An investigation of interfacial behaviour in steel concrete composite system by cohesive zone method

AN INVESTIGATION OF INTERFACIAL BEHAVIOUR IN STEEL-CONCRETE COMPOSITE SYSTEM BY COHESIVE ZONE METHOD Wang Tongyun (B.Eng. CJU, M.Eng. NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL & ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2013 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. _________________ Wang Tongyun 18 January 2013 Acknowledgements Research outcomes, especially for engineering topics, carry genes of both group cooperation and personal endeavour. The birth of this thesis is not possible with the support of following people, whom I would like to extend my appreciation to. Foremost, words could not express my gratitude toward my supervisor, Prof. J. Y. Richard Liew. He encouraged me to pursue PhD study, a never easy but fulfilling and rewarding path. Countless discussion, suggestion, encouragement, support and patience from Prof. Liew have shaped this thesis. Moreover, Prof. Liew has mentored me to think creatively, to design practically and to communicate effectively. I am still trying to meet these standards and believe these will shape my way ahead. Managers and technicians in structural lab of NUS have helped to coordinate, setup and instrument many rounds of tests. They include Lim Huay Bak, Stanley Wong, Ang Beng Ong, Koh Yian Kheng and Annie Tan. Along my PhD study in NUS, friendship was rendered by fellow researchers in Prof. Liew's steel and composite group. K.W. Kang, M.X. Xiong, X. Yu, K.M.A. Sohel, S.C. Lee, D.X. Xiong, X.Y. Gao, J.B. Yan and many more have made my life full of support and laughter. Discussions on wide engineering topics may arise from office, library, laboratory, way to lecture halls and canteens. I have benefited from discussions within this group of friends significantly. I would also like to thank the discussion with Prof. Y.S. Choo, Prof. Andrew Palmer and Prof. Peter Marshall on bond enhancement between steel and concrete. Two final year undergraduate students, M.A.M. Abubucker and K.W.T. Ma have worked with me on experimental investigation. Their assistances in specimen preparation and testing are greatly appreciated. i Lastly but dearest to my heart, this thesis is impossible without love and support from my parents, my wife, and my sons. Their love and encouragement have always brought hope and strength whenever there is a barrier to be overcome. I would like to dedicate this thesis to my family. ii Table of Contents Acknowledgements i Table of Contents .iii Summary ix List of Tables xii List of Figures xiii NOMENCLATURE . xxii Chapter Introduction 1.1 Background 1.1.1 Conventional steel-concrete composite slab and beam 1.1.2 Sandwich structures . 1.2 Significance of steel concrete interface behaviour . 1.3 Objectives and research scopes 11 1.4 Layout of thesis 12 Chapter Literature Review . 15 2.1 Existing design methods for steel concrete composite structures 15 2.2 Full composite method . 16 2.2.1 Material Properties . 18 2.2.2 Partial factor design . 20 2.2.3 Effective modular ratio 21 2.2.4 Member Buckling Effects 24 2.2.5 Determine concrete stress profile . 27 2.2.6 Combined flexural and axial loads . 32 2.2.7 Design of shear connectors 34 2.2.8 Plate buckling Factor . 39 2.2.9 Check of design stresses 41 2.3 Partial composite and its implications 45 iii 2.4 Existing analytical and numerical modeling of deformable mechanical shear connection . 47 2.4.1 Direct 3D solid model for shear connectors . 48 2.4.2 Explicit spring elements to model shear connector or interaction . 53 2.5 Analytical model for bonded joint . 56 2.6 Bonded steel-concrete composite beam . 58 2.6.1 Modeling of bonded joint with epoxy bulk material 59 2.6.2 Modeling of bonded joint with cohesive element 60 2.7 Slip capacity of composite section . 60 2.8 Closing remarks on existing models 61 Chapter Cohesive Zone Method for Steel Concrete Composites 65 3.1 Introduction to CZM 66 3.2 Damage and failure mechanism of CZM . 