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A constitutive model of concrete confined by steel reinforcements and steel jackets Y.-F. Li, S.-H. Chen, K.-C. Chang, and K.-Y. Liu 279 Abstract: In this paper, a total of 60 concrete cylinders 30 cm

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A constitutive model of concrete confined by steel reinforcements and steel jackets Y.-F. Li, S.-H. Chen, K.-C. Chang, and K.-Y. Liu 279 Abstract: In this paper, a total of 60 concrete cylinders 30 cm in diameter and 60 cm in length confined by steel jackets of different thicknesses and different types of lateral steel reinforcements are tested to obtain the stress strain curves of the cylinders. A constitutive model is proposed to describe the behavior of concrete confined by steel reinforcement, steel jackets, and both steel reinforcement and steel jackets used to retrofit and strengthen reinforced concrete structures. The confined concrete stress strain curve of the proposed model is divided into two regions: the curve in the first region is approximated using a second-order polynomial equation, and that in the second region using an nth-order power-law equation, where n is a function of the unconfined concrete strength and the lateral confining stress. The results of the experiments show that different types of lateral steel reinforcement contribute greatly to the compressive strength of concrete cylinders confined by the reinforcement. Comparing the stress strain curves of the uniaxial test with that from the proposed model, we conclude that the proposed model for concrete confined by a steel jacket and lateral steel reinforcement can predict the experimental results very well. Key words: constitutive model, steel jacket, confined concrete. Résumé : Cet article décrit les essais effectués sur 60 cylindres de béton d une dimension de ϕ30 60 cm frettés par des enveloppes en acier de différentes épaisseurs et par différents types de renforcement latéral en acier afin d obtenir les courbes contrainte-déformation des cylindres de béton. Un modèle constitutif est proposé afin de décrire le comportement du béton fretté par un renforcement d acier, une enveloppe d acier, et un renforcement d acier ajouté à une enveloppe en acier utilisés pour post-adapter et renforcir des structures en béton armé. La courbe contrainte-déformation du béton fretté du modèle proposé est divisée en deux zones. La courbe contrainte-déformation de la première zone est approximée en utilisant une équation polynomiale du deuxième ordre, alors que la courbe contrainte-déformation de la seconde zone est approximée en utilisant une équation de loi exponentielle du n e ordre, où n est une fonction de la résistance du béton non fretté et de la contrainte latérale de confinement. Les résultats de ces expériences indiquent que divers types de renforcement latéral en acier contribuent grandement à la résistance en compression des cylindres de béton fretté par un renforcement latéral en acier. En comparant les courbes contrainte-déformation du test uniaxial avec le modèle proposé, nous pouvons conclure que le modèle proposé pour le béton fretté par une enveloppe en acier et un renforcement latéral en acier peut très bien prédire les résultats expérimentaux. Mots clés : modèle constitutif, enveloppe en acier, béton fretté. [Traduit par la Rédaction] Li et al. 288 Introduction Many strong earthquakes, such as the 1990 Luzon earthquake (Philippines), the 1994 Northridge earthquake (USA), the 1995 Kobe earthquake (Japan), and the 1999 Ji-Ji earthquake (Taiwan), have occurred in regions of high seismicity in the last decade. These earthquakes resulted in sgnificant Received 5 November Revision accepted 16 September Published on the NRC Research Press Web site at on 26 March Y.-F. Li 1 and S.-H. Chen. Department of Civil Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Rd., Taipei, , Taiwan, ROC. K.-C. Chang and K.-Y. Liu. Department of Civil Engineering, National Taiwan University, Taipei, , Taiwan, ROC. Written discussion of this article is welcomed and will be received by the Editor until 30 June Corresponding author ( loss of life and property and caused infrastructure damage. The columns are the most important structural members in a structure, and the strength and ductility of columns significantly influence the seismic capacity of a structure. Therefore, the seismic retrofit of a column has become a very important issue in countries subject to earthquake activity. In the 1990s, the steel jacketing technique was developed and experimentally verified to be effective in enhancing the seismic capacity of columns. Therefore, the steel jacketing technique has been widely applied in practical construction, particularly in Japan, Taiwan, and the state of California in the United States. The steel jacketing technique was originally developed for circular-sectioned bridge columns. Two semicircular steel plates larger than the diameter of the column are formed in the factory. The vertical seams between