Elastoplastic Confinement Model for Circular Concrete Columns R. Eid1; A. N. Dancygier, M.ASCE2; and P. Paultre, M.ASCE3 Abstract: This paper presents relatively simple, analytically derived curves to model the axial and the lateral stress-strain relations of circular concrete columns. The analytical curves describe the full elastoplastic behavior of the confined concrete column. The solution to the partially confined reinforced concrete column in the elastoplastic range is derived by replacing the discrete lateral reinforcement with an equivalent tube and the Drucker–Prager 共DP兲 yield criterion is applied to represent the concrete behavior in the plastic range. Application of the DP model does not require an iterative procedure in order to solve the problem and thus an explicit solution is obtained. It is shown that the proposed model properly simulates the behavior of reinforced concrete columns partially confined by steel ties and of columns that are fully confined by fiber-reinforced polymer sheets. DOI: 10.1061/共ASCE兲0733-9445共2007兲133:12共1821兲 CE Database subject headings: Concrete columns; Confinement; Elasticity; Plasticity; Stress strain relations; Fiber reinforced polymers.
Introduction It is well-known that confining concrete with lateral reinforcement increases its strength and ductility in axial compression. Many researchers have put a relatively large amount of effort into understanding the confinement mechanism and introducing a suitable model for the behavior of confined concrete in reinforced concrete 共RC兲 elements. Most such models are empirical or semiempirical and were derived by introducing a uniaxial stress-strain relationship of the confined concrete that fit its strength and strain. In the semiempirical steel-confined concrete models, the concrete strength and strains are a function of the effective lateral confining uniform pressure, whose derivation is based on an arch action assumption between the steel ties and on empirical data 共Sheikh and Uzumeri 1982; Mander et al. 1988b兲. While few researchers include computation of the steel stress at the concrete peak stress 共Cusson and Paultre 1994, 1995; Razvi and Saatcioglu 1999b; Légeron and Paultre 2003兲, most of the models assume that the lateral steel yields before the concrete reaches its confined strength. The apparent advantages of confining concrete have led to relatively new techniques of column confinement, in addition to 1 Research Associate, Dept. of Civil Engineering, Univ. of Sherbrooke, Sherbrooke, QC, Canada J1K 2R1 共corresponding author兲. E-mail:
[email protected] 2 Senior Lecturer, Faculty of Civil and Environmental Engineering, National Building Research Institute, Technion-Israel Institute of Technology, Haifa 32000, Israel 3 Professor, Dept. of Civil Engineering, Univ. of Sherbrooke, Sherbrooke, QC, Canada J1K 2R1 Note. Associate Editor: Jin-Guang Teng. Discussion open until May 1, 2008. 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 April 13, 2006; approved on March 9, 2007. This paper is part of the Journal of Structural Engineering, Vol. 133, No. 12, December 1, 2007. ©ASCE, ISSN 0733-9445/ 2007/12-1821–1831/$25.00.
that commonly used, namely properly spaced reinforcing ties 共or spirals兲. One such technique is to wrap the column with a fiberreinforced polymer 共FRP兲 composite jacket. The models of concrete confined by transverse reinforcement cannot represent the stress-strain curves of FRP-confined concrete. Therefore, several researchers proposed stress-strain models especially for FRPconfined concrete where most of these models are empirically based as well. Teng and Lam 共2004兲 reviewed existing FRPconfined concrete models and classified them into two categories: design-oriented models 共e.g., Karbhari and Gao 1997; Samaan et al. 1998; Toutanji 1999; Saafi et al. 1999; Xiao and Wu 2000; Lam and Teng 2003兲 and analysis-oriented models 共e.g., Mirmiran and Shahawy 1996; Spoelstra and Monti 1999; Harries and Kharel 2002兲. While the design-oriented models are presented in closed form expressions, the analysis-oriented models use numerical procedures, which are incremental and iterative, to predict the stress-strain curve of FRP-confined concrete. In the latter case, most of the models are based on the active confinement model proposed by Mander et al. 共1988b兲 to evaluate the axial stress and strain of FRP-confined concrete at every increment of the confining pressure. Other analysis-oriented approaches have been used for modeling FRP-confined concrete, such as the concept of crack slip and separation in the concrete 共Harmon et al. 1998兲, octahedral stress-strain models 共Becque et al. 2003兲, and the plasticity approach 共Karabinis and Rousakis 2002兲. Except for the active confinement-based analysis-oriented models, most of the available FRP-confined concrete models cannot represent the behavior of steel-confined concrete columns. Thus, a notable disadvantage of most available models is their limitation to one specific confining material. A theoretical model of concrete column confined by steel ties was presented by Eid and Dancygier 共2005兲 and was examined against available test results of steel-confined concrete columns. For the case of reinforced concrete, the model is based on a substitution of the discrete tie confinement by an equivalent uniform confinement. The uniformly confined concrete core is then analyzed with the Imran and Pantazopoulou 共2001兲 plasticity concrete model, which requires an iterative solution and therefore
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b⫽band’s spacing and width, respectively 共Fig. 1兲. The derived expressions of the stress components for a long cylinder acted on by n pairs of pressure bands p, which include the action of an axial pressure q 共see Fig. 1兲, are as follows r =− p
冉 冕冋 冉 冊 冋兺 冉 ⬁
共1 − 2 − 兲I0共k1r1兲 + k1r1 +
0
n
k 1b 1 · ⫻sin 2
cos
i=1
冊
册
 I1共k1r1兲 k31 f 1共k1兲 k 1r 1
k1 共2i − 1兲共s1 + b1兲 2
冊册
cos共k1z1兲dk1 共1兲
Fig. 1. Linear elastic cylinder loaded by axial pressure and confined by: 共a兲 lateral pressure bands 共active confinement兲; 共b兲 elastoplastic rings; and 共c兲 equivalent tube.
