Finite Element Modelling of Steel/Concrete Bond for Corroded Reinforcement

Transcription

1 Finite Element Modelling of Steel/Conrete Bond for Corroded einforement By Qixin Du A thesis Presented to the University of Ottawa in fulfillment of the requirements for Master of Applied Siene in Civil Engineering Deember 015 Department of Civil Engineering University of Ottawa Ottawa, Canada K1N 6N5 Qixin Du, Ottawa, Canada 016

2 Abstrat einforement orrosion is the most ommon deterioration problem observed in reinfored onrete (C) strutures loated at oastal or old regions. The orrosion proess an impat the performane of these strutures by induing damage on the bonding ation between onrete and steel, either by the splitting of the onrete over due to the volumetri expansion of orrosion produts or the lubriant effet at the steel/onrete interfae as the orrosion by-produts aumulate. The urrent researh aims at investigating orrosion-indued deterioration of bond between steel and onrete through finite element (FE) analysis of the flexural behaviour of orroded C omponents. By treating the onrete over as a thik-wall ylinder subjeted to internal pressure, the analytial evaluation of impaired bond apaity is studied first and verified against published bonding tests. Then, the formulation of a numerial model is performed using ABAQUS, wherein a link element to simulate the bond behaviour is formulated and implemented through the ABAQUS user-subroutine (UEL) feature aording to the validated analytial model. By introduing orrosion-indued damages, i.e., smaller ross-setional area of reinforement, splitting of onrete and bond deterioration, in the FE analyses, the results of the numerial model show good agreement with experimental observations. Upon validation of the analytial and FE models, a parametri investigation is onduted, wherein the effets of onrete strength, dimension of reinforing bars, properties of oxide produts, different orrosion damage mehanisms and the orrosion loation along the longitudinal reinforement on the flexural behaviour of C beams are studied. The results show that the analytial evaluation for bond degradation is impated by the seletion of the post-raking material model and the thikness of over that determine the holding apaity after raking initiation. Also, the density of rust by-produts affets the results of the analytial model at high orrosion levels. From the FE model results, it was observed that eah damage ii

3 mehanism due to orrosion ontribute to different levels of flexural degradation, although the flexural strength degradation is mainly due to the loss of bonding ation. The parametri study also demonstrates that flexural members whih have reinforement orrosion initiated near the supports suffer greater deterioration in flexural apaity than those with damages at mid span. Finally, based on these observations, suggestions for the appliation of both analytial and numerial models are made. iii

4 Table of ontents Abstrat... ii List of Tables...viii List of Figures... viii Notation... ix 1. Introdution Bakground Objetive of researh Outline of the thesis...3. Literature review Introdution Corrosion of einforing Steel ate of orrosion Main mehanisms ontributing to bond Influene of orrosion on bond stress Overall bond deterioration Experimental observations Numerial study Effet of orrosion loations Experimental observations Frition Analytial evaluations Experimental observations Confinement Analytial evaluations Interlok...19 iv

5 Experimental observations Finite element simulation of bonding Diret ontat Link or interfae element Bond zone Need for esearh Analytial Model for Bond Stress Introdution Bond mehanism Corrosion pressure t Confining stress from onrete Partially-raked thik wall ylinder Confining stress from inner raked ring P p Confining stress from outer unraked ring P...41 e Fully raked onrete over ( i ) Confining stress from stirrups s Summary for evaluation of onfinement stress Confining stress from onrete Confining stress from stirrups Proedure for onfinement stress alulation Frition oeffiient Adhesive stress Mehanial interlok Bond-slip relation Summary...56 v

6 4. Finite Element Model Introdution Seletion of material models Material behaviour of onrete Compressive behaviour Tensile behaviour Finite element material model of onrete Seletion of parameters Calulation of plasti strain Material behaviour of steel Finite element material model of steel Seletion of element types Formulation of link element Boundary onditions of D FE model Corrosion-indued damage models Damage on steel bar Damage on onrete Damage on bond Summary of development of FE model Model Validation Introdution Validation of analytial evaluation of bond stress Comparison with Al-Sulaimani et al. (1990) Comparison with Lundgren (007) Validation of finite element model Comparison with Mangat (1999) Control beam Appliation of link element for unorroded beam...84 vi

7 Appliation of link element for orroded beam Comparison with Cairns et al. (008) Control speimen Corroded speimen Comparison with Du et al. (007) Control speimen Corroded speimen Summary Parametri Study Introdution Parametri study using the analytial model The effet of onrete strength The effet of the ratio n The effet of over-to-rebar diameter ratio (C/d) Parametri study using the finite element model The effet of different types of orrosion-indued damage The effet of orrosion loation along the reinforement Closure Conlusions eommendations for future work...16 eferenes vii

8 List of Tables Table 4-1 Material properties for different grades of onrete aording to CEB-FIP (010)...59 Table 5-1 Material properties used in the analyses... 7 Table 5- Geometry of FE pullout test speimen...76 Table 5-3 Material properties for FE pullout speimen Table 5-4 Material properties for beams tested by Mangat (1999)... 8 Table 5-5 Material properties for sbf beam...9 Table 5-6 Experimental results for group sbf...95 Table 5-7 Material properties of speimens tested by Du et al. (007) Table 6-1 Material data used in Lundgren (007) Table 6- Material data for onrete as provided by CEB-FIP(010) Table 6-3 Frature energy G f and ultimate strain u for different onretes Table 6-4 Finite element speimens with various orroded regions Table 6-5 Finite element speimens with various exposure regions...10 viii

9 List of Figures Figure -1 Bond fore transfer mehanisms (reprodued from ACI Committee )... 8 Figure - Evolution of maximum bond strength from different tests (reprodued from Yaliner et al. 013)...10 Figure -3 Load-defletion urves for beam Samples 1-7 (reprodued from Masnavi 013)...1 Figure -4 Effet of exposure length on beam strength (reprodued from Masnavi 013) Figure -5 The evaluation of x r k / (reprodued from Lundgren 007) Figure -6 Variation of frition oeffiient with slip at different mass losses (reprodued from Amleh and Ghosh 006)...16 Figure -7 Conrete thik wall ylinder model (reprodued from Tepfers 1979) Figure -8 Configuration of a link element (reprodued from Ngo and Sordelis 1967)... Figure -9 Configuration of interfae element aording to Lundgren and Gylltoft (reprodued from Lundgren and Gylltoft 000)... 3 Figure -10 Prinipal stresses in onrete elements loated in bond region for perfet bond and bond zone models (not to sale) (reprodued from Ziari and Kianoush 014)...5 Figure 3-1 Conrete over idealized as a thik-wall ylinder... 8 Figure 3- Values of n representing different orrosion produts... 9 Figure 3-3 Equilibrium ondition in the ylinder model...30 Figure 3-4 Crak shape in onrete over Figure 3-5 Tensile stresses in the thik-wall ylinder... 3 ix

10 Figure 3-6 Figure 3-7 Comparison of thik wall ylinder with different displaement assumptions...34 Comparison of radial displaement obtained from Wang s evaluation and modified distribution...37 Figure 3-8 Tensile stress-strain model for onrete Figure 3-9 Hoop stress and radial displaement distributions in the thik-wall ylinder model.. 40 Figure 3-10 Equilibrium ondition in the outer region of the thik-wall ylinder...4 Figure 3-11 Thik-wall ylinder model with stirrup Figure 3-1 Equilibrium ondition in the ylinder model reinfored by stirrups Figure 3-13 Linear interpolation for the onfinement stress evaluation Figure 3-14 Proedure for onfinement stress alulation Figure 3-15 The bi-linear bond-slip relationship...55 Figure 4-1 Compressive behaviour for onrete (reprodued from CEB-FIP 010)...58 Figure 4- Bi-linear stress-strain relationship for steel... 6 Figure node plane-stress element and -node truss element (reprodued from ABAQUS User Manual 014) Figure 4-4 Strutural member subjeted to bending Figure 4-5 D finite element model Figure 4-6 Boundary onditions of finite element model...65 Figure 4-7 Corroded reinforing bars in C beam ross setion Figure 4-8 Figure 5-1 Proedure of FE model development...69 Comparison of analytial model to experimental results of speimen with 10-mm x

11 diameter reinforement...73 Figure 5- Comparison of analytial model to experimental results of speimen with 14-mm diameter reinforement...74 Figure 5-3 Comparison of analytial model to experimental results of speimen with 0-mm diameter reinforement...74 Figure 5-4 Setup of finite element model in Lundgren (007) (all dimensions are in mm) Figure 5-5 Maximum bond stress versus orrosion penetration for speimen # Figure 5-6 Maximum bond stress versus orrosion penetration for speimen #...78 Figure 5-7 Maximum bond stress versus orrosion penetration for speimen # Figure 5-8 Maximum bond stress versus orrosion penetration for speimen # Figure 5-9 Geometry of beams tested by Mangat (1999)...83 Figure 5-10 Figure 5-11 Figure 5-1 Figure 5-13 Figure 5-14 FE mesh of ontrol beam as displayed in ABAQUS...83 Load-displaement response for ontrol beam with perfet bond...84 Load-displaement response for ontrol beam with link elements...85 Comparison of load-displaement response for both sets of ontrol beams...86 Load-displaement responses from finite element models...87 Figure 5-15 Load-displaement responses for tested speimens (reprodued from Mangat (1999)) Figure 5-16 Load-displaement response for beams with 1.5% orrosion Figure 5-17 Load-displaement response for beams with.5% orrosion Figure 5-18 Load-displaement response for beams with 5% orrosion xi

12 Figure 5-19 Load-displaement response for beams with 10% orrosion Figure 5-0 Geometry of speimen sbf (in mm) (reprodued from Cairns et al. 008) Figure 5-1 Figure 5- Figure 5-3 Figure 5-4 Figure 5-5 FE mesh of ontrol beam as displayed in ABAQUS...93 Load-displaement response for speimen sbf Load-displaement response for speimen sbf Load-displaement response for speimen sbf Comparison of finite element models...97 Figure 5-6 Analytial evaluation of bond strength for speimens in group sbf Figure 5-7 Bending test setup arried out by Du et al. (007) Figure 5-8 FE mesh of ontrol beam as displayed in ABAQUS Figure 5-9 Load-defetion response for ontrol speimen in Du et al. (007) Figure 5-30 Corrosion-indued damaged region in FEM Figure 5-31 Figure 5-3 Load-defetion response for orroded speimen with 8.8% mass loss along mid-span..104 Comparison of load-defetion responses for FE ontrol and orroded speimens in Du et al. (007) Figure 6-1 Comparison of bond apaity for speimens with different onrete ompressive strength Figure 6- Tensile stress-strain relation for onretes with different ompressive strengths Figure 6-3 The omparison of bond apaity of speimens N, N3.5 and N Figure 6-4 Comparison of bond apaity for speimens with C/d = (), C/d = 4 (4) and C/d = 6 (6) xii

13 Figure 6-5 Load-defletion urves for beams with different types of orrosion-indued damage aounted for Figure 6-6 Figure 6-7 Load-defletion urves for beams with different orrosion regions Load-defletion urves for beams with different exposed regions...11 xiii

14 Notation A s = ross-setional area of one stirrup leg A steel = effetive area of rebar in FE model B = width of the beam d a = maximum oarse aggregate size d b = original diameter of rebar d s = stirrups diameter E = onrete young s modulus E 1 = seant modulus from the origin to the peak ompressive stress E s = steel young s modulus E t = onrete modulus after raks f adh = hemial adhesion ` f = onrete ompressive strength f i = mehanial interlok stress due to rebar against the onrete f s = orresponding stress in the ylinder at rs f ss = tensile stress of stirrups ` f t = onrete tensile strength f _ = maximum interlok stress for unorroded reinforement bars u i f y = steel yield strength Fa = hemial adhesion stress xiv

15 G f = frature energy k = plastiity number r b k x / = deterioration fator for frition oeffiient k t = bond stiffness L s = spaing for adjaent shear bars M loss = perentage of mass loss n = ratio between the volume of oxides produt and virgin steel P e = stress ontribution from the unraked ring P p = stress ontribution from raked ring r = radial displaement r b = original radius of rebar r s = radius of the enter of the stirrups r sb = equivalent steel radius per unit length = outer radius of onrete ylinder model i = radius of the rak front s = radius of the orroded bar u1 = u = s y = radial position in whih the hoop strain reahes ultimate strain if the onrete ylinder is not raked radial position in whih the hoop strain reahes ultimate strain if the onrete ylinder is raked relative slip value when the bonding stress reahes the bond apaity xv

16 t r = thikness of rust layer V r = volume of oxides produt V s = volume of virgin steel V r = volume of aumulated orrosion produts x = attak penetration x r = ritial attak penetration depth r = hoop strain at position r 1 = strain at maximum ompressive stress,lim = ultimate strain r = tensile strain at the tensile strength of the onrete el = elasti strain t = total tensile strain in the raked region T = thermal strain s = strain in the spiral reinforement e = onrete elasti hoop strain pl = plasti strain u = onrete tensile ultimate strain = averaged tensile strain = equivalent expansion strains s = onfinement stress due to transverse reinforement = onfinement stress due to onrete over xvi

17 t = orrosion pressure due to the aumulation of orrosion by-produts around the reinforing bar max = bond apaity up _ max = maximum bonding stress for the plain bar ur _ max = maximum bonding stress for the ribbed bar 0 = initial frition oeffiient = frition oeffiient w = rak width opening around the perimeter of the rust front eah rak r for w = harateristi width of the advaning miro raking zone xvii

18 1. Introdution 1.1. Bakground einfored onrete (C) is one of the most widely used onstrution materials due to its durability, versatility and eonomial prie. However, orrosion of the embedded reinforement in hloride-laden environments is an extended problem that an affet the strutural performane. In general, onrete provides protetion against reinforement orrosion due to the high alkalinity environment it provides (Glass and Buenfeld 000). However, the passivity afforded by onrete alkalinity is broken down by the either the presene of high amounts of hloride in the onrete pore solution or arbonation of the onrete over. From an eonomi perspetive, reinforement orrosion is a ostly problem, where a huge amount of resoures need to be employed for maintenane. The total eonomi impat of orrosion and orrosion ontrol appliations was estimated to be $5.5 billion (estimated in year 1949) or.1% of the annual GNP in the US (Thompson et al. 007), whereas in the UK, around 1,365 million pounds per year (estimated in year 1971) was needed to address the problem of reinforement orrosion, whih represented 3.5 per ent of the GNP of 1970 (Bhaskaran et al. 005). In Canada, deteriorated reinfored onrete infrastruture has ost a total of $74 billion to restore it bak to its original state (NSEC 01). Corrosion-indued degradation of the performane of strutures is also a major problem faed by ivil engineers. einforement orrosion has been onsidered as the major ause of deterioration of reinfored onrete strutures around the world (Broomfield 00, Hansson et al. 007). To ounter or minimize this negative impat, a wide range of monitoring and repairing tehniques have been developed in pratial appliations. Also, numerous experimental and analytial studies have been arried out by researhers (e.g., Al-Sulaimani et al. 1990, Li et al. 014, Lundgren 007 1

19 and Wang et al. 004) to evaluate the effet of reinforement orrosion on strutures, whih showase that orrosion attak an lead to redution of ross-setional area of reinforing bars, splitting of the onrete over and weaker bonding ation, ausing an overall deterioration of the strutural performane, suh as a softer residual response and a dereased apaity. 1.. Objetive of researh The bonding interation between reinforing bars and onrete, whih serves as a vital mehanism in C strutures, an be damaged or even eliminated with inreasing orrosion levels (Al-Sulaimani et al. 1990). Bending tests onduted on orroded speimens (Mangat 1999) have highlighted that this degraded mehanism severely impats the residual response of affeted members. In order to evaluate the strutural deterioration due to reinforement orrosion, finite element modelling, whose validity of performing an evaluation for orroded strutures has been demonstrated by other researhers, e.g., Val and Chernin (009) and Li et al. (014), provides a pratial and aurate way for assessment. Therefore, the aim of this researh is to develop a two-dimensional FE model that simulates the bond deterioration in orroded C flexural members. For this propose, the following tasks are arried out: 1. Literature review of orrosion mehanisms and their effets on bonding.. Development of an analytial evaluation of bond deterioration. 3. Development of a finite element model that inorporates orrosion-indued damages and implements the analytial bond model through the use of link elements. 4. Parametri study to extend the appliation of the numerial model and to give a presriptive guide for the residual evaluation of orroded strutures.

