Stress-Strain Model for Fiber-Reinforced Polymer-Confined Concrete Domingo A. Moran1 and Chris P. Pantelides, M.ASCE2 Abstract: The design of fiber-reinforced polymer 共FRP兲-confined concrete members requires accurate evaluation of the performance enhancement due to the confinement provided by FRP composite jackets. A strain ductility-based model is developed for predicting the compressive behavior of normal strength concrete confined with FRP composite jackets. The model is applicable to both bonded and nonbonded FRP-confined concrete and can be separated into two components: a strain-softening component, which accounts for unrestrained internal crack propagation in the concrete core, and a strain-hardening component, which accounts for strength increase due to confinement provided by the FRP composite jacket. A variable strain ductility ratio described in a companion paper is used to develop the proposed stress-strain model. Equilibrium and strain compatibility are used to obtain the ultimate compressive strength and strain of FRP-confined concrete as a function of the confining stiffness and ultimate strain of the FRP jacket. DOI: 10.1061/共ASCE兲1090-0268共2002兲6:4共233兲 CE Database keywords: Stress strain relations; Fiber reinforced materials; Concrete, reinforced.

Introduction A significant amount of research has been carried out on the use of fiber reinforced polymer 共FRP兲 composite jackets for the retrofit of existing reinforced concrete columns and structural systems 共Saadatmanesh et al. 1994; Seible et al. 1997; Xiao and Ma 1997; Gergely et al. 1998; Pantelides et al. 1999, 2001兲. Despite successful application of FRP jacketing systems, research into the constitutive relationships governing the behavior of FRP-confined concrete has been limited. In the past, several investigators, including Saadatmanesh et al. 共1994兲 and Spoelstra and Monti 共1999兲, have attempted to modify the model proposed by Mander et al. 共1988兲 for steel-confined concrete to represent the behavior of FRP-confined concrete. Studies including those of Mirmiran et al. 共1996兲, Mirmiran and Shahawy 共1996, 1997a,b兲, Mirmiran 共1997兲, Miyauchi et al. 共1997兲, Harmon et al. 共1998a,b兲, Stanton et al. 共1998兲, Wu and Xiao 共2000兲, and Xiao and Wu 共2000兲 have shown that the behavior of concrete encased in FRP composite jackets cannot be accurately captured by the Mander et al. 共1988兲 model for steelconfined concrete. A number of investigators, including Mirmiran and Shahawy 共1997a,b兲, Miyauchi et al. 共1997兲, Gergely et al. 共1998兲, Harmon et al. 共1998a,b兲, Samaan et al. 共1998兲, Stanton et al. 共1998兲, Spoelstra and Monti 共1999兲, Wu and Xiao 共2000兲,

1

Engineer, Reaveley Engineers and Assoc., Salt Lake City, UT 84106. Professor, Dept. of Civil and Environmental Engineering, Univ. of Utah, Salt Lake City, UT 84112. Note. Discussion open until April 1, 2003. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on May 29, 2001; approved on December 3, 2001. This paper is part of the Journal of Composites for Construction, Vol. 6, No. 4, November 1, 2002. ©ASCE, ISSN 1090-0268/2002/4233–240/$8.00⫹$.50 per page. 2

and Xiao and Wu 共2000兲, have introduced stress-strain models for the constitutive relationships of FRP-confined concrete. The present model is a uniaxial model based on the Richard and Abbott 共1975兲 model for general bilinear material behavior and the Popovics 共1973兲 fractional model for concrete. The Richard and Abbott 共1975兲 bilinear model has been implemented by Mirmiran and Shahawy 共1997a,b兲, Mirmiran 共1997兲, Samaan et al. 共1998兲, and Xiao and Wu 共2000兲 in modeling the bilinear compressive behavior of concrete confined by FRP jackets. Harmon et al. 共1998a,b兲 introduced an iterative equilibrium-based model in which the behavior of FRP-confined concrete is governed by a crack-opening path model that describes the relationship between crack separation and crack slip, as confined concrete is subjected to uniaxial monotonic loading. Spoelstra and Monti 共1999兲 introduced an iterative equilibrium-based stress-strain model in which the behavior of FRP-confined concrete is governed by the Mander et al. 共1988兲 model for steel-confined concrete and a constitutive model for concrete by Pantazopoulou and Mills 共1995兲. In the Mander et al. 共1988兲 model for steel-confined concrete, the increase in the peak compressive strength of confined concrete is expressed in terms of a constant effective confining pressure and a constant strain ductility ratio that defines the increase in peak compressive strain relative to the increase in peak compressive strength of confined concrete. However, the model ignores the additional strain energy of the concrete due to confinement offered by the elastic FRP jacket. The present model is based on two novel concepts: 共1兲 the increase in plastic compressive strength of FRP-confined concrete is expressed in terms of the internal damage resulting from dilation of the FRP-confined concrete and the kinematic restraint provided by the FRP jacket, at both the onset of volumetric expansion and in the region of plastic behavior; and 共2兲 a variable strain ductility ratio, in which the increase in plastic compressive strain in the FRP-confined concrete is considered to be a function of the stiffness of the FRP composite jacket, the type of bond between the FRP composite and the concrete, and the extent of internal damage in the concrete core.

JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 2002 / 233

Fig. 1. Compressive stress-strain behavior of FRP-confined concrete

Plastic Behavior of FRP-Confined Concrete The stress-strain model developed herein is based on the compressive stress-strain behavior of FRP-confined concrete exhibiting a slight deviation from bilinear compressive behavior, as shown in Fig. 1. In what follows, subscript m indicates the strain component under consideration: m⫽c indicates an axial strain component, and m⫽␪ indicates a radial 共transverse兲 strain component. The FRP-confined concrete may exhibit a localized unstable compressive behavior that can occur near the peak compressive strength, f co , and strain, ⑀ co , of unconfined concrete, where typically ⑀ co ⬇2.0 mm/m. This behavior is typical of circular 共Demers et al. 1996; Wu and Xiao 2000兲 and rectangular 共Demers et al. 1996; Picher et al. 1996; Rochette and Labossie`re 2000兲 FRP-confined concrete members with low effective jacket stiffness. Consider a circular concrete column of diameter D c , unconfined compressive strength f co , and modulus of elasticity of the concrete core E c , confined by an FRP jacket of thickness t j with an average tangent hoop modulus of elasticity E j . The confining stiffness, C j , and the effective confining stiffness, K je , of the FRP composite jacket are K je ⫽K j k e ;

K j⫽

Cj ; f co

C j⫽

2t j E j Dc

(1)

where k e ⫽confining efficiency, which for continuous circular FRP jackets is equal to 1.0. The following stress-strain model describes the behavior of circular FRP-confined concrete members with an FRP jacket of effective stiffness K je . The stressstrain behavior of FRP-confined concrete is separated into two components, a strain-hardening component and a strain-softening component, designated with the subscripts ch and cs, respectively. From experimental evidence the following applies for the FRP-confined concrete core: f c ⫽ 共 f m 兲 ch ⫹ 共 f m 兲 cs

(2)

The stress components can be defined explicitly in terms of strain, such that

f c⫽

再冋 ⫹

f c ⫽ 关共 E m 兲 ch ⫹ 共 E m 兲 cs 兴 ⑀ m

1⫹

再

冏

⬘ ⫺E mp 兲 共Em

冏册

⬘ ⫺E mp 兲 ⑀ m 共Em f om

冋

nm

冉

where (E m ) ch ⫽variable strain-hardening secant modulus, and (E m ) cs ⫽variable strain-softening secant modulus, evaluated at the concrete strain ⑀ m . Fig. 2 shows the two components of the FRP-confined concrete behavior: Fig. 2共a兲 shows the strainhardening component, and Fig. 2共b兲 the strain-softening component. When combined, the two components exhibit the typical compressive behavior of FRP-confined concrete, shown in Fig. 1. The bilinear component shown in Fig. 2共a兲 is assumed to be given by the variable strain-hardening secant modulus (E m ) ch of Eq. 共3兲, governed by the Richard and Abbott 共1975兲 model: 共 E m 兲 ch ⫽

⑀m 共 ⑀ mp 兲 v

冊 册冎 ␤m

⑀m

共 E ⬘m ⫺E mp 兲

冏册

⬘ ⫺E mp 兲 ⑀ m 共Em f om

n m 1/n m ⫹E mp

E ␪p ⫽ f co ␻ je ␸ ␪ ;

E cp ⫽␮ p E ␪p ;

␮ p⫽

4.635 ; 共 K je 兲 2/3

␻ je ⫽K je 共 k 1 兲 avg ␸ ␪⫽ (3b)

(4)

where n m ⫽strain-hardening curvature parameter in either the axial or radial strain direction; E mp ⫽tangent plastic modulus in either the radial or axial strain direction, previously derived in the companion paper 共Moran and Pantelides 2002兲 and further simplified as

(3a)

冎

冋 冏 1⫹

1/n m ⫹E mp ⑀ m

共 ␦ m 兲 cs ␤ m f co

共 ⑀ mp 兲 v 共 ␤ m ⫺1 兲 ⫹

Fig. 2. Components of compressive stress-strain behavior of FRPconfined concrete: 共a兲 strain-hardening component; 共b兲 strainsoftening component