cannot produce an explicit stress-strain curve. Alternatively, a simpler material model, enabling explicit analysis, can be used to analytically derive the characteristic behavior of confined concrete. This paper presents a simplified analytically derived stressstrain curve that describes the axial and lateral behavior of the concrete in circular columns with transverse reinforcement. The analytical curve is derived based on the full elastoplastic behavior of the confined concrete column and by using the well known Drucker–Prager yield criterion. Moreover, the current model can be applied to describe the behavior of concrete confined by reinforcing-steel ties or by FRP.
= p
First Stage: Elastic Analysis Active Confinement The first stage in developing the analytical confinement model is the analysis of a case of active confinement, where a linear elastic cylinder is confined by a sequence of equally spaced lateral pressure bands 关active confinement, Fig. 1共a兲兴, followed by the analysis of a passive confinement case, where confinement is produced by the passive reaction of lateral elastic-perfectly plastic stiffening rings 关Fig. 1共b兲兴. The analysis of a single active pressure band of magnitude p is carried out by superposition of two uniform radial pressures, acting on the surface of a long cylinder and shifted a distance b from each other along the longitudinal axis. Each of these pressures applies compression to one half of the cylinder 共along z + s / 2 ⬎ 0兲 and tension to its other half of equal magnitudes ±p / 2; this part of the solution is based on the solution to one such radial pressure case, as given by Timoshenko and Goodier 共1970兲. The solution to the case of n pairs of pressure bands applied symmetrically with respect to the origin 关Fig. 1共a兲兴 is obtained by superposition of the solution to the problem of a single pair of pressure bands that act at a distance of 共2i − 1兲共s + b兲 − b from each other, where i⫽number of pairs, and s and
共2 − 1兲I0共k1r1兲 +
0
册
 I1共k1r1兲 k31 f 1共k1兲 k 1r 1
n
k 1b 1 · 2
⫻sin
k1 共2i − 1兲共s1 + b1兲 2
cos
i=1
冊册
cos共k1z1兲dk1 共2兲
z q = + p p
冕 冉 冊 冋兺 冉 ⬁
关共4 − 2 − 兲I0共k1r1兲 + 共k1r1兲I1共k1r1兲兴k31 f 1共k1兲
0
n
k 1b 1 · 2
⫻sin
cos
i=1
k1 共2i − 1兲共s1 + b1兲 2
冊册
cos共k1z1兲dk1 共3兲
rz =− p
Analysis of Confined Concrete Column The full derivation of the equivalent uniform confinement is given by Eid 共2004兲 and by Eid and Dancygier 共2005, 2006兲. For the completeness of this paper, the model is briefly described below.
冕冋 冉 冊 冋兺 冉 ⬁
冕
⬁
0
关共− k1r1兲I0共k1r1兲 + 共2 − 2 + 兲I1共k1r1兲兴k31 f 1共k1兲
冉 冊 冋兺 冉 n
k 1b 1 · ⫻sin 2
cos
i=1
k1 共2i − 1兲共s1 + b1兲 2
冊册
sin共k1z1兲dk1 共4兲
where = cylinder’s Poisson’s ratio; k = constant with a dimension of length−1 共wave number兲; I0共k1r1兲 and I1共k1r1兲 = order zero and one modified Bessel functions of the first kind of the argument k1r1, and f 1共k1兲 and  = constants that depend on the boundary conditions of the problem and are given by  = 2共1 − 兲 + k1
f 1共k1兲 = −
4 k41
冋
I0共k1兲 I1共k1兲
1 共1 − 2 − 兲I0共k1兲 + 共k1 + /k1兲I1共k1兲
共5兲
册
共6兲
The subscript “1” denotes normalized variables with respect to the cylinder’s radius, a 共i.e., r1 = r / a, b1 = b / a, s1 = s / a, z1 = z / a, and k1 = ka兲. Examination of the solution’s convergence with respect to the number of pressure bands showed that the solution of the stresses in a typical confined zone 关i.e., within −共s + b兲 / 2 ⬍ z ⬍ 共s + b兲 / 2, Fig. 1兴 converges for five pairs of lateral bands 关2n = 10 in Eqs. 共1兲–共4兲 共Eid 2004兲兴. Therefore, the case of ten bands 共or five pairs兲 was chosen to appropriately represent an infinite number of bands and the examples in the following sections were calculated for this case.
