Structural response and cracking behaviour of fibre reinforced concrete beams with hybrid flexural reinforcement

Mahsa Taheri February, 2021 UMinho | 2021 Structural Response and Cracking Behaviour of Fibre Reinforced Concrete Beams with Hybrid Flexural Reinforcement Universidade do Minho Escola de Engenharia Mahsa Taheri Structural Response and Cracking Behaviour of Fibre Reinforced Concrete Beams with Hybrid Flexural Reinforcement February, 2021 Dissertation Presented in Partial Fulfilment of the Requirement for the Degree of Doctor of Philosophy of Civil Engineering Work unde the Supervision of the Professor Joaquim António Oliveira de Barros Mahsa Taheri Structural Response and Cracking Behaviour of Fibre Reinforced Concrete Beams with Hybrid Flexural Reinforcement Universidade do Minho Escola de Engenharia i ii Acknowledgements This dissertation presents the scientific research carried out at the Civil Engineering Department of the University of Minho, Guimarães, Portugal, as partial fulfilment of the requirement for the degree of doctor of philosophy of Civil Engineering. I would like to express my sincere gratitude to my supervisor Prof. Joaquim Barros, for providing this research opportunity and for his obligingness and cordial support, encouragement and advice throughout the research. My gratitude is due to the Professors and Staff of the Civil Engineering Department of the University of Minho and the Institute for Sustainability and Innovation in Structural Engineering (ISISE) for their consideration and kindness. My sincere appreciation also goes to the Director and, in particular, the Technicians of the structural laboratory (LEST). Their supports, loyalty and enthusiastic assistance were invaluable. I would like to express my deepest sense of gratitude to the Director, Engineers, and Technicians of the Civitest Company for their valuable collaborations and supports in developing the materials. Finally, I extend my profound gratitude to all my lovely family to whom this thesis is dedicated; my affectionate spouse “Hamidreza” for his genuine love, sacrifices, whose unconditional supports and patience made it possible for me to pursue this degree, my lovely son “Mehrsam” who is a tremendous blessing to my life and his passion is the source of motivation for me; and my compassionate mother, “Mrs Vajiheh”, who taught me the persistence and rectitude in my whole life and who dedicated her life to her children. I would also like to perpetuate the memory of my father who left us very soon when I was a child. He is always in my heart. May God place his soul in peace! iii Statement of Integrity I hereby declare having conducted my thesis with integrity. I confirm that I have not used plagiarism or any form of falsification of results in the process of the thesis elaboration. I further declare that I have fully acknowledged the Code of Ethical Conduct of the University of Minho. iv Abstract In the present research, the structural response of concrete beams with an innovative hybrid flexural reinforcing (HFR) scheme is studied. The adopted HFR comprises an effective reinforcement solution in terms of durability, where noncorrodible fibre reinforced polymer (FRP) bars of significant tensile strength are positioned at near the outer surface of the tensile zone of the cross-section, while ductile steel bars located at an inner level with thicker concrete cover. Additionally, the reinforcing contribution of distinct fibres diffused in fibre reinforced concrete (FRC) is mobilised to suppress the necessity of using steel stirrups and to reduce the percentage of conventional flexural reinforcement. The research includes experimental programs for assessing the post-cracking response of three series of steel fibre reinforced self-compacting concrete (SFRSCC) of different concrete strength class and volume fraction of fibres. This post-cracking behaviour is assessed in term of stress-crack width response, which is evaluated according to the recommendations of fib Model Code 2010 and from inverse analysis, both considering the results obtained in three-point notched beam bending tests and material nonlinear analysis with a finite element method. In the structural level, the flexural capacity and cracking behaviour of SFRSCC beams with HFR scheme were evaluated in the four-point bending test. Additionally, the potentiality of the HFR scheme is evaluated by developing two closed-form models. The first one is developed based on the smeared crack approach, being capable of determining the moment-curvature response of a rectangular crosssection made of FRC with HFR scheme (HFR/FRC), where perfect bond is assumed between reinforcements and surrounding concrete. The second model includes an integrated approach developed for the prediction of crack width and crack spacing HFR/FRC beams supported by the discrete crack approach. This model is capable to take into consideration the post-cracking response of FRC in terms of stress-crack opening relationship and to mobilise the shear bond-sliding characteristics of steeland FRP-toconcrete interaction. Predictive performance of the developed models is evaluated by the test results of present research and the ones represented in literature. v Resumo No presente trabalho é investigada a resposta estrutural de vigas de betão reforçadas com um inovador sistema híbrido (HFR). Este sistema híbrido é composto por varões de polímero reforçado com fibras (FRP) e varões de aço, os quais podem ser aplicados sem ou com pré-tensão, de forma a ser maximizada a durabilidade dos elementos estruturais reforçados com este sistema, com comportamento dúctil em rotura. Os varões de FRP, apesar de terem comportamento linear-elástico com rotura frágil, são imunes a fenómenos de corrosão e têm elevada resistência à tração, pelo que são colocados próximo da extremidade mais tracionada da viga. Por seu lado, devido à sua suscetibilidade a fenómenos de corrosão, as armaduras de aço dispostas com maior recobrimento, mas o seu comportamento elasto-plástico assegura a ductilidade necessária aos elementos estruturais reforçados com este sistema híbrido. Tendo em conta que os estribos em aço são os reforços mais suscetíveis a fenómenos de corrosão, pois encontram mais próximos da superfície externa das vigas, são no presente trabalho substituídos por fibras discretas, através da utilização de betão reforçado com fibras (FRC). As fibras podem ainda reduzir a percentagem de armadura de flexão, bem como melhorar a aderência destas ao betão. A investigação realizada inclui um programa experimental para determinar a lei constitutiva de modo I de fratura dos betões autocompactáveis reforçados com fibras de aço (SFRSCC) desenvolvidos, de diferente classe de resistência e com diferentes percentagens de fibras. Essa lei foi determinada quer recorrendo às recomendações do fib Model Code 2010, como por análise inversa, ambas considerando os resultados obtidos em ensaios de flexão sob três pontos de carga em vigas de SFRSCC com entalhe a meio vão. A investigação experimental inclui ainda um programa de ensaios com vigas esbeltas de SFRSCC sob quatro pontos de carga, de forma a analisar o seu comportamento em serviço (abertura e espaçamento de fissuras; flecha) e em estado último (capacidade de carga e modo de rotura). Para simular o comportamento em serviço e para estados limites últimos, foram desenvolvidos dois modelos analíticos. O primeiro permite determinar a relação momento-curvatura de elementos de FRC com reforço híbrido de flexão, admitindo-se perfeita aderência entre os reforços e o betão envolvente. vi Resumo O segundo modelo estende as potencialidade do anterior, através da simulação do deslizamento entre armaduras e betão envolvente, pelo que é capaz de estimar a abertura e espaçamento entre fissuras. A boa capacidade preditiva destes modelos é demonstrada através da comparação dos resultados previstos pelos modelos e os registados, quer nos ensaios experimentais realizados, como em resultados experimentais obtidos por outros investigadores. xiii List of notations Notations Greek letters  Normalised transition strain - f  Fibre orientation factor -  Normalised tensile strain of FRC at bottom fibre - tu  Normalised ultimate tensile strain - 12 ,  Constant parameter -  Ratio between modulus of elasticity of concrete in compression and tension - F  Ratio between modulus of elasticity of FRP bars and concrete - s  Ratio between modulus of elasticity of steel bar and concrete -  Vertical deflection mm k  Vertical deflection in kth generic step of calculation mm SLS  Vertical deflection corresponding to the serviceability limit states mm c  Strain in concrete - cc  Compressive strain of concrete - ,cc p  Strain corresponding to the concrete compressive strength - , PC cc p  Strain at the compressive strength of the plain concrete - , SFRC cc p  Strain corresponding to the concrete compressive strength in SFRC - ,cc top  Compressive strain of FRC at the top fibre of cross-section - ccy  compressive yield strain of concrete (or FRC) - ccu  Ultimate compressive strain of concrete - cm  Mean strain in concrete (or FRC) - cr  Cracking strain of concrete - ct  Tensile strain of concrete - 1ct  Tensile strain of concrete at section 1 - ctu  Ultimate tensile strain of concrete - Continued on next page xiv List of notations Notations (Continued from previous page) ,ct bot  Tensile strain of FRC at the bottom fibre of cross-section - ,ct p  Strain corresponding post-cracking strength of FRC in tension - ,ef i  Effective strain of the generic ith layer - ,ef r  Effective strain of the reinforcement layer - f  Fracturing strain of concrete - F  Tensile strain of FRP (or GFRP) bar - pr F  pre-stressing strain of FRP bars - Fu  Ultimate tensile strain of FRP (or GFRP) bar - * G  Strain of GFRP bar in hybrid reinforced concrete beam for which steel is yielded while concrete is crushed simultaneously in compression - Gu  Ultimate strain of GFRP - r  Strain of reinforcing bar - 2r  Strain of reinforcing bar at section 2 - ry  Strain at yielding of reinforcing bar - ru  Ultimate strain in reinforcing bar - sh  Shrinkage strain - sy  Yielding strain of steel - su  Ultimate strain of steel - pr s  pre-stressing strain of steel bars - sm  Mean strain in steel reinforcement - trn  Transition strain of FRC -  Normalised yield strain of steel bars -  Normalised post-cracking modulus of FRC -  Overall rotation of pure bending region rad 12 ,  Constant parameters -  Normalised compressive strain of FRC at the top fibre of cross-section - 1  Constant parameter - Continued on next page List of notations xv Notations (Continued from previous page) cu  Normalised ultimate compressive strain of FRC -  Normalised post-cracking residual strength of FRC -  Normalised tensile strain of FRP bars - fu  Normalised ultimate tensile strain of FRP bars -  Distribution coefficient -  Coefficient related to concrete cover and bar spacing - SR b  Balanced reinforcement ratio in steel reinforced concrete beams - F  Percentage of longitudinal FRP reinforcement - G  Percentage of longitudinal GFRP reinforcement - * G  Reinforcing ratio of GFRP bar in hybrid reinforced concrete beam for which steel is yielded while concrete is crushed simultaneously in compression - s  Percentage of longitudinal steel reinforcement - ,s eff  Effective percentage of longitudinal steel reinforcement - c  Stress of concrete N/mm2 cc  Compressive stress of concrete N/mm2 ' cc  Normalised compressive stress of FRC N/mm2 ccy  Compressive yield strength of concrete (or FRC) N/mm2 cr  Cracking stress of concrete N/mm2 ct  Tensile stress of FRC N/mm2 ct   Normalised tensile stress of FRC - ,ct p  Tensile strength of strain hardening fibre reinforced concrete N/mm2 F  Tensile stress of FRP bar N/mm2 ' F  Normalised tensile stress of FRP bar - Fu  Ultimate stress of FRP bar N/mm2 Gu  Ultimate stress of GFRP N/mm2 * G  The actual stress of GFRP bar when steel bar is yielding while concrete is crushed simultaneously in compression N/mm2 Continued on next page xvi List of notations Notations (Continued from previous page) r  Stress of reinforcement N/mm2 2r  Stress of reinforcement at section 2 N/mm2 ru  Ultimate stress in reinforcement N/mm2 ry  Yielding stress of reinforcement N/mm2 Constant residual tensile strength N/mm2 ' s  Normalised stress in steel reinforcement - sr  Stress in steel reinforcement at crack initiation N/mm2 st  Stress in steel reinforcement in cracked section N/mm2 sy  Yielding stress of steel N/mm2 su  Ultimate stress of steel N/mm2 ()w  Post-cracking tensile stress of FRC N/mm2 , ( )x  Shear bond stress N/mm2 bm  Average shear bond strength N/mm2 m  Maximum average bond stress N/mm2 R  Residual bond stress N/mm2 0  Bond stress at null sliding N/mm2  Curvature 1/mm ,cr reg  Curvature of cracked region 1/mm Cracking curvature 1/mm e  Effective curvature 1/mm i  curvature in the ith generic stage 1/mm i   Normalised curvature - ,uncr reg  Curvature of uncracked section 1/mm  Normalised tensile strain of steel bars - su  Normalised ultimate tensile strain of steel bars -  Normalised compressive yield strain of FRC -   Increment of overall rotation of the pure bending region rad Continued on next page R  cr  List of notations xvii Notations (Continued from previous page) F  Normalised cover thickness of FRP bars - L the extension of the pure bending region area mm s  Normalised cover thickness of steel bars - Latin Letter i a Normalised tensile stress corresponding to crack width - b Width of section mm c b Breadth of concrete in layer with reinforcement mm i b Width of ith layer mm ri b Equivalent width of reinforcement of ith layer mm c Concrete cover, central distance of the bar from tensile face of section mm 0 c Clear rib spacing of the steel bars mm eff c Effective cover mm F c Concrete cover of FRP bar mm G c Concrete cover of GFRP bar mm s c Concrete cover of steel bar mm b d Diameter of reinforcing bar mm cr d Depth of crack apex mm * cr d Depth of cracked layer in the beam reinforced by steel and GFRP reinforcing ratio of * G  mm , SR cr b d Depth of layer at which concrete cracking strain )(cr  is attained for the balanced condition for steel reinforced concrete elements mm f d Diameter of the fibre mm F d Depth of FRP layer from the top surface of the cross-section mm G d Depth of GFRP layer from the top surface of the cross-section mm i d Depth of ith layer mm NA d Depth of neutral axes mm Continued on next page i w xviii List of notations Notations (Continued from previous page) * NA d Depth of neutral axes in the beam reinforced by steel and GFRP reinforcing ratio of * G  mm , SR NA b d Depth of neutral axis in balanced condition in steel reinforced concrete elements mm ,ri d Depth of ith layer with longitudinal reinforcing bars mm s d Depth of steel layer from the top surface of the cross-section mm cc f Compressive strength of concrete N/mm2 ck f Characteristic values of compressive strength of concrete N/mm2 cm f Average compressive strength of concrete N/mm2 cr f Cracking stress of the concrete N/mm2 ct f Tensile strength of concrete N/mm2 ctm f Average tensile strength of concrete N/mm2 ,Fts m f Average residual strength of FRC in the serviceability limit state N/mm2 ,Ftu m f Average residual strength of FRC in the ultimate limit state N/mm2 res f Post-cracking residual strength of concrete N/mm2 ,Ri k f Characteristic flexural residual strengths N/mm2 ,Ri m f Average flexural residual strengths corresponding to CMODi N/mm2 y f yield strength of steel bar N/mm2 f G Fracture energy N/mm h Height of cross-section mm cc h Height of concrete compression zone mm ct h Height of concrete tensile zone mm F h Height of FRP bar from the neutral axes mm s h Height of steel bar from the neutral axes mm sp h Height of ligament over the notch apex mm 1 J Coefficient related to the geometry and modulus of elasticity of reinforcing bar and surrounding concrete - k Normalised neutral axis depth - crm Lcs k Coefficient for adjustment of crack spacing - Continued on next page List of notations xix Notations (Continued from previous page) f k Fibre effectiveness factor - 1 16 tokk Constant parameter - cb l Crack bandwidth mm ch l Structural characteristic length mm d l Development length mm e L Embedded length mm f l Length of fibre mm ,maxs l Maximum transmission length mm tr l Transmission length mm n Axial stiffness ratio between reinforcement and surrounding concrete - b n Number of bars - cr n Number of cracks - L n Number of layers constituting the cross-section - c L n Number of concrete layers in compressive zone - r L n Number of layers with longitudinal reinforcing bars - t L n Number of concrete layers in tensile zone - , ( )s s x Slip displacement mm b s Effective longitudinal bar spacing mm fle s Slip at loaded-end of the bar mm le s Slip at free loaded-end of the bar mm mi s Factor taking into account the influence of steel fibres by considering the number of fibres bridging a crack 1/mm2 r s Spacing between each pair of adjacent cracks mm rm s Average crack spacing mm ,maxr s Maximum crack spacing mm ,minr s Minimum crack spacing mm sec.1 s Sliding at Section 1 mm Continued on next page xx List of notations Notations (Continued from previous page) tr s Spacing between transverse reinforcement mm i t Thickness of ith layer mm c u Elongation of concrete mm r u Elongation of reinforcing bar mm w Crack width mm d w design value of crack width mm k i w Crack width if ith layer at kth stage of loading mm k w Characteristic value of crack width mm m w Average crack width mm max w Maximum crack width mm r w Crack width at the level of reinforcement mm SLS w crack width corresponding to the serviceability limit state mm u w Ultimate crack width mm cc y Internal arm of compressive force mm ct y Internal arm of tensile force mm F y Internal arm of tensile force of FRP bar mm s y Internal arm of tensile force of steel bar mm c A Area of cross-section of concrete mm2 ,c eff A Effective area of concrete in tension mm2 () i F Exp A   Area beneath the experimental force-deflection curves up to the central deflection of mm2 F A Area of cross-section of longitudinal FRP reinforcement mm2 G A Area of cross-section of longitudinal GFRP reinforcement mm2 () i F Num A   Area beneath the numerical force-deflection curves up to the central deflection of mm2 r A Area of cross-section of longitudinal reinforcement mm2 ri A Area of cross-section of longitudinal reinforcement of ith layer mm2 s A Area of cross-section of longitudinal steel reinforcement mm2 Continued on next page k  k  List of notations xxi Notations (Continued from previous page) tr A Area of transverse reinforcement mm2 f D Diameters of spread concrete mm F D Deviation history in terms of force - i D Deformation of the generic ith layer mm T D Deviation history in terms of toughness - c E Modulus of elasticity of concrete N/mm2 cm E Average modulus of elasticity of concrete N/mm2 cr E The post-cracking tensile modulus of FRC N/mm2 F E Modulus of elasticity of FRP N/mm2 G E Modulus of elasticity of GFRP bar N/mm2 r E Modulus of elasticity of reinforcement N/mm2 s E Modulus of elasticity of steel bar N/mm2 F Applied force N bond F The load transmitted along with the interaction length N cc F Force of concrete compression zone N , k cc i F Compressive force of the ith layer of FRC at kth stage of loading N ct F Force of concrete tensile zone N , k ct i F Tensile force of the ith layer of FRC at kth stage of loading N cr F Force corresponding to crack initiation N 1ct F Tensile force of concrete at section 1 N 2ct F Tensile force of concrete at section 2 N F F Force of FRP bar N pr F F Pre-stressing load of FRP bars N ,im F Average force corresponding to the i CMOD N P F peak load N m p F Average peak load N Continued on next page xxii List of notations Notations (Continued from previous page) Internal force of reinforcing bar at section 1 N 1, k ri F Tensile force of reinforcement at ith layer and at kth stage of loading at section 1 N Internal force of reinforcing bar at section 2 N 2, k ri F Tensile force of reinforcement at ith layer and at kth stage of loading at section 2 N ,r cr F The internal force of the reinforcement at cracking initiation stage N , k ri F Tensile force of reinforcement at ith layer and at kth stage of loading N ,, k ri F   Tensile force of reinforcement at ith layer and at kth stage of loading determined from stress-strain constitutive law N ,, k r s i F   Tensile force of reinforcement at ith layer and at kth stage of loading determined from the bond N SLS F Force corresponding to the serviceability limit states N m SLS F Average force corresponding to the serviceability limit states N k Exp F Force registered experimentally corresponding to k  N sec.1 (s ) e F The load transmitted along with the elastic interaction length N max e F Maximum force that is transferred through elastic bond region N sec.1 (s ) p F The load transmitted along with the plastic interaction length N max p F Maximum force that is transferred through plastic bond region N sec.1 (s ) s F The load transmitted along with the softening interaction length N max s F Maximum force that is transferred through softening bond region N sec.1 (s ) f F The load transmitted along with the frictional interaction length N k Num F Force determined numerically corresponding to k  N s F Force of steel bar N pr s F Pre-stressing load of steel bars N sy F Force corresponding to yielding of steel reinforcement N m sy F Average force corresponding to yielding of steel reinforcement N ,r F   Internal force of reinforcing bar determined by stress-strain constitutive law N ,rs F   Internal force of reinforcing bar determined by bond-slip constitutive law N Continued on next page 1r F 2r F List of Figures xxix Chapter 4: Moment-Curvature Approach to Evaluate Flexural Response of R/SFRSCC Elements Figure 4.1: Geometry and reinforcing scheme of the cross-section ............................... 