1⫺ 共 ␩ ␪ 兲 0.80 ; 1⫺␩ ␪

␩ ␪⫽

(5) 共 ⑀ ␪p 兲 i⫺1 共 ⑀ ␪p 兲 i

(6)

where ␻ je ⫽effective confinement index; ␸ ␪ ⫽radial strain coefficient; and ␮ p ⫽plastic dilation rate previously defined in the companion paper 共Moran and Pantelides 2002兲. The experimental bond-dependent average confinement coefficient, (k 1 ) avg , was

234 / JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 2002

Fig. 3. Determination of strain-hardening stress-strain parameters of FRP-confined concrete

found as (k 1 ) avg⬇4.14 for bonded and (k 1 ) avg⬇2.33 for nonbonded FRP-confined concrete. The terms with subscript i are evaluated at the limit radial strain ⑀ ␪p ⫽(⑀ ␪p ) i ⫽(⑀ ␪p ) lim , and those with subscript (i⫺1) are evaluated at ⑀ ␪p ⫽(⑀ ␪p ) i⫺1 ⫽␩ ␪ (⑀ ␪p ) lim , where 0.70⭐␩ ␪ ⬍1.0. The limit radial strain, (⑀ ␪p ) lim , is a material-dependent limiting radial strain equal to 12.5 mm/m for glass FRP 共GFRP兲-confined concrete and equal to 8.5 mm/m for carbon FRP 共CFRP兲-confined concrete. In addition, ⑀ ␪o ⫽radial strain in the FRP jacket corresponding to the peak compressive strain, ⑀ co , of the unconfined concrete core. The additional terms in the above model are derived from analysis of Figs. 1, 2共a兲, and 3 and are Em ⬘ ⫽E m 关 1⫺ 共 ␦ m 兲 cs 兴

(7) Ec E ␪⫽ ␮o

E c ⫽4,700冑 f co MPa共 57,000冑 f co psi兲 ; f o␪ ⫽ f co 关 1⫹␻ je 共 ⑀ ␪p 兲 i 共 1⫺␸ ␪ 兲兴

(8)

f oc ⫽ f co 共 1⫹␻ je 兵 共 ⑀ ␪p 兲 i ⫺⑀ co ␸ ␪ 关 ␮ p ⫹ 共 ⌬ ␪ 兲 i 兴 其 兲

(9)

共 ⑀ ␪p 兲 i ⫺k b ⑀ ␪o 共 ⌬ ␪ 兲i⫽ ⑀ co

where E m and E m⬘ are the tangent and effective tangent modulus of elasticity in either the axial or radial strain direction, respectively; ␮ o ⫽initial Poisson’s ratio of the unconfined concrete core 共typically ␮ o ⬇0.18); f om ⫽reference intercept stress in either the axial or radial strain direction; and ⌬ ␪ ⫽radial strain ratio. The experimental bond-dependent constant k b was found as k b ⬇2.36 for bonded and k b ⬇1.76 for nonbonded FRP-confined concrete. All the terms in these relationships are the average bilinear stress-strain properties of FRP-confined concrete, where for convenience only the absolute values of strain and stress are considered. In Eq. 共7兲, (␦ m ) cs ⫽strain-softening coefficient, defined as a step function, which determines if a localized strain-softening behavior occurs at the onset of volumetric expansion when the Poisson’s ratio of FRP-confined concrete ␮⬇0.50, such that 共 ␦ m 兲 cs ⫽␦ v 具 共 k cp 兲 v ⫺ 共 k ch 兲 v 典 ; 共 k cp 兲 v ⫽

共 f cp 兲 v ⫽1⫹␻ je 共 ␭ p ⑀ ␪p 兲 v ; f co

␦ v ⫽ 具 130⫺␻ je 典 0 共 ␭ p 兲v⫽

冋

共 ⑀ ␪p 兲 lim 共 ⑀ ␪p 兲 v

共 ⑀ cp 兲 v 2 共 ⑀ ␪p 兲 v ⫽ ⬇1⫹␣ b 共 K je 兲 2 共 k ⑀p 兲 v ⫽ ⑀ co ⑀ co

册

(10) 0.20

(11) (12)