20 1.3. Outline of the thesis Chapter 1 presents bakground information related to orrosion, the objetives of the researh and the layout of this thesis. Chapter presents a literature review mainly fousing on reinforement orrosion effets on bond behaviour. Bakground knowledge on the orrosion mehanism is briefly introdued first. Then an overview of available literature on the effets of orrosion on bond degradation is summarized to identify the general influene of orrosion-indued damages on bonding and fators that impat bond behaviour as orrosion attak propagates. After that, a more detail review is performed to obtain better understanding on how these mehanial properties for bonding an be altered during orrosion. Finally, finite element models proposed by other researhers to mimi the bonding interation between onrete and steel are also reviewed. Chapters 3 and 4 present the development of the analytial and numerial models, respetively. In Chapter 3, the analytial evaluation of bond apaity is developed aounting for the affeted mehanisms identified in the literature review. For this propose, the approximations and models suggested by other researhers and assumptions based on experimental observations are adopted. For instane, one of the ruial mehanism in bond deterioration, i.e., onfinement, is modelled by the thik-wall ylinder model suggested by Wang et al. (004) with modified hoop strain distribution based on mathematial investigations. Then a losed-form formulation for bond deterioration is obtained by ombining those analytial evaluations aording to the bonding mehanism. The development of the finite element model (FEM) is disussed in Chapter 4. Material behaviour for onrete and steel based on the CEB-FIP model ode and other studies is given, followed by the seletion of material models and element types in ABAQUS, the software used to implement the FEM. A D model with orrosion-indued damage is built, where bond behaviour is modelled 3

21 by link elements, whih are programmed in ABAQUS through the user subroutine UEL aording to the analytial evaluation proposed in Chapter 3, and orrosion-indued raking of the onrete over due to volumetri expansion is simulated by applying an equivalent expansion strain in the onrete. Chapter 5 presents the validation of the analytial and finite element models against published data. The analytial evaluation is verified by omparing experimental and numerial pullout tests, to ensure it an properly simulate the bond deterioration before its appliation in the FE link elements. After suessful verifiation of the analytial model, the finite element model with link elements and damages due to orrosion is tested against three sets of bending tests: beams reinfored by plain bars, beam reinfored by ribbed bars, and beam with loalized orrosion. In Chapter 6, a parametri investigation using the analytial and numerial models is performed. For this propose, analytial speimens with various onrete strengths, various ratios of the volumes of rust produts to sound steel, and different over thiknesses are tested and ompared to obtain a better understanding of the influene of those physial properties on the bond degradation. Also, numerial investigations of C beams with different orrosion-indued damage mehanisms and orrosion regions are also disussed. Finally, Chapter 7 loses the thesis with the onlusions of the researh observations and the reommendations for future work. 4

22 . Literature review.1. Introdution Corrosion of reinforing steel in onrete is one of the main auses of premature deterioration of reinfored onrete (C) strutures (Chaker 199). Bond degradation due to orrosion an signifiantly damage the performane of reinfored onrete elements, sine the bond interation between the steel and the onrete plays a vital role in the mehanial behaviour of C (Sanhez et al. 010). This hapter presents a literature review of the effet of reinforement orrosion on the bond behaviour in C. First, a bakground summary of the orrosion proess is presented, followed by an overview of experiments and finite element analysis of bond degradation to identify mehanial properties influened by orrosion. Then aording to this observation, a detailed review of impated fators is given. Also, the development of analytial evaluation of bond degradation and finite element modelling tehniques that simulate the bonding interation at the onrete/steel interfae are investigated... Corrosion of einforing Steel The proess of orrosion is an eletrohemial reation, whih is similar to that of a battery. Corrosion is ommon in steel strutures if appropriate protetion is absent. However, orrosion in reinforing steel is rare in freshly ast onrete. This is beause a passive oxide film, mainly Fe O, 3 4 is formed due to the alkaline nature of hydrated ement, whih overs the reinforing steel bar and isolates iron from orrosive fators (Glass and Buenfeld 000). However, this protetion only exists in a high alkalinity environment, i.e., when the ph is above 1. There are two situations whih will eliminate the passive effet: (i) arbonation and (ii) hloride ontamination. 5

23 Carbonation in onrete results from the penetration of arbon dioxide from the atmosphere, neutralizing the Ca(OH) that keeps the alkalinity in onrete. This will redue the ph to 8 or 9, in whih the passive film vanishes. The rate of arbonation depends on the porosity of the onrete and an be slowed by an appropriate depth of over and/or a denser onrete. Corrosion aused by hloride ions is ommon in oastal regions or in old regions where de-iing salts are used. It an also result from hloride ontamination in mix ingredients suh as aggregates and water. This kind of orrosion is usually loalized and forms pits on the bar surfae. The orrosion ell is formed between exposed iron (anode) and passive steel (athode), with a path established by the reinforing steel and the moist onrete over. Furthermore, a strong energy potential due to the uneven distribution of hloride aumulation along the reinforing steel bar arises. In order words, the iron onsidered as anode is onsumed rapidly into iron ions, leading to the formation of a pit on the rebar surfae. The anodi reation is given by: Fe Fe 4e Eq -1 At the athodi site, the oxygen is redued into hydroxyl ions by reating with pore water and onsuming the eletrons released at the anode. The athodi reation is given by: O 4 H O 4e OH Eq - The above two reations ourring at the anode and athode, respetively, are essential for the orrosion proess..3. ate of orrosion When a reinforing bar is subjeted to orrosion, the ross setional area is redued, whih an be 6

24 modelled as a funtion of time. In order to evaluate the rate of orrosion, Faraday s Law is applied: where IorrtW m s Eq -3 ZF ms is the mass loss (g), Iorr is the orrosion urrent, t is the time (s), W is the atomi weight of steel given by g / mol, Z is the valene number or ion harge of Fe, whih is equal to for Fe Fe e, and F is Faraday s onstant (96,485 C / mol ). Assuming uniform distribution of orrosion along the perimeter of the reinforement ross-setional area, the mass loss per unit length an be approximated by: m d x Eq -4 s b s where db is the diameter of the rebar, x is the attak penetration and s is the density of steel 3 (7.86 g / m ). The orrosion urrent Iorr an be rewritten as: I orr i d L Eq -5 orr b where L is the length of the reinforing bar (m) and i orr is orrosion urrent density ( A / m ). The value of attak penetration x is obtained by ombining Eq -3, Eq -4 and Eq -5: iorrtw x i ZF s orr t ( mm ) Eq -6 7

25 Thus, the redued diameter of the reinforing steel bar at time t, db, is equal to the original diameter db minus the attak penetration ( x ) and is written as: d b t d 0. 03i t Eq -7 b orr.4. Main mehanisms ontributing to bond Bond an be defined as the interation and transfer of fore between reinforement and onrete, and the strutural performane of reinfored onrete strutures is influened by this bond behaviour, e.g., width and spaing of transverse raks, strength of end anhorages (CEB-FIP 010). Bond in C is ahieved through three mehanisms (Lutz and Gergely 1967): 1) hemial adhesion, ) frition, and 3) mehanial interloking. Figure -1 Bond fore transfer mehanisms (reprodued from ACI Committee ) 1) The adhesive fore results from the deformation of the ementitious layer around the reinforing bar, whih onsists of the mirosopial interation between ement stone partiles 8

26 (Brameshuber 006). However, this mehanism is not reliable as an unreoverable drop ours one slippage starts and the loss of adhesion an be expeted at 0.05 mm slip (CEB-FIP 010). ) Frition is the resistane against a parallel displaement between onrete and reinforing bar surfae, affeted by the ompressive fore perpendiular to the ontat surfae and the surfae roughness of the interfae (Brameshuber 006). The frition in C strutures may result from the onfining fore at the onrete/steel interlayer, external ompression fores perpendiular to the interfae, and/or lamping fores due to reinforement and/or onnetors (CEB-FIP 010). 3) Illustrated by ACI Committee 408 (003), the interloking effet is the mehanial anhorage or bearing of the ribs against the onrete surfae, whih plays the most signifiant role for ribbed bars at higher load levels. The evaluation of the mehanial interlok should involve the onsideration of the surfae profile, e.g., the height and spaing of bar lugs of the reinforing bar. The above mehanisms both have influenes on the bonding behaviour of strutures reinfored by ribbed bars, wherein only adhesion and frition fore ontribute to the main bond resistane of plain bars. The following setions presents a review of the literature on the effet of orrosion on eah of these mehanisms..5. Influene of orrosion on bond stress.5.1.overall bond deterioration When a C struture suffers from orrosion, not only the virgin steel is onsumed, resulting in redued ross-setional area or diminished ultimate strength, but also the physial properties at the interfae are impated, leading to a weaker bond response between the steel and onrete. 9

27 Experimental observations A number of studies using experimental tests and/or finite element analysis have been arried out to investigate the orrosion-indued damage in bond (see Figure -). Most of them found a dereasing trend as the mass loss of reinforement inreased (e.g., A1-Sulaimani et al and Chung et al. 008). This effet an be signifiant, as some tests reported that % loss in diameter of the steel bar aused 80% redution in bond strength (Auyeung et al. 000). It is also observed that at high levels of orrosion, only a residual value of bond strength remains. However, many researhers have also reported an initial inrease in bond strength for low orrosion levels (mass loss). For instane, the inreases were noted at up to % orrosion level in Xu et al. (010), 0.4% in Mangat (1999) and 1~3% in Huang (014), whih result from the enhaned frition around the reinforing bar due to the inreased surfae roughness of bars and onfining stress. Then the elimination of the bar lugs, the splitting of onrete and the lubriating effet due to the aumulation of rust produts lead to ontinuous degradation of bond strength (A1-Sulaimani et al. 1990, Chung et al. 004, Fang et al.004). Figure - Evolution of maximum bond strength from different tests (reprodued from Yaliner et al. 013) 10

28 Fang et al. (004) tested a series of orroded speimens with different bar types (deformed and plain) and onfinement onditions (with and without transverse reinforement). The eletrolyti orrosion method was used to aelerate the orrosion proess, but the transverse reinforing bar was isolated and not subjeted to orrosion. The results from pullout tests show that the bond performane varies. For speimens with ribbed rebars and transverse reinforement, a small influene in bond was observed, whereas for those with ribbed rebars but without onfinement, the orrosion level played an important role in bond degradation, with bond strength at 9% steel mass loss being only one third of that of a non-orroded speimen. For speimens with smooth rebars, although an initial inrease in the bond strength was reported, the speimens with onfining reinforement tended to maintain bond strength at higher levels of orrosion (even at more than 5%), whereas for those without transverse onfinement, a rapid derease in the bond strength was reported at lower orrosion levels (~4% of steel mass loss) Numerial study Lundgren (007) extended the experiment onduted by Fang et al. (004) in axisymmetri finite element modelling, applying interfae elements, developed by Lundgren (00), at the interfae between the steel and the onrete to represent the bond mehanism. Corrosion was taken into onsideration by modifying the interfae element properties. The influene of types of reinforing bars (ribbed and plain), and the onfinement onditions was also investigated. The author found that the finite element analysis was able to apture the observed behaviour in the experimental tests. The results show that the thikness of onrete over impats the bond performane, sine the bond apaity kept inreasing as long as the over does not rak. The ribbed bar speimens experiened less initial bond inrease than plain bar speimens. 11

29 .5..Effet of orrosion loations As mentioned in the previous setion, hloride ontamination is the major ause of orrosion for reinfored onrete strutures loated along the oast or in old limates. Beause orrosion aused by hlorides is affeted by the distribution of hloride aumulation, whih an be unevenly distributed, loalized orrosion and the formation of pits on the steel bar surfae are ommonly observed, whih means some parts of struture an suffer more loss of bonding. Hene, the assumption of uniformly distributed orrosion (usually adopted for orrosion aused by arbonation) may not be applied for all orrosion ases, and the damages on different loations might have a different impat on the residual strutural performane Experimental observations In the bending experiments arried out by Masnavi (013) (see Figure -3), some of the beam speimens with different ombinations of exposure patterns with respet to loation and length were tested to simulate the spalling of onrete and the resulting de-bonding. The mid-span of speimens ~4 had its onrete over removed with varying lengths, whereas speimens 5~7 had the onrete over removed in regions near the supports. Figure -3 Load-defletion urves for beam Samples 1-7 (reprodued from Masnavi 013) 1

30 Figure -4 Effet of exposure length on beam strength (reprodued from Masnavi 013) From the experimental results (Figure -3 and Figure -4), it is observed that the rate of apaity derease for speimens with short exposure lengths at midspan is slightly higher than those whose onrete over is removed near supports. It is also observed from Figure -4 that the deterioration effet is more signifiant if the exposure area is loser to the supports, beause the damaged area in Sample 7 was nearer to the supports ompared to Sample 5, leading to greater degradation of the arh ation and lower flexural apaity. Similar observations an be found in Cousin and Martin-Perez (010). The authors tested three sets of pre-tensioned prestressed onrete beams, whih were orroded with various regions or length prior to 4-point bending. The seond and third sets of speimens both had 00 mm length of orrosion out of a 50-mm span, but the seond set of beams had the orrosion area loated at the enter of the span, whereas the orroded regions of the latter speimens were near one of the supports. From the experimental results, although sets and 3 had the same perentage of orroded length, the speimens with damages near the ends presented weaker residual responses. 13

