AXIALLY LOADED FRP CONFINED REINFORCED CONCRETE CROSS-SECTIONS

Transcription

1 AXIALLY LOADED FRP CONFINED REINFORCED CONCRETE CROSS-SECTIONS PhD Thesis by Bernát Csuka Budapest University o Technology and Economics Department o Mechanics Materials and Structures Supervisor: László P. Kollár

2 CONTENTS 1. INTRODUCTION CONFINED CROSS-SECTIONS MATERIALS FRP CONFINED CIRCULAR CONCRETE CROSS-SECTIONS SUBJECTED TO CONCENTRIC LOADING BEHAVIOR OF CONFINED CONCRETE COLUMNS PROBLEM STATEMENT METHOD OF SOLUTION CONCRETE MATERIAL MODEL MODEL FOR FRP CONFINED CONCRETE....6 VERIFICATION RESULTS FEASIBILITY DISCUSSION FRP CONFINED CIRCULAR CONCRETE CROSS-SECTIONS SUBJECTED TO ECCENTRIC LOADING EXPERIMENTAL RESULTS AND EXISTING MODELS PROBLEM STATEMENT APPROACH THE NEW MODEL CALCULATION OF CAPACITY DIAGRAMS (FAILURE ENVELOPES) VERIFICATION DISCUSSION MATERIAL LAW FOR DESIGN-ORIENTED MODELS FRP CONFINED RECTANGULAR CROSS-SECTIONS SUBJECTED TO CENTRIC LOADING EXPERIMENTAL RESULTS AND EXISTING MODELS PROBLEM STATEMENT METHOD OF SOLUTION NUMERICAL MODEL AND VERIFICATION NUMERICAL CALCULATIONS DESIGN EXPRESSIONS DISCUSSION CONCLUSION CONCENTRICALLY LOADED CIRCULAR CROSS-SECTIONS ECCENTRICALLY LOADED CIRCULAR CROSS-SECTIONS CONCENTRICALLY LOADED RECTANGULAR CROSS-SECTIONS TÉZISEK ACKNOWLEDGEMENT NOMENCLATURE REFERENCES APPENDIX A... 8 APPENDIX B APPENDIX C... 9

3 1. INTRODUCTION Axial resistance o concrete and reinorced concrete columns can be signiicantly increased by using lateral coninement. Frequently used solutions are steel helices, jackets or tubes. In the last years instead o steel jackets the use o FRP (iber reinorced polymer) as coninement has increased due to its high corrosion resistance, high ultimate strength and because it is easy to use or repair and/or reinorcement o damaged columns. FRP coninement can be applied to any type o cross-sections but most requently circular- and rectangular cross-sections (with rounded edges) are used. The relatively high cost o FRP materials is a signiicant disadvantage, but recently new, cheaper manuacturing techniques have appeared, which can give a urther boost to the use o FRP in building industry. In Hungary FRP coninement has already been used or retroitting o existing structures in the nineteen-eighties. Rectangular and hexagonal slag-concrete columns o an oice building in Budapest in the Fı utca have been conined using glass iber textile with epoxy resin. Due to the external loads internal orces develop in the column, which may result in ailure. The internal orces are: - axial orce (due to concentric axial load Figure 1.1a), - axial orce and bending moment (due to eccentric axial or axial and horizontal load Figure 1.1b) and - shear orce (due to horizontal load Figure 1.1c horizontal load also causes bending moment). In this work conined cross-sections are investigated and only the irst two cases are considered. To calculate the load bearing capacity o a reinorced concrete column the second order eects, i. e. the deormations o the column must be taken into account. Due to this eect the eccentricity o the axial load (on the cross-section) may increase signiicantly which reduces the ailure load o the column. In this work only the cross-sectional analysis will be discussed. In real design both the ultimate limit state and the serviceability limit state must be considered. In case o conined RC the deormations may be signiicant, but we will ocus only on the calculation o the ailure load. 3

4 σ l d (a) (b) (c) F e F P N = F M = N = F M = ef V = P Figure 1.1: Typical loads on a column: concentric compression (a), eccentric compression (b) and horizontal load (c). 1.1 CONFINED CROSS-SECTIONS The concrete core o an axially loaded column laterally expands due to the Poissoneect. The coninement hinders this expansion and hence the concrete is subjected to triaxial compression and its axial resistance increases. The conining stress σ l in case o concentrically loaded circular cross-sections can be calculated as ollows (Figure 1.): σ t σ l =, (1.1) where σ is the hoop stress in the coninement; t is the thickness o the coninement and d is the diameter o the cross-section. d σ t σ t Figure 1.: Conined circular cross-section. The behavior o conined materials has been investigated or 1 years. Kármán [33] experimentally investigated rigid materials (marble and sandstone) in triaxial stress-state and ound that with proper coninement plastic or even hardening behavior can be achieved. The experiments were conducted on marble and sandstone specimens with constant lateral conining pressure. The coninement provided by steel jackets (or helices) on the concrete core is similar because o the plastic behavior o the conining material. Contrary to the steel jackets, FRP behaves elastically until ailure which signiicantly aects the behavior o FRP conined concrete. 4

5 Several experimental results and models can be ound in the literature to predict the behavior o concentrically loaded FRP conined circular concrete columns; relatively ew experimental results and models are available or rectangular cross-sections and even ewer or eccentrically loaded columns. In the ollowing three Chapters we summarize these experimental data, models and design equations or - concentrically loaded FRP conined circular concrete columns (Chapter ), - eccentrically loaded FRP conined circular concrete columns (Chapter 3) and - concentrically loaded FRP conined rectangular concrete columns (Chapter 4). A new model will be introduced, with the aid o which contradictory results are explained and open questions are answered. New design methods are also presented which can be used in engineering practice. 1. MATERIALS FRP conined RC columns consist o three dierent materials. The material laws which are used in the ollowing chapters are briely introduced below Fiber reinorced polymer (FRP) FRP behaves in a linearly elastic manner and shows brittle ailure. It consists o two materials: high strength ibers and an element called matrix. The ibers can be made o glass (GFRP), carbon (CFRP) or aramid (AFRP), the matrix material is usually epoxy resin. Typical material properties or FRP coninement can be ound in Appendix A and C. Most o the researchers agree that the ailure o the conined column occurs when the conining FRP ruptures. This can be due to the tensile rupture o the ibers in the hoop direction (usually when the eccentricity is small) or due to axial compression (when the eccentricity is high). Any other mode o ailure (i. e. delamination o layers) is considered as a result o manuacturing error. Note that the ailure o brittle materials is very sudden thus in engineering practice the allowable strain is signiicantly lower than the rupture strain. The thickness o the coninement is very small compared to the diameter o the column (t / d < 5%). Uneven strains may appear especially in case o rectangular cross-sections, but this eect is neglected and constant strains and stresses are assumed along the thickness o the coninement. 5

6 1.. Steel Steel reinorcing bars and links i present are modeled as linearly-elastic, perectly plastic materials. The ailure o steel occurs when the ultimate strain is reached. Note, however, that this is much higher than the ailure strain o FRP (see Appendix A and C) and hence the ailure state o conining steel is never reached Conined concrete When unconined concrete reaches its uniaxial strength ( ), the material is in ailure state. Ater this point axial cracks appear and the concrete sotens and a decreasing path appears in the stress-strain diagram (Figure 1.3a). I coninement is used, this post-ailure behavior can change. I the strength o the conining material is low and it has high deormation capabilities the decrease in the stress-strain diagram is less signiicant (Figure 1.3b). I the strength o the conining material is higher and it has very high deormation capability a new increasing slope can appear on the diagram ater the concrete becomes sand and the coninement starts working again (Figure 1.3c). I the strength o the conining material is high the stress-strain diagram becomes monotonic as the post-ailure slope is increasing (Figure 1.3d). Finally, i the strength o the conining material is again low with very small deormation capability (low rupture strain), the ailure o the coninement can occur beore the concrete reaches its ailure state (Figure 1.3e). I the load is eccentric or the cross-section is not circular, the conining stress provided by the elastic FRP is not uniorm. To ollow the behavior o conined concrete we use the material law or concrete in triaxial compression proposed recently (7) by Papanikolaou and Kappos [5]. This material law is chosen because it is veriied or the case when the lateral conining stresses are not equal. Detailed description o the material law and its behavior in case o concentrically loaded circular columns is given in Chapter. For the case o uneven conining stresses a shorter description is given in Chapter 3. Time-dependent behavior o the materials (creep, shrinkage, relaxation, etc.) is not considered. (Note that all experimental results are rom short-term loading, veriication o time-dependent behavior would need an extensive experimental investigation.) 6