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condition, the tangential strains of the cylinder and of the confining rings are equal at the contact points between them, that is, 共r=a兲 = s, where 共r=a兲 = strain at the cylinder surface 共r = a兲 at z equal to the ring’s centerline and s = tension strain of the ring. The passively developed lateral-to-axially applied pressure ratio p / q has thus been derived and is given by
Passive Confinement The solution to the passive confinement problem of a linearelastic cylinder subjected to axial uniform compressive stress 共q兲 and confined by lateral elastic-perfectly plastic rings 关Fig. 1共b兲兴 is obtained by applying a compatibility condition at the cylinder perimeter to the solution of the active confinement problem 共shown in the previous section兲. According to this compatibility
f3 =
p = q 8共1 − 2兲
冕
⬁
冉 冊冋 兺 冉
k 1b 1 sin 2
0
n
i=1
册 冋
k1 k1 cos 共2i − 1兲共s1 + b1兲 cos 共s1 + b1兲 2 2
I0共k1兲2 k31 + 共2 − k21 − 2兲k1 I1共k1兲2
where t1 = normalized ring’s thickness 共t1 = t / a兲; and m = moduli ratio 共m = Elat / Ec, where Elat and Ec = elasticity moduli of the rings’ and of the cylinder’s materials, respectively兲. Eq. 共7兲 implies that, when the tension strain in the ring s exceeds its rupture strain or its yield strain, the confining lateral pressure p drops to zero or remains at a constant level of f yt1, respectively, where f y = ring yield or rupture stress. Stresses and strains were calculated according to the developed equations. An interesting finding from the results of worked examples was that, within a reduced cylinder radius 共RCR兲, there is a zone of uniformly distributed stresses in which the tangential stress is equal to the radial stress and the axial and shear stresses are equal to zero. This is demonstrated in Fig. 2, which shows contour lines of the normalized radial stress r for test cases of s1 = 0.3, s1 = 0.5, and b1 = 0.1. This behavior within the RCR is typical of a fully confined cylinder, which exhibits uniform lateral pressure along its surface.
册
艋 dk1 +
fy t1 q
共7兲
1 t 1m
共8兲 where z,avg, is derived from a superposition of the strain due to the action of the axial pressure q, and of the average axial stain c due to the action of the lateral confining rings, z,avg , as follows z,avg =
q c + z,avg Ec
共9兲
c is obtained by averaging the axial strain The average strain z,avg z共r , z兲 关obtained from the solution given in Eqs. 共1兲–共3兲兴 over the
Second Stage: Equivalent Confinement Model When the applied axial uniform stress increases, certain parts of the cylinder are no longer elastic. The irregular geometry of the limit surface shape between the elastic and the plastic zones, which also changes as the axial load increases, is one of the main difficulties in obtaining an analytical solution for this problem, within its plastic range 共the irregular boundary is indicated by the shape of the contour lines in Fig. 2兲. This mathematical difficulty leads to the need for a different solution strategy for this problem, which is obtained by applying the concept of the RCR that was observed in the elastic solutions and accounting for the confinement influence near the cylinder surface 共Eid 2004兲. This is done by replacing the discrete steel ties that are used in reinforced concrete columns with an equivalent tube of thickness teq, which confines a cylinder with a radius equal to that of the original problem, as shown in Fig. 1共c兲. Furthermore, since the problem of a concrete column confined by steel ties is usually characterized by the axial stress-strain relationship, the axial strain z was chosen here to set the equivalency criterion. Thus, the equivalent thickness teq, which represents the original steel ties, is computed by equating the average axial strain of the partially confined cylinder, z,avg, to the axial strain of a fully confined equivalent cylinder, as follows
Fig. 2. Contour maps of radial stresses normalized with respect to passive lateral pressure p due to action of rings, for two cases of spacing: 共a兲 s1 = 0.3; 共b兲 s1 = 0.5, in cylinder with = 0.2 and b1 = 0.1
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radius and along the longitudinal axis within a typical confined zone between two pressure bands 关i.e., within the range −共s + b兲 / 2 艋 z 艋 共s + b兲 / 2 and 0 艋 r 艋 a, Fig. 1共b兲兴 共see Eid and Dancygier 2005 for more details兲. Substituting Eq. 共9兲 into Eq. 共8兲 and rearranging terms yields the following expression for the normalized equivalent thickness, teq teq1 =
teq = a
冋
f3
m 共 − 1兲f 3 +
冉 冊册 s1 +1 b1
共10兲
where f 3 = p / q ratio given in Eq. 共7兲. Thus, the problem of concrete confined by lateral ties can be analyzed in its plastic range 共when the confining action is pronounced兲 by solving the equivalent problem of a uniformly confined cylinder, where the elastic solution is used to derive the equivalent parameters 共teq兲.