85 Figure 4.2: Typical stress-strain relationship of FRC proposed by (a) Lim et al. (1987), (b) Lok and Xiao (1998), (c) Soranakom and Mobasher (2008) ............................. 86 Figure 4.3: Tensile constitutive law of FRC ................................................................... 87 Figure 4.4: Constitutive law of FRC in compression ..................................................... 90 Figure 4.5: Tensile constitutive law of steel reinforcement ............................................ 91 Figure 4.6: Tensile constitutive law of FRP reinforcement ............................................ 93 Figure 4.7: Strain profile of the section and intervening normalised parameters ........... 94 Figure 4.8: Profile of strain and stress along with the depth of cross-section for the considered stages ...................................................................................................... 97 Figure 4.9: Iinternal forces of concrete and steel and FRP bars (continued) .................. 99 Figure 4.10: Numerical approach to simulate the force-deflection response of simply supported beams failing in bending ....................................................................... 111 Figure 4.11: Geometry and reinforcing scheme of the cross-section considered in the model appraisal (dimensions in mm) ..................................................................... 112 Figure 4.12: Moment-curvature responses predicted by the proposed model and DOCROS for the cross-section of reinforced FRC of (a) strain-softening and (b) strain-hardening behaviour ..................................................................................... 113 Figure 4.13: (a) Geometry and loading scheme of the beams considered for model appraisal, reinforcement and strengthening configurations of the beams (b) B1 and B2 tested by Badawi and Soudki (2009), and (c) B3 and B4 tested by Xue et al. (2010) (dimensions in mm) .................................................................................... 114 Figure 4.14: Force-deflection relationships determined by the proposed model and the ones registered in the experimental tests for (a) B1, (b) B2, (c) B3, and (d) B4 ... 116 Figure 4.15: Force-deflection relationships determined by the proposed model and the ones registered in the experimental program explained in Chapter 3, (a) xxx List of Figures SR/FRC1545, (b) SGR/FRC1545, (c) SR/FRC2560, (d) SGR/FRC2560, (e) SR/FRC4590, and (f) SGR/FRC4590 .................................................................... 118 Figure 4.16: Geometry and reinforcement data for the beam of the parametric study (dimensions in mm). ............................................................................................... 119 Figure 4.17: Effect of the  parameter on the moment-curvature and load-deflection responses for steel and FRP bars pre-stressed at level of 0.0, 25, and 50% ........... 121 Figure 4.18: Effect of the pre-stress level on the: (a-d) moment-curvature response; (eh) increase in the resisting bending moment; for  =0.4 and  equal to 1.01, 10, 50 and 150............................................................................................................... 124 Figure 4.19: Effect of the pre-stress level on the: (a-d) Load-deflection response; (e-h) increase in the load carrying capacity; for  =0.4 and  equal to 1.01, 10, 50 and 150 .......................................................................................................................... 126 Figure 4.20: Effect of the  parameter on the moment-curvature and load-deflection responses for  =10, and steel and FRP bars pre-stressed at 0.0, 25, 50%. .......... 127 Figure 4.21: Effect of the pre-stress level on the: (a-d) moment-curvature response; (eh) increase in the resisting bending moment; for  = 10 and  equal to 0.0, 0.4, 0.8, 1.2. ................................................................................................................... 129 Figure 4.22: Effect of the pre-stress level on the: (a-d) Load-deflection response; (e-h) increase in the load carrying capacity; for  = 10 and  equal to 0.0, 0.4, 0.8, 1.2. ................................................................................................................................ 130 Chapter 5: Prediction of Crack Width and Spacing in R/FRC Flexural Elements Figure 5.1: (a) Cracking propagation in a pure bending region of R/FRC beams, (b) layer approach to model the cross-section .............................................................. 136 Figure 5.2: Stress-strain diagram for simulating the compressive behaviour of an FRC ................................................................................................................................ 138 Figure 5.3: Tensile behaviour of FRC: (a) stress-strain diagram before macro-cracking localization, (b) post-cracking stress-crack width response ................................... 139 List of Figures xxxi Figure 5.4: Stress-strain relationship for simulating the tension and compression behaviour of longitudinal reinforcements .............................................................. 141 Figure 5.5: Shear bond stress-slip relationship for embedded reinforcement .............. 142 Figure 5.6: (a) Reinforcing bar and surrounding concrete, (b) force equilibrium of reinforcement and surrounding concrete along an infinitesimal bond transference length of dx ............................................................................................................ 143 Figure 5.7: Force equilibrium along bond transference length ..................................... 145 Figure 5.8: Variation of shear bond stress and sliding along with the interaction zone when activated (a) the elastic and (b) the plastic phases of bond .......................... 149 Figure 5.9: Recommendations of fib Model Code 2010 (2011) for the evaluation of the effective tension area of the concrete surrounding reinforcing bar in (a) beams, and (b) slabs .................................................................................................................. 151 Figure 5.10: Variation of shear bond stress and sliding along with the interaction zone when activated (a) the softening and (b) the frictional phases of bond.................. 156 Figure 5.11: Reinforced concrete beam subjected to four-point bending load configuration .......................................................................................................... 159 Figure 5.12: Crack propagation and bond stress-slip distribution between two adjacent cracks just before initiation of a (a) second crack, (b) third crack, (c) fourth crack, and (d) fifth crack ................................................................................................... 162 Figure 5.13: Algorithm of the model ............................................................................ 168 Figure 5.14: Four-point bending test setup (G represents GFRP) ................................ 169 Figure 5.15: Tensile stress vs. crack opening diagram recommended by fib Model Code 2010 (2011) ............................................................................................................ 171 Figure 5.16: Typical crack patterns in the tested beams ( m p F is the average peak load of the corresponding series of beams) ........................................................................ 174 Figure 5.17: Predictive performance of the model for the force-deflection response of (a) B1, (b) B2, (c) B3, (d) B4, (e) B5, (f) B6 ......................................................... 176 xxxii List of Figures Figure 5.18: Predictive performance of the model for the moment-crack width of (a) B1, (b) B2, (c) B3, (d) B4, (e) B5, (f) B6 ...................................................................... 177 Figure 5.19: Predictive performance of the model for the moment-crack width of (a) B7, (b) B8, (c) B9, (d) B10............................................................................................ 178 Figure 5.20: Predictive performance of the model for the moment-average crack spacing of (a) B1, (b) B2, (c) B3, (d) B4, (e) B5, and (f) B6............................................... 179 Appendix A: Crack evolution in the SFRSCC beams reinforced by a steel bar Figure A.1: Crack evolution in SR/FRC1545-1 ............................................................ 210 Figure A.2: Crack evolution in SR/FRC1545-2 ............................................................ 211 Figure A.3: Crack evolution in SR/FRC1545-3 ............................................................ 212 Figure A.4: Crack evolution in SR/FRC2560-1 ............................................................ 213 Figure A.5: Crack evolution in SR/FRC2560-2 ............................................................ 214 Figure A.6: Crack evolution in SR/FRC2560-3 ............................................................ 215 Figure A.7: Crack evolution in SR/FRC4590-1 ............................................................ 216 Figure A.8: Crack evolution in SR/FRC4590-2 ............................................................ 216 Figure A.9: Crack evolution in SR/FRC4590-3 ............................................................ 217 Appendix B: Crack evolution in tested SFRSCC beams reinforced by steel and GFRP bars Figure B.1: Crack evolution in SGR/FRC1545-1 ......................................................... 218 Figure B.2: Crack evolution in SGR/FRC1545-2 ......................................................... 219 Figure B.3: Crack evolution in SGR/FRC1545-3 ......................................................... 220 Figure B.4: Crack evolution in SGR/FRC2560-1 ......................................................... 221 Figure B.5: Crack evolution in SGR/FRC2560-2 ......................................................... 222 Figure B.6: Crack evolution in SGR/FRC2560-3 ......................................................... 223 Figure B.7: Crack evolution in SGR/FRC4590-1 ......................................................... 224 List of Figures xxxiii Figure B.8: Crack evolution in SGR/FRC4590-2 ......................................................... 225 Figure B.9: Crack evolution in SGR/FRC4590-3 ......................................................... 226 xxxiv List of Tables Chapter 2: Literature Review Table 2.1: Typical properties of fibres used in FRC composites (Bentur and Mindess 2007) ......................................................................................................................... 13 Table 2.2: Proposed formulation for the initial branch of bond-slip relationship (adopted from fib Bulletin 10 (2000) ....................................................................................... 37 Chapter 3: Experimental Evaluation of Flexural Response of R/SFRSCC Elements Table 3.1: Composition of the developed SFRSCCs (per 1 m3) ..................................... 48 Table 3.2: Slump flow test results ................................................................................... 49 Table 3.3: Adopted scheme for constructing the samples ............................................... 51 Table 3.4: Material properties of the SFRSCCs .............................................................. 53 Table 3.5: The average and characteristic values of flexural residual strengths of the SFRSCCs .................................................................................................................. 56 Table 3.6: Toughness classes according to the fib Model Code 2010 (2011) ................. 58 Table 3.7: Values of parameters defining the constitutive laws of SFRSCCs ................ 62 Table 3.8: Mechanical properties of steel and GFRP bars .............................................. 65 Table 3.9: Designation and reinforcement details of the beams ..................................... 66 Table 3.10: Reinforcing ratio of SR and SGR series beam ............................................. 69 Table 3.11: Average load bearing of SR and SGR beams .............................................. 73 Chapter 4: Moment-Curvature Approach to Evaluate Flexural Response of R/SFRSCC Elements Table 4.1: Variations of normalised strain parameters of the intervening materials ...... 96 Table 4.2: Normalised height of concrete compression and tension zones, and the normalised distance of the steel and FRP bars with respect to the neutral axes (see Figure 4.9)............................................................................................................... 101 List of Tables xxxv Table 4.3: Normalised stresses of concrete in compression and tensile zones and in the steel and FRP bars (see Figure 4.9) ........................................................................ 102 Table 4.4: Normalised forces of concrete and steel and FRP bars (see Figure 4.9) ..... 103 Table 4.5: Normalised internal arm of force components for each stage (see Figure 4.9). ................................................................................................................................ 104 Table 4.6: Equations for the depth of the neutral axis parameter ()k of each stage ..... 106 Table 4.7: Equations for the evaluation of the normalised moment for each stage ...... 108 Table 4.8: Values of the parameter defining the constitutive laws ............................... 113 Table 4.9: Data to define the geometry, the reinforcement and the strengthening systems of the beams represented in Figure 4.13 ................................................................ 115 Table 4.10: Data to define the constitutive laws of the intervening materials in the beams of Figure 4.11 .............................................................................................. 115 Table 4.11: Values considered for the constitutive parameters for the simulation of the beams ...................................................................................................................... 115 Table 4.12: Values considered for the constitutive parameters of SFRSCCs developed in the present study ..................................................................................................... 117 Table 4.13: Values considered for the constitutive parameters of reinforcing bars adopted in the present study ................................................................................... 117 Table 4.14: Values for the parameters of the materials constitutive laws adopted in the parametric study ..................................................................................................... 120 Chapter 5: Prediction of Crack Width and Spacing in R/FRC Flexural Elements Table 5.1: Geometry and reinforcing scheme of the beams (G: GFRP; dimensions are in mm) ........................................................................................................................ 169 Table 5.2: Mechanical properties of the reinforcing bars ............................................. 170 Table 5.3: Relevant properties of the used SFRCs ....................................................... 171 Table 5.4: Bond-slip parameters adopted in the simulations ........................................ 173 1 Chapter 1 Introduction 1.1 Motivation and objective Concrete is the most-consumed manufactured material in today’s world with a long history return to the when cementitious components were using to construct aqueducts in many ancient civilizations. During the last decades, the demand for concrete has grown with industrialisation, so that the worldwide production of concrete is believed revolving around 20 billion tonnes annually (Deluce and Vecchio 2013). When compared with steel material, concrete is neither as strong nor as tough. Nevertheless, the widespread usage of concrete is ascribed to its some peerless properties such as excellent resistance to water, plastic consistency in fresh state and consequent high potentiality to be formed into a variety of shapes and sizes as well as abundance and availability of its main components, which makes concrete the cheapest and most readily material for engineered structures. The latter, however in its turn is the origin of significant environmental criticisms. Nowadays, concrete dependent industries exploit the vastest amount of natural resource Chapter 1 2 and responsible for emitting up to 8 percent of global greenhouse gases (Scrivener and Kirkpatrick 2008). Nevertheless, the durability of reinforced concrete structures remains a controversial subject. Due to its microstructure, concrete is inherently vulnerable in tension. Over the years, this weakness has been remedied by the use of steel reinforcements. This traditional treatment, however, comprises its own consequences. Steel is susceptible against corrosion and must be protected from environmental hostile agents by allocating a cover of concrete of proper thickness, which leads to the increase in height of cross-section. In this scheme, however, the early stage cracking is practically inevitable in plain concrete cover, which like micro streams facilitate traverse of aggressive agents toward the steel reinforcement leading to corrosion of steel and negative circumstances threaten the serviceability and safety of the structure. The numerous researches conducted so far, to improve durability indexes of reinforced concrete structures can be overviewed generally, into two principal approaches. The first one comprises partially or entirely substitution of steel reinforcement by the ones of more significant resistance against corrosion, such as various types of fibre reinforced polymer (FRP) bars with fundamental differences in material properties or mechanical interaction to concrete. Therefore, substantial modifications should be mobilised in design approach and constitutive laws when FRP reinforced concrete structure is aiming. The second approach to enhance the durability of concrete structures is aiming to control cracking in concrete. It is extensively documented that corrosion of steel reinforcement is considerably overcome when maximum crack width of the concrete cover is kept small. In this regard, the improvement of the post-cracking response of concrete is crucial which can be achieved by the use of discrete fibre in concrete. The randomly distributed fibres confer to the concrete the ability to form multiple cracks, resulting in a significant increase in energy absorption capacity and a pseudo-ductile behaviour owing to the fibre pullout mechanisms. Recent studies (Barros et al. 2015a, Mazaheripour et al. 2016b, Salehian and Barros 2017) have shown the potentialities of FRC as a supplement or even as a replacement of conventional longitudinal steel bars for elements failing in bending, mainly when having a strain-hardening character 3 Introduction (Mazaheripour et al. 2016b, Barros et al. 2015a, Salehian and Barros 2017). In this regard, it is fundamental to develop an integrated approach to mobilise accurately the postcracking response of FRC, in a material point of view, for predicting the load carrying capacity of reinforced FRC (R/FRC) elements. Prediction of cracking behaviour of R/FRC remains a crucial aspect in such a comprehensive approach. Experimental studies have revealed the bar-to-concrete interaction and consequently, the tension stiffening effect tends to be improved by the increase in reinforcing contribution of fibres (Oliveira Júnior, 2016 #503). However, the scarcely available design approaches for prediction of cracking behaviour of R/FRC members are uncomprehensive and in some cases comprise arguable predictive performances. 