7 (a) Axial stress Concrete ailure state Axial strain Axial stress cc (b) Concrete ailure state cu Axial stress cc = cu (c) Concrete ailure state cc cu Axial strain Axial strain cc = cu (d) (e) cc = cu Concrete ailure state Axial stress Concrete ailure state Axial stress cc = cu Axial strain cc = cu Axial strain cc = cu Figure 1.3: σ() diagrams o concrete without coninement (a), with coninement with low strength (b), with coninement with high deormation capability (c), with coninement with high strength (d) and with coninement with low strength and low deormation capability (e). 7

8 . FRP CONFINED CIRCULAR CONCRETE CROSS- SECTIONS SUBJECTED TO CONCENTRIC LOADING In this chapter concentrically loaded circular columns are investigated [9]. Several experimental results and models can be ound in the literature to predict the behavior o concentrically loaded FRP conined circular concrete columns. The models can be divided into two categories: - design-oriented models: empirical models (explicit expressions) based on experimental results, - analysis-oriented models: analytical models based on concrete material models in triaxial compression. Examination o these models showed that axial resistance o concrete depends on the strength but not on the stiness o the conining material..1 BEHAVIOR OF CONFINED CONCRETE COLUMNS The conining stress σ l can be calculated as given by Equation (1.1) In this chapter only columns with unidirectional coninement are investigated. Here the axial resistance o the coninement is negligible and σ E, where E is the elastic modulus in the hoop direction, is the hoop strain. The axial stress-strain diagram o unconined columns decreases ater reaching the peak stress (Figure 1.3a), while or steel coninement the diagram remains nearly constant. The typical stress-strain diagram o FRP conined concrete is monotonic, but with a decreasing slope (Figure 1.3c), the peak axial stress is usually reached at the ailure o the conining FRP. The elastic modulus o FRP especially GFRP is lower than the modulus o steel. As a consequence the strains observed in FRP conined columns are higher than that in the case o steel coninement..1.1 Experiments The experimental results collected rom the available literature are summarized in Figure.1. These results were ound in the ollowing papers: De Lorenzis and Tepers [13], Lam and Teng [36], Jiang and Teng [3], Almusallam [3], Al-Salloum [4], Berthet et al. [5], Harries and Kharel [8], Mirmiran et al. [47], Shahawy et al. [64], Toutanji [7]. The numerical values o the experiments are summarized in a table in Appendix A. 8

9 cc / Almusallam Al-Salloum Berthet et al. Harries and Kharel Lam and Teng Lam et al. Teng et al. Jiang and Teng Watanabe et al. Matthys et al. Kshirsagar et al. Rochette and Labossiére Mirmiran et al. ρ = c l / Figure.1: Experimental results Xiao and Wu De Lorenzis et al. Picher et al. Purba et al. Aire et al. Dias da Silva and Santos Micelli et al. Pessiki et al. Wang and Cheong Shehata et al. Toutanji Shahawy The specimens o all these experiments were prepared in such a way that the ibers are arranged primarily in the hoop direction. This could be achieved by using FRP wraps or winding the unidirectional FRP with a low angle. In case o reeling, the insigniicant axial resistance o the FRP was neglected. The ailure mode o the columns was the tensile rupture o the FRP. (Specimens which ailed due to the debonding o the overlapping region o the FRP or due to the axial compression o FRP were not included in the experimental data.) In the presented experiments the diameter o the specimens is between 76 and mm. The uniaxial compressive strength o concrete varies between 19.4 and N/mm. Carbon, glass or aramid ibers were embedded in epoxy matrix. The ailure mode o the specimens was the rupture o the FRP. Failure As we stated above, the ailure mode was the rupture o the FRP, nevertheless in most o the cases the strains measured in the hoop direction were smaller than the ultimate strain o the FRP (given by the manuacturer or measured by coupon tests). This phenomenon is widely known, a number o authors (De Lorenzis and Tepers [13], Pessiki et al. [53], Shahawy et al. [64], Lam and Teng [36], Harries and Carey [9], Matthys et al. [43]) gave similar explanations. The most important reason is the ollowing: - small vertical concrete cracks appear under the FRP, which result in localized strain-peaks in FRP. In addition the ollowing reasons are given: - due to the axial compression FRP is in biaxial stress state, which decreases its resistance in the hoop direction; - the misalignment o the ibers due to manuacturing (especially in case o hand-layup). 9

10 The ratio o the measured strain at ailure and the ultimate strain (called strain eiciency actor, denoted by κ ) varies between.6 and 1.. Experiments show that the higher the elastic modulus o FRP, the higher the κ ratio. Stress-strain Response According to most o the authors the behavior o FRP conined columns can be divided into two groups: suiciently conined or insuiciently conined concrete. In the case o suiciently conined concrete, the stress-strain diagram is either monotonically increasing, and the shape o the diagram is approximately bi-linear (increasing type, Figure.a) or the stress-strain diagram has a post-peak decreasing branch, where the axial stress at ultimate state is higher than the uniaxial concrete strength (decreasing type with < cu, Figure.b). The stress-strain diagram o insuiciently conined concrete also has a decreasing branch, but the axial stress at ultimate state is smaller than the uniaxial concrete strength (decreasing type with > cu, Figure.c). According to the literature [36] in this case a little strength enhancement can be expected and FRP is likely to rupture at small hoop strain. (a) (b) (c) cc= cu Axial stress σc Axial stress σc cc cu Axial stress σc cc cu cc = cu Axial strain c cc cu Axial strain c cc cu Axial strain c Figure.: σ() diagrams o concrete with suicient coninement with monotonic curve (a), suicient coninement with decreasing second part (b), insuicient coninement (c)..1. Existing Models Models are based either on experimental data (design oriented models) or on triaxial concrete material models (analysis oriented models). Design-oriented Models In these models to calculate the compressive strength o conined concrete based on experimental data the authors give similar expressions (Table.1), which depend on the coninement ratio (ρ c ): ρ = /, (.1) c l 1

11 where l is the conining strength, calculated rom Equation (1.1) ( l = t / d), where is the tensile hoop strength o the conining FRP and is the uniaxial compressive strength o concrete. Table.1: Design-oriented ormulas or the prediction o axial strength. Reerence Eurocode [] Formula l,a 1+ 5, i l.5 cc = l,a , i l >.5 Samaan et al. [6].7 cc = l Saai et al. [61].84 cc = 1+. l Lam and Teng [36] cc cc = 1+ l = l Yousse et al. [8] 1.5 cc = 1+.5 l Wu et al. [76], average stiness coninement high stiness coninement data provided by manuacturer,a,a,a 1.53 cc l = + l.6 cc l = + l 3.5 cc l = + l Xiao and Wu [77] cc l = +, where E l E t El = d The typical orm o design-oriented expressions is as ollows: k / = k + k ρ. (.) cc 1 Here k 1, k, k 3 are constants, the value o k 1 is usually 1, the values o k and k 3 are dierent or each model. Comparing the experimental data and the design oriented models De Lorenzis and Tepers [13] showed that either the ormulas proposed by Samaan et al. [6], Saai et al. [61] or Spoelstra and Monti [67] may be used. 3 c 11

