António Ventura Gouveia

Transcription

1 Uiversidade do Miho Escola de Egeharia Atóio Vetura Gouveia Costitutive models for the material oliear aalysis of coete structures icludig time-depedet effects Novembro de 0

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3 Uiversidade do Miho Escola de Egeharia Atóio Vetura Gouveia Costitutive models for the material oliear aalysis of coete structures icludig time-depedet effects Modelos costitutivos para a aálise ão liear material de estruturas de betão icluido efeitos diferidos Tese de Doutorameto Estruturas /Egeharia Civil Trabalho efectuado sob a orietação de Professor Doutor Joaquim Atóio Oliveira de Barros e co-orietação de Professor Doutor Álvaro Ferreira Marques Azevedo Novembro de 0

4 É AUTORIZADA A REPRODUÇÃO INTEGRAL DESTA TESE APENAS PARA EFEITOS DE INVESTIGAÇÃO, MEDIANTE DECLARAÇÃO ESCRITA DO INTERESSADO, QUE A TAL SE COMPROMETE. Atóio Vetura Gouveia

5 To my lovely wife Cris ad my woderful daughters Raquel, Gabriela ad Matilde

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7 v Ackowledgmets This research was carried out at the Departmet of Civil Egieerig of the Uiversity of Miho, Portugal, uder the supervisio of Prof. Joaquim Barros ad co-supervisio of Prof. Álvaro Azevedo. I express my gratitude to Prof. Joaquim Barros for his support, ecouragemet, advice, several iterestig discussios ad friedship. I also thak all the teachigs haded dow over the years, sice I started as his udergraduate studet. I would also like to express my gratitude to Prof. Álvaro Azevedo for his support, ecouragemet, recommedatios, fruitful discussios ad friedship. The fiacial support provided by the Fudação para a Ciêcia e Tecologia (Portuguese Foudatio for the Sciece ad Techology) ad the Europea Social Fud, Grat umber SFRH/BD/336/005, is gratefully ackowledged. I am grateful to the frieds from the Civil Egieerig Departmet of the Uiversity of Miho, especially to Prof. Sea Cruz, for his support, fruitful discussios about some topics related to this work ad for his friedship ad i a later phase to Prof. Miguel Azeha for the productive discussios ad friedship. The validatio of some developed umerical tools by the researchers Iês Costa, Berardo Neto, Hadi Baghi ad Matteo Breveglieri is appreciated. The support ad friedship of the colleagues from the Civil Egieerig Departmet of the School of Techology & Maagemet of the Polytechic Istitute of Viseu are gratefully valued.

8 vi Ackowledgmets To all my frieds that always supported ad ecouraged me durig the developmet of the preset work I would like to express my gratitude. Fially, my gratitude to all my family, Mael, Lea, Rafa, Pipa, Xiha, Aa, Ferado, Bruo, Paulo ad i particular to my wife, Cris, my daughters, Raquel, Gabrielaa ad Matilde, my father, Atóio ad my mother-i-law, Mariaziha. This work was oly possible because of their ucoditioal support, love ad friedship. Thaks for your patiece. I would also like to perpetuate the memory of my mother, Lauretia, ad my father-i-law, Aguiar. God s presece durig this work was also fudametal. Thaks for the gift of life. This research was fuded by the Natioal Strategic Referece Framework , Operatioal Program Huma Potetial (QREN/POPH) - Tipology 4. - Advaced Traiig [ subsidized by the Europea Social Fud (FSE) ad by Natioal fuds from the Miistry of Sciece, Techology ad Higher Educatio (MCTES-FCT)].

9 vii Abstract I recet years coete techology has bee improved sigificatly due, maily, to the developmet of self-compactig coete, ultra-fluid cemet based materials, high performace fiber reiforced coete ad egieered cemet composites. These developmets have their mai applicability i the pre-castig idustry, where the earliest demouldig of the pre-cast elemets is a importat aim for ecoomic reasos. Due to the relatively high cost of these advaced cemet based materials, optimizatio of the behavior of structural elemets made with these materials is a fudametal issue for their competitiveess. As a cosequece, these materials are, i geeral, applied to relatively thi elemets that require special attetio i terms of shear ad puchig resistace. With the aim of studyig these types of structures, a multi-directioal fixed smeared ack model for plae shells has bee developed. This model implemets a iovative approach for capturig the behavior of lamiar structures failig i puchig, which is based o the adoptio of a softeig diagram to simulate the behavior of the out-of-plae shear stress compoets. Sice most advaced cemet based materials have relatively high bider cotet, the risk of ackig at a early age should be evaluated usig models that ca estimate the heat geerated by the hydratio of pozolaic compoets ad the iduced stress fields. For this purpose, a FEM-based heat trasfer model has bee developed ad itegrated ito a mechaical model that ca simulate the ack iitiatio ad propagatio i structures disetized with solid fiite elemets. The mechaical model is a 3D multi-directioal smeared ack model with the capability of simulatig the behavior of structures failig i puchig ad shear. Shrikage ad eep are also a cocer maily for service limit states due to ack opeig limits. I the last two decades fiber reiforced polymer composite materials have also bee used for the structural rehabilitatio of coete structures, maily for the flexural ad shear stregtheig of reiforced coete beams. The predictio of the behavior of the shear

10 viii Abstract stregtheed beams requires the use of ack costitutive models to simulate the deease of the shear stress degradatio with the ack opeig evolutio, i agreemet. Two umerical approaches are proposed to simulate this pheomeo. Oe is based o the use of a softeig ack shear stress versus ack shear strai diagram to model the fracture mode II, while i the other the total ack shear stress is obtaied from the total ack shear strai adoptig a ack shear modulus that deeases with the ack ormal strai. Fiber reiforcemet mechaisms are more effectively mobilized whe support redudacy of a structure is high, sice the stress redistributio capacity provides to this type of structure a ultimate load that is much higher tha the load at ack iitiatio. However, the supportig coditios of a structure ca chage durig the loadig process, ad eve a loss of cotact ca occur. To simulate accurately these situatios, liear, oliear ad uilateral support coditios are umerically implemeted. To iease the robustess of the developed umerical models, iovative umerical strategies are implemeted i the stress update phase of the oliear fiite elemet aalysis process. Furthermore, to improve the covergece performace of the fiite elemet oliear aalyses a arc-legth algorithm is implemeted. All the umerical models are implemeted i the FEMIX 4.0 FEM package, usig the ANSI-C computer laguage.

11 ix Resumo Nos últimos aos tem havido um escete melhorameto a ível dos processos tecológicos do betão, pricipalmete devido ao desevolvimeto de betão auto-compactável, de materiais de matriz cimetícia ultra-fluídos e de betões de elevado desempeho reforçados com fibras. Estes desevolvimetos têm particular aplicação a idústria de pré-fabricação, ode a rápida desmoldagem dos elemetos pré-fabricados é um objectivo importate por razões ecoómicas. Devido aos custos relativamete elevados destes materiais avaçados de matriz cimetícia, a optimização do comportameto de elemetos estruturais costituídos por esses materiais é fudametal para assegurar a sua competitividade. Em cosequêcia, estes materiais são, em geral, aplicados a elemetos relativamete delgados que podem exigir uma ateção especial em termos de resistêcia ao corte e ao puçoameto. Neste cotexto, o presete trabalho é desevolvido um modelo de fedilhação para cascas plaas com a possibilidade da ocorrêcia de múltiplas fedas fixas distribuídas, itegrado uma abordagem iovadora para simular o feómeo de puçoameto. Esta abordagem é baseada a adopção de um diagrama com amolecimeto que simula o comportameto das compoetes de corte trasversal. Estes materiais avaçados de matriz cimetícia têm uma quatidade de ligate relativamete elevado, pelo que a possibilidade de ocorrêcia de fedilhação as primeiras idades deve ser avaliada usado modelos que permitam estimar o calor gerado pela hidratação do ligate durate o seu processo de cura, bem como a determiação das correspodetes tesões. Para o efeito, o presete trabalho é desevolvido um modelo de trasferêcia de calor baseado o método dos elemetos fiitos, o qual é itegrado um modelo mecâico que permite simular o iício de fedilhação e a sua propagação em estruturas disetizadas por elemetos fiitos de volume. Este modelo de fedilhação 3D tem a possibilidade de simular a ocorrêcia de múltiplas fedas fixas distribuídas, bem como o comportameto de estruturas cujo modo de ruptura é codicioado pelas compoetes de corte. A retracção e a fluêcia também são feómeos de relevâcia em estruturas costituídas por estes tipos de materiais, pricipalmete em estados limites de

12 x Resumo serviço por abertura de feda, pelo que a sua modelação foi também itegrada o modelo termo-mecâico. Nas últimas duas décadas, materiais de matriz polimérica reforçados com fibras cotíuas têm sido utilizados a reabilitação estrutural de estruturas de betão, pricipalmete para o reforço à flexão e ao corte de vigas de betão armado. A previsão do comportameto das vigas reforçadas ao corte requer o uso de modelos costitutivos capazes de simular a dimiuição da capacidade de trasferêcia de tesão de corte com a evolução da abertura de feda. Duas abordages uméricas são propostas para este fim. Uma delas baseia-se a utilização de um diagrama de amolecimeto para a modelação do modo II de fractura. A outra suporta-se uma formulação total para a relação etre a tesão e a extesão de corte a feda, adoptado um módulo de rigidez correspodete ao modo II de fractura que dimiui com a extesão ormal à feda. De forma a aumetar a robustez dos modelos uméricos desevolvidos, foram implemetadas algumas estratégias de actualização do estado de tesão o material durate o processo de aálise ão liear. Com o objectivo de melhorar as características de covergêcia dos métodos uméricos utilizados foram itroduzidos algoritmos baseados a técica arc-legth. Codições de apoio com comportameto liear, ão liear e uilateral também foram umericamete implemetadas. Todos os modelos uméricos foram implemetados o software de elemetos fiitos FEMIX usado a liguagem de programação ANSI-C.

13 xi Cotets Ackowledgmets... v Abstract... vii Resumo... ix Cotets... xi List of symbols ad abbreviatios... xvii List of Figures... xxv List of Tables... xxxi Chapter Itroductio. Itroductio ad motivatio.... Objectives....3 Outlie of the thesis... 3 Chapter A overview o the modelig of the oliear behavior of cemet based materials. Cemet based materials Crack costitutive models for cemet based materials Time-depedet pheomea....4 Solutio procedures for oliear problems Itroductio Arc-legth techique Displacemet cotrol at a specific variable Relative displacemet cotrol betwee two specific variables Summary ad Coclusios... 8 Chapter 3 Numerical model for coete lamiar structures 3. Itroductio Geeral layered approach to disetize the thickess of a lamiar structure... 30

14 xii Cotets 3.. Itroductio Formulatio Crack costitutive model Itroductio Formulatio Liear elastic uacked coete Liear elastic acked coete Out-of-plae shear softeig diagram The puchig problem i lamiar structures Desiptio of the diagram Out-of-plae shear status Improvemets made i iteral algorithms Stress update Critical ack status chage Supports with Liear ad oliear behavior Deformable poit, lie ad surface support system Uilateral supports Liear parabolic diagram Biliear expoetial diagram Parametric study of a slab o grade Numerical simulatio of a puchig test with a module of SFRSCC pael Evaluatio of the mode I fracture properties by iverse aalysis Test setup ad values of the parameters used i the umerical simulatios Aalysis based o the values obtaied from iverse aalysis Ifluece of the out-of-plae shear softeig diagram Ifluece of the through-thickess refiemet of the pael Ifluece of the i-plae mesh refiemet of the pael Ifluece of the fracture eergy ( G III f ) used i the out-of-plae shear softeig diagram Ifluece of the parameters that defie the fracture mode I Numerical simulatio of a puchig test with a flat slab Summary ad Coclusios... 08

15 Cotets xiii Chapter 4 Modelig of the ack shear compoet 4. Itroductio Iemetal vs. total approach for modelig the shear ack compoet Iemetal approach Total approach Numerical simulatios Sigle elemet Stregtheig of RC beams Crack shear softeig diagram Desiptio of the diagram Crack shear status Numerical simulatios Mixed-mode test T oss sectio reiforced coete beam failig i shear Reiforced coete beam stregtheed i shear with embedded through-sectio bars Summary ad Coclusios... 6 Chapter 5 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 5. Itroductio Numerical model Itroductio Formulatio Crack strai ad ack stress Costitutive law of the coete Costitutive law of the ack Costitutive law of the acked coete Model implemetatio Model appraisal Summary ad coclusios... 8

16 xiv Cotets Chapter 6 Thermo-mechaical model 6. Itroductio Thermal problem Itroductio Heat coductio equatio Geeral remarks Iitial ad boudary coditios Fiite elemet method applied to heat trasfer Steady-state liear aalysis Numerical example Trasiet liear aalysis Time-disetizatio Computatioal strategies Numerical examples Trasiet oliear aalysis Numerical examples Time-depedet deformatios Shrikage Eurocode Model B Creep Eurocode Model B Coete maturity Update of the multi-fixed smeared 3D ack model to take ito accout the time-depedet effects Numerical example Summary ad coclusios Chapter 7 - Coclusios 7. Geeral coclusios Recommedatios for future research... 53

17 Cotets xv Refereces Appedix A Appedix B Appedix C... 8

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19 xvii List of symbols ad abbreviatios Abbreviatios CFRP Carbo fibre reiforced polymer CM Coarse mesh CMOD Crack mouth opeig displacemet CMSD Crack mouth slidig displacemet CrCS Crack coordiate system EBR Exterally boded reiforcemet ETS Embedded through-sectio FEM Fiite elemet method FRC Fiber reiforced coete GCS Global coordiate system IP Itegratio poit NSM Near surface mouted RC Reiforced coete RM Refied mesh SCC Self-compactig coete SFRC Steel fiber reiforced coete SFRSCC Steel fiber reiforced self-compactig coete List of symbols Roma letters exp AF δ um AF δ A T areas beeath the experimetal load-deflectio curves areas beeath the umerical load-deflectio curves rate costat

20 xviii List of symbols ad abbreviatios a a displacemet vector vector of the odal displacemets b (, ) 0 scale factor that coverts the magitude of load to the magitude of the displacemet C t t specific eep c D specific heat of the material ack costitutive matrix D D i D I D II D ij III,sec D III,sec D co mf, e co D mf co D s co D co D D ti D d f E a E out taget costitutive matrix opeig fracture mode stiffess modulus for the i th triliear stress-strai softeig brach ack costitutive matrix compoet relative to the ack ormal opeig mode (mode I) ack costitutive matrix compoet relative to the ack i-plae slidig mode (mode II) secat costitutive matrix ij compoet relative to the ack out-of-plae slidig mode (mode III) secat stiffess relative to the ack out-of-plae slidig mode (mode III) costitutive matrix of i-plae membrae ad bedig compoets for coete i elastic regime costitutive matrix of i-plae membrae ad bedig compoets for acked coete costitutive matrix of out-of-plae shear compoets for acked coete costitutive matrix costitutive matrix of the acked coete slidig stiffess modulus i the t ˆi directio fracture mode I stiffess modulus steel fiber diameter apparet activatio eergy rate of eergy coducted out the ifiitesimal cotrol volume

21 List of symbols ad abbreviatios xix E i E g E st E c, E F F F rate of eergy coducted i the ifiitesimal cotrol volume rate of thermal eergy geeratio rate of the eergy stored withi the ifiitesimal cotrol volume coete elasticity modulus vector of the forces that are equivalet to the exteral applied loads load i the curret combiatio vector of the load i the previous combiatio F ( a ) vector of the iteral equivalet odal forces e F Q vector correspodig to the elemet iteral heat geeratio e F q F e qt e F c f ( α T ) f c f ct f G c I G f III G f G h c h r III f, c (, ) J t t K 0 vector correspodig to the boudary where the heat flux is imposed vector correspodig to the heat flux where the temperature is presibed vector cotaiig the values correspodig to the covectio boudary ormalized heat geeratio rate compressive stregth tesile stregth geeric fuctio coete elastic shear modulus mode I (i-plae) fracture eergy mode III (out-of-plae) fracture eergy cosumed mode III (out-of-plae) fracture eergy covectio heat trasfer coefficiet radiatio heat trasfer coefficiet compliace fuctio or eep fuctio stiffess matrix ( ) q K T tagetial stiffess matrix i the q iteratio of the curret combiatio

22 xx List of symbols ad abbreviatios e K c e K cov e K t k l b l f t, OP p p Q Q total Qt ( ) q q x q c q e R s T T dt dx T w x i elemet coductio matrix elemet covectio matrix elemet trasiet matrix material thermal coductivity ack badwidth steel fiber legth geeric combiatio ack local coordiate system out-of-plae shear degradatio factor parameter defiig the mode I fracture eergy available to the ew ack iteral heat geeratio rate per uit volume accumulated heat of the cemet (or bider) hydratio accumulated heat geerated util a certai istat t geeric iteratio heat flux i x directio covective heat flux emitted heat flux uiversal gas costat slidig displacemet temperature at the body surface fluid temperature temperature gradiet trasformatio matrix from the coordiate system of the fiite elemet to the local ack coordiate system opeig displacemet elemet local coordiate system

23 List of symbols ad abbreviatios xxi Greek letters α i α th α β fracture parameters used to defie the triliear stress-strai softeig diagram threshold agle coefficiet of thermal shear retetio factor q a i, th i compoet of the vector q a a j i a i predefied iemetal displacemet betwee these two compoets predefied iemetal magitude ε mf vector cotaiig the i-plae membrae ad bedig strai iemetal compoets ε vector of the ack strai i the local ack coordiate system ε co ε ε ε m γ t ( ) t iemetal ack strai vector vector of the uacked coete betwee the acks strai iemet iemetal mechaical strai vector the iemetal ack shear strai σ mf vector cotaiig the i-plae membrae ad bedig stress iemetal compoets σ iemetal local ack stress vector σ τ t F F T t q δ a stress iemet iemet of the ack shear stress iemetal load vector i the load combiatio iemetal load i combiatio temperature variatio iemetal time step iterative displacemet vector of i the q iteratio of the combiatio

24 xxii List of symbols ad abbreviatios ( t ) i ε iitial strai at loadig 0 () t t > t, c ε eep strai at time 0 s ε T ε ε ε m () t co ε ε co ε ε ε ε i, u, () t () t shrikage strai thermal strai mechaical strai emissivity elasto-acked strai ack strai elastic coete strai ack ormal strai ack ormal strai used to defie poit i i the triliear stress-strai softeig diagram ultimate ack ormal strai ε s ε vector cotaiig the out-of-plae shear strai compoets ack strai vector ε ack strai vector i a previous state prev γ ij OP γ p OP γ γ OP ij, p OP γ u γ OP ij, u OP γ max γ OP ij,max γ t λ out-of-plae shear strai ij compoet out-of-plae peak shear strai out-of-plae shear strai out-of-plae peak shear strai for the ij compoet out-of-plae ultimate shear strai out-of-plae ultimate shear strai for the ij compoet out-of-plae maximum shear strai i the softeig brach out-of-plae maximum shear strai ij compoet i the softeig brach total shear ack strai factor

25 List of symbols ad abbreviatios xxiii ν θ ξ i ρ ρ c σ σ σ prev σ, prev Poisso s ratio agle betwee the elemet local coordiate system ad the ack local coordiate system fracture parameters used to defie the triliear stress-strai softeig diagram mass per uit volume volumetric heat capacity Stefa-Boltzma costat ack ormal stress stress vector i a previous state ack stress vector i a previous state i the local ack coordiate system * σ modified ack stress vector (oly the shear compoet) i a previous state, prev i the local ack coordiate system σ σ s τ t i, OP τ p ack ormal stress used to defie poit i i the triliear stress-trai softeig diagram vector cotaiig the out-of-plae shear stress compoets i-plae ack shear stress out-of-plae shear stregth OP τ τ τ ij OP ij, p OP τ max τ τ ϕ OP ij,max t, prev ( tt, ) 0 Ψ ( a ) out-of-plae shear stress out-of-plae shear stregth for the ij compoet out-of-plae shear stress for the ij compoet out-of-plae maximum shear stress i the softeig brach out-of-plae maximum shear stress ij compoet i the softeig brach ack shear stress i a previous state eep coefficiet vector of the ubalaced forces

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27 xxv List of figures Chapter A overview o the modelig of the oliear behavior of cemet based materials Figure. Crackig of coete due to dryig shrikage ad restrai (ACI 00).... Figure. Ifluece of eep relaxatio o the shrikage ackig of coete (adapted from Weiss 999).... Figure.3 Load cotrol procedure Figure.4 Displacemet cotrol procedure Figure.5 Sap-through ad sap-back Figure.6 Arc-legth techique applied to a system with oe degree of freedom ( b =.0 )... Figure.7 Newto Raphso method without ad with arc-legth techique.... Figure.8 Iterative process associated with the arc-legth techique applied to a system with oe degree of freedom ( b =.0 )... 3 Chapter 3 Numerical model for coete lamiar structures Figure 3. Multi-layer plae shell: displacemets, rotatios ad k layer geometry defiitio Figure 3. Example of a fiite elemet idealizatio of a plae shell elemet accordig to the Multi-layer approach Figure 3.3 Procedure to obtai the taget stiffess matrix of a elemet Figure 3.4 Scheme to obtai the iteral equivalet odal forces of a elemet Figure 3.5 Stress compoets, relative displacemets ad local coordiate system of the ack (Sea-Cruz 004) Figure 3.6 Triliear stress strai diagram to simulate the fracture mode I ack propagatio ( σ = f, σ, ct = α σ, σ,, = α σ, ε, = ξ ε, u, ε,3 = ξ ε, u ) ,3, Figure 3.7 Expoetial stress strai diagram to simulate the fracture mode I ack propagatio Figure 3.8 Puchig failure surfaces of a flat slab (Ngo 00) Figure 3.9 Relatioship betwee out-of-plae (OP) shear stress ad shear strai compoets Figure 3.0 Triliear softeig diagram to simulate the relatioship betwee the out-of-plae shear stress ad shear strai compoets. Oly the positive brach is represeted Figure 3. Expoetial softeig diagram to simulate the relatioship betwee the out-of-plae shear stress ad shear strai compoets. Oly the positive brach is represeted Figure 3. Out-of-plae shear statuses Figure 3.3 Critical chage of the fracture mode I modulus covergece difficulties.... 6

28 xxvi List of figures Figure 3.4 Adopted iteria to: (a) update the ack ormal stress for a ack reope; (b) update the ack ormal strai for a ack close Figure 3.5 Poit sprigs: global coordiate system, g x i, local coordiate system of the structure, x i, ad sprig directio vector, s i Figure 3.6 Lie sprig: global coordiate system, g x i, local coordiate system of the structure, x i, ad lie sprig directio vector, s Figure 3.7 Surface sprig: global coordiate system, g x i, local coordiate system of the structure, x i, ad surface sprig directio vector, s Figure 3.8 Pressure settlemet liear-parabolic diagram Figure 3.9 Force displacemet biliear-expoetial diagram Figure 3.0 Gap simulatio betwee a support ad a structure Figure 3. Parametric study of a SFRC slab o grade Figure 3. Fiite elemet mesh Figure 3.3 Crack patter at the slab top surface for a load level correspodig to a maximum ack opeig of 0.3 mm Figure 3.4 Cocept of a lightweight steel fiber reiforced self-compactig coete pael [all dimesios are i mm] (Barros et al. 005a) Figure 3.5 FEM mesh used i the umerical simulatio of a three-poit otched beam flexural test at 7 days (Pereira et al. 008) Figure 3.6 Experimetal average results ad umerical simulatio of the three-poit otched beam flexural test at 7 days (Pereira et al. 008) Figure 3.7 (a) Test pael module, ad (b) test setup [all dimesios are i mm] (Barros et al. 007a) Figure 3.8 (a) Geometry, mesh, load ad support coditios used i the umerical simulatio of the puchig test Coarse Mesh (CM); (b) Properties of the layered oss sectio Figure 3.9 Relatioship betwee load ad deflectio at the ceter of the test pael Figure 3.30 Puchig test simulatio: (a) top surface acks predicted by the umerical model (usig a FEM mesh with eight ode seredipity plae shell elemets), ad (b) photograph showig the acks at the top surface of the pael, at the ed of the test sequece (Pereira 006) Figure 3.3 Ifluece of fiber reiforcemet i the puchig resistace Figure 3.3 Vertical displacemet field (i mm) for the umerical simulatio with out-of-plae shear softeig (for a deflectio of 0 mm at the ceter of the pael) Figure 3.33 Ifluece of III G f, usig the i-plae coarse mesh ad 3 layers i the lightweight zoe, o the umerical relatioship betwee load ad deflectio at the ceter of the test pael Figure 3.34 Ifluece of the umber of layers disetizig the thickess of the pael i the lightweight zoe (results for 3, 6 ad 0 layers are show) Figure 3.35 Geometry, mesh, load ad support coditios used i the umerical simulatio of the

29 List of figures xxvii puchig test Refied Mesh (RM) Figure 3.36 Ifluece of the i-plae mesh refiemet o the umerical load deflectio respose at the ceter of the test pael Figure 3.37 Ifluece of o the umerical relatioship betwee load ad deflectio at the ceter III G f of the pael, whe usig the i plae coarse mesh ad 0 layers i the lightweight zoe Figure 3.38 Ifluece of o the umerical relatioship betwee load ad deflectio at the ceter III G f of the pael, whe usig the i-plae refied mesh ad 0 layers i the lightweight zoe III Figure 3.39 Represetatio of the cosumed out-of-plae fracture eergy, G, whe usig the f, c i-plae coarse mesh ad 0 layers i the lightweight zoe, for a deflectio of 3.5 mm III Figure 3.40 Represetatio of the cosumed out-of-plae fracture eergy, G, whe usig the f, c i-plae refied mesh ad 0 layers i the lightweight zoe, for a deflectio of 3.5 mm Figure 3.4 Ifluece of the ξ parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael Figure 3.4 Ifluece of the α parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael Figure 3.43 Ifluece of the ξ parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael Figure 3.44 Ifluece of the α parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael I Figure 3.45 Ifluece of G : (a) triliear softeig diagrams ad (b) relatioship betwee f load ad deflectio at the ceter of the test pael Figure 3.46 Geometry ad support coditios [all dimesios are i mm] (Moraes Neto et al. 0) Figure 3.47 Puchig test setup [all dimesios are i mm] (Moraes Neto et al. 0) Figure 3.48 Fiite elemet mesh Figure 3.49 Logitudial steel bars tesile reiforcemet (Moraes Neto et al. 0) Figure 3.50 Relatioship betwee load ad deflectio at the ceter of the FC0 slab Chapter 4 Modelig of the ack shear compoet Figure 4. Example of ack shear stress-strai relatio for the iemetal ad total approaches Figure 4. Geometry, load ad support coditios of the sigle elemet mesh Figure 4.3 Load-horizotal displacemet relatioship.... Figure 4.4 Crack ormal stress-strai relatioship i CrCS.... Figure 4.5 Crack shear stress-strai relatioship i CrCS Figure 4.6 x ormal stress-strai relatioship i GCS Figure 4.7 x ormal stress-strai relatioship i GCS Figure 4.8 xx shear stress-strai relatioship i GCS.... 5

30 xxviii List of figures Figure 4.9 Localizatio of the NSM CFRP lamiates ad of the EBR strip of wet layup CFRP sheet [dimesios i mm] (Barros et al. 0) Figure 4.0 Beam geometry, support ad loadig coditios [dimesios i mm] (Barros et al. 0) Figure 4. Load-deflectio at the loaded sectio for the beams of series (Barros et al. 0) Figure 4. Crack patters of the beams of series (i pik color: ack completely ope ε ε ); i red color: ack i the opeig process; i cya color: ack i the (, u reopeig process) (Barros et al. 0) Figure 4.3 Crack patters at the ed of the tested VL beam (Barros et al. 0) Figure 4.4 Diagram to simulate the relatioship betwee the ack shear stress ad ack shear strai compoet ad possible shear ack statuses Figure 4.5 Diagram to simulate the relatioship betwee the ack shear stress ad ack shear strai compoet with post peak triliear braches Figure 4.6 Diagram to simulate the relatioship betwee the ack shear stress ad ack shear strai compoet with post peak expoetial brach Figure 4.7 Iemet of the ack shear strai γ : possible paths whe the startig poit is positive ad i Stiffeig shear ack status (poit A) Figure 4.8 Iemet of the ack shear strai γ : possible paths whe the startig poit is egative ad i Stiffeig shear ack status (poit A) Figure 4.9 Iemet of the ack shear strai γ : possible paths whe the startig poit is positive ad i Softeig shear ack status (poit A) Figure 4.0 Iemet of the ack shear strai γ : possible paths whe the startig poit is egative ad i Softeig shear ack status (poit A) Figure 4. Iemet of the ack shear strai γ : possible paths whe the startig poit is positive ad i Uloadig or Reloadig shear ack status (poit A) Figure 4. Iemet of the ack shear strai γ : possible paths whe the startig poit is egative ad i Uloadig or Reloadig shear ack status (poit A) Figure 4.3 Differet startig poits i the ack shear softeig diagram whe the mode I ack status chages from Closed to Reopeig Figure 4.4 Geometry, mesh, load ad support coditios of the tested four poit beam (all dimesios are i mm) [based o Rots ad de Borst 987] Figure 4.5 Load at poit B ack mouth slidig displacemet (CMSD) relatioship Figure 4.6 Numerical simulatio with softeig ack shear diagram cosiderig G = 0.0 N mm : deformed mesh (a) before ad (b) after the abrupt load decay II f observed i Figure 4.5. The dashed lie represets the udeformed mesh Figure 4.7 Numerical simulatio with softeig ack shear diagram cosiderig G = 0.0 N mm : ack patter (a) before ad (b) after the abrupt load decay observed i Figure 4.5. II f

31 List of figures xxix I dark quadrilateral, ack fully ope, ad the others i opeig or closig process Figure 4.8 Load at poit B ack mouth opeig displacemet (CMOD) relatioship Figure 4.9 Load - displacemet at poit B relatioship Figure 4.30 Reiforced coete T referece beam: geometry, support, steel reiforcemet scheme ad loadig coditios [dimesios i mm] (Dias ad Barros 00) Figure 4.3 S R reiforced coete T oss sectio beam [dimesios i mm] (Dias ad Barros 00) Figure 4.3 Fiite elemet mesh [dimesios i mm] Figure 4.33 Load deflectio at the loaded sectio for the T beam Figure 4.34 Crack patter of the S-R beam by usig a: (a) shear retetio factor; ad (b) a shear ack softeig diagram (i pik color: ack completely ope ( ε ε, u); i red color: ack i the opeig process; i cya color: ack i the reopeig process; i gree color: ack i the closig process; i blue color: closed ack) Figure 4.35 Crack patter at the ed of the tested S-R beam (Dias ad Barros 00) Figure 4.36 Reiforced coete beam: geometry, support, steel reiforcemet scheme ad loadig coditios [dimesios i mm] (Dalfré et al. 0) Figure 4.37 Reiforced coete ETS stregtheig beam: geometry, support, steel reiforcemet scheme ad loadig coditios [dimesios i mm] (Dalfré et al. 0) Figure 4.38 Fiite elemet mesh [dimesios i mm] Figure 4.39 Load deflectio at the loaded sectio for the ETS stregtheig beam Figure 4.40 Crack patter of the S5.90/E5.90 beam by usig: (a) shear retetio factor; (b) shear ack softeig diagram ad steel diagram A; ad (c) shear ack softeig diagram ad steel diagram B (i pik color: ack completely ope ( ε ε, u); i red color: ack i the opeig process; i cya color: ack i the reopeig process; i gree color: ack i the closig process; i blue color: closed ack) Figure 4.4 Crack patter at the ed of the tested S5.90/E5.90 beam (Dalfré et al. 0) Chapter 5 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures Figure 5. Basic fracture modes: (a) opeig mode (tesile), (b) shearig mode (i-plae shear) ad (c) tearig mode (out-of-plae shear) [Wag 996] Figure 5. Crack stress compoets, displacemets ad local coordiate system of the ack Figure 5.3 Geometry, mesh, load ad support coditios used i the umerical simulatio of the puchig test Figure 5.4 Relatioship betwee the force ad the deflectio at the ceter of the test pael Figure 5.5 Ifluece of G f, s, o the umerical relatioship betwee the force ad the deflectio at the ceter of the test pael.... 8

32 xxx List of figures Chapter 6 Thermo-mechaical model Figure 6. Ifiitesimal cotrol volume of a 3D body Figure 6. Steady state example (adapted from Lewis et al. 004) Figure 6.3 Fiite elemet mesh ad temperature field Figure 6.4 Time-disetizatio for oe dimesioal problem Figure 6.5 Temperature vs. time at poit (0.5, 0.5, 0.5) Figure 6.6 Fiite elemet mesh ad temperature field for differet istats of the trasiet aalysis Figure 6.7 Trasiet aalysis domai ad temperature distributio Figure 6.8 Fiite elemet mesh of the domai show i Figure 6.7 ad temperature field for differet istats of the trasiet aalysis Figure 6.9 Temperature variatio through the depth of the oss sectio at the ceter lie Figure 6.0 Temperature variatio through the depth of the oss sectio at the ceter lie compariso with the data from Zhou ad Vecchio (005) Figure 6. Geometry ad the boudary coditios.... Figure 6. Normalized heat geeratio rate.... Figure 6.3 Temperature evolutio at two poits of the cube durig the first day Figure 6.4 Fiite elemet mesh of the body represeted i Figure 6., ad temperature field for the first ad last time step of the trasiet aalysis Figure 6.5 Geometry ad the boudary coditios Figure 6.6 Normalized heat geeratio rate Figure 6.7 Fiite elemet mesh of the body represeted i Figure 6.5, ad temperature field for differet time steps of the trasiet aalysis Figure 6.8 Temperature evolutio at three poits of edge of the wall Figure 6.9 Temperature evolutio at three poits of edge of the wall (Lura ad va Breugel 00) Figure 6.0 Geometry of the prefabricated reiforced coete bridge beam with a U-shaped oss sectio Figure 6. Heat curig regime ad temperature evolutio at two poits of the symmetry plae of the beam Figure 6. Fiite elemet mesh of the structure represeted i Figure 6.0, ad temperature field for differet time steps of the trasiet aalysis Figure 6.3 Evolutio of the ormal stress i x directio ad the tesile stregth at poit P Figure 6.4 Evolutio of the ormal stress i x directio ad the tesile stregth at poit P Figure 6.5 Crack patter for differet time steps of the trasiet aalysis: (a) opeig ack status; (b) closig ack status; (c) reope ack status Appedix B Figure B. Triliear stress-strai diagram to simulate the fracture mode I ack propagatio Figure B. Expoetial stress-strai diagram to simulate the fracture mode I ack propagatio

33 xxxi List of tables Chapter A overview o the modelig of the oliear behavior of cemet based materials Table. Shrikage classificatio (Bastos ad Cicotto 000)... 3 Chapter 3 Numerical model for coete lamiar structures Table 3. Parameters defiig the triliear diagram of Figure 3.6 for the aalyzed SFRC Table 3. Compositio for m 3 of SFRSCC icludig 30 kg/m 3 of fibers... 8 Table 3.3 Values of the parameters of the costitutive model used i the umerical simulatios of the puchig test Table 3.4 Values of the parameters of the costitutive model used i the umerical simulatio of the flat slab puchig test Chapter 4 Modelig of the ack shear compoet Table 4. Model properties used for the sigle elemet simulatio.... Table 4. Properties of CFRP lamiates ad strips of sheets Table 4.3 Steel properties Table 4.4 Model properties used for the beams simulatio Table 4.5 Update of shear ack status whe a ack reopes Table 4.6 Model properties used for the four poit beam simulatios Table 4.7 Properties of CFRP strips of sheets Table 4.8 Steel properties Table 4.9 Model properties used for the T beam simulatio Table 4.0 Steel properties Table 4. Model properties used for the ETS stregtheig beam simulatio Chapter 5 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures Table 5. Values of the parameters of the costitutive model used i the umerical simulatio of the puchig test Chapter 6 Thermo-mechaical model Table 6. Values for k h.... 0

34 xxxii List of tables Table 6. Values of the parameters of the costitutive model used i the mechaical umerical simulatios

35 Chapter Itroductio. INTRODUCTION AND MOTIVATION Coete structures are still widely used i civil costructio. I the last decades some developmets were made o cemet based materials to improve their resistace ad durability. The itroductio of fibers i the coete mix is presetly commo i may applicatios. The beefits of fiber reiforcemet i the improvemet of the coete post-ackig resistace are well recogized. Fibers also reduce the maximum ack width, deease the ack spacig ad iease the eergy absorptio capacity of cemet based materials. Recet experimetal studies with fiber reiforced cemet composites, i terms of optimizig the toughess of these materials, have coducted to the developmet of high performace fiber reiforced cemet composites of high tesile stregth, ad also egieered cemetitious composites presetig tesile strai stiffeig behavior with high tesile strai at peak tesile stress (Li ad Fischer 00, Naama 007). The developmet of self-compactig coete (SCC) (Okamura 997, Okamura ad Ouchi 003) has ieased the potetialities of cemet based materials. Combiig the SCC capacity to flow ad fill the iterior of the formwork passig through the obstacles, with the beefits of steel fiber reiforcemet, a ew high performace material has resulted, beig desigated steel fiber reiforced self-compactig coete (SFRSCC).

