Modelling dowel action of discrete reinforcing bars in cracked concrete structures

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Title Moelling owel action of iscrete reinforcing bars in cracke concrete structures Author(s) Kwan, AKH; Ng, PL; Lam, JYK Citation The 2n International Symposium on Computational Mechanics an the

1 Title Moelling owel action of iscrete reinforcing bars in cracke concrete structures Author(s) Kwan, AKH; Ng, PL; Lam, JYK Citation The 2n International Symposium on Computational Mechanics an the 12th International Conference on the Enhancement an Promotion of Computational Methos in Engineering an Science (ISCM II an EPMESC XII), Hong Kong-Macau, China, 30 November-3 December In AIP Conference Proceeings, 2009, v. 1233, p Issue Date 2010 URL Rights AIP Conference Proceeings. Copyright American Institute of Physics.; Copyright (2009) American Institute of Physics. This article may be ownloae for personal use only. Any other use requires prior permission of the author an the American Institute of Physics. The following article appeare in AIP Conference Proceeings, 2009, v. 1233, p an may be foun at This work is license uner a Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 International License.

2 Moelling Dowel Action of Discrete Reinforcing Bars in Cracke Concrete Structures A.K.H. Kwan a, P.L. Ng a, *, an J.Y.K. Lam a a Department of Civil Engineering, The University of Hong Kong * aress for corresponing author: irngpl@gmail.com Abstract. Dowel action is one of the component actions for shear force transfer in cracke reinforce concrete. In finite element analysis of concrete structures, the use of iscrete representation of reinforcing bars is consiere avantageous over the smeare representation ue to the relative ease of moelling the bon-slip behaviour. However, there is very limite research on how to simulate the owel action of iscrete reinforcing bars. Herein, a numerical moel for owel action of iscrete reinforcing bars crossing cracks in concrete is evelope. The moel features the erivation of owel stiffness matrix base on beam-on-elasticfounation theory an the irect assemblage of owel stiffness into the concrete element stiffness matrices. The owel action moel is incorporate in a nonlinear finite element programme with secant stiffness formulation. Deep beams teste in the literature are analyse an it is foun that the incorporation of owel action moel improves the accuracy of analysis. Keywors: Cracks; Dowel action; Finite element metho; Reinforce concrete. PACS: De INTRODUCTION Compare to the axial an flexural counterparts, the shear behaviour of concrete structures is less preictable ue to the complexity of shear transfer mechanisms an the ifficulties in numerical moelling, an yet it plays an important role in the overall structural behaviour of reinforce concrete members [1]. The owel action of reinforcing bars is one of the component actions for shear transfer in a cracke concrete structure. Accoring to Park an Paulay [1], the shear resistance of a cracke concrete structure is constitute of: (1) irect transfer of shear force by uncracke concrete; (2) irect tensile forces in stirrups; (3) aggregate interlock at crack surface; an (4) owel action of reinforcing bars crossing the crack. Figure 1 shows the above internal forces pertaining to a cracke concrete beam. Being a major component of the shear transfer in a cracke concrete structure, the owel action of reinforcing bars has been investigate experimentally by many researchers [2-7]. However, espite ecaes of research on the finite element analysis of reinforce concrete structures, there has been basically no explicit consieration of the owel action in finite element analysis. At the most, only a gross allowance with the owel action lumpe together with other components of shear transfer was incorporate [8]. It was only until more recently that the first author evelope a owel action moel for application with smeare representation of reinforcing bars [9]. In this moel, the owel force an owel eformation are expresse in smeare forms an the owel stiffness matrix is assemble into the concrete element stiffness matrix. It has been applie to analyse eep beams an coupling beams [10,11]. The moelling of owel action for smeare reinforcement is only an interim measure so as to be compatible with the existing finite element programs using smeare representation of the reinforcing bars. In the long run, for more realistic moelling of the bon-slip behaviour an owel action of the reinforcing bars, iscrete representation of the reinforcing bars shoul be aopte instea of smeare representation. With iscrete representation, the reinforcing bars are moelle by iscrete one-imensional steel elements. To moel the bon-slip behaviour of the reinforcing bars, the steel elements are connecte to the concrete through bon elements. The most commonly use bon 701