71 3.2.1 Triangular traction separation 72 3.2.2 Tri-linear and trapezoidal traction separation 74 3.2.3 Exponential traction separation 76 3.2.4 Polynomial traction separation laws 80 3.3 Determination of cohesive element stiffness . 83 3.4 Mode interaction for combined tensile and shear traction . 86 3.4.1 Effective opening for mode interaction . 87 3.4.2 Direct modeling of model interaction 90 3.4.3 Damage initiation and evolution based on displacement or energy . 96 3.5 Modeling of compressive normal traction . 100 3.6 Effective damage model with tri-linear law . 103 3.6.1 Effective damage parameter . 103 3.6.2 Damage model for tri-linear traction separation law . 106 3.6.3 Effects of mode interaction parameter β 114 iv 3.6.4 Effects of different traction separation shapes interaction . 115 3.7 Concluding Remarks 122 Chapter Finite Element Formulation of CZM Based On Effective Damage Model125 4.1 Numerical algorithm flow chart . 126 4.2 General finite element formulation of CZM 128 4.3 Formulation of tangent stiffness matrix for CZM based on effective damage130 4.4 Local numerical instability related to damage rule 132 4.5 Choice of tri-linear shape . 135 4.6 Validation of the proposed effective damage model 137 4.6.1 Validation with double cantilever beam test 137 4.6.2 Validation with mixed mode bending test . 143 4.7 Concluding Remarks 152 Chapter Experimental Investigation of Epoxy Aided Bond Strength between Steel and Concrete 153 5.1 Introduction of epoxy aided bond 153 5.2 Experimental investigation of bond between steel and fresh concrete 155 5.2.1 Test setup of push-out tests 156 5.2.2 Test specimens of bond between steel and fresh concrete . 157 5.2.3 Test results and discussion . 160 5.3 Experimental investigation of fiber reinforcement effects . 169 5.3.1 Comparison of fiber steel fiber and PVA fiber 169 5.3.2 Comparison of fiber volume fraction and curing timing . 173 5.4 Summary of failure modes at bonded steel-concrete interface 175 5.5 Characterization of steel-concrete interface with CZM . 178 5.5.1 Numerical model of push-out test and mesh density study . 179 5.5.2 Characterization of steel-fresh concrete interface with fiber reinforced epoxy 181 5.6 Concluding remarks . 183 v Chapter Numerical modeling of steel concrete interface behavior in composite structures 185 6.1 Application of CZM to epoxy bonded steel and concrete interface . 186 6.1.1 Testing scheme and results 186 6.1.2 Simplification of 3D problem into 2D planar problem 189 6.1.3 Material model . 190 6.1.4 Numerical modelling of bonded composite beam . 195 6.1.5 Numerical results and discussion . 197 6.1.6 Numerical simulation of bonded composite beam subjected to uniformly distributed load 200 6.1.7 Effect of bond strength for bonded composite beam . 202 6.2 Numerical study of bonded SCS sandwich beam 209 6.2.1 Effects of bond strength . 209 6.2.2 Numerical model for SCS6-100-6 . 214 6.2.3 Effects of fiber reinforcement at interface . 218 6.2.4 Effects of interfacial friction coefficient 219 6.3 Application of CZM to model shear connectors 222 6.3.1 Choice of proper traction separation law . 223 6.3.2 Characterization of mechanical shear connector by CZM . 225 6.3.3 Numerical simulation of load sharing mechanism . 229 6.4 Numerical model of mechanically connected composite beam . 233 6.5 Numerical modeling of mechanically connected SCS sandwich beam . 239 6.5.1 Numerical model of three point loaded SCS sandwich beam 240 6.5.2 Numerical results and discussion for full composite sandwich beam . 241 6.5.3 Numerical results and discussion for partial composite sandwich beams245 vi 6.6 Proposal of framework to determine and implement traction separation law250 6.7 Concluding Remarks 254 References El-Lobody, E. and D. Lam (2002). "Modelling of headed stud in steel-precast composite beams." Steel & Composite Structures 2(5): 355-378. Ellobody, E. and B. Young (2006). "Performance of shear connection in composite beams with profiled steel sheeting." Journal of Constructional Steel Research 62(7): 682-694. Fabbrocino, G., G. Manfredi, M. Pecce and E. Cosenza (1998). Non-linear behaviour of composite beams under negative bending: a theoretical-experimental comparison. Porceedings of the III Italian Workshop