36 Chapter The durability of coete structures ca be largely affected by early age coete damage, such as the occurrece of mio-acks at a early stage of the developmet of the material mechaical properties. Several causes are behid the early age coete damage, such as the tesile stresses due to restraied shrikage, or the thermal stresses as a result of the heat geerated durig the hydratio process. Coete is a composite material with high oliear behavior, due to its heterogeeous compositio based o costituets of distict stiffess ad stregth, ad o a iterface that is the weakest lik of the coete mio-structure. Coete has a tesile stregth that is about 0 percet of its compressive stregth, exhibitig a almost liear elastic behavior up to peak stress. After ackig, the coete behavior is largely depedet o its eergy absorptio capacity, which is sigificatly improved whe fibers are used. To accurately simulate the behavior of coete structures for serviceability ad ultimate limit states, sophisticated models must be used i order to capture the essetial features of this material. I the preset work, costitutive models for the material oliear aalysis of coete structures are preseted. These models have bee implemeted i versio 4.0 of the FEMIX computer code (Azevedo et al. 003, Sea-Cruz et al. 007). The predictive performace of these models is assessed usig results available i the literature, ad also several sets of results obtaied from experimetal programs carried out with SFRSCC (Barros et al. 005a), ad with reiforced coete beams shear ad flexurally stregtheed with Carbo Fibre Reiforced Polymer (CFRP) lamiates (Barros et al. 0).. OBJECTIVES The recet improvemets made i coete techology ad the study of its rheological ad mechaical properties must be accompaied by the developmet of umerical models capable of simulatig its behavior. Thus, the mai objectives of the preset work are the developmet of umerical tools for the simulatio of coete structures cosiderig the coete oliear behavior ad its time-depedet effects. With this purpose, the mai achievemets of the preset work are:

37 Itroductio 3 developmet of a costitutive model for the simulatio of lamiar structures, with special attetio o the predictio of the shear failure mode that ca occur i this type of structures whe subjected to high cocetrated loads; developmet of a multi-fixed smeared 3D ack model capable of predictig the behavior of structures disetized with solid fiite elemets the three fracture modes ca be simulated with stress-strai diagrams capable of reproducig a softeig or a stiffeig behavior; developmet of a thermal model with geeral purposes, icludig the heat developmet due to the hydratio process durig the coete hardeig phase to aalyze structures sice early ages; additio of ew fuctioalities to the multi-fixed smeared 3D ack to perform trasiet liear ad oliear aalyses takig ito accout the time-depedet effects, such as shrikage, eep ad temperature; implemetatio of ew strategies to avoid the umerical istabilities observed i some simulatios of coete structures uder plae stress state usig the multi-fixed smeared D ack costitutive model available i the FEMIX computer code; implemetatio of additioal umerical solutio procedures ito FEMIX to eable the oliear fiite elemet aalysis of structures with complex behavior..3 OUTLINE OF THE THESIS Chapter cosists o a overview of recet developmets about cemet based materials. Models to simulate ack iitiatio ad propagatio i these materials are preseted, ad their mai characteristics are discussed. Time-depedet pheomea, like shrikage, eep ad temperature variatio are also preseted, ad the importace of their umerical simulatio to predict the ackig risk at early age is aalyzed. Numerical techiques used i oliear fiite elemet aalysis that have bee implemeted i the FEMIX computer code are briefly discussed ad their beefits i terms of umerical robustess are preseted. I chapter 3 the developed umerical model for the simulatio of coete lamiar structures is preseted. The multi-fixed smeared ack model is implemeted uder the

38 4 Chapter framework of the Reisser-Midli theory adapted to the case of layered shells, i order to simulate the damage due to ack iitiatio ad propagatio. Special attetio is dedicated to the simulatio of the out-of-plae shear compoets by proposig softeig stress-strai diagrams to improve the predictive performace of lamiar structures where the out-of-plae shear is the goverig failure mode. A strategy to simulate supports with liear ad oliear behavior is preseted, ad special cosideratios are made o uilateral support coditios. Improvemets i the iteral algorithms associated with the stress update ad the ack status chage are desibed ad their advatages are discussed. The performace ad the accuracy of the developed umerical tools are assessed usig the results from a puchig experimetal test of a pael fabricated with steel fiber reiforced self-compactig coete (SFRSCC) (Barros et al. 005a). I chapter 4 two strategies to improve the degradatio of the shear stress trasfer with the ack opeig evolutio are desibed. Oe is based o the simulatio of the ack shear stress-shear strai relatioship with a total approach istead of a iemetal approach. The other is based o the use of a softeig diagram for the simulatio of the relatioship betwee the ack shear stress ad the ack shear strai compoets. All the relevat aspects related with their implemetatio i the fiite elemet computer code are desibed i detail. These strategies are validated by performig umerical simulatios usig results available i the literature ad results available from a experimetal program with reiforced coete beams shear ad flexurally stregtheed with composite materials (Barros et al. 0). I chapter 5 a multi-fixed smeared 3D ack model developed for the simulatio of the oliear behavior of coete structures disetized with solid fiite elemets is desibed. The pricipal aspects of its implemetatio i the fiite elemet computer code are also detailed. Special attetio is dedicated to the modelig of the shear fracture modes, beig the utilizatio of the simulatio strategies desibed i chapter 4 discussed. The umerical model is appraised usig the results available from a puchig experimetal test with a module of a pael fabricated with SFRSCC. This module is a structurally represetative of this pael for this type of loadig cofiguratio.

39 Itroductio 5 I chapter 6 a thermal model with geeral purposes is preseted. The iclusio of heat developmet due to the hydratio process durig the coete hardeig phase to aalyze structures sice its early ages is desibed. The time-depedet effects are coupled with the multi-fixed smeared 3D ack model desibed i chapter 5. All the relevat aspects related to the trasiet liear ad oliear aalysis ad its implemetatio i the fiite elemet computer code are discussed. The performace ad the accuracy of the developed umerical models are assessed usig results available i the literature. Fially, the major coclusios are preseted i chapter 7 ad some suggestios for future research are also give.

40

41 Chapter A overview o the modelig of the oliear behavior of cemet based materials. CEMENT BASED MATERIALS Coete is the most commo material used i the costructio idustry. I terms of its iteral structure, coete ca be defied as a mixture of cemet, sad (fie aggregate), gravel (coarse aggregate) ad water. The chemical reactio, kow as hydratio, betwee cemet ad water leads to a hardeig process, ad the presece of aggregates (sad ad gravel) supplies the ecessary stregth. I a simple way, the cemet paste bids the aggregates together resultig i a rigid structure similar to a artificial rock. Due to its simple fabricatio ad hardeig process (from a liquid to a solid phase), coete is a material that is adaptable to ay structural form, beig well accepted i the costructio idustry. The mechaical properties of coete ca be obtaied by performig several tests. Coete has a high compressive stregth, but low tesile stregth. Accordig to the ACI (005) the tesile stregth of coete i flexure is about 0 to 5 percet of its compressive stregth. Sice the tesile stregth ad the post-ackig residual stregth of coete plays a importat role i the deflectio ad ack patter of a structure uder service loads, the additio of other materials to improve its relatively low tesile load

42 8 Chapter carryig capacity, such is the case of steel bars, is curret practice. This type of reiforced coete is widely used i the costructio idustry, combiig the beefits of the high compressive stregth of coete with those resultig from the high tesile stregth of steel. The excellet bod behavior betwee these two materials is also a positive factor. No-cotiuous (disete) fibers are a iterestig reiforcemet solutio for cemet based materials, sice they ca iease sigificatly the post-ackig residual stregth of these materials. Fibers also reduce the maximum ack width, which cotributes to iease the durability of the coete ad the life cycle of the structure. If radomly distributed i coete, the fibers ca also cotribute to prevet early age shrikage ackig. Various types of fibers are used i coete mixtures ad they ca be grouped i steel, glass, sythetic ad atural fibers (ACI 996). To restrai the formatio ad propagatio of acks due to shrikage, sythetic fibers are the most used, but steel fibers are also curretly adopted. Both types of fibers ca also be added to coete i order to provide reiforcemet mechaisms to cotrol the ack propagatio for early age ad hardeed coete phases. For the case of structural applicatios steel fibers are still the most widely used due to cost ad reiforcemet level iteria, beig the derived composite desigated steel fiber reiforced coete (SFRC). Depedig o the cotet ad geometric characteristics of the steel fibers, diffuse ack patters ca be formed due to the fiber pullout mechaisms provided by fibers bridgig the ack lips. Whe mio-acks occur due to shrikage, the fibers ca assure a relatively high residual tesile stregth of the SFRC, limitig the ack width to a small value, ad permittig a evetual healig or sealig of the acks (ACI 996). Furthermore, the additio of steel fibers to a coete mix improves the impact resistace, the eergy dissipatio capacity, the shear ad flexural stregth of coete, ad the resistace to coete spallig (ACI 993). I the past two decades, with the developmet of self-compactig coete (SCC), also desigated self-cosolidatig coete (Okamura 997, Okamura ad Ouchi 003), ew advatages have emerged for the coete techology. SCC ca be defied as a coete mix that has the ability to flow ad fill the iterior of the formwork, passig through ay obstacles or reiforcig bars, ad cosolidatig uder its ow weight, without vibratio.

43 A overview o the modelig of the oliear behavior of cemet based materials 9 Accordig to the Precast/Prestressed Coete Istitute (003), a SCC must satisfy the followig rheological requiremets: fillig ability the ability of SCC to flow uder its ow weight, without vibratio, ad fill completely all formwork spaces; passig ability the ability of SCC to flow through complex spaces, e.g., betwee steel reiforcig bars; stability the ability of SCC to remai homogeeous durig its trasport, placemet, ad after placemet, i.e., resistig to segregatio. With the additio of steel fibers to the SCC, a high performace material is obtaied, sice the beefits of the SCC ca be combied with those derived from the reiforcemet mechaisms provided by the steel fibers. Research o the optimizatio of the mix desig of steel fiber reiforced self-compactig coete (SFRSCC) ad o the characterizatio of its properties were made by several authors (Barros et al. 005a, Pereira 006, Dhode et al. 007). Egieered cemetitious composites (ECC) are the most recet advace i the coete techology. This material exhibits a tesile strai-hardeig behavior after ack iitiatio, ad the post-ackig behavior is characterized by the formatio of diffuse ack patters (Li ad Fischer 00). Therefore, the ECC ca be classified as a high performace fiber reiforced cemet composite (HPFRCC) (Naama ad Reihardt 003, Naama 007). To retrieve the beefits of the improvemets itroduced i the coete techology alog the years, computer programs should be able to simulate with high accuracy the behavior of structures built with these ew types of coete. For this purpose, these programs should icorporate costitutive models capable of reproducig the behavior of these materials. I the preset work costitutive models for the aalysis of coete structures are desibed. I the ext sectios a brief overview about ack costitutive models, time-depedet effects ad umerical strategies for the material oliear aalysis of cemet based materials are preseted.

44 0 Chapter. CRACK CONSTITUTIVE MODELS FOR CEMENT BASED MATERIALS Presetly, several fiite elemet approaches are available to aalyze the oliear behavior of complex structures subject to arbitrary loads. The most recet oes are capable of modelig the behavior of coete structures presetig brittle failure modes, ad accurately predict ack formatio ad propagatio. Disete cohesive fracture models (disete) with fragmetatio algorithms, strog discotiuity approaches (cotiuum) with the embedded discotiuities method, ad the exteded fiite elemet method are examples of advaced methodologies that, together with powerful mesh refiemet algorithms, reveal great efficiecy i modelig the coete fracture iitiatio ad propagatio (Yu et al. 007). Alterative methods are based o damage models (de Borst ad Gutiérrez 999), smeared ack models (Bazat ad Oh 983)] ad mioplae models (Bazat 984). These methods are less precise i the predictio of the local pheomea related to ack propagatio, but, i terms of computatioal effectiveess ad assessmet of the global behavior of a coete structure, are more appropriate to aalyze complex structures with a large umber of degrees of freedom. As show by de Borst (00), fixed ad rotatig smeared ack models, but also mioplae models, ca be coceived as a special case of (aisotropic) damage models. These three FEM-based solutios are closely related ad produce similar results. Takig ito accout the mai characteristics of all these approaches, the multi-directioal fixed smeared ack model (de Borst 987, Rots 988, Dahlblom ad Ottose 990), already implemeted i the FEMIX 4.0 computer program (Azevedo et al. 003, Sea-Cruz 007) for plae stress problems, has bee improved ad exteded to plae shells ad structures disetized with solid fiite elemets. I the implemetatio process some iovative aspects have bee developed. Sice the majority of the structures aalyzed i this work are made with SFRC, where diffuse ack patters ca be formed, the optio for the multi-directioal fixed smeared ack model is coceptually justified, as log as a appropriate costitutive law is used to model the post-ackig behavior of these materials.

45 A overview o the modelig of the oliear behavior of cemet based materials.3 TIME-DEPENDENT PHENOMENA The developmet of acks i coete at early ages, caused by shrikage ad temperature variatio, has cocered various researchers over the years. Coete is a brittle composite that exhibits a low tesile stregth i compariso with its compressive stregth. If the magitude of tesile stresses due to shrikage (for example: plastic, dryig or thermal) reaches the tesile stregth of coete, the acks are formed. Prevetig or cotrollig the magitude of these acks, i.e. width ad patter, is importat for the durability, performace ad aesthetic appearace of the structure. Whe shrikage is actuatig, ackig i coete members oly appears if they are ot free to shrik (Koeigsfeld ad Myers 003). I fact, tesile stresses do ot develop if coete shriks freely. I curret coete structures, however, their structural elemets are i geeral partially or totally restraied, ad therefore shrikage ackig has a high probability of occurrece. The degree of shrikage depeds o the water-to-cemet (w/c) ratio, relative humidity associated with the temperature of the eviromet, type of cemet ad geometric characteristics of the elemet. Figure. illustrates the volume chage uder urestraied shrikage, the stress developmet, ad the ackig pheomeo due to restraied shrikage (ACI 00). To cotrol restraied shrikage ackig various methods/techiques ca be used. All have the same goal: prevet that, at a certai age, the tesile stresses due to shrikage reach the tesile stregth of coete. To reduce the w/c ratio ad to assure proper curig coditios are the most commo shrikage mitigatio strategies. Other methods are the use of expasive cemets to couteract the shrikage effect, or the use of Shrikage Reducig Admixtures (Weiss et al. 998, D Ambrosia et al. 00). Fiber reiforcemet has also gaied importace due to its ackig cotrol efficiecy ad facility of distributio i the coete mixture with mior modificatios of the coete productio techology.

46 Chapter Figure. Crackig of coete due to dryig shrikage ad restrai (ACI 00). The pheomeo of shrikage i cemet based materials is importat because it ca be associated with differet types of damages that ca occur i coete structures, especially i early ages. Cracks are udesirable because they reduce the load capacity, ad expose the reiforcemet to evetually aggressive evirometal coditios, resultig i a deease of the service-life of a structure. The residual stress ad coete stregth alog the coete ageig process, represeted i Figure., was preseted by Weiss (999). The coete acks whe the fuctios itersect (poit A). Weiss also cocludes that whe the pheomeo of eep is take ito accout the stresss level deeases due to eep relaxatio, delayig the ackig age (poit B). A B Figure. Ifluecee of eep relaxatio o the shrikage ackig of coete (adapted from Weiss 999).

47 A overview o the modelig of the oliear behavior of cemet based materials 3 I order to predict the stresses related to shrikage it is ecessary to uderstad the correspodig pheomeo, ad, therefore, a overview of the associated termiology ad its meaig is preseted below. Bastos ad Cicotto (000) classify the shrikage accordig to four iteria: physical state, ature, degree of restrictio ad permaece, as idicated i Table.. They also discuss the simultaeous occurrece of these types of shrikage. Table. Shrikage classificatio (Bastos ad Cicotto 000). Physical State Nature of the pheomeo Degree of restrictio Permaece of the pheomeo Shrikage at fresh state (Plastic shrikage) Shrikage at hardeig state Dryig shrikage Hydratio shrikage Autogeous shrikage Carboatio shrikage Thermal shrikage Free shrikage Restraied shrikage Reversible shrikage Irreversible shrikage Weiss (999) has classified the shrikage pheomeo i oly two mai groups: thermal shrikage ad water related shrikage. Thermal shrikage is caused by the hydratio of the cemet or the diural or seasoal temperature chages. The water related shrikage is caused by the loss of water from the coete. The loss of water due to its movemet to the eviromet causes a volumetric chage of the coete structure, ad tesile stresses develop due to partial or total restrai of its movemet. Plastic, dryig, carboatio ad autogeous shrikage are classified as water related shrikage. I thi structures water ca escape more quickly ad, cosequetly, they are more sesitive to shrikage tha thick structures. Therefore, the developmet of stresses due to dryig shrikage is more itesive i thi tha i thick structures. This type of shrikage is especially importat i

48 4 Chapter pavemets, slabs ad bridge decks due to their large area to volume ratio. I cotrast, thermal shrikage is more importat i thick structures tha i thi structures due to the slower trasfer of the cemet hydratio heat. Shrikage occurs as a result of the coolig process, beig more promiet i this type of structures. Plastic Shrikage Plastic shrikage is the loss of water (i.e. evaporatio) i a fresh coete surface. Plastic shrikage occurs at the first hours of coete curig, ad its magitude depeds o the evirometal coditios icludig solar effects, wid speed, temperature ad relative humidity. To prevet ackig due to plastic shrikage, the use of a plastic sheet cover is commo to block the early age evaporatio of water. Wid breaks ad the use of special coete admixtures are also strategies that reduce plastic shrikage. Fibers, especially polypropylee fibers, are beig cosidered by desigers, suppliers ad costructors because they sigificatly reduce the width of acks formed due to plastic shrikage. Dryig Shrikage The dryig shrikage occurs i hardeed coete ad is caused by the loss of water through the surface. The water movemet is affected by the differece betwee the iteral ad exteral relative humidity. Less relative humidity i the atmosphere ieases the dryig shrikage ad the potetial for the occurrece of ackig. The water-to-cemet ratio (w/c) also iflueces this type of shrikage, sice the water cotet i the pores deeases with the w/c ratio. Therefore, the lower the amout of water available to be expelled through the surface is, the smaller the dryig shrikage effect becomes. Autogeous Shrikage It is the shrikage that occurs i a coete volume without iterchages of humidity with the outside evirometal coditios. This ca occur i the core of a thick coete structure or i specimes where the loss of water through the surface is ot allowed. Due to the hydratio reactio of bider materials, there is a iteral cosumptio of water called self-desiccatio ad, i combiatio with its volume reductio as a result of the chemical reactio, the autogeous shrikage occurs.

49 A overview o the modelig of the oliear behavior of cemet based materials 5 Carboatio Shrikage This type of shrikage occurs at the surface of the coete as a result of the reactio betwee the carbo dioxide (CO) preset i the atmosphere ad the hydrated cemet. This reactio leads to the shrikage of the coete. Thermal Shrikage As a result of hydratio of bider materials durig the coete curig phase, a exothermic reactio occurs with the geeratio of a large quatity of heat. The dissipatio of this heat is faster i thi tha i thick structures. Durig the coete coolig process after the curig phase, ad sice heatig/coolig phases ca occur simultaeously i distict parts of a coete elemet, tesile stresses are developed (thermal stresses) i the parts that are shrikig, leadig to the formatio of acks. Thermal stresses also occur i hardeed coete due to diural ad seasoal temperature chages, ad the dimesioal variatio ca also cause the developmet of acks. Creep ad Shrikage Accordig to the ACI (99) eep ca be defied as the time-depedet iease of strai i coete subjected to sustaied stress. Basic eep occurs uder coditios of o moisture movemet to or from the eviromet, while dryig eep is the additioal eep caused by dryig. I Figure. it ca be observed that the eep has a effect of relaxatio of the residual stress, resultig a iease i the coete age whe it acks. The stregth iease i this period of time ca avoid the formatio of acks, which is a favorable cotributio of eep i the cotext of coete shrikage. Shrikage ad eep are time-depedet pheomea that are iterrelated ad caot be completely dissociated. I the past years a special attetio has bee dedicated to the tesile eep of coete, ad its ifluece o shrikage iduced ackig whe the coete is restraied (Altoubat ad Lage 00, D Ambrosia et. al. 00, Bissoette et al. 007). Whe the coete is restraied it is subjected to tesile stresses ad the ackig potetial ieases. Tesile eep stresses ad the correspodig strais couteract the shrikage strais ad have a beeficial effect.

50 6 Chapter For example, durig the first days of a coete slab o grade, shrikage is the itical coditio. The restrai imposed by the coete/soil frictio itroduces tesile stresses i the coete that, depedig o the restrai characteristics, coete ad evirometal coditios, cause the formatio of acks. Sice the coete is submitted to tesile stresses durig this period, eep strai has a beeficial effect sice it deeases the strais due to shrikage. The simulatio of the time-depedet pheomea, such as shrikage, eep ad temperature variatio, is ucial ot oly to predict the ackig risk, but also to cotribute to a more accurate predictio of the global behavior of coete structures from their early ages to the hardeed phase. This approach has bee itegrated i the FEMIX computer code (Azevedo et al. 003, Sea-Cruz 007) ad is exposed i chapter 6..4 SOLUTION PROCEDURES FOR NONLINEAR PROBLEMS.4. Itroductio The use of the fiite elemet method (FEM) to obtai the solutio of civil egieerig problems where o aalytical approach is available is very commo. Accordig to the FEM, the cotiuum is divided i several fiite elemets (Ziekiewicz ad Taylor 000a) ad the displacemet field is based o shape fuctios ad odal displacemets. Whe the material exhibits a oliear behavior, the resultig equatios from the applicatio of the priciple of virtual work are also oliear, ad a iemetal/iterative procedure is used to solve the oliear system of equatios. The Newto-Raphso method is widely used i this framework. The equilibrium equatio of a structure ca be writte as K a = F (.) where K, a ad F are the stiffess matrix, the displacemet vector ad the vector of the forces that are equivalet to the exteral applied loads, respectively, correspodig to the

51 A overview o the modelig of the oliear behavior of cemet based materials 7 odal degrees of freedom of the structure. I the cotext of structural oliear aalysis, equatio (.) is ot liear, due to the depedece of the stiffess matrix o the odal displacemet vector (Ziekiewicz ad Taylor 000b). To obtai the structural respose it is coveiet to apply the load iemetally F = F + F (.) where F is the iemetal load vector i the load combiatio, the previous combiatio, ad F is the load i F is the load i the curret combiatio. I this cotext, each combiatio correspods to a iemet of the load. Thus, for the combiatio the equatio of the ubalaced forces Ψ ( a ) ca be defied by ( a) F F ( a) Ψ = (.3) beig F ( a ) the vector of the iteral equivalet odal forces, ad a the vector of the odal displacemets. For a curret combiatio the vector of ubalaced forces must be ull, i.e., Ψ ( a ) = 0 (.4) Equatio (.4) ca be solved by the Newto-Raphso method. Cosiderig the first two terms of the Taylor series expasio, equatio (.4) ca be approximated as q q q Ψ q δ ( ) ( ) Ψ a Ψ a + a = 0 a (.5) where the subsipt q is the iteratio couter. I equatio (.5)

52 8 Chapter q q Ψ F ' = = a a ( K ) T q (.6) is the Jacobia matrix, which, i the cotext of structural aalysis, correspods to ( ) q multiplied by ( ). ( ) q T K T K is the tagetial stiffess matrix i the q iteratio of the curret combiatio. Substitutig (.6) i (.5) yields ( ) q q ( q KT δ a a ) =Ψ (.7) A iterative procedure is required to obtai the solutio of equatio (.4), ad i each iteratio the vector of the displacemets is updated as follows q q q q δ a = a + a = a + a (.8) with q q i q q δ δ i= a = a = a + a (.9) beig a 0 = ad a 0 a = 0 i the begiig of each iteratio process. Figure.3 represets a load-displacemet relatioship of a system with oe degree of freedom that presets a post-peak softeig. The umerical simulatio ca be made by applyig load iemets F, beig this techique desigated load cotrol procedure. It is observed that with the load cotrol procedure the umerical solutio caot be obtaied for the post-peak phase, i.e., the curve betwee poits A ad B. This ca be overcome if a displacemet cotrol procedure is adopted for the umerical simulatio, i.e., by applyig displacemet iemets, a, istead of load iemets. I this case, as show i Figure.4, the post-peak curve ca be umerically obtaied.

53 A overview o the modelig of the oliear behavior of cemet based materials 9 F Load iemets F A B a Figure.3 Load cotrol procedure. F A B Displacemet iemets a a Figure.4 Displacemet cotrol procedure. Figure.5 represets a load-displacemet respose of a structure with a complex behavior. Applyig the load cotrol procedure, the curve betwee poits A ad D is ot obtaied, i.e., the umerical respose icludes the poits betwee O ad A ad the poits after D. This is referred to i the literature as a sap-through behavior. Whe a displacemet cotrol procedure is used, it is verified that the poits o the curve betwee B ad C are ot obtaied, i.e., the umerical respose oly icludes the poits betwee O ad B ad the poits after C. This pheomeo is kow as sap-back behavior. To overcome these

54 0 Chapter difficulties ad obtai the etire umerical respose show i Figure.5, several researchers have proposed differet umerical strategies, amog which stads out the arc-legth techique. This techique was origially proposed by Riks (970) ad Wemper (97), ad was subsequetly modified by several researchers (Crisfield 983 ad 986, Bashir Ahmed ad Xiao-zu 004). To overcome the difficulties associated with solvig a system of oliear equatios, some iterative techiques, such as the arc-legth ad related methods, itroduce a load factor durig the iterative process correspodig to the Newto Raphso-method. The load level is ow also a ukow ad it is ecessary to cosider a additioal equatio. This equatio costrais the solutio to meet a certai iteria. I the followig sectios the arc-legth techique ad related methods are preseted. These techiques are implemeted i the FEMIX computer code, ad the details ca be foud elsewhere (Vetura-Gouveia et al. 006). F sap-through A B D O sap-back C a Figure.5 Sap-through ad sap-back..4. Arc-legth techique Figure.6 represets a oliear relatioship betwee the load ad displacemet i a structure with oe degree of freedom.

55 A overview o the modelig of the oliear behavior of cemet based materials To simulate the oliear behavior of this structure a iemetal loadig procedure is used. Figure.6 also represets the load ad displacemet variatio correspodig to the load iemet betwee the combiatios ad. The use of a iemetal load leads to a solutio that moves away from poit A, ad bypasses the peak correspodig to poit C, beig the behavior of the structure betwee poits A ad D ot captured. To reproduce the full path, the load iemet is multiplied by a factor λ whose value is set by the followig restrictio F ( ) λ ( ) a + b F = L (.0) I this equatio b is a scale factor that coverts the magitude of load to the magitude of the displacemet. F F - + F F = F - + λ F B C D F λ F L F - A - a - a a a Figure.6 Arc-legth techique applied to a system with oe degree of freedom ( b =.0 ). Accordig to Figure.6 the exteral force is ow a fuctio of λ, ad usig equatio (.3) the ubalaced forces become ( a λ) F ( λ) F ( a ) F λ F F ( a ) Ψ =Ψ, = ' = + ' = 0 (.)

56 Chapter I the q iteratio, equatios (.0) ad (.) must be take ito accout, resultig, for a system of more tha oe degree of freedom, i q q q q q q q ( a λ ) F( λ ) F ( a) F λ F F ( a) Ψ, = ' = + ' = 0 (.a) q T q q q q ( λ ) ( λ ) [ ] T, 0 f a = a a + b F F L = (.b) Accordig to Crisfield (99) the factor b ca be cosidered ull for curret problems. I the preset implemetatio the factor b is take ito accout ad ca be advatageously used i the solutio of oliear problems. I the preset work the Newto-Raphso method ca be used without the arc-legth techique by applyig differet load iemets F up to a predefied combiatio is reached, followed by a set of combiatios i which the arc-legth techique is used with a costat load iemet F. I this cotext the iease of exteral force is desigated by F. Figure.7 represets the Newto-Raphso method without ad with the arc-legth techique. F L L λ F q Newto - Raphso Arc-legth λ F λ F q λ F F a Figure.7 Newto-Raphso method without ad with arc-legth techique.

57 A overview o the modelig of the oliear behavior of cemet based materials 3 The iterative process correspodig to the Newto-Raphso method with the arc-legth techique is represeted i Figure.8. F F 3 F λ F 0 ψ ψ F δλ F 3 δλ F ψ ψ 3 3 λ F λ F L F - - δ a δ a δ a 3 a = a a a a a = a a a a 3 Figure.8 Iterative process associated with the arc-legth techique applied to a system with oe degree of freedom ( b =.0 ). I order to use the Newto-Raphso method to obtai the solutio of (.), the first two terms of the correspodig Taylor series expasio are cosidered (Vetura-Gouveia et al. 006), resultig i ( ) (, λ ) q ( a, λ ) q q q q KT F δ a Ψ a q T = q T q q a b λ [ F] F δλ f (.3)

58 4 Chapter This system of liear equatios has a o-symmetric matrix. To overcome this disadvatage, Crisfield (99) proposes the replacemet of q a i the costrait (.b). This developmet is desibed i detail i Vetura-Gouveia et al. (006). I the preset expositio oly a geeral approach is made. Thus, cosiderig (.a) ad makig some developmets, yields ( T ) (, ) q q q q q K δ a =Ψ a λ + F δλ (.4) q Rewrittig (.4) i terms of the iterative displacemet, δ a, results i q q ( ) q q q (, ) ( ) q = T Ψ + T δ a K a λ K F δλ q = δa + q q δλ δa (.5) beig q ( ) T (, λ ) δa K a q q q = Ψ (.6) ad q q ( T ) δ a = K F (.7) with Ψ q q q q ( a, ) = F + λ F F' ( a ) λ (.8) The successive approximatio to the solutio is made usig equatio (.8), beig the load q factor λ updated with

59 A overview o the modelig of the oliear behavior of cemet based materials 5 λ + q q q = λ δλ (.9) Icludig equatios (.9), (.5) ad (.9) i equatio (.b) yields (see Appedix A) q ( δλ ) a + a + a = (.0) q δλ 3 0 q where δλ is the ukow ad q T q δ [ ] T a = δa a + b F F T ( ) [ ] T ( ) ( ) [ ] q T q q q a = δa a + δa + b λ F F a3 = a + a a + a + b F F L q q q q q T δ δ λ (.).4.3 Displacemet cotrol at a specific variable The applicatio of the arc-legth techique to the umerical simulatio of some structural problems with localized oliearities may cause istabilities i the covergece of the iemetal/iterative process. This deficiecy ca be avoided by followig a strategy proposed by Batoz ad Dhatt (979) that cosists o the restrictio of the iemetal displacemet of a particular variable to a predefied value. This displacemet cotrol is made without the additio of ay support. This procedure is called displacemet cotrol at a specific variable. The equatio (.b) is replaced with a = a q i, i (.) q beig a i, the i th compoet of the vector q a ad ai its predefied iemetal magitude.

60 6 Chapter Durig the iterative process the iemetal value of the i th compoet of the vector q a remais costat ad equal to q ai, i.e., the iterative variatio of this compoet, δ a i,, is ull. Give this fact, equatio (.9) ca be writte for the i th compoet of the vector i the followig format q a a = a + δ a = a = a (.3) q q q q i, i, i, i, i q For a give combiatio, the iterative displacemets, δ a, are obtaied with equatio (.5). Writig this equatio for the i th compoet yields δa = δa + δλ δa (.4) q q q q i, i, i, q Kowig that the iterative variatio δ a i, is ull, ad solvig equatio (.4) i order to q obtai δλ the followig expressio is obtaied q δ a δλ = δ a q i, q i, (.5) beig δ a i q, ad δ a i q, the i th compoet of the first member of equatios (.6) ad (.7), respectively..4.4 Relative displacemet cotrol betwee two specific variables As metioed i the previous sectio, the umerical simulatio of structures where localized oliearities occur becomes sometimes impossible due to equatio (.b). I a attempt to avoid the istability i the covergece of the iemetal/iterative process, de Borst (986) suggests that i equatio (.b) oly some preselected compoets of vector q a should be cosidered i the aalysis. I certai structures, such is the case of

61 A overview o the modelig of the oliear behavior of cemet based materials 7 those govered by the relative movemet of the faces of existig itical failure acks, by selectig two appropriate degrees of freedom (displacemet compoets), oe i each face of the ack, ad imposig a relative displacemet value that represets the movemet of the faces of the ack durig the loadig process of the real structure, these types of istabilities ca be avoided. The relative displacemet cotrol betwee these poits is accomplished without the additio of ay support. I order to implemet this techique, equatio (.b) is replaced with q q a, j a, i= aj i (.6) q q beig a i, ad a, j, respectively, thei ad j compoets of the vector q a, ad a j i the predefied iemetal displacemet betwee these two compoets. Durig the iterative process the relative iemetal value betwee i ad j compoets of the vector q a remais costat ad equal to a j i, i.e., the relative iterative variatio betwee these compoets ( q q δa, j δa, i) writte for the i ad j compoets of the vector is ull. Give this fact, equatio (.9) ca be q a i the followig form ( δ ) ( δ ) a a = a + a a + a q q q q q q, j, i, j, j, i, i = a a = a q q, j, i j i (.7) q For a give combiatio, the iterative displacemets, δ a, are obtaied with equatio (.5). Writig this equatio for the i ad j compoets yields δa = δa + δλ δa (.8a) q q q q i, i, i, δa = δa + δλ δa q q q q, j, j, j (.8b)

62 8 Chapter The relative iterative displacemet betwee the i ad j compoets is defied by ( ) ( ) δa δa = δa + δλ δa δa + δλ δa (.9) q q q q q q q q, j, i, j, j, i, i Kowig that ( q q δa, j δa, i) followig expressio is obtaied is ull, ad solvig equatio (.9) i order to obtai q δλ, the δa q δλ = δa δa q q, j, i q q, j δa, i (.30) I this equatio δ a i q, ad δ a j q, are the i ad j compoets of the first member of q equatio (.6), ad δ a i, equatio (.7). q ad δ a, j are the i ad j compoets of the first member of With this procedure, termed relative displacemet cotrol betwee two specific variables, the umerical respose of a structure that exhibits a sap-back behavior ca be obtaied (see Figure.5). Aother possible applicatio of this techique is the simulatio of tests i which the opeig of the ack is cotrolled (Crack Mouth Opeig Displacemet cotrol - CMOD) (Rots 988)..5 SUMMARY AND CONCLUSIONS I this chapter a overview o the developmets of cemet based materials i the past years is made. Some models to simulate the ack iitiatio ad propagatio of these materials are preseted ad time-depedet pheomea, like shrikage, eep ad temperature variatio are also discussed. Numerical solutios used i oliear fiite elemet aalysis ad implemeted i the scope of the preset work i the FEMIX computer code are briefly itroduced ad their beefits i terms of umerical simulatio robustess are preseted.

63 Chapter 3 Numerical model for coete lamiar structures 3. INTRODUCTION I this chapter a multi-directioal fixed smeared ack costitutive model to simulate the flexural/puchig failure modes of coete lamiar structures is preseted. The costitutive model is implemeted i a computer program based o the fiite elemet method, called FEMIX (Azevedo et al. 003, Sea-Cruz et al. 007), beig the lamiar structures simulated accordig to the Reisser-Midli shell theory (Reisser 945, Midli 95, Barros ad Figueiras 00). The thickess of the lamiar structure is disetized ito layers that are assumed to be subjected to a plae stress state. I this approach, the use of costitutive models to simulate the oliear behavior, after ack iitiatio, for the i-plae fracture modes is appropriate i most cases, ad the deformatioal respose of a structure subjected to load cofiguratios that iduce flexural failure modes ca be predicted with sufficiet accuracy. However, the simulatio of lamiar structures failig i puchig is a much more complex task, beig the treatmet of the out-of-plae shear compoets of paramout importace. A stress-strai softeig diagram is proposed to simulate the mode I fracture propagatio, while the i-plae shear ack compoet depeds o a shear retetio factor, defied as a costat value or by a ack ormal strai depedet law. The i-plae shear ack

64 30 Chapter 3 compoet ca also be determied by a ack shear stress-strai softeig diagram (see Chapter 4). To capture the puchig failure mode, a softeig diagram is proposed to model, after ack iitiatio, the deease of the out-of-plae shear stress compoets with the iease of the correspodig shear strai compoets. With this relatively simple approach, accurate predictios of the behavior of fiber reiforced coete (FRC) structures failig i bedig ad i shear ca be obtaied. Improvemets made i the subalgorithms associated with the stress update ad with the itical chage of ack status are preseted ad their advatages are discussed. The formulatio of elastic supports, such as surface, lie ad poit sprigs, with liear ad oliear stiffess, is also preseted i this chapter, ad a special attetio is made about uilateral support coditios. With this approach, the loss of cotact betwee a structure ad a supportig system, such is the case of a slab supported o groud, ca be correctly simulated. To assess the predictive performace of the model, a experimetal puchig test of a module of a façade pael fabricated with steel fiber reiforced self-compactig coete (SFRSCC) is umerically simulated. The ifluece that some parameters defiig the softeig diagrams have o the predictive performace of the model for this type of simulatios is aalyzed. 3. GENERAL LAYERED APPROACH TO DISCRETIZE THE THICKNESS OF A LAMINAR STRUCTURE 3.. Itroductio The Reisser-Midli theory (Reisser 945, Midli 95, Barros ad Figueiras 00) is widely used to simulate the behavior of lamiar structures. I structural egieerig applicatios, a lamiar structure ca be defied as a three-dimesioal body with two

65 Numerical model for coete lamiar structures 3 dimesios that are cosiderably larger tha the other oe, which is its thickess. Idustrial floors, pavemets of buildigs or roads, ad bridge decks are examples of lamiar structures. Sice the last cetury structures of this type are maily costructed with cemet based materials, such as reiforced coete (RC) ad, more recetly, with fiber reiforced coete (FRC) or steel fiber reiforced self-compactig coete (SFRSCC). It is kow that these materials exhibit a oliear behavior, eve whe subjected to service loads, due to ack formatio ad propagatio. This material oliearity may be accetuated whe early acks appear (e.g., due to shrikage restrai or temperature developmet), which ca compromise the durability of the structure. A accurate predictio of the behavior of such structures is of great importace to improve its service life, ad to prevet its excessive deformability ad early failure. 3.. Formulatio I this sectio a brief overview of the Reisser-Midli formulatio applied to a layered plae shell approach is preseted. The theory of plates ad shells ca be foud i Timosheko ad Woikowsky-Krieger (959), ad its implemetatio usig FEM was doe by several researchers, such as: Ugural 98, Huag 989, Barros 989, Barros 995, Oñate 995. A plae shell is a flat lamiar structure with i-plae ad out-of-plae shear deformatios. The i-plae deformatios ca be caused by membrae forces ad bedig momets. Therefore, a plae shell combies the behavior of a slab (developmet of bedig ad out-of-plae shear deformatios) with those of a wall (developmet of membrae deformatios). The basic assumptios of the Reisser-Midli theory applied to the case of plae shells are: the u ad u displacemets of the shell middle surface (see Figure 3.) are ot eglected (the presece of membrae deformatios is allowed);

66 3 Chapter 3 i compariso with the shell thickess, the displacemet ormal to the shell middle surface, u 3 (see Figure 3.), is small; the stress actig i the directio ormal to the shell middle surface, σ 3, is small whe compared with the other stress compoets, beig eglected; straight fibers ormal to the middle surface of the shell are cosidered to remai straight but ot ecessarily orthogoal to the middle surface durig the deformatio process. x 3 u 3 u θ x u Shell middle surface θ x h k Shell middle surface layer k layer x 3,k t m b x x 3,k 3,k h Figure 3. Multi-layer plae shell: displacemets, rotatios ad k layer geometry defiitio. As metioed before, the behavior of cemet based materials is clearly oliear. This oliearity results primarily from the fact that this material has a relatively small ackig stress. A layered shell model to simulate the oliear behavior of lamiar structures ca simulate the damage resultig from ack propagatio through the thickess of a shell due to i-plae stresses. I this approach the strais at differet levels alog the shell thickess (middle surface of each layer) are obtaied from the displacemets of the fiite elemet

67 Numerical model for coete lamiar structures 33 odes, ad the stresses are determied accordig to the costitutive laws that simulate the behavior of the material of this layer. Thus, for the fiite elemet material oliear aalysis, a plae shell structure is divided ot oly i Midli shell fiite elemets, but also i layers through the thickess (see Figure 3.). For example, as show i Figure 3., a plae shell elemet is disetized ito a eight-ode seredipity 3D Midli shell fiite elemet (five degrees of freedom per ode). Four or ie ode Lagragia 3D Midli shell fiite elemets are also available i the FEMIX computer code. Each 3D fiite elemet is divided ito layers through the thickess. These layers ca have thickess ad material properties differet from each other. The strais ad stresses are evaluated at the middle surface of each layer ad their relatio depeds o the costitutive law assiged to the layer. So, the preset formulatio is implemeted with geeral purposes, ad ay plae shell structure composed of differet materials through the thickess ca be umerically simulated.

68 34 Chapter 3 (a) Plae shell elemet x 3 x x 3 g x g Shell middle surface x g x (b) Midli shell (3D) fiite elemet idealizatio Seredipity 8-ode elemet (c) Cross sectio multi-layer approach (Layers with differet material properties) Figure 3. Example of a fiite elemet idealizatio of a plae shell elemet accordig to the multi-layer approach.