31 .5.3.Frition Analytial evaluations Some authors laim that as iron oxides aumulate at the steel interfae, fritional properties an be altered, sine rust produts form a weaker layer, and a lubriant effet should be antiipated. However, an initial inrease in frition at early orrosion stages due to inreased surfae roughness was also observed in experimental tests (Jin et al. 001). Several empirial models for the oeffiient of frition as a funtion of orrosion have been proposed to be used in numerial analysis. Lundgren (007) assumed a slight inrease of frition until 1% orrosion, whih is written as: x / r 0 k Eq -8 where 0 is the frition oeffiient for the unorroded bar, whih was taken as 0.7 and 1.0 for plain and ribbed bars, respetively, in Lungdren (007), and x r k / is a funtion of the attak penetration x. Values for k as a funtion of x/r, where r = db/, are given in Figure -5. Figure -5 The evaluation of x r k / (reprodued from Lundgren 007) 14

32 A lower limit of 0.5 was set to aount for the fat that, for pull-out failure, shear raks form between the ribs of the rebar, and frition is only ative between onrete surfaes. Therefore, the influene of orrosion is not signifiant for frition at large slips. The frition oeffiient suggested by Coronelli (00) dereases one the orrosion depth x exeeds the attak penetration written as: xr assoiated with through onrete over raking, and it is x tan x Eq -9 x r where is the frition angle between onrete and reinforement. Note that, unlike Lundgren (007), the initial inrease is not inluded in Eq -9, as the inrease ours at low orrosion levels, i.e., at approximate 10 to 15 m attak penetration aording to Coronelli (00). Amleh and Ghosh (006) modelled the frition oeffiient with an exponential expression, wherein the frition oeffiient deays from the stati value to kineti one, and it is given as: k d eq e Eq -10 s k where k is the oeffiient of kineti frition, s is the oeffiient of stati frition, d is an empirial deay oeffiient, and eq is the slip. Therefore, the frition oeffiient is affeted by the slip and the empirial deay oeffiient, whih an be evaluated by mass loss due to orrosion. For a orroded surfae, Amleh and Ghosh (006) suggested that a deay of the stati frition oeffiient with inreasing orrosion levels is expeted, whereas the kineti value of frition oeffiient (0.4) remains onstant. Parametri studies using Eq -10 were performed, and the deay oeffiient and stati frition oeffiient were evaluated as a funtion of mass loss ms as follows: 15

33 d 0.061m 0.45 Eq -11 s exp( 0.035m ) Eq -1 s s Figure -6 Variation of frition oeffiient with slip at different mass losses (reprodued from Amleh and Ghosh 006) Substituting Eq -11 and Eq -1 into Eq -10, the oeffiient of frition is rewritten as: k 0.035m 0.061m s e e s 0.45 eq Eq -13 k where ms is the steel mass loss in %. Figure -6 plots the trend of with respet to varying mass loss as the slip inreases. 16

34 Experimental observations The value of frition oeffiient at the interfae between steel and onrete in unorroded speimens has been investigated by Baltay and Gjelsvik (1990), who tested reinfored onrete speimens made out of Type III Portland ement and 3/4-in maximum size gravel. By applying a normal ompression fore on the onrete prior to the steel sample being pulled by a hydrauli jak, the normal ompression fore and the pullout fore were measured, whih gave the observed value of a frition oeffiient between 0.3 and 0.6. To investigate the frition harateristis at the orroded steel/onrete interfae, an experiment was arried out by Cairns et al. (007). A total of four test series were onduted, whih onsisted of a onrete blok 50-mm thik by 100 mm 130 mm, sandwihed between two mild steel plates 50-mm wide and 0-mm thik. Threaded rods were used at the ends of the speimen to lamp the steel/onrete sandwih. After the speimens were orroded, they were subjeted to loading, with a vertial load applied to lamp the speimen between the steel plates and a horizontal load applied to displae the onrete in the lateral diretion, relative to the steel plates. From the observation of experimental data, Cairns et al. (007) onluded that when a surfae rak width in the over is less than 1.0 mm, the lubriant effet of rust produts may not be signifiant beause an inrease of frition due to the aumulation of oxides is noted until 0. mm orrosion expansion..5.4.confinement The inrease of onfining stress at the steel/onrete interfae, whih mainly results from the surrounding onrete over, the stirrups and a field of ompressive stress, ontributes to larger frition stress, and the importane of different onfinement levels has been illustrated experimentally (Fang et al. 004) and numerially (Berra et al. 003, Lundgren 007). They suggest that C with higher amount of onfinement tends to have stronger resistane to raking 17

35 due to orrosion produts build-up around the reinforement, and, hene, it has less bond degradation, as the appearane of orrosion raks triggers a dereasing trend of bond strength for further orrosion levels Analytial evaluations Tepfers (1979) performed an analytial study to investigate the bond behaviour for unorroded deformed bars. Sine the splitting stress in the onrete over, whih equilibrates the indued normal stresses due to onfinement, auses raks when a reinforing bar is pulled out, the author developed an analytial model to evaluate the onfining apaity. The author modelled the onrete as a thik-wall ylinder subjeted to internal pressure (shown in Figure -7), whose thikness is determined by the over depth. When the expansion of the ylinder inner surfae in initialed due to splitting, the pressure at the interfae starts to build up around the bar. One the onrete is raked, as the internal pressure is larger than the tensile apaity of the over, the thik-wall ylinder an be divided into two parts: (i) an inner plasti ring (from db/ to rak front e) and (ii) outer unraked ring (from e to the outer edge of the ylinder y d / ), where y is the depth of over, original diameter db is the original diameter of reinforing steel bar, and e is the radius of the rak front. Thus, the outer part is viewed as a linear-elasti material, whereas the inner part is in the tensile onrete softening stage. 18

36 Figure -7 Conrete thik wall ylinder model (reprodued from Tepfers 1979) Sine rust has a larger volume than virgin steel, the onrete over is also subjeted to inner expansion due to orrosion produts build-up. Wang et al. (004) adopted the thik-wall ylinder analogy to evaluate the onfinement and splitting stresses due to orrosion. The authors solved the expansion displaement by assuming the rust produt was free to expand and ould diffuse into raks. By using the tensile softening behaviour suggested by Pantazopoulou and Papoulia (001), the analytial results showed the onfining stress also had a similar trend with bond degradation, whih experiened an initial inrease and then a derease as the attak penetration propagated..5.5.interlok For deformed bars, the interlok effet between ribs and onrete plays a signifiant role in 19

37 bonding between steel and onrete and makes the assumption of perfet bond feasible. Some analytial and numerial models onsidering the mehanial interloking have been proposed by researhers, suh as Wang et al. (004) and Lundgren (1999), to evaluate the bond stress Experimental observations At high levels of orrosion, the geometry of the rebar, for instane the height of the rib, is hanged, leading to diminished interlok. Aording to Chung et al. (004), this phenomenon ours before the orrosion in the ross setion is initialized, and it eliminates most of the interlok stress at % of orrosion, as observed in experiments onduted by the author, in whih the rib no longer existed or only a small portion remained. Similar onlusions an be found in Almusallam et al. (1996), where pullout tests were performed on speimens of mm with 1mm diameter ribbed bar. They reported about 45% rib profile loss at about 7% orrosion and around 70% profile loss at 1% rebar orrosion. Also, only a small hange in the bond strength due to the redution in the rib profile was observed in the range of 43 to 100% orrosion, whih indiates that the ribs are degraded to the extent that their interloking ation with the onrete beomes negligible. Therefore, further degradation of the ribs does not affet the bond strength to any notieable extent..6. Finite element simulation of bonding The onstitutive behaviour of bond, i.e., a bond-slip relation, has been investigated by many researhers in finite element (FE) analysis, and different modelling tehniques have been proposed and verified against experimental data. In order to model bond behaviour in FE, the methodology to mimi interation between onrete and steel bar should be investigated. There are several methods summarized below. 0

38 .6.1.Diret ontat In finite element modelling, the bond behaviour between onrete and rebar an be treated as a ontat problem if the interation or ontat properties are defined properly. Berra et al. (003) used axi-symmetri finite element analysis in ABAQUS to investigate the bond degradation orresponding to varying onfinement situations and orrosion expansion levels. The interation was assumed to be perfetly or partially ompatible for nodes at the interfae, and the orrosion effet was onsidered by applying a steel volume expansion, whih introdued damage in the onrete over. Although this approah gave good numerial results in FE pullout tests, the drawbak of this interation assumption is that the frition properties, whih should be altered when orrosion is initialized, annot be taken into aount sine the nodes at the interfae, where the ompatibility is applied, are fored to move together. Amleh and Ghosh (006) developed a nonlinear 3D model for finite element pullout tests. The author assumed the bond stress was attributed to frition and performed parametri studies to obtain the frition oeffiient, whih is a funtion of attak penetration and slip. The ontat pressure due to orrosion was defined by the ontat pressure-overlosure relationships provided by ABAQUS, whih inreases exponentially as learane between the onrete inner surfae and reinforing bar redues. The numerial solution showed promising results when they were validated against some pullout tests onduted by Cabrera and Ghoddoussis (199) and Al-Sulaimani et al. (1990)..6..Link or interfae element The link or interfae element serves as a mehanial medium that transfers the stress between steel and onrete. Link elements are loated at the inter-layer and onnet the nodes of different 1

39 elements. As a onsequene, the onrete and steel elements an be viewed as two separated layers and ooperate through those onnetions while undergoing loading rather than their interfae be onsidered as a ontat problem. The first appliation of a link element in C to model bond was proposed by Ngo and Sordelis (1967). This link element onsists of two springs in orthogonal diretions (Figure -8), representing the normal stress, like orrosion-indued internal pressure or inlined fore from interloking, and tangential stress, i.e., bond stress. However, the authors did not give expressions for the stiffnesses of these links. Figure -8 Configuration of a link element (reprodued from Ngo and Sordelis 1967) Lundgren and Gylltoft (000) developed an interfae element (see Figure -9) for finite element analysis. This non-dimensional element is also used at the onrete-steel interfae, and its stiffness matrix is given by tn D t t 0 11 ut D ut D 1 u u n t Eq -14

40 where tn normal splitting stress t tt bond stress u u u relative n t normal displaement slip at the layer D 11, D1 and D are the stiffness oeffiients for the link element. Figure -9 Configuration of interfae element aording to Lundgren and Gylltoft (reprodued from Lundgren and Gylltoft 000) For deformed bars, this fritional model is able to desribe the radial deformation resulting from ut splitting stresses by speifying the nonzero stiffness term D1. To evaluate this stiffness, the u theory of elasto-plastiity is applied, whih inludes the knowledge of yield lines, flow rules, and hardening laws. For plain bars, this term is set to zero. t The appliation of link and interfae elements was demonstrated by some researhers and their validation was shown by omparing numerial results to experimental tests. Lee et al. (00) built a FE model with link and interfae elements, and onduted pullout tests to 3

41 investigate the degradation of bond apaity and rigidity. They first performed the experiments with a series of speimens, whih had different grades of onrete, over thikness and orrosion levels by using aelerated orrosion. ather than using an analytial solution for bond behaviour, the bond response, suh as apaity and rigidity, was measured from tests and then inputted into the finite element model. The authors reported the validation of the link element was demonstrated by showing a good agreement with their experimental data. Li et al. (014) applied similar elements in the nonlinear finite element software ABAQUS to investigate the degradation of bond strength and stiffness due to orrosion. The onneting element, alled translator element, that onneted onrete and steel at the interfae was established. Only the displaement in the slip diretion was enabled in the element, mimiking the bond response, while all the other degrees of freedom for the two onneting nodes were relatively restrained to eah other. By modifying an expliit bond relation proposed by Almusallam et al. (1996) as neessary input data for the element, they ompared their numerial results against bending experiments, where orroded seawalls with dimensions of mm were subjeted to loading. The results demonstrate the promising appliation of this translator, whih is similar to a link element, as the model is able to apture the observed behaviour in their experimental tests..6.3.bond zone The appliation of link elements may require signifiant amount of time if the struture is large. ather than using a link element, a bond zone is established in Ziari and Kianoush (014). The bond zone is a small onrete region next to the reinforing bar, whose material properties are modified to represent better bond interation. Based on the observation that the formation of diagonal ompression struts aused by the miro 4

42 raking is noted during the pullout of reinforement, and due to high onentration of ompressive stresses against the bar lugs, the authors adopted a smaller value of tensile strength and frature energy in the bond zone. The reason is that if the tensile strength and frature is redued, the stress from the steel bar an be transferred to the surrounding onrete through diagonal ompressive stresses ( 1 ) in Figure -10 (where 1 and are prinipal stresses), as the prinipal tensile stresses ( ) are virtually zero, while in the perfet bond model without material modifiations, the bond fore an be transferred through tensile stresses. The result of this methodology is affeted by the depth of the modified region, sine the larger the zone leads to larger angle of the bond bearing stress. Therefore, half the bar diameter is suggested for the depth of the bond zone. Figure -10 Prinipal stresses in onrete elements loated in bond region for perfet bond and bond zone models (not to sale) (reprodued from Ziari and Kianoush 014) 5

43 .7. Need for esearh The literature review showases bond behaviour an be impaired by reinforement orrosion. As reviewed, eah mehanism ontributing to bond is altered as the orrosion level inreases. Although an initial inrease in bond apaity is notied in some experiments (see Figure -), mainly attributed to the onfinement inrease due to the orrosion volumetri expansion at the onrete/steel interfae, bond deterioration starts after the onrete over raks. From the review of the literature, numerous experiments have ontributed to today s knowledge on the loss of bond interation between onrete and steel due to reinforement orrosion. There have also been several studies to apture this bond deterioration in finite element modelling following different approahes, e.g., diret ontat, linkage element and bond zone. The link element has been suessfully used to model bond deterioration along the reinforement in pullout test simulations, if the bond stress-bond slip onstitutive behaviour is known a priori. However, there has been limited effort in applying link elements wherein the normal and transversal bond omponents are deoupled and evaluated using analytial modelling. This approah thus relates the level of onrete over raking (normal omponent) to bond degradation (transverse omponent). Therefore, this work attempts to implement this FE proedure by using link elements that simulate simultaneously the redution of steel ross setional area, the raking of the onrete over, and the loss of bonding between onrete and steel as a result of reinforement orrosion. Proedures are implemented for a two-dimensional analysis of reinfored onrete flexural members. 6

44 3. Analytial Model for Bond Stress 3.1. Introdution When the orrosion level inreases in reinfored onrete, the physial properties for onrete, steel bar and ontat interfae are altered, whih hanges the bond behaviour between steel and onrete. This hapter presents an analytial model for bond behaviour at different levels of reinforing steel orrosion by first investigating the fritional stress, whih an be expressed by the total orrosion pressure times the frition oeffiient, followed by the evaluation of adhesive and mehanial interloking stresses. 3.. Bond mehanism The bond apaity between onrete and reinforement is mainly from (1) hemial adhesion, () frition, (3) and mehanial interlok due to the ribs of rebars. Aording to the fritional model proposed by Tastani and Pantazopoulou (013), the bond apaity max an be written as: max f f Eq 3-1 adh t i where fadh is the hemial adhesion, is the frition oeffiient, t is the orrosion pressure due to the aumulation of orrosion by-produts around the reinforing bar, and fi is mehanial interlok stress due to the rebar sliding against the onrete. Eah of these mehanisms is explored further in the next setions Corrosion pressure t Corrosion ours at the surfae of the steel bar, and the resulting oxides start to aumulate as the 7