on Compsoite Structures. Feih, S. (2005). Development of a user element in ABAQUS for modelling of cohesive laws in composite structures. Foundoukos, N. and J. C. Chapman (2008). "Finite element analysis of steel-concretesteel sandwich beams." Journal of Constructional Steel Research 64(9): 947961. Franklin, S. E. and A. de Kraker (2003). "Investigation of counterface surface topography effects on the wear and transfer behaviour of a POM-20% PTFE composite." Wear 255: 766-773. Gattesco, N. (1999). "Analytical modeling of nonlinear behavior of composite beams with deformable connection." Journal of Constructional Steel Research 52(2): 195-218. Gattesco, N. and E. Giuriani (1996). "Experimental study on stud shear connectors subjected to cyclic loading." Journal of Constructional Steel Research 38(1): 1-21. Hutchinson, J. W. and Z. Suo (1992). "Mixed-Mode Cracking in Layered Materials." Advances in Applied Mechanics, Vol 29 29: 63-191. J.M., A. and L. A.G. (1982). "Mode`le calcul e`lasto-plastique de pustres mixtes a connexion partielle." Construct Metallique(4): 3-51. Jayas, B. S. and M. U. Hosain (1987). Behaviour of headed studs in composite beams: push-out tests. Proceedings of the Institute of Civil Engineers. 15: 240-253. Johnson, R. P. and I. N. Molenstra (1991). "Partial Shear Connection in Composite Beams for Buildings." Proceedings of the Institution of Civil Engineers Part 2Research and Theory 91: 679-704. Jurkiewiez, B., C. Meaud and L. Michel (2011). "Non linear behaviour of steelconcrete epoxy bonded composite beams." Journal of Constructional Steel Research 67(3): 389-397. Kalfas, C., P. Pavlidis and E. Galoussis (1997). "Inelastic behaviour of shear connection by a method based on FEM." Journal of Constructional Steel Research 44(1-2): 107-114. Kennedy, S. J. (2004). Composite steel structural plastic sandwich plates systems. U.S. 273 References Kim, W. C. and C. K. H. Dharan (1992). "Facesheet Debonding Criteria for Composite Sandwich Panels under Inplane Compression." Engineering Fracture Mechanics 42(4): 643-652. Knox, E. M., M. J. Cowling and I. E. Winkle (1998). "Adhesively bonded steel corrugated core sandwich construction for marine applications." Marine Structures 11: 185-204. Kujala, P. (1998). "Preliminary assessment of sandwich plates subject." Journal of Structural Mechanics 31(1-2): 32-45. Lam, D. and E. El-Lobody (2005). "Behavior of headed stud shear connectors in composite beam." Journal of Structural Engineering-Asce 131(1): 96-107. Lee, S. C., K. M. A. Sohel and J. Y. R. Liew (2008). Numerical Simulations of UltraLightweight Steel-Concrete-Steel Sandwich Composite Panels Subjected to Impact. the 9th International Conference on Computational Structures Technology. Athens, Greece: Paper No. 297. Li, A., T. Assih and Y. Delmas (2000). "Influence of the adhesive thickness and steel plate thickness on the behaviour of strengthened concrete beams." Journal of Adhesion Science and Technology 14(13): 1639-1656. Liew, J. Y. R. and K. M. A. Sohel (2009). "Lightweight steel-concrete-steel sandwich system with J-hook connectors." Engineering Structures 31(5): 1166-1178. Liew, J. Y. R., T. Y. Wang and K. M. A. Sohel (2009). COMPOSITE PANEL ASSEMBLIES INCLUDING SEPARATION PREVENTION SHEAR CONNECTORS, National University of Singapore, Maritime and Port Authority of Singapore. Liew, J. Y. R., T. Y. Wang, K. M. A. Sohel, X. X. Dai, S. C. Lee and K. S. Chia (2008). Report on "STEEL-CONCRETE-STEEL SANDWICH SYSTEM FOR MARINE AND OFFSHORE APPLICATIONS": R-264-000-188-490 NUS/MPA/Keppel. Loh, H. Y., B. Uy and M. A. Bradford (2004). "The effects of partial shear connection in the hogging moment regions of composite beams - Part I - Experimental study." Journal of Constructional Steel Research 60(6): 897-919. Loh, H. Y., B. Uy and M. A. Bradford (2004). "The effects of partial shear connection in the hogging moment regions of composite beams - Part II - Analytical study." Journal of Constructional Steel Research 60(6): 921-962. Luo, Y. J., A. Li and Z. Kang (2012). "Parametric study of bonded steel-concrete composite beams by using finite element analysis." Engineering Structures 34: 40-51. Mays, G. and A. R. Hutchinson (2005). Adhesives in civil engineering. Cambridge England ; New York, Cambridge University Press. 