69 Numerical model for coete lamiar structures 35 Whe the respose of the elemets of a structure becomes oliear, the stiffess begis to deped o the strai state that these elemets are subjected to. I the case of lamiar structures, this strai state ca vary through the thickess, ad i a multi-layer approach, from layer to layer. So, the cotributio of each layer to the elemet stiffess matrix is differet, beig the stiffess matrix thus obtaied called the taget stiffess matrix. To accout for the material oliear behavior, the relatioship betwee the stress ad the strai state is established i a iemetal way, i.e. σ = D ε (3.) T where D T is the taget costitutive matrix, σ the iemetal stress vector ad ε the iemetal strai vector. I the calculatio of the stiffess matrix of each fiite elemet the followig procedure is used (see Figure 3.3): from the kow displacemets i the fiite elemet odes, evaluate the displacemets, a, at the itegratio poits (Gauss Poits); calculate the strais, ε at the middle surface of each layer; calculate the taget costitutive matrix, D T, takig ito accout the costitutive relatio of the material of each layer; calculate the elemet taget stiffess matrix, ( e) T K. 3D plae shell elemet level: a (e) K T Cross sectio level (layered approach): ε DT Figure 3.3 Procedure to obtai the taget stiffess matrix of a elemet.

70 36 Chapter 3 Takig ito accout the through-thickess layer disetizatio (see Figure 3.), the elemet taget stiffess matrix, ( e) K T, ca be obtaied from the submatrices associated with the membrae deformatios, ( e) K ad mb K ( e) bm, bedig deformatios, ( e) K m, membrae-bedig ad bedig-membrae deformatios, ( e) K b, ad out-of-plae shear deformatios, ( e) K. s The submatrix ( e) K m is obtaied from K ( e) = B Dˆ B da T m m m m ( e A ) (3.a) where t b ( ) h/ Dˆ = D dx = D x x N layers m mb 3 mb, k h/ 3, k 3, k k= (3.b) I (3.b) h is the shell thickess, N Layers is the umber of layers of the through-thickess disetizatio, D mb, k, is the costitutive matrix associated with the membrae-bedig deformatio of the k layer ad ( t b x3, k x3, k) is the k layer thickess h k. The submatrices ( e) K ad mb ( e) K are obtaied from bm ( e) K = B Dˆ B da T mb m mb b ( e A ) (3.3a) ( e) K = B Dˆ B da T bm b bm m ( e A ) (3.3b) where

71 Numerical model for coete lamiar structures 37 N h/ layers ˆ m t b mb = bm = mb 3 3 = mb, k h/ 3, k 3, k 3, k k = ( ) Dˆ D D x dx D x x x (3.3c) m beig x 3, k the x 3 coordiate of the middle surface of the k layer. The submatrix ( e) K is obtaied from b K ( e) = B Dˆ B da T b b b b ( e A ) (3.4a) where N h/ layers b = mb 3 3 = mb, k h/ 3, k 3, k 3, k k= m t b ( ) ( ) Dˆ D x dx D x x x (3.4b) The submatrix ( e) K is obtaied from s K ( e) = B Dˆ B da T s s s s ( e A ) (3.5a) where N layers t b ( ) h/ Dˆ = D dx = D x x s s 3 s, k h/ 3, k 3, k k = (3.5b) beig D s, k the costitutive matrix associated with the out-of-plae-shear deformatio of the k layer.

72 38 Chapter 3 To also take ito accout the flexural stiffess of each layer, equatio (3.3c) is substituted with (Barros 989) t b ( x ) ( x ) Nlayers h/ 3, k 3, k Dˆ = Dˆ = D x dx = D (3.6) mb bm mb 3 3 mb, k h/ k = ad equatio (3.4b) is replaced with t b ( x ) ( x ) 3 3 Nlayers h/ 3, k 3, k b = mb 3 3 = mb, k h/ k= 3 Dˆ D x dx D (3.7) Equatios (3.6) ad (3.7) are used i the curret layered model i order to keep it suitable for the aalysis of a lamiar structure with liear elastic behavior usig oly oe layer. The B m, B b ad B s matrices i equatios (3.) to (3.5) are used to obtai the membrae, bedig ad shear deformatios from the correspodig degrees of freedom i the fiite elemet (Barros 995, Oñate 995). The costitutive matrix associated with the membrae-bedig deformatio of the k layer, D mb, k, used i equatios (3.) to (3.4) ad equatios (3.6) ad (3.7), depeds o the material state or regime assiged to this layer, i.e., liear or materially oliear behavior. The defiitio of these matrices ca be foud i sectio 3.3. The costitutive matrix associated with the shear deformatio of the k layer, D s, k, used i equatio (3.5), depeds also o the out-of-plae shear material behavior assiged to the layer. The Ds, k matrix for a liear elastic material is preseted i sectio 3.3, while for a material with oliear behavior, D s, k is detailed i sectio 3.4.

73 Numerical model for coete lamiar structures 39 As already desibed i sectio.4, at a give stage of a oliear aalysis, the odal forces that are equivalet to the exteral applied loads must balace the odal forces that are equivalet to the stress state developed i the structure, amed iteral equivalet odal forces, f it. These iteral equivalet odal forces are itrisically depedet o the material behavior, i.e., they deped o the costitutive model assiged to the material. The calculatio of the iteral equivalet odal forces of each fiite elemet is performed accordig to the followig procedure (see Figure 3.4): from the kow displacemets i the fiite elemet odes, evaluate the displacemets, a, at the itegratio poits (Gauss Poits); calculate the strais, ε at the middle surface of each layer; calculate the stress vector, σ, at the same level of the strai calculatio, takig ito accout the costitutive law of the material where the stress vector is beig calculated; calculate the geeralized forces, F, by itegratig the stresses aoss the thickess; calculate the elemet iteral equivalet odal forces, ( e) f. it 3D plae shell elemet level: Cross sectio level (layered approach): a f ( e) it ε σ F Figure 3.4 Scheme to obtai the iteral equivalet odal forces of a elemet. I a plae shell decomposed ito layers, the elemet iteral equivalet odal forces ca be obtaied from the vectors associated with the membrae forces, ( e f ) it, m, bedig momets, f ( e) it, b, ad out-of-plae shear forces, ( e) f. it, s

74 40 Chapter 3 The vector ( e) f is obtaied from it, m f ( e) T = B NdA ˆ (3.8a) it, m ( e A ) m where T h/ t b [ ] σmb 3 σmb, k ( 3, k 3, k ) Nˆ = N N N = dx = x x (3.8b) h/ N layers k= are the membrae forces. The vector ( e) f is obtaied from it, b f ( e) T = B M ˆ da (3.9a) it, b ( e A ) b where T h / m t b [ ] 3σmb 3 σmb, k 3, k ( 3, k 3, k ) Mˆ = M M M = x dx = x x x (3.9b) h/ N layers k = are the bedig momets. The vector ( e) f is obtaied from it, s f ( e) it, s T = B QdA ˆ (3.0a) ( e A ) s

75 Numerical model for coete lamiar structures 4 where T h/ t b [ 3 3] σ s 3 σs, k ( 3, k 3, k) Qˆ = Q Q = dx = x x (3.0b) h/ N layers k= are the shear forces. Whe a particular layer has liear elastic behavior, the equatios (3.8b) ad (3.9b) are substituted with equatios (3.) ad (3.), respectively (Barros 989). This is importat whe the thickess of the plae shell has oly oe layer (for the case of a liear elastic aalysis). t b ( x3, k) ( x3, k) Nlayers t b mb, k ε m( 3, k 3, k) mb, k ε (3.) f k = Nˆ = D x x + D t b t b ( x3, k) ( x3, k) ( x3, k) ( x3, k) N 3 3 layers mb, k ε m mb, k ε (3.) f k = 3 Mˆ = D + D where ε m ad ε b are, respectively, the membrae ad bedig strais. I the framework of the fiite elemet method, the stiffess submatrices from equatios (3.) to (3.5) ad the vectors from equatios (3.8) to (3.0) are calculated applyig the Gauss-Legedre itegratio rule (Cook 995, Ziekiewicz ad Taylor 000a). 3.3 CRACK CONSTITUTIVE MODEL 3.3. Itroductio Smeared ad disete ack cocepts ca be used to model the ack propagatio i coete structures (de Borst et al. 004). Sice fiber reiforcemet ca assure the formatio of diffuse ack patters, a smeared ack model ca be coceptually more

76 4 Chapter 3 appropriate, ad more effective from the computatioal poit-of-view, for the simulatio of the behavior of fiber reiforced coete structures Formulatio I the cotext of fiite elemet aalysis of materially oliear shell structures, the developed costitutive multi-fixed smeared ack model is implemeted uder the framework of the Reisser-Midli theory adapted to the case of layered shells, i order to simulate the progressive damage iduced by ackig. So the shell elemet is disetized ito layers, ad i each layer a plae stress state is assumed. I this sectio the formulatio of the multi-fixed smeared ack model, implemeted uder the framework of the Reisser-Midli theory, is preseted. Its desiptio refers to a geeric ( k ) coete layer ad to the domai of a itegratio poit (IP) of a fiite elemet. However, to simplify the symbols of the formulatio, the subsipt k is dropped. The adopted costitutive laws ad some model optios are also discussed. A iemetal approach is used for the i-plae compoets, while a total approach is adopted for the out-of-plae compoets. Accordig to the adopted costitutive law, stresses ad strais are related by the followig equatio co σ mb Dmb 0 ε mb co σ = s 0 D ε s s (3.3) beig mb [ ] T σ = σ σ τ (3.4) ad

77 Numerical model for coete lamiar structures 43 mb [ ] T ε = ε ε γ (3.5) the vectors of the i-plae iemetal stress ad iemetal strai, while s [ ] σ = τ τ 3 3 T (3.6) ad s [ ] ε = γ γ 3 3 T (3.7) are the vectors of the out-of-plae total shear stress ad total shear strai. The vector of the total i-plae stress compoets, eeded for the evaluatio of the iteral forces i equatios (3.8) ad (3.9), is obtaied by addig to the previous oe, vector of the i-plae iemetal stress compoets obtaied with equatio (3.3) prev σ mb, the prev mb mb mb σ = σ + σ (3.8) The vector of the i-plae total strai compoets is also updated with prev mb mb mb ε = ε + ε (3.9) I equatio (3.3), co D mb ad out-of-plae shear stiffess matrix. co D s are, respectively, the i-plae stiffess matrix ad the

78 44 Chapter Liear elastic uacked coete For the case of liear elastic uacked coete, co D mb of equatio (3.3) is the costitutive matrix of coete with a liear elastic behavior, desigated by D,, ad defied accordig to the Hooke s law as co mb e D ν 0 E = ν 0 ν 0 0 ( ν ) co mb, e (3.0) while D i equatio (3..3) is desigated with D s, e ad is defied by co s co co 0 Dse, = FG c 0 (3.) beig E the elasticity modulus of coete, ν the Poisso s ratio, ad modulus defied by G c the shear G c E = ( + ν ) (3.) The shear correctio factor F is itroduced i equatio (3.) to take ito accout the ouiform out-of-plae shear stress distributio through the thickess of the shell. Its value is cosidered equal to 5/6 (Barros 995, Oñate 995) Liear elastic acked coete I smeared ack models the iemetal strai vector ε mb, derived from the iemetal odal displacemets obtaied uder the framework of a oliear FEM aalysis, is

79 Numerical model for coete lamiar structures 45 decomposed ito a iemetal ack strai vector, ε mb, ad a iemetal strai vector of the coete betwee acks, co ε mb (Rots 988, Barros 995, Sea-Cruz 004). ε = ε + ε co mb mb mb (3.3) I acked coete, with the coete betwee acks i liear elastic state, D co mb is replaced i equatio (3.3) with the i-plae acked coete costitutive matrix, obtaied with the followig equatio (Sea-Cruz 004) co D mb, T ( ) T co co co co mb, e mb, e mb, e mb, e co Dmb = D D T D + T D T T D (3.4) co where D mb, e is the costitutive matrix defied by equatio (3.0) ad T is the matrix that trasforms the stress compoets from the coordiate system of the fiite elemet to the local ack coordiate system T cos θ si θ siθcosθ = siθ cosθ siθcosθ cos θ si θ (3.5) ad D is the ack costitutive matrix D DI 0 = 0 DII (3.6) I equatio (3.5), θ is the agle betwee x ad (see Figure 3.5). I equatio (3.6) D I ad D II represet, respectively, the costitutive compoets relative to the ack opeig mode I (ormal) ad mode II (i-plae shear).

80 θ 46 Chapter 3 x s t τ t σ Crack σ τ t w x Figure 3.5 Stress compoets, relative displacemets ad local coordiate system of the ack (Sea-Cruz 004). To take ito accout the formatio of several acks at the same IP, the ack costitutive matrix, D, ad the trasformatio matrix, T, of relatio (3.4) are substituted with the matrices that iclude the trasformatio matrix ad the ack costitutive matrix of each ack that ca occur at a specific IP (Sea-Cruz 004). The ack opeig propagatio ca be simulated with the tesile-softeig triliear diagram represeted i Figure 3.6, which is defied by the parameters α i ad ξ i, relatig stress with strai at the trasitios betwee the liear segmets that compose this diagram. The ultimate ack strai, ε u,, is defied as a fuctio of the parameters α i ad ξ i, the fracture eergy, I G f, the tesile stregth, σ, = fct, ad the ack badwidth, b l, as follows (Barros 995, Sea-Cruz 004) ε u, = ξ + αξ α ξ + α G I f f l ct b (3.7)

81 Numerical model for coete lamiar structures 47 beig α = σ / σ (3.8a),, α = σ / σ,3, ξ = ε / ε,, u ξ = ε / ε,3, u (3.8b) (3.8c) (3.8d) σ σ, D I, σ, σ,max σ,3 D I,sec D I, D I,3 G I f l b ε,ε,max ε,3 ε,u ε Figure 3.6 Triliear stress-strai diagram to simulate the fracture mode I ack propagatio ( σ = f, σ = α σ, σ,, = α σ, ε, = ξ ε, u, ε,3 = ξ ε, u).,3,, ct A expoetial tesile-softeig diagram to simulate ack opeig propagatio is also available. This diagram, proposed by Corelisse et al. (986), is represeted i Figure 3.7. The ultimate ack strai, ε u,, is defied as I G f ε u, = k f l ct b (3.9)

82 48 Chapter 3 where k = + + c c c c c c c c c c 3 c exp ( ) ( ) (3.30) beig c = 3.0 ad c =6.93. σ σ, D I σ,max D I,sec G I f l b ε,max ε,u ε Figure 3.7 Expoetial stress-strai diagram to simulate the fracture mode I ack propagatio. The complete equatios that defie the triliear diagram of Figure 3.6 ad the expoetial diagram of Figure 3.7 are exposed i the Appedix B. A secat approach is used to simulate the uloadig ad reloadig braches i both diagrams. The value of the fracture eergy, I G f, ca be obtaied with the equatio proposed i the CEB-FIB (993) or with experimetal tests as desibed i sectio The parameters α i ad ξ i that defie the shape of the triliear tesile-softeig diagram deped sigificatly o the compositio of the cemet based material used, e.g., plai coete, FRC, SFRSCC. The values of these parameters ca be assessed by performig a iverse aalysis, as desibed i sectio 3.7..

83 Numerical model for coete lamiar structures 49 The ack badwidth, l b, associated with the smeared ack approach, must be mesh depedet to assure mesh objectivity. I the literature, several values or methods for its calculatio are preseted (Bazat ad Oh 983, Rots 988, Oliver 989, Hofstetter ad Mag 995, Loureço et al. 997). I the preset work, the ack badwidth ca be calculated as the square root of the area associated with the IP of the fiite elemet, as the square root of the area of the fiite elemet, or ca adopt a supplied costat value. The fracture mode II modulus, D II, is obtaied with D II β = G β c (3.3) where G c is the coete elastic shear modulus (see equatio (3.)) ad β is the shear retetio factor. The parameter β is defied as a costat value or as a fuctio of the curret ack ormal strai, ε, ad of the ultimate ack ormal strai, ε u,, as follows, ε β = ε u, p (3.3) Whe p is uitary, a liear deease of β with the iease of ε is assumed. Larger values of the expoet p correspod to a more proouced deease of the parameter β, i order to simulate a higher i-plae shear stress degradatio with the iease of the ack opeig (Barros et al. 004). A softeig costitutive law to model the i-plae ack shear stress trasfer has also bee developed ad implemeted i the FEMIX computer code. This shear softeig law is desibed i detail i Chapter 4.

84 50 Chapter 3 The out-of-plae shear stiffess matrix, co D s, ca be obtaied with equatio (3.) if a liear elastic behavior for the trasversee shear is assumed, or with a out-of-plae shear softeig diagram exposed i detail i sectio OUT-OF-PLANE SHEAR SOFTENING DIAGRAM 3.4. The puchig problem i lamiar structures Puchig is a complex pheomeo ad oe of the most difficult problems to solve i the desig of RC lamiar structures. Several model approaches have bee developedd ad practically all the desig codes for coete structures have paid specific attetio to this type of failure. The Fib techical report º (Fib 00) is dedicated to the problem of puchig of coete slabs. A examiatio of the puchig problem is preseted, ad special attetio is dedicated to the umerical simulatios of puchig usig FEM. Puchig is fudametally a shear failure mode, ad i the last decades cocers regardig this pheomeo have ieased due to the geeralizatio of the use of thi structures, such as flat plates with smalll thickess ad large dimesios supported by colums (see Figure 3.8). This type of failure ca also be a cocer i idustrial floors subjected to highly cocetrated loads. Figure 3.8 Puchig failure surfaces of a flat slab (Ngo 00). A cocetrated load or reactio actig o a relatively small areaa of a slab or foudatio ca cause puchig shear failure (EC 004) ). This type of failure may occur alog a trucated coe or pyramid aroud the cocetrated load or reactio area (ACI 005).

85 Numerical model for coete lamiar structures 5 The puchig shear failure is usually brittle, ad i the umerical predictio of the coical failure surface whe, for example, a colum suddely perforates the supported slab a tridimesioal model is required (Barzegar ad Maddipudi 997a). A umerical model based o the formulatio of the Reisser-Midli shell theory applied to the case of a multilayer approach has a lower computatioal effort i compariso with a full tridimesioal model (Polak 005). To explore the possibility of usig the former formulatio to predict shear failure modes, a softeig diagram is proposed for the out-of-plae shear compoets of equatio (3.3) Desiptio of the diagram Whe the tesile stregth is reached at a IP of a fiite elemet, the portio of coete icluded i its ifluece area chages from uacked to acked state. This local status chage affects the global behavior of a structure, ad cosequetly the umerical simulatio must be capable of reproducig these pheomea. The use of a multi-fixed smeared ack model to umerically predict the behavior of lamiar shell structures failig i bedig is, i most cases, acceptable as log as the fracture parameters used i the costitutive ack stress-strai relatio are accurately predicted. The predictio of the behavior of a structure that fails i shear or i puchig is, however, a much more difficult task, as already metioed. The proposed out-of-plae (OP) shear diagram is represeted i Figure 3.9. The out-of-plae shear behavior is assumed to be liear elastic for both compoets util the coete tesile stregth, f ct, is reached. Whe the portios of coete associated with the OP OP IP chage from uacked to acked state, the out-of-plae shear stresses ( τ 3, p ad τ 3, p ) are stored for later use, ad the relatio betwee each out-of-plae shear stress-strai τ γ ad τ3 γ3) follows the softeig law depicted i Figure 3.9. ( 3 3

86 5 Chapter 3 τ OP τ p OP liear behavior -γ OP u -γ OP max -γ OP p τ max OP G c γ OP p D III,sec γ OP max G III f l b γ OP u γ OP -τ max OP -τ p OP Figure 3.9 Relatioship betwee out-of-plae (OP) shear stress ad shear strai compoets. The positive braches of the diagram represeted i Figure 3.9 are simulated with the expressios show i equatio (3.33). The egative part ca be obtaied by aalogy. FGc γ 0 < γ γ p OP τ p τ ( γ ) = τ ( ) ( ) p γ γ OP OP p γ p < γ γu γu γ p OP 0 γ > γ u OP OP OP OP OP OP OP OP OP OP OP OP (3.33) I the uloadig or reloadig brach (see sectio for the defiitio of the out-of-plae shear status), the out-of-plae shear stress is calculated with a secat approach, give by τ τ γ = γ (3.34) OP OP OP max OP ( ) OP γ max where τ ad γ OP are the maximum out-of-plae shear stress ad the maximum OP max max out-of-plae shear strai observed i the softeig brach before the start of the uloadig phase.

87 Numerical model for coete lamiar structures 53 The defiitio of the out-of-plae shear stiffess matrix i equatio (3.3), co D s, is ow based o the diagram represeted i Figure 3.9, ad amed out-of-plae shear acked coete costitutive matrix, co D s. Therefore, the co D s matrix is defied by D co s 3 D,sec 0 III = 3 0 DIII,sec (3.35) beig D = τ = τ γ (3.36) OP 3 3,max 3 III,sec, D OP III,sec γ3,max OP 3,max OP 3,max OP OP i accordace with the secat approach show i Figure 3.9, where τ ij, max ad γ ij, max are the maximum out-of-plae shear stress ad shear strai observed i the softeig brach of each shear compoet, respectively. OP OP Each out-of-plae peak shear strai, γ 3, p or γ 3, p, is calculated usig the stored OP OP out-of-plae peak shear stress at ack iitiatio, τ 3, p or τ 3, p, ad the coete elastic shear modulus, G, as follows c γ τ τ = = (3.37) OP OP OP 3, p OP 3, p 3, p, γ3, p Gc Gc OP OP Each out-of-plae ultimate shear strai, γ 3, u or γ 3, u, is defied as a fuctio of the correspodig out-of-plae peak shear strai, γ OP p the mode III (out-of-plae) fracture eergy, III G f, the out-of-plae shear stregth, τ OP p,, ad the ack badwidth, l b, as follows

88 54 Chapter 3 G G γ = γ + γ = γ + (3.38) III III OP OP f OP OP f 3, u 3, p, OP 3, u 3, p OP τ3, p lb τ3, p lb The preset approach assumes that the ack badwidth used to assure mesh idepedece whe modelig fracture mode I ca also be adopted i the out-of-plae fracture process. To improve a faster loss of out-of-plae shear stress with the iease of out-of-plae shear strai, two alterative diagrams are also implemeted i the FEMIX computer code. These diagrams are very similar to the oe preseted i Figure 3.9. The mai differece is located i the softeig brach. Istead of a liear softeig brach, oe of the proposed alteratives is based o the triliear softeig diagram, ad the other is based o the expoetial Corelisse diagram used i the defiitio of ack opeig mode I, as desibed i sectio These two diagrams are represeted i Figure 3.0 ad Figure 3., respectively. Oly the positive brach of the out-of-plae shear stress-strai relatioship is represeted. The out-of-plae shear softeig triliear diagram represeted i Figure 3.0 is simulated with the expressios show i equatio (3.39). As for the case of equatio (3.33), oly the positive brach of the diagram is treated. The egative brach ca be aalogously defied. FGc γ 0 < γ γ p τ τ τ γ γ γ γ γ τ τ τ ( γ ) τ ( γ γ ) γ γ γ τ τ ( γ γ ) γ < γ γ 0 OP OP OP OP OP OP p OP OP OP OP OP p ( OP OP p ) p < γ γ p OP OP OP OP OP OP OP OP OP OP = OP OP < γ γ OP OP OP OP OP OP OP OP OP u γu γ OP OP γ > γ u (3.39) Some costats are defied as follows

89 Numerical model for coete lamiar structures 55 τ = a τ (3.40a) OP OP p τ = a τ OP OP p ( ) γ = b γ γ OP OP OP u p ( ) γ = b γ γ OP OP OP u p (3.40b) (3.40c) (3.40d) τ OP liear behavior τ OP p τ OP τ OP G c D III,sec G III f l b γ OP γ OP γ OP p γ OP u γ OP Figure 3.0 Triliear softeig diagram to simulate the relatioship betwee the out-of-plae shear stress ad shear strai compoets. Oly the positive brach is represeted. OP OP Each out-of-plae ultimate shear strai, γ 3, u or γ 3, u, is defied as a fuctio of the parameters a i ad b i, the correspodig out-of-plae peak shear strai, γ OP p, the out-of-plae shear stregth, ack badwidth, l b, as follows OP τ p, the mode III (out-of-plae) fracture eergy, III G f, ad the γ III OP OP f 3, u = γ3, p + OP b+ ab ab+ a τ 3, p lb G (3.4) ad

90 56 Chapter 3 γ III OP OP f 3, u = γ3, p + OP b+ ab ab+ a τ 3, p lb G (3.4) Some costats are defied as follows a = τ τ (3.43a) OP OP p a = τ τ OP OP p ( ) b = γ γ γ OP OP OP u p ( ) b = γ γ γ OP OP OP u p (3.43b) (3.43c) (3.43d) The out-of-plae shear softeig expoetial diagram is simulated with the expressios show i equatio (3.44). As for the other cases, oly the positive brach of the diagram is defied here. The egative brach ca be aalogously obtaied. τ OP OP ( γ ) = FGc γ 0 < γ γ p = τ ( ( ) 3 ) exp( ) ( 3 p ca ca A c ) exp( c ) + + γ p < γ γu OP 0 γ > γ u OP OP OP OP OP OP OP OP (3.44) I this equatio the parameter A is defied by OP OP p A γ = γ OP OP γ u γ p (3.45) beig c = 3.0 ad c =6.93.

91 Numerical model for coete lamiar structures 57 τ OP liear behavior τ OP p G c D III,sec III Gf l b γ OP p γ OP u γ OP Figure 3. Expoetial softeig diagram to simulate the relatioship betwee the out-of-plae shear stress ad shear strai compoets. Oly the positive brach is represeted. OP OP Each out-of-plae ultimate shear strai, γ 3, u or γ 3, u, of the expoetial softeig diagram represeted i Figure 3. is obtaied with the followig equatio G G γ = γ + γ = γ + (3.46) III III OP OP f OP OP f 3, u 3, p, OP 3, u 3, p OP k τ3, p lb k τ3, p lb where the parameter k is defied by equatio (3.30) Out-of-plae shear status I sectio 3.4. a out-of-plae shear stress-strai diagram is proposed, ad a secat approach for the calculatio of the out-of-plae shear acked coete costitutive matrix, co D s, is preseted. For the calculatio of the iteral forces correspodig to the out-of-plae shear compoets, f ( e) it, s, as desibed i sectio 3.., the shear stresses must be calculated. For the case of a oliear aalysis, the stress history is fudametal i the predictio of the curret behavior of the structure. Therefore, to take this ito accout, the shear strais ad stresses are stored ad five out-of-plae shear statuses are cosidered. With this procedure,

92 58 Chapter 3 the diagram represeted i Figure 3.9 is completely defied ad, at each loadig stage, for a give out-of-plae shear strai the out-of-plae shear stress ca be evaluated ad, cosequetly, the correspodig iteral forces are obtaied. The five ack statuses are represeted i the diagram of Figure 3.. Two sets of ack statuses are stored, oe for each out-of-plae shear compoet, to accout for their idepedet behavior. The explaatio of each status is supplied oly for oe of these compoets. The five out-of-plae shear statuses represeted i Figure 3. take ito accout the followig assumptios: Stiffeig status, whe the ormal stress is smaller tha the tesile stregth. A liear behavior is assumed for the out-of-plae shear stress-strai relatioship; Softeig status, after the ormal stress reaches the coete tesile stregth. A deease of the out-of-plae shear stress is observed with the iease of the out-of-plae shear strai; Uloadig status, whe a deease of the out-of-plae shear strai is observed ad the previous status is softeig. I this case a secat approach is followed; Reloadig status, whe a iease of the out-of-plae shear strai is observed. The brach of the uloadig status is followed; Free-slidig status, whe the out-of-plae shear strai is greater tha the out-of-plae ultimate shear strai. τ OP τ p OP liear behavior -γ OP u Out-of-plae Shear Statuses 5 γ OP u γ OP -τ p OP - Stiffeig - Softeig 3 - Uloadig 4 - Reloadig 5 - Free-slidig Figure 3. Out-of-plae shear statuses.

93 Numerical model for coete lamiar structures 59 Whe the status is "Free-slidig", the out-of-plae shear modulus occurs for both out-of-plae shear compoets, the matrix D III,sec is ull. Whe this D co s (see equatio (3.35)) becomes a ull matrix. To avoid umerical istabilities associated with this occurrece, the matrix co D s is iitialized as follows D = 0 G, D = 0 G (3.47) III,sec c III,sec c As desibed i sectio 3.4., a liear behavior for the out-of-plae shear compoets is cosidered util the portio of coete associated with a IP is assiged a acked status. Afterwards, a softeig behavior is followed usig oe of the diagrams preseted i this sectio. There is o couplig betwee the ormal (tesile) softeig stress-strai diagram (that commads the ack iitiatio) ad the out-of-plae shear softeig diagram. Sice the out-of-plae shear stress trasfer would deease with the ack opeig, a possible strategy to simulate this effect is the activatio of a softeig diagram for the out-of-plae shear stress compoets. The softeig phase for the out-of-plae shear stress compoets ca be activated whe a certai shear stress threshold value is attaied. This strategy was implemeted i versio 4.0 of FEMIX. 3.5 IMPROVEMENTS MADE IN INTERNAL ALGORITHMS 3.5. Stress update As metioed before, the proposed ack costitutive model is implemeted i the FEMIX computer code (Azevedo et al. 003) uder a FEM framework, beig applied to the Reisser-Midli multi-layer shell approach. The computatioal ad algorithmic aspects of this model are similar to the oes implemeted by Sea-Cruz (004) i FEMIX for the case of a multi-fixed smeared ack model used i the cotext of plae stress aalysis.

94 60 Chapter 3 I this sectio some problems associated with the iteral covergece related to the sub-iemetatio of the i-plae iemetal strai vector, ε mb, i order to fit the law of the ack mode I are preseted. The solutio procedure adopted to overcome these problems is also exposed. Firstly, the assumptios made i the algorithm for the stress update of a geeric layer of a IP are preseted. I a oliear problem the stress update is ecessary to obtai the correct evaluatio of the iteral forces of the elemet that cotais the IP. All the data related to each layer at the level of a IP, e.g., stress ad strai history ad iformatio correspodig to the active acks, is stored for later use. Due to the oliear material behavior, a iemetal-iterative procedure must be implemeted to obtai the solutio of the problem, as desibed i sectio.4 of chapter. I a specific iteratio of the Newto-Raphso method (with or without the use of the arc-legth techique or related methods) the iteral forces are calculated ad compared with the exteral forces to verify the equilibrium. This procedure is executed util covergece is achieved. At the level of a IP the iteral forces, f it, are obtaied with the procedure preseted i sectio 3... As stated before, the stress vector, σ, calculated at the middle surface of each layer that disetize the thickess of the shell, is obtaied from the strai vector ε, takig ito accout the costitutive relatio of the material of the correspodig layer. For the i-plae compoets a iemetal approach is used. Whe the material of a certai layer is i acked state, ad is submitted to a iemet iemetal stress vector, ε mb, the correspodig σ mb, must be obtaied takig ito accout the acked state of the material, ad, afterwards, the stress vector σ mb is updated. I the stress update procedure of the i-plae compoets, the followig system of oliear equatios must be solved (Sea-Cruz 004) (to simplify the otatio the subsipt mb is dropped at this stage)

95 Numerical model for coete lamiar structures 6 ( T ε ) σ co co, prev σ σ prev ε = + + ε = 0 f T T D T D T (3.48) I this equatio σ, prev is the ack stress vector of the previous state i the local ack coordiate system (see Figure 3.5), σ is the iemetal ack stress vector that depeds o the curret iemetal ack strai vector equatio (3.5), of the previous state. ε, T is the matrix defied by co D is the matrix defied by equatio (3.0) ad σ prev is the stress vector Two algorithms are available to solve equatio (3.48), amely the Newto-Raphso method ad the fixed-poit iteratio method. The later is oly used whe the covergece is ot achieved with the former. It could be verified that i some umerical simulatios these iteral algorithms did ot achieve covergece, causig the iterruptio of the aalysis ad forcig the user to perform successive restarts with a smaller load iemet. Eve with this restart mechaism i some cases the simulatio could ot proceed. The problems that justify the o covergece of both iterative methods are the followig: the sudde chage i the stiffess of fracture mode I modulus whe the triliear tesile-softeig is used (see Figure 3.6); the presece of two acks i the same IP, with, for example, oe of the acks tryig to close ad the other tryig to become fully ope; i successive iteratios a repeatig patter is observed, i.e., i a specific iteratio the ack ormal strai evolves from poit A to poit B (see Figure 3.3) ad does ot achieve covergece, so i the ext iteratio it evolves from poit A to poit C ad agai fails to achieve covergece. I subsequet iteratios this patter is repeated util the maximum umber of iteratios is reached ad the algorithm stops without covergece;

96 6 Chapter 3 with the itroductio of a softeig costitutive relatio for modelig the i-plae ack shear compoet, as desibed i chapter 4, these problems are greatly aggravated ad also the difficulty i achievig covergece. σ σ, σ, σ,3 A B C ε, ε,3 ε,u ε Figure 3.3 Critical chage of the fracture mode I modulus covergece difficulties. The covergece of each iterative method is cosidered to be reached whe the ifiite orm of the vector f ( ε ) is smaller tha a residual value amed Toler. This parameter is assumed to be equal to f 6 0 c, beig f c the coete compressive stregth. To overcome the lack of covergece due to the previously eumerated problems, the followig procedure is adopted: whe the Newto-Raphso method fails to coverge, the fixed-poit method is activated ad i a first phase the situatio that leads to the smallest ifiite orm of the vector f ( ε ) is stored; i a secod phase the covergece is assumed for this situatio, all the vectors are updated ad the procedure is cotiued. Although the elargemet of the Toler, a maximum value for it is imposed to prevet a excessive error. It could also be verified that i some cases the miimum ifiite orm of the vector f ( ε ) 5 4 was 0 fc or 0 fc ad the covergece was ot achieved due to a very small 6 gap i the iteral assumed covergece iterio ( 0 fc ).

97 Numerical model for coete lamiar structures 63 The relatio betwee the global stress vector, σ, ad the local ack stress vector, give by σ, is σ = T σ (3.49) 6 Whe Toler is assumed to be greater tha 0 fc, the equilibrium imposed by this relatio is also affected. However, it must be emphasized that i subsequet iteratios these quatities are preset i equatio (3.48) ad cosequetly the elargemet of Toler ca be miimized. I coclusio, with this procedure ad i terms of a global aalysis, this iease of Toler at a specific layer of a IP ca be acceptable, beig the robustess of the umerical simulatio greatly improved Critical ack status chage I order to fit the tesile-softeig diagram associated with the ack mode I, the iemetal strai vector, ε mb, must be decomposed whe oe of the itical status chage occurs durig the strai iemet (Barros 995, Sea-Cruz 004). The itical ack statuses are: ew ack iitiatio, closure of a existet ack ad reopeig of a closed ack. Therefore, after the calculatio of the iemetal strai vector, ε mb, a verificatio of the occurrece of a itical status is made. Whe oe of these occurs, the iemetal vector is sub-iemeted. For the calculatio of the trasitio poit correspodig to a ew ack iitiatio, k ew, to a closure of a existet ack, k close, ad to the reopeig of a closed ack, k reope, two algorithms are available. Oe is based o the Newto-Raphso method, ad the other is based o the bisectio method, beig used whe the former does ot coverge (Sea-Cruz 004).

98 64 Chapter 3 I some umerical simulatios it could be verified that, eve with these two algorithms, the value of the trasitio poit could ot be obtaied, i particular the k reope ad the k close values. I this case the umerical simulatio stops ad to overcome this problem a restart procedure with a reduced iemetal load ca be tried, but, i some cases, this restart procedure does ot solve the problem. A exhaustive aalysis has bee made ad it could be verified that additioal covergece problems might occur, as desibed below for a geeric k layer of a specific IP: the trasitio poit for a ack that tries to reope is obtaied, k reope, ad the ew stress vector is calculated usig the curret iemetal strai vector With the remaiig strai vector, ( kreope ) ε mb kreope ε., a ew stress vector is calculated ad the verificatio of the occurrece of a itical status chage is performed. At this momet the ack that reopes i the earlier stage is tryig to close. This patter is repeated ad the umerical solutio caot be obtaied; the trasitio poit for a ack that tries to close is obtaied, k close, ad a ew stress vector is calculated usig the curret iemetal strai vector, remaiig strai vector, ( k ) close ε mb kclose ε mb mb. With the, a ew stress vector is calculated ad the verificatio of the occurrece of itical status chage is made. At this momet the ack that closes i the earlier stage tries to reope. This patter is repeated ad the umerical solutio caot be obtaied; whe two or more acks occur, more tha oe itical ack status chage ca also occur at the sub-iemetatio of the iemetal strai vector, ε mb, ad the covergece becomes more difficult, eve ot attaied i some cases, e.g., a closed ack tries to reope ad a ew ack is iitiated. The solutio ecoutered for these problems is treated separately for the ack reopeig ad for the ack closig processes. The adopted procedure is similar to the oe implemeted by (Sea-Cruz 004) for the case of the iitiatio of a ew ack.