1.2 Scope of the research The present work aims to develop innovative numerical/analytical models to predict the force-deflection relationship and cracking behaviour of concrete elements failing in bending and reinforced by a hybrid-reinforcing scheme. Such an effective reinforcement solution in terms of durability and structural performance is obtained by placing the FRP bars near the outer surface of the tensile zone and steel bars at an inner level of the tensile zone. In this scheme, the steel bar provides a significant contribution in terms of ductility and stiffness and assure the safety of the structure in the case of a fire occurrence and the consequent loss of FRP reinforcing capacities. The reinforcing contribution of fibres is mobilised to eliminate conventional steel stirrups and concurrently to reduce the ratio of longitudinal reinforcements. In the aimed models, the post-cracking response of FRC is taken into account through two fundamental approaches proposed to simulate fracture mechanism of plain and fibrous concrete; the smeared crack and discrete crack approaches mobilised, respectively, into developed moment-curvature and momentrotation models. Predictive performance of the models is assessed in a comprehensive experimental programme. 1.3 Outline of the dissertation Chapter 2 aims to explain concrete morphology and overview various approaches proposed to reinforce concrete. Fundamental aspect relative to fibre reinforced concrete Chapter 2 10 and non-magnetic materials. Nevertheless, the major obstacles of the application of FRP bars as a sole reinforcing material for concrete structures are the relatively high initial costs, low modulus of elasticity, lack of ductility (linear stress-strain diagram up to rupture with no discernible yield point), and absence of well-consolidated design guidelines (Toutanji and Saafi 2009, Abdalla 2002, Alsayed et al. 2000). When compared with steel, the cost-comparative FRPs have a relatively low elastic modulus with considerably different bond properties in concrete (Abdalla 2002, Bakis et al. 2002, Tian and Yuan 2007). These often lead to a larger deflection and wider cracks in FRP reinforced concrete (FRP/RC) beams, so that the serviceability requirements of FRP/RC are often predominant (Almusallam 1997, Masmoudi et al. 1998). In addition, as a result of larger crack width and a smaller compressive stress blocks when using FRP bars for the flexural reinforcement, the shear capacity of FRP/RC beams is smaller than steel reinforced concrete (S/RC) beams of the same reinforcement ratio (ACI 440R-07 2007). 2.5 Hybrid longitudinal reinforcing scheme In an attempt to overcome the drawbacks relative to the sole usage of steel or FRP bar as reinforcement of concrete, a combination of FRP and steel reinforcements is proposed in some literature for concrete elements (Aiello and Ombres 2002, Leung and Balendran 2003, Mazaheripour 2015, Mazaheripour et al. 2016a). In such a hybrid flexural reinforcing (HFR) scheme, an effective solution in terms of durability is obtained by placing the FRP bars near the outer surface of the tensile zone and steel bars at the inner level. Experimental evidence revealed that the deflection, crack width, and crack spacing of HFR/RC beams are typically smaller than that of FRP/RC beams (Tian and Yuan 2007, Aiello and Ombres 2002). In terms of the structural behaviour of the concrete member, the presence of the steel bars in this hybrid reinforcement system contributes significantly to enhance the ductility and stiffness. The few tests that were executed confirmed this idea. In fact, Tian and Yuan (2007) concluded that the deflection of concrete beams reinforced with FRP and steel bars was smaller than that of beams reinforced just with glass fibre reinforced polymer (GFRP). Aiello and Ombres (2002) also verified that, in comparison with beams exclusively reinforced with FRP bars, the presence of steel bars reduces the crack width and crack spacing values. These studies 11 Literature Review also indicate that the hybrid longitudinal reinforcement system (steel and FRP) represent a competitive solution when the long-term costs of repairing activities are also taken into account. In the hybrid longitudinal reinforcing scheme, FRP and steel bars can be applied with a certain pre-stress to optimise their reinforcing capabilities. According to Nordin and Täljsten (2006), for the strengthening of reinforced concrete beams, there are four advantages when using pre-stressed FRP: better utilization of the strengthening material, smaller distance and width of cracks in concrete, unloading of the steel reinforcement, and higher steel yielding loads. Furthermore, with the pre-stress, a significant increase regarding load carrying capacity for deflection levels corresponding to the serviceability limit states can be obtained (Barros 2009). Some models were also developed in the literature to simulate the tension stiffening behaviour of fibre reinforced concrete (FRC) elements reinforced by hybrid fibre reinforced polymer (FRP) and steel bars (Mazaheripour et al. 2016a). 2.6 Fibre reinforced concrete During the last decades, continuous advances have been made in concrete technology, leading to the advent of various kinds of micro-fillers such as fly ash and silica-fume that have been employed to densify the microstructure of concrete, with noticeable enhancements in terms of strength and durability. Furthermore, considerable scientific research has been conducted on the rheological properties of the fresh concrete, in parallel to the development of novel additives, such as superplasticizers and viscous admixture modifier employed to produce flowable concrete with a reduced water-to-cement ratio, currently called as self-compacting concrete (SCC) (Okamura 1997, Okamura and Ouchi 2003). The advent of fibre reinforced concrete (FRC) is also one of the advances in concrete technology, leading to a new generation of cement-based materials with several advantages regarding plain concrete. In particular, by merging the benefits of the fibre reinforcement to those derived from the self-consolidating character of the selfcompacting concrete, a high-performance structural material is obtained designated by fibre reinforced self-compacting concrete (FRSCC) (Groth 2000, Grünewald 2004). The idea of reinforcing the brittle concrete with the addition of steel splinters was firstly patented by Bernard in 1874 (Maidl 1995). This idea, in fact, was inspired by an ancient Chapter 2 12 construction technique with a 3500-year-old history, when brittleness of sun-baked mud bricks was mitigated by the addition of organic or mineral fibres. Such a century-old structural construction technique is still present in some heritage. This technique is, even so, a competitive approach in the case of low-cost rural housing applications (Aziz et al. 1981, Balaguru and Shah 1985). The asbestos cement is the first widespread use of fibre in the cementitious composite, which was developed in about 1900 with the invention of the so-called Hatschek technology for the production of plates for roofing and pipes. The interest in concrete reinforced with fibres grew noticeably when the enhanced properties of FRC were highlighted by Romualdi (Romualdi and Batson 1963, Romualdi and Mandel 1964). The discrete and randomly distributed fibres provide additional resistance to the opening of concrete microcracks by the so-called bridging mechanism which leads to a significant improvement of the fracture toughness, ductility, impact resistance, and fragmentation of concrete (Shah and Rangan 1971, di Prisco et al. 2004, Li et al. 1993, Naaman and Shah 1976, Naaman 2000, Barros et al. 2005). The bridging mechanism of fibres can equally contribute to enhance the shear resistance of concrete elements (Santos et al. 2008, Barros et al. 2004a, Casanova et al. 1997, Meda et al. 2005). In particular, when a high strength concrete is used and when beams are relatively shallow, fibres noticeably reduce the width of shear cracks and improve the durability of concrete. According to Wang and Belarbi (2013), an improvement of 30% of the durability index of RC beams was achieved by the addition of 0.5% volume fraction of fibres. In the structural level, the influence of fibres on the reduction of shear cracks and improvement of the shear resistance of FRC beams is noticeable (Santos et al. 2008, Barros et al. 2004b, Susetyo et al. 2011). In shallow reinforced FRC (R/FRC) beams, steel stirrups can be replaced by reinforcing effects provided by the fibres, which accelerates the construction process and increases the durability of structure (Casanova et al. 1997). The discrete fibres also contribute to reduce the deflection of R/FRC beams under service loads and to increase the load carrying capacity at the ultimate limit states (Taheri et al. 2011, Barros et al. 2012). Experimental and numerical investigations revealed that the post-cracking behaviour of FRCs effectively improves the tensionstiffening behaviour of R/FRC elements (Abrishami and Mitchell 1997, Lee et al. 2013), which reduces the width and spacing of cracks (Bischoff 2003, Oliveira Júnior et al. 2016) 13 Literature Review and can provide a higher post-yielding strength in S/FRC tensile members (Jordon and Frank 2013). In the study carried out by Meda et al. (2005), a reduction of 20% of crack spacing in FRC beams was observed when compared to reference beams of conventional concrete with and without stirrups. Presently, an extensive range of fibres of various geometrical, mechanical, physical and chemical properties have been considered and used for the reinforcement of cementitious materials (Brandt 1994), as summarised in Table 2.1. Nevertheless, steel fibres are the most commonly used type fibres in practical applications. Table 2.1: Typical properties of fibres used in FRC composites (Bentur and Mindess 2007) Fibre Diameter Specific gravity Modulus of elasticity Tensile strength Ultimate elongation [m] [-] [GPa] [GPa] [%] Steel 5-500 7.87 200 0.5-2.0 0.5-3.5 Glass 9-15 2.6 70-80 2-4 2-3.5 Crocidolite Asbestos 0.02-0.4 3.4 196 3.5 2.0-3.0 Chrysotile Asbestos 0.02-0.4 2.6 164 3.1 2.0-3.0 Polypropylene (PP) 20-400 0.9-0.95 3.5-10 0.45-0.76 15-25 Poly Vinyl Alcohol (PVA) 14-600 1.31 25-40 0.88-1.60 6-10 Aramid (Kevlar) 10-12 1.44 63-120 2.3-3.5 2-4.5 Carbon (high strength) 8-9 1.6-1.7 230-380 2.5-4.0 0.5-1.5 The fracture mechanism of fibrous concrete is noticeably affected by the pullout response of fibres through the concrete matrix (Naaman et al. 1991, Taerwe and Gysel 1996). The pullout response of fibres, in turn, is influenced by the shape and geometry of fibres and structure of the matrix. Regarding smooth fibres, the pull out response is significantly affected by the elastic and frictional bond between the fibre and concrete, which are both improved by the refining of the microstructure of the cementitious matrix and consequent decreasing in its porosity. In conventional concrete, however, the chemical adhesion and frictional bond are negligible when a smooth fibre is utilised. In such a case, the mechanical anchorage of deformed fibres is required for offering extra resistance to crack opening (Banthia and Trottier 1994, Li and Stang 1997, Naaman and Chapter 2 14 Najm 1991). In this regard, deformed steel fibres of various types are utilised in concrete, some of which are schematised in Figure 2.4. (a) (b) (c) (d) (e) (f) (g) Figure 2.4: Various types of deformed steel fibres: (a) Indented, etched or roughened, (b) crimped or corrugated, (c) polygonal twisted, (d) flat-ended, (e) buttons-ended, (f) hooked ended, and (g) double hooked ended (adopted from Salehian (2015)) 2.7 Mechanism of crack formation and propagation in the cementitious composites Evaluating the mechanical properties of the cementitious composites based on fracture mechanics is a conventional approach explored in the literature. The advent of fracture mechanics turns back to World War I when Griffith (1921) presented his theory to explain the failure of materials. In this theory, it was assumed that formed crack becomes immediately stress-free up to the crack tip (Figure 2.5a) and the product of the square root of the crack length and the stress at the fracture is assumed constant. Experimental evidence proved the accuracy of this approach in the case of brittle materials, behaving according to the linear elastic fracture mechanics, like glass. Nevertheless, the Griffith theory was ignored for about three decades. The reason was that the energy required to cause a fracture in structural ductile materials is appreciably larger than that predicted by the Griffith theory. Furthermore, the non-linear response of material cannot be explained by this approach. 15 Literature Review (a) (b) (c) Figure 2.5: Typical representation of linear (L), non-linear (N) and fracture process (F) zones in Fracture of (a) brittle, (b) ductile, and (c) quasi-brittle materials (adopted from Bažant (1992) In the late sixties, Irwin (1957) modified the Griffith theory by considering a nonlinear zone at the tip of a crack growing in ductile material, as depicted in Figure 2.5(b) and 2.5(c). The size of this zone extends by the increase in the applied load, while the elastically strained material behind the crack tip is unloading. Unlike the brittle materials, the nonlinear zone is large in ductile materials (Figure 2.5b), leading to plastic hardening or perfect plastic behaviour. In the case of ductile materials, in a very small part of the nonlinear zone, the fracture process zone forms in the vicinity of the crack apex. In the case of quasi-brittle materials, like concrete, the fracture process zone is significantly large and practically occupies the entire nonlinear zone as depicted in Figure (2.5c). This is attributed to the heterogeneity inherent of concrete and the presence of micro-cracks ahead of the crack tip. In fact, the existence of the fracture process zone in front of a crack constitutes the intrinsic reason for the size dependence of the fracture parameters in quasi-brittle materials. In this case, the plastic response of ductile materials is substituted by the strain-softening behaviour in tension where the stress normal to the crack plane decreases with the increase in strain (Zhang and Wu 1999). The fracture process zone of quasi-brittle materials is considered as a so-called “cohesive zone” in the literature (Dugdale 1960, Barenblatt 1959, Barenblatt 1962). Hillerborg et al. (1976) have taken advantage of the concept of the cohesive zone in their “fictitious crack model”. In this approach, the fracture process zone together with the part of the localised crack where aggregate interlock is present, are substituted by a spurious crack as revealed in Figure 2.6(a). Chapter 2 16 (a) (b) Figure 2.6: (a) Fictitious crack at the tip of growing crack and (b) stress-crack opening constitutive laws assigned to the fictitious crack The fictitious crack model of Hillerborg and his collaborators has been widely used in finite element analysis of the concrete fracture by using interface finite elements (IFE) in the simulations based on the finite element method (FEM). The constitutive law of the IFE simulates the crack opening process, by using a stress-crack opening ()w   relationship (Figure 2.6b), where   w  is the traction applied to the crack surfaces as a function of crack opening w , ranging between the tensile strength of concrete () ct f at the crack apex and zero at the ultimate crack widening. According to the literature, a bilinear softening curve suffices to characterise the fracture of concrete (Guinea et al. 1993). The fracture energy of concrete is defined by the area beneath the w   curve: 0() u w f G w dw   (2.1) The fictitious crack model was later used by Hillerborg (1980) to analyse the fracture of fibre reinforced concrete, as adopted by other authors to describe the bridging effects of fibres in fibre reinforced concrete (Lange-Kornbak and Karihaloo 1998, Li 1992). Due to the contribution of fibres in the pullout mechanism, the post-cracking behaviour of FRC is improved. In this regard, the higher the volume fraction of fibre, the higher the fracture energy was obtained in experimental research (Kooiman 2000). Nevertheless, since the length of fibres bridging the crack is significantly larger than the crack opening, the ultimate crack width () u w in FRC is not strictly relevant in a structural point of view. Therefore, the concept of fracture energy defined by Equation (2.1) may lose its meaning 17 Literature Review from a practical point of view. For this reason, the actual shape of w   curve in the range of acceptable crack opening (e.g. 0-1.5 mm) becomes more important than f G (RILEM TC 162-TDF 2002). The fracture of cement-based materials can also be analysed by a stress-strain relationship. This approach, firstly advanced by Bažant (Bažant and Oh 1983), is based on the fact that the microcracks are randomly distributed along with the fracture process zone, and growing the crack is roughly straight. Therefore, concerning smeared cracks band (or crack bandwidth) of cb l in the fracture process zone (Figure 2.7a), the crack widening can be described by an equivalent strain so-called fracturing strain () f  (Figure 2.7b). The concept of stress-strain relationship is quite familiar for the engineer. It is also fairly convenient for computer programming of the finite element method. (a) (b) Figure 2.7: (a) Crack band model at the tip of growing crack and (b) stress-strain diagram assigned to the crack band In the crack band model, it is fundamental to adjust the material parameters controlling the smeared cracking such that the amount of energy dissipated during failure becomes independent of the mesh size. Therefore, the crack bandwidth is determined on the basis of the finite element size projected onto the direction of the maximum principal strain for the case of tensile cracking (Bažant 1984). Consequently, the crack opening displacement ()w is related to the fracturing strain () f  through the following equation: Chapter 2 18 fcb wl   (2.2) For the structural elements, the crack bandwidth can be considered the same as the corresponding structural characteristic length ( ch l ) of the element (RILEM TC 162-TDF 2002, fib Model Code 2010 2011) which is an indication of spacing between two adjacent cracks. ch l is influenced by numerous parameters, such as compressive strength class of concrete, geometric properties of cross-section, percentage of longitudinal or transversal reinforcement, and the prescribed load level. Accordingly, there is not a consensus for determining ch l . For FRC flexural elements without longitudinal reinforcement, a constant value of ch l is often proposed in the literature equal to /2h (RILEM TC 162TDF 2001, Kooiman 2000, Iyengar et al. 1998, Massicotte 2004), 2 /3h (AFGC-SETRA 2002), h (fib Model Code 2010 2011, CNR-DT 204 2006) or 2h (Strack 2008), where h is representative of the height of cross-section. In some literature, ch l in R/FRC flexural elements is related to the level of prescribing load (Casanova and Rossi 1997, UNI 11188 2004) due to the fact that spacing between cracks changes by variation of the load level. Nevertheless, the minimum value between /2h (Massicotte 2004) or depth of neutral axes (fib Model Code 2010 2011), and the average crack spacing () rm s is ordinarily considered for the derivation of characteristic length in R/FRC elements. rm s can be estimated by following the approach represented by the fib Model Code 2010 (2011). The adopted magnitude of the characteristic length in a numerical modelling may affect deformational results and flexural capacity of the simulated element, whereas, by the increase in ch l , the resisting bending moment reduces while a wider crack is predicted (Montaignac et al. 2012). Therefore, a large value ch l is more appropriate when the maximum crack width at a localized crack is aiming, whereas, the smaller value of the characteristic length is recommended when the average crack spacing is concerned. 