共 k ch 兲 v ⫽

关共 f m 兲 ch 兴 v 关共 E c 兲 ch 共 ⑀ cp 兲兴 v 关共 E ␪ 兲 ch 共 ⑀ ␪p 兲兴 v ⫽ ⫽ (13) f co f co f co

where ␭ p ⫽variable plastic confinement coefficient; and ␣ b ⫽bond-dependent coefficient; for bonded FRP-confined concrete ␣ b ⬇3.17⫻10⫺3 , and for nonbonded FRP-confined concrete, ␣ b ⬇2.38⫻10⫺4 . The volumetric coefficient ␦ v of Eq. 共10兲 by definition has only two values: ␦ v ⫽1.0 when the confinement index (␻ je ) of Eq. 共5兲 is less than 130, which indicates that the Poisson’s ratio, ␮, can exceed 0.50, or (⑀ v ⫽2 兩 ⑀ ␪ 兩 ⫺ 兩 ⑀ c 兩 ⬎0); and ␦ v ⫽␦ h ⫽0 when ␻ je ⭓130, which indicates that the FRP jacket has adequate stiffness to inhibit unstable crack growth in the concrete core such that Poisson’s ratio ␮ is less than 0.50 (⑀ v ⫽2 兩 ⑀ ␪ 兩 ⫺ 兩 ⑀ c 兩 ⬍0). The strain-softening coefficient (␦ m ) cs in Eq. 共10兲 by definition can have only two values: (␦ m ) cs ⫽(k cp ) v ⫺(k ch ) v when (k cp ) v ⬎(k ch ) v and ␦ v ⫽1.0, and (␦ m ) cs ⫽0 otherwise. Also, the axial compressive stress ( f cp ) v of Eq. 共11兲 and the axial compressive strain (⑀ cp ) v and radial strain (⑀ ␪p ) v of Eq. 共12兲 are those stresses and strains in the FRP-confined concrete beyond which Poisson’s ratio, ␮, can exceed 0.50. The terms (k ch ) v and 关 ( f m ) ch 兴 v of Eq. 共13兲 are found by evaluating the strain-hardening secant modulus (E m ) ch of Eq. 共4兲 at the axial compressive strain (⑀ cp ) v and radial strain (⑀ ␪p ) v , as shown in Fig. 2共a兲. This behavior of FRP-confined concrete, in which a localized strain-softening behavior occurs near the peak compressive strength and strain of the unconfined concrete, is typical of concrete members confined by FRP jackets having a low effective jacket stiffness, K je . This unstable compressive behavior occurs due to the ineffectiveness of the FRP composite jacket in curtailing the dilation of the concrete core at very low transverse strains and resultant low confinement stresses when the volumetric strain, ⑀ v ⫽2 兩 ⑀ ␪ 兩 ⫺ 兩 ⑀ c 兩 , goes to zero. The FRP jacket does not become effective until further dilation of the concrete core induces an increase in the hoop stresses in the FRP jacket. When the compressive behavior of the FRP-confined concrete deviates from a bilinear compressive behavior, as shown in Fig. 1, a localized strain-softening compressive component can occur at the onset of volumetric expansion, as shown in Figs. 2共b兲 and 4共a兲. This strain-softening behavior can be represented by the variable secant strain-softening modulus (E m ) cs of Eq. 共3兲, defined as 共 E m 兲 cs ⫽

再

共 ␦ m 兲 cs f co ␤ m

共 ⑀ mp 兲 v 共 ␤ m ⫺1 兲 ⫹

冋

⑀m 共 ⑀ mp 兲 v

册冎 ␤m

(14)

JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 2002 / 235

confined concrete, which exhibits a localized strain-softening behavior at the onset of volumetric expansion, is also shown. In Fig. 4共a兲, it can be observed that when the FRP-confined concrete approaches the axial dilation stress ( f cd ) and strain (⑀ cd ) and the corresponding radial dilation strain, ⑀ ␪d ⬇⑀ cd /(1.10␮ o ), the behavior of the FRP-confined concrete and that of the unconfined concrete begin to deviate. This axial dilation stress ( f cd ) is defined herein as the axial stress at which the rate of volume dilation of the concrete core increases, as shown in Fig. 4共b兲; this occurs when the volumetric strain versus axial strain curve deviates from the linear elastic slope (1⫺2␮ o ) due to the unrestrained crack propagation in the concrete core. It is assumed that the FRP-confined concrete and the unconfined concrete core behave identically up to the dilation stress ( f cd ), as shown in Fig. 4共b兲, which occurs at approximately f cd ⫽0.70f co . By considering that the stress-strain curves of both the unconfined and FRP-confined concrete pass through the dilation stress-strain coordinates ( f cd ,⑀ md ) in both the axial and radial strain direction, the strain-hardening curvature parameter n m of Eq. 共4兲 can be determined from iterative solution of

冉再

Em ⬘ ⫺E mp

Fig. 4. Stress-strain relationships for FRP-confined concrete: 共a兲 typical stress-strain curve near the unconfined concrete strength; 共b兲 volumetric and radial strain versus axial strain

