at time t and the initial train at time t and χ ( tt, ) i the aging oeffiient whih an be written a χ ( tt, ) E ( t ) 1 ( ) (, ) φ (, ) = E t R t t t t, (21) 8

9 in whih R(, tt ) i the relaxation funtion. The AEMM i formulated for one tep loading hitory; thu the load i applied at age t and then i aumed to tay ontant until the urrent time t. The tre expreion for onrete i not only a funtion of the train tate at time t but alo a funtion of the tre tate that ourred at time t and of the hrinkage deformation during time t. The repone to multi-tep loading hitorie an be obtained by uperimpoing the reult for everal one-tep loading hitorie [26]. 4.2 Finite element formulation By uing the virtual work priniple in Equation (5), the equilibrium equation for a urved ompoite teel-onrete beam element at the urrent time t an be written a Kd = F ext + F + F, (22) where K i the diplaement tiffne matrix, d i the nodal diplaement vetor, F ext i the external load vetor, F i the peudo-load vetor due to vioelati behaviour repreenting the hitory of the onrete ine being loaded at time t and F i the load vetor due to the hrinkage effet. The tiffne matrix in Equation (22) an be written a T T T T T T T K = N B da + da + Ω Ω Ωdx Ω d S E S B B S E S B B S ρs B N (23) L A A b in whih the matrie of material propertie for onrete E an be written expliitly a 1 E = Ee( t, t ) 1/2( 1 ν. (24) + ) The omponent at the right hand ide of Equation (22) an be written a and F = N A qd + N A Q, (25) T T T T ext q Q L F T T T d = ϕ A L A (26) F T T T = hda d L A (27) where the vetor of the intantaneou tre σ and the hrinkage trainε h an be 9

10 written a and ( t ) τ ( t ) T σ = σ (28) () T ε = ε. (29) h h t In Equation (23) to (27), N i the finite element hape funtion matrix, whih i obtained herein by interpolating the diplaement and rotation field u, v and φ by uing ubi Hermitian funtion, and the tangential defletion w and the lip defletion 2Ω, 2Ω and 2Ω by uing linear funtion. 5 Appliation x 5.1 Comparion with experimental reult κ Bradford and Gilbert [27] onduted a et of experiment to determine the long-term behaviour of imply upported teel-onrete traight ompoite beam onneted with headed hear tud. The teel etion wa a 2 UB 25.4 and the onrete dek had a width B d =1mm and a thikne t d = 7mm whih produed a total ditributed vertial load of q = 1.9N/mm along the beam due to it elf weight. An additional ditributed load of q = 7.52N/mm wa applied along the 59mm total pan and wa maintained for 25 day. The tiffne of the hear tud wa determined from the tandard puh-out tet explained in [13] a k = 84kN/mm. For the firt beam experiment, two hear tud were ued at every 2mm along the beam (B1 beam in [27]), whih orrepond to a hear tiffne of 3 ρ w = 84 2 ( 2 133) = 6.3N/mm in the ompoite beam model ued herein in whih the width of the effetive interfae urfae between the teel girder and onrete dek i taken a 133mm. For the eond beam experiment, the tud 3 interval were inreaed to 6 mm, i.e. ρ = 2.1N/mm (B3 beam in [27]). w (a) Creep oeffiient (b) hrinkage train Figure 2: Experimental and CEB urve 1

11 2 The teel material wa aumed to be elati with a modulu of E = 2 kn/mm. The reep oeffiient and hrinkage urve for the onrete material were determined by mean of eparate tet arried out on ylindrial peimen and determined a hown in Figure 2. The urve fitting thi data were determined onveniently by uing the CEB model ode [28], baed on whih the ode parameter were hoen and hown in Table 1. Steel girder Setion 2 UB 25.4 Setion deignation of the teel girder E 2 kn/mm 2 Elati modulu of the teel G 77 kn/mm 2 Shear modulu of the teel Shear tud ρ w 2.1 N/mm 3 Stiffne modulu of hear onnetion in tangential diretion ρ u 2.1 N/mm 3 Stiffne modulu of hear onnetion in radial diretion Conrete lab B d 1 mm Width of the dek D d 7 mm Depth of the dek f k 31.1 N/mm 2 Charateriti ompreive trength of the onrete E 25.1 kn/mm 2 Elati modulu of the onrete on the tenth day G 1. kn/mm 2 Shear modulu of the onrete on the tenth day ν.2 Poion ratio of the onrete RH 5% Relative humidity of the ambient environment t 1 day Age of the onrete (day) at the firt loading β 5 Cement oeffiient (β = 5) Table 1: Compoite ro-etion and material propertie. Eight element along the pan were ued to model the beam in whih the onrete dek wa divided into 64 retangular area; ixteen along the width and four aro the thikne a hown in Figure 3. Figure 3 Sampling point heme of the teel-onrete ompoite ro-etion. Evolution of the defletion at mid-pan and at the quarter point for both beam and omparion between the experimental reult and thoe baed on the developed model are hown in Figure 4. 11