99 Numerical model for coete lamiar structures 65 For example, after the calculatio of the curret kreope ε mb, the ack is cosidered a potetially reopeed ack. With the remaiig strai vector, ( kreope ) ε mb, the reopeed ack is cosidered i equatio (3.48) ad is coverted ito a defiitive reopeed ack if the ack at this phase does ot close. If the ack closes, the ack that i the first phase has idicated as potetially reopeed is ot allowed to chage its status from closed to reopeig, ad its historical data is restored. To permit a future ack reope, the ack ormal stress stored i the historical data to allow a ack reope is updated with σ 6 reopeew,, σ 0 fc = + if σ + 0 f > σ (3.50) 6 c, reope, prev beig σ reopeprev,, the ack ormal stress stored i the historical data whe the ack has closed, σ reopeew,, is the ew updated ack ormal stress for ack reope, f c is the compressive stregth ad σ is the curret ack ormal stress. A similar treatmet is made for a ack that tries to close. For example, after the calculatio of the curret kclose ε With the remaiig strai vector ( k ) mb, the ack is cosidered oly a potetially closed ack. close ε mb, the closed ack is ot cosidered i the equatio (3.48) ad is coverted ito a defiitive closed ack, oly if at this phase does ot try to reope. If the ack reopes, the ack that i the first phase has idicated as closed is ot allowed to chage its status, ad its historical data is restored. As desibe before, to permit a future ack close, the ack ormal strai stored i the historical data to allow a ack closure is updated with ε 6 closeew,, ε 0 = if ε 6 0 < ε, close, prev (3.5)

100 66 Chapter 3 beig ε close,, prev the ack ormal strai stored i the historical data, ε closeew,, the ew updated ack ormal strai for ack close, ad ε the curret ack ormal strai. With the desibed procedures the startig axes of the ack tesile-softeig diagram for the specific ack where such problems occur are margially moved, as schematically represeted i Figure 3.4. It must be stated that the global accuracy is ot sigificatly affected, beig the robustess of the umerical simulatios sigificatly improved. σ (a) σ (b) σ,reope,ew ε ε,close,ew ε Figure 3.4 Adopted iteria to: (a) update the ack ormal stress for a ack reope; (b) update the ack ormal strai for a ack close. 3.6 SUPPORTS WITH LINEAR AND NONLINEAR BEHAVIOR 3.6. Deformable poit, lie ad surface support system A structure ca have some of its odal poits coected to elemets that ca exhibit a elastic or ielastic behavior. These elemets ca be, for example, the groud supportig system, bars or ay other structural elemet whose deformability is proportioal to the load applied to this elemet, etc. The cotributio of these elemets to the behavior of a structure is take ito accout by addig their stiffess to the stiffess of the structure (Vetura-Gouveia 996). I geeral, they ca be cosidered as poit sprigs, lie sprigs or surface sprigs, i.e., a poit sprig whe oe poit of a structure is coected to a deformable elemet, such as a

101 Numerical model for coete lamiar structures 67 plate supported o a colum, a lie sprig whe the support coditios of a structure are provided cotiuously alog a lie, such as a plate supported o a wall, ad a surface sprig whe the cotact betwee the structure ad the supportig system is a surface, such is the case of a slab o grade. I the framework of the FEM, the cotributio of these supportig elemets to the stiffess of the global system ca be calculated as follows: Poit sprigs I this case, the stiffess of the sprig is directly added to the diagoal terms of the stiffess matrix of the structure i correspodece with the degrees of freedom of the poit that is coected to the sprig. This additio process takes ito accout the coordiate system of the structure ad the directio vector of the sprig. A geeric represetatio of poit sprigs coected to a structure is represeted i Figure 3.5. x 3 s 3 x x 3 g g x g x s s Structure x Poit sprig Figure 3.5 Poit sprigs: global coordiate system, sprig directio vector, s i. g x i, local coordiate system of the structure, x i, ad The poit sprig stiffess poit K s i the global coordiate system is obtaied from K poit s T = T k T (3.5) s

102 68 Chapter 3 where T is the trasformatio vector relatig the global coordiate system ad the sprig directio vector (see Figure 3.5) ad k s the sprig stiffess. Lie sprigs A geeric represetatio of a lie sprig coected to a edge of a structure is represeted i Figure 3.6. x 3 Fiite elemet s x x 3 g Lie sprig g x g x Structure x Figure 3.6 Lie sprig: global coordiate system, g x i, local coordiate system of the structure, x i, ad lie sprig directio vector, s. The lie sprig stiffess lie K s cotributio to the stiffess of the edge of the fiite elemet where the lie sprig is coected is obtaied with K lie T T s L = N T k T N dl (3.53) s where T is the trasformatio vector relatig the global coordiate system ad the lie sprig directio vector (see Figure 3.6), N is the matrix of the shape fuctios of D fiite elemets, ad k s is the sprig stiffess.

103 Numerical model for coete lamiar structures 69 Surface sprigs I Figure 3.7 a surface sprig coected to a face of a structure is represeted. x 3 s Fiite elemet x x 3 g Structure g x g x x Surface sprig Figure 3.7 Surface sprig: global coordiate system, surface sprig directio vector, s. g x i, local coordiate system of the structure, x i, ad The surface sprig stiffess the surface sprig is coected is obtaied with surf K s cotributio to the stiffess of the fiite elemet where K surf T T s A = N T k T N da (3.54) s where T is the trasformatio vector relatig the global coordiate system ad the surface sprig directio vector (see Figure 3.7), N is the matrix of the shape fuctios of D fiite elemets, ad k s is the sprig stiffess. The sprig stiffess, k s, of equatios (3.53) ad (3.54) is obtaied at each IP usig the shape fuctios of the D fiite elemet or the shape fuctios of the D fiite elemet, respectively. With this procedure a lie sprig with differet stiffess values alog its legth ca be umerically simulated, ad, i the same way, a surface sprig with o

104 70 Chapter 3 costat stiffess ca be treated (for example the simulatio of a slab supported o a heterogeeous soil). The defiitio of the shape fuctios used i the above equatios ca be foud elsewhere (Ziekiewicz ad Taylor 000a) Uilateral supports I the previous sectio a geeral approach for the calculatio of the cotributio of differet types of sprigs for the stiffess matrix of the structure is preseted assumig a liear-elastic behavior for the sprigs. These sprigs ca be coected to the various types of fiite elemets available i the FEMIX computer code (Azevedo et al. 003, Sea-Cruz 004). Alteratively, there ca be situatios i which a supportig system ca be idealized by a sprig system with oliear behavior, as for the case of a soil. The evaluatio of the taget soil reactio modulus ca be performed with plate-loadig tests (Barros ad Figueiras 998), ad results of these tests have revealed that the soil pressure-settlemet relatioship ca be simulated with a multiliear or liear-parabolic diagram (Barros ad Figueiras 00). There are other situatios i which the stiffess of the supportig system of a structure depeds o the type of load actig o this supportig system, ad it ca eve be eglected for certai type of loadig, as for the case of a soil subjected to tesile stresses. These types of supports are amed i the preset work as uilateral supports. I the ext sectios details of two diagrams to simulate this type of supports are preseted: a liear-parabolic diagram to predict soil oliear behavior ad its loss of cotact with the supported structure; a biliear-expoetial diagram that ca be assiged to supports cosidered to be active oly i certai circumstaces. A parametric study carried out with a slab o grade is also preseted.

105 Numerical model for coete lamiar structures Liear-parabolic diagram To provide the developed model with the possibility of umerical simulatios of slabs o grade, a liear-parabolic diagram betwee pressure ad settlemet has bee icluded i the model (Barros 995, Barros ad Figueiras 00), beig represeted i Figure 3.8. p (pressure) p su p max αp su k st 3 k sl Statuses -Loadig - Uloadig 3- Reloadig 4 - Iactive k sl 4 a s a s a smax a su (settlemet) a s Figure 3.8 Pressure-settlemet liear-parabolic diagram. The pressure-settlemet liear-parabolic diagram is defied by the followig expressios k a 0 < a a γ ksl asu p ( a ) = a γk a + k ( γ + ) a < a a p a > a sl s s s s s sl s sl s s su asu su s su (3.55) where a a su s = α (3.56a) γ = α α (3.56b) α + 4

106 7 Chapter 3 beig α a parameter that defies the trasitio poit from the liear to the parabolic brach (see Figure 3.8). The values of p su, a su ad k sl are obtaied by curve fittig based o the experimetal results of plate-loadig tests (Barros ad Figueiras 998). A elastic brach is assumed for modelig the uloadig/reloadig phase of the pressure-settlemet relatio, although the experimetal results obtaied from plate-loadig tests (Barros ad Figueiras 998) reveal that the uloadig /reloadig cycles ca be stiffer tha the iitial elastic phase. For the uloadig or reloadig brach (see Figure 3.8) the pressure is give by the followig equatio ( ) ( ) p a = k a a a < a a (3.57) s sl s s s s smax beig a s the residual settlemet obtaied with a s p max = asmax (3.58) ksl where a smax ad p max are the maximum applied settlemet ad the correspodig pressure, pertaiig to the pressure-settlemet evelope curve, whose values are stored for the evaluatio of the uloadig-reloadig brach. The soil reactio taget modulus i the parabolic brach, k st, is obtaied with the followig equatio s ( ) = γ k a k st s sl a a su (3.59)

107 Numerical model for coete lamiar structures 73 I a oliear aalysis, as desibed i sectio.4 of chapter, the iteral forces must be obtaied. With this aim, the elemet odal forces that are equivalet to the soil pressure are calculated with ( ) e T T it, ( e soil A ) ( ) f = N T p a da (3.60) s where T is the trasformatio vector relatig the global coordiate system ad the surface sprig directio vector, N is the matrix of the elemet surface shape fuctios where the surface sprig is coected, ad p( a s ) is defied by the expressios (3.55) or (3.57). I the framework of the fiite elemet method, the itegrals i (3.53), (3.54) ad (3.60) are calculated usig the Gauss-Legedre quadrature rule (Cook 995, Ziekiewicz ad Taylor 000a). The soil cotributio to the stiffess of the global structural system is computed with equatio (3.54), usig the soil reactio taget modulus, k sl or k st, i the place of k s, accordig to the pressure-settlemet liear-parabolic diagram represeted i Figure 3.8. This oliear behavior, idealized by the liear-parabolic diagram of Figure 3.8, is assiged to the surface sprigs that are orthogoal to the middle surface of a lamiar structure. The frictio betwee the lamiar structure ad the soil is eglected. At each stage of a oliear aalysis the history of some parameters, for example pressure ad settlemet, must be kow. For this reaso the FEMIX computer code stores the sprig historical data, idepedet of the historical data related to the elemets of the structure. For this purpose the four statuses idicated i Figure 3.8 are cosidered. With this procedure the diagram is completely defied at each stage of the oliear aalysis ad for the surface sprig associated with each IP.

108 74 Chapter 3 For the case of a slab o grade, whe the coete slab loses cotact with the soil at a IP, the portio of soil that correspods to this IP does ot cotribute to the stiffess of the slab-soil system ad, cosequetly, the surface sprig has a iactive status. Whe cotact is re-established the soil stiffess is take ito accout, beig its value depedet o the brach of the diagram represeted i Figure 3.8 ad o the previous status before the sprig had became iactive. This iformatio is obtaied from the data stored i the sprig historical data. Although the diagram of Figure 3.8 has bee idealized for soil-structure simulatio, it is also implemeted for the simulatio of poit sprigs ad lie sprigs with oliear behavior. The ecessary adaptatios i the calculatio of the stiffess matrix ad iteral forces are icluded i the computer code Biliear-expoetial diagram The biliear-expoetial diagram desibed i this sectio ad represeted i Figure 3.9 =, ca be assiged to poit sprigs. This diagram is defied by three poits, P ( a, F ) = (, ), ad P ( a, F ) P a F =, ad by the parameter p used i the defiitio of the third brach. Whe a uitary value is attributed to this parameter the third brach is liear. F (force) F 3 (a,f) max F F k k 3t 3 k max Statuses -Loadig - Uloadig 3 - Reloadig 4 - Iactive k 4 a a ar a3 (displacemet) a Figure 3.9 Force-displacemet biliear-expoetial diagram.

109 Numerical model for coete lamiar structures 75 The equatios that defie the relatio show i Figure 3.9 are the followig ka 0 < a a k ( a a ) + F a < a a p F( a) = a a F + ( F F ) a < a a F a > a a3 a 3 3 (3.6) where k F = (3.6a) a k = ( F F) ( a a ) (3.6b) ( ) ( ) p F F3 a3 a k3t = p a 3 a a 3 a (3.6c) I aalogy with the diagram desibed i sectio 3.6.., the statuses of this curve are also represeted i Figure 3.9. The poit iteral forces are determied with f poit it T = T F (3.63) beig T the trasformatio vector relatig the global coordiate system ad the sprig directio vector. The stiffess matrix is obtaied with equatio (3.5), cosiderig the substitutio of k s with oe of the values defied by expressio (3.6).

110 76 Chapter 3 This diagram ca simulate a evetual gap betwee the structure ad a support. This is a very importat feature sice i some cases a structure becomes i cotact with a support oly after a certai displacemet has occurred. Figure 3.0 represets the simulatio of a gap. The value of a is the gap betwee the structure ad the poit sprig. A poit sprig simulated with the oliear costitutive relatio proposed i this sectio ca be activated to work oly i compressio or oly i tesio, e.g., if the support of a structure oly works for compressio forces, tha the loss of cotact is activated if a tesile force is applied to the support. This model is available i the FEMIX computer code ad ca be used to simulate oliear uilateral supports. F (force) F 3 (a,f) max k 3t F k max F k a a ar a3 (displacemet) a Figure 3.0 Gap simulatio betwee a support ad a structure. A elastic uloadig/reloadig is assumed i this diagram (Figure 3.0), beig its icliatio, k max, assumed as the maximum stiffess provided by equatios (3.6). For the case represeted i Figure 3.0, k max is equal to k. I the uloadig/reloadig phase the force is obtaied with ( ) ( ) F a = k a a a < a a (3.64) max r r max beig a r the residual displacemet obtaied with

111 Numerical model for coete lamiar structures 77 a r F max = amax (3.65) kmax where a max ad F max are the maximum applied displacemet ad the correspodig force pertaiig to the force-displacemet evelope curve, whose values are stored for the evaluatio of the uloadig/reloadig brach Parametric study of a slab o grade Idustrial floors are oe of the most commo applicatios of steel fiber reiforced coete (SFRC). Crack cotrol joits are built to cocetrate the ack propagatio i these weakess-iduced surfaces, resultig i a floor divided ito paels. The desig of a SFRC floor is, i geeral, restricted to the aalysis of a represetative pael. For the most commo situatios a poit load i a corer of the pael is the most ufavorable load cofiguratio. The model desibed i sectio is used for the soil simulatio i a parametric study of a SFRC slab o grade (Barros et al. 005c). Whe the coete slab loses cotact with the soil at a IP, the part of the soil that correspods to this IP does ot cotribute to the stiffess of the slab soil system. I this study, the ifluece of the slab thickess ( h ), the soil reactio modulus ( k s ) ad the amout of fibers ( Q f ) is take ito accout, as show schematically i Figure 3.. Figure 3. Parametric study of a SFRC slab o grade.

112 78 Chapter 3 The adopted fiite elemet mesh used to aalyze the behavior of the 5 5 m pael is represeted i Figure 3.. Sice the elemets outside the dashed lie square are ot affected by ay coete oliear pheomeo, they are assumed to behave liearly. The elemets located iside this square are assumed to exhibit a oliear material behavior. The pael thickess is decomposed i 0 layers of equal thickess. The SFRC fracture parameters used to defie the σ ε triliear diagram adopted to model the fracture mode I are preseted i Table 3.. A average compressive stregth of 38 MPa ad a Youg's Modulus of 3 GPa are cosidered i the aalysis. The soil is simulated with surface sprigs that are orthogoal to the lamiate structure. For example, Figure 3.3 represets the ack patter for the slab with h = 60 mm, Qf 3 = 5 kg/ m ad ks 3 = 0.0 N/ mm, at a load level correspodig to a maximum ack opeig of 0.3 mm. More details about this study ca be foud elsewhere (Barros et al. 005c). x x x x Liear Elemets Noliear Elemets Load Figure 3. Fiite elemet mesh. Mai ack zoe Figure 3.3 Crack patter at the slab top surface for a load level correspodig to a maximum ack opeig of 0.3 mm.

113 Numerical model for coete lamiar structures 79 Table 3. - Parameters defiig the triliear diagram of Figure 3.6 for the aalyzed SFRC. Q f σ σ,, σ,3 ε, ε I,3 G f (kg/m 3 ) (MPa) σ, σ, ε u, ε u, (N/mm) NUMERICAL SIMULATION OF A PUNCHING TEST WITH A MODULE OF SFRSCC PANEL To access the predictive performace of the developed model, a experimetal puchig test with a module of a steel fiber reiforced self-compactig coete (SFRSCC) pael is umerically simulated i this sectio. The umerical results are compared with the experimetal oes, ad the ifluece of some model parameters i the umerical predictios is discussed. SFRSCC is a relatively recet cemet based material that combies the beefits of the self-compactig coete techology (Okamura 997) with the advatages of the additio of fibers to a brittle cemetitious matrix (Pereira 006). To maufacture the lightweight pael system schematically represeted i Figure 3.4, which ca be applied i buildig façades, a developed SFRSCC was used ad desibed elsewhere (Barros et al. 005a). The mix compositio of the SFRSCC used to maufacture the pael is preseted i Table 3.. I the compositio of the SFRSCC, 30 kg/m 3 of hooked eds steel fibers with a legth ( l f ) of 60 mm, a diameter ( f ) ( f f ) d of 0.75 mm, a aspect ratio l d of 80 ad a yield stress of 00 MPa were used. At seve days the average value of the compressive stregth ad modulus of elasticity of this SFRSCC was 5 MPa ad 3 GPa, respectively.

114 80 Chapter S Polystyree 300 S' S-S' 50 Polystyree Figure 3.4 Cocept of a lightweight steel fiber reiforced self-compactig coete pael [all dimesios are i mm] (Barros et al. 005a). The flexural stregth of this type of structural elemets is a key aspect i their desig, sice, i geeral, the bedig momets of the wid load combiatio are a importat factor i the desig process of the pael. To assess the pael flexural behavior, represetative modules of the SFRSCC pael system were tested, beig the details of the experimetal program desibed elsewhere (Barros et al. 007a). Numerical simulatios of these paels were also made usig the developed model ad ca be foud elsewhere (Barros et al. 007b) The puchig resistace is also a key aspect i the desig of this type of pael, sice its lightweight zoes cosist of a thi layer that is oly 30 mm thick. To evaluate the puchig resistace of these zoes, represetative modules of the pael system are submitted to a load cofiguratio that implies the occurrece of this type of failure mode (Barros et al. 005a, Barros et al. 007a). I the ext sectios the results obtaied i oe of these tests are compared with the umerical simulatios i order to assess the predictive performace of the developed model. Several umerical simulatios are carried out to assess the ifluece of some

115 Numerical model for coete lamiar structures 8 parameters that defie the softeig diagrams (Vetura-Gouveia et al. 0). The objective of these simulatios is to uderstad how each parameter affects the respose of a lamiar FRC structure failig i puchig. The ifluece of the i-plae mesh ad through-thickess refiemet of the simulated structure is also aalyzed. The possibility of defiig the fracture parameters that characterize the fracture mode I strai-softeig diagram by performig a iverse aalysis (Barros et al. 005b) is also discussed. The iverse aalysis is based o the results obtaied i three poit otched beam bedig tests carried out accordig to the RILEM TC 6-TDF recommedatios (Vadewalle et al. 00). Table 3. - Compositio for m 3 of SFRSCC icludig 30 kg/m 3 of fibers Paste total volume Cemet CEM I 4.5R Limestoe filler Water Superplasticizer * Fie sad Coarse sad Crushed aggregates (%) (kg) (kg) (dm 3 ) (dm 3 ) (kg) (kg) (kg) * Third geeratio based o polycarboxilates (Gleium 77SCC) 3.7. Evaluatio of the mode I fracture properties by iverse aalysis This sectio desibes the iverse aalysis methodology adopted to evaluate the fracture mode I parameters of the SFRSCC used i the pael prototype that was experimetally tested ad umerically simulated. Detailed iformatio about this iverse aalysis ca be foud elsewhere (Barros et al. 005b, Sea-Cruz et al. 004). As already metioed, i the implemeted smeared ack costitutive model the post-ackig behavior of SFRSCC uder tesio ca be desibed by a triliear stress-strai softeig diagram (see Figure 3.6). This fuctio is defied by a set of fracture parameters ( α i, ξ i, I G f, f ct ad l b ), beig the accuracy of the FEM modelig largely depedet o the values that are assiged to these parameters. I this cotext, the experimetal behavior of a elemet failed i bedig may be predicted by a FEM model,

116 8 Chapter 3 as log as the correct values of the material fracture parameters are itroduced i the costitutive model. The adopted strategy cosists i the evaluatio of the ξ i, α i ad I G f parameters that defie the shape of the triliear parameter σ ε costitutive law, based o the miimizatio of the error err = A A A (3.66) exp um exp F δ F δ F δ beig exp AF δ ad A um F δ the areas beeath the experimetal ad umerical load-deflectio curves correspodig to a three poit otched beam bedig test (Sea-Cruz et al. 004). The experimetal curve correspods to the average results observed i prismatic SFRSCC otched specimes, tested accordig to the RILEM TC 6-TDF recommedatios at the age of 7 days (Vadewalle et al. 00), while the umerical curve cosists of the results obtaied by FEM aalysis, beig the specime, loadig ad support coditios simulated i agreemet with the experimetal flexural test setup as represeted i Figure 3.5. I this cotext, the specime is modeled with a mesh of 8 ode seredipity plae stress fiite elemets. The Gauss-Legedre itegratio scheme with itegratio poits is used i all elemets, with the exceptio of the elemets at the specime symmetry axis, where itegratio poits are used. With this particular itegratio poit layout, the umerical results have a better agreemet with the experimetal observatios, sice a vertical ack may develop alog the symmetry axis. Liear elastic material behavior is assumed i all the elemets, with the exceptio of those above the otch, alog the symmetry axis. I this regio a elastic-acked material model i tesio is adopted. The ack badwidth, l b, is assumed to be equal to 5 mm, beig this value coicidet with the width of the otch ad of the elemets located above it.

117 Numerical model for coete lamiar structures 83 8 ode seredipity plae stress elemets itegratio poits elastic behaviour 8 ode seredipity plae stress elemets itegratio poits elastic-acked behaviour y x Figure 3.5 FEM mesh used i the umerical simulatio of a three-poit otched beam flexural test at 7 days (Pereira et al. 008). I Figure 3.6 the results experimetally obtaied i the flexural tests are compared with the umerical results. The curve of the umerical simulatio, obtaied with the optimized fracture parameters, is ot perfectly coicidet with the experimetal curve, suggestig that additioal parameters should be cosidered i order to obtai a better fittig. The values of the fracture parameters ξ i, α i ad represeted i Figure 3.6 are listed i Table 3.3. G I f that lead to the umerical results 5 (b) 0 Load [kn] 5 0 scatter of experimetal results 5 experimetal average results umerical simulatio Deflectio [mm] Figure 3.6 Experimetal average results ad umerical simulatio of the three-poit otched beam flexural test at 7 days (Pereira et al. 008).

118 84 Chapter Test setup ad values of the parameters used i the umerical simulatios The puchig test of a module of the developed SFRSCC lightweight pael is used to assess the predictive performace of the proposed multi-fixed smeared ack model. The test layout ad the test setup are represeted i Figure 3.7. More details about the correspodig experimetal program ca be foud elsewhere (Barros et al. 007a). Steel plate (00x00x0) 50 Q Q' (a) Q-Q' Actuator Polystyree Steel plate (00x00x0) SFRSCC (x300) (b) Figure 3.7 (a) Test pael module, ad (b) test setup [all dimesios are i mm] (Barros et al. 007a). The ifluece of mesh refiemet ad some model parameters i the results of the umerical simulatios is assessed ad discussed i the ext sectios, amely: the values adopted for the fracture mode I parameters used to defie the triliear diagram, ad the values used to defie the out-of-plae shear stress-strai diagram. The umerical simulatios are performed usig the Newto-Raphso method, with displacemet cotrol at a specific variable (see sectio.4.3 of chapter ). The values of the parameters of the costitutive model used i the umerical simulatios of the puchig test are listed i Table 3.3.

119 Numerical model for coete lamiar structures 85 Table Values of the parameters of the costitutive model used i the umerical simulatios of the puchig test. Poisso s ratio ν = 0.5 Iitial Youg s modulus Compressive stregth Triliear tesio softeig diagram of SFRSCC (used i the umerical simulatios of sectio Parameters values obtaied from iverse aalysis) Triliear tesio softeig diagram of plai coete (used i a umerical simulatio of sectio Parameters obtaied from the compressive stregth of the SFRSCC accordig to CEB-FIP 993 recommedatios) E = N mm c f = 5.0 N mm c I f ct = 3.5 N mm ; f 4.3 N mm G = ; ξ = ; α = 0.5 ; ξ = 0.5 ; α = 0.59 G = ; I f ct = 3.5 N mm ; f N mm ξ = 0.07 ; α = 0.5 ; ξ = ; α = 0.09 Triliear tesio softeig diagram of SFRSCC (used i the umerical simulatios of sectio Parameter values obtaied by ieasig/deeasig ± 50% those obtaied from iverse aalysis) Fracture eergy (mode III) used i the out-of-plae shear stress-strai diagram Parameter defiig the mode I fracture eergy available to the ew ack f ct = 3.5 N mm ; G = 50% 4.3 N mm ; I f ξ = ± 50% ; α = ± 50% 0.5 ; ξ = ± 50% 0.5 ; α = ± 50% 0.59 ( ±- depeds o the umerical simulatio) III from G f =.0 N mm to G III f = 5.0 N mm (depeds o the umerical simulatio) p = Shear retetio factor Expoetial ( p = ) Crack badwidth Threshold agle α th = 30º Square root of the area of the itegratio poit α = σ, / σ,, α = σ,3 / σ,, ξ = ε, / ε, u, ξ = ε,3 / ε, u (see Figure 3.6) Aalysis based o the values obtaied from iverse aalysis Ifluece of the out-of-plae shear softeig diagram The results of the umerical simulatios are compared with the experimetal data obtaied i the puchig test of the pael module. The fiite elemet idealizatio, load ad support

120 86 Chapter 3 coditios used i the umerical simulatios of the puchig test are show i Figure 3.8a). Due to double symmetry, oly oe quarter of the pael is cosidered i the simulatios. The mesh is composed of 6 6 eight-ode seredipity plae shell elemets. The elemets are divided ito layers, each oe beig 0 mm thick. Sice the pael has lightweight zoes (shaded elemets i Figure 3.8a), materialized by the suppressio of 80 mm of coete i the cetral zoe, ull stiffess is assiged to the 8 bottom layers of the correspodig fiite elemets (see Figure 3.8b). The material of the remaiig three layers has a elastic-acked behavior, as desibed i sectio This model is also used i the elemets located outside the cetral lightweight zoe. x (a) Poit load mm A + Supports A' Thickess - 0 mm Thickess - 30 mm 300 mm x (b) Cross sectio: A-A' 3 0 mm 8 0 mm Layers with SFRSCC properties Layers with ull stiffess Figure 3.8 (a) Geometry, mesh, load ad support coditios used i the umerical simulatio of the puchig test Coarse Mesh (CM); (b) Properties of the layered oss sectio.

121 Numerical model for coete lamiar structures 87 A trial-ad-error procedure is required to estimate reasoable values for the out-of-plae compoets of the elastic-acked costitutive matrix, co D s (see sectio 3.4), sice their experimetal evaluatio is quite complex ad beyod the scope of the preset work. The out-of-plae shear fracture eergy that leads to the best agreemet with the experimetal III results of the puchig tests, G = 3.0 N mm, is determied with this procedure. f The values of the mode I fracture parameters that take part i the i-plae elastic-acked costitutive matrix for coete, sectio co D mb, are obtaied by iverse aalysis, as desibed i I Figure 3.9 resposes obtaied with the umerical model are compared with the experimetal results. A good agreemet ca be observed up to a deflectio of.5 mm. For larger deflectios, a overestimatio of the load carryig capacity of the prototype pael occurs whe a liear elastic behavior is assumed for the out-of-plae shear compoets. At a deflectio of about 3 mm, the experimetal curve suddely falls, idicatig the failure of the pael by puchig, as visually cofirmed i the experimetal test. This load decay that is ot reproduced whe assumig a liear elastic behavior for the out-of-plae shear compoets is, however, well captured whe the biliear diagram represeted i Figure 3.9 is used to model the softeig behavior of the out-of-plae shear compoets, with III Gf = 3.0 N mm, ad assumig a ack badwidth, l b, equal to the square root of the area associated with the correspodig IP. The abrupt load decay from approximately 4 kn to 0 kn, which is observed i the experimetal test, is accurately simulated by the umerical model, as well as the subsequet exteded stage of residual load carryig capacity exhibitig a very small load decay.

122 88 Chapter Load [kn] Experimetal Liear behavior Liear out-of-plae shear (CM_3L) Softeig out-of-plae shear (CM_3L) Deflectio [mm] Figure 3.9 Relatioship betwee load ad deflectio at the ceter of the test pael. Up to a 0 kn load, all the curves depicted i Figure 3.9 are practically coicidet. Afterwards, the straight lie that represets the respose assumig a liear elastic behavior o loger follows the curves that correspod to the experimetal test ad to the umerical aalysis with material oliear model. These results suggest that some acks start to form at a very early stage of the experimetal test. The oliear umerical model accurately captures the formatio of bedig acks at the top surface (see Figure 3.30a), i agreemet with the experimetally observed ack patter. Figure 3.30b shows the ack patter at the top surface observed at the ed of the test sequece. The umerical model also idicates the formatio of bedig acks at the bottom surface of the lightweight zoe. These acks iitiate at the ceter of the pael, beeath the loaded area, ad the progress to the corers of the lightweight zoe, showig some similarities with the classical yield lies formed i square coete slabs failig by flexure. These acks ca also be observed i the experimetal test (Pereira 006).

123 Numerical model for coete lamiar structures 89 (b) Figure 3.30 Puchig test simulatio: (a) top surface acks predicted by the umerical model (usig a FEM mesh with eight-ode seredipity plae shell elemets), ad (b) photograph showig the acks at the top surface of the pael, at the ed of the test sequece (Pereira 006). I coclusio, the results idicate that flexure mechaisms prevail i the deformatioal behavior up to a deflectio of approximately.5 mm. For larger deflectios, the puchig failure mechaisms start to assume a greater relevace, ad the overestimatio of the pael out-of-plae rigidity compoets, whe liear out-of-plae shear behavior is assumed, leads to a divergece betwee the umerical model ad the experimetal observatios. With the adoptio of a softeig law for the out-of-plae shear compoets, the umerical model becomes much more accurate i the predictio of the complete behavior of the pael failig i puchig, capturig the sudde load decay associated with puchig failure mechaisms. To estimate the cotributio of fiber reiforcemet to the puchig resistace, a umerical simulatio was performed adoptig for the fracture mode I the parameters idicated i Table 3.3, which correspod to plai coete with compressive stregth matchig the developed SFRSCC. Comparig the curves i Figure 3.3 it ca be cocluded that fibers ot oly ieased sigificatly the puchig resistace, but also, ad especially, improved the ductility.

124 90 Chapter Load [kn] Experimetal Plai Coete (CM_3L) Softeig out-of-plae shear (CM_3L) Deflectio [mm] Figure 3.3 Ifluece of fiber reiforcemet i the puchig resistace. Figure 3.3 represets the vertical displacemet field for a deflectio of 0 mm at the ceter of the pael for the case of the umerical simulatio cosiderig out-of-plae shear softeig. The obtaied strog gradiet of vertical displacemets matches with high precisio the experimetally observed locatio of the iterceptio of the puchig failure surface with the top pael face (see Figure 3.30b). This evideces the suitability of the developed approach for the simulatio of this complex failure mode. Figure 3.3 Vertical displacemet field (i mm) for the umerical simulatio with out-of-plae shear softeig (for a deflectio of 0 mm at the ceter of the pael).

125 Numerical model for coete lamiar structures 9 As already metioed, the selectio of a value for III G f has o experimetal support. I order to aalyze its ifluece o the results of the umerical simulatio usig a softeig law for both out-of-plae compoets, a parametric aalysis is carried out cosistig i the variatio of its value from.0 to 5.0 N/mm. The results depicted i Figure 3.33 show that a value of G = 3.0 N mm leads to a perfect predictio of the abrupt load decay III f experimetally observed at a deflectio of about 3 mm. Ieasig or deeasig the value of III G f implies the occurrece of the abrupt load decay at a larger or smaller deflectio, respectively. The coclusio of this study is that, idepedetly of the value of III G f, whe usig the model desibed i this work, it is essetial to use a softeig law for the out-of-plae shear compoets i order to simulate the sudde load decay observed i the puchig test. Load [kn] Experimetal Softeig out-of-plae shear (CM_3L_GfIII) Softeig out-of-plae shear (CM_3L_GfIII) Softeig out-of-plae shear (CM_3L_GfIII3) Softeig out-of-plae shear (CM_3L_GfIII4) Softeig out-of-plae shear (CM_3L_GfIII5) Deflectio [mm] Figure 3.33 Ifluece of III G f, usig the i-plae coarse mesh ad 3 layers i the lightweight zoe, o the umerical relatioship betwee load ad deflectio at the ceter of the test pael. Similar results were obtaied i umerical simulatios i which the supports of the pael, represeted i Figure 3.8a by the dashed lie, were simulated with lie sprigs with ifiite stiffess i compressio ad ull stregth i tesio usig the liear-parabolic diagram desibed i sectio 3.6.., i order to simulate the loss of cotact betwee the pael ad the support durig the loadig process (Vetura-Gouveia et al. 007).

126 9 Chapter Ifluece of the through-thickess refiemet of the pael I this sectio, the ifluece of through-thickess refiemet of the pael o the load-deflectio relatioship is aalyzed. The parameters used to simulate the fracture mode I ad the out-of-plae shear softeig diagram are those that have best fitted the experimetal results, accordig to the strategy desibed i the previous sectio. For this purpose, the followig two refiemets are cosidered: 6 layers i the lightweight zoe ad layers i the remaiig parts; 0 layers i the lightweight zoe ad 6 layers i the other zoes. I Figure 3.34 the load-deflectio relatioships of these umerical simulatios are compared with the experimetal oe, idicatig CM_jL the curve obtaied with a j-layer disetizatio i the lightweight zoe Load [kn] Experimetal Softeig out-of-plae shear (CM_3L) Softeig out-of-plae shear (CM_6L) Softeig out-of-plae shear (CM_0L) Deflectio [mm] Figure 3.34 Ifluece of the umber of layers disetizig the thickess of the pael i the lightweight zoe (results for 3, 6 ad 0 layers are show). It ca be observed that by ieasig the umber of layers i the lightweight zoe from 3 to 6, the maximum load ieases approximately 7%, ad the stiffess correspodig to the brach betwee ack iitiatio ad peak load also ieases. This behavior ca be justified by the fact that the flexural stiffess of each layer is ot cosidered i the evaluatio of the iteral forces of the Midli shell fiite elemets whe a material

127 Numerical model for coete lamiar structures 93 oliear aalysis is performed. Therefore, the larger the umber of layers disetizig the elemet, the higher the flexural stiffess of the elemet is, resultig i a smaller deformability of the pael ad a higher load carryig capacity. However, Figure 3.34 also shows that whe the umber of layers ieases from 6 to 0, oly a margial iease of the maximum load is observed, which idicates that the iease ratio of the flexural stiffess ad load carryig capacity of the layered Midli shell elemet deeases with the umber of layers. It is also iterestig to observe that the deflectio at the abrupt load decay, as well as the residual load carryig capacity of the pael, are very similar i all three umerical aalyses Ifluece of the i-plae mesh refiemet of the pael I order to assess the ifluece of the i-plae mesh refiemet o the load-deflectio relatioship, a alterative ad more refied mesh (RM) is cosidered (see Figure 3.35), beig the correspodig results preseted below. Eight-ode seredipity plae shell elemets are used, with 0 layers i the lightweight zoe ad 6 layers i the other zoes. x Poit load 300 mm Supports Thickess - 0 mm Thickess - 30 mm x 300 mm Figure 3.35 Geometry, mesh, load ad support coditios used i the umerical simulatio of the puchig test Refied Mesh (RM).

128 94 Chapter 3 The load-deflectio relatioship for the RM is represeted i Figure 3.36, beig compared with the oe obtaied with the previous coarse mesh (CM), ad with the oe experimetally registered. As expected, the deformability of the pael ieases with the mesh refiemet, causig the abrupt load decay to occur for a larger deflectio (3.3 mm). Due to the higher flexibility of the pael disetized with the RM, a deease of about 5% i terms of load carryig capacity is observed. Therefore, the shape of the load-deflectio (F-u) curve for the RM is approximately the result of the rotatio of the F-u curve for the CM about the poit that correspods to the ack iitiatio. With the iease of the umber of fiite elemets (ad itegratio poits), the coete i acked status ad the correspodig cosumed mode I fracture eergy also iease. This ca be a possible justificatio for the more deformable respose i the umerical simulatio of the i-plae RM Load [kn] Experimetal Softeig out-of-plae shear (CM_0L) Softeig out-of-plae shear (RM_0L) Deflectio [mm] Figure 3.36 Ifluece of the i-plae mesh refiemet o the umerical load-deflectio respose at the ceter of the test pael Ifluece of the fracture eergy ( G III f ) used i the out-of-plae shear softeig diagram To assess the ifluece of the fracture eergy used to defie the out-of-plae shear softeig diagram, III G f, o the load-deflectio relatioship, its value is varied betwee

129 Numerical model for coete lamiar structures 95.0 N/mm ad 5.0 N/mm. I these aalyses the i-plae CM ad the RM are used, with 0 layers disetizig the thickess of the pael i the lightweight zoe. The obtaied umerical curves are represeted i Figure 3.37 ad Figure 3.38, respectively. It is observed that i the RM the III G f value maily affects the residual load carryig capacity after the abrupt load decay. Whe usig the CM, the value attributed to III G f ot oly affects the residual load carryig capacity but also iflueces the value of the deflectio correspodig to the abrupt load decay. This ifluece, however, is less proouced tha whe usig a i-plae CM with 3 layers disetizig the thickess of the pael i the lightweight zoe (see Figure 3.33). Therefore it ca be cocluded that whe a RM is used, suitable predictios ca be obtaied with out for a more reliable estimatio of III G f. G III f = G, but further research eeds to be carried I f III Figure 3.39 ad Figure 3.40 show the cosumed out-of-plae fracture eergy ( G f, c ) up to a deflectio of 3.5 mm for the i-plae CM ad RM, respectively. At each itegratio poit, this cosumed fracture eergy receives the cotributio of the two out-of-plae shear compoets i all layers, ad ca be regarded as a idicator of damage due to the puchig failure mode. It ca be observed that the puchig failure patter is well predicted whe usig the RM. Whe usig the i-plae CM refiemet the shear failure badwidth is larger, thus justifyig the higher sesibility of the deflectio correspodig to the abrupt load decay to the adopted III G f value (see Figure 3.37).

130 96 Chapter 3 Load [kn] Experimetal Softeig out-of-plae shear (CM_0L_GfIII) Softeig out-of-plae shear (CM_0L_GfIII) Softeig out-of-plae shear (CM_0L_GfIII3) Softeig out-of-plae shear (CM_0L_GfIII4) Softeig out-of-plae shear (CM_0L_GfIII5) Deflectio [mm] III Figure 3.37 Ifluece of G o the umerical relatioship betwee load ad deflectio at the ceter of the f pael, whe usig the i-plae coarse mesh ad 0 layers i the lightweight zoe. Load [kn] Experimetal Softeig out-of-plae shear (RM_0L_GfIII) Softeig out-of-plae shear (RM_0L_GfIII) Softeig out-of-plae shear (RM_0L_GfIII3) Softeig out-of-plae shear (RM_0L_GfIII4) Softeig out-of-plae shear (RM_0L_GfIII5) Deflectio [mm] III Figure 3.38 Ifluece of G o the umerical relatioship betwee load ad deflectio at the ceter of the f pael, whe usig the i-plae refied mesh ad 0 layers i the lightweight zoe.

131 Numerical model for coete lamiar structures 97 III Figure 3.39 Represetatio of the cosumed out-of-plae fracture eergy, G, whe usig the i-plae f, c coarse mesh ad 0 layers i the lightweight zoe, for a deflectio of 3.5 mm. III Figure 3.40 Represetatio of the cosumed out-of-plae fracture eergy, G, whe usig the i-plae f, c refied mesh ad 0 layers i the lightweight zoe, for a deflectio of 3.5 mm Ifluece of the parameters that defie the fracture mode I I order to assess the ifluece of the parameters that defie the fracture mode I costitutive law (Figure 3.6) o the load-deflectio relatioship predicted by the umerical model, the values of these parameters are deeased ad ieased by 50 % relatively to those obtaied by iverse aalysis. The ack stress vs. ack strai ( σ ε ) for these aalyses ad the correspodig load-deflectio relatioships are depicted i Figure 3.4 to Figure 3.45.