␤ m⫽

冋 册 再

k ␤E m ; f co E m⫺ 共 ⑀ mp 兲 v

k ␤ ⫽1.0

for ⑀ m ⭐ 共 ⑀ mp 兲 v

k ␤ ⫽ 共 k cp 兲 v

for ⑀ m ⬎ 共 ⑀ mp 兲 v

冎 (15)

where ␤ m ⫽strain-softening curvature parameter in either the axial or radial strain direction, and k ␤ ⫽post-peak curvature modification factor, as defined in Eq. 共15兲. The terms in Eq. 共14兲 consider that the strain-softening behavior is governed by a modified Popovics 共1973兲 fractional model that has a peak compressive strength, f co (␦ m ) cs , at the peak strain ⑀ m ⫽(⑀ cp ) v ⫽(⑀ ␪p ) v , provided that the strain-softening coefficient, (␦ m ) cs , of Eq. 共10兲 is a positive number.

Stress-Strain Model Implementation In the case of circular concrete members confined by either bonded or nonbonded FRP composite jackets, the proposed model given in Eqs. 共1兲 through 共15兲 can be used as follows. By considering the definition of intercept stress f om 共Richard and Abbott 1975兲 in terms of a known stress-strain coordinate in the concrete core ( f c ,⑀ m ), the strain-hardening curvature parameter, n m , of Eq. 共4兲 is found as follows. Assume that both the FRP-confined and unconfined concrete behave identically up to the dilation stress, f cd , as shown in Fig. 4. Fig. 4共a兲 shows a typical axial stress-axial strain curve of an unconfined normal strength concrete cylinder near the peak compressive strength and strain of the unconfined concrete. The axial stress–axial strain curve of an FRP-confined concrete cylinder near the axial stress ( f cp ) v and axial strain (⑀ cp ) v of the FRP-

nm

E md ⫺E mp ⫺ 关共 E m 兲 cs 兴 d ⫺

E md ⫽

再

冎 冊 冎

1/n m

⫺1

Em ⬘ ⫺E mp 0.70f co ⫽0 f om E md ⫺ 关共 E m 兲 cs 兴 d

0.70f co ; ⑀ md

E cd ⫽

0.70f co Ec ⬇ ⑀ cd 1.10

(16)

(17)

0.70f co E cd ⬇ E ␪d ⫽ ⑀ ␪d 1.10␮ o 关共 E m 兲 cs 兴 d ⫽ 共 E m 兲 cs ⑀ md

(18)

where E m⬘ ⫽effective tangent modulus of elasticity defined in Eq. 共9兲; E md ⫽dilation secant modulus in either the axial or radial strain direction evaluated at the axial dilation stress, f cd ⫽0.70f co , and corresponding axial (⑀ cd ) and radial (⑀ ␪d ) dilation strains, as shown in Fig. 4. In both Eqs. 共16兲 and 共18兲 the strainsoftening secant modulus 关 (E m ) cs 兴 d is found by evaluating the strain-softening modulus (E m ) cs of Eq. 共14兲 at the dilation strain, ⑀ m ⫽⑀ md . The effective tangent modulus of elasticity E m⬘ of Eq. 共7兲 and the strain-softening modulus 关 (E m ) cs 兴 d are both a function of the strain-softening coefficient (␦ m ) cs of Eq. 共10兲. The strainsoftening coefficient (␦ m ) cs , by reference to Eqs. 共13兲 and 共4兲, is also a function of the effective tangent modulus of elasticity, E m⬘ . This indicates that the strain-softening coefficient, (␦ m ) cs , and the solution of the strain-hardening curvature parameter, n m , are interdependent and require an iterative solution of both (␦ m ) cs and n m of Eqs. 共10兲 and 共16兲, respectively. The iterative solution of the strain-softening coefficient (␦ m ) cs and the strain-hardening curvature parameter n m can be found by first evaluating the stress-strain properties of the FRP-confined concrete as follows: 共1兲 determine the unconfined properties of the concrete core, f co and ⑀ co ⬇2.0 mm/m; 共2兲 calculate (E c ) and (E ␪ ) of Eq. 共7兲; 共3兲 calculate the dilation stress-strain properties f cd ⫽0.70f co , (E cd ), and radial (E ␪d ) of Eq. 共17兲 and corresponding strains (⑀ cd ) and (⑀ ␪d ); 共4兲 select the bond-dependent constants (k 1 ) avg of Eq. 共5兲, k b of Eq. 共9兲, and ␣ b of Eq. 共12兲; 共5兲 calculate the bond-dependent properties (␻ je ) and (␮ p ) of Eq. 共5兲; 共6兲 based on the FRP jacket material, select the limiting strain (⑀ ␪p ) lim and select ␩ ␪ , where 0.70⭐␩ ␪ ⬍1.0 of Eq. 共6兲; 共7兲 set

