Finite element modeling of confined concrete-II: Plastic-damage model

Engineering Structures 32 (2010) 680–691

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Finite element modeling of confined concrete-II: Plastic-damage model T. Yu a , J.G. Teng a,∗ , Y.L. Wong a , S.L. Dong b a

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

b

Department of Civil Engineering, Zhejiang University, Hangzhou 310027, China

article

info

Article history: Received 14 May 2008 Received in revised form 12 November 2009 Accepted 22 November 2009 Available online 4 January 2010 Keywords: Confinement Concrete FRP Plasticity Damage Finite elements Modeling

abstract This paper presents a modified plastic-damage model within the theoretical framework of the Concrete Damaged Plasticity Model (CDPM) in ABAQUS for the modeling of confined concrete under non-uniform confinement. The modifications proposed for the CDPM include a damage parameter, a strain-hardening/softening rule and a flow rule, all of which are confinement-dependent, and a pressuredependent yield criterion. The distinct characteristics of non-uniformly confined concrete are also included in this model by defining an effective confining pressure. Finite element models incorporating the proposed CDPM model were developed for concrete in a number of confinement scenarios, including active confinement, biaxial compression, FRP-confined circular and square columns, and hybrid FRP-concrete-steel double-skin tubular columns. The finite element predictions are shown to be in close agreement with the existing test results. The limitations of the proposed model are also discussed towards the end of the paper, pointing to future research needs in this area. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Fiber-reinforced polymer (FRP) jackets have been widely used to confine reinforced concrete columns to enhance their structural performance. Consequently, many studies over the past decade have been devoted to the behavior and modeling of FRP-confined concrete. Most of these studies have been concerned with the behavior of concrete in FRP-confined circular columns, where the confinement is uniform around the circumference (e.g. uniform confinement) [1]. Stress–strain models now exist that can provide close predictions of the behavior of such FRP-confined concrete [1]. By contrast, the behavior of FRP-confined concrete in columns of other section forms (e.g. rectangular sections, elliptical sections and annular sections) has received more limited attention and accurate stress–strain models have not been developed yet for such non-uniformly confined concrete. Of the different scenarios of non-uniform FRP confinement, FRP-confined rectangular sections have received the most attention (e.g. [2–6]), and several stress–strain models are now available for FRP-confined concrete in rectangular sections. Since FRP confinement is much less effective for rectangular sections than for circular sections, the possibility of modifying a rectangular section into an elliptical section has been explored (e.g. [7]) but the

∗

Corresponding author. Tel.: +852 2766 6012; fax: +852 2766 1354. E-mail address: [email protected] (J.G. Teng).

0141-0296/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2009.11.013

current understanding of the behavior of concrete in FRP-confined elliptical sections is very limited. Another important form of FRPconfined section is the annular section, found in a number of different column forms. Fam and Rizkalla [8] and Becque et al. [9] presented their attempts to model FRP-confined circular hollow concrete sections and FRP-concrete double-skin tubular columns (DSTCs) composed of concrete sandwiched between two FRP tubes as the skins but they have had limited success [10]. Teng et al. [11] recently proposed a new form of hybrid columns named hybrid FRP-concrete-steel DSTCs which consist of an outer tube made of FRP and an inner tube made of steel, with the space between filled with concrete (Fig. 1). Accurate stress–strain models for FRPconfined concrete in annular sections have not yet been found in the published literature. It is well known that in a non-circular FRP-confined section, the confining pressure provided by the jacket varies around the perimeter and the axial stress in the concrete varies over the section. Existing stress–strain models for non-uniformly confined concrete generally adopt the so-called ‘‘design-oriented approach’’ [12] so that the models provide a relationship between the average axial stress of the concrete to the axial strain, with the prediction of the lateral strains ignored. This design-oriented approach relies on the direct interpretation and regression of test data, and leads to one-dimensional stress–strain models in closedform algebraic expressions. It offers the simplicity needed for such models to be used in design, but is limited in the way that it is empirically based on limited test data. The alternative ‘‘analysisoriented model’’ well developed for concrete with uniform FRP

T. Yu et al. / Engineering Structures 32 (2010) 680–691

681

Fig. 1. Cross-section of new hybrid DSTCs.