19 Literature Review 2.8 Characterization of the post-cracking response of FRC 2.8.1 Uniaxial tensile test The uniaxial tensile test, when displacement controlled, represents the unique approach that directly yields the tensile constitutive laws of plain and fibre reinforced concrete (Hordijk 1991, Van Mier and Van Vliet 2002). According to Naaman and Reinhardt (2006), fibre reinforced concretes can be classified regarding the post-cracking responses in uniaxial tension: strain-softening and strain-hardening FRCs schematically compared in Figure 2.8. In the case of strain-softening FRCs, after initiation of the first crack at the stress that can be considered equal to the tensile strength of the cement matrix ( ct f ), the tensile bearing capacity of the element is reduced abruptly by widening the localised crack until fibres efficiently bridge the crack by undergoing to the pullout mechanism. This mechanism leads to progressive deterioration of the tensile stress by widening of crack. Afterwards, and when debonding of fibres through the concrete matrix is fulfilled, the second drop may occur in stress-strain response up to ultimate separation. In fact, the increase in post-cracking strength of the strain-softening type FRCs is unexpected. Alternatively, significant improvement of toughness and increase in energy dissipation in tension with competitive cost are aimed in the strain-softening type FRCs (Naaman 2008, Bentur and Mindess 2007), in which, the volume fraction of fibres is often limited to 2%. Figure 2.8: Schematic representation of tensile stress-elongation of strain-softening and strainhardening FRC (adopted from Naaman (2008)) Chapter 2 26 Figure 2.13: The concept of toughness class for FRC based on the relationship between the flexural stress and CMOD (fib Model Code 2010 2011) Figure 2.14: Tensile constitutive laws of FRC recommended by RILEM TC 162-TDF (2003) 2.9 Post-cracking behaviour of RC and R/FRC flexural members Cracking in concrete is one of the dominant crucial aspects threaten the durability and structural performance of reinforced concrete (RC) structures. This problem is much more pronounced in the case of steel reinforced concrete (S/RC) structural elements since they are often subjected to tensile stress fields. The stiffness and load carrying capacity of RC elements decrease with the formation and propagation of cracks, which can 27 Literature Review detrimentally affect their design requisites at the serviceability and ultimate limit state conditions (SLS and ULS, respectively). Crack propagation in S/RC elements also increases the permeability of concrete, facilitating the ingress of environmental adverse agents to the concrete zones where steel reinforcement is positioned, which promotes its corrosion as faster as wider are the cracks (Arya and Wood 2015). Number of cracks, is another key parameter that increases the rate of corrosion of reinforcement (Arya and Ofori-Darko 1996, Kaufmann and Marti 1998, Pimentel et al. 2010) (see Figure 2.15). Figure 2.15: Effect of crack frequency on cumulative weight loss due to corrosion (Schiessl and Raupach 1997) Reduction of the cross-section area of the steel reinforcement, deterioration of steelto-concrete bond quality, and cracking and spalling out of the concrete cover, are all consequence of concrete cracking and responsible for the significant reduction of load carrying capacity of the element. Nevertheless, the influence of widening of cracks on the long-term durability of reinforced concrete can be negligible when the width of the crack is kept small according to the environmental conditions (Schiessl and Raupach 1997). When crack width is sufficiently small, penetration of aggressive agents, water and oxygen through the crack is suppressed due to the so-called self-healing phenomenon, resulting from calcium-bearing compounds and inoffensive dirt and dust deposits within the cracks. As a consequent, the rate of the corrosion process decreases. In this regard, the design guidelines often restrict the average crack width in the serviceability limit Chapter 2 28 states to a critical value ranging between 0.1 to 0.4 mm depending on the environment surrounding the structures (ACI 224R-01 2001, fib Model Code 2010 2011). 2.10 Tension Stiffening Along a cracked region of a reinforced concrete member, the intact segments of concrete positioned between pairs of adjacent cracks sustain a significant portion of the total tension force due to the concrete-reinforcement interaction. Consequently, owing to the so-called tension stiffening effect, the average strain of the reinforcement between cracks is significantly lower than that one in the plane of a crack. In structural response point of view, the tension stiffening effect can be explained regarding the force-deflection relationship ()F   of a reinforced concrete beam schematised in Figure 2.16. Figure 2.16: Schematic force-deflection response of the RC element (adopted from Gilbert (2007) and modified) According to Figure 2.16, the force-deflection response of the beam is linear-elastic up to crack initiation, and the stiffness of this stage is proportional to the moment of inertia of the uncracked cross-section and the elasticity modulus of the constituent materials. When the extreme tensile fibre of the RC section attains the concrete tensile strength () ct f , the first crack forms at the cr F load level, followed by an abrupt reduction of the flexural stiffness. The genuine post-cracking response of the beam is ranging between the upper bound, which is based on the assumption that the tensile stress of concrete in cracked region remains equal to ct f , and the lower bound that is determined 29 Literature Review by neglecting the contribution of tensile concrete in cracked region, represented in Figure 2.16 by AC and ABC curves, respectively. In fact, the tension stiffening effect is defined as the difference between the actual response of the reinforced concrete flexural member and its zero-tension response (ABC curve) (Gilbert 2007). It should be, however, remarked that the amplitude of tension stiffening effect decreases up to the attainment of the force corresponding to the yield initiation in steel bar () sy F beyond which it would totally vanish. To be specific, the contribution of tension stiffening increases by the decrease in the quantities of tensile reinforcement. For instance, in slab element with the percentage of longitudinal steel reinforcement () s  lower than 0.3% , more than 50% of the stiffness of the cracked member at service loads may be attained by the contribution of the tension stiffening effect (Gilbert 2007). In such a case, neglecting the tension stiffening may lead to overestimating the flexural deflection by a large proportion. It is, however, remarkable that significant decrease in longitudinal steel bar ratio, lower than the minimum reinforcement ratio recommended by design codes may lead to yielding of steel bar at almost crack initiation stage and frustrate the tension-stiffening effect. Differently from the plain concrete assumed to carry tension just between the cracks only, fibre reinforced concretes are able to transmit significant tensile forces at a crack plane owing to the reinforcing effects provided by the fibres that cross the cracks. Therefore, the tension stiffening in fibre reinforced concrete elements comprises a combination of mechanisms between the cracks and at the cracks (Bischoff 2003). Therefore, the tension stiffening effect is more noticeable in R/FRC elements compared to RC members(Abrishami and Mitchell 1997). In a simplified method, the contribution of tension stiffening to the post-cracking load-deflection response of reinforced concrete beams is taken into account by semiempirical formulations proposed for determining the average effective moment of inertia ( e I ) for a cracked member, such as the case of the equation developed by Branson (1965) for steel reinforced concrete, which is also recommended in ACI 318-05 (2005): Chapter 2 30 33 1 cr cr e g cr g aa MM I I I I MM                     (2.3) being cr M the cracking moment, a M the maximum applied moment, cr I the moment of inertia of the cracked transformed section, and g I the moment of inertia of the gross section. According to some researches, the Branson’s approach (Equation 2.3), however, overestimates noticeably the average stiffness of reinforced concrete members containing low percentages of steel reinforcement (Bischoff 2005), as the same as concrete beams reinforced with longitudinal FRP bars (Benmokrane et al. 1996). Following the approach recommended by Eurocode 2 (1992), a more accurate prediction of the post-cracking response of reinforced concrete members (Gilbert 2007) may be obtained. In this approach, a reinforced concrete member is subdivided into the uncracked region, where the concrete and steel reinforcement both behave elastically and fully cracked region in which the tensile contribution of concrete is overlooked and the tensile force is entirely carried out by the steel bar. In this approach, the effective curvature of the element is obtained by Equation (2.4): ,, (1 ) e uncr reg cr reg        (2.4) where ,uncr reg  and ,cr reg  are the curvatures calculated for the uncracked and fully cracked regions, respectively. For the bending moment acting at the serviceability limit states () s M , the compressive behaviour of concrete and the tensile response of reinforcement can be assumed linearly and elastic, and e  , ,uncr reg  and ,cr reg  are determined from the following equations: s e ce M EI   (2.5) ,s uncr reg c uncr M EI   (2.6) ,s cr reg c cr M EI   (2.7) 31 Literature Review where e I is the average effective moment of inertia and, uncr I and cr I are the moment of inertia of uncracked and cracked cross-section, respectively. In Equation (2.4),  is a distribution coefficient accounting for moment level and degree of cracking, given by: 2 12 1sr s          (2.8) being 1  and 2  constants taking into account the influence of duration of the loading or repeated loading on the average strain. In this equation, sr  is the stress in the tensile reinforcement in the cracked section under the loading conditions leading to the crack initiation (i.e. cr MM ), and s  is the stress in the tensile reinforcement when the service moment () s M is acting. Substituting Equations (2.5) to (2.8) in Equation (2.4), for a flexural member with deformed bars under short-term loading, the effective moment of inertia is determined by the following equation: 2 11 cr e cr cr uncr s I IIM IM           (2.9) Apart from the above-described approaches, the influence of tension stiffening effect on the post-cracking deflection response of reinforced concrete beams can be analysed through theoretical models adopting non-linear constitutive laws. Some of these models consider an average stress-strain relationship which was firstly introduced by Rashid (1968) based on the smeared crack model, which due to its simplicity and flexibility represent desirable methods for finite element simulation. In this approach, the tension stiffening effects are taken into account by modifying the constitutive law of tensile reinforcement (Feenstra and Borst de 1995) or more commonly, tensile concrete surrounding the reinforcement (Barros et al. 2001, Prakhya and Morley 1990). In some other approaches, the tension stiffening effect is mobilised in the bond-slip relationship between the reinforcement and surrounding concrete along an interaction zone, which was first advanced by Saliger (1936). These models are developed for a Chapter 2 32 concrete prism with an embedded reinforcing bar subjected to the tensile force applied on the two protruding ends of the bar, as depicted in Figure 2.17. Figure 2.17: Conventionally reinforced direct tension specimen When a crack initiates in a tensile reinforced concrete element, the bond between the concrete and the bar is stimulated by the movement of the bar ribs, leading to transfer a shear load to the intact concrete positioned between the cracks. This mechanism can be mathematically represented by the differential Equation (2.10), adopted in numerous formulations available in the literature (Balázs 1993, Fehling and Leutbecher 2007, Stang and Aarre 1992, Bianco et al. 2009):   2 1 2 () ( ) 0 d s x J s x dx   (2.10) where ()sx and   ()sx  are, respectively, sliding and corresponding shear bond stress over the bond length, and 1 J is a constant determined from the following: 1() pp r r c c LL JE A E A  (2.11) 33 Literature Review being r A and c A the cross-sectional area of reinforcement and surrounding concrete, respectively (Figure 2.17), and () pb Ld   is the perimeter of the reinforcing bar of diameter b d . r E and c E are also the elastic modulus of reinforcement and surrounding concrete, in turn. To solve the differential Equation (2.10), it is fundamental considering a proper local shear bond stress-slip (or in brief bond-slip) relationship ()s   between the reinforcing bar and surrounding concrete. 2.11 Bond Behaviour The bond between the concrete and reinforcing bar is an integration of chemical adhesion, friction, and mechanical interaction (Lutz and Gergely 1967). For smooth bar, the mechanical interaction is caused by the micro-roughness of the bar surface and therefore, is extremely small. In deformed bars, however, the mechanical interaction is due to the presence of the ribs, which prevents relative movements at the interface (Figure 2.18). (a) (b) Figure 2.18: Schematic deformation of concrete surrounding a deformed steel bar after the formation of internal cracks; (a) longitudinal section of the axially loaded specimen and (b) crosssection (Goto 1971); Chapter 2 34 In this case, the bond action spread out from the bar into surrounding concrete, which can be subdivided into a stress component parallel to the bar axis, denoted the bond stress, and radial components, designated as normal or splitting stress balanced by tensile stress ring in the concrete (Tepfers 1973). In concrete cover, where the ring is in its thinnest region, the tensile stress may exceed the tensile strength of the concrete, leading to the formation of longitudinal splitting cracks in concrete (Figure 2.19). In this regard, the bond strength is noticeably influenced by concrete strength class (fib Model Code 2010 2011, ACI 318-05 2005, Eurocode 2 1992). Figure 2.20: Schematic representation of radial components of the bond forces in an anchorage zone (Tepfers 1973) Presence of the normal stress is quite necessary to assure transference of the load through the mechanical bond. Otherwise, the interaction between the concrete and reinforcement may be lost by the so-called splitting failure in which the concrete surrounding the reinforcement bar is penetrated by longitudinal splitting cracks (Lundgren 2005). On the contrary, if the concrete cover is large enough and concrete surrounding the reinforcement bar is well confined to withstand the normal stresses, splitting cracks do not form and surrounding concrete appears uncracked macroscopically until pullout failure occurs. The pullout failure is characterised by the propagation of shear cracks between the adjacent ribs, which represents the upper limit for the bond strength. The bond-slip relationship can be investigated experimentally through different groups of the test setup. The direct pullout test with a short anchorage (embedment length to bar diameter ratio less than 5) is a relatively convenient and inexpensive way for investigation the primary parameters affecting the bond behaviour. It is a test widely used in the literature for steel bar, FRP bar, or FRP laminates utilised in the near-surface 35 Literature Review mounted (NSM) method (Eligehausen et al. 1983, Hungspreug 1981, Malvar and Warren 1992, Lundgren 2005). Typical setup of the direct pullout test is depicted in Figure 2.20, according to which the tensile force F is applied to the protruding end of the bar of the cross-sectional perimeter p L , while the bar relative slip to concrete is measured at loaded-end and free loaded-end of the bar ( le s and fle s in Figure 2.20). The bond behaviour is then represented by s   curve, where  is the average bond stress over the embedded length () e L given by the following equation: ep F LL   (2.12) In the pullout test, the tensile force applied to the bar is balanced by the compression introduced into concrete, which does not occur in a reality, since the bar and concrete are both in tension. Furthermore, due to confining effects in concrete at the bearing end (Aiello et al. 2007), higher resistance to splitting failure is determined when compared with real situations (Cairns and Plizzari 2003). Figure 2.21: Typical direct pullout test setup Chapter 2 42   ,maxk r sm cm ws   (2.18) where ,maxr s is the maximum crack spacing determined by Equation (2.17) and the term sm cm   is representative of the difference between the mean strain in the reinforcement and concrete between two adjacent cracks, given by the following equation:   5, , 1 st ctm sm cm s s eff s s s eff kf EE            (2.19) In above equation, st  is the stress in the steel reinforcement at a cracked section, ctm f is the mean tensile strength of concrete, s E is the modulus of elasticity of steel reinforcement, s  is the ratio between modulus of elasticity of steel bar and concrete (i.e. / sc EE ), and 5 k is a coefficient depends on short-term and long term loading. The maximum value of crack width max ()w can be considered equal to k w (CEB-FIP 1978 1984) which is correlated to the average crack width m ()w by the following equation: max 6 m w k w (2.20) 6 k is a statistical coefficient recommended to be equal to 1.7 (Borosnyói and Balázs 2005). A numerous experimental researches presented in the literature to investigate crack width and spacing in fibre reinforced concrete (Abrishami and Mitchell 1997, Noghabai 2000, Bischoff 2003, Deluce and Vecchio 2013, Tan et al. 1995, Vandewalle 2000), which reveal a tendency of reduction of crack spacing and crack width by the increase in volume content and aspect ratio of fibres. In this regard, one of the most frequently used formulation for the prediction of crack spacing in fibre reinforced concrete elements is the one proposed by RILEM TC 162-TDF (Vandewalle et al. 2003) given by the following equation: 34 , 50 50 0.25 / b rm s eff f f d s k k ld               (2.21) Equation (2.21) is in fact, a modified feature of EN 1992-1-1 (2004) formulation for non-fibrous concrete (Equation 2.17) in which / ff ld is the fibre aspect ratio (being f l 43 Literature Review and f d the fibre length and diameter). A more or less similar approach is proposed by Moffatt (2001) to estimate the average crack spacing of reinforced strain-softening type of fibre reinforced concrete elements as follows: 34 , 50 0.25 1 b res rm s ef cr df s k k f            (2.22) where res f is the post-cracking residual concrete stress, and cr f is the cracking stress of the concrete. In the approach proposed by RILEM TC 162-TDF (Vandewalle et al. 2003), the design value of the crack width   d w for R/FRC members subjected principally to flexure or tension is determined by the following equation: 7d rm sm w k s ε (2.23) where 7 k is a constant coefficient correlating the average crack width to the design value and sm  is the average strain in the reinforcement. Despite the main affecting parameters of fibre reinforced concrete have been included in the approach proposed by RILEM TC 162-TDF (Vandewalle et al. 2003) (Equation 2.21) and Moffatt (2001) (Equation 2.22), However, the predictive performance of these equations are arguable (Deluce and Vecchio 2013). Deluce et al. (2014) proposed Equation (2.24) for the average crack spacing of steel fibre reinforced concrete under uniaxial strain when cracking is stabilised. 