236 / JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 2002

Ultimate Axial Strain In an FRP-confined concrete member, compressive failure will occur simultaneously with failure of the FRP jacket, be it due to rupture, delamination, lap failure, or shear failure. This failure occurs at an ultimate radial FRP jacket strain, ⑀ ␪u , that may be below the rupture strain of FRP composite tensile coupon tests. Premature failure of the FRP jacket can occur as a result of interaction between the axial shortening and radial dilation, which induces a triaxial 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 FRP-confined concrete core progresses. By considering both equilibrium and strain compatibility, the stress-strain model of Eq. 共3兲 can be modified and adopted, such that the following equilibrium relationship at the ultimate stressstrain condition ( f cu ,⑀ cu ) and ( f cu ,⑀ ␪u ) applies: 关共 E c 兲 ch ⫹ 共 E c 兲 cs 兴 ⑀ cu ⫽ 关共 E ␪ 兲 ch ⫹ 共 E ␪ 兲 cs 兴 ⑀ ␪u

(19)

The plastic region of the modified Richard and Abbott 共1975兲 model, given in Eq. 共4兲, can be approximated by the equation of a linear function with an intercept stress f om and a slope E mp ; the above relationship can be approximated as Fig. 5. Solution algorithm for the strain-hardening curvature parameter and the strain-softening coefficient

⑀ ␪p ⫽(⑀ ␪p ) i ⫽(⑀ ␪p ) lim and ⑀ ␪o ⫽0.50 mm/m and calculate (⌬ ␪ ) i of Eq. 共9兲; 共8兲 set ⑀ ␪p ⫽(⑀ ␪p ) i⫺1 ⫽␩ ␪ (⑀ ␪p ) lim and calculate (␸ ␪ ) of Eq. 共6兲; 共9兲 calculate the plastic modulus E ␪p and E cp of Eq. 共5兲; 共10兲 calculate the intercept stresses f o␪ of Eq. 共8兲 and f oc of Eq. 共9兲; 共11兲 calculate the stress-strain properties at the onset of volumetric expansion, which are (k ⑀p ) v of Eq. 共12兲, the corresponding strains (⑀ cp ) v and (⑀ ␪p ) v of Eq. 共12兲, and (k cp ) v and ( f cp ) v of Eq. 共11兲; and 共12兲 calculate ␤ m of Eq. 共15兲 based on (⑀ cp ) v and (⑀ ␪p ) v . Once the above stress-strain properties of the FRP-confined concrete are evaluated, the iterative solution of the strainsoftening coefficient, (␦ m ) cs , and the strain-hardening curvature parameter, n m , can be found using the flow chart provided in Fig. 5: Convergence of the solution for both the strain-softening coefficient, (␦ m ) cs , and the strain-hardening curvature parameter, n m , as described in Fig. 5, can be obtained within a few iterations. An iterative solution of both the strain-softening coefficient, (␦ m ) cs , and the strain-hardening curvature parameter, n m , is required to ensure that the analytical stress-strain model of Eqs. 共2兲 and 共3兲 passes through two sets of stress-strain coordinates: the dilation stress-strain coordinate, ( f cd ,⑀ md ), at initiation of cracking in the concrete core, and the stress-strain coordinate 关 ( f cp ) v ,(⑀ mp ) v 兴 , at onset of volumetric dilation, as shown in Fig. 4, when ␻ je ⬍130. This approach is in contrast to the models introduced by Spoelstra and Monti 共1999兲 and Harmon et al. 共1998a, b兲, in which the stress-strain curve of the FRP-confined concrete is obtained in an incremental, iterative manner throughout its loading history. In the present approach, the iterative solution is undertaken at both the initiation of microcracking and at the onset of volumetric expansion of the FRP-confined concrete core, which are assumed to occur at the dilation stress-strain coordinate ( f cd ,⑀ md ) and at the stress-strain coordinate 关 ( f cp ) v ,(⑀ mp ) v 兴 , respectively, as shown in Fig. 4. Once the terms in the stress-strain model of Eqs. 共2兲 through 共18兲 have been determined, the stressstrain behavior of FRP-confined concrete can be predicted.

f oc ⫹ 关 E cp ⫹ 共 E c 兲 cs 兴 ⑀ cu ⫽ f o␪ ⫹ 关 E ␪p ⫹ 共 E ␪ 兲 cs 兴 ⑀ ␪u

(20)

Substituting the reference intercept stress in the axial, f oc , or radial, f o␪ , strain direction of Eqs. 共8兲 and 共9兲 into Eq. 共20兲, the ultimate axial compressive strain, ⑀ cu , can be obtained in terms of the ultimate jacket strain, ⑀ ␪u , from the iterative solution of ⑀ cu ⬇

