Damage-Based Stress-Strain Model for Fiber-Reinforced Polymer-Confined Concrete

ACI STRUCTURAL JOURNAL Title no. 02-S06 TECHNICAL PAPER Damage-Based Stress-Strain Model for Fiber-Reinforced Polymer-Confined Concrete by Domingo A. Moran and Chris P. Pantelides A damage-based stress-strain model applicable for both bonded or nonbonded fiber-reinforced polymer (FRP) confined concrete is developed for predicting the compressive behavior of circular FRP-confined concrete members. The model is based on a variable Poisson s ratio formulation. The variable secant and tangent Poisson s ratios used in the model are a function of the mechanical properties of the unconfined concrete and confining FRP jacket, and the extent of internal damage in the confined concrete core. Equilibrium and strain compatibility are used to obtain the ultimate compressive strength and strain of FRP-confined concrete as a function of confining stiffness and radial strain in the FRP jacket. Keywords: column; ductility; rehabilitation. INTRODUCTION The seismic retrofit of existing reinforced concrete columns and structural systems has been carried out using fiber-reinforced polymer (FRP) composite jackets in recent years. -6 Despite successful application of FRP jacketing systems, research into the constitutive relationships governing the behavior of FRP-confined concrete is continuing, since the early work presented by Fardis and Khalili. 7 Recently, Spoelstra and Monti, 8 and Fam and Rizkalla 9 modified the Mander, Priestley, and Park 0 model for steel-confined concrete to represent the behavior of FRP-confined concrete. The Spoelstra and Monti 8 model is an iterative equilibrium-based stress-strain model in which the behavior is governed by the Mander, Priestley, and Park 0 model for steel-confined concrete and the Pantazopoulou and Mills constitutive model for concrete. The Fam and Rizkalla 9 model is an incremental Mander, Priestley, and Park 0 equilibrium and strain compatibility model that utilizes a variable axial secant modulus and a confining pressure-dependent variable secant Poisson s ratio. The term damage-based refers to the damage in the concrete due to the extension, growth, and nucleation of microcracks and/or microvoids, which brings about a reduction in the available load-resisting area that contributes to progressive material deterioration (that is, material damage) during axial compressive deformation of the confined and unconfined concrete. The general concepts of elasticity, damage mechanics and plasticity theory are included in a stress-strain model that considers the macrostructural effects of the increase in internal damage into a simple mechanics model that utilizes simple variable secant and tangent Poisson s ratio formulations to describe the compressive behavior of circular FRP-confined concrete subjected to a uniform state of stress and confinement. With some modifications, the applicability of the proposed damagebased model can be extended to the case of concrete subjected to nonuniform cyclic loading and nonuniform confinement such as rectangular FRP-confined concrete; these modifications are beyond the scope of this paper. RESEARCH SIGNIFICANCE A satisfactory confinement model allows accurate estimates of the displacement and curvature ductility of reinforced concrete columns and required FRP jacket thickness. The confinement stress-strain model developed is based on the fact that the dilation behavior of FRP-confined concrete is dependent on the lateral kinematic restraint provided by the elastic FRP jacket, since concrete is a restraint-sensitive material, rather than a pressure-sensitive material. The confinement model is based on a secant Poisson s ratio formulation, and is applicable to either bonded or nonbonded circular FRP-confined concrete, and to concrete confined with either low- or high-stiffness FRP jackets subjected to uniform axial compression. BEHAVIOR OF FRP-CONFINED CONCRETE The proposed model is based on the compressive stressstrain behavior of circular FRP-confined concrete members exhibiting either localized strain-softening