␣=
2 sin ⌽
冑3共3 − sin ⌽兲 ;
Third Stage: Elastoplastic Analysis
Stress-Strain Curve of Confined Concrete
An explicit solution to the equivalent problem is obtained by using the Drucker–Prager 共DP兲 yield criterion given by f = 冑J2 − ␣I1 − kD
f c0共1 − sin ⌽兲 2 cos ⌽
共13兲
g = 冑J2 − ␣1I1 − k1D
共14兲
where ␣1 and k1D = material constants 共note that the case of ␣1 = ␣ and k1D = kD is that of an associated material model兲. The simple boundary condition of the problem with r = = p = lat reduces the problem to three equations with the axial and lateral stresses f c and lat and the lateral strain lat as the unknown variables. The following two Eqs. 共15兲 and 共16兲 are derived from the incremental elastoplastic constitutive relations and one Eq. 共17兲 is derived from equilibrium at the concrete–steel boundary df c =
dlat =
冋
册
E c Ec dz + dI1 共1 + 兲 共1 − 2兲共1 + 兲 −
冋
册
Ec Sz E c␣ 1 − d 共1 + 兲 2冑J2 共1 − 2兲
冋
共15兲
册
E c Ec dlat + dI1 共1 + 兲 共1 − 2兲共1 + 兲 −
冋
册
Ec Slat E c␣ 1 d − 冑 共1 + 兲 2 J2 共1 − 2兲
dlat = −
dlat Elatteq1
共16兲
共17兲
where I1 = first strain invariant and Sz = f c − 共I1 / 3兲 and Slat = lat − 共I1 / 3兲 = deviatoric stresses in the axial and lateral directions, respectively. The scalar d, which defines the plastic strain magnitude, is derived using the consistency condition 共df = 0兲 and is given by
共11兲
where J2 = second deviatoric stress invariant; I1 = first stress invariant; and ␣ and kD = positive material constants that can be related to the Mohr–Coulomb constants cm and ⌽ 共the cohesion and the internal friction angle of the material, respectively兲. Note that, in the current derivations, a positive sign denotes compressive stresses and strains 共and vice versa兲. In the compressive meridian, ␣ and kD are expressed in the following form 共Chen 1982; Durban and Papanastasiou 1997兲
共12兲
For a nonassociated concrete model, the potential function can be given as
Analyses of various RC columns with the model described above yielded results that were in good agreement with available experimental results 共Eid 2004; Eid and Dancygier 2005兲. The Imran and Pantazopoulou 共2001兲 concrete plasticity material model is, however, a relatively complex one. It includes hardening, failure, and softening surfaces, which are a function of the accumulated plastic strain. Therefore, in order to solve the problem with this material model, a numerical procedure such as the one introduced by Ortiz and Simo 共1986兲 has to be applied. As a result, no explicit expressions can be derived for the confined concrete stressstrain curve or for the variables that defined it. In order to obtain explicit expressions for the confined concrete strength and for its corresponding strain, the equivalent confined cylinder is solved by applying the relatively simple and commonly used Drucker– Prager 共1952兲 yield criterion. Stress-Strain Model Based on Drucker–Prager Yield Criterion
6cm · cos ⌽
冑3共3 − sin ⌽兲
Although the cross-sectional shape of the DP surface is circular in the deviatoric planes, it does not affect the prediction for a confined circular cross section, for which the state of stresses is on the compressive meridian 共uniform confinement兲. Since the crosssectional shapes of the concrete’s failure surface in the deviatoric planes tend to be closer to triangular than to circular under low hydrostatic pressure 关according to test results 共Chen 1982兲兴, the prediction may be less accurate for noncircular concrete cross sections for which the stress state is generally out of the compressive meridian, especially with low levels of confinement. For a given ⌽, cm can be calculated from the uniaxial behavior 共z = f c0, r = = 0兲 by the following expression 共Chen 1982兲 cm =
The equivalent confined cylinder is solved in the elastoplastic range of the concrete material with the relatively simple boundary conditions at the concrete-tube interface by applying appropriate material models to the steel and concrete 共Eid and Dancygier 2005兲. For example, the steel can be represented by a uniaxial multilinear material model, while the concrete cylinder can be represented by a plasticity material model, such as that introduced by Imran and Pantazopoulou 共2001兲.
kD =
d =
f ,C d 2关共1 − 2兲Sijdij − 2共1 + 兲␣dI1冑J2兴冑J2 = f ,Cg, 12共1 + 兲␣␣1J2 + 共1 − 2兲SijSij 共18兲
where C = isotropic material tensor; f , = f / ; g, = g / ; and Sij and dij共=d兲 = deviatoric stress tensor and the incremental strain tensor, respectively. The solution of the three Eqs. 共15兲–共17兲 gives the following expressions for the stress and strain increments
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df c = Fzdz = =