E ␪p 兵 关 ⑀ ␪u ⫺ 共 ⑀ ␪p 兲 i 兴 ⫹⑀ co 关 ␮ p ⫹ 共 ⌬ ␪ 兲 i 兴 其 ⫹ 共 E ␪ 兲 cs ⑀ ␪u E cp ⫹ 共 E c 兲 cs (21)

where (⑀ ␪p ) i ⫽(⑀ ␪p ) lim , and (⌬ ␪ ) i is evaluated using Eq. 共9兲 with (⑀ ␪p ) i ⫽(⑀ ␪p ) lim . Convergence of the above expression is fast and within a few iterations. For FRP-confined concrete members with an FRP jacket having a high effective jacket stiffness K je , where ␻ je ⬎130, a noniterative solution for the ultimate compressive strain, ⑀ cu , can be obtained by assuming that in Eq. 共21兲, (E c ) cs ⫽(E ␪ ) cs ⫽0 at ultimate strains ⑀ cu and ⑀ ␪u , respectively. Using Eqs. 共9兲 and 共21兲 with (E c ) cs ⫽(E ␪ ) cs ⫽0 yields the following expression for the ultimate axial compressive strain ⑀ cu at the ultimate radial strain ⑀ ␪u in the FRP jacket:

冋

⑀ cu ⬇⑀ co 1⫹

册

共 ⌬␪兲u ; ␮p

共 ⌬ ␪ 兲u⫽

共 ⑀ ␪u ⫺k b ⑀ ␪o 兲 ; ⑀ co

␮ p⬇

4.635 共 K je 兲 2/3 (22)

where (⌬ ␪ ) u ⫽ultimate radial strain ratio, and ⑀ ␪o ⬇0.50 mm/m. The above relationship indicates that for concrete members confined by an FRP jacket having a high effective stiffness, K je , the ultimate compressive strain, ⑀ cu , is directly proportional to the extent of internal damage as measured by the ultimate radial strain ratio, (⌬ ␪ ) u , and inversely proportional to the member’s plastic dilation rate, ␮ p . The latter was shown to be a function of both the effective stiffness of the FRP jacket, K je , and the type of construction 共that is, bonded or nonbonded兲. In addition, Eq. 共22兲 is identical to Eq. 共23兲 of the companion paper 共Moran and Pantelides 2002兲; except that it is being evaluated at the ultimate jacket radial strain, where (⑀ ␪p )⫽⑀ ␪u .

Comparison of Model with Experimental Results The proposed stress-strain model is compared with experimental results of concrete cylinder tests confined by nonbonded GFRP-

JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 2002 / 237

Fig. 6. Comparison of FRP-confined concrete model with experiments by Mirmiran 共1997兲

confined concrete cylinders performed by Mirmiran 共1997兲 and bonded CFRP-confined concrete cylinder tests performed by Wu and Xiao 共2000兲, Picher et al. 共1996兲, and Rochette and Labossie`re 共2000兲. The CFRP-confined concrete cylinder tests performed by Picher et al. 共1996兲 and Rochette and Labossie`re 共2000兲 are included in order to verify the effectiveness of the proposed constitutive model of Eq. 共3兲 using Eqs. 共4兲 to 共18兲 by using a strain ratio, ␩ ␪ , equal to 0.70; in addition, the equilibrium equation at failure of the FRP jacket given in Eq. 共19兲 is used. The experimental data from Picher et al. 共1996兲, and Rochette and Labossie`re 共2000兲 were excluded in the analysis of FRP-confined concrete for model calibration performed herein.

Tests by Mirmiran (1997) A total of 22 concrete-filled GFRP composite tubes and 6 plain concrete specimens were tested by Mirmiran 共1997兲. The specimens were cylinders with a 152.5 mm diameter and a 305 mm height. Three batches of concrete with varying water-cement ratios were used in the study. For each batch, three jacket thicknesses of 1.44, 2.20, and 2.97 mm were tested. In Fig. 6, the results of the proposed analytical stress-strain model are compared to the experimental stress-strain curves of the third batch, members DC-11, DC-21, and DC-31 corresponding to 6, 10, and 14 layers of GFRP composite, respectively. From Fig. 6 it can be observed that the proposed model can accurately capture the bilinear compressive behavior of nonbonded GFRPjacketed concrete members.