45 orrosion proesses. Sine iron oxides have lower density (e.g., 5.4 the virgin steel ( g / m 3 g / m for Fe OH 3 ) than ), the rust produt resulting from steel onsumption due to orrosion has a larger volume than the original steel, and this volumetri expansion auses a pressure at the interfae between the reinforement and the onrete. Tepfers (1979) first evaluated the onfining stress at this interlayer by modelling the onrete as a thik-wall ylinder subjeted to pressure at the inner surfae (Figure 3-1), where the thikness orresponds to the onrete over. In Tepfers model, the pressure P starts to build up at the inner surfae of the thik-wall ylinder, whose inner radius is rb and outer radius is. adial raks are reated if the pressure P indues tensile stress above the tensile apaity of the onrete over. One the ylinder is partially raked, the model an be divided into two parts: an elasti unraked part (from i to outer edge of ylinder ) and a raked inner ring (from the edge of the orroded rebar r b - x tr to rak front i ), where rb is the radius of the unorroded steel, tr is the thikness of the rust layer, i is the radius of the rak front, and x is the attak penetration, whih is used to measure the orrosion level and an be alulated if the orrosion urrent and the orrosion period are known, or obtained from on-site observation. Figure 3-1 Conrete over idealized as a thik-wall ylinder 8

46 Wang et al. (004) also adopted this methodology to evaluate the onfinement pressure resulting from orrosion build-up. By assuming orrosion buildup is uniformly distributed around the reinforing bar, the orresponding volume of onsumed steel per unit length is given as: Vs rb s rb x x Eq 3- where r b, s are the radii of the virgin rebar and orroded bar, respetively. As oxides produts have lower density ompared to sound steel, the orresponding volume of aumulated iron oxides an be obtained from: V n Eq 3-3 r V s where n is the ratio between the volume of oxide produts and virgin steel. Aording to osenberg et al. (1989), the ratio n an be taken from to 6.4 for different orrosion produts (see Figure 3-). The approximation of the value n should be based on the assessment of oxide produts, and it is assumed to be in Lundgren (007). Figure 3- Values of n representing different orrosion produts The stress in the onrete over due to orrosion expansion t results from the onfining 9

47 behaviour of the onrete itself, denoted here as, and from the existing stirrups, denoted as s. The following setions present how to alulate eah omponent ontributing to t Confining stress from onrete Partially-raked thik wall ylinder When the ylinder model is partly raked, i.e., the rak front i, the onfining stress ontribution omes from elasti and inelasti regions in the onrete over. Thus, the onfinement stress an be evaluated by establishing the equilibrium ondition aording to Figure 3-3. b rb i rb r dr dr dr P P Eq 3-4 i p e Figure 3-3 Equilibrium ondition in the ylinder model 30

48 where Pp is the stress ontribution from the inner raked ring, Pe is the stress ontribution from the outer unraked ring, and is the tensile stress in onrete Confining stress from inner raked ring Pp One raks in onrete are initiated, rust produts an diffuse and fill the spae. Aording to Pantazopoulou and Papoulia (001), raks an be assumed to form in triangular shape (Figure 3-4), whose volume per unit length is written as w /. 0 i r, where w is the rak width opening around the perimeter of the rust front r for eah rak. Figure 3-4 Crak shape in onrete over It is assumed that the orrosion produts are free to expand, i.e., the volume of aumulated orrosion produts Vr does not redue under pressure P, whih is rewritten as: t w V nv t /. 0 Eq 3-5 r s r s r i r where w is the total amount of rak width openings, given by w ur r, b r b 31

49 and r b u is the radial displaement at r rb aused by orrosion, written as: u r b t r x n 1 r x x i b r b Eq 3-6 To obtain the analytial solution for tensile stresses in the thik-wall ylinder, the theory of elastiity is adopted, whih is given by: r / r 1 / r 1 f Eq 3-7 s s where rs is the radial position, and fs is the orresponding tensile stress in the ylinder at r r s (Figure 3-5). If the tensile stress fs at r rs in the thik-wall ylinder is known, the tensile stress r within the elasti ring an be evaluated. Figure 3-5 Tensile stresses in the thik-wall ylinder Beause the onrete reahes its tensile strength at the rak front i, i.e., r s i and ' fs f t, Eq 3-7 an be rewritten as: 3

50 where ' ft r ' t / r 1 / 1 f Eq 3-8 is the tensile strength of onrete. i In Wang et al. (004), the authors assume elasti behaviour in the inner raked zone and neglet Poisson s effet. Therefore, a radial displaement distribution for an elasti thik-wall ylinder is used for both regions in the onrete over (raked and unraked), i.e., from r b to, and it is given by: r ft ' / r 1 E / 1 r Eq 3-9 E i / r 1 / 1 ft ' ur r r r Eq 3-10 E i for r r b where r is the hoop strain at position r, and E is the initial elasti modulus of onrete. However, in the inner raked ring, the raking strain r has been attained. Therefore, the total tensile strain in the raked region t, whih an be expressed as t r e, where e is the elasti hoop strain, should be larger than the hoop strain (Eq 3-9) evaluated by Wang et al. (004), where only the elasti strain e is onsidered in the inner region ( rb r i ). In other words, for a partially raked ylinder, the radial displaement ur within the raked ring (left side in Figure 3-6) should be larger than the radial displaement at the same radial position r where the inner ylinder is not raked (right side in Figure 3-6) and the deformation of the entire onrete over is assumed to be elasti. 33

51 Figure 3-6 Comparison of thik wall ylinder with different displaement assumptions The drawbak of the elasti distribution assumption for the whole ylinder is that the stiffness of the ylinder is overestimated, leading to a higher onfinement stress. As pointed out by Gambarova et al. (1997, 1998), the assumption made in the hoie of onrete softening model and displaement field along the onrete over is a signifiant modelling onsideration, sine the onfinement apaity in the plasti region depends on the onrete strength after raking. Therefore, other assumptions of radial displaement in the inner ring (from r b to i ) leading to larger displaements should be made. Sine for a raked ylinder model, the only variable in Wang s evaluation of radial displaement (Eq 3-9) is the radial position r, from a mathematial prospetive, it an be simplified as: 34

52 35 r B r A r r E f r u i t 1 / 1 / ' Eq 3-11 where oeffiients 1 / ' i t E f A and 1 / ' i t E f B are onstants derived from the original equation. In order to obtain an approximation of larger displaements in the inner region, the rational funtion whose order is equal to -1, i.e., r C r u, where C is a oeffiient, is used in this researh to desribe r u from b r to i. Therefore, the displaement field is written as: r C r u for i b r r Eq / 1 / ' i t r r r E f r r u for i r Eq 3-13 Due to ontinuity, i.e., Eq 3-1 and Eq 3-13 both pass ( i, i t E f ' ), the oeffiient C an be solved, and Eq 3-1 and Eq 3-13 should be rewritten as: r E f r u i t 1 ' for i b r r Eq / 1 / ' i t r r E f r u for i r Eq 3-15 The hoop strain an then be rewritten as: 1 ' r E f r r u r i t for i b r r Eq 3-16

53 / r 1 / 1 ft ' r for i r Eq 3-17 E i The evaluation of radial displaement aording to Eq 3-14 and Eq 3-15 is plotted on the right side of Figure 3-7, and it is ompared to Eq 3-10 obtained from Wang et al. (004) on the left side of the figure. Combining Eq 3-6 and Eq 3-14, the following equation is obtained: i ft ' E r 0 b n 1 r x x i b r b Eq 3-18 whih an be rearranged as: 3 i i r (1 n)(r x x ) r 0 Eq 3-19 r r b b b The value of the rak front i an be solved from Eq 3-19, and the regions for raked and unraked onrete along the onrete over an be established. The solution of Eq 3-19 is obtained by applying the general formula for the root of a ubi 3 funtion ax bx x d 0 (William et al. 199), i.e., 1 b C 3a C x 0 Eq

54 where C Eq 3-1 b 3a Eq b 9ab 7a b Eq 3-3 Figure 3-7 Comparison of radial displaement obtained from Wang s evaluation and modified distribution 37

55 The solution for Eq 3-19 is therefore written as: i ( x) ia ib ib i ib ib i Eq 3-4 where the orresponding mathematial oeffiients are written as: ib ia rb Eq (1 n)(r x x ) r r 54 3 b b r b Eq 3-6 r i rb Eq To evaluate the stress ontribution from the inelasti region, a bi-linear urve is adopted to desribe the strain-stress relation in tension, i.e., after reahing the tensile strength ' ft at strain r, the tensile stress dereases linearly to zero at ultimate strain u, and the expressions are given by: E for 0 r Eq 3-8 ft E t r for r u Eq for u Eq 3-30 The value of the ultimate tensile strain u an be evaluated from the frature energy G f, whih is the energy dissipation due to loalized raking per unit area of plane, and its evaluation is presented in the CEB-FIP Model Code (010) as: f ' f 0.18 G 73 (N/m) Eq

56 where ' f is the ompressive strength of onrete. Therefore, the frature energy is a material property dependent on the onrete strength. Figure 3-8 Tensile stress-strain model for onrete The frature energy is equal to the area under the uniaxial tensile stress-rak opening urve (Bazant 1986). One Gf is determined, the final tensile strain an be alulated by ensuring the same energy dissipation in the tensile softening stage. Therefore, the value of u is written as: where w G f u Eq 3-3 f w ' t is the harateristi width of the advaning miro raking zone, whih Bazant and Oh (1983) suggested to be equal to 3 d a, where da is the maximum aggregate size in the onrete. One the ultimate strain u is obtained, the softening modulus in Eq 3-9 an be expressed as: E t ' ft u r Eq

57 where ' ft is the tensile strength of onrete, and r is the strain at the maximum tensile strength. Conrete loses its tensile strength if its hoop strain is larger than the ultimate strain u. To estimate the range over whih onrete has lost its tensile strength, u is substituted into Eq 3-14 and, noting that the hoop strain and the displaement are related by u r, the ultimate r i strain in the analytial model is written as u, where u1 u1 is the radial position in whih the hoop strain reahes ultimate strain u, given by: r u1 i Eq 3-34 u Figure 3-9 Hoop stress and radial displaement distributions in the thik-wall ylinder model Therefore, when r u1 b, the region of onrete over from b r to u1 loses its tensile apaity 40

58 41 (Figure 3-9). By setting ), max( 1 u b a r, the onfining stress from the raked region p P an be obtained from: a i a i r t a i r t t r i r t t r r t t p E E f dr r x E f dr E f P i a i a ' ) ( '- '- Eq Confining stress from outer unraked ring e P The onfinement stress e P an be evaluated by solving the equilibrium ondition illustrated in Figure 3-10 based on the ylinder model developed by Wang et al. (004), sine the tensile strain is not modified in the elasti region, where the hoop strain an be evaluated from Eq ) '( 1 ) / ( ' 1 ) / ( 1 ) / ( ' i i i t i t i t e f r r f dr r f dr E dr P i i i i Eq 3-36 Thus, the total onfining stress is obtained by substituting Eq 3-35 and Eq 3-36 into Eq 3-4, whih is rewritten as: b e p r P P / Eq 3-37

59 Figure 3-10 Equilibrium ondition in the outer region of the thik-wall ylinder Fully raked onrete over ( i ) If the onrete over is fully raked, i.e., the rak front i reahes the outer surfae ; there is no elasti ring in the analytial model. The radial displaement at the steel/onrete surfae and the equilibrium equation are respetively rewritten as: u r n 1 r x x b r b Eq 3-38 rb rb r dr P Eq 3-39 p As disussed in the previous setion, the radial displaement field along the raked region is modified the one in the ylinder model proposed by Wang et al. (004) and assumed as a rational funtion, whih is written as: C ur for rb r Eq 3-40 r 4

60 where C an be solved by ombining Eq 3-6 and Eq 3-40, whih results in: u r b C r b n 1 r x x b r b Eq 3-41 Thus, the radial displaement u(r) and hoop (r) strain for the fully raked ylinder are rewritten as: x U ur Eq 3-4 r for rb r u r Ux r Eq 3-43 r r where the oeffiient C is substituted by U x U x, whih is given by: n 1 r x x b r b r b Eq 3-44 Similar to the previous setion, to determine the effetive thikness of the thik-wall ylinder whih has tensile stress apaity (Figure 3-9), the radial position r u where the tensile strain reahes u is obtained. By substituting u into Eq 3-43, x U u is obtained. If the u value of u is less than the radius of the steel bar, there is no loss of onrete tensile strength. When u rb, by setting b max( rb, u ), the onfining stress of the raked thik-wall ylinder Pp with a modified tensile strain is given by: 43

61 P p U( x) ft'-et r rdr f '-E dr t t r r b Et b r b U( x) f ' E t b t b Eq Confining stress from stirrups s If transverse reinforement is provided, the onfinement apaity in the onrete is enhaned. Noghabai (1996) evaluated the onfinement stress from transverse reinforement based on the thik-wall ylinder model proposed by Tepfers (1979). The author idealized and modelled the stirrups as spiral reinforement in the ylinder, as illustrated in Figure 3-11, loated at a distane rs from the enter of the tension rebar. It is realized that in reality the onfinement provided by stirrups in flexural members would be less than the formulation proposed by Noghabai (1996). Figure 3-11 Thik-wall ylinder model with stirrup In order to obtain the onfining stress per unit length due to spiral reinforement, equilibrium as 44

62 shown in Figure 3-1 has to be established, where the stress assumed to be uniformly distributed. fss in the stirrup s ross setion is Figure 3-1 Equilibrium ondition in the ylinder model reinfored by stirrups Thus, equilibrium is written as: 0 sin r d s b r 0.5r s r 0.5r s sb sb f ss dr 0 Eq 3-46 Thus, f r ss sb s Eq 3-47 rb where s is the onfinement stress due to transverse reinforement, and f ss is the tensile stress in the stirrups, whih an be expressed as f ss Es s fy, where Es is the elasti modulus of steel, s is the strain in the spiral reinforement, f y is the yielding stress of steel, rsb is the 45

63 equivalent steel radius per unit length, given by r / sb As Ls, As is the ross-setional area of one stirrup leg, and Ls is the spaing between adjaent stirrups. In this work, assuming the spiral reinforement does not influene the hoop strain distribution along the onrete over in the analytial model, the strain field an be evaluated by Eq 3-16~Eq 3-17 or Eq 3-43 depending on whether or not the thik-wall ylinder model is fully raked and on the position of the stirrups. Therefore, the strain in the stirrups an be obtained by substituting r with the radial distane to the enter of the transverse bar, i.e., r rs. The strain in the stirrups is therefore obtained as: If rs i and i (The ylinder is partially raked, and the stirrups are in the inner i Eq 3-48 s r rs region.) If rs i and i (The ylinder is fully raked, and the stirrups are in the inner region.) If r s i (The stirrups are in the outer region.) U x s Eq 3-49 rs / rs 1 / 1 Eq 3-50 s r i Using Eq 3-48~Eq 3-50, the orresponding expressions for the expansion stress resulting from the transverse reinforement per unit length are given by: 46

