FINITE ELEMENT MODELING OF STEEL BAR BUCKLING IN FRP-CONFINED RC COLUMNS

Transcription

1 FINITE ELEMENT MODELING OF STEEL BAR BUCKLING IN FRP-CONFINED RC COLUMNS Yu-Lei Bai 1 and Jian-Guo Dai 2, * 1. Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China. 2. Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China * cejgdai@polyu.edu.hk. ABSTRACT Fiber-reinforced polymer (FRP) fining jackets offer an attractive solution for the seismic retrofit of reinforced crete (RC) columns. For an accurate prediction of the strength and ductility of FRP-fined RC columns, it is necessary to understand the interaction between the FRP jacket and the RC column at all deformation levels under seismic loading. In particular, when widely-spaced steel stirrups/spirals are used as the transverse steel reinforcement, the longitudinal steel bars are likely to develop buckling deformations, which are however restrained by the FRP-fined crete cover. This paper presents a beam-on-elastic foundation model for simulating the buckling behavior of longitudinal steel reinforcing bars laterally supported by FRP-fined crete using the finite element (FE) approach. In addition, a curved beam approximation is proposed to evaluate the stiffness of the lateral springs that are used to represent the restraining effect offered by the FRP-fined crete cover. The proposed FE model is verified through comparisons between the predicted and the experimental average stress-strain relationships of steel reinforcing bars, the latter of which were obtained from compression tests of FRP-fined RC columns. The proposed FE model provides an effective method for simulating the buckling behavior of laterally supported longitudinal reinforcing bars for a more accurate analysis of the behavior of FRP-fined RC columns. KEYWORDS FRP-fined crete; Bar buckling; FE model; Beam-on-elastic foundation; Curved beam model INTRODUCTION Old reinforced crete (RC) columns, particularly those built prior to the 1970s, often have inadequate transverse steel reinforcement details. As a result, the longitudinal steel reinforcing bars (referred to as steel rebars or rebars for brevity) may buckle at a critical level of compressive strain due to insufficient lateral support. Buckling of rebars can lead to the spalling of crete cover and a significant loss of the load-carrying capacity of RC columns. The ability of external fiber-reinforced polymer (FRP) jackets (with fibers oriented in the hoop direction to provide finement to the column) in enhancing the buckling resistance of steel rebars in FRP-fined RC columns has been proven by previous researchers (e.g., Priestley et al. 1996;Hollaway and Teng 2008; Bournas and Triantafillou 2011). In FRP-fined RC columns, the crete cover is fined by the external FRP jacket and provides much stronger support to the steel rebars, while in ventional RC columns the cover can easily spall when the steel rebars experience significant buckling deformations. Therefore, the buckling of steel rebars in FRP-fined RC columns is generally postponed to a higher strain level due to FRP-finement, which, however, may not completely eliminate the possibility of rebar buckling (Tastani et al. 2006; Bournas and Triantafillou 2011), particularly when the column section is non-circular and FRP finement is not so effective. If buckling of rebars does occur, the growth of inelastic buckling deformation of rebars may lead to additional strains in the FRP jacket, causing its premature rupture (Tastani and Pantazopoulou 2004; Pellegrino and Modena 2010; Rousakis and Karabinis 2012, Bai et al. 2015). This interaction between the steel rebars and the external FRP jacket through the crete cover is an important mechanism governing the behavior of FRP-fined RC columns. For the accurate prediction of strength and post-peak behavior of FRP-fined RC columns under seismic loading, it is important to understand the above-mentioned interaction mechanism between steel rebars and FRP-fined crete at all deformation levels. Although a large amount of research has been ducted on the strength and ductility of FRP-fined crete (e.g., Lam and Teng 2003; Dai et al. 2011), there has been rather limited work on the effect of FRP finement on the buckling of steel rebars (e.g., Tastani and Pantazopoulou 2004; Pellegrino and Modena 2010; Megalooikonomou et al. 2012); this is trast with the situation cerning 466