12 The ormula proposed by Lam and Teng [36] is similar but it also considers the dierence between the measured strain and the ultimate strain o FRP at ailure. Lam and Teng [36] and Yousse et al. [8] also propose a ormula to calculate the stress-strain diagram o conined concrete. Wu et al. [76] propose three dierent expressions: one should be used i the material properties o the FRP are predicted by the manuacturer, the other two should be used when the properties o the FRP are predicted by experiments (Table.1). In the table the expression or high stiness coninement should be used i the elastic modulus o the coninement is 378 kn/mm E 64 kn/mm. The expression recommended by Xiao and Wu [35] contains the eect o the stiness o the coninement, however according to De Lorenzis and Tepers [13] this estimation o the compressive strength is inaccurate. Most o the researchers agree that the eect o the stiness o the conining FRP on the compressive strength o conined concrete is negligible. Analysis-oriented Models The analysis-oriented models are based on the triaxial concrete material models with strain and stress compatibility between the concrete and the FRP. Several models are based on nonlinear elastic material laws (not considering plastiication). In this case there is a direct relationship between stresses and strains: σ = D, (.3) ij ij where σ ij is the stress tensor, ij is the strain tensor and D is the tensor o incremental moduli. The elements in tensor D depend on the current level o stresses. In uniaxial loading with monotonic axial strain the elastic modulus o concrete decreases meanwhile the Poisson s ratio increases. Ater reaching the peak stress elements o D become negative. Based on these models explicit expressions can be derived or the strength o conined concrete. These are summarized in Table.. In these models the actual conining strength ( l,a ) is used, which is calculated rom Equation (1.1) ( l,a = σ t / d), however σ is usually lower than the tensile strength o composite, and (or unidirectional coninement) it is calculated as σ E, where is the experimentally measured hoop strain at rupture ( l ). 1

13 Table.: Analysis-oriented ormulas or the prediction o axial strength. Reerence Spoelstra and Monti [67] Mander et al. [4] Berthet et al. [6] Binici [7] Li et al. [4] Formula =.+ 3 l cc,a.5 cc l,a k cc 7.94 = l = +, where 1 k1 l,a 3.45 =, i 5 1 k ( ) Turgay et al. [71] ( ) =, i 5 cc l,a cc 9.9 = 1+ + l,a l,a ϕ = 1+ tan 45 +, where ϕ = α ξ ξ + ρ k =, where.355 ξ α =.46, l,a l,a k = ρ = ( cc l,a), ξ = 3 + l,a cc 3,a, Elastoplastic concrete models (when the plastiication is taken into account) give more reliable modeling o concrete. The incremental strain tensor is calculated as: el p ij ij ij d = d + d, (.4) where tensor. el d ij is the incremental elastic strain tensor and The stresses are: ij el ij p d ij is the incremental plastic strain σ = D, (.5) i.e. the plastic strains do not indicate stresses. In the models the plastic behavior depends either on the volumetric plastic deormations or on the internal energy. 13

14 These models are reliable, however the calculation is rather complex and requires numerical procedures. They were used by Meláo Barros [44], Karabinis and Rousakis [3] and Deniaud and Neale [16], however none o them investigated the eect o the stiness o the coninement. In all the analysis-oriented models the plastic behavior o concrete is described by a yield criterion (Kaliszky [31]). The yield criteria used by the above-mentioned authors: Drucker- Prager (Turgay et al. [71], Deniaud and Neale [16] and Karabinis and Rousakis [3]), Ottosen (Meláo Barros [44]), Willam-Warnke (Mander et al. [4]), Leon-Pramono (Binici [7]), Mohr-Coulomb (Berthet et al. [6], Li et al. [4]). The yield criterion can be illustrated in a three-dimensional stress space, where the axes are the principal stresses. The yield surace can be invariant during the loading path [6],[7],[16],[4],[76], or it can change its shape or location [3],[44]. The latter models are called hardening-sotening models. In elastoplastic models according to the classical plastiication theory, the direction o incremental plastic strain vector is perpendicular to the yield surace (associated low [16]), however experiments showed that this is not true or concrete. Models taking into account that the incremental plastic strain vector is not perpendicular to the yield surace are non-associated models [3],[44]. The comparison o experimental results and the predicted conined compressive strengths o dierent closed-orm equations can be seen in Figure.3. / cc Wu et al. (common modulus FRP) Wu et al. (high modulus FRP) 6 Wu et al. (data rom manuacturer) Eurocode Saaman et al. 5 Saai et al. Lam and Teng 4 Yousse et al. 3 (a) cc / (b) Spoelstra and Monti Lam and Teng Mander et al. Berthet et al. Binici Li et al. 1 ρ = / c l ρ = / c l,a Figure.3: Comparison o experimental results with design-oriented models using coninement strength measured rom coupon tests (a), analysis-oriented models based on measured hoop strain at rupture (b). (The authors o the experimental results are the same as in Figure.1.) There are dierent recommended ormulas to predict the axial strain at rupture, these are summarized in Table.3. 14

15 Table.3: Formulas or the prediction o axial strain at peak stress. Reerence De Lorenzis and Tepers [13] Lam and Teng [36] Yousse et al. [8] Wu et al. [76] Richart et al. [57] Berthet et al. [6] Li et al. [4] Formula cc cc = l = l E l,a ( ).45 u cc = l ( ).5 u = =, where ν.56 k / u u cc.66 1 l 1. i E 5 N/mm k1 = 5/ E i E> 5 N/mm cc cc cc 1 5 cc 1 = + 3 E ( ν ) = 1 + l l,a ϕ = 1+.4 tan 45 +, where ϕ = u In their comparative study De Lorenzis and Tepers [13] summarized various ormulas. Comparing these to the experimental data they ound that the prediction o axial strain is not satisactory and they suggested a new approximate expression. We summarized the recommended expressions or stress-strain diagrams in Table.4. Note that all the recommended curves are either monotonic or they have a monotonic increasing and a monotonic decreasing part. Insuicient Coninement Low conining pressure leads to small increase in strength, which cannot be reliably predicted and thereore it should be avoided. According to Lam and Teng [36] the coninement is insuicient i the stress-strain diagram decreases with > cu (Figure.c). To reach suicient coninement a minimum value or the coninement ratio is recommended. Mirmiran et al. [46] suggested / >.15, while Spoelstra and Monti [67] gave a lower value / >.7 according to their experimental results. Also based l on experiments Xiao and Wu [77] proposed a ormula which contains the stiness o the l coninement: -1 E t / d >. MPa. Lam and Teng [36] accepted Spoelstra and Monti s 15

16 expression with a small modiication: instead o the conining strength ( l ), the actual conining strength ( l,a ) should be used. Table.4: Formulas or the calculation o the stress-strain diagram. Reerence Formula Almusallam [3] ( E E ) Saenz [63] σ 1 c c = + E 1 c n n ( E1 E) c, where 1+ E 1 irst slope o stress-strain curve E second slope o stress-strain curve reerence plastic stress at intercept o second slope with the stress axis n curve shape parameter that mainly controls the curvature in the transition zone σ E E c c = E c 1+ c E + s cc cc = 475, E Popovics [55] ( c cc) r σ c = cc r 1+ ( ) Lam and Teng [36] Ec r = E c cc cc, c s = cc cc cc, where, where r E c elastic modulus o concrete ( E E ) c c Ec c c 4 σ =, i c t, σ = + E, i c cc, where c c t = E E c, E = t cc Yousse et al. [8] n 1 σ 1 c c Ecc 1 n =, i c t, t ( ) σ = + E, i c cc, where c t c t cc 5 4 ρ Et t = 1+ 3, t = t ρ E t 5 4 ρ volumetric ratio o FRP jacket, t =., n= ( E E ) c t E c t t, E = cc cu t t u, where 16