behavior, as occurs with concrete confined by low-hoop-stiffness FRP jackets, or bilinear compressive behavior as occurs with concrete confined by moderate- to high-stiffness FRP jackets, as shown in Fig.. Consider a circular concrete column confined by a circular FRP composite jacket having an effective confining stiffness Fig. Typical stress-strain behavior of fiber-reinforced polymer (FRP) confined concrete confined by low- and high-stiffness FRP jackets. ACI Structural Journal, V. 02, No., January-February 2005. MS No. 03-286 received July 6, 2003, and reviewed under Institute publication policies. Copyright 2005, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including authors closure, if any, will be published in the November- December 2005 ACI Structural Journal if the discussion is received by July, 2005. 54 ACI Structural Journal/January-February 2005

Domingo A. Moran is a structural engineer for Reaveley Engineers and Associates, Salt Lake City, Utah, and a PhD candidate at the University of Utah. He received his BSc from the U.S. Naval Academy and his MSc from the University of Utah. His research interests include analytical investigations of reinforced concrete columns confined with fiber-reinforced polymer composites, seismic performance of beamcolumn joints, and reinforced concrete design using state-of-the-art materials. ACI member Chris P. Pantelides is a professor of civil and environmental engineering at the University of Utah. He is a member of ACI Committee 374, Performance-Based Seismic Design of Concrete Buildings. His research interests include seismic design, evaluation, and rehabilitation of reinforced concrete building and bridge construction. C je, and normalized effective confining stiffness K je, are defined as K je C -- je f co 2t C j E j k ; e je where k e is confining efficiency, which for continuous circular FRP jackets equals.0. Mirmiran and Shahawy 2 defined the dilation rate µ c, or tangent Poisson s ratio (ν c ) t, of FRP-confined concrete as At the initiation of uniform compressive axial strain ε c, and resultant axial stress f c, both the circular concrete core and confining FRP jacket behave elastically prior to cracking of the concrete core. Note that the sign convention used is that compressive stresses and strains are positive. The resultant radial strain ε θ, in both the thin FRP jacket, where D c >> t j, and confined concrete core are found using Hooke s generalized law 3,4 as ε θ µ o ε c ; µ θ o --- -- (3) c o C + -- je ( 2 ν ci )( + ν ci ) Using Hooke s generalized law, the initial axial E co, and radial E θo, moduli of elasticity of concrete confined by an FRP jacket are D c µ c ( ν c ) θ t --- c E ci ν ci 2ν E co E ci ci µ o C + --- je ; E co Eθo --- E ci µ o () (2) (4) increases as dilation and internal microcracking of the concrete core increase. Due to the elastic kinematic restraint provided by the FRP jacket, the dilation rate µ c of FRPconfined concrete tends to approach an asymptotic plastic value of µ p at high axial compressive strains, as shown in Fig. 2, where µ θ p --- c Note that plastic stress-strain behavior occurs when ε co ε c ε cu and ε θo ε θ ε θu. Dilation behavior of FRP-confined concrete In the stress-strain model introduced herein, the concrete core is considered a restraint-sensitive material whose dilation behavior is dependent on the lateral kinematic restraint provided by the elastic FRP jacket 5 rather than a pressuresensitive material. 9 If the dilation behavior of FRP-confined concrete throughout its compressive behavior is considered to depend solely on the lateral stiffness of the confining FRP jacket and the mechanical properties of the unconfined concrete core, then the compressive stress-strain behavior of FRP-confined concrete can be described by considering the variation of the secant Poisson s ratio ν c, and tangent Poisson s ratio (ν c ) t, as damage progresses (that is, as both ε c and ε θ increase). In Fig. 2, a series of ε θ -versus-ε c curves of concrete cylinders confined by an FRP jacket of low, medium, and high effective jacket stiffness K je, are