confinement cannot be readily extended to FRP-confined concrete in non-circular sections due to stress non-uniformity in such sections. As a result, the mechanism of how the FRP jacket and the concrete interact and how the stresses vary in the section are not yet well understood for concrete with non-uniform FRP confinement. Against the above background, the finite element (FE) method appears to offer an attractive method for the accurate modeling of the three-dimensional behavior of concrete with non-uniform FRP confinement as it is capable of capturing complex stress variations in the concrete. Many studies have been published on the FE modeling of FRP-confined concrete sections, using a variety of different constitutive models for concrete [1]. In the companion paper [1], the performance of existing Drucker–Prager (D–P) type plasticity models for the modeling of confined concrete has been assessed, leading to the conclusion that a D–P type plasticity model providing close predictions of the behavior of confined concrete should include a strain-hardening/softening rule and a flow rule that are both confinement-dependent, and a yield criterion dependent on the third deviatoric stress invariant. Yu et al. [1] also proposed a modified D–P model which takes into consideration the conclusions of the assessment. Although the modified D–P model proposed by Yu et al. [1] was shown to provide accurate predictions for both actively-confined and FRP-confined concrete under uniform confinement, its validity for concrete under nonuniform confinement is uncertain. In addition, it suffers from other limitations including its inability in simulating the reduction of elastic stiffness and the numerical difficulty it may encounter when employed to simulate the strain softening behavior of concrete [1]. This paper presents an improved plastic-damage model for concrete, which removes the deficiencies of the modified D–P type model presented in the companion paper [1]. This plastic-damage model is formulated within the theoretical framework of the Concrete Damaged Plasticity Model (CDPM) provided in ABAQUS. The model employs the concept of damaged elasticity to simulate reductions in elastic stiffness and captures accurately the distinct characteristics of non-uniformly confined concrete. The proposed model is verified with test results of concrete in a number of different stress states, including actively-confined concrete, concrete under biaxial compression, and FRP-confined concrete in circular, rectangular and annular sections respectively.

Fig. 2. Typical stress–strain curve of concrete under compression.

In this section, the Concrete Damaged Plasticity Model (CDPM) provided in ABAQUS is briefly presented before the proposed modifications are introduced in the next section. The CDPM in ABAQUS uses concepts of isotropic damage in combination with isotropic tensile and compressive plasticity to represent the inelastic behavior of concrete [16]. The key aspects of this model in terms of the compressive behavior of concrete, including the damage variable, the yield criterion, the hardening/softening rule, and the flow rule, are summarized as follows. In this paper, compressive stresses/strains are defined to be positive while tensile strains/stresses are defined to be negative, unless otherwise specified. 2.2. Damage The scalar damaged elasticity equation is adopted, which takes the following form:

σij = (1 − d)Deijkl (εij − εijp )

(1) p ij

where σij is the stress tensor, εij and ε are the strain tensor and the plastic strain tensor respectively, Deijkl is the initial (undamaged) elasticity matrix and d is the damage variable which characterizes the degradation of the elastic stiffness. When concrete is subjected to uniaxial monotonic compression, Eq. (1) is simplified to:

σ1 = (1 − d)Ec (ε1 − ε1p )

(2)

2. Concrete damaged plasticity model (CDPM)

where σ1 and ε1 are the compressive stress and strain of concrete p in the loading direction respectively; ε1 is the plastic strain in the loading direction; and Ec is the initial elastic modulus of concrete. The effective stress σ 1 is defined as:

2.1. General

σ1 =

In general, the nonlinearity of concrete under compression can be modeled by approaches based on the concept of either damage or plasticity, or both [13,14]. Plasticity is generally defined as the unrecoverable deformation after all loads have been removed. Damage is generally characterized by the reduction of elastic constants. Both the reduction of unloading stiffness and unrecoverable deformation have been clearly observed in concrete compression tests [13,15] as illustrated in Fig. 2, which suggests that the concept of plasticity should be combined with the concept of damage to correctly represent the nonlinear behavior of concrete.

σ1 1−d

.

(3)

Similarly, the first effective stress invariant I 1 and the second effective deviatoric stress invariant J 2 are defined in terms of the effective stress tensor. I 1 = σ ii J2 =

1 2

S ij S ij

where S ij is the effective deviatoric stress tensor.

(4) (5)

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T. Yu et al. / Engineering Structures 32 (2010) 680–691

(a) Yield surfaces in the I 1 −

q

J 2 plane for triaxial compression.

(b) Yield surfaces in the deviatoric plane.

Fig. 3. Yield surfaces of the concrete damaged plasticity model in ABAQUS.

2.3. Yield criterion The yield function proposed by Lubliner et al. [17] and modified by Lee and Fenves [18] is adopted. In terms of effective stresses, the yield function takes the following form. F =

q

1



3J 2 − AI 1 + B(˜εp ) h−σ min i − C hσ min i

1−A

− σ cn (˜εpc ) = 0

(6)

with fb0 /fco0 − 1

A=

0 ≤ A ≤ 0.5,

(7)

σ cn (˜εpc ) (1 − A) − (1 + A), σ tn (˜εpt )

(8)

2fb0 /fco0 − 1

B= C =

;

Fig. 4. D–P hyperbolic flow potential in the I 1 −

3(1 − K )

1

2K − 1

3

q C +1

3J 2 −

J 2 plane.

(9)

where σ min is the minimum principal effective stress; fb0 is the concrete strength under equal biaxial compression; σ cn and σ tn are the effective compressive and tensile cohesion stresses respectively; ε˜ pc and ε˜ pt are the equivalent compressive and tensile plastic strains respectively; and K is the strength ratio of concrete under equal biaxial compression to triaxial compression (i.e. axial stress in combination with equal lateral stresses). Typical yield surfaces in the deviatoric plane are shown in Fig. 3(b) for different values of K . For the case of triaxial compression, Eq. (6) reduces to the Drucker–Prager yield condition given by the following equation:



q

(C + 3A) 3

I 1 = (1 − A)σ cn .