132 98 Chapter 3 All these umerical aalyses utilize the refied mesh ad use 0 layers for the disetizatio of the thickess of the lightweight part of the pael. From the aalysis of these graphs it ca be cocluded that the icliatio of the first brach of the diagram ( D i Figure 3.6) govers the poit correspodig to the first drop i the load-deflectio relatioship. I fact, the less abrupt this brach is, the higher the load at this poit becomes. Cosequetly, it is observed that the load carryig capacity of the pael is quite sesible to the slope of this brach. Direct tesile tests with SFRSCC similar to the oe used i the tested paels showed, i fact, a abrupt stress decay immediately after ack formatio. σ ε Figure 3.4b evideces that the umerically predicted load carryig capacity of the pael is quite depedet o the α parameter, sice a proouced softeig ad a sigificat hardeig deflectio are estimated whe a value of α smaller or larger tha the oe obtaied by iverse aalysis is used (see Figure 3.4a). The higher stregth ( ) secod brach of σ ε, whe adoptig higher values for the σ ε of the α parameter (see Figure 3.4a), also cotributes to iease both the load carryig capacity of the pael ad the deflectio correspodig to the puchig failure. However, Figure 3.44 reveals that the σ ε correspodig to the first brach of σ stregth ( ) ε diagram has a much higher ifluece o the load carryig capacity of the pael tha the stregth ( ) σ ε of the secod brach. Nevertheless, Figure 3.44 ad Figure 3.45 also demostrate that the slope of the load-deflectio brach before the puchig failure grows with the value of (Figure 3.6). Fially, the deease of the fracture eergy is maily reflected at the poit correspodig to the first drop of the load-deflectio relatioship (Figure 3.45b). This deease leads to a more abrupt decay of the first brach of the σ 3.45a), resultig i a deease of the load at this poit. D ε diagram (Figure

133 Numerical model for coete lamiar structures 99 Crak stress [MPa] Iverse aalysis Modify Czi Modify Czi Crack strai (a) 60 Load [kn] Experimetal Softeig out-of-plae shear (RM_0L_GfIII3) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Czi ) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Czi-0.035) Deflectio [mm] (b) Figure 3.4 Ifluece of the ξ parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael.

134 00 Chapter 3 Crak stress [MPa] Iverse aalysis Modify Alpha Modify Alpha Crack strai (a) 70 Load ]kn] Experimetal Softeig out-of-plae shear (RM_0L_GfIII3) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Alpha-0.5) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Alpha-0.75) Deflectio [mm] (b) Figure 3.4 Ifluece of the α parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael.

135 Numerical model for coete lamiar structures 0 Crak stress [MPa] Iverse aalysis Modify Czi Modify Czi Crack strai (a) 60 Load [kn] Experimetal Softeig out-of-plae shear (RM_0L_GfIII3) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Czi-0.075) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Czi-0.5) Deflectio [mm] (b) Figure 3.43 Ifluece of the ξ parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael.

136 0 Chapter 3 Crak stress [MPa] Iverse aalysis Modify Alpha Modify Alpha Crack strai (a) 60 Load [kn] Experimetal Softeig out-of-plae shear (RM_0L_GfIII3) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Alpha-0.95) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_Alpha-0.885) Deflectio [mm] (b) Figure 3.44 Ifluece of the α parameter: (a) triliear softeig diagrams ad (b) relatioship betwee load ad deflectio at the ceter of the test pael.

137 Numerical model for coete lamiar structures 03 Crak stress [MPa] Iverse aalysis Modify GfI Crack strai (a) 60 Load [kn] Experimetal Softeig out-of-plae shear (RM_0L_GfIII3) Softeig out-of-plae shear (RM_0L_GfIII3_Modify_GfI-.5) Deflectio [mm] (b) I Figure 3.45 Ifluece of G : (a) triliear softeig diagrams ad (b) relatioship betwee load ad f deflectio at the ceter of the test pael. 3.8 NUMERICAL SIMULATION OF A PUNCHING TEST WITH A FLAT SLAB A experimetal program was carried out by Afoso (00) with reiforced coete slabs subjected to a test cofiguratio that coducts to the puchig failure of the tested slabs. I two slabs steel fibers were itroduced i the coete mix ad oe of them, FC0 (Afoso 00), is i this sectio umerically simulated usig the developed model.

138 04 Chapter 3 The geometry, supports, load coditios ad test setup are represeted i Figure 3.46 ad Figure The square slab of 500 mm edge has a thickess of 80 mm ad the load is applied to a pile ( mm 3 ) casted i the cetral zoe of the slab to simulate a real field case. More details about the correspodig experimetal program ca be foud elsewhere (Afoso 0) Figure 3.46 Geometry ad support coditios [all dimesios are i mm] (Moraes Neto et al. 0). FC0 flat slab Load DYWIDAG bar Reactio slab Figure 3.47 Puchig test setup [all dimesios are i mm] (Moraes Neto et al. 0).

139 Numerical model for coete lamiar structures 05 The fiite elemet idealizatio, load ad support coditios used i the umerical simulatios of the flat slab puchig test are show i Figure Due to double symmetry, oly oe quarter of the pael is cosidered i the simulatios. The mesh is composed of eight-ode seredipity plae shell elemets. A Gauss-Legedre itegratio scheme with 3 3 IP is used. The elemets are divided i 6 layers, beig the first layers.79 mm thick, followed by a layer with.6848 mm thick (to simulate the tesile reiforcemet see Figure 3.49) ad 3 layers with.998 mm thick to simulate the coete cover. The surface load is applied i the shaded elemets show i Figure 3.48, ad the supports are simulated by poit sprigs with a liear elastic behavior. The Newto-Raphso method with displacemet cotrol at a specific variable (cetral vertical displacemet) is used i the oliear aalysis (see sectio.4.3 of chapter ). x 586 mm 50 mm 50 mm 586 mm 50 mm Supports Loaded area x 50 mm Figure 3.48 Fiite elemet mesh.

140 06 Chapter 3 9φ6 c/5mm φ6 c/5mm Figure 3.49 Logitudial steel bars tesile reiforcemet (Moraes Neto et al. 0). The values of the parameters of the costitutive model used i the umerical simulatios are idicated i Table 3.4. The compressive stregth, f c, the coete tesile stregth, f ct, ad the modulus of elasticity, E c, are determied experimetally (Afoso 00), beig the values for characterizig the triliear tesio softeig diagram (see Figure 3.6) ad out-of-plae shear stress-strai diagram (see Figure 3.9) obtaied by back-fittig aalysis i order to approximate as much as possible the experimetal curve i the post-ackig phase, sice o experimetal data was available. Tree umerical simulatios are performed. Oe uses a liear elastic behavior for the coete, ad the others uses the proposed costitutive model varyig oly i the treatmet of the ou-of-plae shear compoets. Figure 3.50 represets the experimetal ad umerical relatioships betwee the load ad the deflectio at the cetral poit for the tested FC0 flat slab. Usig a liear elastic behavior for coete the umerical respose diverges from the experimetal oe just after ack iitiatio, idicatig that some acks start to form at a very early stage of the experimetal test. Whe a liear elastic behavior is assumed for the out-of-plae shear compoets a good agreemet ca be observed up to a deflectio of.8 mm. After this deflectio the experimetal curve idicates a deease i the load carryig capacity ad the suddely falls, suggestig the failure of the flat slab by puchig. This load decay is ot well reproduced whe assumig a liear elastic behavior for the out-of-plae shear compoets, ad this umerical simulatio has predicted a slight high load carryig capacity. However, if the biliear diagram represeted i Figure 3.9 is used to model the softeig behavior of the out-of-plae shear compoets, the abrupt decay i the load carryig capacity is better captured.

141 Numerical model for coete lamiar structures 07 Table Values of the parameters of the costitutive model used i the umerical simulatio of the flat slab puchig test. Poisso s ratio ν = 0.5 Iitial Youg s modulus Compressive stregth Triliear tesio softeig diagram (assiged to the coete layers) E = N mm c f = 39.3 N mm c I f ct = 3.5 N mm ; f 0.05N mm G = ; ξ = 0.3; α = 0.5 ; ξ = 0.5 ; α = 0.5 Fictitious parameters assiged to the layer to simulate the tesile reiforcemet Fracture eergy (mode III) used i the out-of-plae shear stress-strai diagram Parameter defiig the mode I fracture p = eergy available to the ew ack G = ; I f ct = N mm ; f 0 N mm ξ = 0.; α =.0 ; ξ = 0.5 ; α =.05 E s = N mm G = 0.05 N mm Shear retetio factor Expoetial ( p = ) Crack badwidth Threshold agle α th = 30º Maximum umber of acks per each IP III f Square root of the area of the IP α = σ, / σ,, α = σ,3 / σ,, ξ = ε, / ε, u, ξ = ε,3 / ε, u (see Figure 3.6) Load [kn] Experimetal Liear behavior Liear out-of-plae shear Softeig out-of-plae shear Deflectio [mm] Figure 3.50 Relatioship betwee load ad deflectio at the ceter of the FC0 slab.

142 08 Chapter SUMMARY AND CONCLUSIONS I the preset work a model based o the fiite elemet method is proposed to simulate coete lamiar structures failig i bedig ad shear. The Reisser-Midli theory i the cotext of layered shells is preseted ad special emphasis is placed o the treatmet of the shear behavior. The proposed model is based o a multi-directioal ad fixed smeared ack cocept. By cosiderig the oliear behavior of each shell layer, ack propagatio through the thickess of these structures ca be simulated. Fracture mode I is modeled with a ack stress vs. ack strai triliear diagram, whose defiig parameters ca be obtaied by iverse aalysis usig the load-deflectio relatioship obtaied with three-poit otched beam tests, carried out accordig to the RILEM TC 6-TDF recommedatios. With this strategy the values of the fracture parameters that defie the ormal stress-strai ack costitutive relatioship are obtaied. Sice this type of test is much simpler ad faster to execute, it becomes a advatageous alterative to the direct tesile tests recommeded to evaluate the fracture mode I parameters of cemet based materials. The adopted iverse aalysis strategy is preseted ad discussed i sectio To simulate the out-of-plae strai gradiet that occurs i puchig tests, a softeig diagram is proposed to model, after ack iitiatio, the out-of-plae shear compoets. The formulatio of liear ad oliear support coditios, such as surface, lie ad poit sprigs, is preseted, ad a special attetio is dedicated to uilateral support coditios. With this approach, the loss of cotact betwee the structure ad the supportig system ca be simulated, e.g., for the case of a slab supported o groud. A parametric study based o a steel fiber reiforced coete slab supported o soil, usig a oliear model for the simulatio of the slab support coditios, is also preseted.

143 Numerical model for coete lamiar structures 09 The improvemets made i the iteral algorithms associated with the stress update ad with the itical ack status chages are preseted ad their advatages are discussed. The adequacy ad accuracy of the proposed model is appraised usig the results obtaied i a puchig test of a pael prototype built with steel fiber reiforced self-compactig coete (SFRSCC) ad i a puchig test of a reiforced coete flat slab that also icluded steel fibers i the coete matrix. The proposed umerical strategy allows for a accurate simulatio of the load-deformatioal process of the experimetally SFRSCC tested pael, which exhibited a brittle puchig failure. Several umerical simulatios are preseted ad discussed. Mesh refiemet, data obtaied with iverse aalysis to defie the triliear diagram ad a softeig out-of-plae shear diagram are alteratives whose ifluece o the predictio of the experimetal pael respose is ivestigated. The load-deformatioal process of the experimetally tested reiforced coete flat slab was also well predicted. The use of softeig laws to simulate the mode I ack opeig ad the out-of-plae shear compoets is ucial i order to obtai accurate umerical simulatios The umerical simulatios carried out with the proposed model ad its compariso with the results of the experimetal tests used i this work lead to the coclusio that the behavior of lamiar structures failig i puchig ca be umerically predicted by a FEM-based Reisser-Midli shell approach as log as a ack costitutive model that icludes a softeig diagram for modelig both out-of-plae shear costitutive laws is used.

144

145 Chapter 4 Modelig of the ack shear compoet 4. INTRODUCTION I the previous chapter a model for coete lamiar structures based o the formulatio of the Reisser-Midli layered approach is desibed. Each layer is cosidered to be i a state of plae stress. To simulate the oliear behavior of the iterveig materials, as occurs, for example, i the ack propagatio through the thickess of a shell, a ack costitutive model is proposed ad explaied i sectio 3.3. To improve the predictive accuracy of the model for the simulatio of the behavior of lamiar structures failig i puchig, special attetio is dedicated to the treatmet of the out-of-plae shear compoets by proposig a softeig diagram after ack iitiatio (see sectio 3.4). For the case of acked coete, stress ad strai i-plae compoets are related by a acked coete costitutive matrix, co D mb, defied by equatio (3.4). For the simulatio of the mode I ack opeig, two tesile-softeig diagrams ca be used, beig the ack mode II (i-plae shear) modulus calculated usig the cocept of shear retetio factor, β, defied as a costat value or as a fuctio of the ack ormal strai, as show i equatios (3.3) ad (3.3). As observed, the adoptio of a softeig tesile-diagram to simulate the ack opeig propagatio is a suitable strategy to accurately assess the behavior of structures govered by flexural failure mode. However, a accurate simulatio of structures failig i shear or i flexural/shear is still a challege i the computatioal mechaics domai. To improve the predictive accuracy of the model for the simulatio of

146 Chapter 4 the behavior of structures govered by this type of failure, two strategies to simulate the ack shear compoet that appears i the formulatio of the smeared ack costitutive model desibed i sectio are preseted ad discussed i this chapter. These strategies are exposed for the i-plae compoets of a specific k layer of the lamiar structure, but they were also implemeted i the already available smeared multi-fixed ack model developed for plae stress, as well as i the three-dimesioal multi-fixed smeared ack model desibed i chapter 5. Therefore, for the expositio of these strategies the subsipt mb used i the formulatio preseted i sectio 3.3 is ot used. The mai purpose of the implemetatio of these two strategies is to improve the simulatio of the degradatio of the shear stress trasfer with the ack opeig evolutio. Oe of the strategies is based o the adoptio of a total approach for the ack shear stress-shear strai relatioship. The other strategy is based o the itroductio of a softeig diagram for the relatio betwee the ack shear stress-shear strai compoets. Numerical simulatios are performed to evidece the mai differeces provided by both strategies whe they are applied to shear-failure structures. 4. INCREMENTAL VS. TOTAL APPROACH FOR MODELING THE SHEAR CRACK COMPONENT Whe the material correspodig to a specific itegratio poit (IP) is assumed to be i a acked state ad is submitted to a iemetal strai, ε, the strai field i this IP is modified ad the stress state must be updated. I the followig sectios two formulatios are preseted for the stress update. The first oe is a iemetal approach for both, ormal ad shear ack compoets, ad the secod oe is a iemetal approach for the ormal ack compoet ad a total approach for the ack shear compoet. The formulatios are restricted to oe ack, but its geeralizatio to the case of multiple acks at each IP is a straightforward process.

147 Modelig of the ack shear compoet 3 The cocept of the iemetal ad total approach for the ack shear stress-shear strai relatioship is represeted i Figure 4.. τt D II,3 D II,4 D II,5 D II, D II, D II, D II,3 Total approach Iemetal approach D II,4 D II,5 γ t Figure 4. Example of ack shear stress-strai relatio for the iemetal ad total approaches. I the iemetal approach the ack shear stress τ t provided by the aggregate iterlock effect at a certai ack shear strai stage is obtaied with τ = τ + τ = τ + D γ (4.) t t, prev t t, prev II t where τ t, prev is the ack shear stress i a previous state at the same IP, τ t is the iemet of the ack shear stress, D II is the mode II stiffess modulus, defied by equatio (3.3), ad γ t is the iemetal ack shear strai. It ca be observed i Figure 4. that the iemet of the ack shear stress τ t is oly ull whe D = 0, i.e., for ε ε, u (see equatios (3.3) ad (3.3)). Therefore, eve II

148 4 Chapter 4 whe the ack width is ieasig, the ack shear stress τ t ca also iease up to a asymptotic value, regardless the ack is o loger capable of trasferrig ormal tesile stresses. This ca lead to the formatio of a ew ack, resultig i several acks at a IP, which itroduces severe difficulties i accomplishig the costitutive laws of the acks formed at the IP, eve whe a rigorous strai-decompositio cocept is adopted for this purpose (Sea-Cruz 004). The occurrece of quite high ack shear stresses ca also cotribute to umerical predictios with higher stiffess ad load carryig capacity tha the values registered experimetally, maily i elemets failig i shear. I the total approach the ack shear stress τ t at a certai ack shear strai stage is obtaied with ( ) τ = D γ = D γ + γ = D γ + D γ (4.) t II t II t, prev t II t, prev II t By observig Figure 4., it ca be stated that for a iemet of the ack shear strai γ t a deease i the ack shear stress τ t ca occur due to a sigificat deease of D II. Therefore, the aim of the total approach, proposed i the preset work for modelig the fracture mode II, is to reproduce umerically a deease of ack shear stress trasfer with the iease of the ack shear strai γ t γ t (Abaqus 00), as is expected whe ack opeig τ t, after a first phase where τ t ieases with ε is also ieasig. 4.. Iemetal approach The relatioship betwee the i-plae stress compoets i the coordiate system of the fiite elemet, σ, ad the ack stress compoets i the local ack coordiate system, σ, is obtaied with equatio (3.49), rewritte at this phase for coveiece. σ = T σ (4.3)

149 Modelig of the ack shear compoet 5 where T σ σ τ = t (4.4) ad [ ] T σ = σ σ τ (4.5) beig σ ad τ t the ack ormal ad shear stresses i the ack, respectively, ad is the matrix that trasforms the stress compoets from the coordiate system of the fiite elemet to the local ack coordiate system (see Figure 3.5) for a specific ack i the IP. T Takig ito accout equatio (3.8), equatio (4.3) ca be rewritte as follows ( prev ) σ, prev+ σ = T σ + σ (4.6) where the subsipt prev idicates, i this case, the stress i a previous state i the IP, σ the iemetal stress vector for the i-plae compoets i the coordiate system of the fiite elemet defied by equatio (3.4) ad σ the iemetal local ack stress vector defied by T σ = σ τ t (4.7) beig σ ad τ t the iemetal ack ormal ad shear stress compoets, respectively.

150 6 Chapter 4 Usig the strai decompositio cocept, co ε = ε + ε ad kowig the relatioship betwee the iemetal ack strai compoets i the coordiate system of the fiite elemet ε ad the ack strai compoets i the local ack coordiate system ε T ε = T ε (4.8) where T ε = ε γ t (4.9) ad [ ] T ε = ε ε γ (4.0) the iemetal stress is obtaied with T ( ) co co co σ = D ε = D ε T ε (4.) beig co D defied by equatio (3.0). Icludig equatio (4.) i equatio (4.6) ad makig some arragemets, this equatio ca be writte as T co co, + prev + = 0 σ prev σ T σ T D ε T D T ε (4.) where σ depeds o ε, beig obtaied with this equatio

151 Modelig of the ack shear compoet 7 σ DI 0 ε σ = = (4.3) τ t 0 DII γ t I equatio (4.3) D I ad D II represets, respectively, the costitutive compoets of the ack opeig mode I (ormal) ad ack slidig mode II (shear). The D I ca be obtaied with a diagram characterizig the ack fracture mode I propagatio, while the D II ca be determied from shear retetio factor cocept defied i the sectio of chapter 3. As referred i sectio 3.5., to solve the system of oliear equatios represeted i equatio (4.), where the ukows are the compoets of the ack strai vector ε, two algorithms for the stress update are available, beig oe based o the Newto-Raphso method, ad the other based o the fixed poit iteratio method. Both these methods are desibed elsewhere (Sea-Cruz 004). 4.. Total approach The total approach is applied oly to the shear compoets. I this case equatio (4.6) ca be writte as σ prev, τ t + σ = σ σ cos θ si θ siθcosθ = σ σ siθcosθ siθcosθ cos θ si θ + τ τ prev (4.4) where σ = D ε (4.5) I

152 8 Chapter 4 ad τ t is defied by (4.). Substitutig equatios (4.5) ad (4.) i the left term of the equatio (4.4) σ + = D prev, DI ε II γ t, prev DII γ t σ σ cos θ si θ siθ cosθ = σ σ siθcosθ siθ cosθ cos θ si θ + τ τ prev (4.6) or i matrix form ( prev ) * σ, prev+ σ = T σ + σ (4.7) ad itroducig equatio (4.) i (4.7) results * T co co, + prev + = 0 σ prev σ T σ T D ε T D T ε (4.8) beig * σ, prev σ, prev DII γ = t, prev T (4.9) the modified ack stress vector (oly the shear compoet) from a previous state. The compoets of σ are obtaied with (4.3). Similarly to the iemetal approach, the ack strai vector the Newto-Raphso or the fixed-poit iteratio methods. ε ca be obtaied with

153 Modelig of the ack shear compoet 9 Equatio (4.8) ca be reduced to f ( ε ) = 0 (4.0) At iteratio q of the Newto-Raphso method the first derivative of f i order to the iemetal ack strai vector ε must be obtaied, beig defied by (see Appedix C) f ( ε ) ε = D + D + T D T co T (4.) where 0 0 D = DII t 0 γ ε (4.) beig D II obtaied with equatio (3.3). Equatio (4.) is similar to the oe used i the iemetal approach desibed i (Sea-Cruz 004), beig the mai differece the replacemet of the iemetal shear ack strai, γ t, with the total shear ack strai, γ t Numerical simulatios As metioed above, the total approach for the ack shear compoet is also icluded i the smeared ack model previously implemeted for plae stress aalysis with the iemetal approach. I this sectio, to validate the exposed approaches, two umerical simulatios are made. The first oe is a more theoretical umerical simulatio, whose mai purpose is to show the differeces betwee both approaches ad the capabilities of the total approach to simulate the degradatio of the stresses at a IP. The secod example is

154 0 Chapter 4 dedicated to the umerical simulatio of reiforced coete beams shear ad flexurally stregtheed with composite materials Sigle elemet To compare the capabilities of the total ad iemetal approaches two umerical simulatios are made i this sectio. Oe with a iemetal approach for both ack ormal ad shear compoets, ad the other with a iemetal approach for the ack ormal compoet ad a total approach for the ack shear compoet. A sigle elemet cosidered to be subjected to a plae stress state is used for both these umerical simulatios. Figure 4. represets the geometry, load ad support coditios of the elemet, ad Table 4. icludes the values of some parameters required by the model. A four-oded plae stress fiite elemet with oe IP (Gauss poit) is used, ad a tesile-softeig triliear diagram is adopted to simulate the mode I fracture propagatio. The Newto-Raphso method with displacemet cotrol at a specific variable is utilized i this example. x,u Poit load 5 mm Thickess - 5 mm Supports (u = u =0) 5 mm x,u Figure 4. Geometry, load ad support coditios of the sigle elemet mesh.

155 Modelig of the ack shear compoet Table 4. - Model properties used i the sigle elemet simulatio. Poisso s ratio ν = 0. Iitial Youg s modulus Compressive stregth E = N mm c f = 38.0 N mm c Triliear tesio softeig diagram Parameter defiig the mode I fracture eergy available to the ew ack G = ; I f ct =.9 N mm ; f N mm ξ = 0. ; α = 0.7 ; ξ = 0.75 ; α = 0.5 p = Shear retetio factor Expoetial ( p = ) Crack badwidth Threshold agle α th = 30º Square root of the area of the elemet Load [N] Iemetal approach Total approach Horizotal displacemet [mm] Figure 4.3 Load-horizotal displacemet relatioship. The load-horizotal displacemet relatioship is represeted i Figure 4.3, usig the total ad the iemetal approach for the ack shear compoet. Util a load of 6 N, a liear-elastic behavior is observed for both simulatios. At this level ackig is iitiated i the coete ad the curves follow differet paths, maily after 0.0 mm, which is where the iemetal respose presets a hardeig behavior leadig to a higher load carryig

156 Chapter 4 capacity, maitaiig a high residual load i compariso with the total curve that presets a softeig behavior up to the complete loss of load carryig capacity. This behavior ca be justified by the aalysis of the stress-strai relatioship for fracture mode I ad mode II obtaied i the IP for each approach, ad represeted i Figure 4.4 ad Figure 4.5. Figure 4.4 represets the ack ormal stress-strai relatio i the ack coordiate system (CrCS), ad Figure 4.5 the ack shear stress-strai relatio i the same CrCS. As expected, both approaches provide similar ack ormal stress strai diagram, because the total approach is oly applied to the ack shear compoet. I fact, Figure 4.5 shows that quite differet ack shear stress-strai diagrams ca be obtaied for both approaches. I the total approach the ack shear stress deeases with the ack opeig process, while i the iemetal approach the ack shear stress ieases up to a asymptotic value. 3 Crack ormal stress - CrCS [N/mm ] 0 Iemetal approach Total approach Crack ormal strai - CrCS Figure 4.4 Crack ormal stress-strai relatioship i CrCS.

157 Modelig of the ack shear compoet 3 4 Crack shear stress - CrCS [N/mm ] Crack shear strai - CrCS Iemetal approach Total approach Figure 4.5 Crack shear stress-strai relatioship i CrCS. The stresses ad strais developed at the IP i the global coordiate system (GCS) are represeted i Figure 4.6 to Figure 4.8. It ca be observed i all of these charts that the behavior of each approach is similar up to ack iitiatio ad, the, they gradually follow a differet path, leadig, for the cases of the x ormal stress ad x x shear stress, to a very differet post-peak residual value (see Figure 4.7 ad Figure 4.8). This differece ca be justified by the fact that i the iemetal approach the ack shear stress has a asymptotic residual value, as show i Figure 4.5. Due to its cotributio to the stresses i the global coordiate system, higher stress compoets are obtaied usig the iemetal approach. A umerical simulatio of the sigle elemet with four itegratio poits was also performed, ad similar coclusios were obtaied.

158 4 Chapter 4 x ormal stress - GCS [N/mm ] Iemetal approach Total approach x ormal strai - GCS Figure 4.6 x ormal stress-strai relatioship i GCS. -8E-5 0E+0 8E-5 E-4 E-4 3E-4 4E-4 0 x ormal stress - GCS [N/mm ] x ormal strai - GCS Iemetal approach Total approach Figure 4.7 x ormal stress-strai relatioship i GCS.

159 Modelig of the ack shear compoet 5 5 x x shear stress - GCS [N/mm ] Iemetal approach Total approach x x shear strai - GCS Figure 4.8 xx shear stress-strai relatioship i GCS Stregtheig of RC beams The flexural stregtheig of reiforced coete (RC) beams with small coete cover thickess by usig the Near Surface Mouted (NSM) techique demads to cut the bottom arm of the still stirrups for the istallatio of the Carbo Fiber Reiforced Polymer (CFRP) lamiates. To assess the ifluece of this itervetio o the load carryig capacity of the beams, a experimetal research program was carried out. The details of the experimetal program composed of three series of beams of distict oss sectio depth ca be foud elsewhere (Costa ad Barros 00). The NSM techique for the flexural stregtheig of RC beams or slabs cosists of istallig CFRP lamiates ito thi slits ope oto the coete cover of the RC elemets to stregthe boded with a epoxy adhesive to the surroudig coete. The efficiecy of this techique ca be foud elsewhere (Täljste et al. 003, El-Hacha ad Rizkalla 004, Sea-Cruz 004, Barros ad Fortes 005, Boaldo et al. 008, Barros et al. 008). I a tetative of avoidig the occurrece of shear failure i the NSM-flexurally stregtheed beams, wet layup CFRP strips of sheet of U cofiguratio were also applied accordig to the Exterally Boded Reiforcemet (EBR) techique (ACI Committee 440, 007). The U shape CFRP strips were placed betwee the existig steel stirrups (see Figure 4.9).

160 6 Chapter 4 f f f Existig steel stirrups A f 00 sf sf 00 L 00 Figure 4.9 Localizatio of the NSM CFRP lamiates ad of the EBR strip of wet layup CFRP sheet [dimesios i mm] (Barros et al. 0). I the umerical simulatio of these beams, the total ad iemetal approaches for the ack shear compoet were used. I this sectio, oly the umerical results of the beams of the first series are preseted. The complete umerical research ca be foud i Barros et al. (0). The desigatios of the beams used i the experimetal research are the followig: VR - reiforced coete referece beam; VE - VR beam with the bottom arm of the steel stirrups cut; VL - VE beam flexurally stregtheed with NSM CFRP lamiates; VLM - VL beam shear stregtheed with EBR strips of wet layup CFRP sheets of U cofiguratio. The geometry, support ad loadig coditios of the beams are represeted i Figure 4.0. For the case of the beams of series : L = 550mm ; L = 950mm ; b= 00mm; h= 50mm logitudial steel bars at bottom surface As 0 6( 85mm ) bars at top surface As 0( 57mm ) = φ. + = φ + φ ad logitudial steel 0 0 A + s b A - s h 00 L L Cotrol LVDT 00 Figure 4.0 Beam geometry, support ad loadig coditios [dimesios i mm] (Barros et al. 0).

161 Modelig of the ack shear compoet 7 Uder the framework of the fiite elemet aalysis, the tested beams are cosidered as a plae stress problem. Therefore, the beams are modeled with a mesh of 8-oded seredipity plae stress fiite elemets. A Gauss-Legedre itegratio scheme with IP is used i all the coete elemets. The steel bars, the NSM lamiates ad the EBR CFRP strips are modeled with -oded perfectly boded embedded cables (oe degree-offreedom per ode). For modelig the behavior of the steel bars, the stress-strai relatioship available i the FEMIX computer code is used (Sea-Cruz 004). The curve (uder compressive or tesile loadig) is defied by the poits PT = ( sy, sy) ε σ, PT = ( ε, σ ) ad PT3 = ( ε, σ ) ad a parameter p that defies the shape of the last brach of the curve. Uloadig ad reloadig liear braches with slope Es = σ sy εsy are assumed i the preset approach. For modelig both the NSM lamiates ad EBR strips of sheet, a liear elastic stress-strai relatioship is adopted. sh sh su su The values of the parameters of the costitutive model used i the umerical simulatios are idicated i Table 4., Table 4.3 ad Table 4.4. Usig the average compressive stregth f c, determied experimetally, ad the equatios proposed by CEB-FIP (993), the coete tesile stregth f ct ad the fracture eergy I Gf are obtaied. As suggested by Steves (987), the tesile yield stress ad the stress values correspodig to strais higher tha the tesile yield strai of the steel bars are reduced by the term σ = 75 f / φ, beig f ct the coete tesile stregth i MPa, ad φ s the diameter of y ct s the steel bar i mm. This reductio is to take ito accout that the stress i the steel reiforcemet at the coete ack plae is higher tha the average stress determied i the IP of the correspodig embedded cable elemet. This stress is obtaied from the displacemets of the mother elemet of the embedded cable.

162 8 Chapter 4 Table 4. - Properties of CFRP lamiates ad strips of sheets. CFK Lamiate C Sheet 40 f fu (MPa) E f (GPa) ε fu ( ) t f (mm) Table Steel properties. φ s =6 mm φ s =0 mm f sym (MPa) f sum (MPa) ε sy ( ) σ sy (MPa) ε sh ( ) 5 5 σ sh (MPa) ε su ( ) σ su (MPa) p Table Model properties used for the beams simulatio. Poisso s ratio ν = 0. Iitial Youg s modulus Compressive stregth E = N mm c f = 3.0 N mm c Triliear tesio softeig diagram Parameter defiig the mode I fracture eergy available to the ew ack G = ; I f ct =.5 N mm ; f N mm ξ = 0.; α = 0.5 ; ξ = 0.3 ; α = 0. p = Shear retetio factor Cubic ( p = 3 ) Crack badwidth Threshold agle α th = 30º Maximum umber of acks per each IP Square root of the area of the IP I Figure 4. the experimetal relatioship betwee the load ad the deflectio at the loaded sectio for the tested beams ad the umerical oes obtaied with the iemetal

163 Modelig of the ack shear compoet 9 ad the total approaches are compared. The dash-dot horizotal lie correspods to the maximum experimetal load of each beam. Figure 4. represets, for each beam ad for the umerical simulatios ivolvig both approaches, the ack patters at the ed of the aalysis, i.e., for the last coverged load iemet. The acks are represeted by quadrilateral 4-oded fiite elemets cetered at the IP, beig draw with a width that is proportioal to the ack ormal strai, details about the status of the acks are supplied i the captio of the Figure 4.. ε. More Load [kn] Experimetal INCREMENTAL TOTAL Load [kn] Experimetal INCREMENTAL TOTAL Deflectio at loaded sectio [mm] (a) VR Deflectio at loaded sectio [mm] (b) VE Load [kn] Deflectio at loaded sectio [mm] (c) VL Experimetal INCREMENTAL TOTAL Load [kn] Deflectio at loaded sectio [mm] (d) VLM Experimetal INCREMENTAL TOTAL Figure 4. Load-deflectio at the loaded sectio for the beams of series (Barros et al. 0). From the aalysis of Figure 4. ad Figure 4., it ca be cocluded that both umerical approaches simulated accurately the deformatioal respose of the VR ad VE beams,

164 30 Chapter 4 ad predicted with good accuracy the deformatioal respose of the VL ad VLM beams. However, comparig the ack patter of VL beam represeted i Figure 4.3 with the ack patters obtaied umerically ad represeted i Figure 4., it ca be cocluded that oly the total approach captured with high precisio the localizatio ad profile of the shear failure ack. The logitudial steel bars of the VL ad VLM beams have already yielded at the momet of the shear failure. This is well predicted by the umerical simulatios usig the total ad iemetal approaches, sice vertical completely ope acks are observed ear the loaded sectio (see Figure 4.). Iemetal approach Total approach VR VE VL VLM Figure 4. Crack patters of the beams of series (i pik color: ack completely ope ( ε ε, u); i red color: ack i the opeig process; i cya color: ack i the reopeig process) (Barros et al. 0).

165 Modelig of the ack shear compoet 3 Figure 4.3 Crack patters at the ed of the tested VL beam (Barros et al. 0). 4.3 CRACK SHEAR SOFTENING DIAGRAM The use of softeig diagrams to reproduce the fracture mode I process is commo i smeared ad disete ack models, but the use of softeig diagrams to model the shear stress trasfer aoss the ack is lesss usual. To simulate the fracture mode II process a shear retetio factor is curretly used (Rots ad de Borst 987). Accordig to this last approach, the shear stress trasfer betwee the ack plaes deeases with the iease of the ormal ack strai (seee equatio (3.3)). I most structures assumed to be i a state of plae stress, this strategy leads to simulatios with reasoable accuracy. Exceptios occur i structures that fail by the formatio of a itical shear ack. For these cases the simulatio of the structural softeig with high accuracy requires the adoptio of a softeig ack shear stresss vs. ack shear strai relatioship. I this sectio a softeig diagram to simulate the ack shear stress-strai behavior is desibed. This costitutive model is implemeted i FEMIX, ad ca be used i the simulatio of plae stress structures, Reisser-Midli shellss ad 3D structures disetized with solid elemets. Its capabilities are assessed by performig a umerical simulatio of a experimetal test available i the literature (Arrea ad Igraffea 98) Desiptio of the diagram The proposed ack shear diagram is represeted i Figure 4.4. The shear softeig diagram starts at the origi because, accordig to the ack iitiatio iterio, whe a ack iitiates the ack shear stresss is ull. As a cosequece of the rotatio of the directios of pricipal stresses, shear stresses ca develop aoss the surfaces of the ack

166 3 Chapter 4 (Rots ad de Borst 987). The ack shear stress ieases liearly util the ack shear stregth is reached (first brach of the shear ack diagram), followed by a deease i the shear residual stregth (softeig brach). τ t -γ t,u 5 -γ t,max 4 -γ t,p τ t,p τ t,max 3 D t, 3 γ t,p -τ t,max -τ t,p D t, 4 D t,3-4 costat shear retetio factor γ t,max Shear Crack Statuses - Stiffeig - Softeig 3- Uloadig 4- Reloadig 5 - Free-slidig G f,s l b 5 γ t,u γ t Figure 4.4 Diagram to simulate the relatioship betwee the ack shear stress ad ack shear strai compoet ad possible shear ack statuses. The diagram represeted i Figure 4.4 (based o Rots ad de Borst 987) is defied by the expressios show i equatio (4.3). The positive part of the diagram is explaied here, beig the treatmet of the egative part straightforward. D γ 0 < γ γ τ t, p τ ( γ ) τ ( ) ( γ γ ) γ γ γ γtu, γt, p 0 γ t > γ t, t t t, p t t = t, p t t, p t, p < t t, u t, u (4.3) The iitial shear fracture modulus, D t,, is defied by D t, β = G β c (4.4)

167 Modelig of the ack shear compoet 33 where G c is the coete elastic shear modulus ad β is the shear retetio factor, defied as a costat value i the rage ]0,[, i this case. The peak ack shear strai, γ, t p, is obtaied usig the ack shear stregth (from the iput data), τ t, p, ad the ack shear modulus is obtaied with equatio (4.4) τ γ = (4.5) t, p t, p D t, The ultimate ack shear strai, γ tu,, depeds o the ack shear stregth, τ, t p, o the shear fracture eergy (mode II fracture eergy for the case of i-plae shear), G f, s, ad o the ack badwidth, l b, as follows γ tu, G f, s t, plb = τ (4.6) I the preset approach it is assumed that the ack badwidth, used to assure that the results are idepedet of the mesh refiemet, is the same for both fracture mode I ad mode II processes. Whe the softeig costitutive law represeted i Figure 4.4 is used to evaluate the fracture mode II stiffess modulus D II of equatio (3.6), its value depeds o the braches defiig the diagram. For this reaso five shear ack statuses are proposed ad their meaig is explaied i the ext sectio. The ack mode II stiffess modulus of the first liear brach of the diagram is defied by equatio (4.4), the secod liear (softeig) brach is defied by

168 34 Chapter 4 D τ t, p II = Dt, = γ tu, γ t, p (4.7) ad the ack shear modulus of the uloadig ad reloadig braches is obtaied from D τ = = (4.8) t,max II Dt,3 4 γ t,max beig γ t,max ad t,max τ the maximum ack shear strai already attaied ad the correspodig ack shear stress determied from the softeig liear brach. Both compoets are stored to defie the uloadig/reloadig brach (see Figure 4.4). To iease the geerality of the simulatio of the post-peak ack shear stress, two alterative ack shear stress-strai diagrams are proposed, beig their shapes represeted i Figure 4.5 ad Figure 4.6. For the case of Figure 4.5, the post-peak phase is simulated with a triliear diagram, while i Figure 4.6 this brach is expoetial. The ack shear softeig diagram of Figure 4.5 is defied by the expressios show i equatio (4.9). Similarly to equatio (4.3), oly the positive part of the diagram is treated i D γ 0 < γ γ τ τ τ γ γ γ γ γ τ τ τ ( γ ) τ ( γ γ ) γ γ γ τ τ ( γ γ ) γ γ γ 0 γ t > γ t, t t t, p t, p t, p ( t t, p) t, p< t γ γt, p t t = t < t γ γ t < t t, u γtu, γ t, u (4.9) beig

169 Modelig of the ack shear compoet 35 τ = c τ (4.30a) t,p τ = c τ tp ( ) γ = d γ γ t,u t,p ( ) γ = d γ γ t,u t,p (4.30b) (4.30c) (4.30d) -γ t,u -γ -γ τ t τ t,p τ τ -γ t,p D t, γ t,p -τ -τ * D t, ** D t, D t,3-4 γ γ costat shear retetio factor *** D t, G f,s l b γ t,u γ t -τ t,p Figure 4.5 Diagram to simulate the relatioship betwee the ack shear stress ad ack shear strai compoet with post-peak triliear braches. The ultimate ack shear strai, γ, the peak ack shear strai, γ t, p, the ack shear stregth, τ, tu, is defied as a fuctio of the parameters c i ad d i, II shear) fracture eergy, Gf = Gf, s, ad the ack badwidth, l b, as follows t p, the mode II (for i-plae γ II f, aux tu, = γt, p+ d+ cd cd+ c τ t, plb G (4.3) beig G II f, aux G τ = lb γ II f t, p t, p l b (4.3)

170 36 Chapter 4 ad c = τ τ (4.33a) t,p c = τ τ t,p ( ) d = γ γ γ t,u t,p ( ) d = γ γ γ t,u t,p (4.33b) (4.33c) (4.33d) The ack mode II stiffess modulus of the first liear brach of the diagram is defied by equatio (4.4), while for the post peak braches it is obtaied from the followig equatios D ( )( + + ) c d cd c d c l f = (4.34a) * b ct t, II d Gf, aux D ( )( + + ) ( ) c c d cd c d c l f ** b ct t, = II d d Gf, aux (4.34b) D ( ξ + + ) ( d ) c cd c d c l f *** b ct t, = II Gf, aux (4.34b) For the uloadig ad reloadig braches, the ack mode II stiffess modulus is determied by equatio (4.8). The expoetial brach of the ack shear softeig diagram represeted i Figure 4.6 is based o the Corelisse diagram used i the defiitio of the ack opeig mode I. Equatio (4.35) gives expressios of the positive part of the proposed softeig diagram

171 Modelig of the ack shear compoet 37 τ t ( γt ) = D γ 0 < γ γ 3 3 = τ ( ( cb) ) exp( cb) B( c ) exp( c ) + + γ < γ γ 0 γ t > γ t, t t t, p t, p t, p t t, u t, u (4.35) beig B γ = γ γ t t, p tu, γ t, p (4.36) ad c = 3.0 ; c =6.93. τ t τ t,p costat shear retetio factor -γ t,u -γ t,p D t, D t,3-4 γ t,p D t, G f,s l b γ t,u γ t -τ t,p Figure 4.6 Diagram to simulate the relatioship betwee the ack shear stress ad ack shear strai compoet with post-peak expoetial brach. The ultimate ack shear strai, γ, Figure 4.6 is defied by tu, of the ack shear softeig diagram represeted i

172 38 Chapter 4 γ G = + (4.37) II f, aux tu, γt, p k τ t, plb II where the parameter k is defied by equatio (3.30) ad G f, aux by equatio (4.3). The ack mode II stiffess modulus of the first liear brach of the diagram is defied by equatio (4.4), for the uloadig ad reloadig braches by equatio (4.8) ad for the expoetial (softeig) brach by γt c γt t, τ t, p 3 exp γ tu, γ tu, γ tu, D = c c γ t c γ t + c + exp c + c exp ( c) γ tu, γ tu, γ tu, γtu, (4.38) beig γ tu, obtaied with equatio (4.37) Crack shear status As a cosequece of the formatio of other acks i the eighborhood of existig acks, these existig acks ca close or reope. The model must take ito accout this chage of ack status. For the opeig mode I the model takes this ito accout (Sea-Cruz 004) ad for the ack shear compoet a similar approach is used. The shear ack status is show i Figure 4.4 ad its defiitio takes ito accout the followig assumptios: Stiffeig status, if the ack shear strai is less tha the ack shear strai at peak ack shear stress, γ, t p, obtaied with equatio (4.5) ad assumig a costat shear retetio factor for the evaluatio of the fracture mode II stiffess modulus;

173 Modelig of the ack shear compoet 39 Softeig status, after the ack shear strai has reached γ, t p. A deease i the ack shear stress is observed with the iease of the ack shear strai; Uloadig status, whe a shear softeig ack experieces a deease of ack shear strai. I this case a secat approach is followed; Reloadig status, whe a ack with a uloadig status experieces a iease of ack shear strai. The same brach of the uloadig status is followed; Free-slidig status, whe the ack shear strai is greater tha the ultimate ack shear strai. These ack shear statuses are stored to be used i the subsequet steps of the oliear aalysis. With this procedure, at each istat the shear softeig diagram is well defied. For example, Figure 4.7 to Figure 4. illustrate the possible paths that ca be followed at each iemet of ack shear strai, ad, cosequetly, the complexity of the associated ack status chages. τ τ τ p B C τ p A A -γ u -γ p γ p γ u D γ H -γ u -γ p E γ p γ u γ F -τ p G -τ p a) γ > 0 b) γ < 0 Figure 4.7 Iemet of the ack shear strai γ : possible paths whe the startig poit is positive ad i Stiffeig shear ack status (poit A).