shown. Based on analysis of normal-strength FRP-confined concrete circular 6 and square sections, 7 the transition point for members exhibiting strain-softening or strain-hardening behavior appears to occur when the value of the plastic dilation rate µ p approaches unity (that is, when µ p.0). As a result, the behavior of circular concrete members confined by thin FRP jackets can be separated into two distinct types: a) quasi-bilinear strain-hardening compressive stress-strain behavior typical of concrete members confined by moderate to high effective stiffness FRP jackets for which µ p <.0 as shown in Fig. 3(a); and b) strain-softening compressive stress-strain behavior typical of concrete members confined by low effective stiffness FRP jackets for which µ p.0 as shown in Fig. 3(b). In Fig. 4, axial versus radial strain curves of concrete members confined by low- and high-stiffness p (7) E ci ( 3320 f co + 6900)MPa (5) ( 40, 000 f co +.0 0 6 E )psi; E ci θi -- In what follows, it is assumed that C je << E ci and C je << E θi. As a result of microcracking of the concrete core, the transverse strains increase at a faster rate than those predicted by elasticity theory. The secant Poisson s ratio of the concrete given by ν ci ν c ε θ ε c (6) Fig. 2 Radial strain versus axial strain curves of concrete members confined by low-, medium-, and high-stiffness fiber-reinforced jackets. ACI Structural Journal/January-February 2005 55

FRP jackets are shown; an inverse secant Poisson s ratio is defined as µ ν p vol -for µ 2µ p p ν vol ν c ε c ε θ E - θ E c (8) ε µ p µ p + µ oc p --- ( ε θ ) lim Ω µ ; ( ε θ ) pk α pk ε co ν pk (2) These quasi-bilinear curves are described by a variable inverse secant Poisson s ratio model for FRP-confined concrete as ν pk ε θ ε c pk 2µ o µ p µ - o + µ ; ν -- µ ν p pk pk p µ p ν pk (3) ν c -- µ o Ω µ ( Ω - µ ) + --- ε + -- 2 -- ε 2 θ + - 2 θ ( ε µ ε θ ) pk o oc + -- µ p (9) α vol µ ε oc α pk ε p µ o co - ; ( εc ) µ p + µ o vol α vol ε co ( ε c ) -- vol ε co 2µ --- p µ p µ - o for µ p ν 2µ p µ p + µ o vol (4) (5) Ω µ µ p 0 E ; ν c sec - E θ o ε θo ε co -- ; ν sec µ p ν --- sec µ p ν sec µ µ p µ o E o --; ν c µ p µ vol - 0.50; o E θ vol ε c vol ε θ (0) () The term Ω µ in Eq. (9), (0), and (2) is a step function, where Ω µ.0 when µ p >.0, as typically occurs in low stiffness FRP-confined concrete, and Ω µ 0 otherwise. Note that when µ p.0, the compressive behavior of the FRPconfined concrete is similar to that of an elasto-plastic compressive material. (a) (a) (b) Fig. 3 Typical stress-strain behavior of FRP-confined concrete confined by: (a) high-stiffness FRP jackets; and (b) low-stiffness FRP jackets. (b) Fig. 4 Axial strain versus radial strain curves and strain parameters for concrete members confined by: (a) lowstiffness FRP jackets; and (b) moderate- to high-stiffness FRP jackets. 56 ACI Structural Journal/January-February 2005

The peak axial strain ratio α pk (ε c ) pk /ε co of Eq. (2) and (4) can be determined as follows. When the circular FRP jacket has sufficient effective stiffness, C je or K je, to curtail the volumetric expansion of the circular FRP-confined concrete core, such that µ p 0.50 α pk.0. When 0.5 < µ p <.0, α pk is given by α α vol ( µ p + µ o ) pk 4µ p µ o µ o ( 2µ p ) - µ p µ o ( 2µ p ) (6) When µ p.0, α pk, can be found iteratively by solving for the following relationship ν vol --- 2 µ o ν vol -- 2 --- ν vol --- 0 ; (7) α pk µ o µ --- p for µ 2µ p p ν vol α p ν vol µ --- o --- 2 ν vol µ o (8) An iterative procedure that converges within 2 to 5 iterations using an initial value of α pk.0 in conjunction with Eq. (0) through (2), (4), and (8) can be used to solve for α pk in Eq. (7). Note that in the damage-based secant Poisson s ratio formulation of Eq. (8) through (8) no consideration is given to the passive confining pressure provided by the confining elastic FRP jacket as is typically assumed in the analysis of passively confined concrete. The dilation behavior of circular FRP-confined concrete depends only on the lateral kinematic restraint provided by the circular confining elastic FRP jacket and the