12 (a) Mid-pan defletion of B1 (b) Mid-pan defletion of B3 () Quarter point defletion of B1 (d) Quarter point defletion of B3 Figure 4: Time-dependent vertial defletion. It an be een from Figure 4 that baed on the parameter hoen in Table 1 the reult are in good agreement after 25 day, however a greater gap an be oberved for the intantaneou repone of the beam and the model depit lightly more flexible behaviour than the atual behaviour at the initiation of loading. 5.2 Comparion with ABAQUS model In order to validate the developed method, an ABAQUS hell element model ha been developed for omparion purpoe, in whih the reep and hrinkage effet of onrete a well a the partial interation between the teel girder and onrete dek are taken into aount. The example beam i imply upported with a total 3 length of 5.9 m and the hear onnetion tiffne of ρw = ρu = 2.1 N/mm. The beam i urved in plan with an inluded angle of θ = 15 (Figure 1). The detail of the ompoite ro-etion and material propertie are ame a in the previou example (given in Table 1), with the applied ditributed load being q = 9.42 N/mm. Similarly, eight element were ued to model the beam and the onrete dek wa divided into 64 retangular area. On the other hand, for the ABAQUS model a total of 13 (S4) hell element were ued a hown in Figure 5. 12

13 Figure 5: ABAQUS model for the urved ompoite beam. Figure 6 how the vertial and radial defletion and angle of twit at the mid-pan of the beam during 25 day baed on the developed model (DM) and the ABAUQS model. (a) Vertial defletion (b) Radial defletion () Angle of twit Figure 6: Time-dependent mid-pan defletion baed on the ABAQUS model and the developed method. 13

14 Spring element were ued in the tangential and radial diretion to onnet the onrete dek and the top flange of the teel girder in order to model the partial interation at the dek and girder interfae. Three pring element with equal tiffne were ued every 118 mm along the beam. Baed on the tiffne modulu of hear onnetion ρ w = ρ u = 2.1 N/mm 3, the tiffne of the pring were determined a k = 11 N/mm. Tru element with large ro-etion were ued to onnet the onrete dek and the top flange of the teel girder in the vertial diretion in order to prevent uplifting. The vertial and radial defletion a well a the rotation were retrained at the entroid of the girder at both end (u = v = φ = ), while the tangential diplaement of the entroid i retrained only at one end (w = ). Five other point of the ro-etion at both end were retrained in the vertial diretion to prevent the twiting of the ro-etion while allowing for warping deformation. The ditributed vertial loading of q = 9.42 N/mm along the pan wa applied by uing P =1.11 kn nodal vertial load at the entroid of the girder every 118 mm along the beam. The vioelati time analyi wa performed in the ABAUQS model by defining the vioelati material parameter of the onrete uing the VISCOELSTIC, TIME = CREEP TEST DATA ommand [3]. The hrinkage effet of the onrete dek wa onidered by uing the EXPANSION ommand [29], in whih the expanion oeffiient have negative value. 5.3 Slip due to warping A indiated in Equation (3), by uing the funtion Ω κ, the lip effet in the onrete lab and teel flange interfae due to the warping of the ro-etion i onidered in thi paper. In Figure 7, two ae are illutrated to how the relationhip between the lip, the flange rotation angle and the funtion Ω. κ Figure 7: Warping and lip due to warping. 14