174 40 Chapter 4 τ τ p τ τ p G F D -γ u -γ p γ p γ u γ -γ u -γ p E γ p γ u H γ A A C B -τ p -τ p a) γ < 0 b) γ > 0 Figure 4.8 Iemet of the ack shear strai γ : possible paths whe the startig poit is egative ad i Stiffeig shear ack status (poit A). τ τ τ p τ max A p τ max A -γ u -γ p γ p γ max B C γ γ u G -γ u -γ p D γ p γ max γ u γ F E -τ p -τ p a) γ > 0 b) γ < 0 Figure 4.9 Iemet of the ack shear strai γ : possible paths whe the startig poit is positive ad i Softeig shear ack status (poit A).

175 Modelig of the ack shear compoet 4 τ τ p τ τ p E F C -γ u - γ max -γ p -γ max -γ p G B γ p γ u γ -γ u D γ p γ u γ A -τ max -τ p A -τ max -τ p a) γ < 0 b) γ > 0 Figure 4.0 Iemet of the ack shear strai γ : possible paths whe the startig poit is egative ad i Softeig shear ack status (poit A). τ τ τ p τ p τ max τ max F E -γ u -γ p B A γ p γ max γ u γ -γ u -γ p A γ p γ max G H γ γ u D C -τ p -τ p a) γ < 0 b) γ > 0 Figure 4. Iemet of the ack shear strai γ : possible paths whe the startig poit is positive ad i Uloadig or Reloadig shear ack status (poit A).

176 4 Chapter 4 τ τ τ p τ p C D -γ u -γ max -γ p A B γ p -τ max γ u E γ H -γ u G -γ max F -γ p A -τ max γ p γ u γ -τ p -τ p a) γ > 0 b) γ < 0 Figure 4. Iemet of the ack shear strai γ : possible paths whe the startig poit is egative ad i Uloadig or Reloadig shear ack status (poit A). I Free-slidig status the ack mode II stiffess modulus, DII = Dt,5, is ull. To avoid umerical istabilities i the calculatio of the stiffess matrix ad i the calculatio of the iteral forces, whe the ack shear status is Free-slidig, a residual value is assiged to this term. This topic is desibed i chapter 5. The cotrol of the ack evolutio is made by the opeig mode I, whose behavior is defied by oe of the tesile-softeig diagrams discussed i sectio Whe a softeig diagram is also used for the ack shear compoet some assumptio must be made sice there is o couplig betwee the two softeig diagrams used for the ormal compoet ad for the shear compoet. Therefore, the ack ca be i a Softeig status for the opeig mode I ad i a Uloadig status for the mode II. The followig problem occurs i some aalyses: the ack is Fully Ope ( ε ε, u) > ad the ack shear compoet, defied by oe of the diagrams preseted i the previous sectios, is i Stiffeig or i Softeig status. It is obvious that if the ack is Fully Ope the shear stress trasfer betwee the ack plaes is ull. Therefore, the solutio adopted is to assig a Free-slidig status to the shear ack status of this ack. Aother problem that sometimes occurs cosists o the reopeig of a closed ack. At the momet of closure, the ack adopts a liear elastic behavior i compressio. If this closed

177 Modelig of the ack shear compoet 43 ack reopes at a subsequet step, i.e., the ormal stress of the ack becomes tesile, the opeig mode I behavior follows the uloadig/reloadig brach of the tesile-softeig diagram (see for example Figure 3.6) ad the data relative to strai ad stress is updated. A more complex task is the update of the ack shear strai ad stress whe a softeig diagram is used, because the ack shear stress at ack closure ca be very differet from the ack shear stress whe the ack reopes. This is similar to the treatmet of the rotatio of the pricipal axes whe the status of the ack is Closed. This problem was reported by Rots ad de Borst (987) ad the proposed solutio was to impose that the shear ack compoet eters immediately i a softeig phase. For this purpose, the τ, ad γ t, p compoets (see Figure 4.4) that defie the ew shear softeig diagram is the ack shear stress upo reope ad the stored ack shear strai at closure of the ack, respectively. The ultimate ack shear strai of the ew diagram, γ tu,, is updated i order to maitai the same mode II fracture eergy. This procedure was implemeted i a first phase, but it could be observed that i some cases the value of the shear ack stress was very low upo reope of the ack, implyig that the ew softeig diagram could have a abormal high ultimate ack shear strai because of the coditio of preservig the mode II fracture eergy. Therefore, a ew approach is adopted ad the correspodig assumptios are commeted i Table 4.5 ad illustrated i Figure 4.3. This approach takes ito accout the shear ack status before the closure of the ack, beig the diagram updated i some cases, ad maitaied i others. t p Figure 4.3 represets the differet startig poits i the ack shear softeig diagram whe the mode I ack status chages from Closed to Reopeig. The value of τ i is the curret ack shear stress at ack reope, ad γ i is the correspodig calculated ack shear strai. The symbol γ i represets the ack shear strai before the closure of the ack. If the ack shear stress, τ i, is egative the startig poits are located o the egative brach of the ack shear softeig diagram (see Figure 4.4). I accordace with Figure 4.3, Table 4.5 represets the coditios for the update of the shear ack status whe a ack reopes.

178 44 Chapter 4 τ τ τ p,ew B (γ i,τ i ) τ p τ p τ i A D II D II γ i γ p γ γ γ u p γ p,ew a) b) γ u,ew γ u γ τ τ τ p τ i D II C τ p τ i D II D D II 3,4 γ p γ γ γ i u γ p γ i γ max c) d) γ u γ τ τ τ p τ p τ i E D II 3,4 τ i F D II 3,4 γ γ p γ i γ γ i- max γ u γ γ γ p γ γ i- i γ max u e) f) Figure 4.3 Differet startig poits i the ack shear softeig diagram whe the mode I ack status chages from Closed to Reopeig.

179 Modelig of the ack shear compoet 45 Shear ack status before ack closure Stiffeig Softeig Uloadig or Reloadig Table 4.5 Update of shear ack status whe a ack reopes Coditios τ τ τ τ τ i i i i τ p Start Poit Shear ack status whe ack reopes Update shear ack stregth < A Stiffeig * - τ B Softeig *, τ p p τ p ew < C Softeig * - τ B Softeig *, < τ i τ i II D3,4 τi τ i II D3,4 τ i II D3,4 τi τ i II D3,4 τ i II D3,4 τ i > γ < τ γ < τ γ > p p max p max γ i p max γ i τ p τ p ew D Softeig * - E Uloadig - F Reloadig - B Softeig *, τ p ew = τ = τ = τ i i i Update ack shear strai ** γ γ γ i τ γ i = D pew, II D p II τ = = γ τ p, ew τ i u i = γ p + II D τ γ = i pew, II D τ τ i u i = γ p + II D τ γ i = D τ γ i = D τ γ = i i II 3,4 i II 3,4 pew, II D Free-slidig - - Free-slidig - = max *** γ i γ * γ max - the maximum ack shear strai is updated; II β II ** D = G c ; D β τ p = γ γ u p ; D II 3,4 τ = ; γ max max *** the ack shear stress is assumed to be ull: τ = 0. i

180 46 Chapter Numerical simulatios To validate the implemeted softeig diagram of the shear ack compoet three umerical simulatios are performed. The first oe is dedicated to the simulatio of a mixed-mode test (Arrea ad Igraffea 98), while the secod deals with the simulatio of a reiforced coete T oss sectio beam that has failed i shear (Dias ad Barros 00). The last example is dedicated to the umerical simulatio of a reiforced coete beam stregtheed i shear with embedded through-sectio bars (Dalfré et al. 0) Mixed-mode test I this sectio the experimetal test from Arrea ad Igraffea (98) is umerically simulated. This test is a bechmark for the validatio of models takig ito accout the flexural ad shear failure modes, beig this pheomeo usually desigated mixed-mode failure. May researchers simulated this test umerically, beig the results available elsewhere (Rots et al. 985, de Borst 986, Rots ad de Borst 987, Rots 988, Ozbolt ad Reihardt 000, Jirásek ad Zimmerma 00, Most ad Bucher 006). Some material parameters used i the umerical simulatios coicide with those used by Rots ad de Borst 987. The geometry, mesh, support ad load coditios of the tested four poit beam are illustrated i Figure 4.4. The beam is modeled with 8-oded plae stress fiite elemets. The elemets outside the refied zoe are assumed to exhibit a liear-elastic behavior. The multi-fixed smeared ack model is adopted for the elemets of the refied zoe thus takig ito accout the oliear behavior due to ack iitiatio ad propagatio. The Gauss-Legedre itegratio scheme with 3 3 IP is used i all the elemets. Three umerical simulatios are performed, varyig i the treatmet of the ack shear compoet. The mode I fracture behavior is simulated with a bi-liear softeig diagram, adapted from the triliear diagram preseted i chapter 3. The parameters defiig this diagram are listed i Table 4.6. Oe of the umerical simulatios cosiders a costat shear retetio factor to obtai the mode II stiffess modulus, while the others use the softeig diagram of Figure 4.4 to determie the ack shear modulus. The differece

181 Modelig of the ack shear compoet 47 betwee these last umerical simulatios resides o the value used for the mode II fracture eergy. The parameters defiig these diagrams are preseted i Table F F A B otch Thickess Figure 4.4 Geometry, mesh, load ad support coditios of the tested four poit beam (all dimesios are i mm) [based o Rots ad de Borst 987]. Table Model properties used i the four poit beam simulatios. Poisso s ratio ν = 0.8 Iitial Youg s modulus Compressive stregth Triliear tesio softeig diagram Shear ack softeig diagram E = N mm c f c = 48.0 N mm I f ct =.8 N mm ; f N mm G = ; ξ = 0.5 ; α = 0.3 ; ξ = 0.4 ; α = 0. β = 0. ; τ t, p= 0.5 N mm ; G = N mm or G II = 0.0 N mm II f f Parameter defiig the mode I fracture eergy available to the ew ack Shear retetio factor 0. Crack badwidth p = 0.3 mm Threshold agle α th = 60º Maximum umber of acks i each IP

182 48 Chapter 4 The umerical simulatios are performed usig the Newto-Raphso method, cotrolled by the relative displacemet betwee two specific variables (see sectio.4.4). The cotrollig variables cosist o a pair of vertical displacemets, located at each tip of the otch. This type of cotrol is amed Crack Mouth Slidig Displacemet or CMSD. With this procedure a evetual sap-back ca be captured (de Borst 986, Vetura-Gouveia et al. 006). Figure 4.5 shows the relatio betwee the load at poit B ad the CMSD. The experimetal patter was obtaied by Rots ad de Borst 987. It is observed that the three umerical simulatios have similar behavior up to the peak load, ad afterwards differet paths are followed. With the costat shear factor the residual stress is close to the peak stress ad practically o softeig is predicted. With the itroductio of the shear ack softeig diagram a softeig behavior is observed ad a good agreemet with the experimetal results ca be obtaied, especially with the simulatio that cosiders the mode II fracture eergy equal to N/mm. The abrupt load decay, from 47.7 kn to 9 kn, i the simulatio with mode II fracture eergy equal to 0.0 N/mm ca be justified by the deformed mesh observed before ad after this occurrece, as represeted i Figure Experimetal results shear retetio - β = 0. Softeig - G f II = 0.075N/mm Softeig - G f II = 0.00N/mm Load at poit B [kn] CMSD [mm] Figure 4.5 Load at poit B ack mouth slidig displacemet (CMSD) relatioship.

183 Modelig of the ack shear compoet 49 (a) (b) Figure 4.6 Numerical simulatio with softeig ack shear diagram cosiderig G = 0.0 N mm : deformed mesh (a) before ad (b) after the abrupt load decay observed i Figure 4.5. The dashed lie represets the udeformed mesh. II f The ack patter observed immediately before ad immediately after the abrupt load decay is show i Figure 4.7. (a) (b) Figure 4.7 Numerical simulatio with softeig ack shear diagram cosiderig G = 0.0 N mm : ack patter (a) before ad (b) after the abrupt load decay observed i Figure 4.5. I dark quadrilateral, ack fully ope, ad the others i opeig or closig process. II f From the aalysis of the two deformed meshes ad respective ack patter, it ca be stated that the model is capable to reproduce umerically the deease of the ack shear stress trasfer durig the ack opeig process, ad is able to evaluate the residual load carryig capacity. This residual load capacity is i most cases due to the aggregate iterlock occurrig at the ack. I Figure 4.8 the load at poit B is plotted agaist the ack mouth opeig displacemet (CMOD). Coclusios idetical to the aforemetioed oes ca be extracted i terms of

184 50 Chapter 4 usig a softeig diagram or the shear retetio factor to simulate the degradatio of the shear stress trasfer durig the ack opeig process. A iterestig pheomeo, captured umerically for the curve with G = 0.0 N mm, is the sap-back of the CMOD II f at the sudde load decay. The explaatio for this behavior ca be foud by examiig the deformed meshes i Figure 4.6. Due to the rotatio of the left side of the beam about the support poit, the CMOD deeases at this istat ad the progressively reopes. This reopeig process i a residual phase is umerically reproduced ad show i Figure shear retetio - β = 0. Softeig - G f II = 0.075N/mm Softeig - G f II = 0.00N/mm 40 Load at poit B [kn] CMOD [mm] Figure 4.8 Load at poit B ack mouth opeig displacemet (CMOD) relatioship. As observed by de Borst (986) ad Rots ad de Borst (987), a sap-back occurs after the peak load ad is oly achieved whe the CMSD cotrol is used i the oliear aalysis. This sap-back is more proouced whe a ack softeig diagram is used (see Figure 4.9). A iterestig sap-back is observed after the abrupt load decay that occurs whe the softeig G = 0.0 N mm curve is used. II f

185 Modelig of the ack shear compoet 5 60 shear retetio - β = 0. Softeig - G f II = 0.075N/mm Softeig - G f II = 0.00N/mm 40 Load at poit B [kn] Vertical poit B displacemet [mm] Figure 4.9 Load - displacemet at poit B relatioship T oss sectio reiforced coete beam failig i shear A experimetal program was carried out by Dias ad Barros (00) with reiforced coete (RC) T oss sectio beams i the scope of a research project for the assessmet of the effectiveess of the ear surface mouted (NSM) techique by usig carbo fiber reiforced polymer (CFRP) lamiates for the shear stregtheig of RC T beams. Several RC T beams with differet percetage of NSM CFRP lamiates were tested. The geometry, support ad loadig coditios of the RC T referece beams are represeted i Figure C-R 50 A L i = º B 3 layers of U strips of wet lay-up CFRP sheet with the fibers directio at 0º F L r = //50 i L 6 i ad 6//75 i L r 00 d = //75 i L r (lateral coete cover = mm) Figure 4.30 Reiforced coete T referece beam: geometry, support, steel reiforcemet scheme ad loadig coditios [dimesios i mm] (Dias ad Barros 00).

186 5 Chapter 4 The RC tested beam amed S-R (see Figure 4.3) is umerically simulated, which correspods to the cotrol T beam of the series with two steel stirrups i the L i spa, i.e., without NSM CFRP lamiates. To prevet coete spallig ear the most loaded support, a cofiemet system based o the use of wet layup CFRP strips of sheets of U cofiguratio is applied as represeted i Figure S-R F x300 8x75 00 Figure 4.3 S-R reiforced coete T oss sectio beam [dimesios i mm] (Dias ad Barros 00). For the fiite elemet aalysis, the beam is cosidered as a plae stress problem. Therefore, the beam is modeled with a mesh of 4-oded seredipity plae stress fiite elemets (see Figure 4.3). A Gauss-Legedre itegratio scheme with IP is used i all coete elemets ad CFRP strips of sheet elemets. The steel bars are modeled with -oded perfectly boded embedded cables (oe degree-of-freedom per ode). The Newto-Raphso method with displacemet cotrol at a specific variable is used i the oliear aalysis (see sectio.4.3 of chapter ). Figure 4.3 Fiite elemet mesh [dimesios i mm].

187 Modelig of the ack shear compoet 53 For modelig the behavior of the steel bars, the stress-strai relatioship available i the FEMIX computer code is used (Sea-Cruz 004). A liear elastic behavior is adopted for modelig CFRP strips of sheets. The values of the parameters of the costitutive model used i the umerical simulatios are idicated i Table 4.7, Table 4.8 ad Table 4.9. Usig the average compressive stregth, f c, determied experimetally, the coete tesile stregth, f ct, ad the mode I fracture eergy, I G f, were iitially obtaied from equatios proposed by CEB-FIP (993), ad the were slightly adjusted i order to fit with high accuracy the load at ack iitiatio. The values for characterizig the softeig diagram of Figure 4.4 were obtaied by back-fittig aalysis i order to approximate as much as possible the experimetal curve i the post-ackig phase. It ca be observed that the mode II fracture eergy value is similar to the value assiged to the mode I fracture eergy. Table Properties of CFRP strips of sheets. C Sheet 40 f fu (MPa) 863 E f (GPa) 8 ε fu ( ) 3.3 t f (mm) 0.76 Table Steel properties. φ s =6 mm φ s = mm φ s=6 mm φ s =3 mm ε sy ( ) σ sy (MPa) ε sh ( ) σ sh (MPa) ε su ( ) σ su (MPa) p

188 54 Chapter 4 Table Model properties used for the T beam simulatio. Poisso s ratio ν = 0.5 Iitial Youg s modulus Compressive stregth Triliear tesio softeig diagram E = N mm c f = 39.7 N mm c I f ct =. N mm ; f N mm G = ; ξ = ; α = 0.4 ; ξ = 0. ; α = 0.3 Shear ack softeig diagram β = 0.5 ; Parameter defiig the mode I fracture p = 3 eergy available to the ew ack Shear retetio factor 0.5 Crack badwidth Threshold agle α th = 30º Maximum umber of acks per each IP τ t, p=.0 N mm ; G II f = 0.08 N mm Square root of the area of the IP Two umerical simulatios are performed that differ i the treatmet of the ack shear compoet. Oe uses the cocept of shear retetio factor with a costat value ( β = 0.5) ad the other uses the shear ack softeig diagram of Figure 4.4. Figure 4.33 represets the experimetal ad umerical relatioships betwee the load ad the deflectio at the loaded sectio for the tested S-R beam. It ca be observed that the use of the cocept of the shear retetio factor coducts to a overestimatio of the load carryig capacity. The curve derived from this umerical aalysis starts divergig sigificatly from the experimetal oe for deflectios larger tha.5 mm. I the aalysis where a shear ack softeig diagram was used, a quite accurate simulatio was obtaied up to a load level that is 9.6 % of the ultimate load registered experimetally. I this case the umerical simulatio was iterrupted because covergece was ever possible to attai due to the formatio of a shear failure ack, as show i Figure This figure represets the ack patters at a deflectio of about 4.9 mm for both umerical simulatios. The use of the shear ack softeig diagram reproduces very well the shear failure observed at the ed of the experimetal test ad represeted Figure 4.35.

189 Modelig of the ack shear compoet 55 Load [kn] Experimetal Shear retetio factor Shear ack softeig diagram Deflectio [mm] Figure 4.33 Load-deflectio at the loaded sectio for the T beam. F F (a) (b) Figure 4.34 Crack patter of the S-R beam by usig a: (a) shear retetio factor; ad (b) a shear ack softeig diagram (i pik color: ack completely ope ( ε ε, u); i red color: ack i the opeig process; i cya color: ack i the reopeig process; i gree color: ack i the closig process; i blue color: closed ack). Figure 4.35 Crack patter at the ed of the tested S-R beam (Dias ad Barros 00).

190

191 Modelig of the ack shear compoet 57 embedded cables (oe degree-of-freedom per ode) ad a Gauss-Legedre itegratio scheme with 3 IP is used. Figure 4.38 Fiite elemet mesh [dimesios i mm]. For modelig the behavior of the steel bars, the stress-strai relatioship available i the FEMIX computer code is used (Sea-Cruz 004). Two relatioships are used i the umerical simulatios ad the values are preseted i Table 4.0, sice the strais were ot measured i the experimetal tests after the yield iitiatio of the steel bars. Therefore, for the strai ad its correspodig stress that defie the ed of the secod brach of the stress-strai diagram of the steel bars two pairs of values were cosidered, leadig to the diagrams desigated by A ad B. The values of the parameters of the costitutive model for the coete elemets used i the umerical simulatios are idicated i Table 4.. Usig the average compressive stregth, f c, determied experimetally, the coete tesile stregth, f ct, ad the mode I fracture eergy, I G f, were iitially obtaied from equatios proposed by CEB-FIP (993), ad the were adjusted i order to fit with high accuracy the load at ack iitiatio. The values for characterizig the softeig diagram of Figure 4.4 were obtaied by back-fittig aalysis i order to approximate as much as possible the experimetal curve i the post-ackig phase.

192 58 Chapter 4 ε sy ( ) σ sy (MPa) ε sh ( ) σ sh (MPa) ε su Table Steel properties. φ s =6 mm φ s =0 mm φ s = mm φ s =5 mm A B A B A B A B ( ) σ su (MPa) p Table 4. - Model properties used for the ETS stregtheig beam simulatio. Poisso s ratio ν = 0.5 Iitial Youg s modulus E = N mm c Compressive stregth Triliear tesio softeig diagram (coete elemets ear the bottom logitudial steel bars) Triliear tesio softeig diagram (other coete elemets) Shear ack softeig diagram (coete elemets ear the bottom logitudial steel bars) Shear ack softeig diagram (other coete elemets) f = N mm c G = ; I f ct =.0 N mm ; f 0.06 N mm ξ = 0.0; α = 0.5 ; ξ = 0.5 ; α = 0. G = ; I f ct =.8 N mm ; f 0.05N mm ξ = 0.0; α = 0.4 ; ξ = 0. ; α = 0. β = 0. ; β = 0. ; Parameter defiig the mode I fracture p = eergy available to the ew ack Shear retetio factor 0. Crack badwidth Threshold agle α th = 30º Maximum umber of acks per each IP τ t, p=.38 N mm ; G II f = 0.5 N mm τ t, p=.38 N mm ; G II f = 0.3 N mm Square root of the area of the IP

193 Modelig of the ack shear compoet 59 Tree umerical simulatios are performed, varyig i the treatmet of the ack shear compoet ad i the modelig of the behavior of the steel bars ad the ETS stregtheig bars. Oe uses the cocept of shear retetio factor with a costat value ( β = 0.) ad the other two uses the shear ack softeig diagram of Figure 4.4, differig oly i the stress-strai relatioship used for the simulatio of the steel ad the ETS stregtheig bars (see Table 4.0). Figure 4.39 represets the experimetal ad umerical relatioships betwee the load ad the deflectio at the loaded sectio for the tested S5.90/E5.90 beam. I compariso to the experimetal curve the use of the cocept of the shear retetio factor coducts to a more stiff respose ad the load carryig capacity is attaied for a smaller deflectio. The curve derived from this umerical aalysis starts divergig from the experimetal oe for deflectios larger tha 3.5 mm. The aalyses performed usig the shear ack softeig diagram improves sigificatly the umerical resposes. A better accurate simulatio is obtaied usig the properties B (see Table 4.0) for the steel bars ad the ETS stregtheig bars but the load carryig capacity is uderestimated. Adoptig the properties A (see Table 4.0) the ultimate load is better predicted with a respose that is a little bit more stiff tha the behavior recorded experimetally. I these umerical simulatios the aalyses were iterrupted because covergece was ever possible to attai due to the formatio of shear failure acks, as show i Figure This figure represets, for each umerical simulatio, the ack patters at maximum attaied load. The use of the shear ack softeig diagram improves sigificatly the shear failure observed at the ed of the experimetal test ad schematically represeted Figure 4.4.

194 60 Chapter Load [kn] Experimetal Shear retetio factor Shear ack softeig diagram ad steel diagram A Shear ack softeig diagram ad steel diagram B Deflectio [mm] Figure 4.39 Load-deflectio at the loaded sectio for the ETS stregtheig beam. F F (a) (b) F (c) Figure 4.40 Crack patter of the S5.90/E5.90 beam by usig: (a) shear retetio factor; (b) shear ack softeig diagram ad steel diagram A; ad (c) shear ack softeig diagram ad steel diagram B (i pik color: ack completely ope ( ε ε, u); i red color: ack i the opeig process; i cya color: ack i the reopeig process; i gree color: ack i the closig process; i blue color: closed ack). Figure 4.4 Crack patter at the ed of the tested S5.90/E5.90 beam (Dalfré et al. 0).

195 Modelig of the ack shear compoet SUMMARY AND CONCLUSIONS To accurately simulate the deformatioal behavior of the shear ad flexural/shear failure modes, two alterative approaches are proposed for the treatmet of the ack shear compoet. The former is the implemetatio of a total ack shear stress-shear strai approach to simulate the degradatio of the shear stress trasfer with the ack opeig evolutio, ad the latter is based o the use of a costitutive softeig relatio betwee the ack shear stress ad the ack shear strai compoet. Each of these strategies is desibed ad their capabilities are assessed by performig several umerical simulatios. The results obtaied are preseted ad discussed. I coclusio, it ca be said that the implemetatio of these ew tools i the multi-fixed smeared ack model improves its capabilities to predict with higher accuracy the behavior of structures failig i shear or i flexural/shear.

196

197 Chapter 5 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 5. INTRODUCTION The type of model to be selected for the aalysis of a certai structure depeds o the specificities of this structure. Sometimes some simplificatios ca be adopted without compromise the relevace of the coclusios that ca be extracted from the aalysis, such is the case of assumig a structure like a bi-dimesioal body. However, i some cases, due to complex loadig ad/or geometry coditios these simplificatios are ot appropriate, ad to simulate the complex failure modes that ca be formed i this type of structures three-dimesioal (3D) approaches must be used. I the preset chapter a multi-fixed smeared 3D ack model, uder the oliear FEM framework, is proposed to simulate the behavior of coete structures. This model is suitable to be used with 3D solid elemets available i the FEMIX computer code (Azevedo et al. 003, Sea-Cruz et al. 007).

198 64 Chapter 5 The 3D model is desibed below ad its performace is appraised by simulatig a puchig test with lightweight paels of steel fiber reiforced self-compactig coete (SFRSCC). 5. NUMERICAL MODEL 5.. Itroductio I the last decades the developmet of sophisticated 3D models to simulate the complex oliear behavior of coete structures has bee sigificat, ad the applicability of these models to real structures is becomig possible due to the cotiuous progress i high-performace computig hardware (Barzegar ad Maddipudi 997a, 997b). I the preset sectio the multi-fixed smeared D ack model, previously implemeted i the FEMIX computer code by Sea-Cruz (004), is geeralized to a multi-fixed smeared 3D ack model, implemeted with solid fiite elemets (Vetura-Gouveia et al. 008). The fracture mode I is simulated with oe of the tesile-softeig stress-strai diagrams preseted i sectio of chapter 3. The characterizatio of the shear fracture modes is much more complex, sice the shear stress trasfer betwee the ack surfaces depeds o several parameters, such as coete lateral cofiemet, ack opeig, roughess of the ack surfaces, coete stregth class, umber of acks ad its relative orietatio, etc. For its simulatio, the classical shear ack retetio factor cocept ca be used, associated with a iemetal or total approach for the shear ack compoets, whose formulatio is detailed i sectio 4. of chapter 4. Alteratively, the softeig ack shear stress-strai diagram desibed i sectio 4.3 of the previous chapter ca also be used i the simulatio of the shear stress trasfer betwee the ack surfaces. 5.. Formulatio 5... Crack strai ad ack stress As metioed before, i a material oliear aalysis the costitutive matrix depeds o the stress or strai levels at a give stage of the loadig process. To obtai a solutio, the

199 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 65 exteral load is applied iemetally ad a iterative techique is used to solve the resultig system of oliear equatios (Ziekiewicz ad Taylor 000b, Vetura-Gouveia 000, Vetura-Gouveia et al. 006). The relatioship betwee iemetal strai ad iemetal stress is i this case give by σ = D ε (5.) where σ represets the stress iemet, ε is the strai iemet ad D is the taget costitutive matrix. The iemetal strai associated with the acked material is, i smeared ack models, decomposed ito a iemetal strai vector of the ack, vector of the uacked coete betwee the acks, ε, ad a iemetal strai co ε (de Borst ad Nauta 985, Rots et al. 985, de Borst 986, Rots 988, Barros 995). This iemetal strai decompositio is useful for the iclusio of other pheomea, such as temperature, eep or shrikage (de Borst ad Nauta 985, Hofstetter ad Mag 995, va Zijl et al. 00). The iclusio of these time-depedet pheomea is treated i chapter 6. co ε = ε + ε (5.) For the three-dimesioal case the iemetal local ack strai vector, by ε, is defied T ε = ε γt γ t (5.3) ad, i the global coordiate system, by ε = ε ε ε3 γ3 γ3 γ (5.4) T

200 66 Chapter 5 Equatio (5.5) represets the relatioship betwee ε ad ε T ε = T ε (5.5) where T is the trasformatio matrix (see Figure 5.) defied by a a a3 aa3 aa3 aa T = a a a a a a a a + a a a a + a a a a + a a a a a a a a a a + a a a a + a a a a + a a (5.6) The a, a ad a 3 compoets are the cosie directors of the uit vector of the axis ormal to the ack plae, ; a, a ad a 3 are the cosie directors of the uit vector of the t axis; fially, a 3, a 3 ad a 33 are the cosie directors of the uit vectors of the t axis. The, t t axes form the local ack coordiate system. I matrix form the compoets of a are defied by a a a3 cos(, x ) cos(, x) cos(, x3) a = a a a = cos( t, x ) cos( t, x ) cos( t, x ) 3 3 a3 a3 a 33 cos( t, x ) cos( t, x ) cos( t, x3 ) (5.7) The T matrix trasforms the stress compoets from the global coordiate system to the local ack coordiate system ad its terms are extracted from the matrix that trasforms the stress tesor betwee coordiate systems (Azevedo 996, Silva 006). It ca be T demostrated that T trasforms the strai compoets from the local ack coordiate system to the global coordiate system (Azevedo 985).

201 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 67 Accordig to the fracture mechaics priciples, three differet types of fracture modes ca be cosidered. The ack opeig mode (fracture mode I), the shearig mode (fracture mode II - i-plae shear), ad the tearig mode (fracture mode III - out-of-plae shear). A acked body ca be loaded i oe of these modes or i a combiatio of the three modes (Wag 996). (a) (b) (c) Figure 5. Basic fracture modes: (a) opeig mode (tesile), (b) shearig mode (i-plae shear) ad (c) tearig mode (out-of-plae shear) [Wag 996]. For the case of 3D solids, the distictio betwee mode II ad mode III ca be dropped (Hofstetter ad Mag 995). For this reaso, i the preset model mode II ad mode III are desigated as slidig modes i ˆt ad ˆt directios, respectively. The iemetal local ack stress vector, σ, ca be defied by T σ = σ τt τ t (5.8) where σ is the mode I iemetal ack ormal stress, is the slidig mode τ t iemetal ack shear stress actig i ˆt directio ad is the slidig mode τ t iemetal ack shear stress actig i ˆt directio.

202 68 Chapter 5 Figure 5. represets the ack stress compoets i the local coordiate system of the ack ad the correspodig displacemets: the opeig displacemet, w, the slidig displacemet i ˆt directio, s, ad the slidig displacemet i ˆt directio, s. x 3 x 3 ack plae s τ t σ τ t s t ack plae t w x x x x Figure 5. Crack stress compoets, displacemets ad local coordiate system of the ack. I the global coordiate system the iemetal stress compoets are T (5.9) σ = σ σ σ τ τ τ The relatioship betwee σ ad σ is σ = T σ (5.0) 5... Costitutive law of the coete A liear elastic behavior is assumed for the coete betwee acks, beig the relatio betwee co ε ad σ give by

203 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 69 co σ = D ε co (5.) with D co ( ) ν ν ν ν ( ν) ν ν ν ( ν) E ν = ( ν)( ν) + ν ν (5.) beig E the Youg s modulus ad ν the Poisso s ratio of the udamaged coete Costitutive law of the ack At the ack zoe (damaged coete), the relatioship betwee ε ad σ is give by σ = D ε (5.3) where D D 0 0 = 0 Dt Dt (5.4) is the ack costitutive matrix.