17 . PROBLEM STATEMENT As we stated in.1. according to the existing models the ailure strength o FRP conined column is hardly aected by the stiness o the coninement. We may observe, however, that or a very sot coninement the concrete might ail beore the development o the conining stresses (Figure 1.3b), and or a very rigid FRP the coninement may ail beore the concrete reaches its plastic state (Figure 1.3e). The ollowing questions arise: - How does the stiness o FRP coninement aect the behavior o the conined concrete column? - Under what conditions can it be assumed, that the strength o the conined concrete is not aected by the stiness o the coninement? These questions have practical importance as the stiness o FRP may strongly vary and, in addition, in the uture new materials may also be applied as FRP coninement..3 METHOD OF SOLUTION To answer our questions and to understand the behavior o FRP conined circular columns we introduce an analysis oriented model, which is based on a new, quite accurate (coninement-sensitive, non-associated) concrete material law proposed by Papanikolaou and Kappos [5]. The FRP coninement is modeled with the classical laminate plate theory and it is assumed that it behaves in a linearly elastic manner..4 CONCRETE MATERIAL MODEL In this section we present the coninement-sensitive plasticity constitutive model or concrete in triaxial compression based on the work o Papanikolaou and Kappos [5]. The incremental strain vector consists o an elastic and a plastic component (Equation.4), and the elastic strain increments are related to the stress increments by Equation (.5). The plastic (irreversible) incremental strains ollow a non-associated low-rule described in Table.5. Both ailure and plastic potential suraces are ormulated in the Haigh-Westergaard stress-space, which is described by the hydrostatic length (ξ), deviatoric length (ρ) and lode angle (θ). These coordinates are calculated as given in Table.5. 17

18 Table.5: Formulas used in the concrete material law proposed by Papanikolaou and Kappos [5]. Description Non-associated low rule Coordinates o Haigh- Westergaard stress space σ 1 > σ > σ 3, (compression negative) Parameters o yield criterion: elliptic unction (r) and riction parameter (m) Hardening unction (k) Sotening unction (c) Plastic potential surace Formula d p ij g = dλ σ ij I1 ξ =, I1 = σ1+ σ + σ 3 3 ρ =, J = ( σ σ ) + ( σ σ ) + ( σ σ ) J θ = cos 3 r ( θ, e) m = / J J 6 ( σ / 3 )( σ / 3 )( σ / 3) J = I I I ( 1 e ) cos θ + ( e 1) ( ) θ ( ) ( ) = 1 e cos + e e cos θ + 5e 4e ( ) k t e k e+ 1, where t k = k(κ) hardening unction described below, c = c(κ) sotening unction described below, pl pl pl 1 3 κ = + + plastic volumetric strain e parameter o out-o-roundness (e.5, see [5]), t uniaxial tensile strength o concrete. p p v,t v = v = p v,t p ( κ) ( ) ( ) k k k k k c = σ /, p v,t c σ = / 6,, where = ( 1 ν) limit value or volumetric plastic strain. E p ( κ) c( ) = = p p 1 v / v,t n n =, p ( t ) 1 v 1+ p v,t s / v,t = +, t s = / 15. n n 1 n 1 1, where ρ 1 ρ ξ g = A + C+ ( B C)( 1 cos 3θ) + a, where k c k c k c A, B, C plastic potential coeicients, detailed calculation is described in [5], n plastic potential unction order, can be, 3,4 or 5 (in our calculations we use n = 3 as recommended in [5]). 1/, 18

19 The triaxial stress state o concrete during plastic low is described by the Menétrey - Willam yield criterion: (,, ) 1,5 ρ m ρ r(, e) ξ ξ ρ θ = + θ + c=, (.6) k 6k 3k where the riction parameter (m), the elliptic unction (r), the eccentricity parameter o outo roundness (e), the hardening parameter (k) and the sotening parameter (c) are again deined in Table.5. The material model is pressure sensitive, the hardening and sotening behavior o concrete is controlled by the plastic volumetric strain (κ, see Table.5). This model is non-associated, the plastic strain vector is perpendicular to the plastic potential surace (g) (Table.5), which is dierent rom the yield surace (). The yield surace when σ 1 = σ is shown in Figure.4 or three dierent hardening and sotening parameters. σ = σ 3 c cc,max( k = 1, c = 1, ailure state) 5 4 cc,min ( k = 1, c = ) 3 ( k = k, c =1, initial state) 1 σ = σ = σ 1 l Figure.4: Three yield suraces (solid lines) and a typical loading path (dashed line). The numerical method recommended by Papanikolaou and Kappos [5] is based on a backward-euler algorithm, which can be used when the conining stresses are constant. In case o FRP coninement due to the elastic behavior o the FRP the conining stresses increase with increasing axial and hoop strains. To ollow this phenomena a straightorward method is used. (Further details about the algorithm can be ound in Appendix B.) 19

20 .5 MODEL FOR FRP CONFINED CONCRETE During the derivation o our model we assumed that both the axial and the hoop strains o the concrete and the conining FRP are identical, while the relationship between the hoop stress in the FRP and the in-plane stress in concrete is given by Equation (1.1). We used the concrete material model described in the previous section, which requires a numerical solution. We applied an incremental method: increased the strains step by step and evaluated the stress, the plastic strain and the yield surace in each step. The calculation was terminated at the ailure o the FRP. To understand the behavior o conined concrete we irst present and explain a typical loading path: At the beginning o the loading history (point 1 in Figure.4) the concrete is in elastic state, the value o the hardening parameter is k = k (smaller than one), the sotening parameter c is equal to one (c = 1). In the elastic state the relationship between the axial and the hoop stress is linear and the slope depends on the elastic modulus and Poisson s ratio o both materials. The soter the coninement, the steeper the loading curve. As the load increases the stress state o concrete reaches the yield surace (point ) and the value o the hardening parameter k starts to increase, and the yield surace is opening. When the plastic volumetric strain reaches a certain value, the hardening process is terminated (k = 1, point 3). At this stage the yield surace reaches its most expanded shape, which is reerred to as ailure state (k = 1, c = 1). As the loading is carried on the sotening region is initiated and the value o the sotening parameter c decreases, while the value o the hardening parameter remains k = 1. The lower limit or c is zero, which can never be reached. In uniaxial loading the axial stress o concrete is reached at the end o the hardening region (Figure 1.3a), and then in the sotening region the stress-strain diagram o concrete decreases. In triaxial loading or suicient and increasing conining pressure as in the case o FRP coninement the axial stresses can increase in the sotening region o the loading history (point 4). The slope o the loading curve depends on the stiness o the conining material. For typical coninements the loading path is between the two curves that belong to the ailure state and the state o ull plasticity (Figure.4), and hence the concrete strength is also between these curves (denoted by cc,max and cc,min, respectively). The condition when the loading curve is below cc,min will be discussed later in Section.7. The exact value o concrete strength depends on the endpoint o the loading curve (point 5), which belongs to the ailure o the coninement.

21 The experimental results were compared to the theoretical lower and upper limit based on cc,min and cc,max, respectively. The results are shown in Figure.5. 6 cc / Proposed cc,max Proposed cc,min 1 ρ c,a= l,a / Figure.5: Relation between the experimental results and the upper and lower limits or the axial strength. (The authors o the experimental results are the same as in Figure, the neglected results are not indicated.).6 VERIFICATION The accurate concrete strengths and strains at ailure were also calculated numerically or all the experimental cases, the accuracy is shown in Tables.6 and.7, respectively. (Experiments, where the dierence between the measured axial stress and the calculated stress was more than 3% or more than hal o the equations suggested by dierent authors were neglected.) The average absolute error o the dierent models or the prediction o axial stress was calculated as: 1 n i= 1 exp cc cc exp cc error =. (.7) n The best results are achieved by the (design oriented) model recommended by Lam and Teng [36], which was itted to experimental results. Our proposed model, which is based on theoretical investigation (and most o the material properties are unknown) is reasonably accurate. The prediction o axial strain at rupture is less accurate. The average error o the best expression (suggested by Richart et al. [57]) is 35%. The error o our model is even higher, because the applied concrete material model is based on a our parameter yield criterion (with our more parameters or the yielding behavior), and in most experiments only one or two (rarely three) parameters were available. The unknown parameters were approximated as recommended by Papanikolaou and Kappos [5]. 1

22 Table.6: Comparison o dierent ormulas and results or conined strength. Reerence Average error [%] r Eurocode [] Samaan et al. [6] Saai et al. [61] Spoelstra and Monti [67] Lam and Teng ( l ) [36] Lam and Teng ( l,a ) [36] Yousse et al. [8] Wu et al. [76] Xiao and Wu [77] Mander et al. [4] Berthet et al. [6] Binici [7] Li et al. [4] Turgay et al. [71] Csuka and Kollár Table.7: Comparison o dierent ormulas and results or axial strain at peak stress. Reerence Average error [%] r De Lorenzis and Tepers [13] Lam and Teng [36] Yousse et al. [8] Wu et al. [76] Richart et al. [57] Berthet et al. [6] Li et al. [4] Csuka and Kollár The ormula proposed by Richart et al. [57] was used by several authors [4],[76]. The ormula contains cc and in this calculation, we used the experimental result. I we use the ormula or design, we will need to calculate cc as well and the error will be higher. We also compared the stress-strain diagrams calculated by the expressions recommended by dierent authors (Table.4) with the experimental results o Deniaud and Neale [16] (Figure.6) and Mirmiran et al. [47] (Figure.7). It can be observed that our proposed model can ollow the shape o the diagram, however the accuracy is poor because o the lack o material properties as stated above. By calibrating the unknown properties a much higher accuracy can be reached.