3 element is the 4-noe interface element evelope by Gooman et al. [12]. Each such bon element is assume to have an infinitesimally small thickness. It has two pairs of uplicate noes. The two noes in each pair of uplicate noes have the same coorinates but ifferent egrees of freeom. Between them, one is connecte to the steel reinforcement while the other is connecte to the concrete. The ifference in isplacement of the uplicate noes in the irection of the steel-concrete interface is taken as the slip. Herein, a numerical metho of incorporating the owel stiffness of iscrete reinforcing bar into the concrete element stiffness matrix is propose. Main bar Stirrups Aggregate interlock at crack surface Tensile force Compressive force Shear force resiste by uncracke part of section Tensile force of stirrup Dowel force Reaction Stirrup istance FIGURE 1. Internal forces in cracke beam MODELLING OF CONCRETE, STEEL REINFORCEMENT AND BOND To account for the biaxial behaviour of the concrete, the biaxial stress-strain relation is escribe in terms of equivalent uniaxial strains, an the tensile an compressive strengths in the principal irections are etermine using the biaxial strength envelope evelope by Kupfer an Gerstle [13]. For any principal irection uner tension, the stress-strain curve propose by Guo an Zhang [14] is aopte. For any principal irection uner compression, the stress-strain curve propose by Saenz [15] is aopte. From the stress-strain curves, the secant stiffness values in the two principal irections E c1 an E c2 are evaluate an use to erive the constitutive matrix of concrete [ ] c ' D in the local coorinate system [16]: Ec1 v2ec1 0 1 [ ] ( ) Dc ' = v1ec2 Ec2 0 (1) 1 v1v v1v2 G where v 1 an v 2 are the Poisson s ratios in two principal irections an G is the shear moulus. Before cracking, the shear moulus is taken as the initial elastic shear moulus G o. After cracking, the shear moulus is taken as γ G o, in which γ is the shear retention factor to account for the aggregate interlock effect. The elastic, plastic an strain harening behaviour of the steel reinforcement is moelle by an appropriate stressstrain curve, base on which the secant stiffness of the steel reinforcement E s is evaluate an the constitutive matrix of steel [ D s '] in the local coorinate system is erive as: Es 0 0 [ D s '] = (2) For the bon between the steel reinforcement an concrete, the bon stress-slip relation recommene by Moel Coe 1990 [17] is employe. The secant bon stiffness is evaluate as the bon stress to bon slip ratio an the stiffness matrix of bon element in the local coorinate system is erive with the area of interface taken as the length of bon element times the total perimeter of steel reinforcement. From the above erive stiffness matrices in the local coorinate system, the corresponing stiffness matrices in the global coorinate system are obtaine by coorinate transformation. 702