204 70 Chapter 5 I equatio (5.4), D, D t ad D represet, respectively, the fracture mode I stiffess t modulus, the slidig stiffess modulus i the ˆt directio ad the slidig stiffess modulus i the ˆt directio, ad their values deped o the law assumed to simulate the ack behavior. I the preset approach the direct shear-ormal couplig is igored, thus justifyig the diagoality of the ack costitutive matrix. Its effect may be idirectly obtaied by allowig o-orthogoal acks to form ad defiig the slidig fracture modes as a fuctio of the ack ormal strai (Rots 988). The fracture mode I modulus, D desibed i sectio of chapter 3., is defied by oe of the tesile-softeig diagrams The slidig fracture mode modulus, D or t D t, ca be obtaied with, β D = D = G t t β c (5.5) where G c is the coete elastic shear modulus ad β is the shear retetio factor, defied by a costat value or by equatio (3.3) i sectio of chapter 3. Alteratively, to model the ack shear stress trasfer i ˆt ad ˆt directios, ad to improve the accuracy of the simulatios of structures failig i shear, D or t D t ca be obtaied with oe of the shear ack softeig diagrams desibed i sectio 4.3 of chapter 4. I this case, each compoet follows a idepedet ack softeig diagram Costitutive law of the acked coete Takig ito accout equatio (5.) ad equatio (5.5), equatio (5.) ca be writte as follows

205 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 7 T ( T ) co σ = D ε ε (5.6) Multiplyig this equatio by the trasformatio matrix T ad takig ito accout equatio (5.0), equatio (5.6) becomes co T co σ T D T ε T D ε + = (5.7) ad substitutig σ usig equatio (5.3) co T co D ε T D T + ε = T D ε (5.8) or T ( ) co co = D + T D T T D ε ε (5.9) Equatio (5.9) establishes a relatioship betwee the iemetal local ack strai vector ad the iemetal strai vector i the global coordiate system. Icludig equatio (5.6), the followig costitutive law for the acked coete ca be obtaied T ( ) T co co co co σ = D D T D T D T + T D ε (5.0) or co = D (5.) σ ε beig

206 7 Chapter 5 T ( ) T co co co co co D = D D T D T D T + T D (5.) the costitutive matrix of the acked coete. Although the precedig equatios are obtaied assumig oly oe ack per itegratio poit (IP), the model ca be applied to the case of acks beig formed at each IP. For this purpose, the ack costitutive matrix, D, ad the trasformatio matrix, T, iclude the iformatio that correspods to these acks D D D... 0 = D (5.3) ad T T = T T... T (5.4) I equatios (5.3) ad (5.4), D i ad T i correspod to the ack costitutive matrix ad to the ack trasformatio matrix of the i th ack Model implemetatio A simple Rakie iterio is used to detect ack iitiatio. Whe the maximum pricipal tesile stress reaches the coete tesile stregth at a IP of a fiite elemet, the material cotaied i its ifluece volume chages from uacked to acked state. The ack plae is cosidered to be ormal to the directio of the maximum pricipal stress. The ack ormal tesile stress follows the tesile-softeig diagram characterized by the

207 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 73 fracture mode I modulus, D. For the ack shear compoets two hypothesis are available: each ack shear stress follows a shear softeig diagram characterized by the slidig fracture modes D ad t D t, which ca be obtaied from oe of the diagrams represeted i sectio 4.3 of chapter 4; or the shear stresses ad the slidig fracture modes are determied by meas of the shear retetio factor cocept, which is based o equatio (5.5). As a cosequece of the rotatio of the axes of the pricipal stresses durig the subsequet load iemets, a ew smeared ack patter ca be iitiated. Two coditios must be satisfied for the ew ack iitiatio: the maximum pricipal tesile stress reaches the coete tesile stregth; the agle betwee the directio vector of the maximum pricipal tesile stress ad the directio vector of the existig acks is greater tha a predefied agle, amed threshold agle, α th. Values betwee 30º ad 60º are recommeded for the threshold agle (de Borst ad Nauta 985, de Borst 986). To maitai the cosistecy of the ack iitiatio process, whe the secod coditio is ot satisfied, the tesile stregth must be updated. This value ca be sigificatly differet from the iitial coete tesile stregth, especially for large values of α th (Rots 988). The formatio of additioal acks i a previously acked coete ca cotribute to the modificatio of the ack status of these existig acks. The previously developed multi-fixed D smeared ack model takes ito accout these ack status modificatios. I the preset 3D model a similar approach is used for the opeig mode I (Sea-Cruz 004), ad for both slidig modes whe the ack shear softeig diagram is selected (see sectio 4.3 of chapter 4). A evetual couplig betwee the ormal ad shear modes is ot cosidered i the 3D model. Therefore, a ack ca, for example, uload i mode I, softe i ˆt slidig mode, ad reload i ˆt slidig mode. For this reaso, the model treats separately the ormal ack status, the shear ack status i ˆt directio, ad the shear ack

208 74 Chapter 5 status i ˆt directio. For the case of the slidig modes the shear ack statuses are desibed i sectio 4.3. of chapter 4. As a cosequece of the material oliear behavior, the stress must be updated i order to satisfy the laws of the material respose. This stress update is performed with the followig equatio σ = σ prev+ σ (5.5) where the subsipt prev meas the stress at a previous state of the IP. Multiplyig equatio (5.5) by the trasformatio matrix T results T ( prev ) σ = T σ + σ (5.6) ad takig ito accout the iemetal relatio from equatio (5.0), established i terms of total stresses, equatio (5.6) results i ( prev ) σ = σ, prev+ σ = T σ + σ (5.7) Combiig equatio (5.0) with this equatio yields, after some arragemets, σ, prev + σ = T ( ) T co co co co prev T σ + T D ε T D T D + T D T T D ε (5.8) ad takig ito accout equatio (5.9) the followig equatio is obtaied

209 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 75 T co co, prev+ T prev T D T D T + = 0 σ σ σ ε ε (5.9) Sice σ is a fuctio of the followig form ε (see equatio (5.3)), equatio (5.9) ca be writte i T ( ) ( prev prev ) co co, 0 f ε = σ T σ T D ε + D + T D T ε = (5.30) The Newto-Raphso method is used to solve equatio (5.30), where the vector of ukows is the local iemetal ack strai vector, ε. The first derivative of equatio (5.30) i order to the iemetal local ack strai vector is required i the q iteratio of this method. Depedig o the strategy assumed for the ack shear compoets, i.e., the use of the cocept of shear retetio factor or the use of a ack shear softeig diagram, this derivative is defied by equatio (5.3) or equatio (5.37), respectively. Whe the cocept of shear retetio factor is used to obtai the slidig fracture modulus, D t or D t, the derivative of f ( ε ) is f ( ε ) ε = D + D + T D T co T (5.3) where D D D... 0 = D (5.3)

210 76 Chapter 5 beig D D 0 0 Dt γ t 0 0 ε i t i = γ t ε ì (5.33) if a iemetal approach is used for the ack shear compoets or D D = 0 0 Dt γ t 0 0 ε i t i γ t ε i (5.34) if a total approach is used for the ack shear compoets (see sectio 4. of chapter 4). Whe equatio (3.3) is adopted to defie the shear retetio factor used i equatio (5.5), the derivative of the i th slidig fracture modulus D i order to ti ε is p ( ) c ( ) ( ) p ( ) p (( A) ) Dt A G p Gc p i = ε ε ε p ( ) ( ) ( ), u A A u, A A (5.35) beig A ε + = ε (5.36) prev, ε u,

211 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 77 Whe a costat shear retetio factor is used, the matrix D becomes a ull matrix. If a ack shear softeig diagram is used to obtai the slidig fracture modulus, D t, the derivative of f ( ε ) is D or t f ( ε ) ε = D + T D T co T (5.37) Whe the covergece of the Newto-Raphso method caot be achieved, the fixed-poit iteratio method is used to attempt the solutio of the system (5.30). The solutio of equatio (5.30) is the iemetal local strai vector of the acks, ε. The ack strai vector i the global coordiate system is calculated usig equatio (5.5) ad the ack strai is updated with prev ε = ε + ε (5.38) From equatio (5.3) the iemetal local ack stress vector is obtaied ad the local ack stress vector is updated with =, prev + σ σ σ (5.39) From equatio (5.6) the iemetal stress is calculated ad the stress is updated usig equatio (5.5). To verify the cosistecy of the solutio a verificatio is made, usig equatio (5.0), i terms of total stresses with

212 78 Chapter 5 σ = T σ (5.40) The improvemets made i the stress update ad i the itical ack status chage, explaied i sectio 3.5 of chapter 3 for the D multi-fixed smeared ack model, are also implemeted i the 3D versio of this model. The evaluatio of the iteral forces requires the solutio of (5.30), ad the calculatio of the stiffess matrix of a elemet depeds o the costitutive matrix of the acked coete (5.). For both these purposes the iversio of the matrix defied by (5.4) is required. co T D = D + T D T (5.4) Whe D is a ull matrix, i.e., whe the acks are i Fully Ope status or for the case of usig a ack shear softeig diagram, the shear ack statuses are i Free-slidig, resultig i a ull stiffess for the slidig fractures modes, evetual problems might arise i the iversio of D. To circumvet these difficulties the followig residual values are assiged to the diagoal of D (see equatio (5.4)) D 6 = 0 fc ; D ti 6 = 0 Gc (5.4) 5.3 MODEL APPRAISAL The performace of the proposed costitutive model is assessed by simulatig the behavior observed i a puchig test of a lightweight SFRSCC pael. The test layout ad the test setup are represeted i Figure 3.7 of chapter 3. The fiite elemet model, the load ad the support coditios used i the umerical simulatio of the puchig test are show i Figure 5.3.

213 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 79 Oly oe quarter of the pael is used i the simulatio due to double symmetry. Seredipity 0 ode solid elemets with 3 Gauss-Legedre itegratio scheme are used (three itegratio poits i the through-thickess directio). Three solid elemets are used through the thickess of 0 mm, while i the lightweight zoe (shaded elemets i Figure 5.3) oly oe solid elemet is used through the thickess. The dashed lie represets the support of the pael. The values of the parameters of the costitutive model used i the simulatio of the puchig test are listed i Table 5.. The results of two umerical simulatios are desibed below. I the first oe, the shear stress trasfer i both slidig modes is simulated accordig to equatio (5.5) with the shear retetio factor defied by equatio (3.3) assumig p =. I the secod oe, the ack shear stress vs. ack shear strai diagram represeted i Figure 4.4 (see sectio 4.3. of chapter 4) is used to model both shear slidig modes. I both simulatios the values of the properties associated with the fracture mode I are obtaied with the iverse aalysis desibed i sectio 3.7. of chapter 3. The force-deflectio relatioship registered experimetally ad those obtaied from both umerical simulatios are preseted i Figure 5.4. x,u (u =0) Poit load 300 mm Supports (u =0) 3 (u =0) Thickess - 0 mm Thickess - 30 mm x 3 x x (x 3,u ) mm x,u Figure 5.3 Geometry, mesh, load ad support coditios used i the umerical simulatio of the puchig test.

214 80 Chapter Load [kn] Experimetal Shear retetio factor Softeig shear stress-strai diagram Deflectio [mm] Figure 5.4 Relatioship betwee the force ad the deflectio at the ceter of the test pael. Table 5. - Values of the parameters of the costitutive model used i the umerical simulatio of the puchig test. Poisso s ratio ν = 0.5 Iitial Youg s modulus E = N mm c Compressive stregth Triliear tesio softeig diagram of SFRSCC (see Figure 3.6 of sectio of chapter 3) Parameter defiig the mode I fracture eergy available to the ew ack Softeig ack shear stress-strai diagram (see Figure 4.4 of sectio 4.3. of chapter 4) f = 5.0 N mm c G = ; I f ct = 3.5 N mm ; f 4.3 N mm ξ = ; α = 0.5 ; ξ = 0.5 ; α = 0.59 p = τ t, p=.0 N mm ; f, s 5.0 N mm Shear retetio factor Expoetial ( p = ) Crack badwidth Threshold agle α th = 30º G = ; β = 0.5 Cube root of the volume of the itegratio poit Figure 5.4 shows that, i the experimetal test, for a deflectio of about. mm at the cetral poit, the pael load carryig capacity is almost retaied up to a deflectio of about 3 mm. Afterwards a abrupt load decay occurs due to the formatio of a shear failure surface, typical of a puchig rupture. However, usig the cocept of shear retetio

215 Multi-fixed smeared 3D ack model to simulate the behavior of coete structures 8 factor, the model predicts a iease of the load carryig capacity with the iease of the pael deflectio. Accordig to this approach, the shear stresses i ˆt ad ˆt slidig directios ( τ, τ ) iease with the iease of the correspodig shear strais ( γ, γ ). t t Oly after the exhaustio of the mode I fracture eergy ( ε ε, u, see Figure 3.6 of sectio 3.3..), both ack shear stress compoets become costat (the iemetal values are ull, = 0, τ t = 0 ). However, for this type of problem, where the ack τ t shear costitutive compoets have a fudametal ifluece o the behavior the structure, maily after a certai deflectio, the formulatio of these compoets should allow the shear stress compoets to deease with the iease of the ack opeig, which requires the use of a shear softeig diagram, similar to that represeted i Figure 4.4 of sectio I fact, the secod simulatio, which is based o this approach, captured the plateau registered experimetally (see Figure 5.4), sice the parameters of this diagram are evaluated by back-fittig aalysis i order to reproduce this phase of the pael respose. However, the abrupt load decay observed i the experimetal test, at a deflectio of about 3 mm, is ot captured by this secod approach, sice the umerical simulatio predicts a cotiuous, but smooth, degradatio of the pael load carryig capacity. It could be verified that a deease of G f, s (e.g G f, s= 4.0 N mm or G f, s=.0 N mm ) causes a deease o the pael load carryig capacity, maily i its structural softeig phase, but the abrupt load decay is still ot captured as show i Figure 5.5. t t Load [kn] Experimetal Softeig shear stress-strai diagram - Gf,s = 5.0N/mm Softeig shear stress-strai diagram - Gf,s = 4.0N/mm Softeig shear stress-strai diagram - Gf,s =.0N/mm Deflectio [mm] Figure 5.5 Ifluece of G f, s, o the umerical relatioship betwee the force ad the deflectio at the ceter of the test pael.

216 8 Chapter SUMMARY AND CONCLUSIONS I the preset chapter a multi-fixed smeared 3D ack model is proposed. The developed model is based o the fiite elemet method ad is implemeted i the FEMIX computer code. I a attempt to simulate the shear strai gradiet that occurs i puchig tests, a shear softeig diagram is proposed i order to make both ack shear stress compoets depedet o the correspodig ack shear strais. The ack shear stress trasfer ca also be simulated usig the cocept of shear retetio factor, which ca be defied as a costat value or as a fuctio of the ack ormal strai. I this case the shear retetio factor assumes a uitary value at ack iitiatio ad a ull value whe the mode I fracture eergy is exhausted. The performace of the model is appraised by usig the results obtaied i a puchig test with a lightweight pael prototype of steel fiber reiforced self-compactig coete. Two umerical simulatios are performed ad discussed: oe usig the cocept of shear retetio factor ad the other usig a softeig diagram to model both ack shear compoets. It is observed that the use of a softeig diagram for the ack shear compoets improves the umerical simulatio of the tested pael.

217 Chapter 6 Thermo-mechaical model 6. INTRODUCTION I this chapter all relevat aspects related to the heat trasfer ad its implemetatio i the FEMIX computer code are desibed. The heat trasfer problem is preseted, ad a umerical model is developed to simulate the heat trasfer i structures built with materials whose mechaical behavior ca be cosidered to be liear or oliear. The heat developmet due to the hydratio process durig the coete hardeig phase ad its iclusio i the heat trasfer model is also treated. The performace ad the accuracy of the developed umerical model are assessed usig results available i the literature. The formulatios of the time-depedet deformatios due to shrikage, eep ad temperature variatio are also preseted, ad the multi-fixed smeared 3D ack model proposed to simulate the behavior of coete structures, desibed i chapter 5, is adapted to iclude these time-depedet effects. 6. THERMAL PROBLEM 6.. Itroductio Heat trasfer ca be defied as the eergy trasferred betwee material bodies due to a temperature differece (Holma 986, Lewis et al. 004, Iopera et al. 006). The heat flows from hot to cold mediums util a equilibrium state is reached (Silveira 993), beig

218 84 Chapter 6 the process of heat trasfer divided i three modes: coductio, covectio ad radiatio. A briefly desiptio of each mode, icludig its goverig equatios, is preseted i this sectio. More details ca be foud elsewhere (Holma 986, Silveira 993, Lewis et al. 004, Iopera et al. 006, Liehard IV ad Liehard V 005, Azeha 009). Coductio heat trasfer From a simple poit of view, this is the heat trasfer that occurs iside a solid. The coductio heat trasfer ca occur i a steady-state regime, whe the temperature field i a solid does ot chage with time or i a trasiet regime if the temperature field chages with respect to time. The heat coductio rate equatio is defied by the Fourier s law. For the case of oe dimesioal coductio this equatio has the followig form q q A dt dx x x = = k (6.) where q x is the heat flux i x directio, A is the area perpedicular to the directio of heat trasfer, k is the material thermal coductivity ad dt dx is the temperature gradiet. The mius sig sigifies that the heat flows i the directio of the deeasig temperature. Covectio heat trasfer Covectio is a process of heat trasfer that occurs betwee a body surface ad a fluid i movemet whe the temperatures differ betwee the two domais. The heat covectio rate equatio is defied by the Newto s law of coolig, ( ) = (6.) qc hc T T where q c is the covective heat flux, h c is the covectio heat trasfer coefficiet, T is the temperature at the body surface ad T is the fluid temperature.

219 Thermo-mechaical model 85 Radiatio heat trasfer Radiatio is the mode of heat trasfer that occurs whe o cotact exists betwee the body that emits heat ad the body that receives it (this kid of heat trasfer ca occur i vacuum coditios). The maximum flux emitted by radiatio from a black body is defied by the Stefa-Boltzma law equatio, q 4 e = σt (6.3) where q e is the emitted heat flux, σ is the Stefa-Boltzma costat, defied as σ = Wm K, ad T is the surface temperature. Itroducig i equatio (6.3) the cocept of emissivity, ε, to take ito accout that, i reality, the bodies emit less eergy tha black bodies, this equatio is trasformed ito q 4 e = εσt (6.4) where 0< ε <. The et rate radiat exchage, for the case of a heat trasfer surface at temperature T completely eclosed by a much larger surface maitaied at temperature T, is obtaied with (Holma 986) r 4 4 ( ) q = εσ T T (6.5) This equatio ca be rewritte as r r ( ) q = h T T (6.6)

220 86 Chapter 6 where h r is the radiatio heat trasfer coefficiet that is defied by r ( )( ) h = εσ T + T T + T (6.7) Equatio (6.6) is ow aalogous to the heat covectio rate equatio (6.), which ca be useful whe covectio ad radiatio occur i the same body surface. 6.. Heat coductio equatio 6... Geeral remarks I this sectio the heat coductio equatio for the case of three-dimesioal (3D) heat coductio aalysis is directly derived by applyig the coservatio of eergy priciple to a ifiitesimal cotrol volume of a 3D body as represeted i Figure 6.. z y 3D body x q z+dz q x E st q y q y+dy d z E g q x+dx q z d x d y Figure 6. Ifiitesimal cotrol volume of a 3D body.

221 Thermo-mechaical model 87 The eergy balace for the geeral case, represeted i Figure 6., where the temperature may be chagig with time ad iteral heat sources may be preset, ca be assured by (Iopera et al. 006, Holma 986) E + E = E + E (6.8) i g st out beig, E i the rate of eergy coducted i the ifiitesimal cotrol volume, defied by E = q + q + q (6.9) i x y z E g i equatio (6.8) is the rate of thermal eergy geeratio, give by E = Qdxdydz (6.0) g where Q is the iteral heat geeratio rate per uit volume of the ifiitesimal cotrol volume, E st i equatio (6.8) is the rate of the eergy stored withi the ifiitesimal cotrol volume, expressed as T E st = ρc dxdydz (6.) t where ρ is the mass per uit volume ad c is the specific heat of the material. The ρ c quatity represets the volumetric heat capacity ad measures the capacity of a material to store thermal eergy.

222 88 Chapter 6 Fially, E out i equatio (6.8) is the rate of eergy coducted out the ifiitesimal cotrol volume, defied by E = q + q + q (6.) out x+ dx y+ dy z+ dz By cosiderig the first two terms of the Taylor series expasio, each term o the right-had side of the previous equatio becomes q = + x x qx+ dx qx dx q = + y y qy+ dy qy dy q = + z z qz+ dz qz dz (6.3a) (6.3b) (6.3c) Substitutig equatios (6.9) to (6.) ito equatio (6.8), ad cosiderig equatio (6.3), yields T q q q = ρ t x y z (6.4) x y z Qdxdydz c dxdydz dx dy dz From equatio (6.), it ca be deducted that qx = kxdydz x T ; T qy = kydxdz y ; T qz = kzdxdz z (6.5) Substitutig (6.5) ito (6.4) ad makig some arragemets, leads to the geeral three-dimesioal heat coductio equatio i Cartesia coordiates, as follows

223 Thermo-mechaical model 89 T k T T T x + ky + kz + Q = ρc (6.6) x x y y z z t For the case of isotropic materials, the thermal coductivity is the same i all directios, i.e., kx = ky = kz = k Iitial ad boudary coditios To obtai the temperature distributio i a body, the heat coductio equatio must be solved cosiderig appropriated boudary coditios, ad, for the case of time depedet temperature pheomea, the iitial coditios must also be kow. The Dirichlet coditio, or boudary coditio of first order, is T = T o S T (6.7) where T is the presibed temperature i the boudary, ad S T is the boudary surface where the temperature is imposed. The Neuma coditios, or boudary coditios of secod order, are the followig Costat heat flux i the boudary: T q = k = q o S (6.8) q where q is the heat flux imposed o the boudary, S q is the boudary surface, ad is the directio vector ormal to the boudary surface, defied by its directio cosies. Therefore, equatio (6.8) ca be writte as

224 90 Chapter 6 T T T q = kx x + ky y + kz z = q x y z o S q (6.9) where x, y ad y are the directio cosies of the ormal to the S q boudary surface. Isulated or adiabatic boudary: T T T q = kx x + ky y + kz z = 0 x y z o S q (6.0) The covectio coditio o the boudary surface is T T T q = kx x + ky y + kz z = hc T T x y z ( ) o S c (6.) beig c h the covectio heat trasfer coefficiet ad ( T ) betwee the surface ad the fluid (e.g. the eviromet temperature). the temperature differece T The radiatio heat trasfer ca be take ito accout by substitutig the covectio heat trasfer coefficiet by a appropriate covectio-radiatio heat trasfer coefficiet, h, (Silveira 993, Azeha 009). For the case of a trasiet aalysis the iitial coditios must be kow a priori. Thus, for a specific time t 0 the temperatures, T, of the etire 3D body must be supplied. This coditio ca be defied by T = T iit (6.) beig T iit the iitial temperature imposed to the 3D body at time t 0.

225 Thermo-mechaical model Fiite elemet method applied to heat trasfer By applyig the method of weighted residuals to the heat coductio equatio (6.6) wrdv = 0 (6.3) V where the residual fuctio is defied by T r k T T = T x + ky + kz + Q ρc (6.4) x x y y z z t ad w is the weight fuctio wxyx, (,, ) yields T T T w k V x + ky + kz dv x x y y z z T + w QdV wρc dv = 0 V V t (6.5) By performig a itegratio by parts of the first term of equatio (6.5) usig the Gree-Gauss theorem (Ottose ad Petersso 99), results i V T T T w kx + ky + kz dv = x x y y z z w T w T w T kx ky kz dv V + + x x y y z z T T T + w kx x + ky y + kz z ds S x y z (6.6) Itroducig the boudary coditios desibed i sectio 6... i the secod itegral of (6.6) yields

226 9 Chapter 6 S T T T w kx x + ky y + kz z ds x y z ( ) ( ) ( ) = w q ds + w q ds + w h T T ds S T T T S q c c q S c (6.7) I this equatio q T is the ukow heat flux o the boudary where the temperature is presibed, q is the imposed boudary heat flux (whe q = 0 a isulated or adiabatic boudary is assumed), ad h ( T ) c is the boudary covectio heat flux (by T substitutig h c with h, covectio-radiatio heat trasfer ca be take ito accout). By cosiderig (6.6) ad (6.7), equatio (6.5) becomes w T w T w T T k V x + ky + kz dv + whtds S c c + wρc dv x x y y z z c V t = w QdV w q ds w q ds + wh T ds ( ) ( ) V S T T q c c T Sq S c (6.8) The first itegral of (6.8) ca be writte as V ( ) T w D T dv (6.9) where kx 0 0 D= 0 ky k z (6.30) The temperatures ca be approximated usig shape fuctios as follows T = T x, y, z, t = N x, y, z T t (6.3) ( ) ( ) ( ) i= i i

227 Thermo-mechaical model 93 where is the umber of odes of the elemet, Ni is the shape fuctio of elemet ode i ad T i is the temperature of elemet ode i. I a trasiet aalysis this temperature field is a fuctio of time. I matrix form equatio (6.3) ca be writte as T e = NT (6.3) beig [ ] N = N N N (6.33)... ad e [ ] T T = T T... T (6.34) By usig the Galerki method, the weight fuctios, w, are chose to coicide with the fuctios that defie the ukow variables (temperatures). I the preset case these fuctios are the shape fuctios, N. Thus, equatio (6.8) ca be writte for a specific fiite elemet, cosiderig (6.3) ad (6.9), as ( ) ( ) e ( NT ) T e T e T ( N ) D NT dv + N h V S cnt dsc + N ρc dv c V t = + T T T T N QdV N qt dst N q ds V S q N hct c T Sq S ds c (6.35) or ( ) K K T K T F (6.36) e e e e e e c + cov + t = where e K c is the elemet coductio matrix defied by

228 94 Chapter 6 e c ( ) T T V (6.37) V K = N D NdV = B DBdV beig B N N N... x x x N N N y y y N N N... z z z =... (6.38) e K cov is the elemet covectio matrix K e T cov hcn NdS S c c = (6.39) ad e K t is the elemet trasiet matrix K e t V T = ρcn NdV (6.40) The vector e F ca be divided ito e e e L q F = F + F (6.4) T where e F L is defied by e e e e L = Q + q + c F F F F (6.4) beig e F Q the vector correspodig to the elemet iteral heat geeratio,

229 Thermo-mechaical model 95 e T F Q = NQdV (6.43) V e F q is the vector correspodig to the boudary where the heat flux is imposed, e q T = ( ) q (6.44) F N q ds Sq ad e F c is the vector cotaiig the values correspodig to the covectio (or covectio-radiatio) boudary F e T c N ht S c dsc c = (6.45) I equatio (6.4) e qt F is the vector correspodig to the heat flux where the temperature is presibed, T = ( ) (6.46) e qt ST T T F N q ds By cosiderig a domai disetized ito several fiite elemets, equatio (6.36) is writte i a global form as ( ) K + K T+ K T = F (6.47) c cov t I this equatio the matrices ad vectors take ito accout the cotributio of each fiite elemet of the domai.

230 96 Chapter Steady-state liear aalysis I this sectio the implemetatio of the heat coductio problem i the FEMIX computer code, for the case of a steady-state liear aalysis, is preseted. The performace ad the accuracy of this implemetatio are appraised by performig a umerical simulatio. I a steady-state liear aalysis equatio (6.47) becomes ( ) K + K T = F (6.48) c cov By takig ito accout (6.4), ad cosiderig K = Kc + Kcov, results i KT = F L + Fq T (6.49) For its solutio equatio (6.49) is writte as K K T F FF FP F LF, KPF K PP T = P F + LP, F qt, P 0 (6.50) where the subsipt F (free) correspods to the odes of the domai where the temperature is ot kow, ad the subsipt P (presibed) correspods to the odes of the domai where the temperature is presibed. I a first phase the temperatures are obtaied by solvig the followig system of equatios, usig the Gauss elimiatio method K T = F K T (6.5) FF F L, F FP P ad i a secod phase, with the obtaied temperatures, temperature is presibed is calculated with T F, the heat flux where the

231 Thermo-mechaical model 97 F = K T + K T F (6.5) qt, P PF F PP P L, P Numerical example The performace of the model is appraised usig a example from Lewis et al. (004), as show i Figure 6.. z 500 ºC x y 00 ºC 00 ºC.0 m Isulated 00 ºC.0 m.0 m Figure 6. Steady-state example (adapted from Lewis et al. 004). For the umerical aalysis the domai is desitized with seredipity 8 ode solid elemets with a Gauss-Legedre itegratio scheme. The thermal coductivity of the material, k, is costat ad is equal to 0 Wm - ºC -. Figure 6.3 represets the mesh ad temperature field after the steady-state aalysis. The temperature i the ceter of the cube (0.5, 0.5, 0.5) is 00.37ºC. Similar results were obtaied by Lewis et al. (004).

232 98 Chapter 6 z x y Figure 6.3 Fiite elemet mesh ad temperature field Trasiet liear aalysis Time-disetizatio Figure 6.4 represets the temperature variatio with time for a oe dimesioal problem. I the iemetal time step, derivative of temperature is approximated with t, a liear variatio of the temperature is assumed. Thus, the T T T t t + = (6.53) T T T + temperature T T +θ T θ t t t +θ t + t t Figure 6.4 Time-disetizatio for oe dimesioal problem.

233 Thermo-mechaical model 99 Cosiderig a multi-dimesioal problem, the temperature at the time t + θ is calculated with T T T θ T θ t θt ( θ) T t + + = + = + + (6.54) ad its derivative with (see Figure 6.4) T T T T θ t t + θ + + θ = = T (6.55) Writig equatio (6.47) for the time t + θ results i ( ) K + K T + K T = F (6.56) c cov + θ t + θ + θ ad by substitutig (6.54) ad (6.55) yields T T K + K T + T + K = F t + ( ) θ ( θ ) c cov + t + θ (6.57) Cosiderig for vector temperature, equatio (6.57) ca be writte as F + θ the same type of approximatio that is adopted for the T T K c Kcov T+ T Kt F+ F t + ( ) ( ) + θ + θ + = θ + ( θ) (6.58) or

234 00 Chapter 6 ( ) + ( θ) ( ) θ ( θ) K + θ t K + K T t c cov = K t K + K T + t F + F t c cov + (6.59) By adoptig differet values for θ several time-steppig schemes ca be obtaied (Vila Real 988, Lewis et al. 996, Foseca 998, Lewis et al. 004), ad equatio (6.59) ca take the followig formats: Forward-Euler ( θ = 0) ( ) K tt+ = Kt t Kc + Kcov T + tf (6.60) Backward-Euler ( θ = ) ( ) + + K + t K + K T = K T + tf t c cov t (6.6) Crak-Nicolso θ = K + t K + K T = K t K + K T + t F + F ( ) ( ) ( ) t c cov + t c cov + (6.6) Galerki θ = 3 ( ) ( ) K t + t Kc + K cov T Kt t Kc K cov T t + F+ F 3 = (6.63) If θ 0 the time-steppig schemes are called implicit, ad if θ = 0 the time-steppig scheme is called explicit. It is demostrated i Lewis et al. (996) that for θ the

235 Thermo-mechaical model 0 solutio is ucoditioally stable, ad whe 0 θ < the solutio is coditioally stable ad the time step must be limited, such is the case of the Forward-Euler scheme. It is also cocluded that the Galerki time-steppig scheme leads to less oscillatory errors tha the Crak-Nicolso time-steppig scheme, although this last oe provides a higher accuracy (Ziekiewicz 000a) Computatioal strategies The implemetatio of the heat coductio problem i the FEMIX computer code for the case of the trasiet liear aalysis is preseted i this sectio. Cosiderig the domai disetized ito several fiite elemets ad writig (6.59) i order to group the odes where the temperature is ukow, subsipt F (free), ad i odes where the temperature is presibed, subsipt P (presibed), results i E T = F (6.64) beig E + ( ) θ ( ) ( ) θ ( ) Kt, FF + θ t Kc, FF + Kcov, FF Kt, FP + t Kc, FP + K cov, FP = Kt, PF + θ t Kc, PF + Kcov, PF Kt, PP + t Kc, PP + Kcov, PP (6.65) T + T +, F = T +, P (6.66) F + ( θ) ( ) ( θ) ( ) ( θ) ( ) ( θ) ( ) Kt, FF t Kc, FF + Kcov, FF Kt, FP t Kc, FP + K cov, FP = Kt, PF t Kc, PF + Kcov, PF Kt, PP t Kc, PP + Kcov, PP ( θ) ( θ) T F, tθfl, +, F+ t FLF,, T + tθf + t F P, L, +, P LP,, 0 + tθ Fq,, T + P + t θ ( ),, F q T P (6.67)

236 0 Chapter 6 For the curret time step, temperatures T +, F are obtaied by solvig the followig system of equatios usig the Gauss elimiatio method E T = F (6.68) +, FF +, F +, F E + where, FF is the free effective thermal trasiet matrix defied by ( ) E + = K + θ t K + K (6.69), FF t, FF c, FF cov, FF ad the F +, is the free load vector for the curret time step defied by F ( θ ) ( ) ( ) ( ) ( θ) ( θ) F = K t K + K T +, F t, FF c, FF cov, FF, F + K T T + K + K tt θ tt tfp, P, +, P cfp, covfp, P, +, P + tθf + t F L, +, F LF,, (6.70) For θ 0 the heat flux where the temperature is presibed is calculated with tpp, +, P P, ( θ ) c, PP cov, PP +, P, P ( θ) ( θ) ( θ ) Fq,,,,, (,, ) T + P = Kt PF T+ F T F + Kc PF + Kcov PF T+, F + T, F θ t θ t θ + K T T θ t θ t + ( K + K ) T + T θ + FL, +, P FLP,, + Fq T, P, θ θ (6.7)

237 Thermo-mechaical model Numerical examples To validate the implemeted trasiet heat coductio model two umerical simulatios have bee performed, beig the correspodig results preseted below. The first simulatio is a trasiet aalysis based o the problem show i Figure 6.. The same elemet mesh ad material properties are used. The iitial temperature is set to 0ºC ad a Crak-Nicolso scheme is used. The evolutio of temperature with time at the ceter poit (0.5, 0.5, 0.5) is plotted i Figure 6.5. It is verified that the temperature ieases rapidly util a steady-state is reached. The temperature field at differet times is represeted i Figure 6.6, i the yz plae (see Figure 6.) Temperature [ºC] Time [s] Figure 6.5 Temperature vs. time at poit (0.5, 0.5, 0.5). The secod example for model appraisal is a bi-dimesioal trasiet aalysis proposed by Zhou ad Vecchio (005). I the preset case a three-dimesioal represetatio of the domai is performed, beig represeted i Figure 6.7. The domai is disetized with 8-oded hexahedral elemets, ad a Gauss-Legedre itegratio scheme is used. The coductivity of the material is costat ad equal to 5 Jm - s - K -, the specific heat is Jkg - K -, ad the desity is 7850 kg/m 3. The iitial temperature is set to 0ºC ad a Crak-Nicolso scheme is used. The temperature field at differet istats is represeted i Figure 6.8.

238 04 Chapter 6 t = 0.00s t = 0.0s t = 0.05s t = 0.05s t = 0.s t =.0s Figure 6.6 Fiite elemet mesh ad temperature field for differet istats of the trasiet aalysis.

239 Thermo-mechaical model 05 z 00 ºC x y 0 ºC 0 ºC.0 m Isulated 0 ºC.0 m.0 m Figure 6.7 Trasiet aalysis domai ad temperature distributio. Figure 6.9 represets the evolutio of temperature through the depth of the oss sectio at the ceter lie of the cube. It is verified that as time evolves the curves gradually ted to the steady-state respose. Similar results were obtaied by Zhou ad Vecchio (005) for the bi-dimesioal aalysis of the problem, ad a compariso for differet istats of the trasiet aalysis is represeted i Figure 6.0. The differeces betwee the two types of aalyses deease with time, but eve for the first istats the disepacy is quite acceptable takig ito accout that the aalyses have bee performed assumig a D ad a 3D disetizatio of the body.

240 06 Chapter 6 z x y t = 50s t = 00s t = 500s t = 000s t = 5000s t = 50000s Figure 6.8 Fiite elemet mesh of the domai show i Figure 6.7 ad temperature field for differet istats of the trasiet aalysis.

241 Thermo-mechaical model 07 Depth from the bottom [m] Temperature [ºC] Steady-State Trasiet - 50s Trasiet - 00s Trasiet - 500s Trasiet - 000s Trasiet - 000s Trasiet s Trasiet s Trasiet s Trasiet s Trasiet s Trasiet s Figure 6.9 Temperature variatio through the depth of the oss sectio at the ceter lie Depth from the bottom [m] Steady-State ZhouVecchio - Steady-State ZhouVecchio - 00s ZhouVecchio - 500s ZhouVecchio - 000s ZhouVecchio - 000s ZhouVecchio s Trasiet - 00s Trasiet - 500s Trasiet - 000s Trasiet - 000s Trasiet s Temperature [ºC] Figure 6.0 Temperature variatio through the depth of the oss sectio at the ceter lie compariso with the data from Zhou ad Vecchio (005).

242 08 Chapter Trasiet oliear aalysis I the presece of early age heat developmet, the heat geeratio rate of cemet based materials ca be obtaied with the mathematical formulatio proposed by Reihardt et al. (98) ad based o the Arrheius type relatio, beig defied by equatio (6.7) (Azeha 009). E a R ( T ) ( α ) Q = f A e (6.7) T T I this equatio Q is the heat geeratio rate to be itroduced i equatio (6.43), f ( αt ) is the ormalized heat geeratio rate directly obtaied through experimets (de Borst ad va de Boogaard 984, Azeha 009), A T is a rate costat, E a (Jmol - ) is the apparet activatio eergy that depeds o the type of cemet, R is the uiversal gas costat (8.34 Jmol - K - ), ad T is the temperature i ºC. I this case the secod member of equatio (6.64) depeds o the temperature ad, for that reaso, a iterative process is required at each time step to solve the oliear system of equatios. The Newto-Raphso method is used for this iterative process. For the curret time step +, the equatio of the ubalaced heat fluxes ca be defied by ( ) Ψ T = F E T (6.73) beig F + ad E + defied by equatio (6.67) ad equatio (6.65), respectively. For the curret time step T + +, it is iteded that the vector ( ) Ψ is ull, i.e., ( ) Ψ = (6.74) T + 0

243 Thermo-mechaical model 09 Equatio (6.74) ca be solved by applyig the Newto-Raphso method. Cosiderig the first two terms of the Taylor series expasio, equatio (6.74) ca be approximated as q q q Ψ q + + δ + ( ) ( ) Ψ T Ψ T + T = 0 T + (6.75) where the subsipt q is the iteratio couter. I equatio (6.75) ( F ET) q q Ψ = = T T + + ( E ) T q + (6.76) is the Jacobia matrix. For simplificatio, the term q q + T + F is dropped i the preset formulatio, beig the Jacobia matrix equal to the effective tagetial matrix, q iteratio of the curret time step, +. Substitutig (6.76) ito (6.75) yields E T of the q q q ( ET) δt + ( T + ) + =Ψ (6.77) A iterative procedure is executed up to the solutio of equatio (6.74), ad i each iteratio the vector of the temperatures is updated as follows q q q q + + δ + + T = T + T = T + T (6.78) with q q i q q + δ + + δ + i= T = T = T + T (6.79) beig T 0 + = T ad T + = at the begiig of the iterative process. 0 0

244 0 Chapter 6 The ormalized heat geeratio rate f ( α T ) is obtaied directly from experimets, ad is a fuctio of the degree of heat developmet α T. This parameter desibes the relative amout of heat geeratio due to the cemet hydratio (Ferreira et al. 008, Azeha 009) () Qt α T = (6.80) Q total where Qt ( ) is the accumulated heat geerated util a certai istat t, ad Q total is the fial accumulated heat of the cemet (or bider) hydratio. A iitial value for the degree of heat developmet α Tiit, is ecessary to umerically activate the oliear trasiet aalysis due to early age heat developmet Numerical examples A experimetal program has bee executed by Azeha (009) with a coete cube of 40 cm edge legth. The cube was moitored with temperature sesors placed at several poits withi the cube. A thermography camera has also bee used to obtai the temperature field (thermography pictures) durig the hardeig of coete. Figure 6. represets schematically the geometry ad the boudary coditios of the cube. Oly oe face is i cotact with the eviromet, while i the others a woode formwork separates the coete from the eviromet. The value of the heat trasfer coefficiet assiged to the face that is i cotact with the eviromet is 0.0 Wm - K -, ad for the other faces a equivalet heat trasfer coefficiet, h eq, is used to accout for the woode formwork. The value is obtaied with the electrical aalogy (Holma 986, Iopera et al. 006, Azeha 004) h eq L i (6.8) i= ki = + h

245 Thermo-mechaical model beig L i ad k i the thickess ad the thermal coductivity, respectively, of the i th layer of the material located betwee the coete ad the eviromet. I the preset case, the woode formwork has a thickess of.85 cm ad a coductivity of 0. Wm - K -, ad a value of 5. Wm - K - for h eq is obtaied. The woode formwork of the lateral faces is removed 8.6 h after castig, ad the correspodig faces of the coete cube are made i cotact with the eviromet, beig a value of 0.0 Wm - K - assiged to the heat trasfer coefficiet. z T =0 ºC amb 40.0 cm y x 40.0 cm 40.0 cm Woode formwork - - h eq = 5. Wm K Coete - - h = 0.0 Wm K T =6 ºC iit Figure 6. Geometry ad the boudary coditios. I the umerical aalysis the domai is disetized with 8-oded hexahedral elemets with a Gauss-Legedre itegratio scheme. The material coductivity, k, is costat ad equal to.6 Wm - K -, the volumetric heat capacity, ρ c, is kjm -3 K - ad the iitial temperature is set to 6 ºC. The ambiet temperature is 0 ºC ad a Backward-Euler time itegratio scheme is used with a iemetal time step of 864s. The values that characterize the heat geeratio rate defied by equatio (6.7) are depedet o the type of cemet used i the experimet. A type I 5.5R cemet cotet of

246 Chapter 6 A =, E = 47.5 kj mol 3 430kg m has bee used (Azeha 009), with T.053E ad 3 Q total = kj m. The ormalized heat geeratio rate ( T ) a f α used i the aalysis is represeted i Figure 6., ad a value of 0.05 is cosidered for α Tiit,. f (α T ) α T Figure 6. Normalized heat geeratio rate. α f ( α T ) T As metioed before, the cube is moitored by temperature sesors placed at several poits. Azeha (009) has compared the experimetal results with umerical simulatios ad a good agreemet was achieved. To assess the predictive performace of the developed thermal trasiet oliear model, the temperature due to heat geeratio durig the first 4 h is compared with the umerical simulatios of Azeha (009) at two poits (see Figure 6.3). Oe poit (TP) is located at the ceter of the top surface of the cube (0., 0., 0.4) ad the other (TP9) is located at the ceter of the lateral surface (0., 0.4, 0.). From Figure 6.3 it ca be cocluded that the preset model matches perfectly the umerical simulatios performed by Azeha (009) at the predefied poits. This figure also shows that, due to the presece of the lateral woode formwork, a greater temperature gradiet due to the heat geeratio has occurred at the poit located o the face that cotacts directly with the eviromet. Whe the lateral formwork is removed (8.6h after castig) a iflectio poit is observed i the TP9_ThermalModel curve, ad its temperature curve teds to match the results of the TP_ThermalModel. Both curves are almost coicidet after a period of.0h.