23 Axial stress Experimental data Saenz Popovics Lam and Teng Yousse Proposed Lateral strain Axial strain Figure.6: Comparison o experimental results rom Deniaud and Neale [16] and calculated σ() diagrams rom Table.4. Experimental Data (DA11) Experimental Data (DA13) Richart et al. Samaan et al. Mander et al. Rochette and Labossiére Mirmiran et al. Proposed (no calibration) Proposed (with calibration) Figure.7: Comparison o experimental σ() diagram by Mirmiran et al. [47] and the calculated curves. 3

24 .7 RESULTS.7.1 Eect o the stiness o the coninement With the aid o the developed model we investigated the eect o the stiness o the coninement. An example can be seen in Figure.8 or a C3 concrete with unidirectional coninement. Each solid line belongs to a given stiness ratio (ρ s ) deined as: E t ρ s =, (.8) dec where E c is the elastic modulus o concrete. Note that both the stiness- and the coninement ratio are linear unctions o the thickness o the coninement Stiness ratios ( ρ s ): Axial stress 4 Coninement ratios: ρ c,a =.5 ρ c,a =. ρ c,a =.15 ρ c,a =.1 ρ c,a = Lateral strain Figure.8: Eect o stiness and strength o coninement. Axial strain We may observe (Figure.8) that or higher stiness the stress-strain curve is monotonic (as in Figure.a), while or lower stiness the diagram has one local maximum point and one local minimum point. The rupture o coninement (and hence the concrete strength at ailure) depends on the coninement ratio (ρ c, see Equation.1). Identical conining ratios are indicated by identical marks on the solid lines (Figure.8). Depending on the stiness and coninement ratios there are three types o stress-strain diagrams as illustrated in Figure., were Figure.a belongs to high-stiness coninement and.b belongs to low stiness coninement. Figure.8 shows that the stiness aects the strength o the conined concrete. This is urther illustrated in Figure.9, where eight stress-strain diagrams are shown, which belong to dierent stinesses, however to the same strength (coninement ratio). The dots show the end o the stress-strain curves. 4

25 σ [N/mm ] 75 cc,max cc,min c 5 5 (a) E = 765 kn/mm (b) E = 153 (c) E = 51 E = 76 E = 38 E = 5 E u = 38,91 N/mm Constant! E = 19 E = (d) Low-stiness coninement High stiness coninement Overconined concrete Figure.9: Eect o the stiness o coninement on the stress-strain diagram o C3 concrete. When the stiness is very high the FRP ruptures beore the concrete can reach the ailure state (curve a, E = 765 kn/mm ). This is deined as overstiened concrete, and must be avoided. The loading curve is also shown in Figure.1 (curve a). c σ 3 = σ c cc,max( k = 1, c =1, ailure state) 1.5 (c) cc,min ( k = 1, c = ) (d) ( k = k, c = 1, initial state) (b) (a) overconined.5 (a) (b) optimal coninement (c) high-stiness coninement (d) low-stiness coninement σ = σ = σ 1 l Figure.1: Eect o the stiness o coninement on the loading path o concrete. By decreasing the stiness o FRP, at a certain point we obtain the case, when the ailure strength is maximum (curve b in Figure.1), this case is deined as optimum coninement. Figure.1 shows that the FRP ails, when the concrete is in ailure state (k = c = 1). Further drop in the stiness results in lower ailure strength (curve c in Figure.9). The stress-strain curve and the loading path (curve c, Figure.1) are monotonic. This case is reerred to as high stiness coninement. As the stiness is reduced again (curve d, Figure.9), the stress-strain curve has a decreasing part. A similar shape can be observed in Figure.1. This is deined as low stiness coninement. Note that the lower the stiness the lower the ailure strength. In the limit we may reach the ailure strength, which belongs to cc,min. 5

26 Based on the above observations the ollowing statements can be made: - overstiened concrete must be avoided; - or low stiness coninement calculation based on cc,min is realistic; - high stiness coninement results in higher ailure strength than low stiness coninement; - a urther advantage o high-stiness coninement is that there is no decreasing part o the stress-strain diagram and the strains at ailure are signiicantly lower..7. Achievements The limit between high-stiness and low-stiness coninement was investigated numerically. The results are shown in Figure.11. These data can be approximated by the ollowing expression: ρ s,limit = , i 4.195, i 4. (.9).3 ρ s..1 Numerical calculation Eq. (.8) Figure.11: Limit between high-stiness and low-stiness coninement. The optimal stiness (which is the limit between the overstiened concrete and the high-stiness coninement) was also investigated numerically (Figure.1), and the ollowing curve was itted on these results: ρ s,opt. =.1+.ρc,a.3. (.1) 6

27 ρ s Values o : Eq. (.9) ρ c,a= l,a/ Figure.1: Optimal stiness as a unction o uniaxial concrete strength ( ) and coninement ratio (ρ c,a ). As we stated above cc,min is a reasonable approximation or low stiness coninement and it is also a conservative approximation or high-stiness coninement. From Equation (.6) an analytical expression can be derived or the ailure strength. It yields to = + m, (.11) cc,min l,a l,a where m is given in Table.5. By introducing the recommended values o the parameters in the expression o m, we obtain: = (.1) cc,min l,a l,a Note that this value (m = 1.16) belongs to = 4 N/mm. In the concrete strength range, where the material law is veriied ( N/mm 1 N/mm ) the dierence is within ±% o The value o m depends approximately linearly on the unconined concrete tensile strength ( t ), in our calculations the recommended t = / 1 value is used. The maximum possible axial strength, cc,max (which belongs to the optimum stiness) can also be calculated rom Equation (.6). It results in the ollowing expression: cc,max l,a l,a = + m +, (.13) or, by introducing the same parameters as beore: cc,max l,a 1.16 l,a = + +, (.14) Equations (.11) or (.1) can be used to determine the limit o insuicient coninement. Several authors, such as Lam and Teng [36], Mirmiran et al. [46] and Spoelstra and Monti [67] deined the suicient coninement as cu. (.15) 7

28 Introducing Equation (.1) into Equation (.15) we obtain a second order equation, which yields l,a /.83. (.16) This value, obtained theoretically is close to the recommended value o Spoelstra and Monti [67] (also used by Lam and Teng [36]); based on experimental results they gave.7. (Note that or high-stiness coninement the stress-strain diagram is monotonic and condition Equation (.15) is always satisied.) In all the expressions above the actual conining strength ( l,a ) and the actual coninement ratio (ρ c,a ) should be used. These values can be calculated rom the conining strength ( l ) and the coninement ratio (ρ c ) by using a reduction actor κ, which is usually between.6 and.9 ( l,a = κ l, ρ c,a = κ ρ c ). There are several ormulas available in the literature or the calculation o κ [13],[9],[36],[43],[53],[67], the details are not discussed in this work..7.3 Simpliied model The limit values cc,min and cc,max calculated in the previous section can be used as rough approximations. A more accurate approximation or cc and cu can be achieved with the ollowing method. In case o high-stiness coninement or the calculation o the conined concrete strength a linear interpolation between cc,min and cc,max is recommended with respect to the stiness o the coninement: cc cc,min ( cc,max cc,min) = + ρ ρ ρ s s,opt ρ s,limit s,limit, (.17) where ρ s, ρ s,limit, ρ s,opt, cc,min and cc,max are calculated rom Equations (.8,.9,.1,.1 and.14) respectively. In case o low-stiness coninement i the coninement ratio is high, the conined strength can be approximated with cc,min. I the coninement ratio is low, the stress-strain diagram becomes decreasing and the eect o the coninement is negligible. This leads to the ollowing equation: cc cc,min = max. (.18) The approximation o the axial strength at ailure ( cu ) is more simple. In case o highstiness coninement cu = cc ; and in case o low-stiness coninement cu = cc,min. 8