64 If If If rs i and i rs i and i r s i Es rsb r i s Eq 3-51 r r b E r s U x s sb s Eq 3-5 rb rs Es rsb r / rs 1 s Eq 3-53 r / 1 b i Summary for evaluation of onfinement stress The stress in the onrete over due to orrosion expansion results from the onfining behaviour from the onrete itself and the existing stirrups, i.e., t Eq 3-54 s where and s are the onfinement stresses from onrete and stirrups, respetively, whose analytial solutions are summarized below.(check) Confining stress from onrete In order to evaluate the onfining stress at the inner surfae of the onrete, the raked front i in the analytial model needs to be determined first: where i Eq 3-55 ia 3 3 ib ib i ia 3 3 ib ib i rb Eq (1 n)(r x x ) r r 3 b b r b ib Eq r 47

65 i rb Eq From the above equations, the rak front an be viewed as a funtion of x, i.e., x i i. Note that the drawbak of the analytial evaluation developed in the previous setions is the assumption that the ylinder is raked. Therefore, to approximate the stress when the onrete is not raked, x r, a linear interpolation starting from zero pressure at 0% orrosion to the point where for i b the ylinder is just raked ( i x rb ) is assumed (Figure 3-13). Figure 3-13 Linear interpolation for the onfinement stress evaluation When r ( x) (partially-raked onrete over): b i P p Pe / rb Eq 3-59 P p i i ft '( ) i Pe Eq 3-60 f E t r i a i t ' t r i a Eq 3-61 a E 48

66 a max( rb, u1) Eq 3-6 r u1 i x Eq 3-63 u When ( x) (fully-raked onrete over): i P / r Eq 3-64 p b where Et b Pp ft' Et r b U( x) Eq 3-65 b b max( rb, u ) Eq 3-66 x U u Eq 3-67 Parameters in Eq 3-59~Eq 3-67 have already been defined in Setion u Confining stress from stirrups The evaluation of expansion fore resulting from the transverse reinforement depends on the radial position of the stirrups, r s, and the rak front, i, i.e., If If If rs i and i rs i and i r s where i x Es rsb r i s Eq 3-68 rb rs E r U x s sb s Eq 3-69 rb rs / x Es rsb r rs 1 s Eq 3-70 r / 1 U x b n 1 r x x i b r b r b Eq

67 Parameters in Eq 3-68~Eq 3-71 have already been defined in Setion Proedure for onfinement stress alulation The algorithm followed to evaluate the onfinement stress provided by the onrete over and transverse reinforement and based on the proedures presented is shown in Figure 3-14, whih an be briefly summarized as follows: 1) The attak penetration x (mm) due to orrosion needs to be determined first as input data to alulate the rak front i (Eq 3-55~Eq 3-58). If the rak front i is not greater than the radius of the outer layer of the ylinder model ( ), i.e., the ylinder is partially raked, the proedure for the partly-raked model should be seleted. Otherwise ), the fully-raked model should be utilized. ( i i ) For the partly- and fully-raked models, the onfinement stress from the onrete an be evaluated aording to Eq 3-59~Eq 3-63 or Eq 3-64~Eq 3-67, respetively. 3) If there are no stirrups, the onfining stress from stirrups s is equal to 0. If there are stirrups, s should be evaluated. For the fully-raked ylinder model, s an be obtained from Eq 3-69, whereas for the partly-raked model,s should be evaluated from Eq 3-68 if the transverse reinforement is loated within the elasti region ( r ),or from Eq 3-70 if the stirrups are in the region of raked onrete. s i 4) The total onfinement stress t ontribution from the onrete over an be obtained by ombining the onfinement and the transverse reinforement s. 50

68 Figure 3-14 Proedure for onfinement stress alulation 51

69 3.4. Frition oeffiient In order to evaluate the bond apaity aording to Eq 3-1, the frition oeffiient needs to be determined. For this work, the oeffiient is defined aording to the model proposed by Lundgren (007) (see Eq 3-7), beause this model aptures a slight inrease of frition at low levels of orrosion due to the inrease of roughness when the orrosion produts just start to aumulate, and a derease again as the orrosion level inreases, whih results from the lubriant effet when the orrosion layer is formed. A lower limit was also set, whih is not affeted by the degree of reinforement orrosion. where 0 kx / rb 0 x Eq 3-7 is the frition oeffiient for an unorroded bar, and x k / is a funtion of the r b attak penetration x. For a value of -5). x / rb greater then 0.0, parameter k is lower than 1 (see Figure As mentioned in the literature review hapter, the initial frition oeffiient 0 obtained in experimental studies has been measured as 0.3~0.6. In the finite element pullout test performed by Lundgren (007), the value of 0 was taken as 0.9 and 1.0 for plain and ribbed bars, respetively, showing good agreement with experimental observations. Therefore, based on these published values, it is reasonable to hoose the initial frition oeffiient 0 between 0.3 and Adhesive stress Chemial adhesion between the steel reinforement and onrete is not a reliable soure of bond apaity. However, it is the main soure of bond apaity of plain bars when there is no orrosion 5

70 aording to Eq 3-1. In the finite element pullout test performed by Lundrgen (007), this stress was assumed not to vary with the orrosion level but hanged with an inrease in slippage between the steel and the onrete. In this work, a small onstant value equal to 0.8 MPa (obtained from the speimen with orrosion levels lose to 100 μm tested by Youlin 199) is used if this value is not provided by published experimental data. Note that this value is assumed to remain onstant for all levels of orrosion Mehanial interlok Approximating the stress resulting from the mehanial interlok between the reinforement and the onrete should onsider the geometry of the rebar rib, suh as the rib height and the distane between adjaent ribs, but these properties are not always available. The bond apaity for unorroded ribbed and hot-rolled plain bars, as suggested by the CEB-FIP Model Code (010), are respetively given by: f Eq 3-73 ur _ max.5 ' f Eq 3-74 up _ max 0.3 ' where ur _max and up _max are the maximum bonding stress for the ribbed bar and plain bar, respetively, and ' f is the ompressive strength of onrete. Aording to the fritional model, for an unorroded speimen reinfored with smooth bars, the only ontribution to bond is that from hemial adhesion, whereas there is an extra bond ontribution from the mehanial interloking of ribbed rebars, i.e., f f Eq 3-75 ur _ max adh u _ i 53

71 Eq 3-76 up _ max f adh where f u _ i is the maximum interlok stress for unorroded reinforement bars. Therefore, it is reasonable to obtain the initial interloking stress by subtrating Eq 3-74 from Eq 3-75, resulting in: f u _ i. f ' Eq 3-77 ur _ max up _ max Chung et al. (004) have reported that the elimination of mehanial interlok stress was observed at relatively low levels of orrosion (around % diameter loss), where the rib no longer existed or only a small portion remained. Therefore, this work assumes that the mehanial interlok ontribution to bond apaity dereases linearly from f u_ i to zero at % orrosion, i.e., where f i 0.0 ( x / rb ) fu _ i Eq fu _ i is the maximum interlok stress for a orroded ribbed rebar, x is the attak penetration due to orrosion, and rb is the radius of the virgin reinforement. It is realized that this assumption is based on the experimental results reported by Chung et al. (004), and further investigation of the effet of orrosion on mehanial interloking is warranted Bond-slip relation For simpliity, in this work, the bond-slip behaviour of unorroded and orroded reinforement is assumed to adopt a bi-linear onstitutive relation, whih was also utilized in finite element tests arried out by Lee et al. (00) and is illustrated in Figure

72 max / s max s y for s sy Eq 3-79 max for s y s Eq 3-80 Figure 3-15 The bi-linear bond-slip relationship where sy is the relative slip when the bonding stress reahes the bond apaity max. Note that the value of sy is obtained from experimental observation onduted on speimens with similar physial properties. As the C beams seleted for validation in Chapter 5 had 10-mm diameter rebar and 0-mm thikness onrete over, giving a ratio of onrete over-to-rebar diameter of, the value of sy 0. provided by Tastani and Pantazopoulou. (013) with similar onrete over-to-rebar diameter ratio is adopted for the finite element validation. By ombining the fritional model expressed in Eq 3-1 and the bond-slip onstitutive relation given in Eq 3-79, the bond stiffness kt along the reinforement is written as: k t for max Eq 3-81 s 55

73 3.8. Summary This hapter presents the development of an analytial evaluation of bond apaity of orroded reinforement by aounting for the affeted mehanisms identified in the literature review, i.e., hemial adhesion, mehanial interlok and fritional stress. The analytial evaluation is based on the following: 1) If adhesive stress is not known, it is assumed to be a small value (0.8 MPa). Also, the adhesive stress is assumed to be onstant for all orrosion levels. ) The initial interlok stress is obtained by ombining the analytial expressions of bond apaity for ribbed and plain bars, as given by the CEB-FIP Model Code (010). A linear degradation is adopted for the mehanial interlok stress as the orrosion level inreases. 3) The fritional stress is expressed as the produt of the frition oeffiient, whose deterioration an be desribed by the model provided by Lundgren (007), and the onfining stress, whih is evaluated by a thik-walled ylinder model. The ylinder model is developed based on Wang et al. (004). However, the tensile strain distribution in the plasti region, i.e., that orresponding the raked onrete, is modified to improve the evaluation of onfinement stress due to orrosion. 56

74 4. Finite Element Model 4.1. Introdution This hapter presents the implementation of orrosion-indued damages in a D finite element (FE) model to simulate the deterioration of flexural behaviour due to reinforement orrosion. The finite element software ABAQUS has been used for this purpose due to the availablity of its user subroutine (UEL), where the analytial model that desribes bond behaviour an be programmed. First, the hapter presents the FE model of a D ontrol beam, with an investigation of the material models to be used and the seletion of FE types. The link element representing the bonding interation between the steel and the onrete is formulated aording to the analytial evaluation of bond degradation presented in Chapter 3. It is followed by a presentation of the implementation of orrosion-indued damages (redued ross-setional area of steel, onrete raking and degraded bond) in the FE modelling of affeted C flexural members. 4.. Seletion of material models To simulate the material behaviour in the numerial model, the stress-strain relationships of onrete and steel were studied first, in order to input data for the FE material model in ABAQUS Material behaviour of onrete Compressive behaviour The model for ompressive behaviour of onrete given in the CEB-FIP Model Code (010) has been used in this work. The relation between stress ompression behaviour is expressed as: and strain for short term uniaxial 57

75 k ' f 1 k for, lim Eq 4-1 where / 1, k E i / E 1, 1 is the strain at maximum ompressive stress, E 1 is the seant modulus from the origin to the peak ompressive stress, k is the plastiity number,, lim is the ultimate strain, ' f is the uniaxial ompressive strength, and Ei is modulus of elastiity. The shemati representation of Eq 4-1 is shown in Figure 4-1. Figure 4-1 Compressive behaviour for onrete (reprodued from CEB-FIP 010) In the CEB-FIP Model Code (010), the grades of onrete are lassified by their ompressive strength. For different grades of onrete, the material data are given in Table

76 Table 4-1 Material properties for different grades of onrete aording to CEB-FIP (010) Tensile behaviour The bi-linear stress-strain relations (Eq 3-8~Eq 3-30) that were presented in Chapter 3 are adopted in the FE model to model onrete under uniaxial tension. 4...Finite element material model of onrete There are three types of onrete models available in ABAQUS. In this work, the onrete damaged plastiity model, designed for appliations in whih onrete is subjeted to monotoni load, is used in the numerial simulations. In the onrete damaged plastiity model, onrete is modelled using a ontinuum, plastiity-based, damage model (Abaqus Analysis User's Manual 014). The model assumes that the main two failure mehanisms are tensile raking and ompressive rushing of the onrete material. The evolution of the yield (or failure) surfae is ontrolled by two hardening variables, pl t and pl, linked to the failure mehanisms under tension and ompression, respetively, where pl t and pl are the tensile and ompressive equivalent plasti strains, respetively. In this material model, five parameters need to be speified: (1) the dilation angle, () the flow 59

77 potential eentriity, (3) the ratio of initial biaxial ompressive yield stress to initial uniaxial ompressive yield stress, (4) the ratio of the seond stress invariant on the tensile meridian, and (5) the visosity parameter. Also, to desribe the stress-strain relationship for onrete, the plasti strain and orresponding stress are needed as input data to ABAQUS Seletion of parameters Sine the alulations for these parameters involve omplex mathematial derivations and assumptions using the yield surfae of the onrete damage plastiity model, the proposed values in other researhers work have been used here, and the verifiation of the model is performed by omparing the numerial results with published experimental data and is presented in Chapter 5. (1) Dilation angle A parametri study arried out by Malm et al. (006) suggested that there is not signifiant differene between 0 and 40 dilation angle if a reinfored onrete beam is subjeted to bending. The best agreement with experimental data was reahed for a dilation angle between 30 and 40. Lee and Fenves (1998) also performed a verifiation of this material model with experimental data from Kupfer et al. (1969), where a dilation angle of 31 was adjusted for uniaxial tensile and ompressive failures, whereas a value of 5 was used for biaxial ompressive failure. Hene, the dilation angle is seleted as 31 in this work. () Flow potential eentriity The flow potential eentriity is a small positive number, whih defines the rate at whih the hyperboli flow potential approahes its asymptote (Abaqus Analysis User's Manual 014). In the finite element simulation done by en et al. (014), the default value of 0.1 was used, whih fitted the experimental test. (3) atio of biaxial to uniaxial ompressive yield stress The biaxial stress ratio and the tensile-to-ompressive meridian ratio were assumed to be equal to 1.16 and 0.667, respetively, 60

78 based on reommendations of Chen and Han (1995). (4) atio of the seond stress invariant on the tensile meridian The ratio of the seond stress invariant on the tensile meridian to that on the ompressive meridian was taken as /3 (en et al. 014). (5) Visosity parameter Sine nonlinear material models with stiffness degradation often have onvergene issues in ABAQUS, applying the tehnique of visoplasti regularization is a ommon method to overome this diffiulty, whih for suffiiently small amount of time inrements, auses the onsistent tangent stiffness of the softening material to beome positive (Abaqus Analysis User's Manual 014 ). However, as suggested by Lapzyk et al. (007), the seletion of the value for the visosity parameter affets the stiffness model, and a low value of the visosity parameter should be seleted (small ompared to the harateristi time inrement). The authors also performed parametri studies, and they found that the speimens with a visosity parameter ranging from to had good agreement with experimental results. Thus, the above parameters are tested and alibrated on a ontrol beam before the orrosion-indued damage models are implemented, in order to apture experimental observations appropriately Calulation of plasti strain In ABAQUS, the hardening or softening behaviour after yielding is desribed by the plasti strain. Thus, a set of a plasti strain and orresponding stress is needed to speify the plasti region of the elasti-plasti materials that use the Mises or Hill yield surfae (ABAQUS User Manual 014). One the urve of stress-strain relationships in tension and ompression are defined aording to the previous setions, the plasti strain an be alulated by using the following equations: 61