amount of damage in the concrete core rather than on the confining pressure provided by the confining FRP jacket. In this model, the variable inverse secant Poisson s ratio of FRP-confined concrete is given in terms of the initial tangent dilation rate µ o of Eq. (3), the plastic dilation rate µ p of Eq. (7), and the extent of internal damage as measured by the radial concrete strain ε θ. -- 2 (a) (c) (b) (d) Fig. 5 Comparison of analytical and experimental curves of Specimen H-3p-2 by Xiao and Wu (2000): (a) stress-strain curves; (b) axial versus radial strain curves; (c) volumetric strain versus axial and radial strain curves; and (d) secant Poisson s ratio versus axial and radial strain curves. ACI Structural Journal/January-February 2005 57

The inverse dilation rate /µ c, or inverse tangent Poisson s ratio (/ν c ) t, can be derived directly from the secant Poisson s ratio formulation of Eq. (9) and (0) as of µ p (ε θ ) lim /ε oc is recommended for the case of unconfined normal-strength concrete and concrete confined by very low effective stiffness FRP jackets. µ c ν c t --- c θ (9) ( Ω µ ) ε - θ 2 -- Ω - µ ( ε θ ) pk + - µ 3 o ε + -- 2 -- 2 ε θ + - 2 θ 2 + -- µ p ( ε µ o ε θ ) pk oc STRESS-STRAIN MODEL FOR FRP-CONFINED CONCRETE In what follows, it will be demonstrated that the previously mentioned damage-based secant Poisson s ratio formulation can be easily implemented into a damage-based secant modulus type stress-strain model that can accurately capture the behavior of FRP-confined concrete. Two types of compressive stress-strain curves of circular FRP-confined concrete members are shown in Fig. 4, for low- and high-stiffness FRP jackets, which can be described by the following damagebased stress-strain model For high-stiffness FRP-jacketed sections (that is, µ p.0), the above inverse dilation rate formulation is different from the four-parameter fractional dilation rate formulation proposed by Mirmiran and Shahawy 2,8 in which damage is accounted for indirectly by considering the effects of the axial strain ratio (ε c /ε co ). In the approach considered herein, the dilation rate is considered to be directly dependent on the amount of damage in the FRP-confined section, as measured by the expansive hoop strain ε θ, at a given axial strain ε c. In addition, the maximum dilation rate (µ c ) max can be obtained by setting δµ c /δε θ 0 for ε θ 0 in Eq. (9), and is found to occur at a hoop strain ratio of ε θ /(ε θ ) pk 3, and its magnitude is found by setting ε θ /(ε θ ) pk 3 in Eq. (9). The tangent volumetric strain (ε v ) t, or volumetric dilation rate ψ v, in both the radial and axial strain directions is defined based on the inverse dilation rate /µ c, using Eq. (9), (0), and (4), respectively, as ( ψ v ) v ( θ --- c + 2 θ ) - θ θ 2 (20) (2) Determination of the plastic dilation rate In the mechanics-based model presented herein, with the exception of the plastic dilation rate µ p of the FRP-confined concrete, the terms utilized herein to describe the relationship between the imposed axial strain ε c and resultant expansive strain ε θ of the circular FRP-confined section were all defined both analytically and graphically in the previous section. The authors 9,20 have previously shown that, as a result of the kinematic restraint provided by the confining elastic FRP jacket, the asymptotic plastic dilation rate µ p of the FRP-confined concrete section is dependent on the normalized effective confining stiffness of the FRP jacket K je, and can be approximated as (22) The above empirical approximation for the plastic dilation rate µ p can be used in the mathematically derived expressions introduced in Eq. (8) through (2), for which the equality ν pk 2µ o was assumed. In addition, an upper bound value µ c ( ψ v ) v ( c --- c + 2 θ ) - ( 2µ c c ) c µ θ p --- c p.05 0 2 µ ε co ( K je ) 0.667 o E θ f c E θ ε θ E c ε c Ω µ E --+ co ν c 2 E co --- -- ε θ ( Ω µ ) E sec E θo (23) (24) E θp f co ω je ϕ θ ; f oθ f co [ ω je ( ε θ ) lim ( ϕ θ )]; (25) (26) In the aforementioned model, the radial curvature parameter n θ of Eq. (24) can be conservatively