29 The simpliied method described above was compared to the experimental results ound in the literature and showed a good agreement. The average absolute error or cc and cu is 11.7% and 16.85% respectively (or cc r =.88 and or cu r =.851). Comparison o the experimental and calculated results is shown in Figure cc / (a) 5 cu / (b) 4 4 Experiment 3 Experiment Simpliied model cc / Simpliied model Figure.13: Comparison o experimental results and the simpliied model or cc (a) and cu (b). cu /.8 FEASIBILITY In practice the actual coninement ratio (ρ c,a ) or glass or carbon iber coninement is usually between.5 and.5 (it goes up rarely to 1.). The ultimate strain or GFRP is usually between.15 and.3, while or CFRP it varies between.8 and.. In Figure.14 we show the stiness ratio or typical glass and carbon iber coninement as a unction o the coninement ratio or C, C3, C5 and C1 concrete. The limit between high-stiness and low-stiness coninement (Equation.9) and the optimal stiness (Equation.1) (which is the limit between the overstiened concrete and the high-stiness coninement) are also shown. In the case o GFRP the coninement is usually low-stiness, and in the case o CFRP the coninement is typically high-stiness. It is important to emphasize that overstiening is very unlikely. (An extreme example, when overstiening occurs, when a C column with a diameter o 5 mm is reinorced with 8 layers o.167 mm thick CFRP (E = 39 GPa and u = 3 N/mm ). The coninement ratio in this case is very high: ρ c = 1.6.) 9

30 .9 DISCUSSION In this chapter a new model was presented to calculate the stress-strain curve o axially loaded FRP conined, circular concrete columns, based on a sophisticated (coninementsensitive plasticity constitutive) concrete material law. Based on this model we also derived a new analytical expression (Equations.17 and.18) to determine the strength o conined concrete. We also derived analytically the limit or the insuicient coninement (Equation.16). Both analytical results agree well with the experimental data. ρ s.5 (a) ρ s.5 (b) OVERSTIFFENED CFRP ρ s,opt HIGH-STIFFNESS GFRP LOW-STIFFNESS ρ s,limit ρ c,insu ρ c,a= l,a / OVERSTIFFENED CFRP ρ s,opt HIGH-STIFFNESS GFRP LOW-STIFFNESS ρ s,limit ρc,insu ρ c,a= l,a / ρ s CFRP (c) OVERSTIFFENED ρ s,opt HIGH-STIFFNESS ρ s OVERSTIFFENED (d) CFRP ρ s,opt HIGH-STIFFNESS.5 GFRP LOW-STIFFNESS ρ s,limit ρ c,insu ρ = l,a 3.5 GFRP ρ s,limit LOW-STIFFNESS ρ c,insu c,a l,a c,a / ρ = / Figure.14: Typical glass and carbon iber coninement in practice in case o C (a), C3 (b), C5 (c) and C1 (d) concrete. In the Problem statement the question was stated, how does the stiness o the FRP coninement aect the load bearing capacity o conined concrete. Based on the new model we showed that the strength depends on the stiness, however there is a parameter range, where the eect is negligible. We also showed that (under the limit o optimal stiness ) the higher the stiness the higher the concrete strength, and an about 4% increase in strength can be reached by increasing the stiness. When the stiness o the coninement is very high the concrete is overstiened, and there is a drop in the concrete strength, and hence this case must be avoided.

31 By investigating the material properties o commonly used materials, we ound that (i) in the case o glass iber coninement the stiness has a minor eect on the concrete strength, (ii) in the case o graphite ibers an about % gain in concrete strength can be reached by taking into account the coninement stiness, and (iii) the overstiening is not realistic. (It should be mentioned however, that prestressing o the coninement which is useul or decreasing strains can lead to overstiening and to a drop in concrete strength even or conventional materials.) 31

32 3. FRP CONFINED CIRCULAR CONCRETE CROSS- SECTIONS SUBJECTED TO ECCENTRIC LOADING As we stated in the Introduction dierent internal orces develop in the column. Concentrically loaded columns were discussed in the previous chapter. We recall, however, that depending on the stiness and strength o the coninement there are three types o stress-strain responses or conined concrete, which were shown in Figure.. In this chapter eccentrically loaded circular columns are considered [1]. 3.1 EXPERIMENTAL RESULTS AND EXISTING MODELS Relatively ew documented experiments on eccentrically loaded FRP conined circular columns can be ound in the literature. Hadi [3] presented data or unreinorced concrete columns with unidirectional coninement, where the ibers are arranged in the hoop direction and the FRP provides no axial resistance. Hadi [4],[5] published data or reinorced concrete columns with unidirectional (hoop) coninement. Fam and Rizkalla [1] ran experiments where FRP tubes (with both axial and circumerential resistance) were illed with concrete. Their results are summarized in Figure 3.1. The diagrams show the normalized orce and moment, where the normal orce is divided by the axial resistance o the unconined cross-section and the moment is divided by the maximum moment resistance o the unconined cross-section. There are a ew urther experimental results on eccentrically loaded FRP conined columns, where the coninement did not play an important role: the coninement is insuicient or the GFRP conined specimens o Hadi [3],[4]; the eccentricity is very high [19],[6],[66] and hence the coninement hardly inluenced the load carrying capacity (as it is stated by Bisby and Ranger [8]). These results are also shown in Figure 3.1, but they are not considered or urther investigations. There are two types o models o conined columns: design-oriented models, where the Bernaulli-Navier hypothesis (plane cross-section) is combined with an axial stress-strain curve o concrete (which contains the eect o coninement); and analysis-oriented models, which are based on a triaxial material model or concrete. The axial stress-strain curves o design-oriented models are built on the axial strength and ailure strain ( cc, cc, Figure.) o concentrically loaded conined columns. 3

33 (a) (b) Normalized axial orce 1.5 Hadi CFRP [3] Hadi GFRP [3] 1.5 Normalized axial orce Fam and Rizkalla [1] (tube no. 6) Fam and Rizkalla [1] (tube no. 5) Normalized moment Normalized moment Normalized axial orce Hadi CFRP [4] Hadi GFRP [4] Hadi [4] control specimens Hadi CF [5] Hadi CF [5] control specimens Hadi CFS [5] Hadi CFS [5] control specimens Saadatmanesh et al. [6] Saadatmanesh et al. [6] control specimens Sheikh and Yau GFRP [66] Sheikh and Yau CFRP [66] Sheikh and Yau [66] control specimens Elnabelsy and Saatcioglu [19] Ghali et al. [] Ghali et al. [] control specimens Bisby and Ranger [8] (c) Normalized moment Figure 3.1: Experimental results or circular columns with dierent arrangements: concrete columns with unidirectional FRP coninement (a), concrete illed FRP tubes (b) and reinorced concrete columns with unidirectional FRP coninement (c). For example, according to Lam and Teng [36]: 4.5 cc,a = l, (3.1) cc u ( ).45,a = l. (3.) We also show the ormula used in Eurocode []: cc 1 + 5, i.5 l,a l,a = l,a + l,a > , i.5. (3.3) Here is the uniaxial compressive strength o concrete, cc is the compressive strength o conined concrete, l,a is the actual conining stress at ailure, cc is the axial strain at maximal strength o conined concrete, is the axial strain at maximal strength o 33