79 pl Eq 4- t el whe f ' el Eq 4-3 E re t is the total strain, el is the elasti strain, pl is the plasti strain, f ' is the ompressive stress, and E is the elasti modulus of onrete Material behaviour of steel For simpliity, the onstitutive behaviour for steel is assumed to follow a bi-linear relationship in the finite element model. The asending urve has a slope whih is equal to Young s modulus E s. One the stress reahes the yielding stress f y, the stress is maintained onstant. Steel is assumed to behave the same in tension and ompression (Figure 4-). Figure 4- Bi-linear stress-strain relationship for steel 6

80 4..4.Finite element material model of steel The plasti model is adopted for steel elements. Similar to the onrete model, the definition of plasti strain is required. One the material properties for steel are obtained from experimental data, the input data are also alulated aording to Eq 4-and Eq Seletion of element types Sine a strutural member under bending does not experiene external loading in the z diretion, the D 4-node plane stress elements (CPS4) are utilized to model onrete (Figure 4-3). einforing bars are modelled as -node D truss elements (TD), whih only have stiffness in one diretion, rather than beam elements with stiffness in the x and y diretions (see Figure 4-4). It is assumed that tension bars are not taking any shear fore during the loading, i.e., dowel ation is not modelled. Figure node plane-stress element and -node truss element (reprodued from ABAQUS User Manual 014) Figure 4-4 Strutural member subjeted to bending 63

81 4.4. Formulation of link element To model the bond interation between onrete and reinforing steel, link elements are used at the interlayer (Figure 4-5). The finite element link element to model bond behaviour in C was first proposed by Ngo and Sordelis (1967), and it behaves similar to a nonlinear spring with stiffness kt in the longitudinal diretion, providing the bond stiffness along the reinforement. Figure 4-5 D finite element model The user-defined element subroutine (UEL) available in ABAQUS, whih needs to be programmed in Fortran, is used to model the proposed link element, whose bond-slip behaviour, i.e., bond apaity max (Eq 3-1) and stiffness k t (Eq 3-81), has been developed in Chapter Boundary onditions of D FE model After the seletion of the finite element material models and the element types, the D FE beam an be built to evaluate the flexural response of an C flexural member. Boundary onditions orresponding to the numerial simulation are presented in Figure 4-6. Due to symmetry, only half of the speimen is analyzed to redue the problem size. In order to model the boundary ondition of a full length speimen, the symmetry plane was restrained 64

82 against translation in the x diretion as shown in Figure 4-6, where H is the beam height, B is the beam width and L is the span of the beam. Figure 4-6 Boundary onditions of finite element model 4.6. Corrosion-indued damage models Damage on steel bar The ross setion of the reinforement dereases when the attak penetration x inreases, leading to degraded flexural stiffness and apaity. The expression for the effetive area of rebar is given by: A steel r x Eq 4-4 b where r b is the radius of the virgin bar and x is the attak penetration. The redued area of steel 65

83 an be speified for the truss elements (TD) in the ABAQUS model Damage on onrete In the D FE model, orrosion-indued raking in the onrete due to the volumetri expansion of rust produts is simulated by applying an equivalent expansion strain at the onrete elements surrounding the reinforement. As disussed in Chapter 3, the hoop strains for the partly- and fully-raked thik-wall ylinder an be evaluated from Eq 3-48~Eq Therefore, if the thik-wall ylinder is not fully raked, i.e.,, the average tensile strain along the ylinder model an be evaluated from: i 1 1 rb r i 1 i rdr rdr r 1 1 rb i rb r / i i dr 1 1 i i Eq 4-5 where is the radius of the thik-wall ylinder, rb is the radius of the reinforing bar, i is the radial displaement of the rak front, is the tensile strain, and r is the tensile strain orresponding to the tensile strength of the onrete. For the ase where the thik-wall ylinder is fully raked, i.e.,, the average tensile strain i is written as: x 1 U 1 1 rdr Eq 4-6 rb rb 66

84 Figure 4-7 Corroded reinforing bars in C beam ross setion In the D model, it is assumed that the reinforing bars at the same ross setion have the same orrosion level. Thus, by averaging the tensile strains over the width B of the beam (Figure 4-7), the resulting equivalent expansion strain an be obtained from: n n Eq 4-7 B i1 where is the average strain for eah reinforing bar and is alulated aording to Eq 4-5 or Eq 4-6, and n is the number of steel bars in the ross setion. The simulation of expansion damage is aomplished by using thermal expansion elements, whih are modelled as D truss elements (TD) with mehanial expansion. Those elements an expand 67

85 and ause damage in onrete if the temperature in the elements rises. Thus, by setting the thermal strain t equal to the equivalent expansion strain, the simulation orrosion produts build-up an be done, and the expansion elements an ause damage in the onrete surrounding the reinforing bar Damage on bond The deterioration of bond has been investigated in Chapter 3. By programming the relationship between attak penetration x and degraded bond apaity max (Eq 3-59~Eq 3-71) in the UEL subroutine, the deteriorated bonding behaviour orresponding to a ertain attak penetration x an be modelled in the FE model Summary of development of FE model The proedure for developing a FE model of an C flexural member with orrosion-indued damage inorporated is summarized in the flow hart shown in Figure 4-8, whih proeeds as follows: 1) The redution of ross-setional area of reinforement is simulated by reduing the area of D truss elements aording to Eq 4-4, when the attak penetration x is given. ) By applying an equivalent expansion strain, the raking in onrete due to volumetri expansion is modelled. The equivalent expansion strain is obtained by averaging the tensile strain in onrete ross-setion width (see Eq 4-7). 68

86 Figure 4-8 Proedure of FE model development 69

87 3) The loss of bond is modelled in the finite element model by utilizing the link element, in whih the analytial evaluation of bond degradation is programmed. The bi-linear bond-slip relation is adopted to desribe bond behaviour. Note that the orrosion stage and loading stage are two separate steps defined in ABAQUS, wherein the loading is applied after the orrosion stage is ompleted. 70

88 5. Model Validation 5.1. Introdution This hapter presents the validation of the analytial evaluation and D finite element models developed in Chapters 3 and 4. The verifiation of the analytial approximation of bond apaity is onduted first against the experimental and numerial pullout tests performed by Al-Sulaimani et al. (1990) and Lundgren (007), respetively, to ensure the bond model an simulate the experimental and numerial observations properly. Then, the appliation of the link element is tested by omparing three sets of bending experiments: (i) beams reinfored with ribbed bars (Mangat 1999), (ii) beams reinfored with plain bars (Cairns et al. 008), and (iii) beams affeted by uneven distribution of reinforement orrosion (Du et al. 007). Note that modelling bond for plain bars is the same as ribbed bars without aounting for mehanial interlok. To do so, ontrol finite element models without orrosion-indued damage are first built. After the ontrol models suessfully apture the experimental responses, the link elements are applied for different orrosion levels, and the load-displaement urves from FE results are plotted to determine if the link element an reprodue the orrosion-indued bond damage realistially. 5.. Validation of analytial evaluation of bond stress 5..1.Comparison with Al-Sulaimani et al. (1990) In this setion, the analytial bond model is validated against the experimental pullout tests performed by Al-Sulaimani et al. (1990), whih were onduted on 150-mm ubi onrete speimens with 10, 14, and 0-mm diameter ribbed bars embedded in the entre of the speimens 71

89 to obtain a over-to-rebar diameter (C/d) ratios of 7.50, 5.36, and 3.75, respetively. Diret urrent was impressed on the embedded bar to simulate the orrosion proess prior to the pullout test. The properties of materials used in the validation are listed in Table 5-1. Note that not all material properties needed for the analytial model are provided in Al-Sulaimani et al. (1990) (e.g., properties at the interfae). Thus, in order to determine those parameters, the material data suggested by the CEB-FIP Model Code (010) and observations from other researhers were adopted, whih were reviewed in Chapter. It is realized that some of these parameters are assumed due to the lak of available data. Further researh on quantifying them is needed Conrete Table 5-1 Material properties used in the analyses Young s Modulus E 3 GPa Tensile strength Steel ' ft.6 MPa Young s Modulus Interfae Es 00 GPa *Initial frition oeffiient *Chemial adhesion stress Fa 0.8 MPa ust produt *atio between the volume of oxides produt and virgin steel n Note: Properties marked by * are determined aording to the literature review. The analytial alulations were obtained by inputting the material properties provided in Table 7

90 5-1 into the analytial evaluation developed in Chapter 3. The results are ompared with experimental data in Figure 5-1 to Figure 5-3. From the figures, it is observed that there is a good agreement between the analytial model and the experimental observation. Similar to the experimental results in Al-Sulaimani et al. (1990), it is observed that there is an initial inrease in the bond apaity at low levels of orrosion for all results alulated from the analytial model, followed by ontinuously dereasing bond apaity as the perentage of orrosion-indued mass loss is inreased. Both analytial and experimental results have similar bond degradation trend. Figure 5-1 Comparison of analytial model to experimental results of speimen with 10-mm diameter reinforement 73

91 Figure 5- Comparison of analytial model to experimental results of speimen with 14-mm diameter reinforement Figure 5-3 Comparison of analytial model to experimental results of speimen with 0-mm diameter reinforement 74

92 In addition, the speimens with higher over-to-rebar diameter ratio (C/d) tend to have a larger bonding apaity, sine the thik-wall ylinder model used in Chapter 3 leads to higher raking resistane when the onrete over inreases, providing, therefore, a higher onfining stress and bonding resistane when the rust produts aumulate around the reinforing bar. Notwithstanding, an overestimation of the maximum bond apaity predited by the analytial model is noted at high levels of orrosion. Note that these divergenes result from the seletion of material models, suh as the linear post-raking model of onrete, sine the onfining apaity of the thik-wall ylinder model is signifiantly affeted by the residual tensile stress after the raking in onrete is initiated, or the assumption of radial displaement along the onrete over, whih is a ruial fator for the evaluation of strain in onrete. 5...Comparison with Lundgren (007) The validation for speimens reinfored with plain rebars was performed on the finite element (FE) pullout test onduted by Lundgren (007), whih was designed to investigate the degrading bond response of models with varying physial properties when the orrosion attak penetration inreases. Speimens with different onfinement onditions (with or without transverse reinforement) and over thikness (all summarized in Table 5-) were modelled by axisymmetri elements in the FE tests onduted by Lundgren (007). The analysis setup, shown in Figure 5-4, is a ylinder speimen with a 10-mm radius reinforing bar entrally embedded in the speimen. The reinforing bar was pulled out while the onrete was restrained, as illustrated in Figure

93 Figure 5-4 Setup of finite element model in Lundgren (007) (all dimensions are in mm) Table 5- Geometry of FE pullout test speimen Bar type Smooth bar Cover thikness b (mm) Distane from the edge of tension bar to stirrups a (mm) N/A * N/A * Speimen ref. # * N/A means no stirrup provided The material properties used in the finite element analysis onduted by Lundgren (007) are summarized in Table 5-3. As the yield strength for reinforement was not speified in the finite element test, the yielding of the steel bar was taken as 500 MPa. Sine physial properties in the 76

94 interfae, suh as adhesive stress and initial frition oeffiient, were provided by Lundgren (007), those properties were used rather than being assumed aording to the literature review. Conrete Table 5-3 Material properties for FE pullout speimen Young s Modulus E 34. GPa Tensile strength ' ft 3 MPa Frature energy G f 79. N / mm Stirrups Yielding strength Young s Modulus f y Es 500 MPa 00 GPa Diameter db 6 mm Interfae Initial frition oeffiient Chemial adhesion stress Fa MPa ust produt atio between the volume of oxides produt and virgin steel n Figure 5-5 to Figure 5-8 show the omparisons between the bond apaity degradation obtained from the analytial model and the numerial results obtained by Lundgren (007) for eah of the four types of speimens. Figure 5-5 and Figure 5-6 show results for unonfined speimens, whereas results for onfined speimens are illustrated in Figure 5-7 and Figure

95 Figure 5-5 Maximum bond stress versus orrosion penetration for speimen #1 Figure 5-6 Maximum bond stress versus orrosion penetration for speimen # 78

96 Figure 5-7 Maximum bond stress versus orrosion penetration for speimen #3 Figure 5-8 Maximum bond stress versus orrosion penetration for speimen #4 79

97 Overall, the omparisons shown in Figure 5-5 to Figure 5-8 indiate that the analytial approximations apture the finite element results obtained by Lundgren (007), sine similar values of maximum bond apaity and inreasing/dereasing trends are noted when the attak penetration x inreases. Also, the models that are able to aount for the onfining stress, i.e., speimens with higher over-to-diameter ratio (C/d) and transverse reinforement, tend to predit higher maximum bond apaity. However, some differenes between the two sets of results are also found in the omparisons. In groups #1 and #, both sets of analytial results reah their maximum values at lower levels of attak penetration than the numerial results. The FE results for speimen # degrade more rapidly as the orrosion level inreases ompared to the analytial model s results. As for results of groups #3 and #4, where speimens were reinfored by transverse rebars, although the differenes are smaller than in groups #1 and #, some divergenes are notied, for instane, in groups #3 bond degradation as given by the analytial model starts at 10 m of attak penetration as opposed to that predited by the FE analysis, in whih the bond apaity keeps inreasing up to 00 m of attak penetration. Like the validation against the experimental results reported by Al-Sulaimani et al. (1990), the differenes between the two sets of results in Figures 5-8 an result from the different material models used in the analytial evaluation. Hene, the major fators that affet the results of the analytial model, suh as the tensile strength of the onrete and the seletion of the volume ratio between oxides and virgin steel n, are investigated through a parametri study in Chapter 6. 80

98 5.3. Validation of finite element model Comparison with Mangat (1999) Mangat (1999) onduted 4-point flexural testing of a total of 111 under-reinfored onrete beams reinfored with ribbed bars and subjeted to different attak penetrations. The beams were 150 mm 100 mm in ross setion, and 910 mm in length. Two ribbed bars (10-mm diameter) were used as tension reinforement, proteted by a 0-mm onrete over (Figure 5-9). The reinforement in the beams was subjeted to an aelerated orrosion proess before loading, wherein a mass loss ranging from 1.5 to 10% was indued. The authors reported the load-displaement responses for eah of the tested beams. In order to develop full flexural apaity, one beam was reinfored with stirrups, while the other beams were reinfored in shear with external steel ollars. The required material parameters for the finite element analysis are summarized in Table 5-4. Note that the average ompressive strength of the onrete ubes after 8 days was 40 MPa in the experiment. Other material properties, e.g., the uniaxial tensile strength for onrete, were determined from the CEP-FIP Model Code (010) Control beam A ontrol beam without orrosion-indued damage and without link elements was first analyzed, as illustrated in Figure The FE mesh onsists of 1,904 nodes, 1,73 onrete elements (plane-stress elements CPS4), whose dimension is mm, and 68 -node truss elements to represent the reinforing bars. The external steel ollar was modelled using 180 -node truss (TD) elements. 81