15 The ro-etion for the firt ae i not ompoite and the flange rotation angle i related to the longitudinal diplaement at the tip of the flange due to warping. In the eond ae, warping indued differene between the longitudinal diplaement of the top flange and the top dek i hown for the ompoite ro-etion whih require the funtion Ω κ introdued in the beam model. A traight ompoite beam model with doubly ymmetri ro-etion i analyed under torional load to generate torional behaviour only, and thu to aue lip between the omponent due to warping of the ro-etion only. A hown in Figure 8, an ABAQUS model i alo developed for omparion purpoe. The beam i under uniformly ditributed torional load of m =1kN mm/mm, implyupported and ha a total length of 5.9 m. The top and bottom onrete lab of the beam are onneted to the top and bottom flange of teel girder, with flexible hear onnetion with a tiffne of ρ w =ρ u =.5N/mm 3. The ro-etion i fored to behave rigid in it plane during deformation in the ABAQUS model whih i onitent with the kinemati aumption of the developed beam formulation. Other detail of the ompoite ro-etion and material propertie of the beam are a ame a in the previou example (given in Table 1). Figure 8: Straight ompoite beam. Figure 9 how the lip at the tip of the teel flange due to warping along the length of the beam baed on the ABAQUS model and the developed method (DM), from whih it an be een that the reult baed on the developed method are in very good agreement with the one baed on the ABAQUS model. Figure 9: Warping indued lip at the tip of the flange in the longitudinal diretion baed on the ABAQUS model and the developed method. On the other hand, warping indued lip at the right and left tip of the top flange are equal in value but in oppoite diretion, and thu average lip due to warping i zero. Thu, an independent parameter Ω κ hould be aigned to apture the lip effet due to warping. 15

16 5.4 Effet of partial interation In order to illutrate the effet of partial interation on the time-dependent behaviour of ompoite urved beam, a group of beam with different onnetion tiffnee are analyed uing the developed model. Figure 1 how the vertial and radial diplaement, angle of twit, and radial and tangential lip along the beam at the initiation of the loading and after 1 day due to external loading only and without the hrinkage effet. (a) Vertial defletion (b) Radial defletion () Angle of twit (d) Tangential lip (e) Radial lip Figure 1: Deformed hape at the initiation of loading and after1 day for different onnetion tiffnee without hrinkage effet. 16

17 The tiffne modulu of hear onnetion are eleted a ρ w = ρ u =.5N/mm 3, ρ w = ρ u = 2.1N/mm 3 and ρ w = ρ u =5 N/mm 3 for omparion. It hould be noted that ρ w = ρ u =.5N/mm 3 orrepond to hear onnetor modulu of 68 N/mm 2 whih i a pratial value ued for hear tud onnetor [3]. Figure 11 depit the orreponding deformed hape due to the hrinkage effet only. The reult how that the diretion of radial and tangential lip aued by external load are oppoite to thoe due to hrinkage effet. (a) Vertial defletion (b) Radial defletion () Angle of twit (d) Tangential lip (e) Radial lip Figure 11: Deformed hape after 1 day for different onnetion tiffnee due to hrinkage effet. 17

18 It an alo be oberved that, the lip in radial and tangential diretion in the beam with hear onnetion tiffne ρ w = ρ u =5N/mm 3 are negligible omparion to the other two beam, and that the hear onnetion tiffne ha a lear effet on the vertial defletion and angle of twit. A hown in Figure 11, the mid-pan vertial diplaement at the initiation of loading, dereae 39% when the tiffne of hear onnetion are inreaed from ρ w = ρ u =.5N/mm 3 to ρ w = ρ u =5N/mm 3 and after 1 day thi differene redue to 22%. Thu, the effet of partial interation between the ompoite omponent i more ignifiant at the initiation of loading; however, they hould be taken into aount for aurate time-dependent analyi ine the aumption of full interation may underetimate the defletion. 6 Conluion An effiient numerial method ha been developed for the reep and hrinkage analyi of teel-onrete ompoite beam that are urved in plan by uing the ageadjuted effetive modulu method. Through omparion with available experimental reult and a vioelati ABAQUS hell element model, it wa hown that the developed formulation apture the time-dependent reep and hrinkage behaviour of ompoite beam. The developed method onider the effet of partial interation between the onrete dek and the teel girder in the radial diretion a well a in the tangential diretion and their effet on the time-dependent behaviour of urved ompoite beam wa invetigated. It wa demontrated that the initial urvature of the beam ha a ignifiant influene on the time-dependent behaviour of ompoite beam. It wa alo hown that the effet of partial interation between the two ompoite omponent hould be taken into onideration for an aurate time-dependent analyi, ine the aumption of full interation may mietimate the deformation. In addition, the warping indued lip effet ha been tudied. In thi tudy, it i learly hown that an independent parameter i required to apture the warping indued tangential lip a onidered by the beam model adopted herein. Aknowledgement The work in thi paper wa upported by the Autralian Reearh Counil through a Diovery Projet award to the third author. The upport i gratefully aknowledged. Referene [1] F. Giuani, F. Mola, Servie-tage analyi of urved ompoite teelonrete bridge beam, Journal of Strutural Engineering, ASCE, 132(12), , 26. [2] D.J. Oehler, M.A. Bradford, Compoite teel and onrete trutural member: Fundamental behaviour, Pergamon, Oxford, [3] D.J. Oehler, M.A. Bradford, Elementary behaviour of ompoite teel and onrete trutural member, Butterworth-Heinemann, Oxford,