estimated as n θ 2.0 or can be found more accurately from the iterative solution of (27) The behavior of FRP-confined concrete when µ p >.0 is shown in Fig. 4(a), in which a localized strain-softening behavior occurs near the peak compressive strength of unconfined concrete. This unstable compressive behavior occurs because of ineffectiveness of the low-stiffness FRP jacket in curtailing the concrete dilation at very low transverse strains, as the volumetric strain ε v goes to zero (that is, when ν c ν vol 0.50), where ε co ( E θp ) --- + E θp + ( E θo E θp )ε - θ f oθ E sec f -- co ε co n θ n θ H ω je K je ( k ) avg ; ϕ ( ) 0.80 θ θ --- ( H θ ) -- E θ E θd E θp n θ E θp E θd n θ 0.70f co Eθ --- E θp 0; f cθ E θd 0.70f co E - θo ε θd.0 58 ACI Structural Journal/January-February 2005

ε ν ( ε c + 2ε θ ) ε c ( 2ν c ) ε θ 2 (28) The low-stiffness FRP jacket does not become effective until further dilation of the concrete core induces an increase in hoop stresses in the FRP jacket. It should be noted that in the damage-based model developed herein, the plastic dilation rate µ p of Eq. (22) is the only experimentally determined parameter used in the damagebased secant Poisson s ratio model of Eq. (9) through (2). In addition, the average bond-dependent confinement coefficient (k ) avg and the exponent of 0.80 given in the definition of the radial strain coefficient ϕ θ of Eq. (26) are the only experimentally determined parameters utilized in the damage-based secant modulus type stress-strain model of Eq. (23) through (26). STRESS-STRAIN MODEL IMPLEMENTATION In an FRP-confined concrete member, compressive failure occurs simultaneously with failure of the FRP jacket, be it failure due to rupture, delamination, lap failure or shear failure. This failure occurs at an ultimate radial FRP jacket strain ε θu that may be below the rupture strain of FRP composite tensile coupons. Premature failure of the FRP jacket occurs as a result of interaction between axial shortening ν c and radial dilation, which induces a biaxial state of stress and strain in the FRP jacket, in addition to stress concentrations at the jacket-to-concrete interface that occur as dilation of the FRPconfined concrete core progresses. The damage-based stressstrain model for circular FRP-confined concrete is implemented using one of the following methods: Method A: Incremental secant solution; and Method B: iterative secant solution. Method A: Incremental secant solution. Set (ν c ) j ν o ; (ε θ ) j (ε θ ) cr 0.000; (ε c ) j (ε θ ) j /(ν c ) j ; (E c ) j E co ; and (E θ ) j (E c ) j /(ν c ) j (E co )/(ν o ); find (f c ) j using Eq. (23) and (24); 2. Select an incremental axial strain δε c. Set (ε c ) j+ (ε c ) j + δε c, set (ν c ) j+ (ν c ) j and (ε θ ) j+ (ε c ) j+ (ν c ) j+. Set (ε θ ) j (ε θ ) j+ ; find (ν c ) j using Eq. (9) and evaluate (ε c ) j (ε θ ) j /(ν c ) j. Find (f c ) j using Eq. (23) and (24); and 3. If (ε θ ) j < ε θu, repeat Step 2. Otherwise, if (ε θ ) j > ε θu, set (ε θ ) j ε θu ; find ν cu using Eq. (9) and evaluate ε cu ε θu /ν cu. Find f cu using Eq. (23) and (24). Method B: Iterative secant solution For a given FRP-confined concrete subjected to a given axial strain ε c, the resultant axial stress f c and radial strain ε θ in the FRP-confined concrete are found as follows: (a) (c) (b) Fig. 6 Comparison of analytical curves of concrete members confined by fiber-reinforced polymer jacket of varying stiffness: (a) stress-strain curves; (b) axial versus radial strain curves; (c) volumetric strain versus axial and radial strain curves; and (d) secant Poisson s ratio versus axial and radial strains. ACI Structural Journal/January-February 2005 59 (d)