99 Table 5-4 Material properties for beams tested by Mangat (1999) Conrete Compressive strength ' f 40 MPa Young s modulus Tensile strength ' ft E 33.6 GPa.9 MPa Maximum oarse aggregate size Steel da 10 mm Yielding strength Young s modulus Interfae f y Es 50 MPa 06 GPa *Initial frition oeffiient *Chemial adhesion stress Fa 0.8 MPa ust produt *atio between the volume of oxides produt and virgin steel n Note: Properties marked by * are seleted aording to the literature review. Perfet bond was assumed in the analysis of the ontrol beam. The assumption of perfet bond is widely applied in normal analysis and design of C, so it is reasonable to use it to represent the bond behaviour between onrete and unorroded ribbed bars. The load vs mid-span displaement urve from the experimental and numerial results is highlighted in Figure As it is observed from the figure, the finite element analysis suessfully models the experimental results of the ontrol beam reported by Mangat (1999), although the overestimation of flexural strength at large defletions (around 8%) is notied. 8

100 Figure 5-9 Geometry of beams tested by Mangat (1999) Figure 5-10 FE mesh of ontrol beam as displayed in ABAQUS 83

101 Figure 5-11 Load-displaement response for ontrol beam with perfet bond Appliation of link element for unorroded beam One the verifiation of the finite element model had been done, the validation of the link element for an unorroded beam was onduted. The link elements, where 0% orrosion was speified, were plaed at the interfae between the onrete elements and the truss elements representing the reinforing bars (Figures 4-5 in Chapter 4). Using the same meshing as shown in Figure 5-10, the FE speimen was subjeted to a 4-point load analysis. Figure 5-1 shows the resulting flexural response. Comparing both experimental and numerial responses, the FE results aptures the overall load-deformation response, although the FE model slightly underestimates the strength and stiffness up to a 6-mm mid-span defletion. 84

102 Figure 5-1 Load-displaement response for ontrol beam with link elements Figure 5-13 shows the omparison of the load-displaement urves for FE results with the assumption of perfet bond and using link elements to simulate bonding ation. The use of link elements results in a softer response, with a slightly lower stiffness and strength. The reason for this variation is that at 0% orrosion, the link elements still have limited stiffness along the reinforing bar, differing from perfet bond where no slip is allowed between nodes, and this results in a lower bond stress than a speimen modelled under the assumption of perfet bond. 85

103 Figure 5-13 Comparison of load-displaement response for both sets of ontrol beams Appliation of link element for orroded beam Corrosion-indued damage was modelled aording to the development of analytial and numerial model presented in Chapters 3 and 4, respetively. The redued ross-setion area was onsidered aording to Eq 4-4. Bond behaviour was modelled with the link elements. Finally, orrosion-indued raking of the onrete over resulting from the orrosion produts aumulation around the reinforing bar was modelled using expansive elements (see Setion 4.6.). Figure 5-14 illustrates the load-displaement urves from the numerial model using link elements at a level of orrosion of 0%, 1.5%,.5%, 5% and 10% (measured by the ratio of attak penetration x to radius of reinforement r b ). 86

104 Figure 5-14 Load-displaement responses from finite element models Figure 5-15 Load-displaement responses for tested speimens (reprodued from Mangat (1999)) From Figure 5-14, it is observed that flexural behaviour in beams degrades as the orrosion level 87

105 in the tension reinforement inreases. Under the same level of loading, the orroded speimens displayed a larger mid-span defletion, this degradation being more severe in beams with higher orrosion attak penetration. Similar observations were noted in Mangat (1999), as shown in Figure 5-15, whih indiates the FE model is able to simulate orrosion-indued flexural deterioration, and the FE model with link elements is sensitive to the degree of tensile reinforement orrosion. The omparisons of the finite element model results against the experimental results by Mangat (1999) are also presented in Figure 5-16 to Figure 5-19 for 0%, 1.5%,.5%, 5% and 10% orrosion, respetively. Both numerial results orrelate well with the experimental flexural response, although the speimen with 10% orrosion shows the largest differenes, wherein the flexural strength is overestimated, whih might result from the overestimation of bond apaity at high orrosion levels. Sine the omparisons show that the finite element model using link elements an reprodue the degradation of the overall strutural behaviour of the orroded beams reasonably well, it an also be used as a pratial tool to predit the flexural strength of C beams with orroded reinforement. Figure 5-16 Load-displaement response for beams with 1.5% orrosion 88

106 Figure 5-17 Load-displaement response for beams with.5% orrosion Figure 5-18 Load-displaement response for beams with 5% orrosion 89

107 Figure 5-19 Load-displaement response for beams with 10% orrosion 5.3..Comparison with Cairns et al. (008) Sine some aged buildings were onstruted and reinfored with plain bars before the appliation of ribbed bars, further validation was onduted by analyzing tests performed by Cairns et al. (008), in whih the speimens were reinfored with plain round steel bars and stirrups. The bonding apaity of orroded plain reinforement rises to a notieable extent even at relatively large attak penetrations (Lundgren 007), although it is still lower than that of deformed bars due to the lak of mehanial interloking. This is beause for unorroded plain rebars, the frition stress is equal to zero as no onfinement stress is introdued aording to the analytial model in Chapter 3, and the bond apaity only results from the hemial adhesion between the steel and the onrete, whereas for orroded plain rebars, the bond apaity inreases as the frition inreases due to the aumulation of orrosion produts at the steel/onrete interfae. This results in an inrease in the overall flexural stiffness and strength, even though orrosion-indued 90

108 damages, suh as redued ross-setional area of steel and orrosion produts expansion, are introdued. Cairns et al. (008) observed this enhaned response for orroded beams as well. Therefore, the verifiation of the link element with plain reinforement an be done by observing if the finite element model is able to apture this inreased in flexural response as the orrosion level inreases. Cairns et al. (008) tested four types of beams: (i) simply-supported beams lightly reinfored (0.6% reinforement ratio) in flexure (denoted as sbf), (ii) simply-supported beams heavily reinfored (.3% reinforement ratio) in flexure (denoted as sbs), (iii) two-span ontinuous beams with ontinuous reinforement (bf), and (iv) two-span ontinuous beams with reinforement lapped near points of ontra flexure (dbf). All these beams were subjeted to different orrosion levels before being tested in bending. The validation is onduted by modelling the speimen type sbf, whose geometri details are illustrated in Figure 5-0. The beam speimen had overall setion dimensions of 150 mm 00 mm, and it was reinfored with plain round mild steel. The onrete over to flexural reinforement was 0 mm. There were two 10-mm diameter top bars and two 10-mm diameter bottom bars plaed at the orners of the speimens. Six-mm diameter shear reinforing bars with a spaing of 15 mm were also provided to prevent shear failure. Note that onrete ube strength reahed 43. MPa by the time the first load tests were arried out. Other material parameters used in the FE analysis are obtained from the CEB-FIP ode (identified as C35 grade onrete), and they are listed in Table

109 Table 5-5 Material properties for sbf beam Conrete Compressive strength ' f 43. MPa Young s modulus Tensile strength Steel Tension bar Yielding strength Young s modulus Stirrups Yielding strength Young s modulus Interfae ' ft E f y Es f y Es 35 GPa 3.1 MPa 34.7 MPa 00 GPa 38.5 MPa 00 GPa *Initial frition oeffiient *Chemial adhesion stress Fa 0.8 MPa ust produt *atio between the volume of oxides produt and virgin steel n Note: Properties marked by * are seleted aording to the literature review. 9

110 Figure 5-0 Geometry of speimen sbf (in mm) (reprodued from Cairns et al. 008) Control speimen A similar modelling tehnique to the one used in the validation of beams with ribbed reinforement is also adopted here. The D ontrol beam with 0% orrosion is modelled first. However, sine the beam reinfored by plain bars has lower bond apaity, the perfet bond assumption annot be used. Therefore, the ontrol speimen is modelled using the link elements but without any orrosion-indued damage (Figure 5-1). Figure 5-1 FE mesh of ontrol beam as displayed in ABAQUS The omparison between experimental and numerial results for the ontrol beam is presented (see Figure 5-). Note that only two signifiant points in the load-defletion urve were provided by Cairns et al. (008): yielding of reinforement and ultimate apaity. In Figure 5-, the FE model 93

111 underestimates the flexural apaity at yielding point. However, as the mid-span defletion inreases, the response stiffens, whih leads to a small overestimation of flexural strength (around 5%) at large levels of deformation. Figure 5- Load-displaement response for speimen sbf Corroded speimen The experimental results for group sbf are presented in Table 5-6. In this table, the orroded beams, i.e., sbf-01 and sbf-05, show similar responses, although the latter beam (sbf-05) has slightly stiffer performane, sine larger flexural stiffness is observed. Also, the orroded speimens have higher flexural stiffness than the unorroded one (sbf-00), whih may result from the initial inrease of bond strength due to the aumulation of orrosion by-produts at the onrete/steel interfae. 94

112 Table 5-6 Experimental results for group sbf Speimen *Flexural Yield Defletion Maximum Defletion at Mean attak ref. stiffness load at yield load max load penetration ( kn / mm ) (kn) (mm) (kn) (mm) (mm) sbf sbf sbf *measured by overall deformations Using the FE model that aounts for orrosion-indued damages, the load-defletion responses for sbf-01 and sbf-05 are illustrated in Figure 5-3 and Figure 5-3, respetively. Figure 5-3 showases good agreement between the numerial and experimental results of speimen sbf-01, sine the FE model for sbf-01 aptures the two signifiant points provided by the experimental results. Also, both sets of results have a similar flexural stiffness before yielding of the speimen. In Figure 5-4, although a reasonably good orrelation is noted in the omparison, the FE analysis of speimen with 0.3mm attak penetration (sbf-05) tends to slightly underestimate the flexural response (for both yielding and maximum points). Overall, the finite element models were able to reprodue the experimental data. 95

113 Figure 5-3 Load-displaement response for speimen sbf-01 Figure 5-4 Load-displaement response for speimen sbf-05 96

114 The three sets of numerial results (sbf-00, sbf-01 and sbf-05) are ompared in Figure 5-5. Although orrosion-indued damages are inluded in the analyses, a stiffer flexural response as orrosion inreases is still observed; this is due to a higher bond stress resulting from the enhanement of onfining stress at the interfae and reflets the experimental observations reported by Cairns et al. (008) Figure 5-5 Comparison of finite element models However, unlike the results observed in the experiments, wherein the speimen with the highest level of orrosion (sbf-05) had the highest flexural stiffness, the numerial results of speimen sbf-05 presented a softer response ompared to sbf-01. It an be explained by the fat that aording to the analytial bond evaluation for speimens tested in Cairns et al. (008) (Figure 5-6), the bond deterioration starts at 0.1-mm attak penetration, leading to a lower bond apaity in sbf

115 Figure 5-6 Analytial evaluation of bond strength for speimens in group sbf This senario may result from the seletion of the volume ratio between oxides and virgin steel n, sine it an impat the rate of bond deterioration. For instane, a higher value of n an ontribute to an earlier bond deterioration and a shorter period of initial inrease in bond strength. In this validation, the ratio n is hosen as aording to the literature review. However, based on the results from the numerial model, a smaller n should have been seleted for this experiment, whih would have set the attak penetration for bond deterioration to a higher value than 0.1 mm. Thus, to study the influene of the seletion of this parameter, a parametri investigation using the FE model with varying ratios of n is performed in Chapter 6. 98

116 5.3.3.Comparison with Du et al. (007) The assumption of uniformly distributed orrosion along the reinforing bars may not be appliable for all orrosion ases. For instane, orrosion-indued damage is ommonly found in strutures in oastal or old regions, where onrete over ontamination by hlorides is the major ause of reinforement orrosion. It is ommonly observed in C strutures affeted by hloride-indued orrosion that damage is aused by loalized orrosion reated by formation of pits on the steel bar surfae. In order to evaluate the strutural performane of onrete strutures suffering from hloride-indued orrosion, the influene of loalized orrosion should be onsidered. To validate the appliability of the finite element model to beams with loalized orrosion, the bending tests of orroded beams performed by Du et al. (007) were analyzed. Only the mid-span regions of these beams were orroded artifiially before being loaded. The beam speimens had dimensions of ,100 mm, and the length of the orroded tension bar was 600 mm (Figure 5-7). There were 19 sets of speimens tested in this experimental study; they were identified by perentage of tension reinforement and the loation where orrosion was indued (ompression bars, tension bars and/or stirrups) and. The speimens whose tension bars were orroded, i.e., ontrol beam (T680) and 8.8% orroded beam (T68) (measured by mass loss) were seleted in this study to further validate the FEM. As the onrete ube strength is given as 36 MPa, the remaining material parameters an be evaluated from CEB-FIP (010) and are summarized in Table

117 Table 5-7 Material properties of speimens tested by Du et al. (007) Conrete Compressive strength ' f 36 MPa Young s modulus Tensile strength Steel Tension bar ' ft E 33.6 GPa 3.0 MPa Diameter db 16 mm Yielding strength f y Young s modulus E s 59 MPa 01 GPa Stirrup and ompression bar Diameter ds 8 mm Yielding strength Young s modulus Interfae f y Es 56 MPa 03 GPa *Initial frition oeffiient *Chemial adhesion stress Fa 0.8 MPa ust produt *atio between the volume of oxides produts and virgin steel n Note: Properties marked by * are seleted aording to the literature review. 100

118 Figure 5-7 Bending test setup arried out by Du et al. (007) Control speimen The ontrol beam was modelled prior to the introdution of orrosion-indued damages. The FE meshing is plane-stress elements and shown in Figure 5-8. As in the analysis of the beams tested by Mangat (1999), the differene between the FE model with perfet bond and with 0% orrosion level link elements is not signifiant. Hene, outside the orrosion region, perfet bond is assumed to redue the required time for modelling. As a result, the link elements were not used in modelling and analyzing the ontrol speimen. Figure 5-8 FE mesh of ontrol beam as displayed in ABAQUS The flexural response of the ontrol beam is illustrated in Figure 5-9. The finite element model is able to simulate the experimental result reasonably well, although it is observed from the figure 101

119 that the numerial results had an initial stiffer response and a slightly higher flexural strength. Figure 5-9 Load-defetion response for ontrol speimen in Du et al. (007) Corroded speimen The orrosion level in the experiment was measured by mass loss, whih an be evaluated by: M loss b rb x 1 Eq 5-1 r where M loss is the perentage of mass loss, rb is the original radius of rebar and x is the attak penetration. Hene, a orrosion attak penetration is alulated as 0.36 mm from an 8.8% mass loss. In the experiment, although the reinforing steel bars were orroded separately, the 10