dlat = Flatdz =
dlat = −
冋 冋 冋
− 共12␣␣1 + 2冑3␣ + 2␣1冑3 + 1兲Elatteq1
6teq1m − 18teq1m␣␣1 + 2冑3␣ − 2 + 2␣1冑3 − 6␣␣1 − 3teq1m − 18teq1m␣␣1 共6␣␣1 + 冑3␣ − 2␣1冑3 − 1兲Elatteq1
6teq1m − 18teq1m␣␣1 + 2冑3␣ − 2 + 2␣1冑3 − 6␣␣1 − 3teq1m − 18teq1m␣␣1
再 再
lat = p =
冦
z ⬍ z,ep zE cA D f c,ep + Fz共z − z,ep兲 z,ep 艋 z ⬍ cc
冎
z ⬍ z,ep zE cA DB D lat,ep + Flat共z − z,ep兲 z,ep 艋 z ⬍ cc
zE cA DB D z ⬍ z,ep Elatteq1 lat = Flat lat,ep − 共z − z,ep兲 z,ep 艋 z ⬍ cc Elatteq1 −
共22兲
冎
冧
共23兲
共24兲
where 1 + 共1 − 兲teq1m AD = 1 + 共1 + 兲共1 − 2兲teq1m
z,ep =
共19兲
dz
共20兲
BD =
1−+
1
共25兲
teq1m
− 3kD
共3␣ + 6␣BD − 冑3 + 冑3BD兲EcAD
共26兲
− 3kD
3␣ + 6␣BD − 冑3 + 冑3BD
共27兲
lat,ep =
− 3kDBD
3␣ + 6␣BD − 冑3 + 冑3BD
共28兲
3kDBD
共29兲
共3␣ + 6␣BD − 冑3 + 冑3BD兲Elatteq1
The parameters Fz and Flat are defined in Eqs. 共19兲 and 共20兲. Note that Eqs. 共22兲–共24兲 are suitable for describing the behavior of a transverse confining material in its linear-elastic range 共i.e., for 兩s 兩 ⬍ y兲. For an elastic-perfectly plastic confining material that has yielded 共兩lat兩 艌 y兲, Eqs. 共15兲 and 共16兲 are solved with dlat = 0, which also leads to df c = 0. Furthermore, it is noted that z,ep, lat,ep, f c,ep, and lat,ep 关Eqs. 共26兲–共29兲兴 are the axial and lateral strains and stresses when the state of stresses reaches the yield criterion. At this state, increasing the lateral stress will require an increase of the axial stress in order to satisfy the yield criterion 共Drucker–Prager without hardening/softening surfaces兲. Hence, according to this material model, the concrete strength is reached when the confining material yields or ruptures. This condition
共21兲
共p = f yteq1兲, together with Eqs. 共20兲, 共23兲, and 共26兲, gives the following expression for the confined concrete strain at peak stress, cc cc =
Flatz,ep + f yteq1 − lat,ep Flat
共30兲
Substituting Eq. 共30兲 into Eq. 共22兲 or solving the yield criterion 关Eq. 共11兲兴 with p = f yteq1 关and the expressions given in Eq. 共12兲兴 yields the expression for concrete confined strength f cc 冑3 + 6␣ evf + 1 = kDPevf + 1 = f c0 冑3 − 3␣
共31兲
where kDP = strength enhancement factor and evf = equivalent mechanical volumetric transverse reinforcement ratio evf = teq1
f c,ep =
lat,ep =
dz
− 共6␣␣1 + 冑3␣ − 2␣1冑3 − 1兲 Flatdz = dz Elatteq1 6teq1m − 18teq1m␣␣1 + 2冑3␣ − 2 + 2␣1冑3 − 6␣␣1 − 3teq1m − 18teq1m␣␣1
It should be mentioned that these expressions are valid only for elastoplastic concrete behavior. The full range of confinedconcrete stresses and strains are derived by integrating the expressions in Eqs. 共19兲–共21兲 and by adding the elastic parts as follows fc =
册 册 册
fy f c0
共32兲
The Drucker–Prager model, which has been used here, does not include softening surfaces. This implies that, for concrete confined by elastic-perfectly plastic material, the stress will be constant and equal to the concrete strength f cc for strains that are larger than cc 共postpeak branch of the stress-strain curve兲. An alternative for this case can be the incorporation of a postpeak softening behavior into the current explicit solution by using the following expressions for the residual concrete strength, f cr, and for the ultimate concrete strain, cu, which are based on the expressions proposed by Eid 共2004兲 f cr = 共vf兲0.4 艋 1 f cc
共33兲
cu 共0.0035f y /f c0+0.17兲 = 共0.58f y/f c0 + 14兲vf +3 c0
共34兲
where vf = mechanical volumetric transverse reinforcement ratio. This ratio is defined as vf = v f y / f c0 where v = volumetric lateral reinforcement ratio defined as v = 2t / 2a共s + t兲. It should be noted that Eq. 共34兲 is a simplification of the expression derived by equating the volumetric strain at the ultimate condition 共Imran and Pantazopoulou 1996兲 to cu + 2sf, where sf = fracture strain of the elastic-perfectly plastic material 关see Eq. 共40兲 in the Appendix兴. On the basis of the above derivations, the axial and lateral stress-strain relations of confined concrete 共in circular columns兲 are described by trilinear curves, which are defined by four points, as follows
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fc =
冦
z = 0 f c,ep , z = z,ep f cc , z = cc f cr , z = cu
0,
冧 冦
lat = 0 lat,ep , lat = lat,ep lat = f yteq1 , lat = − y f yteq1 , lat = − sf 0,
冧
FRP-Confined Concrete 共35兲
where y = rupture or yield strain of the elastic or the elastoplastic lateral material, respectively. It should be noted that, for the case of a concrete column confined by an elastic lateral material, the concrete confined strength and strain are defined by the rupture of the lateral material. Therefore, the axial and lateral stress-strain curves of such confined columns are obtained by the first three expressions of Eq. 共35兲.