both half steel shells are welded in situ to become a continuous steel tube with a small annular gap between the bridge column and the steel plate. The gap is filled with pure cement or epoxy matrix to transfer the stress in the bridge column to the steel plates (Priestley et al. 1996). Therefore, concrete confined by a Can. J. Civ. Eng. 32: (2005) doi: /L04-093 280 Can. J. Civ. Eng. Vol. 32, 2005 steel jacket can be seen as that confined by continuous lateral steel reinforcement. Steel jacketing has proven to be effective because none of the bridges retrofitted with a steel jacket suffered damage during the 1994 Northridge earthquake (CALTRANS 1994). In Taiwan, the steel jacketing technique has also become a very popular seismic retrofit technique after the 1999 Ji-Ji earthquake. The steel jacket mounted around the column can increase the compressive strength, shear strength, and ductility of the column. By doing so, the constitutive behavior of the concrete is changed due to the increase in the confinement stress of the concrete (Moehle 1992; Priestley and Seible 1991; Priestley et al. 1996). Therefore, it is necessary to develop a suitable constitutive model for concrete confined by a steel jacket in the structural analysis and also the retrofit design. In this paper, a constitutive model of concrete confined by both steel reinforcement and a steel jacket in the use of retrofitting and strengthening reinforced concrete structures is proposed, and test results are also recorded from 60 concrete cylinders, 30 cm in diameter and 60 cm in length, confined by steel jackets of different thicknesses and different types of lateral steel reinforcement. The stress strain curves of the test results are compared with that of the proposed constitutive model to show that the proposed model is effective. Constitutive models for confined concrete The constitutive model for concrete plays an important role in the analysis and design of concrete structures. The frequently cited models to predict the peak stress or the stress strain curve of confined concrete are introduced in this section. Richart et al. (1928) were the first to study the nominal strength of concrete confined by either hydrostatic pressure or spiral reinforcement. The peak stress formula proposed by Richart et al. was used for a long time and was also cited in many textbooks. Kent and Park (1971) proposed a constitutive model for confined concrete using a parabolic stress strain curve for the ascending branch and a linear stress strain curve for the descending branch. Muguruma et al. (1978) later proposed a model of two second-order parabolic stress strain curves. Park et al. (1982) modified the Kent and Park stress strain model by adding the affect factor of the unconfined concrete strength, the volume ratio, and the yield strength of lateral steel reinforcement. Mander et al. (1988a, 1988b) proposed a parabolic functional expression to represent the stress strain behavior of concrete confined by lateral (hoop) reinforcement. This model was widely used in the last decade to calculate the required thickness of steel plate for the retrofit design of columns (Chai et al. 1991). Saatcioglu and Razvi (1992) proposed a stress strain curve, which was formulated by a parabolic ascending branch followed by a linear descending branch for confined concrete with circular and rectangular hoop types. According to their experimental data, Hoshikuma et al. (1997) proposed a general form of an nth-order polynomial equation for the ascending branch of the stress strain curve. The aforementioned stress strain models were developed for concrete confined by lateral steel reinforcements. The following models are developed for concrete confined by carbon fiber reinforced plastics (CFRP) or glass fiber reinforced plastics (GFRP). Mirmiran and Shahawy (1997), based on test results from 30 concrete cylinders and compared with other confined concrete models, proposed an equation to predict the peak stress of concrete confined by GFRP. Hosotani et al. (1998) and Hosotani and Kawashima (1999) proposed a series of stress strain models for concrete confined by CFRP and by both steel reinforcement and CFRP together. They used the regression analysis of the experimental results and modified the model proposed by Hoshikuma et al. (1997) to extend the application to different confinement materials by adjusting the coefficients. Daudey and Filiatrault (2000) introduced the seismic evaluation and retrofit of reinforced concrete (RC) bridge piers using steel jackets. Karabinis and Rousakis (2001) used the low volume ratio of CFRP to replace the steel reinforcement in confining concrete cylinders and found that the strength and ductility of confined concrete increased. Li et al. (2003), based on the results of tests on 108 concrete cylinders confined by CFRP material, proposed a theoretically based constitutive model for concrete confined by CFRP. The peak strength of this constitutive model is derived from the Mohr Columb failure envelope theory