274 References Mi, Y., M. A. Crisfield, G. A. O. Davies and H. B. Hellweg (1998). "Progressive delamination using interface elements." Journal of Composite Materials 32(14): 1246-1272. Miklofsky, H. A., M. R. Brpown and M. Gonsior, J. (1962). Epxoy bending compounds as shear connectors in comosite beams. Engineering Research Series, State of New York: Department of Public Works. Narayan, R., F. J. Naji and T. M. Roberts (1994). Design Guide for Steel-ConcreteSteel Sandwich Construction U.K., Steel Construction Institute. Needleman, A. (1987). "A Continuum Model for Void Nucleation by Inclusion Debonding." Journal of Applied Mechanics-Transactions of the Asme 54(3): 525-531. Nethercot, D. A., T. Q. Li and B. S. Choo (1995). "Required Rotations and Moment Redistribution for Composite Frames and Continuous Beams." Journal of Constructional Steel Research 35(2): 121-163. Newmark, N. M., C. P. Siess and I. M. Viest (1951). "Tests and analysis of composite beams with incomplete interaction." Proceedings of Society of experimental Stress Analysis 9(1): 75-92. Oduyemi, T. O. S. and H. D. Wright (1989). "An Experimental Investigation into the Behavior of Double-Skin Sandwich Beams." Journal of Constructional Steel Research 14(3): 197-220. Oehlers, D. J. (1990). "Behavior of Headed Studs in Composite Beams - Push-out Tests - Discussion." Canadian Journal of Civil Engineering 17(3): 497-499. Oehlers, D. J. (1996). "Experimental study on stud shear connectors subjected to cyclic loading - Discussion." Journal of Constructional Steel Research 40(2): 165-166. Oehlers, D. J. and G. Sved (1995). "Composite Beams with Limited-Slip-Capacity Shear Connectors." Journal of Structural Engineering-Asce 121(6): 932-938. Ollgaard, J. G., R. G. Slutter and J. W. Fisher (1971). "Shear Strength of Stud Connectors in Lightweight and Normal-Weight Concrete." Engineering Journal-American Institute of Steel Construction Inc 8(2): 55-&. Park, K. and G. H. Paulino (2012). "Computational implementation of the PPR potential-based cohesive model in ABAQUS: Educational perspective." Engineering Fracture Mechanics 93: 239-262. Park, K., G. H. Paulino and J. R. Roesler (2009). "A unified potential-based cohesive model of mixed-mode fracture." Journal of the Mechanics and Physics of Solids 57(6): 891-908. 275 References Pascault, J.-P. and R. J. J. Williams (2010). Epoxy polymers : new materials and innovations. Weinheim, Wiley-VCH. Ranzi, G., M. A. Bradford and B. Uy (2004). "A direct stiffness analysis of a composite beam with partial interaction." International Journal for Numerical Methods in Engineering 61(5): 657-672. Reeder, J. R. and J. H. Crews (1990). "Mixed-Mode Bending Method for Delamination Testing." Aiaa Journal 28(7): 1270-1276. SCI (1997). Bi-Steel Design & Construction Guide. U. K. , Steel Construction Institute. Smith, S. T. and J. G. Teng (2001). "Interfacial stresses in plated beams." Engineering Structures 23(7): 857-871. Song, S. H., G. H. Paulino and W. G. Buttlar (2006). Influence of the Cohesive Zone Model Shape Parameter on Asphalt Concrete Fracture Behavior. MULTISCALE AND FUNCTIONALLY GRADED MATERIALS 2006. G. H. Paulino, M.-J. Pmdera, R. H. Doddset al. Oahu Island Hawai, U.S.A: 730735. Souici, A., J. F. Berthet, A. Li and N. Rahal (2013). "Behaviour of both mechanically connected and bonded steel-concrete composite beams." Engineering Structures 49: 11-23. Taljsten, B. (1997). "Defining anchor lengths of steel and CFRP plates bonded to concrete." International Journal of Adhesion and Adhesives 17(4): 319-327. Topkaya, C., J. A. Yura and E. B. Williamson (2004). "Composite shear stud strength at early concrete ages." Journal of Structural Engineering-Asce 130(6): 952960. Turon, A., C. G. Davila, P. P. Camanho and J. Costa (2007). "An engineering solution for mesh size effects in the simulation of delamination using cohesive zone models." Engineering Fracture Mechanics 74(10): 1665-1682. Tvergaard, V. (1990). "Effect of Fiber Debonding in a Whisker-Reinforced Metal." Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing 125(2): 203-213. Tvergaard, V. and J. W. Hutchinson (1993). "The Influence of Plasticity on MixedMode Interface Toughness." Journal of the Mechanics and Physics of Solids 41(6): 1119-1135. van den Bosch, M. J., P. J. G. Schreurs and M. G. D. Geers (2006). "An. improved description of the exponential Xu and Needleman cohesive zone law for mixed-mode decohesion." Engineering Fracture Mechanics 73(9): 1220-1234. Volokh, K. Y. (2004). "Comparison between cohesive zone models." Communications in Numerical Methods in Engineering 20(11): 845-856. 276 References Wang, A. J. (2011). "Numerical Investigation into the Structural Behaviour of Shear Connectors in Composite Beams." Progress in Steel Building Structures 13(3): 20-28. Xie, M., N. Foundoukos and J. C. Chapman (2004). "Experimental and numerical investigation on the shear behaviour of friction-welded bar-plate connections embedded in concrete." Journal of Constructional Steel Research 61(5): 625649. Xie, M., N. Foundoukos and J. C. Chapman (2007). "Static tests on steel-concretesteel sandwich beams." Journal of Constructional Steel Research 63(6): 735750. Xu, X. P. and A. Needleman (1993). "Void Nucleation by Inclusion Debonding in a Crystal Matrix." Modelling and Simulation in Materials Science and Engineering 1(2): 111-132. Xu, X. P. and A. Needleman (1996). "Numerical simulations of dynamic crack growth along an interface." International Journal of Fracture 74(4): 289-324. Xue, Z. Y. and J. W. Hutchinson (2003). "Preliminary assessment of sandwich plates subject to blast loads." International Journal of Mechanical Sciences 45(4): 687-705. Yan, J. B. (2012). Ultimate strength behavior of steel-concrete-steel sandwich composite beams and shells. Ph.D., National University of Singapore. Yang, Q. D. and M. D. Thouless (2001). "Mixed-mode fracture analyses of plastically-deforming adhesive joints." International Journal of Fracture 110(2): 175-187. Zhao, G. Z. and A. Li (2008). "Numerical study of a bonded steel and concrete composite beam." Computers & Structures 86(19-20): 1830-1838. Zou, Z., S. R. Reid, S. Li and P. D. Soden (2002). "Modelling interlaminar and intralaminar damage in filament-wound pipes under quasi-static indentation." Journal of Composite Materials 36(4): 477-499. 277 References 278 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS In this appendix, a user element (UEL) developed employing cohesive element formulation with tri-linear traction separation law is presented. Linear Gauss quadrature is used for numerical simulation. For the nodes cohesive element, two Gauss points are used at reference level with position and weights given below: = p1 1/= 0.577350269= w1 p2 = −1/ = −0.577350269 w2 = The UEL requires 13 properties defined by user and are given as follow: σ =PROPS(1) τ =PROPS(2) σ =PROPS(3) τ =PROPS(4) δ 0,n =PROPS(5) δ 0,t =PROPS(6) δ1,n =PROPS(7) δ1,t =PROPS(8) δ f ,n =PROPS(9) δ f ,t =PROPS(10) β =PROPS(11) telement =PROPS(12) K n , penalty =PROPS(13) Most of the parameters are as shown in Figure 3.6. In order to approximate the interfacial behaviours such as push-out tests results of shear connectors, these parameters need to be defined within the ABAQUS input file to define various shapes. 279 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS C*************************************************************************** C C Two dimensional user element for tri-linear traction separation law C C Effective Damage for mode mix C C Developed by Wang Tongyun at CEE National University of Singapore C C Supervisor: Prof. J.Y. Richard Liew C C All Rights Reserved. C C*************************************************************************** SUBROUTINE UEL(RHS,AMATRX,SVARS,ENERGY,NDOFEL,NRHS,NSVARS, PROPS,NPROPS,COORDS,MCRD,NNODE,U,DU,V,A,JTYPE,TIME,DTIME, KSTEP,KINC,JELEM,PARAMS,NDLOAD,JDLTYP,ADLMAG,PREDEF,NPREDF, LFLAGS,MLVARX,DDLMAG,MDLOAD,PNEWDT,JPROPS,NJPROP,PERIOD) INCLUDE 'ABA_PARAM.INC' C DIMENSION