9 10 8 210 b rm eff mi s k k s c k s       (2.24) The above formulation is based on the approach recommended by CEB-FIP (1978), in which eff c is the effective concrete cover taken 1.5 times the maximum aggregate size, b s is representative of the effective longitudinal bar spacing given as follows: 2 , 0.5 15 b bb s eff d sd    (2.25) Chapter 2 44 being b d the bar diameter, and ,s eff  the effective reinforcement ratio. In Equation (2.26), mi s is a factor taking into account the influence of steel fibres by considering the number of fibres bridging a crack determined by the following equation: , 2 s eff f f mi f bf V sk dd   (2.26) where f d is the fibre diameter and f V is the volume fraction of fibres, restricted to a maximum value of 0.015. Parameter f  is also the fibre orientation factor can be taken equal to 0.5 for the random three-dimensional orientation of fibres in infinite elements (Stroeven and Hu 2006). In Equation (2.26), f k is the fibre effectiveness factor given by: 1.0 50 ff f f lV kd  (2.27) In Equation (2.24), the beneficial effects of steel fibres is also taken into consideration by the factor 8 k determined from the following equation:   1 8 min( ,0.015) 11 0.015 ff V kk        (2.28) Furthermore, in this equation, the bond characteristics of the reinforcing bars are accounted by 9 k factor of 0.4 or 0.8 for deformed bar and plain bars or pre-stressing tendons, respectively. 10 k also account the strain conditions in the concrete member determined by Equation (2.29):   12 10 1 0.25 2 k     (2.29) being 1  and 2  the largest and smallest tensile strains in the concrete, respectively. In the approach proposed by fib Model Code 2010 (2011) the contribution of fibres on the cracking characteristics is taken into account using the concept of average residual strength of the fibre reinforced concrete at serviceability limit states , () Fts m f (see Figure 45 Literature Review 2.12). In this approach, the average spacing between cracks () rm s is estimated by multiplying a factor of 1.5 to the maximum transmission length given by the following equation:   , ,max 11 , 4 ctm Fts m b s bm s eff ff d l k c    (2.30) where 11 k is an empirical coefficient for simulating the influence of the concrete cover and bm  is the average bond strength between reinforcing bars and surrounding concreteRemark that Equation (2.30) is valid for FRCs whose average residual strength at serviceability limit states , () Fts m f is less than the crack strength of the concrete matrix () ctm f . Regarding fib Model Code 2010 (2011) recommendations, the design values of crack width () d w is determined by equation below: ,max 12 13 2() s d st sr sh s s l w k k E E       (2.31) where st  is the stress of steel reinforcement in a crack, sh  is the shrinkage strain, and sr  is the steel stress at the crack section in the crack formation stage. 12 k is also an empirical coefficient to assess the mean strain over ,maxs l , and 13 k is a coefficient takes into account the shrinkage contribution. 2.13 Concluding remarks In this chapter, the vulnerable of concrete in tension was explained by detailing its morphology and the different approaches employed to overcome the tensile weakness of concrete, including taking advantage of longitudinal steel or FRP bar lonely or simultaneously in a hybrid-reinforcing scheme were discussed. The decrease in sound cross-sectional area, spalling the concrete cover, diminishing the concrete cross-sectional area, and deterioration of bond between reinforcing bar and concrete are the main eventuates of corrosion of steel bar affect negatively the load carrying capacity and the structural safety indexes. Although substitution of steel Chapter 2 46 reinforcement by the one made of FRP is an effective remedy against corrosion, the lack of ductility and the large deflection of FRP reinforced concrete (FRP/RC) elements are the main obstacles for using the FRP bar as the sole reinforcement. These drawbacks can be overcome by a hybrid flexural reinforcing (HFR) scheme obtained by placing the FRP bars near the outer surface of the tensile zone and steel bars at the inner level. In this chapter, taking advantage of fibre in concrete was discussed as a novel approach to mitigate the tensile vulnerability in tension. Cracking mechanism of plain and fibre reinforced concrete was discussed and different approaches to simulate cracking behaviour of concrete were reviewed among which the fictitious crack model Hillerborg et al. (1976) and the crack band model of Bažant (Bažant and Oh 1983) were explained. The various approaches proposed in the literature to identify the post-cracking behaviour of fibre reinforced concrete were also presented. The chapter also includes a literature review on relevant aspects affecting the interaction between reinforcing bar and plain and fibre reinforced concrete, and its correlation to the tension-stiffening phenomenon was described. It was illustrated that the contribution of tension stiffening on the overall response of RC flexural elements is amplified by the reduction of tensile longitudinal reinforcement of the cross-section. In addition, the noticeable effect of fibres on the tension stiffening of R/FRC is highlighted. Regarding the literature, the local bond models for steel/RC, FRP/RC, and R/FRC were presented and compared. Finally, the different approach proposed in the literature for predicting crack width and crack spacing of RC and R/FRC flexural elements were detailed. 47 Chapter 3 Experimental Evaluation of Flexural Response of R/SFRSCC Elements 3.1 Introduction This chapter describes a comprehensive experimental programme conducted to evaluate the load carrying capacity and cracking behaviour of flexural elements reinforced with a hybrid system composed by discrete hooked-end steel fibres and longitudinal steel and fibre reinforced polymer (FRP) bars. In this evaluation, the influence of the key parameters affecting the material properties of SFRSCC, including the concrete strength class and volume fraction of fibres is assessed. Furthermore, the scheme and percentages of longitudinal reinforcement is studied. 48 Chapter 3 3.2 Composition and Development of SFRSCC The test programme was carried out on three series of steel fibre reinforced selfcompacting concrete (SFRSCC) designated by C15-f45, C25-f60, and C45-f90, where the numbers after letters “C” and “f ” represent, respectively, the average compressive strength of SFRSCC at 28 days in Mega Pascal, and the content of hooked-end steel fibres in kilograms per concrete cubic meter. The adopted contents of fibres of 45, 60, and 90 kg/m3 are conventionally used in pavements, prefabricated elements, and elevatedSFRSCC slab systems, respectively. Table 3.1: Composition of the developed SFRSCCs (per 1 m3) SFRSCC indication C15-f45 C25-f60 C45-f90 Compressive strength a [MPa] 15 25 45 Cement [kg] 220 350 423 Water [kg] 105 160 144 Water-to-cement ratio [-] 0.48 0.46 0.34 Superplasticizer [kg] 6.08 9.50 5.92 Limestone filler [kg] - - 362 Fly-ash [kg] 100 150 - Fine river sand [kg] 437 233 220 Coarse river sand [kg] 693 698 671 Crushed granite [kg] 615 580 491 VMA b [g] 22 22 - Fibre type - HESF1c HESF1 HESF1 Supplier - IBERMIX IBERMIX IBERMIX Content [kg/m3] 45 60 90 Volume fraction [%] 0.6 0.8 1.1 Fibre’s length [mm] 35 35 35 Fibre’s diameter [mm] 0.55 0.55 0.55 Aspect ratio - 63 63 63 Tensile Strength [MPa] 1300 1300 1300 a Nominal b VMA: Viscosity modifying admixture c HESF: Hooked end steel fibres The compressive strength of SFRSCC of C15-f45 series is lower than the minimum strength often recommended in design guidelines for structural applications, and hence it was unplanned in the predefined test programme. Nevertheless, to evaluate the influence of variables of the research in a broader domain, this series of SFRSCC was additionally included. The SFRSCCs series were developed by using the following materials: Cement Cem 42.5R type I, water, three types of aggregates including fine river sand of maximum Experimental Evaluation of Flexural Response of R/SFRSCC Elements 49 diameter of 0.59 mm, coarse river sand of maximum diameter of 4.76 mm, crushed granite of maximum diameter of 12 mm, superplasticizer of third-generation based on polycarboxylates (SikaViscoCrete 3005), limestone filler, fly-ash, viscosity modifying admixture (Chryso®PlastV70), and hooked end steel fibres. The proportions of the constituents of the series of the developed SFRSCCs are detailed in Table 3.1. 3.2.1 Mixing procedure To blend the components, a planetary mixer of the vertical axis and 360 litres capacity was used. The aggregates were put into the mixer from the highest to the lowest size (i.e. first the crushed granite, then the coarse river sand, and finally the fine river sand) and they were mixed for about one minute. Subsequently, a share of the water was added to the mixed aggregates to saturate the aggregates and mixing continued for one minute. Then the cement, fly ash or limestone filler, the viscosity modifying admixture, if any, the superplasticiser, and the remaining water were added. Mixing was continued for one more minute. Afterwards, the fibres were slowly added to the mixture for preventing fibres blockage in the mixture. The mixing process continued until proper homogeneity of the mixture was attained. Before casting, the flowability and the segregation resistance of the mixtures were examined through the slump flow test executed according to the recommendation of RILEM TC 188-CSC (2006) to assess the self-compacting nature of the developed concretes whose results are summarised in Table 3.2. Table 3.2: Slump flow test results C15-f45 C25-f60 C45-f90 f D [mm] 740 660 570 50 T [s] 1.5 2.1 2.9 3.2.2 Specimens Regarding the shape, geometry, and reinforcing scheme, four types of specimen were built as shown in Figure 3.1, including standard cylinders of 150 mm diameter and 300 mm height and prismatic beams of 150×150 mm cross-section and 600 mm length used to evaluate, respectively, the compressive strength and post-cracking flexural tensile capacity of the developed SFRSCCs. Additionally, beams of 2500 mm length and 50 Chapter 3 100×150 cross-section with two longitudinal reinforcing schemes were constructed to assess the influence of the investigated variables on the flexural capacity and cracking behaviour of SFRSCC flexurally reinforced beams. These beams were tested under fourpoint bending test configuration. Figure 3.1: Geometry of the specimens: (a) standard cylinder for the uniaxial compression test; (b) SFRSCC beams used in three-point notched beam bending test; and beam reinforced longitudinally with (c) steel bar and (d) steel and GFRP bars used in four-point bending test (dimensions in mm) The samples of each series of the developed SFRSCCs were built by three batches with the same composition according to the scheme detailed in Table 3.2. Experimental Evaluation of Flexural Response of R/SFRSCC Elements 51 Table 3.3: Adopted scheme for constructing the samples SFRSCC series C15-f45 C25-f60 C45-f90 Batch number 1 2 3 1 2 3 1 2 3 Number of samples Cylinder 2 2 3 2 2 3 3 2 2 Beam of 600 mm length 2 2 6 2 2 6 2 2 6 Beam of 2500 mm length 3a 3b - 3a 3b - 3a 3b - a with longitudinal steel bar b with longitudinal steel and glass fibre reinforced polymer (GFRP) bar 3.2.3 Curing and preparation of specimens After casting, the samples were covered by wet sack for a day to ensure proper moisture for the curing process. The samples were subsequently demoulded and kept covered with wet sack until the day before testing. In the day before testing, the irregular surface of the cylinders was rectified by a wet saw machine, and a transversal notch was introduced at mid-length of the prismatic beams of 600 mm. 3.3 Mechanical properties of the constituent materials 3.3.1 Compressive behaviour of the SFRSCCs The compressive strength of each series of SFRSCCs at 28 days ( cm f ) was obtained through the uniaxial compressive test detailed in Figure 3.2, where the compressive force was provided by a servo-controlled actuator of 3000 kN load carrying capacity. The test was displacement controlled by the internal linear variable displacement transducer (LVDT) of the loading machine. The test methodology was consistent with the specifications advanced by RILEM TC 148-SSC (2000) except, the prescribed rate of the deformation was increased to 10 μm/s as recommended in JSCE-SF4 (1984). This is due to the increase in compressive deformation of concrete by the addition of fibres (Barros and Figueiras 1999b, Cunha 2010). 58 Chapter 3 Figure 3.8: Post-cracking stress-crack width relationship of SFRSCCs determined by following the recommendations of fib Model Code 2010 (2011) Figure 3.9: The concept of toughness class for FRC based on the relationship between the flexural stress and CMOD (fib Model Code 2010 2011) Table 3.6: Toughness classes according to the fib Model Code 2010 (2011) Case study Interval of 1Rk f [MPa] Interval of 31 / R k R k ff Toughness classification C15-f45 [2-3] [0.5-0.7] 2a C25-f60 [4-5] [0.7-0.9] 4b C45-f90 [8-9] [0.7-0.9] 8b 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 Tensile stress, σct [MPa] Crack opening, w[mm] C15-f45 C25-f60 C45-f90 Experimental Evaluation of Flexural Response of R/SFRSCC Elements 59 3.3.2.2 Characterisation of the post-cracking of the SFRSCCs by inverse analysis The mode I fracture parameters of the developed SFRSCCs were also determined by the inverse analysis (IA), by using the force-deflection results registered in the performed 3PNBBT. For this purpose, the 3PNBBT was simulated with a constitutive model implemented in a software based on the finite element method (FEM), FEMIX V4.0, which is described elsewhere (Ventura-Gouveia et al. 2011). By using the finite element mesh shown in Figure 3.10, a notched FRC beam with a span length of 500 mm, a crosssection of 150×150 mm2, and a notch of 5 mm width and 25 mm depth was simulated. Figure 3.10: Finite element mesh, loading and support conditions of the specimen adopted for the simulation of three-point notched beam bending tests (dimensions in mm) This beam was modelled by 8-node serendipity plane stress finite elements with Gauss-Legendre integration scheme of 2×2 integration points (IP). The opening of the crack over the notch was also simulated by introducing in the plane of symmetry of the specimen interface finite elements of six nodes with Gauss–Lobatto integration scheme of 1×3 IP for assuring the crack progresses along this plane. Apart from the interface finite elements, the remaining finite elements were considered with linear-elastic behaviour in compression and tension depicted, respectively, in Figure 3.11(a) and (b). The compression and the tensile behaviour up to crack initiation of the material over the notch (simulated by the interface finite elements) 60 Chapter 3 were also assumed linear-elastic, while its post-cracking tensile stage was simulated by a trilinear stress-crack opening relationship depicted in Figure 3.11(c). In the performed IA, a specific value of the tensile strength ( ctm f ) was assumed, determined by Equation (3.1) (Table 3.3). (a) (b) (c) Figure 3.11: Constitutive laws of FRC adopted in the performed FEM simulation; (a) and (b) linear elastic stress-strain behaviour of intact concrete in compression and tension, respectively, and (c) mode I fracture property of cracked FRC in terms of the stress-crack opening relationship In a trial computation, the ct w   of the fracture mode I constitutive law of the interface finite elements is obtained by best fitting the force-deflection relationship registered experimentally in the 3PNBBT. In a generic kth step of the incremental procedure of the notched beam mid-span deflection () k  , the numerical force () k Num F determined by the FEM is compared to the force registered experimentally () k Exp F to evaluate the deviation history in terms of force ( F D ) and toughness () T D during the loading process of the beam, by adopting the following equations: / k k k F Exp Num Exp D F F F (3.6)       / k k k F F F T Exp Num Exp D A A A        (3.7) Experimental Evaluation of Flexural Response of R/SFRSCC Elements 61 where   k F Exp A   and   k F Num A   are the area beneath, respectively, the experimental and numerical force-deflection curves up to the central deflection of k  . In each step of the calculation, if F D or T D exceeds the considered tolerance (2%), the procedure is interrupted, and a new simulation is automatically started by adopting new values for the post-cracking parameters of the FRC. Figure 3.12 represents the accuracy of performed inverse analysis by comparing the force-deflection responses obtained from the inverse approach with the ones registered in the experiment. Figure 3.13 represents the post-cracking response of the three series of SFRSCCs determined by the inverse approach in terms of stress-crack opening displacement, whose intervening parameters are summarised in Table 3.7. (a) (b) (c) Figure 3.12: Predictive performance of the inverse analysis of the three-point notched beam bending tests performed on the (a) C15-f45, (b) C25-f60, and (c) C45-f90 series of specimens 0 5 10 15 20 25 30 35 40 0.0 1.0 2.0 3.0 4.0 Force, F[kN] Deflection, δ[mm] Experimental (average) Inverse analysis 0 5 10 15 20 25 30 35 40 0.0 1.0 2.0 3.0 4.0 Force, F[kN] Deflection, δ[mm] Experimental (average) Inverse analysis 0 5 10 15 20 25 30 35 40 0.0 1.0 2.0 3.0 4.0 Force, F[kN] Deflection, δ[mm] Experimental (average) Inverse analysis 62 Chapter 3 Figure 3.13: Post-cracking stress-crack width relationship of SFRSCCs determined by inverse analysis Table 3.7: Values of parameters defining the constitutive laws of SFRSCCs SFRSCC series [MPa] [-] [-] [mm] [mm] [mm] C15-f45 0.89 1.90 1.60 0.01 0.20 5.00 C25-f60 1.87 1.45 0.20 0.10 4.90 5.00 C45-f90 3.27 1.35 0.60 0.20 3.00 5.00 3.3.2.3 Comparison between the prediction of fib model code 2010 and the results of inverse analysis In Figure 3.14, the stress-crack opening relationship of the SFRSCCs obtained by the inverse analysis is compared to the ones determined by following the recommendations of fib Model Code 2010 (2011). This figure reveals following the recommendation of fib Model Code 2010 (2011) may lead to relatively higher post-cracking constitutive laws when compared to the inverse approach. 