4 DOWEL FORCE-DISPLACEMENT RELATIONSHIP Let the owel force be enote by V (kn) an the owel isplacement be enote by Δ (mm). The relationship between V an Δ has been erive from experimental results [2,3,6]. Herein, the linearly elastic-perfectly plastic owel force-isplacement relation erive by He an Kwan [9] is aopte. Mathematically, it is given by: V = k 0 Δ for Δ Δ o (3a) V = V o for Δ > Δ o (3b) where k 0 is the initial owel stiffness, V o is the peak owel force (or owel strength), an Δ o is the owel isplacement at peak owel force, Δ o = V o /k 0. Δ V Dowel bar Dowel bar Δ /2 Concrete Δ /2 Concrete Δ /2 Winkler spring xˆ = 0 Fixe be xˆ Crack (a) Contraflexural eformation of owel bar against elastic founation (b) Moelling of elastic founation by Winkler springs FIGURE 2. Visualisation of owel action as beam on elastic founation The owel stiffness k (kn/mm) is establishe base on the beam-on-elastic-founation theory with the reinforcing bar treate as a beam to eal with the interaction between the reinforcing bar an the surrouning concrete. Accoring to the beam-on-elastic-founation theory, the founation may be treate as a be of Winkler springs so that the reaction force from the founation at any point may be assume to be proportional to the eflection of the beam at that point. Cutting the reinforcing bar subjecte to owel action at the point of contraflexure, the bar may be treate as a semi-infinite beam resting on the founation subjecte to concentrate loa at one en, as shown in Figure 2. From the analytical solutions of the beam-on-elastic-founation problem [18], the relationship between the eflection of owel bar at any point Δ an the concentrate owel force can be erive as ( λ xˆ ) cos( λ xˆ ) V Δ xˆ = exp f f (4) 3 Es0I sλ f where xˆ is the istance of the point being consiere from the owel force, E s0 is the initial elastic moulus of steel, I s is the moment of inertia of the bar (for a reinforcing bar with iameter φ s, I s is equal to πφ s 64 ), λ f is a parameter (mm -1 ) representing the relative stiffness of the founation (i.e. the surrouning concrete), as given by: f k f φs 4 4Es0I s xˆ λ = (5) In Equation (5), k f (MPa/mm) is the founation moulus of the surrouning concrete. The following ata-fitting expression for k f propose by Soroushian et al. [19] is use: 127c f fc k f = (6) 2 3 φs where c f is a coefficient ranging from 0.6 for a clear bar spacing of 25 mm to 1.0 for larger bar spacing, an f c is the uniaxial compressive strength of concrete. Equations (4) an (5) represent classical solution for the problem of beam-on-elastic-founation, where flexural but not shear eformation of the beam is consiere. For the case of steel bars embee in concrete, shear eformation of bars is sufficiently small to be neglecte. Substitute x ˆ = 0 into Equation (4), the relationship between the owel force an the owel isplacement uner elastic conition is obtaine as 3 V = E 0I λ Δ (7) s s f 4 703

5 from which the initial owel stiffness may be erive as 3 k0 Es0I sλ f = (8) The peak owel force V o is evaluate using the expression by Vintzeleou an Tassios [4]: Vo = 2.3 s 1 φ (9) fc f y where f y is the yiel strength of owel bar. With Equations (3), (8) an (9), the owel force-isplacement relationship is well-efine. To formulate the owel stiffness matrices, the secant owel stiffness k is evaluate as per Equation (8) if the owel action is linearly elastic, an is evaluate as V o /Δ if the owel action becomes plastic. MODELLING OF DOWEL ACTION Herein, it is propose to incorporate the owel stiffness of the steel bar elements into the ajoining concrete elements so that the steel elements o not nee to have rotational egrees of freeom. This is one by ientifying the concrete elements ajoining the steel element (for each steel element, there shall be two ajoining concrete elements), an then superimposing the owel stiffness matrix [ ] i K (i = 1, 2) onto the stiffness matrices of these concrete elements. The owel stiffness matrix is evise base on the energy principle, as epicte hereuner. The work one U to cause the owel eformation may be represente as: U = 0. 5Δ k Δ (10) ( ) The energy associate with the work one is istribute over the two concrete elements ajoining each steel element on a pro-rata area basis. Let α i (i = 1, 2) be the istribution coefficient such that for the two concrete elements whose areas are A 1 an 2 = A1 A1 imparte in each concrete element is expresse as: U =.5α A, α ( ) an α = A ( A ) 1 + A2 i ( k Δ ) Denoting the strain matrix of the ajoining concrete element by [ ] i T T 2 [ K i ] = [ B] i [ TΔ ] ( i k s )[ TΔ ][ B] i is the length of steel bar element an [ ] where s A2. The strain energy U i (i = 1, 2) 0 i Δ (11) B (i = 1, 2), the owel stiffness matrix is given by: α (12) T is the transformation matrix of owel isplacement, efine by: Δ 2 2 c s cs 2 2 [ T Δ ] = s c cs (13) 2 2 2cs 2cs c s with c an s enoting respectively the cosine an sine of the angle between axial irection of reinforcement an global x-irection. It shoul be note from Equation (12) that the concrete element shape function has been integrate into the owel stiffness matrix. Hence, the owel stiffness of reinforcing bars can be irectly assemble into the concrete element stiffness matrix. APPLICATIONS TO THE ANALYSIS OF DEEP BEAMS The reinforce concrete eep beams analyse inclue the Specimens NNN-1 an NHN-1 teste by Xie et al. [20]. These two eep beams have been analyse by He [10] base on the smeare reinforcement an smeare crack approaches. A sketch of the beams is epicte in Figure 3. The eep beam specimens belonge to the same test series an they have similar configurations. The two eep beam specimens have uniform cross-section of mm breath by mm epth an the effective epth is mm. They are subjecte to single point loas at mi-span position. The shear span to effective epth ratio is fixe at 1.0. NNN-1 is cast of normal-strength concrete (f c = 44.6 MPa); whereas NHN-1 is cast of high-strength concrete (f c = 98.6 MPa). The beams are singly reinforce with tension steel ratio of 1.8%. Yiel stress of reinforcement is MPa. The elastic moulus, strain at start of strain harening, ultimate tensile strength an ultimate strain of reinforcing steel are taken as 200 GPa, 1%, 740 MPa an 12%, respectively. 704