and can be explicitly expressed as a function of the unconfined concrete strength, the lateral confining stress, and the angle of internal friction of concrete. The strain at the peak strength in this model is obtained from the regression analysis of the experimental results. A second-order polynomial equation is used to present the stress strain curve of the model. Li and Fang (2004) then modified their model and extended the application of this model to concrete confined by both steel reinforcement and CFRP. Thirty-six concrete cylinders 30 cm in diameter and 60 cm in length were tested to verify the effectiveness of the proposed model. A review of earlier literature indicates that the existing constitutive models for confined concrete were developed for lateral steel reinforcement and fiber reinforced plastics; they might not be suitable for concrete confined by a steel jacket. Only a few researchers have explored the constitutive model for concrete confined by a steel jacket. If we look into the mechanism of concrete confined by a steel jacket, we see that the concrete inside the steel jacket expands outward and extrudes the steel jacket; in other words, the steel jacket applies confining stress to the concrete under the uniaxial load (Nataraja et al. 1999). The expansion of the steel jacket is therefore caused by the failure of the concrete and then results in the redistribution of the axial stress to the steel jacket. The strength and ductility of concrete will increase due to this triaxial state of stress. Walter et al. (2001) proposed a stress strain behavior of epoxy polymer concrete confined by a steel tube under uniaxial loading. The research results show that epoxy polymer concrete with low Young s modulus and low strength behaves like a fluid when a uniaxial load is applied. The stress strain behavior of the steel tube infilled with low Young s modulus and low-strength epoxy polymer concrete is different from that infilled with normalstrength concrete or high-strength epoxy polymer concrete. Proposed constitutive model As discussed previously, the stress strain curve of concrete confined by steel reinforcement and a steel jacket depends extensively on the lateral confining stress and the core Li et al. 281 concrete strength. It is not easy to use a single equation to represent the entire stress strain curve of concrete confined by a steel jacket. Therefore, we divide the stress strain curve into two regions: region I represents the steep ascending branch of the stress strain curve, and region II the less steep ascending or descending branch of the stress strain curve. Figure 1 illustrates the stress strain curve of concrete confined by a steel jacket under uniaxial loading. In the stress strain coordinate system, the position (ε c1, f c1 )isthe intersection point of regions I and II, and it is evident that the intersection point of the stress strain curve plays an important role when modeling the curve in regions I and II. In the next section, the axial stress and strain at the intersection point and the stress strain curves in regions I and II are discussed in detail. Stress at the intersection point It is known that the increase in strength of confined concrete is a result of the combination of lateral pressure and axial compression, which put the concrete in a triaxial stress state. The lateral pressure is provided by lateral steel reinforcement and a steel jacket. In general, we assume that the strength of confined concrete is related to the contribution of the confinement pressure. Therefore, the strength of confined concrete can be expressed as the sum of the strength of unconfined concrete and the strength increase due to the confining stress: a [1] fc1 = fco + fl where f c1 is the compressive strength of confined concrete at the intersection point, f co is the compressive strength of unconfined concrete, f l is the effective lateral confining stress, and the constitutive parameter a is the coefficient of the power-law equation which can be obtained from experimental data using regression analysis. If the effective lateral confining strength is provided by both lateral steel reinforcement and a steel jacket, eq. [1] can be rewritten as the following equation to express the strength at the intersection point: a [2] fc1 = fco + ( fl1 + fl2 ) where f l1 is the effective lateral confining strength provided by the lateral steel reinforcement, and f l2 is the effective lateral confining strength provided by the steel jacket. In eq. [2], f l1 and f l2 can be represented as follows: [3] fl1 = keρ s fyh /2 [4] fl2 =2kctEsε s / D In eq. [3], k e is the confinement effectiveness coefficient and depends on the type of lateral steel reinforcement. Equations [5] and [6] show the formulas for the coefficient k e with different kinds of circular hoop and spiral steel reinforcement, respectively: [5] k [6] k e e Ae [ 1 ( s / 2d = = s)] A 1 ρ cc 1 s d = ( / 2 s) 1 ρ cc cc 2 (for a circular hoop) (for a circular spiral) Fig. 1. Illustration of the stress strain