(10)

The yield surface of concrete under compression (with the minimumq principal stress larger than zero) can be represented in the I 1 − J 2 plane by a linear curve for the case of triaxial compression (Fig. 3(a)) and in the deviatoric plane by a non-circular curve (Fig. 3(b)).

2.4. Hardening/softening rule For concrete under uniaxial monotonic compression, the strainhardening/softening function can be defined in the CDPM by

σ cn = σ cn (˜εp ).

(11)

2.5. Flow rule A non-associated flow rule is assumed in the CDPM. The flow potential adopted in this model is the Drucker–Prager hyperbolic function shown below:

∂G dε = λ ; ∂σij p ij

q I1 2 G = ( σto tan ψ)2 + 3J 2 − tan ψ 3

(12)

where ψ depends on the potential function parameter [1]; σto is the uniaxial tensile stress at failure; , referred to as the eccentricity, defines the rate at which the function approaches the asymptote defined by ψ (see Fig. 4). The flow potential tends to a straight line when the eccentricity  is close to zero. A typical curve of the flow

T. Yu et al. / Engineering Structures 32 (2010) 680–691

potential in the I 1 −

q

J 2 plane is shown in Fig. 4. It is seen (Fig. 4) √

that the slope of the function is very close to tan ψ when the concrete is under compression in all three directions. 3 9

3. Proposed modifications 3.1. General It was concluded by Yu et al. [1] that a plasticity model leading to close predictions for FRP-confined concrete should include a strain-hardening/softening rule and a flow rule that are both confinement-dependent, and a yield criterion that is dependent on the third deviatoric stress invariant. A confinement-dependent strain-hardening/softening rule is necessary to reflect the difference in the experimental stress–strain curve between confined concrete and unconfined concrete. A confinement-dependent flow rule, which is related not only to the confining pressure but also to the rate of confining pressure increment, is required to capture the unique lateral expansion behavior of passively-confined concrete. A yield criterion dependent on the third deviatoric stress invariant is necessary to simulate the experimental observation that the shear strength is different for concrete under equal biaxial compression and triaxial compression, even when the hydrostatic pressure is the same for both cases. In a plastic-damage model, a confinement-dependent damage variable also needs to be included, as strain softening is at least partially simulated by the scalar damage in such models. The CDPM implicitly includes the effects of hydrostatic pressure and the third deviatoric stress invariant on the shear strength of concrete, but the hardening/softening rule, the flow rule and the damage variable are not confinement-dependent. Therefore, modifications are proposed herein for the three aspects based on the above discussions. Teng et al.’s analysis-oriented model [19] for uniformly-confined concrete is adopted herein to produce the necessary material parameters for this constitutive model. Teng et al.’s model [19] is able to predict the entire axial stress–axial strain curve and lateral strain–axial strain curve, and thus provides sufficient information to define the hardening/softening rule and the flow rule for the special case of concrete under uniform confinement. In addition, the unique properties of concrete under nonuniform confinement are appropriately included, as discussed later in this section. 3.2. Damage variable As explained earlier, concrete nonlinearity can be modeled as either damage or plasticity, or both. Different definitions of damage and plasticity lead to different plastic-damage models. When suitable material parameters are used, these different models may lead to the same prediction for concrete under monotonic loadings. However, predictions of cyclic loading tests depend significantly on how to differentiate the effects of damage and plasticity, as the definition of damage directly determines the stiffness of the unloading curve (Fig. 2). To accurately reflect experimental observations, it is desirable to isolate the effect of damage using test results of concrete under cyclic loadings. In the present research, as only the simulation of monotonic loading tests is concerned, the following assumption is adopted for simplicity: concrete nonlinearity before the peak stress is due only to concrete plasticity and strain-hardening and the softening of concrete after the peak stress is due only to concrete damage. A similar assumption was adopted by Murray and Lewis [20] and Schwer [21] in their plastic-damage models. Besides its simplicity, the advantage of this assumption is that it simulates stress reductions after the peak stress by reductions in elastic constants instead of retractions of the yield surface

683

in the stress space. Retractions of the yield surface are necessary to simulate the strain softening behavior in a concrete plasticity model and may cause numerical problems. Similar to the strain-hardening/softening rule, the damage variable is assumed to be dependent on the confining pressure, as the descending branches of the stress–strain curves of confined and unconfined concrete have different slopes [1]. Based on the assumption stated earlier, the damage variable is equal to zero before the peak stress and is given by the equations below after the peak stress. For concrete under uniaxial compression, the damage parameter is obviously given by

σc

d=1− 0 fco

(13)

in which σc is the axial stress of concrete on the descending branch and fco0 is the stress of concrete at the peak point. For concrete with a constant confining pressure, the corresponding damage parameter becomes