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 Tensile stress, σct [MPa] Crack width, w[mm] C15-f45 C25-f60 C45-f90 ct cr f   1 a 2 a 1 w 2 w u w Experimental Evaluation of Flexural Response of R/SFRSCC Elements 63 (a) (b) (c) Figure 3.14: Comparison between the post-cracking response of SFRSCCs determined by inverse analysis and the ones recommended by fib Model Code 2010 (2011) for the (a) C15-f45, (b) C25-f60, and (c) C45-f90 series of specimens 3.3.3 Mechanical properties of longitudinal reinforcement The tensile behaviour of steel and glass fibre reinforced polymer (GFRP) bars used in the test programme to reinforce the R/FRC beams were determined by executing the direct tensile test according to the ASTM A370 (2014) recommendation. The adopted test setup is depicted in Figure 3.15 according to which a pair of self-gripping clamps was utilised to transfer the tensile force to the bars. The bottom grip was fixed on the bottom surface of the loading machine while the top one, was pulled up by a hydraulic actuator of 100 kN. The clamping system consisted of interchangeable wedge jaws moved down and fixed into the bar by the initial gripping force generated by hydraulic pressure. The 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 Tensile stress, σct [MPa] Crack width, w[mm] fib Model Code 2010 (2011) Inverse analysis 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 Tensile stress, σct [MPa] Crack width, w[mm] fib Model Code 2010 (2011) Inverse analysis 0 1 2 3 4 5 6 0.0 0.5 1.0 1.5 2.0 2.5 Tensile stress, σct [MPa] Crack width, w[mm] fib Model Code 2010 (2011) Inverse analysis 64 Chapter 3 direct tensile test was executed on three representatives steel and GFRP bars of 300 mm and 1200 mm length, respectively. The preliminary test conducted on GFRP bars presented undesirable slippage of GFRP bar because of its smooth surface. Furthermore, a premature failure was observed at clamped extremities of GFRP bars due to localisation of tensile stresses. To overcome these drawbacks, two extremities of the GFRP bars were covered by a pair of steel tubes of 300 mm length, 15 mm outer diameter, and 2 mm thickness (Figure 3.15b). The fixed connection between GFRP bar and steel tube was ensured by a high strength epoxy resin with a tensile strength of 30 MPa, which was cured for two weeks (Figure 3.15c). The uniaxial tensile test executed on bars was displacement controlled with a constant rate of 3 m/s. During loading, elongation of the bars was measured by a clip-on extensometer with a measuring length of 50 mm, mounted at mid-length of the specimens as shown in Figure 3.15(a) and (b). The test continued until the failure of specimens. (a) (b) (c) Figure 3.15: Direct tensile test conducted on (a) steel, (b) GFRP bars, and (c) details of transferring the applied load to GFRP Experimental Evaluation of Flexural Response of R/SFRSCC Elements 65 The average tensile stress-strain relationship of the steel and GFRP bars are depicted in Figure 3.16(a) and (b), respectively, where the stress and strain are determined, respectively, by dividing the total imposed axial force to the bar cross-section (=50.27 mm2), and dividing the elongation measured by the extensometer to the initial measuring length (=50 mm). Considering diagrams of Figure 3.16, mechanical properties of reinforcing bars including, modulus of elasticity of steel and GFRP bars (s E and ), G E stress and strain of steel bars at yielding ( sy  and sy  ) and ultimate stress and strain of steel bars ( su  and su  ), and ultimate stress and strain of GFRP bars ( Gu  and Gu  ) were determined, and are summarised in Table 3.8. (a) (b) Figure 3.16: Average tensile stress-strain relationship of (a) steel and (b) GFRP bars Table 3.8: Mechanical properties of steel and GFRP bars Steel bar GFRP bar s E sy  sy  su  su  G E Gu  Gu  [GPa] [MPa] [‰] [MPa] [‰] [GPa] [MPa] [‰] 205 575 2.8 605 32 58 1058 18.0 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 35 Tensile stress, σt [MPa] Tensile strain, εt [‰] 0 200 400 600 800 1000 1200 0 5 10 15 20 25 30 35 Tensile stress, σt [MPa] Tensile strain, εt [‰] 66 Chapter 3 3.4 Flexural behaviour of SFRSCC beams hybrid reinforced longitudinally The flexural behaviour of reinforced FRC (R/FRC) beams was evaluated in a fourpoint bending test configuration. The test specimens were beams of 2500 mm length, with a cross-section of 100 mm height and 150 mm width, reinforced longitudinally by two different schemes. Three series of which were reinforced with a 8 mm diameter steel bar (SR beams) with a 40 mm concrete cover, and three others were reinforced by either steel and GFRP bars (SGR beams) both of 8 mm diameter, with a concrete cover of 40 mm and 20 mm, respectively (see Figures 3.1c and 3.1d). The adopted geometric properties of the R/FRC beams reduce the flexural capacity of the beams and prevent the occurrence of shear failure without using any transversal reinforcement. Table 3.9: Designation and reinforcement details of the beams Beam’s designation Num. of samples Constituent SFRSCC Long. steel bar Long. GFRP bar s  [%] G  [%] SR/FRC1545 3 C15-f45 1Φ8 - 0.56 - SGR/FRC1545 3 C15-f45 1Φ8 1Φ8 0.56 0.42 SR/FRC2560 3 C25-f60 1Φ8 - 0.56 - SGR/FRC2560 3 C25-f60 1Φ8 1Φ8 0.56 0.42 SR/FRC4590 3 C45-f90 1Φ8 - 0.56 - SGR/FRC4590 3 C45-f90 1Φ8 1Φ8 0.56 0.42 Each series of beams includes three samples of similar geometric properties and reinforcement scheme. The beams are generally designated by xR/FRCyz (Table 3.9), where x represents the longitudinal reinforcement scheme, which can be “S” and “SG” in the case of beams reinforced exclusively with a steel bar, and with steel and GFRP bar, respectively. The letters “y” and “z” represent, respectively, the nominal compressive strength (in MPa) of the beam’s SFRSCC and the content of fibres (in kg/m3) in this composite (see Table 3.1). It is well acknowledged that the contribution of the post-cracking response of FRC in R/FRC elements is much more pronounced when a lower percentage of longitudinal Experimental Evaluation of Flexural Response of R/SFRSCC Elements 67 reinforcement is utilised (Taheri et al. 2012). Therefore, a relatively small reinforcing ratio ( / s s s A bd   and / G G G A bd   for steel and GFRP bar, respectively) were adopted for the SR and SGR series beams to assure yielding of steel bar before the ultimate failure. In this regard, s  of lower than the balanced reinforcement ratio )(SR b  was utilised in the case of SR beams, which is obtained in a condition that the failure mode is attained by concrete crashing in compression while steel bar is yielded in tension, simultaneously, as depicted in Figure 3.17. Neglecting the contribution of precracking tensile concrete SR b  can be determined from Equation (3.8):   1 2 , , 1() SR SR SR b cm NA b Ftu cr b s sy f d f h d d     (3.8) where cm f and sy  are, respectively, the mean concrete compressive strength and the yield strength of steel bar, and , SR NA b d and , SR cr b d are depth of neutral axis for the balanced condition and depth of layer in which concrete cracking strain )(cr  is attained, respectively, which are obtained from the following equations: , SR NA b cu s cu sy dd     (3.9) ,, SR SR cr b NA b cu cr cu dd     (3.10) being cu  the ultimate compressive strain of concrete considered equal to 3.5‰ (fib Model Code 2010 2011), sy  the strain of steel bar at yield initiation represented in Table 3.8, and / ct ccr fE   cracking strain of concrete. In Equation (3.9), s d = 60 mm is the depth of steel reinforcement with respect to the upper face of the cross-section (Figure 3.17). In Equation (3.8) 1  and 2  are two constants defining, respectively, the equivalent concrete compressive strength and height of the compression zone (see Figure 3.17). For the concrete classes utilised in the present research, 1  and 2  are considered equal to 1.0 and 0.8, respectively (fib Model Code 2010 2011). Furthermore, ,Ftu m f is 74 Chapter 3 ) adopted in SR series beams, yielding of steel bar occurs in these series beams, which leads to the crack localisation phenomenon, with the widening of one or two distinct cracks at the ultimate limit state, which is shown in Figures 3.22 and 3.23. Figure 3.23: Crack failure localisation in SR/FRC beam (a) (b) (c) Figure 3.24: Ultimate crack pattern of (a) SR/FRC1545, (b) SR/FRC2560, and (c) SR/FRC4590 series beam Experimental Evaluation of Flexural Response of R/SFRSCC Elements 75 The crack localisation phenomenon is often accompanied by premature rupture of reinforcement at the localised failure crack, and reduction of the ductility of R/FRC elements as reported by Redaelli and Muttoni (2007), Deluce and Vecchio (2013) and Yang et al. (2009) in the case of R/FRC tensile element and Dancygier and Savir (2006), Yang et al. (2010), and Georgiadi-Stefanidi et al. (2011) for the R/FRC flexural beams. However, in the case of SR beams tested in the present research, rupture of steel bar did not occur while a large deflection was prescribed to the beams. This can be ascribed to the fact that the beams are quite shallow and the steel bar is located close to the neutral axes. Regarding the experimental results, it is revealed that the load decay after the localisation of the crack is more significant in the case of SR/FRC4590 (Figure 3.21e) where the higher dosage of fibres is utilised as emphasised by Yang et al. (2009). The load decay due to crack localisation after yield initiation of steel bar is unobserved in the case of SGR series beams due to the increase in longitudinal reinforcement ratio. Accordingly, after yield initiation in steel bar, the load bearing capacity of the SGR beams continues to increases up to a large deflection (>100 mm) which is more than ten times the deflection corresponding to the serviceability limit states. The average force-deflection response of SR and SGR beams are compared in Figure (3.24). (a) (b) Figure 3.25: Average force-central deflection relationship of FRC beams reinforced with (a) steel bar, and (b) steel plus GFRP bars 0 2 4 6 8 10 12 14 16 18 20 010 20 30 40 50 60 Force, F[kN] Central Deflection, δc[mm] SR/FRC1545 SR/FRC2560 SR/FRC4590 Yield initiation δSLS = 9.2 mm 0 2 4 6 8 10 12 14 16 18 20 010 20 30 40 50 60 Force, F[kN] Central Deflection, δc[mm] SGR/FRC1545 SGR/FRC2560 SGR/FRC4590 Yield initiation δSLS = 9.2 mm 76 Chapter 3 Due to the influence of fibres, an increase of 8% and 86% in the load corresponding to the serviceability limit states was obtained for the SR beams, when the content of fibres increases from 45 to 60 kg/m3, and to 90 kg/m3, respectively (Table 3.11). Correspondingly, the increasing rate of in the case of SGR beams is 35% and 59%. According to the test results, comparing to SR/FRC1545 beams, an increase of 11% and 13% was obtained for the load corresponding to the yield initiation )(sy F and peak load )(P F , respectively for SR/FRC2560 beams, while in the case of SR/FRC4590 series beams, the increase in terms of sy F and P F was 48% and 45%, respectively. In the case of SGR series beams, the increase in fibre content from 45 kg/m3 to 60 kg/m3 conducted to an increase of 27% and 25% of sy F and P F , respectively, while an increase of 42% and 27% of sy F and P F , respectively, is obtained for the SGR/FRC4590 beams with respect to the SGR/FRC1545 series beams. Considering the test results depicted in Figure 3.24, using GFRP bar increases the flexural capacity of the SGR beams, which is more pronounced in the case of the load at yield initiation ( sy F ) and peak load ( P F ) of SGR beams. Regarding Table 3.7, an increase of 39%, 59%, and 33% of sy F , and 121%, 143%, and 93% of P F was obtained in the case of SGR/FRC1545, SGR/FRC2560, and SGR/FRC4590, respectively, when compared to corresponding SR beams with the same fibre reinforced concrete constituent. Moreover, an increase of 16% and 44% in service load of SGR/FRC1545 and SGR/FRC2560 was obtained with respect to SR/FRC1545 and SR/FRC2560, respectively. Nevertheless, the use of GFRP longitudinal bar had a negligible influence on SLS F of the SGR/FRC4590 when compared to that of SR/FRC4590. To be specific, the reinforcing effects of GFRP bar of the relatively low modulus of elasticity (=58 GPa), is mobilised in a relatively large deflection (higher than the one considered for the serviceability limit states). () SLS F SLS F Experimental Evaluation of Flexural Response of R/SFRSCC Elements 77 3.4.3 Crack spacing In Figure 3.25 is shown the force versus average crack spacing relationship () rm Fs measured along with the pure bending region (i.e. the 500 mm segment between the two applied forces, see Figures 3.19 and 3.20). The rm s is determined at the level of steel bars in SR and SGR beams by the following equation: 1 cr rm cr L sn  (3.14) where cr L and cr n are, respectively, the distance between two furthest crack at the level of steel reinforcement in the pure bending region and the total number of cracks formed at this level. cr L and cr n were obtained regarding the crack evolution in SR and SGR series beam are represented in Appendix A and B respectively. Figure 3.25 reveals that when cracking is stabilised the average crack spacing along with the pure bending region decreases marginally with the increase in fibres dosage as well as the addition of a longitudinal reinforcing bar. In the crack stabilised stage of SR series beams, the average crack spacing of 62.2, 58.6, and 58.4 mm is determined for SR/FRC1546, SR/FRC2560, and SR/FRC4590, respectively. rm s in the crack stabilisation stage of SGR series beams ranges between 56.2 mm in SGR/FRC1545, 52.2 mm in SGR/FRC2560, and 46.8 mm in SGR/FRC4590. (a) (b) Figure 3.26: Force-average crack spacing measured at steel bar level of the pure bending region in (a) SR and (b) SGR series beams 0 2 4 6 8 10 12 14 16 18 20 050 100 150 200 250 300 350 Force, F[kN] Average crack spacing, srm [mm] SR/FRC1545 SR/FRC2560 SR/FRC4590 Trendline (SR/FRC1545) Trendline (SR/FRC2560) Trendline (SR/FRC4590) 0 2 4 6 8 10 12 14 16 18 20 050 100 150 200 250 300 350 Force, F[kN] Average crack spacing, srm [mm] SGR/FRC1545 SGR/FRC2560 SGR/FRC4590 Trendline (SGR/FRC1545) Trendline (SGR/FRC2560) Trendline (SGR/FRC4590) 78 Chapter 3 3.4.4 Crack width The moment-average crack width relationships of SR and SGR beams are compared in Figure 3.26, in which the average crack opening was determined by the following equation:   // PBR ctm c cr wL L f E n   (3.15) where L is the extension of the pure bending region area with a length () PBR L of 500 mm measured at level of steel bar by the readout of LVDT7 depicted in Figure 3.20b. In this equation, ctm f and c E are the average tensile strength and modulus of elasticity of concrete, respectively, and cr n is the total number of cracks counted in the pure bending region at steel bar level. In Figure 3.26, 0.1mm SLS w is indicated by a dashed line as the average crack width corresponding to the serviceability limit state recommended by EN 1992-1-1 (2004). (a) (b) Figure 3.27: Comparison between moment-crack opening response (a) SR and (b) SGR beams As Figure 3.26 shows, for a certain load level, the average crack width has decreased with the increase in the fibre content, in consequence of the improvement of the postcracking response of FRCs. According to this figure, an increase of 27% was obtained in the bending moment of SR/FRC2560 at crack width of 0.1 mm (the average crack width corresponding to the serviceability limit states) when compared to SR/FRC1545. The moment-average crack width relationship of SR/FRC4590 beam increased up to a crack 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 Moment, M [kN.m] Average crack width, wm[mm] SR/FRC1545 SR/FRC2560 SR/FRC4590 Yield initiation wSLS = 0.1 mm 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 Moment, M [kN.m] Average crack width, wm[mm] SGR/FRC1545 SGR/FRC2560 SGR/FRC4590 Yield initiation wSLS = 0.1 mm Experimental Evaluation of Flexural Response of R/SFRSCC Elements 79 width of around 0.07 mm and then the load and corresponding crack width reduces. This behaviour is due to the widening of failure crack outside the pure bending region (see Figure 3.23c) and consequent unloading of this region. This phenomenon is also observed in the case of SR/FRC2560 (see Figure 3.23b). The influence of the improved postcracking response of FRCs is also noticeable in the case of SGR beams, where an increase of 29% and 46% in bending moment corresponding to the crack width of 0.1 mm was obtained in SGR/FRC2560 and SGR/FRC4590 when compared to SGR/FRC1545 series beam. Figure 3.27 depicts the influence of additional GFRP longitudinal reinforcement on the moment-crack width response of SGR beams. (a) (b) (c) Figure 3.28: Comparison between moment-crack opening response of (a) SR/FRC1545 and SGR/FRC1545, (b) SR/FRC2560 and SGR/FRC2560, and (c) SR/FRC4590 and SGR/FRC4590 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 Moment, M [kN.m] Average crack width, wm[mm] SR/FRC1545 SGR/FRC1545 Yield initiation 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 Moment, M [kN.m] Average crack width, wm[mm] SR/FRC2560 SGR/FRC2560 Yield initiation wSLS = 0.1 mm 0 1 2 3 4 5 6 7 0.0 0.1 0.2 0.3 0.4 0.5 Moment, M [kN.m] Average crack width, wm[mm] SR/FRC4590 SGR/FRC4590 Yield initiation wSLS = 0.1 mm 80 Chapter 3 This figure reveals the noticeable contribution of GFRP bar to improve the flexural capacity of SGR/FRC1545 and SGR/FRC2560 series beam. An increase of 36% and 38% in bending moment was obtained at the crack opening of 0.1 mm (the considered crack width for the serviceability limit state), when compared to the corresponding SR/FRC1545 and SR/FRC2560 beams, respectively. In the case of SGR/FRC4590 beam, the initial flexural stiffness is lower than the one of SR/FRC4590, which is quite unexpected. 3.5 Concluding remarks An experimental programme was carried out by developing three types of hookedend steel fibre reinforced self-compacting concrete (SFRSCC) of nominal compressive strength of 15, 25, and 45 MPa reinforced, respectively, by 45, 60, and 90 kg/m3 of fibres. The material properties of the SFRSCCs and reinforcing steel and GFRP bars were experimentally determined. In particular, the post-cracking response of the SFRCs was evaluated through three-point notched beam bending test (3PNBBT) by adopting the recommendation of fib Model Code 2010 (2011) and by inverse analysis of the forcedeflection responses recorded in the 3PNBBT. It was revealed that the approach recommended by fib Model Code 2010 (2011) overestimates the stress-crack width response of the SFRCs when compared to the one determined through the inverse analysis. The developed SFRSCCs were then utilised to build flexural beams reinforced longitudinally by one steel bar, designated by SR beams, and by one steel and one GFRP bars, designated by SGR series beam. The force-deflection response and cracking behaviour of the beams were evaluated at four-point bending test conditions. Due to the low percentage of longitudinal reinforcement utilised, failure crack localisation has occurred in SR series beam after yielding of the steel bar, followed by deflectionsoftening stage. This phenomenon became more pronounced when higher contents of fibres were utilised. Consequently, the peak load of the SR series beams was more or less equal to the load corresponding to the yield initiation in the steel bar. The load decay, however, was not observed in the case of SGR series beam. Furthermore, an increase of 8% and 86% were obtained in SR beams, and 35% and 59% in SGR beams, in terms of the load corresponding to the serviceability limit state Experimental Evaluation of Flexural Response of R/SFRSCC Elements 81 )(SLS F , when the content of fibres increases from 45 to 60 kg/m3, and to 90 kg/m3, respectively and corresponding increase in strength class of concrete. For such an increase in the fibre content, the load corresponding to the yield initiation of the steel bar )(sy F has increased 11% and 48% in SR series, and 27% and 42% in SGR series beam. Furthermore, according to the experimental results, it was revealed that the contribution of GFRP bar was more pronounced on the load at yielding of steel bar as well as the peak load of SGR beams. The test results showed an increase of 39%, 59%, and 33% of sy F , and 121%, 143%, and 93% of P F in the case of SGR/FRC1545, SGR/FRC2560, and SGR/FRC4590, respectively, when compared to the SR series beams made of the same fibre reinforced concrete constituent. It was also revealed when a higher dosage of fibre was used (i.e. 90 kg/m3, the use of additional GFRP longitudinal bar in SGR/FRC4590 beams, had a negligible influence on the load corresponding to the serviceability limit states. Finally, it was revealed that the average crack spacing and crack width along with the pure bending region of the beams have tended to decrease with the increase in fibre dosage, as well as with the addition of GFRP bar. 82 Chapter 4 Moment-Curvature Approach to Evaluate Flexural Response of R/SFRSCC Elements 4.1 Introduction The reinforcing contribution of discrete fibres diffused in concrete can decrease the crack opening and crack spacing in concrete elements, contributing to the increase in durability and integrity of concrete structures (ACI 544.5R-10 2010). Owing to the crack arrestment of fibres bridging the crack surfaces, the load carrying capacity, the energy dissipation, and the ductility at the serviceability and ultimate limit design states are increased (Barros 2008, Cunha et al. 2010). The nonlinear analysis is a fundamental approach for evaluating the influence of fibres on the increase in the post-cracking response of fibre reinforced concrete (FRC) and consequent improvement in the loadcarrying capacity of reinforced FRC (R/FRC) flexural elements. For this aim, the finite element method (FEM) is indisputably a comprehensive approach with the capability of simulation of sophisticated structures and taking into account the intervening mechanisms (Tiejiong and Theodore 2019, Häussler-Combe and Hartig 2008). However, even if Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 83 implemented in computer software, FEM often comprises cumbersome computation when higher accuracy of the simulation is needed for which due to additional input data plugged into the approach the complexity of the simulation may be amplified significantly. The nonlinear response of R/FRC flexural elements can be alternatively analysed by employing the principles of beam theory, according to which assuming plane sections remaining plane after bending, the strain at each layer of the cross-section is proportional to the distance of the layer with respect to the neutral axes. Accordingly, the deflectional response of the flexural element is dependent on the moment-curvature relationship of the element’s cross-section. The moment to curvature ratio represents a macroscopic modified flexural toughness of the cross-section (Chen and Atsuta 1976), in which all nonlinearity aspects due to cracking of concrete and yielding of steel reinforcement is entirely included. The moment-curvature relationship of a reinforced concrete (RC) cross-section can be determined by a direct method, where the strains over the section and curvature are resulted from the imposed external bending moment and axial force (Pfeiffer and Quast 2003). In the other approach, the moment-curvature is determined in an inverse method in which a traditional iterative procedure