curve of concrete confined by a steel jacket under uniaxial loading and notations of the proposed model. Also in eq. [3], ρ s is the ratio of the volume of the lateral confining steel reinforcement to the volume of the confined concrete core, and f yh is the yield strength of the transverse reinforcement. In eq. [4], k c is the coefficient of the sectional shape (Priestley et al. 1996), t is the thickness of the steel jacket, E s is the elastic modulus of the steel jacket, ε s is the yield strain of the steel jacket, and D is the diameter of the cylinder. As the steel jacket reaches its yield strain, the steel reinforcement also reaches yielding strain. Therefore, eqs. [3] and [4] are reasonable equations to express the confining stress of the concrete. In eqs. [5] and [6], A e is the area of the effectively confined core concrete, A cc is the area of the core of the column section within center lines of the perimeter spiral, s is the clear spacing between the spiral or hoop bars, d s is the diameter of the spiral, and ρ cc is the ratio of the area of the axial steel to the area of the core of the section. Strain at the intersection point When axial stress reaches the compressive strength f c1, the corresponding strain of the confined concrete is ε c1, and the strain of the intersection point depends on the lateral confining stress, which is provided by lateral steel reinforcement and the steel jacket. The form of eq. [7] was suggested by previous researchers (Balmer 1949; Mander et al. 1998b; Li et al. 2003a), who found that it could produce good predictions of experimental values, shown as follows: ( l1 + l 2 [7] εc1 = εco 1 + b f f ) fco where ε co is the strain of the unconfined concrete, usually set as ε co = 0.002; and the parameter b can be obtained from experimental data. Stress strain relationship in region I Based on previous studies (Muguruma et al. 1978; Li et al. 2003), the ascending branch of most constitutive models of confined concrete is usually formulated by a second-order 282 Can. J. Civ. Eng. Vol. 32, 2005 parabolic equation. In region I, where the strain ε c falls between 0 and ε c1, the stress f c and strain ε c relationship of the confined concrete can be expressed as follows: [8] f = Aε 2 + Bε + C 0 ε ε c c c Three boundary conditions are used to determine the coefficients A, B, and C in eq. [8]. The three boundary conditions are the initial point, the stress continuity, and the firstorder differential continuity conditions at the intersection point, shown as follows: [9] f c =0 atε c =0 [10] f = f at ε = ε c c1 c c1 [11] dfc/ dε = fc1 ( n/ εc1 ) at εc = εc1 where n is the constitutive parameter of the power-law equation, shown in eq. [13]. Substituting the three boundary conditions into eq. [8], the three coefficients can be determined. Equation [8] can be rewritten as follows: [12] fc = fc1 [( n 1)( εc/ εc1 ) + ( 2 n)( εc/ εc1 ) ] where 0 ε ε where f c1 is the stress of the intersection point, and ε c1 is its corresponding strain; these two terms can be calculated from eqs. [2] and [7], respectively. Stress strain curve of region II As seen in Fig. 1, the stress strain curve in region II is an ascending or descending curve with smaller slope than that of the curve in region I, and it is due to the difference in the unconfined concrete strength. We use an nth-order polynomial equation to model the stress strain relationship in region II as follows: n ε [13] fc = f c εc1 c1 1 + εc εc1 εc1 where n is a function of the unconfined concrete strength and lateral confining stress. Experimental program For RC buildings and bridges in Taiwan that need to be seismically retrofitted using steel jackets, the compressive concrete strength is between 7.85 MPa (80 kgf/cm 2 ) and 20.6 MPa (210 kgf/cm 2 ), which is obtained from core and nondestructive inspection of the existing aged RC buildings and bridges. In this paper, 60 concrete cylinders were made with two different strengths. The target concrete compression strengths of 7.85 MPa (80 kgf/cm 2 ) and 20.6 MPa (210 kgf/cm 2 ) at 28 d were used for groups A and B, respectively. Ready-mixed concrete was used, and the average strengths of concrete cylinders at 28 d are 7.06 MPa (72 kgf/cm 2 ) for group A and MPa (203 kgf/cm 2 ) for group B. Table 1 shows the design parameters of the 60 concrete cylinders, such as the strength of unconfined concrete, the type of lateral reinforcement, and the thickness of the steel jacket. Each group contains concrete cylinders with no 2 c c1 c c1 lateral reinforcement, spiral steel reinforcement, circular (hoop) steel reinforcement, and spiral steel wire cable. It should be noted that the lap length of the circular steel reinforcement is 30 cm. Each group also has 2 and 5 mm thick steel jackets. For group A, three concrete cylinders for each design parameter are used for the uniaxial compression test. Since the test results of the three cylinders for each design parameter in group A showed little variation, we d

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