. Using Eq. (8) and (9), iterate the radial strain ε θ of the damaged concrete section until ε c ε θ /ν c. Find f c using Eq. (23) and (24); and 2. If ε θ > ε θu, set f c 0; to find f c f cu, set ε θ ε θu ; find ν cu using Eq. (9) and evaluate ε cu ε θu /ν cu. Find f cu using Eq. (23) and (24). COMPARISON OF MODEL WITH EXPERIMENTAL RESULTS In Fig. 5, experimental curves of a bonded CFRP-confined concrete cylinder, test (H-3P-2) by Xiao and Wu, 6 are compared with analytical curves obtained using the incremental secant solution; the experimental curves are shown as heavy bold curves, whereas the analytical curves are shown as light curves. Figure 5(a) shows the stress-strain curve and Fig. 5(b) the axial versus radial strain curve. The volumetric strain curves are plotted versus axial and radial strain in Fig. 5(c). The secant Poisson s ratio curves are plotted versus axial and radial strain in Fig. 5(d). From Fig. 5, it can be observed that the proposed analytical damage-based model of Eq. () through (28) can accurately capture the compressive behavior of the bonded CFRPconfined concrete cylinder test by Xiao and Wu. 6 The analytical constitutive stress-strain model using the incremental secant solution was also compared with experimental results of concrete cylinder tests confined by nonbonded GFRP jackets performed by Mirmiran, 2 and bonded CFRP-confined concrete cylinder tests performed by Picher, Rochette, and Labossière, 22 Wu and Xiao, 4 and Rochette and Labossière. 7 The proposed constitutive stressstrain model was found to accurately capture the bilinear compressive behavior of FRP-confined concrete in all cases; the results were similar to those presented by the writers in Moran and Pantelides. 9,20 ANALYTICAL RESULTS In Fig. 6, the analytical curves of bonded CFRP-confined concrete columns obtained using the incremental secant solution are compared to the analytical curves of an unconfined concrete column having an unconfined peak compressive strength f co 34.5 MPa and an unconfined peak compressive strain ε co 2.0 mm/m. Note that the selected FRP jacket stiffnesses of K je 0.0, K je 20.0, and K je 50.0 represent the typical range of FRP jacketed concrete cylinder tests found in the literature. Figure 6(a) shows the stress-strain curves and Fig. 6(b) the axial versus radial strain curves. Note that as the stiffness of the FRP jacket increases, the average slope of the plastic region of the stress-strain curve increases. In the axial versus radial strain curves, the average slope of the plastic region increases as a result of the increase in the kinematic restraint provided by the stiffer FRP jacket. The volumetric strain is plotted versus axial and radial strain in Fig. 6(c). In this figure, it can be observed that as the FRP jacket stiffness increases, the occurrence of volumetric expansion is delayed. For the case of high-stiffness FRP jacketed members, volumetric expansion is inhibited. The secant Poisson s ratio curves are plotted versus axial and radial strain in Fig. 6(d). Note that as the FRP jacket stiffness increases, the plastic secant Poisson s ratio decreases. The stress-strain behavior shown in Fig. 6 is typical of circular FRP-confined concrete cylinder tests found in the literature, and this indicates that the analytical stressstrain model of Eq. (0) through (28) developed herein can predict accurately the behavior of circular FRPconfined concrete members. CONCLUSIONS An analytical damage-based model for describing the compressive behavior of circular FRP-confined concrete is presented, which is based on principles of mechanics, and is applicable to bonded or nonbonded FRP-confined concrete. The distinguishing feature of the model is the introduction of a damage-based variable secant Poisson s ratio formulation in which the secant Poisson s ratio of the FRP jacket is demonstrated to be a function of the stiffness of confining FRP composite jacket, the properties of unconfined concrete, and the extent of internal damage as measured by the expansive hoop strain in the FRP-confined concrete. The proposed damage-based stress-strain model governs the behavior of circular FRP-confined concrete throughout its compressive behavior and can be easily implemented into a spreadsheet, finite element, or other computer language program for the analysis of FRP-confined concrete members. Comparisons with experimental results of FRPconfined concrete cylinders indicate excellent agreement. ACKNOWLEDGMENTS The authors would like to acknowledge financial support from the