is generally employed by incrementing the concrete strain at the extreme compression (or tension) fibre of the cross-section, whereas a numerical integration is used to determine the internal forces (Park and Paulay 1975). The potentialities of fibres as a reinforcement system are not well explored in the moment-curvature models proposed in the literature. Some idealised moment-curvature relationships was employed for non-fibrous concrete elements with longitudinal reinforcement, such as the ones proposed by Charkas et al. (2002) and El-Mihilmy and Tedesco (2000) for FRP reinforced concrete elements. A few available models in a design practice format for R/FRC elements are not capable to embrace the variety of postcracking response of FRC categorised as strain-softening or strain-hardening FRCs (Olesen 2001, Stang and Olesen 1998, Barragán 2002). The closed-form solution developed by Soranakom (2008) is capable of determining the moment-curvature relationship of a cross-section of a beam reinforced longitudinally with steel bars and made by strain- 90 Chapter 4 4.3.2 Compressive behaviour of FRC Experimental research shows that for both SSand SH-FRC the compressive strength is marginally affected by the presence of fibres unless a significant content of fibres is utilised (Barros and Figueiras 1999a). In fact, the benefits of fibre reinforcement for the compressive behaviour are most reflected in the compression softening phase, with an increase in the energy absorption that can be quite significant, depending on the characteristics of the fibres and surrounding matrix (Cunha et al. 2008). Figure 4.4: Constitutive law of FRC in compression In the developed model the simplified bilinear stress-strain relationship of Figure 4.4 is considered for FRC in compression can be represented by the following equation:   c cc cc ccy cc cc c ccy ccy cc ccu E E                 (4.12a) (4-12b) The adopted constitutive law for the compression behaviour of a FRC is composed by an initial linear branch characterised by the elasticity modulus ( ) c E up to the compressive “yield” strain ( ) ccy  , and is continued with a constant value of compressive yield stress ( ) ccy c ccy E   up to the ultimate compressive strain ( ) ccu  , after which it is assumed that FRC loses the capacity of supporting compressive loads. The adopted constitutive law can be normalised by utilising the cracking stress ( ) c cr E   : Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 91 cc cu               (4.13a) (4.13b) where the dimensionless parameters are determined from the following relations:   cc cc cc c cr E    (4.14) cc cr    (4.15) ccy cr    (4.16) ccu cu cr    (4.17) In the above equations, cc   and  are the normalised compressive stress and strain, respectively,  is the normalised compressive “yield” strain, and cu  is the normalised ultimate compressive strain of FRC. 4.3.3 Tensile behaviour of steel bar Figure 4.5 depicts the tensile behaviour of the steel bars considered in the model simulated by a bilinear stress-strain diagram composed of a linear-elastic branch up to the yield strain ( ) sy  , followed by a perfect plastic branch up to attain the ultimate tensile strain ( ) su  , after which the steel tensile strength capacity is assumed null. Figure 4.5: Tensile constitutive law of steel reinforcement 92 Chapter 4 The constitutive law of steel reinforcement is defined by Equation (4.18), in which s E is the elastic modulus of steel. The constitutive law of steel bar is represented in a normalised format by Equation (4.19).   s ss s s sy s sy sy s su E E              (4.18a) (4.18b) s s ssu               (4.19a) (4.19b) In Equation (4.19) s   and  are the normalised tensile stress and corresponding normalised strain of steel reinforcement, respectively, s  is the normalised stiffness factor of steel, and  and su  are, respectively, the normalised yield strain and ultimate tensile strain of steel reinforcement. These parameters are determined from the following equations: (4.20) (4.21) (4.22) (4.23) (4.24)   cr s c ss E    s s c E E   s cr    sy cr    su u cr s    Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 93 4.3.4 Tensile behaviour of FRP bar The linear stress-strain relationship ( ) FF    schematised in Figure 4.6 is considered to represent the tensile behaviour of FRP reinforcing bar, which is represented by Equation (4.25) Figure 4.6: Tensile constitutive law of FRP reinforcement   ( ) FFF FF Fu E       (4.25) In Equation (4.25) F E is the elastic modulus and Fu  represents the ultimate tensile strain of the FRP bar. The normalised format of the tensile constitutive law of FRP reinforcement is represented in the following equation: '() F F Fu       (4.26) where F   and  are the normalised tensile stress and corresponding stain of FRP reinforcement, respectively, F  is the normalised stiffness factor, and Fu  represents the normalised ultimate tensile strain of FRP reinforcement, respectively, that can be determined from the following equations:   cr F c FF E    (4.27) F F c E E   (4.28) 94 Chapter 4 F cr    (4.29) F u cr u F    (4.30) 4.4 The closed-form solution to estimate the moment-curvature relationship Assuming that a plane section remains plane after bending, and shear deformation of the section can be ignored, the strain profile along with the height of the cross-section is depicted in Figure 4.7 in which, the depth of the neutral axis is designated by NA d . In Figure 4.7, ( / ) NA k d h represents the normalised depth of the neutral axis, while ( / ) ss ch and ( / ) FF ch are the normalised concrete cover thickness of steel and FRP reinforcement, respectively. Figure 4.7: Strain profile of the section and intervening normalised parameters Regarding Figure 4.7, the normalised tensile strain at the concrete bottom fibre ()  , the normalised compressive strain at the concrete top fibre ()  , and the normalised tensile strain of the steel ()  and FRP bar ()  are defined by the following equations: ,ct bot cr    (4.31) ,cc top cr    (4.32) Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 95 s cr    (4.33) cr F    (4.34) In the proposed closed-form solution, a gradual increment is applied to the normalised tensile strain at the concrete bottom fibre ()  , and corresponding values of the normalised compressive strain at the concrete top fibre ()  , and the normalised tensile strain of the steel ()  and FRP ()  are obtained from Equation (4.35) to (4.37): 1 k k   (4.35) 1- - 1s k k    (4.36) 1- - 1F k k    (4.37) By assuming a perfect bond between the longitudinal bars and surrounding FRC, an independent pre-stress level can be assigned to steel and/or FRP bars. The pre-stress levels are defined by prescribing initial tensile strain to the steel and FRP bars, designated by pr s  and pr F  , respectively. The pre-stress level of steel and FRP bars, are respectively defined as the ratio of pr s  to the steel tensile yield strain () sy ε and the ratio of pr F  to the ultimate tensile strain of FRP () fu  , which depend on the type of bars and loading conditions, and should be in agreement with the recommendations of ACI 440R-07 (2007), ACI 440.1R-06 (2007), CAN/CSA-S806-02 (2007), and fib Bulletin No. 40 (2007). Assuming that variation of pre-stress levels ranging between zero and unitary value, the pre-stress strains are restricted to the linear elastic region of steel and FRP tensile stress-strain response. Therefore, the pre-stress loads for the steel ( pr s F ) and FRP ( pr F F ) are obtained from the following equations: pr pr s s s c s s F E bd     (4.38) prpr F F FF c F F E bd     (4.39) 96 Chapter 4 Regarding the depth of the neutral axis ( kh ), the bending moments corresponding to pre-stress loads are calculated by the following equations: (1 ) pr pr s s s M F k h   (4.40) (1 ) pr pr F F F M F k h   (4.41) Due to the constitutive laws of the intervening materials considered in the model, (Figure 4.3 to 4.6), nine different stages should be considered for the range of strains as summarised in Table 4.1. According to this table, the stages are designated by a four digits number that, from the left to the right, representing the condition of the concrete in tension, concrete in compression, steel bar, and FRP bar, respectively. Table 4.1: Variations of normalised strain parameters of the intervening materials Stage Concrete Steel FRP Tension Compression 1.1.1.1 2.1.1.1 2.1.2.1 2.2.1.1 2.2.2.1 3.1.1.1 3.1.2.1 3.2.1.1 3.2.2.1 Regarding Table 4.1, there are three possible main configurations for tensile strain at bottom fibre: , , and . Each configuration 2 and 3 has four possible conditions due to the value of concrete compressive strain at top fibre in either elastic or plastic behaviour in compression, and regarding the value of steel tensile strain in either elastic or plastic behaviour, and also due to the value of FRP tensile strain . For the nine considered stages, the profile of strain and stress along with the depth of cross-section is depicted in Figure 4.8. 01   0   0   0fu   1   0   0   0fu   1   0   su     0fu   1   cu     0   0fu   1   cu     su     0fu   tu     0   0 ψζ  0fu   tu     0 λω  su     0fu   tu     cu     0   0fu νν  tu     cu     su     0fu   01   1α   tu     (0 )   )( cu       0   )( su       0fu   Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 97 Figure 4.8: Profile of strain and stress along with the depth of cross-section for the considered stages 98 Chapter 4 In Figure 4.9 the internal forces of compressive and tensile zone of concrete and the ones for steel and FRP bars are depicted for the adopted stages, according to which and corresponding to the priscribed tensile strain at bottom fibre , the normalised height of compression and tensile zones of the corss-section, the normalised stresses in concrete, and compressive and tensile force in concrete can be determined by the equations summarised represented in Tables 4.2, 4.3, and 4.4, respectively, whose deduction is presented in Appendix C, D, E, in turn. It is remarked that the force components for the steel and FRP bars are obtained multiplying their cross sectional area by the stress in these bars. In Table 4.5 is summarised the Normalised distance of the concrete forces in compression and tension and steel and FRP bars with respect to the neutral axess. The deduction of theses equations can also be found in Appendix F. ()  Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 99 Figure 4.9: (Continued) 106 Chapter 4 Table 4.6: Equations for the depth of the neutral axis parameter ()k of each stage Stage k 1.1.1.1             1111 1111 1111 2 1 2 1 1 2 / 1 2 /1/ 1 1 // / s F F pr Fp s s F s s F r B d h d h kB BF for D B F D for d h d h                      2 1111 ( / / 1 (1/ ) ) (1 )(1 2 / (1 2 / (1 (2/ ) ) ) ) s s F F pr s s s F F f pr D B d h B d h F B d h B d h F                2.1.1.1                      2111 2111 2 2111 2 2111 2 2111 0.5 2 2111 2111 // 1// 2 / 1 / 1 s s F F pr s s F F pr s s s F F f pr D B d h B d h F kD D B d h B d h F D D D B d h B d h F                                      2 2111 1 2 1D        2.1.2.1               2121 2121 2 2121 2 2121 2 2121 0.5 2 2121 2121 // 1// 2 / / 1 pr s s F F F pr s s F F F pr s s F F F F D B d h B d h F kD D B d h B d h F D D D B d h B d h F                                        2 2121 1 2 1D        2.2.1.1                     2211 2211 2211 2 2211 2211 0.5 2211 2211 // 2 1// 2 2 2 / 1 / 1 s s F F pr s s F F pr s s s F F prF D B d h B d h F kD D B d h B d h F D D D B d h B d h F                                           22 2211 1 2 1 D           2.2.2.1               2221 2221 2221 2 2221 2221 0.5 2221 2221 // 2 1// 2 2 2 / / 1 pr F F s s F pr F F s s F pr s s F F F F D B d h B d h F kD D B d h B d h F D D D B d h B d h F                                           22 2221 1 2 1 D           Continued ,, FF pr pr s s s F pr s F B B F F F    γ ρ γ ρ Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 107 Table 4.6: Continue Stage k 3.1.1.1                       3111 3111 2 3111 2 3111 2 3111 0.5 2 3111 3111 // / 1 / / 1 1/2 sF sF s s F s F pr s F pr s F prF d h d h k dh D B B F D D B B F D D D B B dh d Fd h h                                         2 3111 1 2 1 1D            3.1.2.1               3121 3121 2 3121 2 3121 2 3121 0.5 2 3121 3121 // 2 / 1 1 / // pr F s F pr FF Fs F s s pr s F F F s F d h d k D h d D h d h d h d B B F D D B B F D D B B Fh                                            2 3121 1 2 1 1D            3.2.1.1                       3211 3211 3211 2 3211 3211 0.5 3211 3211 // 2 // / 2 1 2 2 1 1/ s F pr s sF sF s s F F pr s F prF B d h B d h Bd DF kD DF D h B d h dh Fd D DBhB                                             22 3211 1 2 1 1D               3.2.2.1               2 3221 3221 3221 2 2 3221 3221 0.5 3221 3221 2 1 2 2 // / /2 / / 1 pr s F F pr s sF s F F s F pr s F FFF D B F kD D B F D D d h B d h dh D B B F B d h d h d h                                             22 3221 1 2 1 1D               ,, FF pr pr s s s F pr s F B B F F F    γ ρ γ ρ 108 Chapter 4 Table 4.7: Equations for the evaluation of the normalised moment for each stage Stage (i) i M i   1.1.1.1   33 1111 1111 2 1111 1111 2' 1111 1111 1111 2 (1 ) (1 ) 6 (1 ) (1 ) 6 (1 ) () (1 ) s s s F F F pr k h h k kh B d k kh B d k M kh        βγ β β Δ Δ     '1111 1111 1111 6(1 ) (1 ) pr pr pr s s F F c cr M F k F k bhE       ε ΔΔ , s s s F F F BB      1111 2(1 )k   2.1.1.1   3 3 3 3 2111 2111 2111 22111 3 2 3 2 2111 2111 ' 22111 2111 2 2 (1 ) (1 ) ( 1) (1 ) 6 (1 ) 6 (1 ) (1 ) s s s F F F pr k h h k h k C kh B d k B d k M kh                  β γ β β ββ β ΔΔ 2 2111 2 3 3C              '2111 2111 2111 6(1 ) (1 ) pr pr pr s s F F c cr M F k F k bhE          , s s s F F F BB      2111 2(1 )k   2.1.2.1   3 3 3 3 2121 2121 2121 22121 22121 2121 22121 32 ' 2121 22121 2121 2 2 (1 ) (1 ) ( 1) (1 ) 6 (1 )(1 ) (1 ) 6 (1 ) (1 ) s s s F F F pr k h h k h k C kh B d k k kh B d k M kh                β γ β β βξ β β β Δ Δ 2 2121 2 3 3C            '2121 2121 6 (1 ) pr FF pr c cr Fk MbhE      , s s s F F F BB      2121 2(1 )k   Continued Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 109 Table 4.7: Continue Stage (i) i M i   2.2.1.1   3 3 3 2211 2211 22211 2211 2211 2211 2211 2211 22211 3 3 2 2211 2211 2211 22211 32 ' 2211 22211 2211 2 (1 ) 2 (1 ) (1 ) 3 ( )( (1 ))(1 ) (1 ) (1 ) ( 1) 6 (1 ) (1 ) 6 (1 ) (1 ) s s s F F F pr h k h k kh h k k k k k kh h k C B d k kh B d k M kh                    ωγ β ωγ β ω ω β ω β ββ β β β Δ Δ 2 2211 2 3 3C              '2211 2211 2211 6(1 ) (1 ) pr pr pr s s F F c cr M F k F k bhE       ε ΔΔ , s s s F F F BB      2211 2(1 )k   2.2.2.1 3 3 3 2221 2221 22221 2221 2221 2221 2221 2221 22221 32 2221 2221 2221 2221 22221 32 2221 22221 2 (1 ) 2 (1 ) (1 ) 3 ( )( (1 ))(1 ) (1 ) (1 ) ( 1) 6 (1 )(1 ) (1 ) 6 (1 ) (1 ) s s s F F F h k h k kh h k k k k k kh h k C B d k k kh B d k kh                     ωγ β ωγ β ω ω β ω β β β ξ β β β Δ Δ   ' 2221 pr M 2 2221 2 3 3C            '2221 2221 6 (1 ) pr FF pr c cr Fk MbhE   ε Δ , s s s F F F BB      2221 2(1 )k   3.1.1.1   3 3 3 3 3111 3111 3111 3111 23111 3 3111 23111 3 2 3 2 ' 3111 3111 23111 3111 2 2(1 ) (1 ) ( 1) (1 ) 3(1 ) ( )( ) (1 ) 6 (1 ) 6 (1 ) (1 ) s s s F F F pr k h k h k C h kh kh kh B d k B d k M kh                  γβ α β β α α β μ β ββ β ΔΔ     2 3111 '3111 3111 3111 2 3 3 6(1 ) (1 ) pr pr pr s s F F c cr C M F k F k bhE            ηα ηα η α ε ΔΔ , s s s F F F BB      3111 2(1 )k   Continued 110 Chapter 4 Table 4.7: Continue Stage (i) i M i   3.1.2.1 3121 3 3 3 3 3121 3121 3121 23121 32 3121 3121 3121 23121 32 ' 3121 3121 23121 2 2(1 ) (1 ) ( 1) (1 ) 3(1 ) ( )( ) 6 (1 )(1 ) (1 ) 6 (1 ) () (1 ) s s s F F F pr k h k h k C h kh k h B d k k kh B d k M kh                  γβ α β β α α β μ β ξ β β β Δ Δ   2 3121 '3121 3121 2 3 3 6 (1 ) pr FF pr c cr C Fk MbhE        ηα ηα η α ε Δ , s s s F F F BB      3121 2(1 )k   3.2.1.1   3 3 3 3211 3211 23211 33 3211 3211 3211 23211 3 2 3 2 ' 3211 3211 23211 3211 2 (1 ) 2(1 ) (1 ) (1 ) ( 1) 3(1 ) ( )( ) (1 ) 6 (1 ) 6 (1 ) (1 ) s s s F F F pr h k k h kh k C h k h kh B d k B d k M kh                   ωγ β α β α α β μ β ββ β ΔΔ     2 3211 '3211 3211 3211 2 3 3 6(1 ) (1 ) pr pr pr s s F F c cr C M F k F k bhE            ηα ηα η α ε ΔΔ , s s s F F F BB      3211 2(1 )k   3.2.2.1 3221 3221 3221 3221 3221 23221 3 3 3 3 3221 3221 3221 3221 23221 32 3221 3221 3221 23221 3 3 ( )( (1 ))(1 ) (1 ) 2 (1 ) 2(1 ) (1 ) ( 1) (1 ) 3(1 ) ( )( ) 6 (1 )(1 ) (1 ) 6 (1 s s s FF h k k k k k kh h k k h k C h kh k h B d k k kh Bd                       ωγ β ω ω β ω β ω γ α β β α α β μ β ξ β β Δ 2 3221 '3221 23221 )() (1 ) F pr kM kh      β Δ   2 3221 '3221 3221 2 3 3 6 (1 ) pr FF pr c cr C Fk MbhE        ηα ηα η α ε Δ , s s s F F F BB      3221 2(1 )k   Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 111 4.5 Model to estimate the force-deflection relationship The force-deflection response of a statically determinate beam failing in bending is obtained by the algorithm schematically represented in Figure 4.10. According to this approach, for the successive increment of k  of the M   relationship of the beam mid-span cross-section, the corresponding k M is read, and the total applied force k F ( 2 / k ss ML , see Figure 4.10) is determined. The bending moment at the middle of the ith segment () i M is also obtained according to the beam bending diagram. Figure 4.10: Numerical approach to simulate the force-deflection response of simply supported beams failing in bending Decomposing the beam in small segments, the bending moment in a generic crosssection at a distance x can be designated by   i Mx . From the M   relationship of this cross-section, the corresponding flexural stiffness () i EI x is obtained, as well as the bending moment in this section for the base system corresponding to the evaluation 112 Chapter 4 of the deflection at the beam mid-span   Mx . By applying the Virtual Work Method, the mid-span deflection of the beam for the kth loading step   mid k  , is determined, which, jointly with k F provides a point of the F   curve. 