19 [4] N.M. Newmark, C.P. Sie, I.M. Viet, Tet and analyi of ompoite beam with inomplete interation, Proeeding of the Soiety for Experimental Stre Analyi, 9(1), 75-92, [5] R.E. Erkmen, M.A. Bradford, Nonlinear elati analyi of ompoite beam urved in-plan, Engineering Struture, 31, , 29. [6] Standard Autralia (SA), Autralian Standard AS Compoite truture Part 1: Simply upported beam, Sydney, Autralia, 23. [7] M.A. Bradford, B. Uy, Y.L. Pi, Behaviour of unpropped ompoite girder urved in plan under ontrution loading, Engineering Struture, 23, , 21. [8] A.M. Tarantino, L. Dezi, Creep effet in ompoite beam with flexible hear onnetor, Journal of Strutural Engineering, ASCE, 118(8), , [9] L. Dezi, A.M. Tarantino, Creep in ompoite ontinuou beam Ι: Theoretial treatment, Journal of Strutural Engineering, ASCE, 119(7), , [1] L. Dezi, A.M. Tarantino, Creep in ompoite ontinuou beam ΙI: Parametri tudy, Journal of Strutural Engineering, ASCE, 119(7), , [11] L. Dezi, F. Gara, G. Leoni, A.M. Tarantino, Time-dependent analyi of hear-lag effet in ompoite beam, Journal of Engineering Mehani, 127(1), 71-79, 21. [12] Z.P. Bazant, Numerial determination of long-range tre hitory from train hitory in onrete, Material and Struture, 5(27), , [13] M.A. Bradford, R.I. Gilbert, Compoite beam with partial interation under utained load, Journal of Strutural Engineering, ASCE, 118(7), , [14] R.I. Gilbert, M.A. Bradford, Time-dependent behaviour of ontinuou ompoite beam at ervie load, Journal of Strutural Engineering, ASCE, 121(2), , [15] L. Dezi, G. Leoni, A.M. Tarantino, Algebrai method for reep analyi of ontinuou ompoite beam, Journal of Strutural Engineering, ASCE, 122(4), , [16] G. Ranzi, M.A. Bradford, Analyi olution for the time-dependent behaviour of ompoite beam with partial interation, International Journal of Solid and Struture, 43, , 25. [17] C. Amadio, M. Fragiaomo, Simplified approah to evaluate reep and hrinkage effet in teel-onrete ompoite beam, Journal of Strutural Engineering, ASCE, 123(9), , [18] R.E. Erkmen, M.A. Bradford, Time-dependent reep and hrinkage analyi of ompoite beam urved in-plan, Computer & Struture, 89, 67-77, 211. [19] R.E. Erkmen, M.A. Bradford, Non-linear quai-vioelati behaviour of ompoite beam urved in-plan, Journal of Engineering Mehani, ASCE, 137, ,

20 [2] R.T. Gilbert, Time Effet in Conrete Struture, Elevier, Amterdam, [21] Z.P. Bazant, B.H. Oh, Deformation of progreively raking reinfored onrete beam, Journal of ACI, 81, , [22] D. MHenry, A new apet of reep in onrete and it appliation to deign, Proeeding of ASTM, 43, , [23] SA (Standard Autralia), Autralian Standard AS 36 Conrete Struture, Sydney, Autralia, 21. [24] Z.P. Bazant, Predition of onrete reep effet uing age-adjuted effetive modulu method, Journal of ACI, 69, , [25] CEB (Comité Euro-International du Béton), CEB Deign manual on trutural effet of time dependent behaviour of onrete, Georgi Publihing, Saint-Saphorin, Switzerland, [26] M. Jiraek, Z.P. Bazant, Inelati analyi of truture, John Wiley & Son, New York, 22. [27] M.A. Bradford, R.I. Gilbert, Time-dependent behaviour of imply-upported teel onrete ompoite beam, Magazine of Conrete Reearh, 43(157), , [28] CEB-FIB (Comité Euro-International du Béton Fédération International de la Préontrainte), Mode Code 199: Deign Code, Thoma Telford, London, UK, [29] H.D. Hibbit, B.I. Karlon, E.P. Sorenen, ABAQUS Uer manual Verion 6.8, Hibbitt, Karlon & Sorenen In., Pawtuket, USA, 28. 2