√ d=1− √

J2 − θ I1

J2c − θ I1c

=1−

σc − 0 fcc∗

−

σ

1+C +2A l 1 −A 1+C +2A l 1−A

σ

(14)

0

in which fcc∗ is the peak stress of concrete under a constant confining pressure; σl is the confining pressure; J2c and J2 are the second deviatoric stress invariants corresponding to the peak stress point and a point on the descending branch respectively; I1c and I1 are the first stress invariants corresponding to the peak stress point and a point on the descending branch respectively. The confining pressure-dependent damage variable as defined above was implemented in ABAQUS through the following procedure: (1) obtain a series of axial stress–strain curves of concrete for various constant confining pressures using Teng et al.’s model [19]; (2) find the values of the damage variable corresponding to different axial strains and confining pressures using Eqs. (13) and (14); (3) input these values for the damage variable into ABAQUS in the required format, in which the association of the damage variable with the confining pressure is defined through the SDFV option. The SDFV (solution-dependent field variables) option allows the material properties to be set to be dependent on field variables that vary throughout the solution process [1]. A computer program was developed to produce the input material data.

3.3. Yield criterion It is evident that the CDPM implicitly includes the effects of the first stress invariant and the third deviatoric stress invariant. The two controlling parameters of this model for concrete in compression are A and C defined by Eqs. (7) and (9). The constant A can be determined using Eq. (7) based on the experimental concrete strength under equal biaxial compression. Kupfer et al. [22] found from their tests that the ratio fb0 /fco0 is approximately 1.16, yielding a value of 0.12 for A. This value is adopted in the present research. The constant C can be determined using Eq. (9) based on the experimental shear strength ratio of concrete between biaxial compression and triaxial compression. It has been shown by Yu et al. [1] that this ratio is equal to 0.725 based on empirical equations, yielding a value of 1.83 for C , which is used in the present research.

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3.4. Hardening/softening rule It has been made evident by Yu et al. [1] that the hardening/ softening rule should be related to the confining pressure. The following equation is adopted in the present research:

σ cn = σ cn (˜εp , σl ).

(15)

According to the assumption stated in Section 3.2, the definition of strain-hardening is the same as in a plasticity model [1] before the peak stress of concrete. After the peak stress has been reached, no strain-hardening/softening is defined and the yield surface remains unchanged (i.e. perfectly-plastic behavior is assumed). 3.5. Flow rule The procedure to calculate the flow rule is similar to that explained by Yu et al. [1] for a concrete plasticity model, except that the equivalent plastic strain should be calculated based on Eq. (1) and the damage variable obtained using Eqs. (13) and (14). In addition, the Drucker–Prager hyperbolic function is adopted as the flow potential (Eq. (12)). It is shown in Fig. 4 that the slope √ of the function is close to 93 tan ψ when I 1 > 0. Therefore, the potential function parameter of the D–P model [1] is approximated √

as 93 tan ψ and is used to calculate the ψ value for the CDPM. This approximation is believed to have only minor effects on the calculated results. 3.6. Concrete under non-uniform confinement Up to now, all the material parameters have been found from Teng et al.’s analysis-oriented model [19], which is for concrete under uniform FRP confinement. It has been pointed out by Yu et al. [1] that the effect of the third deviatoric stress invariant should be taken into account in the yield criterion so that the strength of concrete under equal biaxial compression and non-uniform confinement can be accurately predicted. The CDPM implicitly includes this effect. The inclusion of this effect, however, does not necessarily mean that the deformation (e.g. the axial stress–strain curve and the lateral strain–axial strain curve) of concrete under non-uniform confinement can be closely simulated. The prediction of the axial stress–strain behavior depends significantly on the definition of the hardening/softening rule, and the prediction of the lateral strain–axial strain behavior depends significantly on the definition of the flow rule, instead of the yield criterion. The hardening/softening rule is related to the confining pressure as explained earlier, but the definition of the confining pressure, which is the same as the two equal principal lateral stresses for concrete under uniform confinement, is not straightforward for concrete under non-uniform confinement. In the present study, an effective confining pressure σl,eff is proposed to achieve accurate predictions of the deformation of concrete under non-uniform confinement. This effective confining pressure is proposed to be of the following form:

σl,eff =

2 σ2 + afco0


σ3 + afco0
− afco0 σ2 + σ3 + 2afco0


(16)

where σ2 and σ3 are the two principal lateral stresses respectively; fco0 is the cylinder compressive strength of concrete; and a is a constant to be determined based on test results. Eq. (16) can be regarded as a special caseof the generalized f  mean expressed by σl,eff = f −1

f (σ2 )+f (σ3 ) 2

with f (x) =

1 0 . x+afco

It is easy to see that Eq. (16) refers to the well-known harmonic mean when a = 0. In addition, σl,eff = σ2 = σ3 when the

two lateral stresses are equal regardless of the value of a, so uniform confinement is covered as a special case. The inclusion of an additional term afco0 in Eq. (16) for the effective confining pressure is mainly to reflect the experimental observation that the effectiveness of confinement depends partially on the unconfined concrete strength. Kupfer et al. [22] provided stress–strain curves for concrete under biaxial compression with different stress ratios. The test results provided by Kupfer et al. [22] have been extensively cited (e.g. [13,23]) and were also employed in the present research for the calibration of the value of a in Eq. (16). Based on Kupfer et al.’s test results [22], the best-fit value for a is 0.039. Eq. (16) is then rewritten as