Model Verification The proposed model is based mainly on two concrete variables: the friction angle, ⌽, and the dilatancy angle, . The friction angle determines the slope of the yield function, ␣ 关see Eq. 共11兲兴, and the dilatancy angle determines the slope of the potential function, ␣1 关see Eq. 共14兲兴, according to the same relation 关that in Eq. 共12兲兴. Previous studies used different values for these variables. Karabinis and Kiousis 共1994兲 used the values ⌽ = 48° and ␣ / ␣1 = 6, while Mirmiran et al. 共2000兲 have concluded, based on sensitivity analysis using the finite-element method, that the best values to fit their experimental data are ⌽ = 28° and ␣ / ␣1 = ⬁ 共 = 0 ° 兲. Chen 共1982兲 used a friction angle range of 30–56.6° for different cases of concrete elements. Nielsen 共1999兲 concluded that the friction angle varies from 37° for low-strength concrete to a constant value of 28° for concrete strengths greater than 65 MPa. Nielsen 共1999兲 also pointed out that, for higher confinement levels, the relation between the concrete strength and lateral pressure is linear, and therefore a more accurate failure condition must contain two linear curves with two different slopes. In view of the above, the friction angle, ⌽, and the yield-topotential surface slopes ratio, ␣ / ␣1, are taken in the following two sections, where the current model is verified against experimental results, as 40° and 18, respectively 关with the exception of Fig. 3共a兲, which was derived with the parameters that were used for the finite element method 共FEM兲 calculations 共see following text兲, ⌽ = 42° and ␣ / ␣1 = 6兴.
Fig. 4 shows the axial and lateral stress-strain curves of the proposed model and of FRP-confined concrete cylinder tests taken from Lam and Teng 共2004兲 and Xiao and Wu 共2000兲. The confined axial concrete strain is obtained when the FRP ultimate tensile strain is reached. Research has shown, however, that the ultimate lateral strain of FRP-confined concrete columns is smaller than the ultimate tensile strain obtained from a flat coupon test ASTM D 3039 共ASTM 1995兲. Lam and Teng 共2003兲 concluded from published test results that the ratios between the ultimate lateral strain of FRP-confined concrete cylinders and the FRP tensile strain obtained from a flat coupon test 共referred to as the FRP efficiency factor兲 were 0.586 and 0.624 for 52 carbon FRP 共CFRP兲-wrapped concrete cylinders and for 9 glass FRP 共GFRP兲-wrapped concrete cylinders, respectively. These values are used for determining the FRP rupture strain. Fig. 4 shows that the model properly simulates the overall behavior of the confined concrete, and that the confined concrete strength and its corresponding strains are in fair agreement with published experimental results. Sensitivity Analysis The explicit solution obtained for circular confined concrete columns can be used to perform sensitivity analysis to obtain the best values of the friction and dilatancy angles to fit experimental data. The experimental data include 207 test results, including 114 steel-confined concrete specimens and 93 FRP-confined concrete specimens 共see Table 1兲. According to the proposed model, the confined concrete strength is influenced only by the friction angle, while its corresponding strain is mainly influenced by the dilatancy angle 共or ␣ / ␣1兲. It is well known that the friction angle and therefore the parameter ␣ decrease as confining pressure increases 共Nielsen 1999兲. This behavior was also indicated by the experimental results summarized in Table 1. The data in Table 1 lead to a friction angle that is greater than 50° for very small lateral pressure 共caused by small amount of lateral confinement evf ⬇ 0兲 and that decreases to a constant average value of 32° for relatively high lateral pressure 共caused by relatively high amount of lateral confinement evf 艌 0.2兲. According to these results the following values are suggested for the frictional angle
Steel-Confined Concrete Fig. 3 shows the stress-strain curves of the proposed model and of several published test results, which cover a wide range of the problem’s variables. The figure shows that the model simulates well the overall behavior of the confined concrete and that the confined concrete strength and strain are in good agreement with the test results. It is noted that the use of the DP yield criterion without hardening surfaces allows the derivation of the simplified model but results in confined strains that are in lesser agreement with test results. Nonetheless, it is also noted that very good agreement was obtained between the equivalent cylinder solution 共with the DP model兲 and that of FEM analysis of a partially confined RC column, which was carried out with ATENA 2D 共2003兲 by Eid 共2004兲. For example, Fig. 3共a兲 shows the results obtained with the current model and with the FEM analysis where, for clarity, the model is described by solid and dashed curves, which represent the behavior of the concrete when analyzed with and without softening, respectively.