4.6 Model appraisal To evaluate the accuracy of the proposed model, the results obtained with it are compared to the ones determined from DOCROS software (Basto and Barros 2008). The model implemented in DOCROS assumes that a plane section remains plane after deformation, and the bond between materials is perfect. The section is divided into layers, and the thickness and width of each layer are user-defined and depend on the cross-section geometry. DOCROS can analyse sections of irregular shape and size, composed of various types of materials subjected to an axial force. DOCROS provides a wide database of constitutive laws for the simulation of monotonic and cyclic behaviour of cementbased materials, polymer-based materials and steel bars. The predictive performance of the model was assessed by evaluating the momentcurvature relationship for a rectangular cross-section of 250 mm width and 500 mm depth depicted in Figure 4.11. The cross-section is reinforced longitudinally with steel and FRP bars whose reinforcing ratio is 0.2% and 0.1%, respectively. Furthermore, a concrete cover of 30 mm is considered for FRP bars, while steel bars are positioned deeper, at a distance of 80 mm from the tensile face of the section. The steel bars and FRP bars are applied with a pre-stress percentage of 50%. Figure 4.11: Geometry and reinforcing scheme of the cross-section considered in the model appraisal (dimensions in mm) Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 113 Concerning material properties of steel and GFRP bars commonly used in practical applications, the considered values of the parameters defining the constitutive laws of the intervening materials of Figures 4.3 to 4.6 are included in Table 4.8. Table 4.8: Values of the parameter defining the constitutive laws FRC Steel bar FRP bar Tension Compression cr  c E  tu    cu   s  su  Fu  F  [‰] [GPa] [-] [-] [-] [-] [-] [-] [-] [-] [-] [-] 0.1 30 10 150 0.33a /1.10b 10 40 20 6.67 120 300 1 a Strain-softening / b Strain-hardening The moment-curvature relationships predicted by the proposed model and those obtained from DOCROS software are compared in Figures 4.12a and 4.12b for the crosssection of beams made by strain-softening and strain-hardening FRC, respectively, according to which the excellent accuracy of the developed model is verified. (a) (b) Figure 4.12: Moment-curvature responses predicted by the proposed model and DOCROS for the cross-section of reinforced FRC of (a) strain-softening and (b) strainhardening behaviour The predictive performance of the model was also evaluated by simulating two experimental tests carried out by Badawi and Soudki (2009), and by Xue et al. (2010). As Figure 4.13 shows the two various strengthening techniques have been employed in the 0 25 50 75 100 125 150 175 200 0 2 4 6 8 10 12 14 16 18 20 Moment, M[kN.m] Curvature, χ[m-1] Proposed model DOCROS 0 25 50 75 100 125 150 175 200 0 2 4 6 8 10 12 14 16 18 20 Moment, M[kN.m] Curvature, χ[m-1] Proposed model DOCROS 114 Chapter 4 considered test programs. The first one applying a pre-stressed longitudinal GFRP (glass fibres reinforced polymer) bar placed into a groove open on the concrete cover of the beam, in agreement with the procedures of the near-surface mounted (NSM) technique (Badawi and Soudki 2009); and the second one applying a pre-stressed carbon fibre reinforced polymer (CFRP) laminate according to the externally bonded reinforcement (EBR) technique (Badawi and Soudki 2009). (a) (b) (c) Figure 4.13: (a) Geometry and loading scheme of the beams considered for model appraisal, reinforcement and strengthening configurations of the beams (b) B1 and B2 tested by Badawi and Soudki (2009), and (c) B3 and B4 tested by Xue et al. (2010) (dimensions in mm) The geometric properties of the beams and strengthening arrangements are included in Table 4.9. In Tables 4.10 and 4.11 are presented the relevant values of the parameters that define the constitutive laws of the intervening materials. Since non-fibrous concrete was used in these RC beams, the contribution of the post-cracking residual strength of the concrete was neglected by assigning a null value to  parameter. Moment-Curvature Approach to Evaluate Flexural Response of R/FRC Elements 115 Table 4.9: Data to define the geometry, the reinforcement and the strengthening systems of the beams represented in Figure 4.13 Beam Designation L ss l PBR l Section type W 1 D 2 D Prestressing [mm] [mm] [mm] [mm] [mm] [mm] B1 3500 1100 1100 T1 - - - 40.0% B2 3500 1100 1100 T1 - - - 60.0% B3 2700 950 600 T2 50 12 14 42.1% B4 2700 950 600 T2 20 12 12 50.6% Table 4.10: Data to define the constitutive laws of the intervening materials in the beams of Figure 4.11 Beam Designation cy  cr  c E sy  s E Fu  F E [MPa] [MPa] [GPa] [MPa] [GPa] [MPa] [GPa] B1 53 3.79 30.20 440 190 1970 136 B2 53 3.79 30.20 440 190 1970 136 B3 50.3 3.60 32.50 383 142 2500 150 B4 50.3 3.60 32.50 429 145 2500 150 Table 4.11: Values considered for the constitutive parameters for the simulation of the beams Beams FRC a Steel bar GFRP bar cr    tu   cu  s  s  ζ su  pr s  F  F  Fu v pr F  [‰] [-] [-] [-] [-] [-] [%] [-] [-] [-] [%] [%] [-] [-] [%] B1 0.125 2 0 150 13.98 28 1.04 6.29 18.53 120 0 0.19 4.5 116 0.56 B2 0.125 2 0 150 13.98 28 1.04 6.29 18.53 120 0 0.19 4.5 116 0.84 B3 0.111 2 0 150 13.97 32 1.25 4.36 24.30 120 0 0.186 4.62 150 0.7 B4 0.111 2 0 150 13.97 32 1.00 4.46 26.65 120 0 0.074 4.62 150 0.84 a 1   218 Appendix B Crack evolution in tested SFRSCC beams reinforced by steel and GFRP bars Figure B.1: Crack evolution in SGR/FRC1545-1 Crack evolution in the SFRSCC beams reinforced by steel and GFRP bars bar 219 Figure B.2: Crack evolution in SGR/FRC1545-2 220 Appendix B Figure B.3: Crack evolution in SGR/FRC1545-3 Crack evolution in the SFRSCC beams reinforced by steel and GFRP bars bar 221 Figure B.4: Crack evolution in SGR/FRC2560-1 222 Appendix B Figure B.5: Crack evolution in SGR/FRC2560-2 Crack evolution in the SFRSCC beams reinforced by steel and GFRP bars bar 223 Figure B.6: Crack evolution in SGR/FRC2560-3 224 Appendix B Figure B.7: Crack evolution in SGR/FRC4590-1 Crack evolution in the SFRSCC beams reinforced by steel and GFRP bars bar 225 Figure B.8: Crack evolution in SGR/FRC4590-2 226 Appendix B Figure B.9: Crack evolution in SGR/FRC4590-3 227 Appendix C Deduction of the normalised height of compression and tension zones Stage 1.1.1.1 1 1 cc cc h h kh k h      11 ct h h kh h k      1 1 t hk h        Δ 1 Δ   1 Δ s s s s s h h h kh h h k k h               Δ 1 Δ   1 Δ F F F F F h h h kh h h k k h           Stage 2.1.1.1 1 1 cc cc h h kh k h    1 ct cr cr h h kh         11 1 1 1 ct ct k hh h k h           2 2 1 1 11 ct ct cc ct k h h h h h h             Δ 1 Δ   1 Δ s s s s s h h h kh h h k k h               Δ 1 Δ   1 Δ F F F F F h h h kh h h k k h           Stage 2.1.2.1 1 1 cc cc h h kh k h    234 Appendix C Stage 3.1.2.1 1 1 1 1 cc cc c cr c cr c cr kk EE k E k                1ct cr   1 1 ct c cr c cr EE      2 ( ct c cr c trn cr c cr c cr E E E E                 2 ) 1 1 ct cr c cr E          3 3 ct ct cst c cr c cr EE            s s c cr E     ss c cr E        1Δ 1 Δ 11 cF F FF FF c F cr c rF c cr kk EE k E k                   Stage 3.2.1.1 2 2 cc cc cr c c cr EE          1 cc cr c E     1 cc c cr E    1ct cr   1 1 ct c cr c cr EE      2 ( ct c cr c trn cr c cr c cr E E E E                 2 ) 1 1 ct cr c cr E          3 3 ct ct cst c cr c cr EE            Deduction of the normalised height of compression and tension zones 235     1Δ 1 Δ 11 s s s s s s c cr s c cr c cr kk EE k E k                      1Δ 1 Δ 11 cF F FF FF c F cr c rF c cr kk EE k E k                   Stage 3.2.2.1 2 2 cc cc cr c c cr EE          1 cc cr c E     1 cc c cr E    1ct cr   1 1 ct c cr c cr EE      2 ( ct c cr c trn cr c cr c cr E E E E                 2 ) 1 1 ct cr c cr E          3 3 ct ct cst c cr c cr EE            s s c cr E     ss c cr E        1Δ 1 Δ 11 cF F FF FF c F cr c rF c cr kk EE k E k                   236 Appendix E Deduction of the normalised forces Stage 1.1.1.1 1 1 1 2 1 2 cc cc cc c cr b k b F h E kh k            2 1 21 cc c cr Fk bhE k         1 1 1 1 1 1 2 2 2 ct ct ct ct cr c cr k F bb F h E k h bhE                      1Δ 1 Δ 11 s s s s s s s s s s s c cr s s c cr k d k hF F bd E bd k h bhE k h                           1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 2.1.1.1 1 1 1 2 1 2 cc cc cc c cr b k b F h E kh k            2 1 21 cc c cr Fk bhE k     1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct ct c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE          Deduction of the normalized forces 237       1Δ 1 Δ 11 s s s s s s s s s s s c cr s s c cr k d k hF F bd E bd k h bhE k h                           1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 2.1.2.1 1 1 1 2 1 2 cc cc cc c cr b k b F h E kh k            2 1 21 cc c cr Fk bhE k     1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct t c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE          s s s s s sy s s s c cr s s c cr h F d F bd E bd h bhE h                      1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 2.2.1.1   1 1 1 1 22 cc cc cc c cr k bb F h E h                2 11 2 cc c cr k F bhE         2 2 2 2 1 cc cc cc cc c cr c cr kk F F h b E bh k k bhE                      1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct ct c cr k bb F h E h                  238 Appendix E     21 1 2 2 ct c cr k F bhE                1Δ 1 Δ 11 s s s s s s s s s s s c cr s s c cr k d k hF F bd E bd k h bhE k h                           1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 2.2.2.1   1 1 1 1 22 cc cc cc c cr k bb F h E h                2 11 2 cc c cr k F bhE         2 2 2 2 1 cc cc cc cc c cr c cr kk F F h b E bh k k bhE                      1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct t c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE          s s s s s sy s s s c cr s s c cr h F d F bd E bd h bhE h                      1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 3.1.1.1 1 1 1 2 1 2 cc cc cc c cr b k b F h E kh k            2 1 21 cc c cr Fk bhE k     1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE              Deduction of the normalized forces 239        2 1 2 2 11 ( ) 1 1 1 22 ct ct ct ct c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE                3 3 3 3 11 ct ct ct ct c cr c cr kk F F h b E h b bhE                              1Δ 1 Δ 11 s s s s s s s s s s s c cr s s c cr k d k hF F bd E bd k h bhE k h                           1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 3.1.2.1 1 1 1 2 1 2 cc cc cc c cr b k b F h E kh k            2 1 21 cc c cr Fk bhE k     1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct ct c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE                3 3 3 3 11 ct ct ct ct c cr c cr kk F F h b E h b bhE                        s s s s s sy s s s c cr s s c cr h F d F bd E bd h bhE h                      1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 3.2.1.1 240 Appendix E   1 1 1 1 22 cc cc cc c cr k bb F h E h                2 11 2 cc c cr k F bhE         2 2 2 2 1 cc cc cc cc c cr c cr kk F F h b E bh k k bhE                      1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct ct c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE                3 3 3 3 11 ct ct ct ct c cr c cr kk F F h b E h b bhE                              1Δ 1 Δ 11 s s s s s s s s s s s c cr s s c cr k d k hF F bd E bd k h bhE k h                           1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     Stage 3.2.2.1   1 1 1 1 22 cc cc cc c cr k bb F h E h                2 11 2 cc c cr k F bhE         2 2 2 2 1 cc cc cc cc c cr c cr kk F F h b E bh k k bhE                      1 1 1 1 11 2 2 2 ct ct ct ct c cr c cr F b k b k F h E h bhE                     2 1 2 2 11 ( ) 1 1 1 22 ct ct ct ct c cr k bb F h E h                      21 1 2 2 ct c cr k F bhE          Deduction of the normalized forces 241       3 3 3 3 11 ct ct ct ct c cr c cr kk F F h b E h b bhE                        s s s s s sy s s s c cr s s c cr h F d F bd E bd h bhE h                      1Δ 1 Δ 11 F F F F F F c c F F FF rF FF c cr k d k hF F bd E bd k h bhE k h                     242 Appendix F Deduction of the normalised arm of the forces with respect to the neutral axis Stage 1.1.1.1 1 11 2 2 2 3 3 3 cc cc cc y h y h kh k        1 11 2 2 2 1 1 3 3 3 ct ct ct h y y h h k k          1Δ   1 Δ s s s s s yh y h h k k           1 Δ   1 Δ FF F FF y y h h k k h         Stage 2.1.1.1 1 11 2 2 2 3 3 3 cc cc cc y h y h kh k      1 11 1 22 33 ct ct ct y hh k y      ' 2 1 2ct ct ct y h y         22 22 2 1 2 2 '2 2 1 2 2 2 3 3 62 1 132 2 ct ct ct ct ct ct ct ct ct ct hh y yk h h                           1Δ   1 Δ s s s s s yh y h h k k       Deduction of the normalized arm of forces with respect to the neutral axis 243     1 Δ   1 Δ FF F FF y y h h k k h         Stage 2.1.2.1 1 11 2 2 2 3 3 3 cc cc cc y h y h kh k        1 11 11 2 2 2 3 3 3 ct ct ct k h k y y h h       ' 2 1 2ct ct ct y h y         22 22 2 1 2 2 '2 2 1 2 2 2 3 3 62 1 132 2 ct ct ct ct ct ct ct ct ct ct hh y yk h h                           1Δ   1 Δ s s s s s yh y h h k k           1 Δ   1 Δ FF F FF y y h h k k h         Stage 2.2.1.1   1 11 1 22 33 cc cc cc yh h k y             22 21 1 1 1 2 2 2 cc cc cc cc k k k k k hy h y h h h                        1 11 11 2 2 2 3 3 3 ct ct ct kk y y h h h       ' 2 1 2ct ct ct y h y         22 22 2 1 2 2 '2 2 1 2 2 2 3 3 62 1 132 2 ct ct ct ct ct ct ct ct ct ct hh y yk h h                           1Δ   1 Δ s s s s s yh y h h k k       250 Appendix G               2221 2221 2221 2 2221 2221 0.5 2221 2221 // 2 1// 2 2 2 / / 1 pr F F s s F pr F F s s F pr s s F F F F D B d h B d h F kD D B d h B d h F D D D B d h B d h F                                            22 2221 1 2 1 D           , , s s s F F F BB      Stage 3.1.1.1 1 1 2 3 00 pr pr cc ct ct ct s F s F F F F F F F FFF                              23111 3111 3111 3111 3111 3111 s 3111 3111 F 3111 1 1 1 2 2 1 2 2 11Δ 1 1Δ0 1 s s ss pr pr FF F F kk k k kk kh kF k d dF h                                                          3111 3111 2 3111 2 3111 2 3111 0.5 2 3111 3111 // 1 21 / 1 / // sF s s F pr s F pr sFs F FprsF d h d h d D B B F kD D B B F D DD h d h Fd Bh d hB                                          2 3111 1 2 1 1D            , ,, pr pr s s s F F F pr s F B B F F F         Deduction of the depth of the neutral axis parameter 251 Stage 3.1.2.1 1 1 2 3 00 pr cc ct ct ct sy F F F F F F F F F F                        23121 3121 3121 3121 3121 3121 F 3121 1 1 1 2 2 1 2 2 11Δ0 1 F spr FF ss F kk k k kk k d dF h h                                            3121 3121 2 3121 2 3121 2 3121 0.5 2 3121 3121 // / 12 / // 1 pr F s F pr F F s s Fs s sr F F p F F F dhD B B F B dh kD D B F D D D B B F d h d h d h d h                                             2 3121 1 2 1 1D            , , s s s F F F BB      Stage 3.2.1.1 1 2 1 2 3 00 pr pr cc cc ct ct ct s F s F F F F F F F F F F F                              23211 3211 3211 3211 3211 3211 3211 s 3211 F 3211 3211 11 22 1 1 2 1 2 1Δ 1 Δ 11 0 s s F F pr pr s F F s k k k k kk kk k h k h dd FF                                                                 3211 3211 3211 2 3211 3211 0.5 3211 3211 // 2 / 1 2 2 2 1 / 1 / / s F pr s F pr sF sF sF s s F prF B d h B d h B B d h B d h DF kD DF D DD d h dBFh                                               22 3211 1 2 1 1D               , ,, pr pr s s s F F F pr s F B B F F F         Stage 3.2.2.1 252 Appendix G 1 2 1 2 3 00 pr cc cc ct ct ct sy F F F F F F F F F F F                          23221 3221 3221 3221 3221 3221 3221 F 3221 11 22 1 1 2 1 2 1Δ 1 0 FFF s pr F ss k k k k kk k k d d h h F                                                         2 3221 3221 3221 2 2 3221 3221 0.5 3221 3221 / 2 1 2 22 / // //1 pr s F F pr sF pr s F F sF s F F s F F D B F kD DB d h B d h d h B d h d D h F D D B B dh F                                               22 3221 1 2 1 1D               , s s s F F F BB      253 Appendix H Deduction of the normalised bending moment and corresponding curvature Stage 1.1.1.1               1111 1 1 1 1 1111 1111 21111 1111 1111 1111 1111 s 1111 s 1111 F 1111 (1 ) (1 ) 1 2 1 2 2 1 3 2 3 1Δ1Δ 1 1Δ () 1 pr pr cc cc ct ct s s F F s s F F c cr c cr ss c cr c s FFFc M F y F y F y F y F k F k kk kbhE k h bhE h k kbhE d k kh kbhE k dh h                               1111 F 1111 1111 1Δ (1 ) (1 ) () r pr pr s s F F k F k F h k              '1111 1111 1111 2 3 3 2 1111 1111 1111 2' 1111 1111 1111 '1111 1111 1111 /6 2 (1 ) 3 (1 ) (1 ) 6 (1 ) () (1 ) 6(1 ) (1 ) cr c cr s s s F F F pr pr pr pr s s F F c cr MM MM E bh k h h k B d k kh B d k M kh M F k F k bhE                              254 Appendix H Stage 2.1.1.1               2111 1 1 1 1 2 2 2111 2111 22111 2111 2111 2111 2 2111 2111 (1 ) (1 ) 1 2 1 2 2 1 3 2 3 1 1 2 2 3 3 1 2 3 2 1 pr pr cc cc ct ct ct ct s s s s F F c cr c cr c cr FF sss M F y F y F y F y F y F k F k kk kbhE k h bhE h k kbh h d Ek                                                         2111 s 2111 s 2111 2111 F 2111 F 2111 2111 2111 Δ1Δ 1 1Δ1Δ 1 (1 ) (1 ) () () c cr FF c cr pr p F r s s F F h kbhE k kh kbhE k kh Fk dh Fk                       '2111 2111 2111 2 3 3 3 3 2111 2111 2111 2111 22111 3 2 3 2 2111 2111 '2111 22111 '2111 2111 /6 2 2 (1 ) (1 ) ( 1) (1 ) 6 (1 ) 6 (1 ) () (1 ) 6(1 ) cr c cr s s s F F F pr pr p pr s s F c cr MM MM E bh k h h k h k C kh B d k B d k M kh M F k F bhE                                    2111 (1 ) rF k   Stage 2.1.2.1               2121 1 1 1 1 2 2 2121 2121 22121 2121 2121 2121 2121 2 2121 2121 (1 ) (1 ) 1 2 1 2 2 1 3 2 3 1 1 2 2 3 3 1 2 3 2 pr pr cc cc ct ct ct ct s s s s c FFF c cr c r c cr F s M F y F y F y F y F y F k F k kk kbhE k h bhE h k kbhE k h                                                   2121 F 2121 s 2121 F 2121 2121 2121 ( ) ( ) 1Δ 1Δ 1 Δ 1 (1 ) (1 ) FF sc cr c cr pr pr sF s s F F k bhE k bhE k kh F k F d dhh k h                   Deduction of the normalised bending moment and corresponding curvature 255   '2121 2111 2121 2 3 3 3 3 2121 2121 2121 22121 2 3 2 2121 2121 2121 '2121 22121 '2121 2121 /6 2 2 (1 ) (1 ) ( 1) (1 ) 6 (1 )(1 ) 6 (1 ) () (1 ) 6(1 cr c cr s s s F F F pr pr pr s s c cr MM MM E bh k h h k h k C kh B d k k B d k M kh M F k bhE                                      2121 ) (1 ) pr FF Fk    Stage 2.2.1.1               2211 1 1 2 2 1 1 2 2 2211 2211 22211 2211 2211 2211 2211 2211 2211 2211 2211 (1 ) (1 ) 1 2 1 23 1 2 1 2 1 23 1 pr cc cc cc cc ct ct ct ct s s f f s s pr FF c cr c cr c cr M F y F y F y F y F y F y F k Fk kk bhE h k k k k bhE h kk bhE h k                                                        2 2211 2211 s 2211 s 2211 2211 F 2211 F 2211 2211 2211 122 3 3 1 2 3 2 1Δ1Δ 1 1Δ1Δ 1 (1 ) (1 ) () () c cr ss c cr FF c cr pr pr s s F s F F bhE k h kbhE k kh kbhE k kh F dh kk dh F                                           256 Appendix H   '2211 2211 2211 2 2211 2211 2211 2211 2211 22211 3 3 3 3 2211 2211 2211 2211 22 2211 2211 32 2211 /6 3 ( )( (1 ))(1 ) (1 ) 2 (1 ) 2 (1 ) (1 ) ( 1) (1 ) (1 ) 6 (1 ) 6 cr c cr s s s MM MM E bh h k k k k k kh h k h k h k C k h k h B d k B                                             32 2211 22211 '2211 '2211 2211 2211 (1 ) (1 ) () 6(1 ) (1 ) F F F pr pr pr pr s s F F c cr dk kh M M F k F k bhE                   Stage 2.2.2.1                 2221 1 1 2 2 1 1 2 2 2221 2221 22221 2221 2 (1 ) (1 ) 1 2 1 1 2 3 2 1 2 1 1 1 2 2 2 3 2 pr pr cc cc cc cc ct ct ct ct s s s s F F c cr c cr c cr c c F r F M F y F y F y F y F y F y F k F k k k k k k k bhE h bhE h k k k bhE h bhE                                                                 F sF 2221 2221 33 1 32 1Δ 1Δ 1 Δ 1 (1 ) (1 ) ( ) ( ) FF ss c cr c cr pr pr ss F F s F kh k bhE k bhE k kh h F k F k d dh h                              '2221 2221 2221 2 33 2221 2221 2221 2221 2221 2221 22221 3 3 2 2221 2221 2221 2221 2221 22221 322 /6 2 (1 ) 3 ( )( (1 ))(1 ) (1 ) 2 (1 ) (1 ) ( 1) 6 (1 )(1 ) (1 ) 6 (1 cr c cr s s s FF MM MM E bh h k h k k k k k kh h k h k C B d k k kh B d k                                         2' 21 2221 22221 '2221 2221 2211 )() (1 ) 6(1 ) (1 ) Fpr pr pr pr s s F F c cr M kh M F k F k bhE              Deduction of the normalised bending moment and corresponding curvature 257 Stage 3.1.1.1             3111 1 1 1 1 2 2 3 3 3111 3111 23111 3111 3111 3111 2 3111 311 (1 ) (1 ) 1 2 1 2 2 1 3 2 3 1 1 2 2 3 3 1 2 3 2 pr cc cc ct ct ct ct ct ct s s s s pr FF c cr c cr c FF cr M F y F y F y F y F y F y F k Fk kk kbhE k h bhE h k kbhE k                                                              1 3111 3111 3111 s 3111 s 3111 3111 F 3111 F 3111 3111 3111 11 2 1Δ1Δ 1 1Δ1Δ 1 (1 ) (1 ) () () c cr ss c cr FF c cr pr pr s s F F s F h h dh kk bhE kbhE k kh kbhE k kh F k F k dh                                   '3111 3111 3111 2 3 3 3 3 3111 3111 3111 3111 23111 3 3 3 3 3111 3111 3111 23111 33 3111 3111 3111 2 /6 2 2(1 ) (1 ) ( 1) (1 ) 3(1 ) ( )( ) 2 2(1 ) (1 ) (1 ) ( 1) 3(1 ) ( )( ) ( cr c cr MM MM E bh k h k h k C h kh k h k h k h kh k C h k h                                            3111 3 2 3 2 ' 3111 3111 3111 23111 '3111 3111 3111 1) 6 (1 ) 6 (1 ) () (1 ) 6(1 ) (1 ) s s s F F F pr pr pr pr s s F F c cr kh B d k B d k M kh M F k F k bhE                      258 Appendix H Stage 3.1.2.1             3121 1 1 1 1 2 2 3 3 3121 3121 23121 3121 3121 3121 3121 2 312 (1 ) (1 ) 1 2 1 k 2 2 1 3 2 3 1 1 2 2 3 3 1 2 3 2 pr cc cc ct ct ct ct ct ct s s s s pr FF c cr c cr c FF cr M F y F y F y F y F y F y F k Fk kk kbhE k h bhE h k kbhE k                                                      1 3121 3121 11 2 c cr h h kk bhE          3121 '3121 3121 3121 2 3 3 3 3 3121 3121 3121 23121 32 3121 3121 3121 23121 32 ' 3121 3121 23121 /6 2 2(1 ) (1 ) ( 1) (1 ) 3(1 ) ( )( ) 6 (1 )(1 ) (1 ) 6 (1 ) () (1 ) cr c cr s s s F F F pr MM MM E bh k h k h k C h kh k h B d k k kh B d k M kh                                        2 3 2 3121 3121 3121 '3121 23121 '3121 3121 3121 6 (1 )(1 ) 6 (1 ) () (1 ) 6(1 ) (1 ) s s s F F F pr pr pr pr s s F F c cr B d k k B d k M kh M F k F k bhE                          Deduction of the normalised bending moment and corresponding curvature 259 Stage 3.2.1.1             3211 1 1 2 2 1 1 2 2 3 3 3121 3121 23121 3121 3121 3121 3121 3121 3121 3121 (1 ) (1 ) 11 2 23 1 1 2 1 22 pr Fcc cc cc cc ct ct ct ct ct c c Ft s s s s pr FF c cr c cr cr M F y F y F y F y F y F y F y F k Fk k k k k bhE h bhE k k k k k h bhE                                                             2 3121 3121 3121 3121 3121 3121 s 3121 s 3121 3121 F 3121 F 3121 3 1 1 2 2 3 3 1 2 3 2 11 2 1Δ1Δ 1 1Δ1Δ 1 () () (1 c cr c cr sc s F scr FF c cr pr s h kbhE k k kk bhE kbhE k kh kbhE k kh Fk h h dh dh                                                    3121 3121 ) (1 ) pr s F F Fk      '3211 3121 3211 2 3 3 3 3211 3211 23211 3211 3211 3211 3211 3211 23211 33 3211 3211 3211 23211 3 /6 2 (1 ) 2(1 ) (1 ) 3 ( )( (1 ))(1 ) (1 ) (1 ) ( 1) 3(1 ) ( )( ) (1 ) 6 cr c cr s MM MM E bh h k k h kh h k k k k k kh k C h k h kh Bd                                           2 3 2 ' 3211 3211 3211 23211 '3211 3211 3211 (1 ) 6 (1 ) () (1 ) 6(1 ) (1 ) s s F F F pr pr pr pr s s F F c cr k B d k M kh M F k F k bhE                    