σl,eff =


σ3 + 0.039fco0
− 0.039fco0 . σ2 + σ3 + 0.078fco0

2 σ2 + 0.039fco0




(17)

For actively-confined concrete, the flow rule is related to the confining pressure [1]. σl,eff given by Eq. (17) is adopted in the present study as the effective confining pressure. For FRP-confined concrete, the flow rule needs to be related to the ratio between the confining pressure and the lateral strain [1]. For FRP-confined σ concrete in a solid circular section, this ratio (i.e. ε l ) can be l expressed in terms of the properties of the FRP tube (or jacket) as σl εl

=

Efrp tfrp

, where Efrp and tfrp are respectively the elastic modulus Ro and the thickness of the FRP tube and Ro is the diameter of the section. For FRP-confined concrete in non-circular sections, this ratio is however not readily available as the lateral stresses and the strains are unequal in different directions. In the present study, two methods were explored for the flow rule of confined concrete in such cases, as detailed below.

3.6.1. Method I The first method is to make use of the flow rule for concrete in an equivalent FRP-confined circular section. For FRP-confined annular sections, the flow rule for concrete in an FRP-confined circular section with the same outer diameter and FRP tube is adopted E

t

for the concrete, as for both cases the term frpR frp directly relates the o hoop expansion to the confining pressure provided by the FRP tube. For FRP-confined rectangular sections, the Ro value of an equivalent FRP-confined circular specimen, in which the thickness of and the area contained by the FRP tube remain the same as those in the original rectangular section, is used. It is obvious that when adopting this method, the flow rule is assumed to be the same for concrete over the whole section.

3.6.2. Method II The second method is to make use of the effective confining pressure given by Eq. (17), together with the area strain, which is defined as the average of the two lateral strains, as the effective lateral strain. By doing so, ψ in Eq. (12) can be expressed as

ψ =ψ



2σl,eff

ε2 + ε3

, ε˜ p

 (18)

where ε2 and ε3 are the two principal lateral strains and ε˜ p is the equivalent plastic strain. It is obvious that when adopting this method, the flow rule for concrete may be different for each point over a non-circular section.

T. Yu et al. / Engineering Structures 32 (2010) 680–691

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Fig. 5. FE model for hybrid DSTCs.

4. Numerical verification

4.2. Actively-confined concrete

4.1. Finite element modeling

Fig. 6 shows a comparison between the predictions obtained from the FE model and the test results reported by Sfer et al. [24] for actively-confined concrete. The concrete had an unconfined strength of 32.8 MPa and a corresponding strain of 0.0018, and was confined by a constant pressure of 9 MPa. Predictions from Teng et al.’s analysis-oriented model [19] are also shown in Fig. 6 for reference. It is evident that the FE analysis provides almost the same predictions as Teng et al.’s model [19], and the predictions of both models are reasonably close to the experimental stress–strain curve. The minor difference in the lateral–axial strain curve between the FE results and the predictions of Teng et al.’s model [19] is due to the limitation of the allowable values of ψ in the CDPM in ABAQUS. The allowable values of ψ are limited between 0◦ and 56◦ in ABAQUS, while the ψ value calculated from Teng et al.’s model [19] for use in the proposed model is negative in the early loading stage. In order to make use of the theoretical framework of the CDPM model in ABAQUS, the negative values were replaced with 0◦ in the implementation. This consequently caused a small overestimation of the lateral strain in the early loading stage.