RHS(MLVARX,*),AMATRX(NDOFEL,NDOFEL),PROPS(*), SVARS(*),ENERGY(8),COORDS(MCRD,NNODE),U(NDOFEL), DU(MLVARX,*),V(NDOFEL),A(NDOFEL),TIME(2),PARAMS(*), JDLTYP(MDLOAD,*),ADLMAG(MDLOAD,*),DDLMAG(MDLOAD,*), PREDEF(2,NPREDF,NNODE),LFLAGS(*),JPROPS(*) DIMENSION Uloc(ndofel), Sc(ndofel,ndofel), Fc(ndofel,nrhs), & T(mcrd,nrhs), T_d(mcrd,mcrd), R(mcrd, mcrd), & Bc(mcrd,ndofel), Bct(ndofel,mcrd), ShapeN(nnode), & sep(mcrd), GP(2), GP_Weight(2), tmp(ndofel,mcrd) INTEGER nGP DOUBLE PRECISION nTmax, tTmax, n0Delta, t0Delta, n1Delta, & t1Delta, nfDelta, tfDelta, n1Trac, t1Trac, th, nSep, tSep, & dn, dt, m, n, Gam_n, Gam_t, nKpen, nKori, tKori, Deff, & N1, N2, sep1, sep2, sep3, sep4, nSepMax, tSepMax, el_length, & beta, rbeta, thick, tDamage, nDamage, nDD, tDD, DnDt, DnDtB, & thv C Read the properties of tri-linear traction separation rule nTmax = PROPS(1) tTmax = PROPS(2) n1Trac = PROPS(3) t1Trac = PROPS(4) n0Delta = PROPS(5) t0Delta = PROPS(6) n1Delta = PROPS(7) 280 C APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS t1Delta = PROPS(8) nfDelta = PROPS(9) tfDelta = PROPS(10) beta = PROPS(11) th = PROPS(12) nKpen = PROPS(13) nKori = nTmax/n0Delta tKori = tTmax/t0Delta rbeta = 1.0/beta nGP = data GP / 0.577350269189626 , -0.577350269189626 / data GP_Weight / 1.0 , 1.0 / call matZeros (RHS,ndofel,nrhs) call matZeros (AMATRX,ndofel,ndofel) C Change from the global coordinates to the local coordinates call coordsTrans (R, COORDS, el_length, U, ndofel, & nnode, mcrd) i = 0, nnode-1 Uloc(1+i*mcrd) = R(1,1)*U(1+i*mcrd) + R(1,2)*U(2+i*mcrd) Uloc(2+i*mcrd) = R(2,1)*U(1+i*mcrd) + R(2,2)*U(2+i*mcrd) end sep1 = Uloc(7) - Uloc(1) sep2 = Uloc(8) - Uloc(2) sep3 = Uloc(5) - Uloc(3) sep4 = Uloc(6) - Uloc(4) C Numerical integration to compute RHS and AMATRX SVARS(5) = i = 1, nGP N1 = 0.5*(1 - GP(i)) N2 = 0.5*(1 + GP(i)) sep(1) = N1*sep1 + N2*sep3 sep(2) = N1*sep2 + N2*sep4 tSep=abs(sep(1)) nSep=sep(2) if (sep(1) .LT. 0) then sign_t = -1 else 281 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS sign_t = endif tSepMax = SVARS(nGP*(i-1)+1) nSepMax = SVARS(nGP*(i-1)+2) if (nSep .GE. nSepMax) then if(nSep .LE. n0Delta) then nDamage = 0.0 nDD = 0.0 elseif(nSep .LE. n1Delta) then nDamage = (nTmax*n1Delta-n1Trac*n0Delta)/(n1Delta-n0Delta) & /nTmax*(1.0-n0Delta/nSep) nDD = n0Delta*(nTmax*n1Delta-n1Trac*n0Delta)/(n1Delta & -n0Delta)/nTmax/nSep**2.0 elseif(nSep .LE. nfDelta) then nDamage = n1Trac*n0Delta*(nfDelta*(nSep-n1Delta)-nSep* & (nfDelta-n1Delta))/(nTmax*n1Delta*nSep*(nfDelta-n1Delta))+1.0 nDD=n1Trac*n0Delta*nfDelta/(nTmax*(nfDelta-n1Delta)*nSep**2.0) else nDamage = 1.0 nDD = 0.0 endif else if(nSepMax .LE. n0Delta) then nDamage = 0.0 nDD = 0.0 elseif(nSepMax .LE. n1Delta) then nDamage = (nTmax*n1Delta-n1Trac*n0Delta)/(n1Delta-n0Delta) & /nTmax*(1.0-n0Delta/nSepMax) nDD = n0Delta*(nTmax*n1Delta-n1Trac*n0Delta)/(n1Delta & -n0Delta)/nTmax/nSepMax**2.0 elseif(nSepMax .LE. nfDelta) then nDamage = n1Trac*n0Delta*(nfDelta*(nSepMax-n1Delta)-nSepMax* & (nfDelta-n1Delta))/(nTmax*n1Delta*nSepMax*(nfDelta-n1Delta))+1.0 nDD=n1Trac*n0Delta*nfDelta/(nTmax*(nfDelta-n1Delta)*nSepMax**2.0) else nDamage = 1.0 nDD = 0.0 282 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS endif endif if (tSep .GE. tSepMax) then if(tSep .LE. t0Delta) then tDamage = 0.0 tDD = 0.0 elseif(tSep .LE. t1Delta) then tDamage = (tTmax*t1Delta-t1Trac*t0Delta)/(t1Delta-t0Delta) & /tTmax*(1.0-t0Delta/tSep) tDD = t0Delta*(tTmax*t1Delta-t1Trac*t0Delta)/(t1Delta & -t0Delta)/tTmax/tSep**2.0*sign_t elseif(tSep .LE. tfDelta) then tDamage = t1Trac*t0Delta*(tfDelta*(tSep-t1Delta)-tSep* & (tfDelta-t1Delta))/(tTmax*t1Delta*tSep*(tfDelta-t1Delta))+1.0 tDD=t1Trac*t0Delta*tfDelta/(tTmax*(tfDelta-t1Delta) & *tSep**2.0)*sign_t else tDamage = 1.0 tDD = 0.0 endif else if(tSepMax .LE. t0Delta) then tDamage = 0.0 tDD = 0.0 elseif(tSepMax .LE. t1Delta) then tDamage = (tTmax*t1Delta-t1Trac*t0Delta)/(t1Delta-t0Delta) & /tTmax*(1.0-t0Delta/tSepMax) tDD = t0Delta*(tTmax*t1Delta-t1Trac*t0Delta)/(t1Delta & -t0Delta)/tTmax/tSepMax**2.0*sign_t elseif(tSepMax .LE. tfDelta) then tDamage = t1Trac*t0Delta*(tfDelta*(tSepMax-t1Delta)-tSepMax* & (tfDelta-t1Delta))/(tTmax*t1Delta*tSepMax*(tfDelta-t1Delta))+1.0 tDD = t1Trac*t0Delta*tfDelta/(tTmax*(tfDelta-t1Delta) & *tSepMax**2.0)*sign_t