2 ventional RC columns, for which steel rebar buckling has received extensive research attention (e.g., Priestley et al. 1996). Against the above background, this paper presents a finite element (FE) method for the full-range compressive stress-strain behavior of steel rebars in FRP-fined circular RC columns. The interaction between the steel rebars and the FRP jacket is represented using a beam-on-elastic-foundation model, in which the stiffness of the Winkler springs, which are used to represent the lateral support offered by the FRP-fined crete cover, is derived from a curved-beam approximation of the FRP-fined crete cover layer. DESCRIPTION OF THE MODEL CONCEPT When an RC member is subjected to flexure, the cover crete in the compression zone may spall at the ultimate state, leading to a sudden loss of the load-carrying capacity (e.g., Dhakal and Maekawa 2002). In FRP-fined RC columns, the cover crete is kept in position until the rupture of the external FRP jacket due to the finement in the hoop direction; this rupture usually signifies the ultimate state of the column. A few researchers (e.g., Tastani and Pantazopoulou 2004; Tastani et al. 2006; Bournas and Triantafillou 2011) have paid special attention to the dilation behavior of the core crete and the cover crete which is accompanied by the buckling of steel rebars before the rupture of FRP jacket. It can be expected as the axial strain increases, the steel rebars expand together with the core crete, but this propensity to move away from the column center is restrained by the crete cover layer which is in turn supported by the FRP jacket. During this process, the FRP jacket provides increasing fining pressures through the cover crete to restrain the steel rebars from buckling. Figs.1a and 1b illustrate a beam-on-elastic foundation model to describe this interaction mechanism: the steel rebar functions as a beam supported by a set of Winkler springs representing the lateral support from the FRP-fined crete cover layer. The stiffness of the Winkler springs can be determined based on a curved-beam approximation as described in the next section. This idealized curved beam is composed of two materials: the fined crete and the FRP jacket. It is important to realize that the FRP jacket has two important functions in the curved-beam approximation: as tensile reinforcement for the curved beam and as the fining device to modify the properties of the crete. Reinforcing bar Transverse stirrup Cover crete (FRP-fined crete) Concrete h c Transverse stirrup (a) Physical model (b) Simplification of the interaction mechanism Fig. 1 Description of model cept EVALUATION OF THE SPRING STIFFNES: CURVED BEAM APPOXIMATION A simple curved beam model is proposed to estimate the stiffness of the springs between the longitudinal steel reinforcing bar and the FRP jacket. In the model, the circular column section can be divided into several equivalent domains according to the number of longitudinal reinforcing bars due to the symmetry (Fig. 2a). If the middle point of the curved beam is subjected to a centric load P, the stiffness of the spring (i.e., the cover crete) can be obtained using K=P/Δ, where Δ is the resultant displacement over there, which can be solved using the flexibility (force) method and the principle of virtual work in structural mechanics (e.g., Laible 1985; Leet 1988). [please refer to Bai (2014) for details]. The boundary ditions of the two ends of the curved beam can be defined as fixed due to the symmetry even though the cover crete and longitudinal reinforcing bars move outward simultaneously due to the dilation of crete with the axial shortening of FRP-fined RC column. This is an approximation by which only the relative displacement between the longitudinal steel reinforcing bars and FRP-fined crete is sidered. The simultaneous lateral movement caused by the 467

3 dilation of core crete has a marginal effect on the relative movement. The angle φ formed by the two ends of the curved beam (Fig. 2b) can be determined using 2π/n, where n is the number of reinforcing bars in the whole column section. The radius of the curved beam r has the value of (R-c+h c), where R is the radius of the circular column, c is the thickness of cover crete and h c is the height of neutral axis of the curved beam (Fig. 2b). c R φ r neutral axis (a) Column section (b) Curved beam model SECTION PROPERTIES OF THE CURVED BEAM Fig. 2 Curved beam model As mentioned before, the curved beam is a beam of two materials: FRP-fined cover crete and FRP jacket. It is well known that the compressive stress-strain curve of FRP-fined crete with effective finement stiffness can be described as a combination of two significant portions: a parabolic portion first and a linear ascending portion afterwards. The slope of the linear ascending portion E 2 can be determined by the following equation (Lam and Teng 2003): ' ' fcc fco E2 (1) where f cc is the compressive strength of fined crete, f co is the compressive strength of unfined crete, and ε cu is the ultimate strain of fined crete (Lam and Teng 2003). During the loading process, FRP-fined crete enters into the sed linear portion of the compressive stress-strain relationship once the strength of unfined crete is reached, after which the FRP finement effect is activated. Since the buckling of steel reinforcement is unlikely occur before the strength of unfined crete is reached, for the buckling analysis of reinforcing bars FRP-fined crete can be assumed to lie in the sed linear portion of the compressive stress-strain curves. Thus, for the buckling analysis of reinforcing bars in FRP-fined crete, the equivalent elastic modulus of the FRP-fined cover crete E is assumed for simplicity as: E E (2) 2 This is an approximation by which the FRP-fined cover crete is regarded as elastic isotropic material, as E 2 is originally defined for the axial direction. The application of E 2 for the transverse direction of the fined crete cover layer may need further justifications when more test data are available. For a composite section the position of the neutral axis is determined as follows (Kaw 1997; Altenbach et al. 2004; Daniel and Ishai 2006; Chen et al. 2009), A h1 Aequ h2 hc A A equ cu t wh t w h t wt w 1 frp equ 2 frp equ t wh t E / E wh t wt E / E w 1 frp frp 2 frp frp where w is the width of the curved beam; t frp and t are the thickness of FRP jacket and cover crete, respectively (Fig. 2b); E frp is the elastic modulus of FRP; h 1 and h 2 are the height of the centroid of the crete area and the FRP area, respectively. It should be noted that the FRP area is transformed to an equivalent area with the same elastic modulus of crete. Thus, the centroid of the composite section can be calculated based on an inverse T section (Fig. 2b). Once the neutral axis of the composite section is determined, the section properties of the composite beam can be determined as follow: (3) 468