Comparisons between FE predictions obtained using the proposed constitutive model and test results are presented in this section for actively-confined concrete, concrete under biaxial compression, and FRP-confined concrete in circular section and rectangular sections as well in hybrid DSTCs with an annular section. For the two cases of concrete under biaxial compression and a uniform active confining pressure, only a single 8-node solid element was employed in the numerical analysis. For FRPconfined columns, although the two ends of a short column were constrained by the loading platens in the test, these constraints were assumed to have little effects on the behavior in the midheight region of the column with a length equal to twice the diameter. Consequently, for both FRP-confined square and circular concrete specimens, the FE model included only one-fourth of a vertical slice of the specimen and consisted of a single layer of 8-node solid elements for the concrete tied to 4-node shell elements for the FRP jacket (see Fig. 11). For hybrid DSTCs (Fig. 1), the FE model employed consisted of one layer of finite elements spanning a one-degree circumferential segment (Fig. 5). The finite element model was assigned boundary conditions representing axis-symmetric behavior (Fig. 5). These simple models were employed to provide predictions of the behavior of the mid-height region of columns. For FRP-confined columns, the FRP tube was assumed to behave in a linear elastic manner with stiffness in the hoop direction only. For hybrid DSTCs, the average stress–strain curve of the steel from tensile tests was represented with a number of data points. The well-known J2 flow theory was employed to model the plastic behavior of the steel. The concrete was modeled by the modified CDPM presented in the preceding sections. The predicted curves are terminated when the experimental ultimate hoop strain is reached. For FRP-confined columns, the Mesh Tie Constraint option of ABAQUS was adopted in the model to simulate the interaction between the FRP and the concrete. Using these constraints, a node on the FRP tube was tied to a corresponding node on the outer edge of the concrete infill so that the two nodes were forced to experience the same translations. For hybrid DSTCs, the Contact Pairs option of ABAQUS was adopted to simulate the interaction between the steel tube and the concrete. In the radial direction, the so-called ‘‘hard’’ contact, which allows the two surfaces to separate from each other, was specified. The contact pressure was automatically calculated by the program when the two surfaces were in contact. On the contrary, if the surfaces were not in contact, the pressure became zero. In the tangential direction, a friction coefficient was specified but this friction coefficient was not expected to affect the predictions as no slips were expected between the steel tube and the concrete infill due to the axis-symmetric nature of the FE model. Mesh convergence studies were conducted for each type of FRPconfined columns, arriving at appropriate meshes which provided almost the same axial stress–strain curves as those from further refined meshes.

4.3. Concrete under biaxial compression Fig. 7 shows a comparison between the FE results and the results of biaxial compression tests by Kupfer et al. [22]. The concrete had a uniaxial compressive strength of 32.0 MPa and a corresponding strain of 0.0021, and was subjected to biaxial compression with two different axial-to-lateral stress ratios, namely, σ1 /σ2 = 1 and 0.5 respectively. Fig. 7 shows that the FE results agree well with the test results. 4.4. FRP-confined circular concrete column Fig. 8 shows a comparison between the predictions of the FE model and the test results of an FRP-confined circular concrete column tested by Wong et al. [10]. The concrete in this specimen had an unconfined strength of 39.6 MPa and a corresponding strain of 0.00263. The cylinder had a diameter of 152.5 mm and a height of 305 mm, and was confined by a two-ply FRP tube with an elastic modulus of 80.1 GPa based on a nominal thickness of 0.17 mm per ply. As the concrete is subjected to uniform confinement in this case, there is no difference between Method I and Method II in specifying the flow rule of concrete. Results from Teng et al.’s analysis-oriented model [19] are also shown for reference. The FE results are almost the same as the predictions from Teng et al.’s model [19], and the results from both methods are reasonably close to the test stress–strain curve (the three curves are almost identical and the error in the concrete strength is around 10%). 4.5. FRP-confined square concrete columns 4.5.1. Comparison with test results Figs. 9 and 10 show comparisons between the FE predictions and the test results for two FRP-confined square concrete columns.

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T. Yu et al. / Engineering Structures 32 (2010) 680–691

(a) Axial stress–strain curves.

(b) Axial stress–lateral strain curves.

(c) Lateral strain–axial strain curves. Fig. 6. Actively-confined concrete.

Fig. 7. Concrete under biaxial compression.

The test results shown in Fig. 9 are for a square specimen (Specimen I) with an unconfined concrete strength of 46.0 MPa and a corresponding strain of 0.0026 tested at The Hong Kong Polytechnic University (PolyU). The specimen had a width of 150 mm, with the four corners rounded into a radius of 24 mm. It was wrapped with a two-ply carbon fiber-reinforced polymer (CFRP) jacket with an elastic modulus of 250,000 MPa based on a nominal thickness of 0.165 mm per ply. The test results shown in Fig. 10 are for another square specimen (Specimen II) tested at PolyU with an uncon-

fined concrete strength of 37.5 MPa and a corresponding strain of 0.0031. The specimen had a width of 150 mm, with the four corners rounded into a radius of 25 mm. It was wrapped with a three-ply glass fiber-reinforced polymer (GFRP) jacket with an elastic modulus of 80,100 MPa based on a nominal thickness of 0.17 mm per ply. The comparisons are for both the average axial stress–axial strain curve and the corner hoop strain–axial strain curve. The average axial stress is defined as the load divided by the cross-sectional area of the concrete. The experimental corner hoop strains were averaged from the readings of four strain gauges at the centers of the four corners. The curves denoted as ‘‘FE results I’’ in Figs. 9 and 10 were obtained by using Method I explained in Section 3.6 for the flow rule of concrete while those denoted as ‘‘FE results II’’ were obtained by using Method II for the flow rule. Figs. 9 and 10 show that the predictions are in reasonably close agreement with the test results in terms of the average axial stress–strain curve, regardless of the method used to define the flow rule. Method I led to closer predictions in terms of the corner hoop strain–axial strain curve, thus leading to more accurate predictions for the ultimate state (the errors in the concrete strength is within 10%). 4.5.2. Stress variations over the section Fig. 11 shows the predicted axial stress distributions over the section for Specimen I, obtained using the two methods for specifying the flow rule respectively. The negative values of S33 in the legend box are the axial compressive stresses. It is obvious from Fig. 11 that the axial stress varies significantly over the section,

T. Yu et al. / Engineering Structures 32 (2010) 680–691

(a) Axial stress–strain curves.