6 φ Elevation Cross-section FIGURE 3. Geometric imensions of eep beam specimens Loa (kn) Mi-span eflection (mm) Experiment Analysis by He [10] Analysis (with D.A.) Analysis (with no D.A.) Loa (kn) Mi-span eflection (mm) Experiment Analysis by He [10] Analysis (with D.A.) Analysis (with no D.A.) (a) Specimen NNN-1 (b) Specimen NHN-1 FIGURE 4. Loa-eflection curves of eep beam specimens (D.A. enotes owel action) Base on the propose numerical moel for owel action of iscrete bars, the eep beams are analyse with owel action accounte for. The loa-eflection curves of eep beam Specimens NNN-1 an NHN-1 are presente in Figure 4. In the same figure, the analytical curves by He [10] are presente. The eep beams are re-analyse with the exclusion of owel action, an the resulting loa-eflection curves are epicte in Figure 4. It can be observe that when the owel action is neglecte, the compute peak loa is slightly lower, an the shear uctility of beams is impaire. Moreover, the eviations from the experimental curves are aggravate. It can be conclue that in the numerical analysis of shear critical beams, the incorporation of owel action in the structural moelling is certainly esirable. CONCLUSIONS In the finite element analysis of concrete structures, iscrete representation of reinforcing bars is a more realistic way to reflect the interaction between concrete an reinforcement as compare to smeare representation. A new strategy for moelling the owel action of iscrete reinforcing bars has been evise in this stuy. Base on the energy principle an the beam-on-elastic-founation theory, the owel stiffness matrix an the owel forceisplacement relationship have been formulate. Since the owel stiffness can be incorporate irectly into the concrete elements, no special type of finite element for owel action is require. The propose owel action moel has been applie to analyse reinforce concrete eep beams. Numerical results have verifie that the propose owel action moel coul improve preiction of the full range loa-eflection behaviour of shear critical concrete members. The results have also correctly reflecte the enhancement in shear uctility of eep beams by incorporation of owel action, which constitutes one of the major components of shear resistance in post-peak regime. 705