4 E I E wt wt h t 3 2 CB CB ( /12 ( c /2) ) E wt wt t t h 3 2 frp ( frp /12 frp ( frp /2 c) ) E A E wt E wt CB CB frp frp G A G wt G wt CB CB frp frp (4) where G and G frp are the shear moduli of the FRP-fined crete and the FRP jacket, respectively. BEAM-ON-ELASTIC FOUNDATION MODEL Once the stiffness of the springs is known, the compressive behaviour of longitudinal reinforcing bars in an FRP-fined RC column can be simulated using the Beam-on-elastic foundation model as shown in Fig. 3a. The role of springs is to provide lateral support to the longitudinal reinforcing bars, whose behaviour is simulated using the FE approach based on the software ABAQUS (2008). In the FE model, the reinforcing bar is modelled as an assemblage of 2-node Timoshenko beam elements (B21 element) with appropriate cross-section integration. The elastic springs are modelled using an elastic spring element (i.e. the Spring2 element) with one end nected to the adjacent beam element node and the other end fixed. For the two end nodes of the reinforcing bar, both rotational and translational degrees of freedom are fixed, except for the axial direction. To initiate lateral deformation, a tiny initial imperfection is imposed by cosine-shaped offset satisfying boundary ditions with a mid-length amplitude, which is in the range of 0.001~0.005% of the length of the longitudinal reinforcing bar model (Zong et al. 2013). The axial compression is applied on the steel bar by imposing displacements of equal magnitude but opposite direction at the two ends. Fig.3b shows the deformation shape of the longitudinal reinforcing bar from the FE analysis. K h K (a) FE model (b) Deformed shape Fig. 3 Beam-on-elastic foundation model Fig. 4 Convergence study The stress and strain values obtained from the FE analyses represented the average response. The average stress was calculated by dividing the axial force by the cross section area of the longitudinal reinforcing bar and the average strain was measured by dividing the axial displacement of one end node by the half length of the model due to the symmetry of the load and geometry. A vergence study was also ducted to investigate the effect of element size on the global compressive behaviour of the longitudinal reinforcing bar. Beam elements with the lengths of 10 mm, 5 mm and 2 mm were used in the vergence study and the analytical stress-strain curves are plotted in Fig. 5. It can be seen that when the element length is shorter than 5 mm, the stress-strain curves reached vergence. Therefore an element size of 5 mm was used in all subsequent analyses. FE PREDICTIONS VS. TEST RESULTS In this paper, three cases of stress-strain curves of reinforcing bars in FRP-fined RC columns as reported in Bai (2014) were used here to verify the proposed FE model. Details of the experiment can be referred to Bai (2014). The spring stiffness values for the three cases calculated based on the flexibility method and the principle of virtual work were 39.4 N/mm, N/mm, N/mm per unit height, respectively. Figures 5a~5c show the comparisons of FE predictions and test results of the compressive stress-strain curves of longitudinal 469