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(b) Hoop strain–axial strain curves.

Fig. 8. FRP-confined circular concrete cylinder.

(a) Average axial stress–axial strain curves.

(b) Corner hoop strain–axial strain curves. Fig. 9. FRP-confined concrete in square specimen I.

with the largest values at the corners and the smallest values near the edges. It is also obvious that a different definition of the flow rule leads to significant differences in the axial stress distribution over the section. 4.6. Hybrid DSTCs 4.6.1. Comparison with test results Fig. 12 shows comparisons between the FE results and the experimental results for six of the hybrid DSTC specimens presented in Wong et al. [10], in terms of the axial stress–strain curve. Some simple descriptions of the specimens are also included in Fig. 12, while further specimen details are summarized in Table 1. All the specimens had an outer diameter of 152.5 mm and a height of 305 mm, and were confined by one-ply, two-ply and three-ply FRP tubes respectively. These FRP tubes were reinforced with hoop fibers only and had an elastic modulus of 80.1 GPa in the hoop direction based on a nominal thickness of 0.17 mm per ply. The inner steel tube had an outer diameter of 76 mm, a thickness of 3.3 mm, a yield stress of 352.7 MPa, a tensile strength of 380.4 MPa, and a Young’s modulus of 207.3 GPa. The void ratio φ in Table 1 is defined as the ratio between the inner diameter and the outer diameter of

the annular concrete section. The experimental axial stress of the concrete in a DSTC is defined as the load carried by the annular concrete section divided by its cross-sectional area. The load carried by the concrete section is assumed to be the difference between the load carried by the DSTC specimen and the load carried by the steel tube at the same axial strain. The latter was found from the compression tests of hollow steel tubes. When the axial strain of a DSTC specimen exceeds the buckling strain of the corresponding hollow steel tube, it is assumed that the load resisted by the steel inner tube is equal to Ps which is the ultimate load from the compression tests of hollow steel tubes. Similarly, the axial stress of concrete from the FE results is defined as the predicted load (i.e. the total axial load subtracted by the load taken by the steel tube)carried by the annular concrete section divided by its cross-sectional area. Again, the curves denoted as ‘‘FE results I’’ in Fig. 12 were obtained using Method I defined in Section 3.6 for determining the flow rule for the concrete while those denoted as ‘‘FE results II’’ were obtained using Method II. It is evident from Fig. 12 that the FE models based on both methods of specifying the flow rule for concrete provide reasonably close predictions for all specimens except the one-ply specimens. For the one-ply specimens (Fig. 12(a)), both FE models overestimate the test results by predicting a monotonically ascending

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(a) Average axial stress–axial strain curves.

(b) Corner hoop strain–axial strain curves. Fig. 10. FRP-confined concrete in square specimen II.

(a) Flow rule based on Method I.

(b) Flow rule based on Method II.

Fig. 11. Axial stress distributions over the section.

curve. This overestimation is believed to be due to Teng et al.’s analysis-oriented model [19] which was adopted to produce material parameters for the constitutive model. Teng et al. [19] pointed out that although their model provided accurate predictions for numerous independent tests of FRP-confined concrete, it might overestimate the axial resistance of concrete confined by a weak FRP jacket (such as a one-ply FRP jacket herein). The FE results indicate that for such DSTC sections subjected to axial compression, the steel tube and the concrete are in contact in the beginning as the initial Poisson’s ratio of concrete is smaller than that of steel but later separate from each other as the dilation of concrete increases. The two components come into contact again later if the FRP tube is sufficiently stiff. Such third-stage interaction was predicted by the FE models to occur only for specimens D40-B3. This interaction explains the difference in the predicted stress–strain curve using the two methods for specifying the flow rule for concrete. For specimens D40-B3, the curve predicted using Method II becomes higher in the late stage as the interaction between the concrete and the steel tube was predicted to begin at a smaller axial strain than that predicted using Method I. When using Method I, such interaction was predicted to begin at an axial strain slightly beyond the experimental ultimate strain. Fig. 13 shows comparisons between the FE and the experimental results, in terms of the hoop strain–axial strain curve. The experimental strains were obtained from several strain rosettes located

Table 1 Details of DSTC specimens tested by Wong et al. [10]. Specimen

FRP outer tube thickness

D40-B1-I, II D40-B2-I, II D40-B3-I, II

1 ply

Void diameter (mm) (void ratio φ )