34 unconined concrete, u is the hoop strain at ultimate state o the conining FRP. Note that l,a is smaller than the conining strength l, which belongs to the ailure strength o FRP (see Equation 3.7). The ratio o l,a and l is the strain eiciency actor [36], which is denoted by κ : l,a κ =. (3.4) The simplest model (also used in Eurocode []) applies a similar stress-strain curve as or unconined concrete, where the strength is replaced by cc. This curve is identiied as Eurocode in Figure 3.. Lam and Teng [36] recommended a parabolic-linear curve or conined concrete: l σ c ( E E ) E =, where c c c c, i c t 4 + E i, c t c cc t = E E c, E cc =. (3.5) cc This curve (see Figure 3.) was recommended by Bisby and Ranger [8] and also by Rocca et al. [58] or eccentric loading. σ c cc Eurocode cc,ec Lam and Teng Unconined t cu c,c cc cc,ec Figure 3.: Concrete material models. Based on the three stress-strain curves o Figure 3. we calculated the normalized N-M ailure envelopes (capacity diagrams) o conined columns, which were experimentally investigated. Three typical curves are shown in Figure 3.3. Note that according to Bisby and Ranger [8] the conining eect o steel stirrups should be neglected; however or the sake o better comparability we took the eect o steel stirrups into account by increasing the conining stress in the area inside the stirrups. c 34

35 . N (a) Eurocode. N (b) Eurocode 1.5 Lam and Teng 1.5 Lam and Teng Unconined.5 Unconined M M. N (c) Eurocode Lam and Teng.5 Unconined Figure 3.3: Comparison o experimental results and simpliied models: concrete columns with unidirectional CFRP coninement (Hadi [3]) (a), reinorced concrete columns with unidirectional FRP coninement (Bisby and Ranger [8]) (b) and concrete illed FRP tubes (Fam and Rizkalla [1], tube no. 5) (c). Bisby and Ranger [8] and Rocca et al. [58] suggested that due to the strain eiciency actor the top part o ailure envelope must be cut o by a horizontal line and the maximum normal orce is reduced by the actor κ (Equation 3.4). This is the reason o the plateaus in Figures 3.3a and 3.3b. In the case presented in Figure 3.3c κ 1. (The calculation o concrete illed composite tubes with design oriented expressions was complex. The tube enhances the load bearing capacity in two ways: (i) it has axial resistance and (ii) through circumerential coninement it increases the concrete (axial) strength. The inluence o these eects and hence the ailure load depends on the ratio o the axial and the circumerential strain. Unortunately, the design oriented expressions do not predict the circumerential strains. In the calculation we have chosen this ratio in such a way that the experimental data or concentric load could be obtained, and then, this ratio was applied or eccentric loading as well.) Some o the results are acceptable (the calculated curves are close to the experimental results), however in some cases the models seem to overestimate the eect o the coninement at high eccentricity. M 35

36 Analysis-oriented concrete models or concentric loading were discussed in the previous chapter and are not reiterated here. We ound only one article in the literature which applied an analysis-oriented model or the eccentrically loaded FRP conined concrete columns. Parvin and Wang [5] used the MARC TM nonlinear inite element sotware and applied the built-in Mohr-Coulomb yield criterion with isotropic hardening rule or concrete. They demonstrated the applicability o the 3D model or conined columns, however only qualitative comparisons with experiments were made. 3. PROBLEM STATEMENT As we stated in the Introduction there are models or concentrically and eccentrically loaded FRP conined columns, however all the existing models ail to properly predict the behavior o eccentrically loaded conined columns. Authors also admit [8] that urther research in this area is needed. Our aim here is to develop a new model or eccentrically loaded (Figure 1.1b) FRP conined concrete or reinorced concrete columns (Figure 3.4). With the aid o this model we wish to predict the experimental data and to explain the behavior o conined columns. (a) e F (b) e F concrete FRP stirrup rebar concrete FRP y y t t d d Figure 3.4. Cross-section o conined columns without reinorcement (a) and with reinorcement (b). 3.3 APPROACH While or concentric loading both design-oriented and analysis-oriented approaches are easible, or eccentric loading design-oriented models which are based on the conined concrete strength (e.g. Equations ) seem unacceptable, because the eect o coninement is dierent or concentric- and or eccentric loading. This is illustrated in Figure 3.5. Under concentric loading the in-plane stresses in the hoop and radial directions are identical (Figure 3.5a). For pure bending assuming a linearly elastic material law the highest conining stress in the hoop direction is three times bigger than in the radial 36

37 direction (Figure 3.5b). In addition, shear stresses arise between the concrete and the conining FRP. For eccentric loading the stress state (again or linearly elastic behavior) is between the previous two cases, as it is illustrated in Figure 3.5c. (When concrete starts yielding the behavior is similar, however the calculation is more complex.) (a) (b) (c) N M e N concrete FRP concrete FRP concrete FRP ax ax ax l l l σ l σ l σ l 3σ l σ l σ l σ l 3σ l σ l ~σ l ~σ l σ l Figure 3.5: Elastic strains and stresses o conined columns subjected to concentric load (a), bending (b) and eccentric load (c). Because o the signiicant dierence in the radial and hoop conining stresses we decided to use an analysis oriented model, which is based on a new, sophisticated 3D concrete material law proposed by Papanikolaou and Kappos [5]. 3.4 THE NEW MODEL A D inite element model was developed or the calculation o the cross-section. Note that the inite element mesh is two-dimensional, however the strains and stresses are threedimensional: it is assumed that the axial strain varies linearly through the cross-section i.e. we assume the Bernoulli-Navier hypothesis (and the nonlinearly varying axial stress is calculated by the FE code). A triangular inite element mesh was applied (Figure 3.6). The concrete was modeled by the coninement sensitive plasticity constitutive model o Papanikolaou and Kappos [5], which was also discussed in the previous chapter. The FRP coninement was calculated by the laminated plate theory assuming a linearly elastic behavior until ailure. It was modeled by boundary elements along the circumerence (Figure 3.6). 37

38 (a) z FRP boundary element (b) z FRP boundary element stirrup boundary element y concrete inite element concrete inite element y rebar elements Figure 3.6: The inite element mesh o the cross-section without (a) and with reinorcement (b). Axial rebars and stirrups i present were calculated by an elastic-plastic material law. The eect o stirrups was taken into account by assuming continuous layer (with zero axial resistance), and applying a boundary layer within the cross-section (Figure 3.6b). Due to the high nonlinearity o the concrete material law an incremental calculation was implemented, the block diagram is shown in Figure 3.7. I the eccentricity is high, the neutral axis will be inside the cross-section. At the tensionpart o the cross-section, where the axial strains are positive, we assumed that the concrete is cracked and has no axial strength (however the in-plane stinesses are non-zero and hence the cross-section remains approximately circular). As we stated above, shear stresses arise at the concrete surace. These shear stresses can lead to principal tensile stresses, which can cause rupture in concrete even in the compressed region o the cross-section. This eect was also considered in our model. Input data materials, geometry Increase strains yield criterion reached in some points Plastic strains added in points reaching yield criterion Creating FEM mesh In-plane tensile ailure? no FEM calculation with considered plastic strains FEM calculation materials in elastic state yes Tensile concrete modeling FRP ailure? no Output data yes Figure 3.7: Block diagram o the calculation. 38

39 As an example o the numerical calculation a strain-orce diagram (and the corresponding N-M curve) is shown in Figure 3.8, assuming that the neutral axis is ixed. The calculation is terminated, when the FRP breaks due to hoop tension or axial compression. (The behavior o concrete (plastic hardening, sotening, etc.) is explained in the previous chapter). N (a) N (b) Coninement hardening σ σ Plastic sotening σ Plastic hardening x σ Elastic M Figure 3.8: Load path or ixed neutral axis. 3.5 CALCULATION OF CAPACITY DIAGRAMS (FAILURE ENVELOPES) For a given cross-section the strain-orce (and the corresponding orce-moment) diagrams (Figure 3.8) were calculated or dierent load paths. For each load path the position o the neutral axis was ixed. The envelope o all N-M curves is identical to the capacity diagram (or ailure envelope) as it is shown in Figure 3.9. N Capacity diagram Figure 3.9: Capacity diagram obtained rom envelope o loading paths. M 39