⌽=
再
40° evf 艋 0.2 32° evf ⬎ 0.2
冎
共36兲
According to Eq. 共36兲, the strength enhancement factor 关Eq. 共31兲兴 will be kDP = 4.6 and 3.25 for concrete columns with relatively low and high confinement levels, respectively. This is consistent with the factors 4.1 and 3.3 obtained by Richart et al. 共1928兲 and Lam and Teng 共2003兲 for steel- and for FRP-confined concrete columns, respectively. Moreover, based on the proposed model 关Eq. 共30兲兴, the dilatancy angle and the potential surface slope ␣1 are mainly functions of the lateral stiffness ratio t1m. By introducing the experimental axial strains, cc 共taken from the data in Table 1兲, into Eq. 共30兲, it was found that increasing the lateral stiffness ratio decreases the dilatancy angle. Based on these test results 共Table 1兲, the following values of the dilatancy angle are suggested
1826 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2007
Fig. 3. Analytical and experimental axial behavior of several published steel-confined concrete test results
冦
10° t1m ⬍ 0.035 = 0° 0.035 艋 t1m ⬍ 0.05 − 10° t1m 艌 0.05
冧
共37兲
Note that, using the flow rule, the incremental volumetric plastic p strain dzp + 2dlat can be written as −3␣1d. This relation shows that, for positive values of ␣1 共or 兲, the plastic deformation will be accompanied by a volume increase 共volume expansion兲. Researchers 共Mirmiran and Shahawy 1997; Shahawy et al. 2000兲 have shown, however, that for concrete columns, which are confined by a high level of confinement, the concrete expansion is curtailed and the plastic deformation will be accompanied by a decrease in volume 共volume compaction兲. The proposed model can account for this behavior by applying a negative value to the
parameter ␣1 or to for high confinement levels 关t1m ⬎ 0.05, Eq. 共37兲兴. Fig. 5 compares the theoretical confined strengths to the published test results. Since it was shown that, for practical cases of RC columns 共i.e., reinforcement ratios, column diameter, etc.兲, Eq. 共10兲 yields approximate values of the equivalent thickness of 0.47v 共Eid 2004兲, this relation 共i.e., teq1 = 0.47v兲 was used here for the comparison described in Fig. 5. Also note that, for the continuously applied FRP, teq is indeed the actual thickness of the FRP jacket and that the estimated FRP ultimate tensile strain was taken as 0.586fu and 0.624fu for CFRP-wrapped concrete specimens and for GFRP-wrapped concrete specimens, respectively 共according to Lam and Teng 2003兲. These values are based on FRP ultimate tensile strain, fu, obtained from flat coupon tests.
JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2007 / 1827
Fig. 4. Analytical and experimental axial behavior of several published FRP-confined concrete test results
However, for the simulation of the Samaan 共1997兲 results, the FRP ultimate tensile strain was taken as being equal to the actual strain, because the FRP ultimate tensile strain in these experiments was derived from ring splitting tests ASTM D 2290 共ASTM 1992兲. Fig. 5 shows that there is good agreement between the proposed model 关Eq. 共31兲兴 and the experimental results with a predicted-to-experimental mean error 共error= 关Pred.-Exp.兴 / Exp.兲 of 0.07% and a mean absolute error 共error = abs兵关Pred.-Exp.兴 / Exp.其兲 of 10.3%. Fig. 6 shows a comparison between the predicted and experimental confined concrete strain at peak stress. The figure shows that there is a fair to good agreement between the predicted confined concrete strain 关Eq. 共30兲兴 and the test results of 138 columns with elastic or elastic-perfectly plastic confining material and various concrete strengths. In this figure, the predicted-toexperimental average mean error is 12.0% and the average absolute mean error is 27.6%.
Conclusions This paper presents a relatively simple yet analytically derived curve to model the axial and the lateral stress-strain relations of
circular concrete columns. This model is applied here for cases of concrete columns that are confined by lateral reinforcement or external FRP sheets. The explicit analytical curve is based on the full elastoplastic behavior of the confined concrete column. The solution to the partially confined RC column in the elastoplastic range is derived by replacing the discrete lateral reinforcement with an equivalent tube. The equivalent tube’s properties are obtained from an equivalency criterion, which is determined from the solution of an elastic cylinder confined by passive elasticperfectly plastic ties. It has been shown that these properties are suitable for an analysis of the behavior of the equivalently confined column in its full elastic and plastic range. The derived model is based on elastic or elastic-perfectly plastic confining material behavior and on the Drucker–Prager yield criterion without hardening surfaces. While the confined concrete strength is not influenced by these properties, they do affect the confined concrete strain at peak stress due to the path dependency of the concrete deformations. Therefore, this model is more appropriate for concrete columns confined by elastic or elasticperfectly plastic material. The model was verified against available experimental results of concrete columns partially confined by steel ties or fully confined by FRP sheets. It was shown that the proposed model prop-
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Table 1. Circular Specimens Tested under Concentric Compression
Reference
Number of specimens
Lateral steel: v 共%兲 FRP: t 共mm兲
D 共mm兲
f c0 共MPa兲
fy 共MPa兲
Elat 共GPa兲
24–32 35.2–82.5 25–85 35–36 16–34 51–105 18.5–28.8 30–50
307–340 445–1,318 909–1,296 452–629 318.7 400–1,000 235–295 400
210 210 210 210 210 210 210 210
33.7–55.2 36–38.5 30.2 34.9 42–43 26.2 18 30–32 31
1,577 522–3,758 1,285–2,873 1,100–2,600 230–1,265 383–580 513–1,353 524–641 1,518–2,940
105 22–250 87.3–629.6 200–420 13.6–82.7 38.1–21.6 36–150 37–41 73–373
共a兲 Steel-confined concrete Mander et al. 共1988a兲 Li et al. 共2001兲 Assa et al. 共2001兲 Sheikh and Toklucu 共1993兲 Iyengar et al. 共1970兲 Razvi and Saatcioglu 共1999a兲 Hoshikuma et al. 共1997兲 Eid et al. 共2006兲
15 16 11 27 12 16 11 6
500 240 145 203–356 150 250 200–500 303
v = 0.6– 2.5 v = 0.9– 3.2 v = 1.2– 4.3 v = 0.6– 2.4 v = 0.8– 3.1 v = 0.4– 3.1 v = 0.2– 4.7 v = 1.5– 2.3
共b兲 FRP–confined Concrete Xiao and Wu 共2000兲 Lam and Teng 共2004兲 Watanable et al. 共1997兲 Matthys et al. 共1999兲 Rochette and Labossiere 共2000兲 Pessiki et al. 共2001兲 Karbhari and Gao 共1997兲 Samaan 共1997兲 Toutanji 共1999兲
27 13 9 4 7 4 4 22 3
152 152 100 150 100–150 152 152 152 76
erly simulates the overall behavior of the confined concrete and that the confined concrete strength and its corresponding strain are in good agreement with the test results.