Steel tube diameter Do (thickness t) (mm)

Concrete cylinder strength fco0 (MPa)

76 (0.50)

76 (3.3)

39.6

2 plies 3 plies

at the mid-height of the outer FRP tube. The predictions are seen to be in close agreement with all test results. 4.6.2. Stress variations in the radial direction Fig. 14 shows the predicted axial stress distributions in the radial direction for Specimen D37-B2-I using the two different methods for specifying the flow rule for concrete. It is again obvious that, although a different definition of the flow rule makes only small differences in the predicted axial stress–strain curve and hoop strain–axial strain curve (Figs. 12 and 13), it leads to significant differences in the axial stress distribution over the section. When Method II is used to define the flow rule, the stress variation is more

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(a) Void ratio = 0.5, fco0 = 39.6 MPa, one-ply FRP tube.

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(b) Void ratio = 0.5, fco0 = 39.6 MPa, two-ply FRP tube.

(c) Void ratio = 0.5, fco0 = 39.6 MPa, three-ply FRP tube. Fig. 12. Axial stress–strain curves of concrete in DSTCs.

rapid near the inner edge but slower near the outer edge, compared with the results obtained on using Method I. It is also obvious from Fig. 14 that the axial stress varies significantly in the radial direction. The axial stress reduces with the distance from the outer edge, and the rate of variation is more significant near the inner edge. This could be explained by the fact that the two lateral stresses are more non-uniform near the inner edge, leading to a smaller effective confining stress and a less significant confining effect. The ability to predict stress variations over the section is one of the advantages of a three-dimensional FE model over a one-dimensional analysis-oriented model such as Teng et al.’s model [19]. 5. Conclusions This paper has presented a modified plastic-damage model within the theoretical framework of the Concrete Damaged Plasticity Model (CDPM) in ABAQUS [16] for the modeling of confined concrete. The modifications proposed for the CDPM include a damage parameter, a strain-hardening/softening rule and a flow rule, all of which are confinement-dependent, and a pressuredependent yield criterion. The distinct characteristics of nonuniformly confined concrete are also included in this model by defining an effective confining pressure. Two methods of defining the flow rule are proposed for non-uniformly confined concrete; the first method is simpler while the second method

appears to be more reasonable in that it takes into account the difference in the flow rule between different points over a noncircular section. Finite element models incorporating the proposed CDPM model were developed for concrete in a number of stress states, including active confinement, biaxial compression, FRPconfined circular and square columns, and hybrid DSTCs [11]. FE predictions, obtained using either method for the definition of the flow rule for concrete, have been found to be in close agreement with existing test results in terms of the overall behavior, including the axial stress–strain behavior and the hoop strain–axial strain behavior. The differences due to these two different methods lie mainly in the axial stress distribution. Despite the close agreement between the FE and the test results, it is important to note that while the proposed constitutive model has a rigorous basis for uniformly-confined concrete (confined concrete in solid circular sections), it needs to rely on certain assumptions derived from empirical evidence for confined concrete in non-circular sections, particularly in the specification of the flow rule. Nevertheless, the present CDPM model still represents an important improvement to existing approaches for modeling FRP-confined concrete in non-circular sections which is based on the assumption of either a constant potential function parameter (e.g. [25,26]) or potential function parameters that imply volume compaction of concrete throughout the deformation process

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(a) Void ratio = 0.5, fco0 = 39.6 MPa, one-ply FRP tube.

(b) Void ratio = 0.5, fco0 = 39.6 MPa, two-ply FRP tube.

(c) Void ratio = 0.5, fco0 = 39.6 MPa, three-ply FRP tube. Fig. 13. Hoop strain–axial strain curves of concrete in DSTCs.

(a) Flow rule based on Method I.

(b) Flow rule based on Method II. Fig. 14. Axial stress distributions in the radial direction.

(e.g. [27]). In the review process of the present paper, Karabinis et al. [28] published a study in which they varied the potential function parameter with the FRP jacket stiffness in their FE model. However, their model still suffers from the following

shortcomings: (a) the characteristics of FRP-confined concrete is only partially reflected [1]; (b) the variation of the potential function parameter with the plastic strain is not accounted for; (c) the shear strength ratio adopted is higher than experimental values;

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(d) the hardening/softening rule was derived from the uniaxial stress–strain curve of unconfined concrete, which leads to underestimation of the stress in the confined concrete [1]. The present CDPM model, however, have appropriately addressed all these issues. A thorough discussion of the existing approaches presented in [25–28] is given the companion paper [1]. Clearly, further research is still needed to verify the proposed model more fully and to enhance the theoretical rigor of the model. The proposed model can also be very usefully exploited in FE models to investigate the behavior of confined concrete in various forms of columns to develop a better understanding of structural behavior and to develop design methods for practical use. Acknowledgements The authors are grateful for the financial support from The Hong Kong Polytechnic University (Project Codes: RG7E and BBZH) and the Natural Science Foundation of China (Project No. 50329802).

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