120 non-urrent-impressed bars were also orroded within the intended orrosion region, whih means the transverse reinforement in the seleted speimen T68 was also damaged. However, the study of strutural deterioration due to orrosion of stirrups is not the fous of this work. Therefore, the influene of orrosion-indued damage on the stirrups was not taken into onsideration in the finite element model. Sine the strutural damage resulting from orrosion was only introdued within the orroded region of the tension bar, the link elements and the equivalent expansion strain were applied at the highlighted (orroded) region in Figure Figure 5-30 Corrosion-indued damaged region in FEM Figure 5-31 presents the load-defletion urve for the FE analysis of the beam with loalized orrosion along its mid-span. As shown in the figure, the numerial results ompare well with published data in Du et al. (007), as the damaged FE model an simulate the same degree of flexural deterioration. The finite element model with orrosion-indued damage slightly overestimates the initial stiffness and flexural apaity. This might be explained on the basis of the overlooked shear reinforement orrosion, whih an lead to further deteriorated flexural response. 103

121 Figure 5-31 Load-defetion response for orroded speimen with 8.8% mass loss along mid-span The omparison of the load-deformation responses of the FE analysis without and with orrosion is presented in Figure 5-3. Although both models estimate the same flexural stiffness before yielding, there is flexural stiffness and strength deterioration in the orroded speimen, beoming inreasingly notieable at large mid-span defletions. In Du et al. (007), orrosion-indued damage was only introdued along the mid-span region. However, the effet of different loations for orroded reinforement has not been investigated. Therefore, it is worthwhile to extent this study by analyzing the FE beams with orrosion ourring along different regions of the reinforement. esults of this analysis are presented in Chapter

122 Figure 5-3 Comparison of load-defetion responses for FE ontrol and orroded speimens in Du et al. (007) 5.4. Summary The analytial and finite element models are verified against experimental observations and previous numerial studies. The following onlusions an be drawn: 1) The analytial model for bond is able to apture degradation experiened with inreasing orrosion levels. ) The appliation of link elements that mimi the bonding interation between onrete and rebar suessfully models the strutural response of ontrol flexural members. 3) Good agreement has been showased between the finite element model with orrosion-indued damages aounted for and artifiially orroded speimens tested by other researhers. 105

123 After the validation of both models, they are utilized to ondut a parametri investigation to study how the important effets identified in the verifiation study impat the deterioration of an C flexural member affeted by reinforement orrosion. The results are presented in the following Chapter

124 6. Parametri Study 6.1. Introdution This hapter presents the results of a parametri study onduted using the analytial and finite element models developed and validated in previous hapters, in order to investigate the influene of various physial properties on the bond apaity and residual flexural response of beams when orrosion-indued damage is present. Based on the results from the validation analyses presented in Chapter 5, a parametri study of the effet of (i) onrete strength, (ii) over-to-bar diameter ratio, and (iii) volume ratio between oxides and virgin steel was arried out using the analytial model introdued in Chapter 3. The parametri study was performed on the numerial pullout test onduted by Lundgren (007), used for validation in Chapter 5. The finite element model presented in Chapter 4 was then used to study (i) the influene of orrosion-indued damage (i.e., the damage on the reinforing bar, onrete and bonding) and (ii) orrosion loation on the residual behaviour of C beams affeted by flexural reinforement orrosion. The parametri analysis using the FE model used as a referene the C beams tested by Mangat (1999). 6.. Parametri study using the analytial model Sine the analytial evaluation of the bond apaity of group#1 speimens analyzed by Lundgren (007), whose material parameters are summarized in Table 6-1, has been validated in Chapter 5, this group of speimens was hosen as the ontrol speimen in the parametri study. An illustration of this ontrol speimen is shown in Figure

125 Table 6-1 Material data used in Lundgren (007) Conrete Compressive strength ' f 40 MPa Cover-to-bar diameter ratio (C/d) 1 (the thikness of onrete over C = 0 mm) Interfae Initial frition oeffiient Chemial adhesion stress Fa MPa ust produt atio between the volume of oxides produt and virgin steel n 6..1.The effet of onrete strength As mentioned in Chapter 5, if any of the material input parameters needed for the analysis is not provided by Lundgren (007), it is estimated from the material models suggested by the CEB-FIP Model Code (010), whih provides important mehanial properties for onretes with different ompressive strengths. However, the models found in CEB-FIP (010) were still used to investigate the influene of onrete strength by estimating the orresponding tensile strength and elasti modulus (see Table 6-). The bond apaity as alulated by the analytial model is illustrated in Figure 6-1 for speimens made of onrete with ompressive strengths of 30 MPa, 40 MPa and 50 MPa. 108

126 Table 6- Material data for onrete as provided by CEB-FIP(010) Compressive strength f Tensile strength f ' Elasti modulus E ' t 30 MPa.9 MPa 33.6 GPa 40 MPa 3.5 MPa 36.3 GPa 50 MPa 4.1 MPa 38.6 GPa Figure 6-1 Comparison of bond apaity for speimens with different onrete ompressive strength It is observed from Figure 6-1 that the model with higher onrete strength and stiffness tends to have higher bond apaity at low levels of reinforement orrosion in omparison to the model with lower onrete grade. This phenomenon is antiipated, sine the ultimate onfinement stress is evaluated by the thik-wall ylinder analogy in the analytial model, and the tensile apaity of the onrete over is diretly related to the tensile strength of the onrete. The higher this apaity is, the higher the onfining stress and resistane to raking provided by the onrete 109

127 over are, leading to a higher bonding apaity of the reinforing bar. However, a more rapid bonding deterioration is also noted in the speimen with higher onrete grade, and it results from the seletion of post-raking material models. In this work, the post-raking model for onrete was introdued in Chapter 4, wherein the ultimate strain u is influened by the frature energy of onrete (Eq 3-31) and alulated using Eq 3-3, where the harateristi width w is determined by the maximum aggregate size in onrete, whih is assumed to be 10 mm for all speimens. For omparison, the frature energies Gf and ultimate strains u for different onretes are listed in Table 6-3, and the orresponding tensile stress-tensile strain urves are plotted in Figure 6-. Table 6-3 Frature energy G f and ultimate strain u for different onretes Compressive strength (MPa) ` Frature energy G f 73 f ( N / m ) Ultimate strain u G f ' t f w In Figure 6-, although the higher grade onrete has higher tensile strength ` ft and frature energy G f, its ultimate strain u is smaller aording to the softening model for tensile strength suggested by Bazant (1986). Therefore, the tensile stress of lower grade onrete delines to zero with inreasing tensile strain at a slower rate. For instane, at a strain of in Figure 6-, the tensile strength of onrete with 30 MPa ompressive strength exeeds the tensile strength in a onrete with 50 MPa ompressive strength, and it ontinues to provide higher tensile resistane at 110

128 larger levels of tensile strain. Figure 6- Tensile stress-strain relation for onretes with different ompressive strengths 6...The effet of the ratio n Aording to the analytial model presented in Chapter 3, the approximation of radial displaements along the onrete over is important for evaluating the tensile strain in the thik-walled ylinder, and it is greatly impated by the ratio between the volume of oxide produts and virgin steel n. The ratio n orresponding to different orrosion produts is shown in Figure 3-. To investigate the influene of this parameter, the analytial model was used to alulate the bond apaity of speimens with a ratio n equal to,.5 and 3, whih are marked as N, N3.5 and N3, respetively. The analytial results are plotted in Figure

129 Figure 6-3 The omparison of bond apaity of speimens N, N3.5 and N5 Figure 6-3 indiates that models with different values of n have similar bond apaities but different rates of deterioration. The speimen with the highest value of n (N3 speimen) reahes its maximum value at a smaller value of attak penetration (10μm ) but degrades to zero at a more rapid rate, whereas the bond apaities of models with lower n ( and.5) peak at a larger attak penetration (μm and 18μm, respetively). In addition, the fat that similar bond apaities are observed in Figure 6-3 implies that the ratio between the volume of oxides and virgin steel only ontrols the rate of bond degradation. It an be explained by the fat that in the analytial model, the value of n is used to evaluate the volume of oxide produts, i.e., Vr nv, where V s r is the volume of oxides and Vs is the virgin steel. A higher volume of rust produts results in larger radial displaement at the onrete/steel interfae, initiating raking earlier on, but it does not affet the onfining strength in the thik wall ylinder as no onfinement ondition is altered. Lundgren (00) suggested that the value of n varies based on the different orrosion produts 11

130 formed around the reinforing steel. It is realized that, although the analytial model is highly sensitive to this parameter, the assessment of the atual orrosion produts formed might not be pratial for an existing affeted struture. Therefore, evaluations performed with the analytial model an be based on a range of reasonable assumptions for n if this kind of assessment is not feasible The effet of over-to-rebar diameter ratio (C/d) The ratio of the minimum thikness of the onrete over to the rebar diameter, C/d, is a parameter that provides an indiation of the protetion provided against orrosion, and its effet on bonding apaity has been investigated by many researhers (e.g., Al-Sulaimani 1990). Minimum onrete overs required by CAN/CSA-A are 40 mm for C beams exposed to freezing and thawing in a saturated ondition but not to hlorides, and 60 mm for C beams exposed to more severe environments where hlorides are antiipated. The strutural members with thiker over not only have a higher physial protetion against orrosion initiation, but also provide stronger mehanial resistane to bond degradation. Therefore, to study the bond deterioration of C members with various protetion overs, speimens with a onrete over of 0 mm, 40 mm and 60 mm were modelled and analyzed with the analytial model, resulting in over-to-rebar diameter (C/d) ratios of, 4, and 6, respetively, denoted as, 4 and 6. Figure 6-4 illustrates that the bond apaity of speimens with a higher C/d ratio have a longer initial build up to ultimate bond stress, whih ours at a higher level of orrosion attak. The improvement in bond apaity results from a higher resistane to raking and larger onfining stress when the onrete over inreases. 113

131 Figure 6-4 Comparison of bond apaity for speimens with C/d = (), C/d = 4 (4) and C/d = 6 (6) Parametri study using the finite element model The effet of different types of orrosion-indued damage As mentioned in Chapters 3 and 4, there are three kinds of orrosion-indued damage implemented in the finite element model: (1) the redution of the ross-setional area of the reinforing steel bar, () onrete raking due to orrosion produts expansion, and (3) bond deterioration. In order to investigate how eah of those 3 types of orrosion-indued damage impat the flexural behaviour of an affeted C beam, finite element analyses were onduted aounting for only one of the models above at a time. 114

132 The C beam with 5% orrosion mass loss in the bending test performed by Mangat (1999), validated in Chapter 5, was seleted for this study. The following finite element analyses were onduted: (1) FE model with only 5% redution of ross-setional area of the reinforing steel bar element (denoted as FE5%S), () FE model with only onrete raking due to 5% orrosion expansion (denoted as FE5%C), and (3) FE model with only the bond deterioration orresponding to a 5% steel mass loss (denoted as FE5%B). Figure 6-5 shows the experimental results obtained by Mangat (1999) with no steel orrosion (ontrol speimen), finite element results FE5%S, FE5%C and FE5%B, and the finite element results when all orrosion-indued damages are inluded in the analysis (FE5%A). From Figure 6-5, it is observed that the flexural behaviour is very similar up to steel yielding for the ontrol beam and a beam with only 5% redution in the reinforement ross setion. As yielding of the reinforement starts to take plae, the inlusion of ross-setional redution results in a derease in the flexural apaity. The results of the FE model with only orrosion-indued raking aounted for (FE5%C) show a notieable redution in stiffness at the initial loading stage, although its residual strength is not greatly impated by the indued damage at large deformations. Compared to the numerial results of FE5%S and FE5%C, the FE analysis of FE5%B displays the most signifiant deterioration in bending apaity, whih implies that bond redution is the main ause of orrosion-indued strutural degradation of C beams. Also shown in the figure are the results of the FE analysis when all orrosion-indued damage models are aounted for (FE5%A). It is noted that the results indiate deterioration in both residual apaity and stiffness. Therefore, in order to model realistially the degradation of flexural members due to reinforement orrosion, the damage indued in the reinforing steel, onrete and bonding should all be inluded in the finite element analysis. 115

133 Figure 6-5 Load-defletion urves for beams with different types of orrosion-indued damage aounted for 6.3..The effet of orrosion loation along the reinforement The assumption of uniform distribution of orrosion along the reinforement is not always valid due to different types of orrosion (general or loal) and expeted orroded regions. Some strutural omponents may only suffer severe orrosion at the mid-span region, whereas the effet of orrosion at the supports maybe more pronouned in other beams. The deterioration of flexural response may vary depending on the loation and extent of orrosion-indued damage in an C 116

134 beam. In order to investigate the influene of orrosion loation and extent along the reinforement, the speimen tested by Mangat (1999) was hosen as a ontrol speimen for the FE analyses. There were three finite element models with different orroded regions analyzed (Table 6-4): (1) FELM, whose tension reinforement was orroded 400 mm at mid-span, () FELS, whih had a length of 00 mm of orroded tension reinforement near eah support, and (3) FELA, where the tension bar was orroded along the entire span. The orrosion level used in the analysis was that orresponding to 5% of steel mass loss. Note that no orrosion-indued damage model was applied to regions outside that where reinforement was assumed to orrode. The load-defletion urves for the three FE models presented in Table 6-4 are presented in Figure 6-6. Comparing the three responses, it is noted that the most severe flexural deterioration ours in speimen FELA, whih had the longest length of orroded tension reinforement. The FE results for FELM and FELS, where only the mid-span or support regions were orroded, respetively, show that the latter speimen suffers more degradation of stiffness and apaity. In fat, by omparing the results of the FELS and FELA analyses, it is observed that most of the deterioration arises from the orroded reinforement near the supports. This phenomenon shows that strutural deterioration due to orrosion beomes more severe if reinforement orrosion is initiated near the supports of the C beam. 117

135 Table 6-4 Finite element speimens with various orroded regions Speimen ref# Corroded region FELM FELS FELA 118

136 Figure 6-6 Load-defletion urves for beams with different orrosion regions As disussed before, the volumetri expansion due to orrosion produts build-up at the onrete/steel interfae auses splitting of the onrete over and bond deterioration. If the flexural omponents suffer very high levels of orrosion, the onrete over might spall off, leading to no bond interation remaining at the damaged region. To investigate the orrosion loation effet in these extreme ases, finite element analysis were onduted in whih the FE models of the beams had regions of onrete removed, simulating the severe damage state just disussed (see Table 6-5). Two senarios were onsidered: (1) onrete was removed along the mid-span of the beam, and () onrete was removed near supports. The height of the removed onrete was equal to the diameter of the thik-wall ylinder model (50 mm), and the total length of exposed region was 400 mm for both models, with ase () having 00 mm of onrete removed near eah support (Table 6-5)..10% of steel mass loss was used in the area where onrete was removed. Note that no orrosion-indued damage model was applied to regions where the speimen was not orroded. 119

137 Table 6-5 Finite element speimens with various exposure regions Speimen ref# Corroded region FECM FECS Figure 6-7 shows the resulting load-deformation urves for the FE analyses of speimens FECM and FECS. Like the results presented in the previous setion (Figure 6-6), the analysis of the speimen with mid-span damage (FECM) presents stiffer response and higher residual strength ompared to beam FECS. Also, in this extreme ase where the onrete over has spalled off near supports, the effet of orrosion loation is amplified in omparison to Figure 6-6, where the onrete is not removed, implying that the influene of orrosion loation is more pronouned as the orrosion level inreases. 10