else tDamage = 1.0 tDD = 0.0 283 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS endif endif C Calculation of Effective Damage DnDt = nDamage**beta+tDamage**beta Deff = DnDt**rbeta if (Deff .GT. 1) then Deff=1 elseif (Deff .LT. 0) then WRITE (7,*)'Deff=',Deff,'Something is wrong! Exit Now' CALL XIT endif if (DnDt .EQ. 0) then DnDtB=0 else DnDtB=DnDt**(rbeta-1) endif T(2,1) = nKori*nSep*(1-Deff) T(1,1) = tKori*tSep*(1-Deff)*sign_t T_d(1,1) = tKori*(1-Deff)-tKori*tSep*(tDamage**(beta-1)) & *DnDtB*tDD*sign_t T_d(1,2) = -tKori*tSep*(nDamage**(beta-1))*DnDtB*nDD*sign_t T_d(2,1) = -nKori*nSep*(tDamage**(beta-1))*DnDtB*tDD T_d(2,2) = nKori*(1-Deff)-nKori*nSep*(nDamage**(beta-1)) & *DnDtB*nDD if (nSep .LT. 0) then T_d(2,2) = nKpen*nKori T(2,1) = nKpen*nKori*nSep endif if (Deff .GE. 1) then T_d(1,1)=0 T_d(1,2)=0 T_d(2,1)=0 T_d(2,2)=0 T(1,1) = T(2,1) = endif ShapeN(1) = -N1 284 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS ShapeN(2) = -N2 ShapeN(3) = N2 ShapeN(4) = N1 j = 1, nnode k = 1, mcrd l = 1, mcrd Bc(k,l+(j-1)*mcrd) = ShapeN(j)*R(k,l) end end end CALL matTrans(Bc,Bct,mcrd,ndofel) CALL matDotMult(Bct,T_d,tmp,ndofel,mcrd,mcrd) CALL matDotMult(tmp,Bc,Sc,ndofel,mcrd,ndofel) CALL matDotMult(Bct,T,Fc,ndofel,mcrd,nrhs) thick = 0.5 * el_length * GP_Weight(i) * th CALL matPlus(AMATRX,Sc,thick,ndofel,ndofel) CALL matPlus(RHS,-Fc,thick,ndofel,nrhs) C Update the state variables: SVARS if(tSepMax .LE. abs(sep(1))) then SVARS(nGP*(i-1)+1) = abs(sep(1)) end if if (nSepMax .LE. sep(2)) then SVARS(nGP*(i-1)+2) = sep(2) end if if (Deff .GE. 1) then SVARS(5) = endif end if (SVARS(5).EQ.1) then m=1, ndofel n=1, ndofel AMATRX(m,n)=0 end end endif RETURN END 285 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS c ********************************************************************* c Coordinates Transformation c ********************************************************************* SUBROUTINE coordsTrans (R, COORDS, el_length, U, ndofel, & nnode, mcrd) INCLUDE 'ABA_PARAM.INC' DIMENSION R(mcrd,mcrd), COORDS(mcrd,nnode), U(ndofel) DIMENSION dCoord(mcrd,nnode), dCoordR(2,2) i = 1, mcrd j = 1, nnode dCoord(i,j) = COORDS(i,j) + U(2*(j-1)+i) end end do i = 1, dCoordR(i,1) = (dCoord(i,1)+dCoord(i,4))*0.5 dCoordR(i,2) = (dCoord(i,2)+dCoord(i,3))*0.5 end c Calculate the directional cosine & the rotation matrix R (THETA in thesis) d_x = dCoordR(1,2) - dCoordR(1,1) d_y = dCoordR(2,2) - dCoordR(2,1) el_length = (d_x**2.0 + d_y**2.0)**0.5 cos_a = d_x / el_length sin_a = d_y / el_length R(1,1) = cos_a R(1,2) = sin_a R(2,1) = -sin_a R(2,2) = cos_a RETURN END C ********************************************************************** C Subroutines for matrix operation C ********************************************************************** SUBROUTINE matZeros (A,n,m) INCLUDE 'ABA_PARAM.INC' DIMENSION A(n,m) i = 1, n j = 1, m 286 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS A(i,j) = 0.0 end end RETURN END C SUBROUTINE matTrans (A,B,n,m) INCLUDE 'ABA_PARAM.INC' DIMENSION A(n,m), B(m,n) CALL matZeros (B,m,n) i = 1, n j = 1, m B(j,i) = A(i,j) end end RETURN END C SUBROUTINE matPlus (A,B,c,n,m) INCLUDE 'ABA_PARAM.INC' DIMENSION A(n,m), B(n,m) i = 1, n j = 1, m A(i,j) = A(i,j) + c*B(i,j) end end RETURN END C SUBROUTINE matDotMult (A,B,C,l,n,m) INCLUDE 'ABA_PARAM.INC' DIMENSION A(l,n), B(n,m), C(l,m) CALL matZeros (C,l,m) i = 1, l j = 1, m k = 1, n C(i,j) = C(i,j) + A(i,k) * B (k,j) 287 APPENDIX A. Source code of tri-linear traction separation law user element for ABAQUS end end end RETURN END C ******************************************************************** C 288 [...]