7 REFERENCES 1. R. Park an T. Paulay, Reinforce Concrete Structures, New York: John Wiley & Sons, 1975, 769pp. 2. H. Dulacska, Dowel action of reinforcement crossing cracks in concrete, ACI Journal 69(12), (1972). 3. S. G. Millar an R. P. Johnson, Shear transfer across cracks in reinforce concrete ue to aggregate interlock an owel action, Magazine of Concrete Research 36(126), 9-21 (1984). 4. E. N. Vintzeleou an T. P. Tassios, Behavior of owels uner cyclic eformations, ACI Structural Journal 84(1), (1987). 5. S. Dei Poli, M. Di Prisco an P. G. Gambarova, Shear response, eformations, an subgrae stiffness of a owel bar embee in concrete, ACI Structural Journal 89(6), (1992). 6. S. Dei Poli, M. Di Prisco an P. G. Gambarova, Cover an stirrup effects on the shear response of owel bar embee in concrete, ACI Structural Journal 90(4), (1993). 7. S. S. Mannava, T. D. Bush an A. R. Kukreti, Loa-eflection behavior of smooth owels, ACI Structural Journal 96(6), (1999). 8. ASCE Task Committee on Finite Element Analysis of Reinforce Concrete Structures, State-of-the-Art Report on Finite Element Analysis of Reinforce Concrete Structures, American Society of Civil Engineers, New York, 1982, 545pp. 9. X. G. He an A. K. H. Kwan, Moeling owel action of reinforcement bars for finite element analysis of concrete structures, Computers an Structures 79(6), (2001). 10. X. G. He, Constitutive Moeling of Reinforce Concrete for Nonlinear Finite Element Analysis, Ph.D. Thesis, The University of Hong Kong, 1999, 217pp. 11. Z. Z. Zhao, A. K. H. Kwan an X. G. He, Nonlinear finite element analysis of eep reinforce concrete coupling beams, Engineering Structures 26(1), (2004). 12. R. E. Gooman, R. L. Taylor an T. L. Brekke, A moel for the mechanics of jointe rock, Journal of the Soil Mechanics an Founations Division, ASCE 94(3), (1968). 13. H. B. Kupfer an K. H. Gerstle, Behavior of concrete uner biaxial stresses, Journal of the Engineering Mechanics Division, ASCE 99(4), (1973). 14. Z. H. Guo an X. Q. Zhang, Investigation of complete stress-eformation curves for concrete in tension, ACI Materials Journal 84(4), (1987). 15. L. P. Saenz, Discussion of the paper Equation for the stress-strain curve of concrete by Prakash Desayi an S. Krishman, ACI Journal 61(9), (1964). 16. D. Darwin an D. A. Pecknol, Nonlinear biaxial stress-strain law for concrete, Journal of the Engineering Mechanics Division, ASCE 103(2), (1977). 17. Comite Euro-International u Beton, CEB-FIP Moel Coe 1990: Moel Coe for Concrete Structures, Lonon: Thomas Telfor, 1993, 437pp. 18. M. I. Hetényi, Beams on Elastic Founation: Theory with Applications in the Fiels of Civil an Mechanical Engineering, Ann Arbor: The University of Michigan Press, 1958, 256pp. 19. P. Soroushian, K. Obaseki an M. C. Rojas, Bearing strength an stiffness of concrete uner reinforcing bars, ACI Materials Journal 84(3), (1987). 20. Y. L. Xie, S. H. Ahma, T. J. Yu, S. Hino an W. Chung, Shear uctility of reinforce concrete beams of normal an highstrength concrete, ACI Structural Journal 91(2), (1994). 706

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OF CHS. associated. indicate. the need. Rio de Janeiro, Brazil. a) Footbridge Rio. d) Maria Lenk. CHS K joints EUROSTEEL 2, August 3 September 2, 2, Buapest, Hungary A NUMERICAL EVALUATION OF CHS T JOINTS UNDER AXIAL LOADS Raphael S. a Silva a, Luciano R. O. e Lima b, Pero C. G. a S. Vellasco b, José G. S. a Silva