Fig. 7. Comparison of FRP-confined concrete model with experiments by Wu and Xiao 共2000兲

different configurations of bonded CFRP jackets. The specimens were FRP-confined concrete cylinders with a 152 mm diameter and a 305 mm height. The jackets consisted of three layers of CFRP wrapped around the concrete specimens. Figs. 8共a and b兲 compared the results of the proposed analytical stress-strain model to the experimental stress-strain curves of specimens C0 and C12, respectively. From these figures it can be observed that the proposed model can accurately capture the compressive behavior of these bonded CFRP-jacketed concrete members.

Tests by Rochette and Labossie`re (2000) Three concrete cylinders confined with bonded CFRP jackets were tested by Rochette and Labossie`re 共2000兲. The specimens were FRP-confined concrete cylinders with a 100 mm diameter

Tests by Wu and Xiao (2000) A total of 18 concrete cylinders confined by bonded CFRP 共type 1兲 jackets were tested by Wu and Xiao 共2000兲 and are considered in this analysis. The specimens were cylinders with a 152 mm diameter and a 304 mm height. Three batches of concrete with varying water-cement ratios were used in the study. For each concrete batch, three jacket thicknesses of one, two, and three layers of CFRP composite were tested. Fig. 7 compares the results of the proposed analytical stressstrain model to the experimental stress-strain curves of the low 共L兲 and medium 共M兲 concrete strength, corresponding to specimens L1-3P-3 and M1-3P-3. From Fig. 7 it can be observed that the proposed model can accurately capture the bilinear compressive behavior of bonded CFRP-jacketed concrete members.

Tests by Picher et al. (1996) Fifteen concrete cylinders were tested by Picher et al. 共1996兲: three unconfined and three sets of four cylinders confined with

Fig. 8. Comparison of FRP-confined concrete model with experiments by Picher et al. 共1996兲: 共a兲 specimen C0; 共b兲 specimen C12

238 / JOURNAL OF COMPOSITES FOR CONSTRUCTION / NOVEMBER 2002

equilibrium; however, the same analytical expression was derived based on a plasticity analysis presented in the companion paper 共Moran and Pantelides 2002兲. Comparisons with experimental results indicate good agreement. The stress-strain model proposed herein can be easily implemented in a spreadsheet or other computer language program for the moment-curvature or finiteelement analysis of FRP-confined concrete members.

Acknowledgments

Fig. 9. Comparison of FRP-confined concrete model with experiments by Rochette and Labossie`re 共2000兲

and a 200 mm height. The FRP jackets consisted of two layers of CFRP composite wrapped around the concrete cylinder specimens. Fig. 9 compares the results of the analytical stress-strain model proposed herein to the experimental stress-strain curves of specimen C100-C2. From Fig. 9 it can be observed that the proposed model can accurately capture the compressive behavior of bonded CFRP-jacketed concrete specimens. In Figs. 6 through 9, the proposed constitutive stress-strain model of Eq. 共3兲 using Eqs. 共4兲 to 共18兲, and a strain ratio, ␩ ␪ , equal to 0.70 was implemented; the equilibrium equation at failure of the FRP jacket given in Eq. 共19兲 was used. The proposed constitutive stress-strain model captures the bilinear compressive behavior of FRP-confined concrete and most of the experimental results, with some deviation at the onset of plastic behavior and at strains near failure.

Conclusions A model for describing the compressive behavior of concrete members confined by FRP composite jackets is presented. The proposed model is based on accepted concrete and FRP composite behavior and fundamental principles of mechanics and is applicable to both bonded and nonbonded FRP-confined concrete. The distinguishing feature of the proposed model is a variable strain ductility ratio, which was demonstrated to be a function of the stiffness of the confining FRP composite jacket and the extent of internal damage, rather than a constant, as is typically assumed for steel confined concrete. It is shown that the compressive behavior of FRP-confined concrete can be separated into two components: 共1兲 a strainsoftening component, which accounts for the nonlinear stressstrain behavior that results from unrestrained crack propagation near the peak compressive strength and strain of unconfined concrete; and 共2兲 a bilinear strain-hardening component, which accounts for the increase in strength due to confinement provided by the elastic FRP jacket after the jacket becomes effective in curtailing the dilation of the concrete core. An expression was obtained for predicting the ultimate compressive strength and strain of the FRP-confined concrete based on equilibrium. The ultimate compressive strength and strain of the FRP-confined concrete were found to be a function of the effective jacket stiffness, type of jacket construction 共bonded or nonbonded兲, and ultimate strain in the FRP jacket. The expression for the ultimate axial compressive strain was derived based on

The writers would like to acknowledge the financial support of the Utah Department of Transportation, the Federal Highway Administration, the Idaho National Engineering and Environmental Laboratories, and the National Science Foundation under contract No. CMS 9712761. The writers would also like to thank the reviewers for their comments. The opinions expressed in this article are those of the writers and do not necessarily reflect the opinions of the sponsoring organizations.

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