National Science Foundation under contract No. CMS 0099792, the Utah Department of Transportation, and the Federal Highway Administration. The authors wish to thank Y. Xiao of the University of Southern California for providing the experimental data on CFRP-confined concrete cylinders. The opinions expressed in this article are those of the writers, and do not necessarily reflect the opinions of the sponsoring organizations. NOTATION C je effective confining stiffness of FRP jacket D c column diameter E c variable secant axial modulus E ci axial modulus of elasticity of concrete formula (E c ) i axial secant modulus of elasticity of concrete at (ε c ) i E co initial axial modulus of elasticity of confined concrete (E c ) o axial secant modulus of elasticity of confined concrete at ε co (E c ) pk peak axial secant modulus of concrete at (ε c ) pk E j average tangent hoop modulus of elasticity of FRP composite E sec axial secant modulus of unconfined concrete core (E c ) vol axial secant modulus of concrete at ν vol 0.50 E θ variable secant radial modulus E θ d dilation radial secant modulus of concrete E θ i initial radial modulus of elasticity of unconfined concrete formula (E θ ) i radial secant modulus of elasticity of concrete at (ε θ ) i E θ o initial radial modulus of elasticity of concrete (E θ ) o radial secant modulus concrete at ε θ o E θ p radial tangent plastic modulus (E θ ) pk peak radial secant modulus of concrete at (ε θ ) pk (E θ ) vol radial secant modulus of concrete at ν vol 0.50 f c axial compressive concrete stress f cd dilation axial stress of concrete (f c ) i axial compressive concrete stress at (ε c ) i f co unconfined concrete compressive strength f cp plastic axial compressive stress f o θ reference intercept axial stress H θ radial strain ratio, where 0.70 H θ 0.90 K je normalized effective confining stiffness of FRP jacket (k ) avg bond-dependent average confinement coefficient; (k ) avg 4.0 for bonded and (k ) avg 2.35 for nonbonded FRP jacket construction k c confinement effectiveness of jacket k e confining efficiency of FRP jacket k cp plastic confinement effectiveness n θ radial curvature parameter t j FRP jacket thickness α pk peak axial strain ratio α vol volumetric axial strain ratio δε c incremental axial concrete strain δε v incremental volumetric strain δε θ incremental radial strain 60 ACI Structural Journal/January-February 2005

ε c axial concrete strain ε co axial concrete strain at unconfined compressive strength f co ε cp plastic axial compressive strain (ε c ) pk localized peak axial strain (ε c ) pk effective localized peak axial strain ε cu ultimate compressive axial strain of FRP-confined concrete (ε c ) vol volumetric axial strain ε oc reference intercept axial strain ε v volumetric strain ε θ radial concrete strain ε θd radial dilation strain (ε θ ) lim material-dependent limiting radial strain; (ε θ ) lim 0.0085 for CFRP and (ε θ ) lim 0.025 for GFRP ε θo radial strain corresponding to ε co ε θp plastic radial strain ε θu ultimate radial strain of FRP-confined concrete (ε θ ) pk localized peak radial strain (ε θ ) vol volumetric radial strain (ε v ) t tangent volumetric strain µ c dilation rate of FRP-confined concrete µ o initial tangent dilation rate of the FRP-confined concrete µ o effective initial dilation rate µ p plastic dilation rate of the FRP-confined concrete µ p effective plastic dilation rate ν c secant Poisson s ratio of concrete (ν c ) t tangent Poisson s ratio ν o given value of secant Poisson s ratio of concrete ν p plastic Poisson s ratio of concrete ν ci initial secant Poisson s ratio of concrete ν pk localized peak Poisson s ratio ν pk effective localized peak Poisson s ratio ν sec peak secant Poisson s ratio ν sec effective peak secant Poisson s ratio ν vol volumetric secant Poisson s ratio ϕ θ radial strain coefficient ψ v volumetric dilation rate (ψ v ) c axial volumetric dilation rate (ψ v ) θ radial volumetric dilation rate ω je effective confinement index of FRP-confined concrete Ω µ dilation step function REFERENCES. 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