247 Thermo-mechaical model 3 Temperature [ºC] TP_ThermalModel TP_Azeha009 TP9_ThermalModel TP9_Azeha Time [days] Figure 6.3 Temperature evolutio at two poits of the cube durig the first day. Figure 6.4 represets the temperature field at two differet times of the trasiet aalysis. Oe is the temperature field at the first time step (t=0.4h) ad the other is the temperature field at the last time step (t=4h). Some coclusios ca be draw by observig these temperature fields. At t=0.4h, oly the top face of the cube is i direct cotact with the eviromet, ad it ca be observed that the temperature gradiet is smaller compared with the gradiet of the faces that cotact the woode formwork. The observatio of the temperature field at time t=4h leads to the coclusio that the lateral faces ted to exhibit a temperature distributio that is similar to the oe observed o the top surface. However, the bottom face has a sigificatly differet temperature field, with a greater temperature gradiet. This is justified by the presece of the woode formwork o this face durig the whole test. Aother example for assessig the predictive performace of the implemeted model is a coete wall that is cast o a hardeed coete foudatio. The aalysis of this structure assumig plae strai coditios was aalyzed by Lura ad Breugel (00). I the preset work a three-dimesioal represetatio of the domai is performed, assumig for the wall a legth of 8 m, as show i Figure 6.5. The domai is disetized with 0-oded hexahedral fiite elemets (see Figure 6.7), ad a Gauss-Legedre itegratio

248 4 Chapter 6 scheme is used. The coductivity of the material is costat ad equal to.6 Wm - K -, the volumetric heat capacity, ρ c, is kjm -3 K - ad the iitial temperature is set to 0 ºC. The ambiet temperature is also 0 ºC ad a Backward-Euler time itegratio scheme is used with a iemetal time step of 800s, beig the total time of the aalysis 0 h. t = 0.4h x z y x z y t = 4h Figure 6.4 Fiite elemet mesh of the body represeted i Figure 6., ad temperature field for the first ad last time step of the trasiet aalysis.

249 Thermo-mechaical model 5 Figure 6.5 represets schematically the geometry ad the boudary coditios of the wall ad the foudatio. The top face of the wall is i cotact with the eviromet, while a woode formwork separates the others coete surfaces from the eviromet util a curig period of 7h. After this time the woode formwork is removed. A equivalet heat trasfer coefficiet, h eq, is used to accout for the woode formwork, which is obtaied with the electrical aalogy by usig equatio (6.8). 0.4 m z T =0 ºC amb y x Top Woode formwork - - h eq = 3. Wm K 3.0 m Ceter 8.0 m Hardeed coete - - h = 5.0 Wm K.0 m Bottom Fresh coete h = 5.0 Wm K m T =0 ºC iit Figure 6.5 Geometry ad the boudary coditios. As stated before, the values that characterize the heat geeratio rate defied by equatio (6.7) are depedet o the type of cemet used i the experimet. By the aalysis of the 3 type of cemet used i Lura ad Breugel (00), cemet cotet ( 400kg m ) ad activatio eergy (45.7 kj mol ), the followig data was derived, takig for this purpose iformatio available i (Azeha 009): a type I 5.5R cemet with A T = 7.400E , E = 46.8 kj mol ad a 3 Q total = kj m. The ormalized heat geeratio rate

250 6 Chapter 6 f ( α T ) used i the aalysis is represeted i Figure 6.6, ad a value of 0.05 is cosidered for α Tiit,. f (α T ) α T α f ( α T ) T Figure 6.6 Normalized heat geeratio rate. Oly oe quarter of the wall-foudatio system is used i the aalysis, due to double symmetry of the problem. The temperature field for differet time steps of the trasiet aalysis is represeted i Figure 6.7. It ca be observed that the temperature is higher i the iterior of the wall ad teds, with time, of attaiig the ambiet temperature. Due to the high covectio heat trasfer i the exteral surfaces, the temperature field deeases from the core of the wall to these lateral surfaces, beig the lowest temperature registered i the corers betwee the top ad the frot-lateral surface. This is more proouced after a time of 7h, whe the woode formwork is removed. Figure 6.8 represets the temperature evolutio at three poits of frot edge of the wall (see Figure 6.5). It ca be observed that the highest temperature developmet is i the ceter poit ad the lowest is at the bottom poit i cotact with the foudatio. At a time of 7 h a smooth iflectio poit is observed i the curves due the removal of the woode formwork. Similar results were obtaied by Lura ad Breugel (00) ad are preseted i Figure 6.9. The mai differece is the time correspodig to the peak temperature. By the observatio of the adiabatic curve of hydratio ad heat release preseted i Lura ad Breugel (00), the heat release up to 8h is abormally very low.

251 Thermo-mechaical model 7 x z y t = 4h z x y t = 7h t = 0h Figure 6.7 Fiite elemet mesh of the body represeted i Figure 6.5, ad temperature field for differet time steps of the trasiet aalysis.

252 8 Chapter 6 60 Temperature [ºC] Bottom Ceter Top Time [h] Figure 6.8 Temperature evolutio at three poits of edge of the wall. T [ o C] Bottom 45 Ceter 40 Top Age [hrs] Figure 6.9 Temperature evolutio at three poits of edge of the wall (Lura ad va Breugel 00).

253 Thermo-mechaical model TIME-DEPENDENT DEFORMATIONS Usig the cocept of strai decompositio, the total strai at time t, ε () t, of a coete member uiaxially loaded at time t 0 with costat stress σ ( t0) ca be expressed by (Póvoas 99, CEB-FIP 993, Hofstetter ad Mag 995, Heriques 998) i c s T () t ( t ) () t () t ( t) ε = ε + ε + ε + ε (6.8) 0 i c where ε ( t ) is the iitial strai at loadig, ( t) 0 T is the shrikage strai ad ε () t is the thermal strai. ε is the eep strai at time t t0 s >, ε ( t) i The strais ( t ) c ε ad ε () t are caused by applied stresses, beig thus called mechaical 0 s T strais, while the other two compoets ε ( t) ad ( t) field (Bazat 988). ε are idepedet from the stress T The thermal strai ε () t ca be obtaied from the temperature field at a certai istat, e.g., usig the results of the thermal model desibed i sectio 6. ad performig the followig calculatio T ε () t = α T (6.83) beig α the coefficiet of thermal expasio ad T the temperature variatio. s The procedures required to evaluate the shrikage, ( t) c ε, ad eep, ε () t, strais are desibed i the followig sectios, which are based o the approaches proposed by the Eurocode (EC 004), ad by Bazat ad Baweja 000 (B3 model).

254 0 Chapter Shrikage Eurocode The total shrikage ca be calculated with () t () t ( t, t ) s ε εca εcd s = + (6.84) where ε () t is the autogeous shrikage strai at time t defied by equatio (6.9), ad cd ca ( tt, ) ε is the dryig shrikage strai at time t, which ca be determied with the s followig equatio ( tt, ) ( tt, ) k,0 ε = β ε (6.85) cd s ds s h cd where β ds ( tt, ) s = ( t ts ) ( t t ) + 0,04 3 h s 0 (6.86) ad k h is a coefficiet that depeds o the otioal size, h 0, ad takes the values accordig to Table 6.. Table 6. Values for k h. h 0 [mm] k h I equatio (6.85) ε cd,0 is the otioal dryig shrikage coefficiet defied by

255 Thermo-mechaical model fcm 6 ε = 0.85 ( 0 + 0α ) exp α 0 β 0 cd,0 ds ds RH (6.87) where α ds ad α ds are coefficiets that deped o the type of cemet α α ds ds 3 = = for cemet Class S for cemet Class N for cemet Class R for cemet Class S for cemet Class N for cemet Class R (6.88) (6.89) ad β RH is the coefficiet that itroduces the effect of the relative humidity o dryig β RH 3 RH = (6.90) beig RH the ambiet relative humidity (%). I the above equatios, t is the age of coete i days whe the pheomeo is evaluated, t s is the age of coete i days at the begiig of the dryig shrikage, ad i equatio (6.86) h 0 is the otioal size of the oss sectio, i mm, defied by h 0 A c = (6.9) u where A c is the coete oss sectioal area, i mm, ad u is the perimeter of the part of the oss sectio which is exposed to dryig, i mm.

256 Chapter 6 I equatio (6.87) f cm is the mea compressive stregth of coete at the age of 8 days i MPa. The autogeous shrikage strai, ε ( t) ca, is defied by () t () t ( ) ε = β ε (6.9) ca as ca where β () t is a fuctio defied by as 0.5 () exp( 0. ) β as t = t (6.93) ad ε ca 6 ( ).5( f 0) 0 = (6.94) ck where f ck is the characteristic compressive stregth of coete at the age of 8 days i MPa. Its value ca be estimated with f ck = f 8 (6.95) cm Model B3 Accordig to this model the total shrikage strai is calculated with () t () t ( t, t ) s ε εa εsh s = + (6.96) where ( t) sh ( tt, ) s ε is the autogeous shrikage strai obtaied with equatio (6.0), ad a ε is the dryig shrikage strai defied by

257 Thermo-mechaical model 3 ε ( tt, ) ks( t) sh s εsh h = (6.97) where S( t ) represets the time-depedece t ts S() t = tah (6.98) τ sh k h itroduces the humidity depedece 3 h for h 0.98 kh = 0. for h =.0 ( swellig i water) liear iterpolatio for 0.98 < h <.0 (6.99) ad ε sh cosiders the time-depedece of ultimate shrikage ε sh = ε s ( 607) ( + τ ) E E t s sh (6.00) beig t = t () E8 E t (6.0) where E 8 is the give by E8 = 4734 fcm (6.0)

258 4 Chapter 6 I the above equatios, t, t s ad f cm have the meaig already preseted i the previous sectio, ad h is the ambiet relative humidity, expressed as a decimal umber i the rage 0 h. I equatios (6.98) ad (6.00) τ sh is the shrikage half-time i days give by sh t s ( k D) τ = k (6.03) where k t is a factor defied by k = 8.5t f (6.04) t cm The parameter k s is the oss sectio shape factor, calculated accordig to k s.00.5 = for a ifiite slab for a ifiite cylider for a ifiite square prism for a sphere for a cube (6.05) ad D represets the effective oss sectio thickess, obtaied with V D = (6.06) S beig V S the volume to surface ratio i cm. I equatio (6.00) ε s is calculated with

259 Thermo-mechaical model s f cm 70 0 = + ε αα ω (6.07) where ω is the water cotet of the coete mix, i kg/m -3. The parameters α ad α are defied by.00 α = for type I cemet for type II cemet for type III cemet (6.08) ad α =.00 for steam-curig for sealed or ormal curig i air with iitial protectio agaist dryig for curig i water or at 00% relative humidity (6.09) The autogeous shrikage strai, ε ( t) a, is defied by () t ( 0.99 h ) S () t εa = εa a a (6.0) beig ha the fial self-desiccatio humidity, that ca be assumed to be about 80% (Bazat ad Baweja 000), ε a the fial autogeous shrikage strai ad S a t tset () t = tah (6.) τ a I this equatio t set is the time of fial set of cemet i days, ad τ a is the half-time of autogeous shrikage, depedig o the rate of hardeig of the type of coete.

260 6 Chapter 6 Accordig to Bazat ad Baweja (000) the material parameters of equatios (6.0) ad (6.) may be calibrated by performig shrikage measuremets i sealed specimes (autogeous shrikage) ad dryig specimes (total shrikage) Creep Eurocode c The eep strai at time t, ε () t, for a costat stress, σ c, applied at a coete age t 0 is give by c ε () t ϕ( t, t ) σ c = 0 (6.) Ec where E c is the taget modulus of coete, that ca be cosidered equal to.05e cm, beig E cm the secat modulus of elasticity of coete at a age of 8 days. I Equatio (6.) the eep coefficiet ( tt, ) ϕ is defied by 0 ϕ ( tt, ) ϕ β ( tt, ) = (6.3) 0 0 c 0 where 0 ϕ is the otioal eep coefficiet determied from equatio (6.4); ( tt, ) β is the coefficiet that desibes the time depedet evolutio of eep after loadig, defied by equatio (6.6), beig t the age, i days, whe the eep is evaluated, ad t 0 the age of coete at loadig i days. c 0 ( f ) ( t ) ϕ = ϕ β β (6.4) 0 RH cm 0 where

261 Thermo-mechaical model 7 ϕ RH RH 00 = + for f cm 35 MPa (6.5a) 0.3 h 0 RH 00 ϕrh = + α α 0.3 h0 for f cm > 35 MPa (6.5b) 6.8 β ( fcm ) = (6.5c) f cm β ( t ) = 0.+ ( t ) (6.5d) beig RH the relative humidity of the ambiet eviromet, i %, h 0 is defied by equatio (6.9) ad f cm is the mea compressive stregth of coete at the age of 8 days i MPa. The term ( tt, ) β i equatio (6.3) is determied with c 0 β c ( tt, ) 0 ( t t ) 0 = βh + t t0 0.3 (6.6) beig ( ) 8 0 βh = RH h ( ) βh = RH h + 50α 500α for f cm 35 MPa for f cm > 35 MPa (6.7a) (6.7b) The coefficiets α, α ad α 3 i equatios (6.5) ad (6.7) deped o the mea compressive stregth of coete, ad are defied by 35 α = f cm α = f cm α3 = f cm 0.5 (6.8)

262 8 Chapter 6 The effects of temperature durig the curig phase ad type of cemet o the eep coefficiet of coete may be take ito accout by modifyig the value of the parameter that cosiders the age at loadig t 0 9 t0 = t 0, T + 0.5days. + ( t0, T ) α (6.9) where α is a parameter that depeds o the type of cemet α = 0 for cemet Class S for cemet Class N for cemet Class R (6.0) ad t 0,T is the age of coete at loadig, i days, adjusted accordig to the followig equatio t 0, T T( ti ) + e t (6.) i i= = beig ( ) T t i the temperature i ºC durig the time period ti, i days. Whe the compressive stress of coete at age t 0 exceeds the value 0.45 fck ( t 0 ) the oliearity of eep may be take ito accout by cosiderig (, ) = (, ) exp.5( 0.45) ϕl tt0 ϕ tt0 k σ (6.)

263 Thermo-mechaical model 9 where ϕ ( tt, ) is the oliear otioal eep coefficiet ad replaces ( tt, ) l 0 0 ϕ ad k = σ f ( t ), which is the stress-stregth ratio, beig f ( t 0) the characteristic σ c ck compressive stregth of coete at time of loadig. ck Model B3 The compliace fuctio (, ) J t t or eep fuctio, represets the strai (elastic plus 0 eep) at time t, caused by a uit costat stress applied at age t 0, ad is give by (, ) (, ) (,, ) J t t = q + C t t + C t t t (6.3) d 0 s where q is the istataeous strai due to uit stress q = (6.4) E 8 beig 8 E obtaied from equatio (6.0). The parameter (, ) C t t is the compliace 0 0 fuctio for basic eep, i.e., eep at costat moisture cotet ad o moisture movemet through the material (Bazat ad Baweja 000), obtaied with t C0( t, t0) = qq( t, t0) + q3l + ( t t0) + q4l t0 (6.5) where = ; q ( ) w c q q 85.4c f cm = ; q ( ) = 0.3 a c (6.6) ad

264 30 Chapter 6 Qf ( t ) 0 Qtt (, 0) = Qf ( t0) + Z( t, t0 ) rt0 0 ( ) rt ( ) (6.7) i which ( ) ( t ) 0. r t = (6.8a) 0 0 (, ) = ( ) m l + ( ) Z t t t t t ( ) = 0.086( ) +.( ) 9 49 Qf t t t (6.8b) (6.8c) I equatio (6.3) (,, ) by C t t t is the additioal compliace fuctio due to dryig give d 0 s ( ) ( ) { } ' { } ( ) Cd t, t, t = q exp 8H t exp 8H t 0 s 5 0 (6.9) where ' 0 ad ' t is the time at which dryig ad loadig first act simultaeously, t max ( t t ) 0 0, s =, () ( h) S() t H t = (6.30) beig S( t ) give by equatio (6.98). The meaig of the parameter h is desibed i sectio 6.3.., ad t s is the age of coete i days whe dryig begis. The value of q 5 i equatio (6.9) is give by q5 = fcm ε sh (6.3)

265 Thermo-mechaical model 3 where ε sh is obtaied with equatio (6.00). I the above equatios =.0, m = 0.5, c is the cemet cotet, ( wc ) the water to cemet ratio, ( ac ) is the aggregate to cemet ratio ad f cm is the mea compressive stregth of coete at the age of 8 days i MPa. Accordig to (Bazat ad Baweja 000) the eep coefficiet should be calculated from the compliace fuctio as follows ( tt) E( t) J( tt) ϕ, =, (6.3) where E( t 0 ) is the modulus of elasticity at loadig age t 0, beig obtaied with equatio (6.0). 6.4 CONCRETE MATURITY The coete mechaical properties ieases sigificatly with time ad cosequetly these chages must be take ito accout. The maturity of coete is a cosequece of the hydratio process of the cemet paste ad its evolutio is strogly affected by temperature, curig coditios ad type of cemet (Heriques 998). The recommedatios of Eurocode (EC 004) to simulate the evolutio of the compressive stregth, tesile stregth ad modulus of elasticity are the followig () β () f t = t f (6.33a) cm cc cm () β () α f t = t f (6.33b) ctm cc ctm () β () 0.3 E t = t E (6.33c) cm cc cm

266 3 Chapter 6 where f cm, f ctm ad E cm are the mea compressive stregth, the mea value of the tesile stregth, ad the secat modulus of elasticity of coete at a age of 8 days. I equatio (6.33b) α is a parameter whose value depeds o the cosidered age t for t < 8 days α = for t 8 day 3 (6.34) I equatio (6.33) ( t) β is determied from cc 8 βcc () t = exp s t (6.35) beig 0.38 s = for cemet Class S for cemet Class N for cemet Class R (6.36) To defie the tesile diagrams desibed i sectio of chapter 3, the mode I fracture eergy must be supplied. A experimetal program is ecessary to obtai data to propose a equatio that defies the evolutio of the fracture eergy with time. Sice this experimetal work is out of the scope of this thesis, a relatio is proposed by usig the coefficiet of equatio (6.35), which depeds o the age of the coete, t () β () α G t = t G (6.37) I I f cc f

267 Thermo-mechaical model 33 where I G f is the mode I fracture eergy of coete at a age of 8 days, ad α is a value that defies the evolutio of ( t) β While o more reliable iformatio is available, the cc values idicated i equatio (6.34) will be used i the preset work. If the mea temperature differs from the referece temperature, 0 ºC, the cocept of equivalet age is commoly used. This cocept ca be defied as the age at which the hydratio at the referece temperature has reached the same stage (Bosjak 000, Azeha 009), ad ca be determied by teq t 0 E a R 73.5 T() t 73.5 T + + ref = e dt (6.38) or i a iemetal form t eq R 73.5 T( ti) 73.5 T + + ref e t (6.39) i i= E a = beig E a (Jmol - ) the apparet activatio eergy that depeds o the type of cemet, R the ideal gas costat (8.34 Jmol - K - ), ref temperature i ºC durig the time period T the referece temperature (0 ºC) ad ( ) ti i days. T t i the Equatio (6.39) is very similar to equatio (6.) proposed by EC (004) for the age adjustmet. Makig some calculatios, the equatio of EC (004) assumes a apparet activatio eergy of about 3356 Jmol - for a referece temperature of 0 ºC. These two approaches are available to the equivalet age calculatio. The equivalet age is itroduced i the above equatios for the evaluatio of the mechaical properties of coete.

268 34 Chapter UPDATE OF THE MULTI-FIXED SMEARED 3D CRACK MODEL TO TAKE INTO ACCOUNT THE TIME DEPENDENT EFFECTS I chapter 5 a multi-fixed smeared 3D ack model, uder the oliear FEM framework, is proposed to simulate the behavior of coete structures. I this sectio this model is adapted to iclude the time depedet effects. I this smeared ack model the strai compoets of the acked material is the additio of the strai compoets i the smeared acks, ε uacked coete betwee acks, co ε, with the strai compoets i the co = + (6.40) ε ε ε Takig ito accout the time depedet effects desibed i sectio 6.3, the strai vector of the uacked coete is decomposed i order to iclude these effects. So, equatio (6.8) is adapted ad results i e c s T () t ( t ) () t () t ( t) ( t) ε = ε + ε + ε + ε + ε (6.4) 0 where e ε, c ε, s ε ad, T ε are the elastic, eep, shrikage ad thermal strai vectors, ad ε is the ack strai vector. A oliear trasiet aalysis must be performed, sice the total strai is time depedet, beig its compoets evaluated durig the time. If the maximum compressive stress is less tha 0.4 of the compressive stregth of the coete, the mechaical strai for a uiaxial loaded coete specime ca be obtaied with () t ( t ) () t J( t, t ) ( t ) ε = ε + ε = σ (6.4) m e c 0 0 0

269 Thermo-mechaical model 35 where (, ) J t t is the compliace fuctio or eep fuctio, represetig the strai, elastic 0 plus eep, at time t caused by a uit costat uiaxial stress actig sice time t 0. Sice the elastic strai is defied by ( t ) σ ( t0 ) ( ) e ε 0 = (6.43) E t0 where ( ) give by E t is the modulus of elasticity at the time t 0, the compliace fuctio ca be c 0 J( t, t ) = + C( t, t ) (6.44) ( ) 0 0 E t0 beig (, ) C t t the specific eep. Therefore, the eep strai ca be obtaied with 0 c ε ( t) C( tt, ) σ ( t) = = J( tt, ) σ t ( ) ( ) E t0 (6.45) or itroducig the eep coefficiet ( tt, ) E( t) C( tt, ) determied from ϕ =, the eep strai ca be c ε () t σ ( t ) ( tt, 0 ) ( ) ϕ = 0 (6.46) E t 0 Itroducig equatio (6.44) i equatio (6.4) the mechaical strai is give by ε m () t = + C t t t E( t0 ) (, ) σ ( ) 0 0 (6.47)

270 36 Chapter 6 Assumig that the strai caused by a geeric stress history σ ( t) ca be determied by the decompositio of the stress history ito small iemets dσ ( t ) applied at times t (Bazat 988), the mechaical strai is t () (, ) σ ( ) ε m t = J t t d t (6.48) 0 By performig ( t ) dσ dσ ( t ) = dt = σ ( t ) dt (6.49) dt equatio (6.48) ca be rewritte as t () (, ) σ ( ) ε m t = J t t t dt (6.50) 0 or, for a three-dimesioal stress state, t () (, ) σ ( ) ε m t = J t t C t dt (6.5) 0 co where ( ) C = E D, beig co D defied by equatio (5.) of chapter 5. For a iemetal time step t, the iemetal mechaical strai vector is obtaied with ( t ) ( t ) ( t ) m m m + ε = ε ε (6.5)

271 Thermo-mechaical model 37 m m beig ε ( + ) ad ( ) respectively. t ε the mechaical strai vector at the time t + ad t, t Usig equatio (6.5) i (6.5) the mechaical strai iemet vector is give by t t+ t ( ) (, ) ( ) (, ) ( ) (, ) ( ) m ε t = J t t C σ t dt + + J t t C + σ t dt J t t C σ t dt (6.53) 0 t 0 or t ( ) (, ) (, ) ( ) (, ) ( ) m ε t = J t t + J t t C σ t dt + J t+ t C σ t dt (6.54) t+ 0 t Multiplyig this equatio by the matrix C, ad takig ito accout thatc = D, equatio (6.54) becomes t t+ m co = ( ) (, ) (, ) ( ) (, ) ( ) D ε t J t t J t t σ t dt J t t σ t dt (6.55) 0 t where D = E D. co Cosiderig that i the iemetal time step, assumed, its derivative is approximated by t, a liear variatio of the stress is ( t ) ( t ) ( t ) ( t ) σ σ + σ σ σ ( t ) = (6.56) t t t t + Substitutig this relatio i the secod itegral of equatio (6.55) ad makig some arragemets, results i

272 38 Chapter 6 ( t ) σ = t ( ) (, ) (, ) σ ( ) m D ε t J t t + J t t t dt 0 t t+ t (, ) J t t dt + (6.57) This equatio ca be writte as (de Borst ad va de Boogaard 984) * m ( ) ( ) σ t = E( t ) D ε t + σ( t ) (6.58) co where * ( ) Et = t t+ t (, ) J t t dt + (6.59) ad t * ( ) = ( ) ( ) ( ) σ t,, E t J t t + J t t σdt (6.60) 0 with * t+ t t (usig a geeralized midpoit rule) To overcome the icoveiet of storig the etire load history, the use of Dirichlet series is commoly used, so the approximatio of the compliace fuctio is made by a series of real expoetials (Bazat ad Wu 973, Bazat 988) ( t t ) 0 N τ J( t, t ) = + e µ E( t ) (6.6) µ = Eµ ( t )

273 Thermo-mechaical model 39 where τ µ are costats desigated by retardatio times, E µ are coefficiets oly depedig o t that have the dimesios of a elastic moduli, ad E( t ) is the istataeous elastic modulus. The process for the determiatio of the retardatios times, τ µ, ad the coefficiets E µ ca be foud elsewhere (Bazat ad Wu 973, Bazat 988, Póvoas 99). Dirichlet series expasio of the compliace fuctio, as preseted i equatio (6.6), ca be iterpreted as a Kelvi chai of N uits (Bazat ad Wu 973). The iemetal mechaical strai vector, m ε equatio (6.4), rewritte i a iemetal form for the a time step, ca be related to the other compoets of t ( t ) ( t ) ( t ) ( t ) ( t ) m s T ε = ε ε ε ε (6.6) Itroducig this equatio i equatio (6.58) results i * s T ( ) ( ) ( ) ( ) ( ) σ t = E( t ) D ε t ε t ε t ε t + σ( t) (6.63) Cosiderig equatios (5.5), (5.0) ad (5.3) of chapter 5, writte for the time t, results i ( t ) T σ ( t ) σ (6.64a) = T ( t ) T ε ( t ) (6.64b) ε = ( t ) D ε ( t ) σ (6.64c) = Itroducig equatio (6.63) i the secod member of equatio (6.64a), results i

274 40 Chapter 6 ( ( ) ( ) ( ) ) * s T ( ) ( ) ( ) ( ) ( ) σ t = T E t D ε t ε t + ε t + ε t + σ t (6.65) Itroducig equatios (6.64b) ad (6.64c) i equatio (6.65), ad makig some arragemets, yields ε = + * T ( t ) ( D T E( t ) D T ) * s T ( T E( t ) D ε( t) ε ( t) ε ( t) + T σ( t) ) (6.66) By icludig equatio (6.66) i (6.64b), it ca be obtaied T T * ( t ) T ( D T E( t ) D T ) ε = + * s T ( T E( t ) D ε( t) ε ( t) ε ( t) + T σ( t) ) (6.67) ad the itroducig equatio (6.67) i equatio (6.63) ad makig some operatios, the iemetal stress vector is obtaied with σ t = I E t D T D + T E t D T T T T * * ( ) ( ) ( ( ) ) * s T ( Et ( ) D( ε( t) ε ( t) ε ( t) ) + σ( t) ) (6.68) Equatio (6.68) replaces equatio (5.0) i the multi-fixed 3D ack model preseted i chapter 5. The iemetal shrikage strai is obtaied by oe of the models desibed i sectio 6.3, ad the iemetal thermal strai is determied from the temperature field, e.g., usig the results of the thermal model desibed i sectio 6.. For the three-dimesioal case, these vectors are defied by

275 Thermo-mechaical model 4 ( ) T s s s s ε t = ε ε ε (6.69) ( ) T T T T T ε t = ε ε ε (6.70) ad are obtaied with s s s T T T ε ( t ) = ε ( t ) ε ( t ) ; ε ( t ) ε ( t ) ε ( t ) + = (6.7) Numerical example The performace of the model is appraised by performig a thermo-mechaical aalysis of a prefabricated reiforced coete bridge beam with a U-shaped oss sectio (Ferreira et al. 008), as represeted schematically i Figure 6.0. z z x y P 30.0 m 43 Figure 6.0 Geometry of the prefabricated reiforced coete bridge beam with a U-shaped oss sectio. 0 7 (x) P 00 [cm] y I the precast idustry differet heat curig regimes (Ferreira et al. 008) are frequetly used to provide a early age stregth developmet capable of aticipatig the process of demoldig as much as possible. I the preset umerical simulatio, the beam is subjected to oe of these heat curig regimes, ad its cosequece i the stregth developmet ad a evetual ack formatio is assessed. I the aalysis carried out, a beam segmet of

276 4 Chapter m legth is cosidered sice Ferreira et al. (008) has verified that legths greater tha 0.0 m have o ifluece o the results. Due to double symmetry of the problem, oly oe quarter of the beam is modeled i the thermal ad mechaical aalysis (see Figure 6.). The ordiary rebars ad the prestressig cables are ot takig ito accout sice Azeha (009) has verified their margial ifluece o the thermal aalysis. For the mechaical aalysis the reiforcemet has also a reduced ifluece up to the hardeed phase of coete. However, if ackig occurs the ifluece of the reiforcemet i the ackig process ca be sigificat, but the computig time required for the iclusio of the rebars o the simulatio has supported the decisio to postpoe this study for a future publicatio. The values that characterize the heat geeratio rate defied by equatio (6.7) are depedet o the type of cemet used i the coete of the beam. The followig data was used to characterize the C50/60 self-compactig coete (Ferreira et al. 008): cemet 3 type I 5.5R ( 33kg m ) with A =.053E+09 33, E = 47.5 kj mol ad T a 3 Q total = kj m. The ormalized heat geeratio rate ( T ) f α used i the aalysis is similar to that represeted i Figure 6., ad a value of 0.05 is cosidered for α Tiit,. The domai is disetized with 0-oded hexahedral fiite elemets (see Figure 6.), ad a Gauss-Legedre itegratio scheme is used. The coductivity of the material is costat ad equal to.6 Wm - K -, the volumetric heat capacity, ρ c, is kjm -3 K - ad the iitial temperature is set to 5 ºC. The ambiet temperature is defied by the heat curig regime imposed to the beam, ad has the followig developmet (see Figure 6.): 30 ºC durig h, followed by a iease of 0 ºC/h util a temperature of 80 ºC is reached, the this temperature is maitaied durig 3h, followed by a deease of 0 ºC/h util the temperature of 0 ºC is attaied. A equivalet heat trasfer coefficiet of.0 Wm - K - is assiged to all exposed faces of the beam. A Backward-Euler time itegratio scheme is used with a iemetal time step of 3600s, beig the total time of the aalysis 7 h. Figure 6. represets the temperature evolutio at two poits located i the oss sectio of the beam coicidig with the logitudial symmetry plae, oe at the top flage,

277 Thermo-mechaical model 43 P (0.0, 0.0, 0.0), ad the other at the iterior of the bottom flage, P (0.0, 66.0, 8.0) (see Figure 6.0 ad Figure 6.). The temperature curig regime is also represeted ad it ca be observed that the temperature at P ad P poits rapidly ieases i the first hours, ad the deeases up to reach the ambiet temperature. Temperature developmet i these poits has similar shape format, but poit P located at the top flage preseted a higher temperature deease rate tha poit P located at the iterior of the bottom flage. Similar results were obtaied by Ferreira et al. (008), ad the mai differece betwee these two studies was registered i the peak temperature observed at about hours after castig. A justificatio ca reside o evetual small differeces o the locatio of the poits ad o the boudary coditios adopted i both aalyses, sice this iformatio is ot clearly idicated i the referece Ferreira et al. (008). The temperature field for differet time steps of the trasiet aalysis is represeted i Figure 6.. It ca be observed that due to the high covectio heat trasfer i the exteral surfaces, the temperature field deeases from the ceter of the U-shape beam walls to the exteral surfaces, ad teds, with time, for the ambiet temperature of 0 ºC. 90 Temperature [ºC] P P Heat curig regime Time [h] Figure 6. Heat curig regime ad temperature evolutio at two poits of the symmetry plae of the beam.

278 44 Chapter 6 P Free ed of the beam P Cross-sectio symmetry plae P t = h Logitudial symmetry plae x z z x y y t = 4h t = 7h Figure 6. Fiite elemet mesh of the structure represeted i Figure 6.0, ad temperature field for differet time steps of the trasiet aalysis.

279 Thermo-mechaical model 45 The temperature field from the thermal trasiet aalysis was used i the mechaical trasiet model desibed i the previous sectio, i order to predict the correspodig stress field. The evolutio of the material properties, such as compressive stregth, tesile stregth ad modulus of elasticity, was simulated by usig equatio (6.33). The equivalet age cocept ( t eq substitutig the time t by t eq ) obtaied by equatio (6.39) was used i equatio (6.33), by. The support coditios cosist i presibed displacemets i z directio i all poits of the bottom flage of the beam i order to simulate the vertical support provided by the formwork, ad presibed displacemets to take ito accout the double symmetry of the beam. The material properties used i the umerical simulatios are preseted i Table 6.. The values of the compressive stregth, tesile stregth ad modulus of elasticity correspod to a age of 8 days for a C50/60 coete stregth class, ad the value of the fracture eergy for the same age is obtaied accordig to CEB-FIP (993). The value of the parameters that characterize the tesile softeig diagram was also obtaied accordig to CEB-FIP (993) recommedatios. The same fiite elemet mesh ad Gauss-Legedre itegratio scheme used i the thermal aalysis is adopted i the mechaical trasiet aalysis. Table 6. - Values of the parameters of the costitutive model used i the mechaical umerical simulatios. Poisso s ratio ν = 0. Thermal coefficiet Youg s modulus Compressive stregth -5 α =.0 0 ºC E = 37.0 GPa cm f = 58.0 MPa cm Tesio softeig diagram Parameter defiig the mode I fracture eergy available to the ew ack -6 f = 4. MPa ; G = MN m ; ctm I f ξ = 0.06; α = 0.5 ; ξ = ; α = 0.09 p = 0 Shear retetio factor Expoetial ( p = ) Crack badwidth Threshold agle α th = 30º Cubic root of the volume of the itegratio poit α = σ, / σ,, α = σ,3 / σ,, ξ = ε, / ε, u, ξ = ε,3 / ε, u (see Figure 3.6)

280 46 Chapter 6 Three umerical aalysis were performed, oe cosiderig the coete with elastic behavior, a secod oe usig the ack costitutive model, ad the last oe usig the ack costitutive model ad takig ito accout the autogeous shrikage. For the evaluatio of the autogeous shrikage the Eurocode model (EC 004) was cosidered. The coete maturity was preset i all umerical simulatios. Figure 6.3 ad Figure 6.4 preset the evolutio of the ormal stress i the x directio ad the tesile stregth developmet for poits P ad P, respectively. From the aalysis of the curves of Figure 6.3 it ca be stated that util a age of about 3 h after castig, the stress developmet is similar i all the umerical simulatios. A iitial compressio util a age of 9 h is observed, which is directly associated with the high imposed exteral heat curig that has coducted to a expasio of the coete developig compressio stresses i poit P, located ear the surface. A quite differet behavior is observed after a age of 3 h for the aalysis that assumes a elastic behavior for the coete ad for the aalysis that simulates ack formatio ad propagatio. The aalysis assumig elastic behavior does ot take ito accout that at this age the tesile stress is greater tha the tesile stregth ad coducts to a urealistic evolutio of the stress field. Usig the proposed updated 3D multi-fixed smeared ack model the real stress developmet at this poit is captured. It is verified that at the momet of the iterceptio of the stress developmet curve ad the tesile stregth curve, the coete acks ad the stress starts deeasig immediately. It is also observed that the autogeous shrikage has margial effect o the stress evolutio at this poit. 5 4 Normal strsess i x directio [MPa] Elastic Crack model Crack model ad shrikage Tesile stregth Time [h] Figure 6.3 Evolutio of the ormal stress i x directio ad the tesile stregth at poit P.

281 Thermo-mechaical model 47 From the aalysis of the curves of Figure 6.4, which correspod to the stress evolutio of the ormal stress i x directio ad the tesile stregth developmet at poit P, it ca be cocluded that all the simulatios provide similar results. Oly whe the autogeous shrikage is take ito accout the umerical respose has a small differece after a age of 8 h after castig. Up 9 h poit P is subjected to tesile stresses, ad the to compressive stresses as observed i Figure 6.4. However, the tesile stress is always smaller tha the tesile stregth developmet, so coete does ot ack. 5 4 Normal strsess i x directio [MPa] Elastic Crack model Crack model ad shrikage Tesile stregth Time [h] Figure 6.4 Evolutio of the ormal stress i x directio ad the tesile stregth at poit P. The ack patter for differet times of the trasiet mechaical aalysis usig the ack costitutive model is represeted i Figure 6.5. It is observed that for a age of 4 h, several acks are formed, maily i the exterior of the top flage ad i the iterceptio of the horizotal ad lateral oss sectio walls ear to the free ed of the beam. I cosequece of temperature deeasig, these acks ted to close, as represeted i the Figure 6.5 for a age of 4 h. However, for later stages the acks reope (see Figure 6.5 for t=7 h), which idicate that for the heat curig regime imposed to the beam, visible acks ca be formed, compromisig the durability of the structure durig its service life.

282 48 Chapter 6 (a) t = 4h (b) t = 4h z x y (c) t = 7h Figure 6.5 Crack patter for differet time steps of the trasiet aalysis: (a) opeig ack status; (b) closig ack status; (c) reope ack status. 6.6 SUMMARY AND CONCLUSIONS I the preset chapter a thermal model with geeral purposes was desibed i detail, ad all the relevat aspects for its implemetatio i the FEMIX computer code were discussed, i order to eable steady-state thermal aalyses, trasiet liear thermal aalyses or oliear thermal aalyses. The heat developmet due to the hydratio process durig the coete hardeig phase was coupled to the thermal model, resultig i a model capable of simulatig the behavior of a coete structure sice its early age. The