t = 0.385– 1.14 t = 0.165– 2.54 t = 0.14– 0.67 t = 0.12– 0.24 t = 0.6– 5.21 t=1–2 t = 1.55– 5.31 t = 1.44– 2.97 t = 0.22– 0.33
v = 共1 − 2兲
冋
冉 冋
z − zlim 2lat z + z0 0 − 0 Ec z z − zlim
册 冊册 2
共38兲
The concrete ultimate strain can be derived from the following expression of the volumetric strain 共Imran and Pantazopoulou1996兲
where lat = 0.47vf f c0 = 0.47v f y = lateral 共radial兲 stress 共see also Eid 2004兲; z0 = axial strain at zero volumetric strain 共taken as equal to the axial compressive strain at peak stress, z0 = cc兲, and zlim = 共1 − 兲 / 共Ec兲lat − 共cr / 兲 = axial strain at which cracking occurs in the lateral direction, where cr = cracking strain of the concrete in direct tension. By assuming that the ultimate concrete strain is reached at lateral steel fracture, the volumetric strain for
Fig. 5. Comparison between predicted and experimental confined concrete strength
Fig. 6. Comparison between predicted and experimental confined concrete strain
Appendix
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axisymmetric members at ultimate condition can be written as follows v = cu + 2sf
共39兲
where sf = fracture strain of the lateral steel. Equating Eqs. 共38兲 共with z = cu兲 and 共39兲 yields the following expression for the ultimate concrete strain cu = zlim −
+ B2
冑冉
− zlim B2
冊
2
−
B3 B2
共40兲
where B2 =
共1 − 2兲cc 共cc − zlim兲2
B3 = 2sf −
0.94共1 − 2兲 vy f c0 + B2共zlim兲2 Ec 共41兲
Eq. 共34兲 is a simplification of Eq. 共40兲.
Notation The following symbols are used in this paper: a ⫽ cylinder radius; b ⫽ width of lateral ties 共b1 = b / a兲; cm ⫽ cohesion of material; D ⫽ gross diameter of RC column; Ec ⫽ cylinder elastic modulus; E f ⫽ FRP elastic modulus; Elat ⫽ ring elastic modulus; Es ⫽ steel elastic modulus; Flat ⫽ constant parameter; Fz ⫽ constant parameter; f ⫽ yield criterion; f c0 ⫽ unconfined concrete compressive strength; f cc ⫽ confined concrete compressive strength; f c,ep ⫽ axial stress when state of stresses reaches yield criterion; f cr ⫽ residual concrete compressive strength; f 1共k1兲 ⫽ function depend on boundary conditions of confined cylinder; f y ⫽ reinforcement yield or FRP rupture stress; f 3 ⫽ p / q ratio; g ⫽ plastic potential function; I1 ⫽ first invariant of stress tensor; J2 ⫽ second invariant of deviatoric stress tensor; k ⫽ constant with dimension of length−1 共k1 = ka兲; kD ⫽ positive material constant; kDP ⫽ strength enhancement factor; kD1 ⫽ positive material constant; p ⫽ lateral pressure; q ⫽ axial pressure; s ⫽ clear spacing of lateral ties 共s1 = s / a兲; sb ⫽ center to center ties spacing; t, t1 ⫽ ties thickness 共t1 = t / a兲; teq, teq1 ⫽ ties equivalent thickness 共teq1 = teq / a兲; ␣ ⫽ slope of Drucker–Prager function; ␣1 ⫽ slope of Drucker–Prager plastic potential function;  ⫽ function depend on boundary conditions of confined cylinder; c0 ⫽ concrete compressive strain corresponding to unconfined concrete strength;
cc ⫽ concrete compressive strain corresponding to confined concrete strength; cu ⫽ concrete ultimate axial strain; fu ⫽ FRP ultimate tensile strain; lat ⫽ lateral strain; lat,ep ⫽ lateral strain when state of stresses reaches yield criterion; sf ⫽ fracture strain of lateral reinforcement; y ⫽ steel yield of FRP rupture strain; z ⫽ axial strain; z,avg ⫽ total axial strain averaged over radius and along axial direction within typical zone; c z,avg ⫽ axial strain due to action of rings, averaged over radial and axial directions within typical zone; z,ep ⫽ axial strain when state of stresses reaches yield criterion; ⫽ Poisson’s ratio; evf ⫽ equivalent mechanical volumetric lateral reinforcement ratio= teq1 f y / f c0; lat ⫽ lateral 共radial兲 stress= p; lat,ep ⫽ lateral stress when state of stresses reaches yield criterion; v ⫽ volumetric ratio of lateral reinforcement; ⌽ ⫽ internal friction angle of material; and t ⫽ cross-section diameter of reinforcing ties.
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