40 The capacity diagram depends on the load path. We obtain dierent envelopes i or example we assume ixed neutral axis or ixed eccentricity o the orce. According to our calculations these dierences are small. An example o calculation with dierent load paths is shown in Figure 3.1. N Capacity diagram Load path with ixed neutral axis Load path with ixed eccentricity M Figure 3.1: Results o calculation with dierent load paths. An axial stress distribution at ailure is shown in Figure 3.11a. The axial stress varies slightly perpendicular to the plane o eccentricity; the stresses due to the shear stresses are smaller at the edges. The average stresses ( σ x = σ x dz / ( b) ) are presented in Figure 3.11b. b b (a) (b) σ x σ x b z y z y Figure 3.11: The stress distribution at ailure (a) and the average stresses (b). The average stress curves at dierent eccentricities are given in Figure 3.1a or a highstiness coninement and in Figure 3.1b or a low-stiness coninement. 4

41 N (a) 1 3 Stresses at ailure state: σ 1 z 3 4 M 4 d N (b) M Stresses at ailure state: σ 1 z 3 4 d Figure 3.1: Average stress curves at ailure. High-stiness coninement (a) and low-stiness coninement (b). 3.6 VERIFICATION We compared our results with available experimental data and also with our own tests and a good correlation was ound Existing test results The available experimental data were compared to our numerical results and also to the diagrams based on design-oriented models (Figure 3.). We recall that or concentric loading instead o the conining stress due to the FRP ailure strength only a reduced stress must be taken into account: l,a = κ, (3.6) l where according to Equation (1.1) l t d =, (3.7) is the ailure strength o FRP (measured on coupon tests), t is the thickness o FRP, d is the diameter o the cross-section (Figure 3.4) and l,a is the radial stress measured at the ailure. We recall that or concentric loading the strain eiciency actor κ is typically 41

42 between.6 and 1. [36]. This is due to the uneven hoop stresses in FRP because o the axial cracks in concrete. Unortunately, or eccentric loading very ew experimental data on κ are available. The results o Bisby and Ranger [8] showed that or eccentric loading the reduction is much less (κ is closer to unity) than or concentric loading. This statement can be explained by investigating the stresses presented in Figure 3.5, which shows that under eccentric loading the hoop compressions are higher than or concentric loading, which reduces the likelihood o axial concrete cracks. Because o the lack o reliable data we simply apply the ollowing κ values (Figure 3.13): 1, i ρκ ρβ ρβ ρκ κ = 1 ( 1 κ), i < ρκ < ρβ, (3.8) ρβ κ, i ρκ = where κ is the strain eiciency actor or concentric loading and ρ κ is the curvature (Figure 3.13b). ρ κ = is the case o concentric loading and ρ β = cc / d is the case when the neutral axis is at the edge o the cross-section. Accordingly, only or the cases o small eccentricity κ plays an important role. We must admit, however that urther experiments are needed to clariy the eect o κ under higher eccentricities. κ 1. (a) (b) ρ = cc x cc κ x d ρ ρ β Figure 3.13: Assumed variation o the strain eiciency actor (κ ) as a unction o the curvature (ρ). The comparison o experimental results and numerical calculations are shown in Figure The values o κ are given in the igures. Only in one case (Figure 3.14d) was κ measured (κ =.49), in all the other cases we have chosen its value in such a way that cc matches the data or concentric loading. The results o the new model including the eect o the strain eiciency actor through Equation (3.8) are shown by solid lines. The results without this correction are plotted by dashed lines. 4

43 N (a) N (b) 3 Eurocode Lam and Teng 4 Eurocode Lam and Teng 3 1 Proposed model κ =.6 1 Proposed model κ = M M N (c) N (d) 4 3 Eurocode Lam and Teng 1 8 Eurocode Lam and Teng 1 Proposed model κ = 1. 4 Proposed model κ = M M N (e) N () 6 4 Eurocode Lam and Teng Eurocode Lam and Teng Proposed model κ = M 4 Proposed model κ = Figure 3.14: Comparison o experimental results and models: concrete columns with unidirectional CFRP arrangement (Hadi [3]) (a), reinorced concrete columns with unidirectional FRP arrangement (Hadi [4]) (b), reinorced concrete columns with unidirectional FRP arrangement (Hadi [5]) (c), reinorced concrete columns with unidirectional FRP arrangement (Bisby and Ranger [8]) (d), concrete illed FRP tubes (Fam and Rizkalla [1], tube no. 5) (e) and concrete illed FRP tubes (Fam and Rizkalla [1], tube no. 6) (). M In Figure 3.14 it seems that or concentric loading our model signiicantly overestimates the ailure load. Due to the act that all the other points are reasonable and considering the calculated load-strain curve (Figure 3.15), there is an other explanation: this case is a lowstiness coninement, which means that there is a local maximum on the orce-strain 43

44 curve. This local maximum agrees well with the ailure load measured in the experiment. It is possible that the increasing branch was not measured. 8 Failure predicted by the model Axial orce [kn] 6 4 Experimental result Axial strain Figure 3.15: Comparison o calculated axial orce-axial strain diagram and experimentally measured axial orce or concentrically loaded specimen o Fam and Rizkalla [1], tube no New test results [11] Fiteen circular column specimens were prepared and tested in our experimental program running in 6 at the Budapest University o Technology and Economics. All columns had the same size: 154 mm diameter and 1 mm height. The columns were reinorced with six axial 8 mm diameter reinorcing bars and with Ø6 mm links placed at 15 mm distance. The concrete was supplied by a local irm; the specimens were cured or at least 8 days beore the FRP coninement was applied. Three columns were conined with carbon ibers with epoxy resin, three with carbon ibers with 3P resin. Six specimens were conined using glass ibers, again, three applied with epoxy- and three with 3P resin. 3P resin has similar properties as the epoxy and was supplied by the manuacturer. The three remaining columns were used as control specimens. FRP wrapping: The surace o the specimen was cleaned. First a layer o the resin was applied and it was ollowed by a layer o the unidirectional glass or carbon sheet. The laminates were continuously wrapped on the surace o the specimen with 1 angle with respect to the hoop direction. Another layer o the resin was applied and the next laminate was wrapped with -1 angle. For GFRP conined columns three layers and or CFRP conined columns two layers o sheet were used. Ater the last layer o laminate a inal layer o resin was applied. The top and bottom 1mm o the column was urther reinorced with two layers o GFRP to avoid the splitting o the head during loading. The wrapped specimens were kept at room temperature or at least the time suggested by the manuacturer or the CFRP and GFRP to cure. 44

45 Instrumentation and loading test: six strain gauges were applied on the surace o each column at the middle cross-section, three in the axial and three in the hoop direction. The axial and hoop gauges were arranged with equal distances. The horizontal displacement o some o the specimen was also measured. The specimens were tested with or 1 mm initial eccentricity. Hinges were placed at both ends o the columns and the theoretical length, l o the columns (distance between the centers o the hinges) was 15 mm. The load was applied using a universal testing machine. The instrumentation or a CFRP conined specimen can be seen in Figure (a) (b) Figure 3.16: Instrumentation o specimen 11 (a) and the specimen curved due to the load (b). Material properties: The average cylinder compressive strength o unconined concrete, was very low, 5.9 N/mm with a high standard deviation. The reinorcing steel was B5 with ultimate strength 6 N/mm. To determine the material properties o the FRP conining materials twelve lat coupon tests were conducted. Three or each arrangement: glass ibers with epoxy and 3P resin and carbon ibers with epoxy and 3P resin. The tests showed insigniicant eect o the matrix on the material properties. The thickness, number o layers and matrix to ibers ratio was equal to the FRP applied on the columns. For the GFRP 35 N/mm nominal elastic modulus (elastic modulus multiplied by the nominal thickness) and.3 % maximal strain was ound. For the CFRP laminates the nominal elastic modulus was 148 N/mm and the maximal strain was.8%. Experimental results: The specimen details and test results are shown in Table 3.1. For column 4 two results are available: one or the state when the strain gauges were lost (their measure limit was reached) and the second is the axial orce measured at the rupture o the ibers in the hoop direction. This was the only column that ailed with the rupture o the FRP, all other experiments were stopped when a plateau was reached in the load path and 45