... utilize steel and concrete materials The key to the development of a novel steel- concrete composite system is to ensure an effective load transferring mechanism at steel- concrete interface This is achieved by either discrete interfacial connection such as mechanical connectors; or continual interfacial connection, e.g structural adhesive; or a combination of preceding two Diverse interfacial failure mechanisms... Experimental investigation of bond performance between steel and fresh concrete aided by epoxy 260 7.1.7 Effect of fiber reinforcement in epoxy bond line 261 7.1.8 Proposed numerical model for steel- concrete composite structure 262 7.1.9 Numerical modeling and parametric study of bonded composite and sandwich beam 263 7.1.10 Numerical study of mechanical connected composite and sandwich... xxiii Chapter 1 Introduction 1.1 Background Steel and concrete are two most widely adopted building materials in the construction industry Composite structural design optimally utilizes the material properties of both steel and concrete when properly designed For certain applications of composite structures such as marine and offshore applications, steel- concrete -steel sandwich composite system provides... proposed to contain damage at interfacial layer Fictitious cohesive zone is employed to eliminate definition of contact Advanced finite element analyses have been carried out using CZM to study various steel- concrete composite systems The proposed numerical model achieves good agreement with testing results including: push-out test involving a hybrid Expamet-hook connector system; epoxy bonded composite beam... Mechanical joint, laser welding and adhesive bond are three major options The following sections summarize existing technologies in sandwich constructions Figure 1.4 Honeycomb sandwich panel In construction industry, light weight sandwich panels are generally not serving as load carrying member Insulation is the main function until the introduction of double skin composite (Oduyemi et al 1989), Bi -Steel. .. formwork for cast in- situ concrete as shown in Figure 1.2 (a) Precast concrete slab can also be joined to steel section as shown in Figure 1.2 (b) Composite slabs and beams are suitable for commercial and industrial buildings require long span and speed of construction In addition, composite slab also provide improved fire resistance to steel skeleton Composite beam is also an important structural form for... resistance Several research projects have been carried out on novel composite deck and sandwich composite systems In this section, the background of current research is discussed 1.1.1 Conventional steel- concrete composite slab and beam Research on steel- concrete composite construction originated from Canada since 1920s aiming to optimally utilize expensive steel section and cheap but tensile-weak concrete. .. shear strength and tensile strength making both face plates working compositely Steel Face Plate Concrete Core Concrete Core Shear Connector Steel Face Plate Figure 1.8 SCS sandwich system developed in NUS (Liew et al 2009) 7 Chapter 1 Introduction 1.2 Significance of steel concrete interface behaviour In order to fully utilize the material strength and achieve desired composite strength and ductility,... stress corresponding to δ1 for tri-linear model Compressive stress in steel Tensile stress in steel Compressive stress in concrete Critical buckling stress Separation in the direction of traction; In DCB testing, opening of specimen crack at load point Crack initiation separation Characteristic separation corresponding to σ 1 and τ 1 (different in 2 modes) Characteristic separation corresponding to full... 3.21 Mode mix of PPR model in terms of normal traction Tn considering interfacial compression 102 Figure 3.22 Mode mix of PPR model in terms of potential energy Φ considering interfacial compression 103 Figure 3.23 Modes interaction defined by β 105 Figure 3.24 Different typical unloading and reloading behaviors for adhesive bond (Alfano et al 2009) and mechanical shear . AN INVESTIGATION OF INTERFACIAL BEHAVIOUR IN STEEL-CONCRETE COMPOSITE SYSTEM BY COHESIVE ZONE METHOD Wang Tongyun (B.Eng. CJU, M.Eng. NUS) A THESIS SUBMITTED FOR THE DEGREE OF. coordinate, setup and instrument many rounds of tests. They include Lim Huay Bak, Stanley Wong, Ang Beng Ong, Koh Yian Kheng and Annie Tan. Along my PhD study in NUS, friendship was rendered by fellow. Numerical modeling and parametric study of bonded composite and sandwich beam 263 7.1.10 Numerical study of mechanical connected composite and sandwich beams264 7.1.11 Proposal of standard framework