5 reinforcing bars in FRP-fined RC columns. The compressive stress-strain curves of the reinforcing bars without buckling effects are also presented for reference. In general, the FE model predictions agree well with the test results. The analytical stress-strain response follows exactly the same path with the reference curve until a certain strain value, where the buckling initiates. Afterwards, the compressive stress decreases gradually with the increase of the strain. The main differences between the FE predictions and the test results are their yield stresses: in the experimentally observed compressive stress-strain curves, the longitudinal reinforcing bars in FRP-fined RC columns in the tests yielded before the actual material yield strength of steel, while the stress-strain curves obtained from the FE analyses exhibited the same yield point as the actual one. This is mainly because the compressive stress-strain relationships obtained from the tests reflected the average response of the longitudinal reinforcing bars over the whole measured height (i.e., 250 mm). The observed premature yielding of reinforcing bars was due to the strain localization mechanism. In other words, the yielding of reinforcing bars always occurs at a local position rather than the whole height in the tests, leading to a lower average yield stress of the reinforcing bar when the yielding occurred locally. Figures 5b and 5c show that the FE predictions tend to underestimate the compressive stress of longitudinal reinforcing bars, particularly when the axial deformation becomes large (Fig. 5c). This was mainly because that the experimental stress-strain curves of reinforcing bars were obtained from the compressive tests on FRP-fined RC columns and represent the average response of four longitudinal reinforcing bars in the columns. In reality, it was less possible that four reinforcing bars buckled simultaneously and then proceeded with the same rate. Usually, one or two bars buckled relatively earlier than others (Bai 2014). As a sequence, the buckling response of the reinforcing bars might be postponed in terms of the average stress-strain relationships. It should be also noted that the interaction between the longitudinal reinforcing bars and the external FRP jacket in FRP-fined RC columns is very complicated. The assumptions adopted in the present study, such as the use of equivalent elastic modulus of FRP-fined crete (e.g., Eq. 2) and the linear characteristics of the lateral spring, need further experimental verifications. Fig. 5 also presents FE predicted compressive stress-strain response of a bare bar (i.e., without lateral support, K = 0) for comparison purposes. It is very clear that the existence of lateral support influences significantly the compressive stress-strain response of longitudinal reinforcing bars Test: CFRP-RC-1-m-a Test: CFRP-RC-1-m-b Compressive stress-strain curve without buckling FE analysis ( laterally supported reinforcing bar ) FE analysis ( bare bar ) Test: PET-RC-1-m-a Test: PET-RC-1-m-b Compressive stress-strain curve without buckling FE analysis ( laterally supported reinforcing bar ) FE analysis ( bare bar ) Stress ( MPa ) FRP rupture, test stopped here Stress ( MPa ) Strain (a) Reinforcing bars in RC columnsfined with one ply of CFRP sheet Strain (b) Reinforcing bars in RC columns fined with one ply of PET FRP sheet 470

6 1000 Test: PET-RC-2-m-a Test: PET-RC-2-m-b Compressive stress-strain curve without buckling FE analysis ( laterally supported reinforcing bar ) FE analysis ( bare bar ) Stress (MPa) CONCLUDING REMARKS (c) Reinforcing bars in RC columns fined with two plies of PET FRP sheets Fig. 5 Stress-strain responses: FE predictions vs. test results This paper has presented an FE model to simulate the compressive stress-strain response of longitudinal reinforcing bars in FRP-fined RC columns with the incorporation of a Beam-on-elastic foundation model. In the model, the longitudinal reinforcing bars is simulated as an assembly of beam elements and the finement effect provided by external FRP jackets is modelled as a series of elastic Wrinkle springs to provide lateral support. A simple curved beam model has been proposed to approximate the stiffness of the springs through the flexibility method and the principle of virtual work. The curved beam is treated as a composite beam accounting for the tributions from both the external FRP jacket and the FRP-fined cover crete. Once the spring stiffness is determined the FE model with the employment of the beam-on-elastic foundation cept can be used to quantify the compressive stress-strain response of laterally supported longitudinal reinforcing bars. Comparisons between the FE predictions and test results have validated the suitability of using the beam-on-elastic foundation model for simulating the compressive stress-strain response (i.e., progressive buckling.) of reinforcing bars in FRP-fined RC columns. ACKNOWLEDGEMENTS The authors are grateful for the financial support received from the Hong Kong General Research Fund (GRF) (Project Code: ), the National Natural Science Fund of China (Grants No ), and The Hong Kong Polytechnic University through a research project (Project code: 4-ZZCH). The work reported in this paper was co-supervised by Prof. J.G. Teng, whose significant tributions to this work are gratefully acknowledged. REFERENCES Strain ABAQUS. (2008). ABAQUS standard user s manual, Volumes I-III, Version 6.8. Hibbitt, Karlsson & Sorensen, Inc., Pawtucket, America. Altenbach, H., Altenbach, J., and Kissing, W. (2004). Mechanics of Composite Structural Elements. Berlin; New York: Springer-Verlag. Bournas, D. A., Lontou, P., Papanicolaou, C. G. and Triantafillou, T. C. (2007). Textile-reinforced mortar versus fiber-reinforced polymer finement in reinforced crete columns. ACI Structural Journal, 104(6), Bournas, D. A. and Triantafillou, T. C. (2011). Bar buckling in RC columns fined with composite materials. Journal of Composites for Construction, ASCE, 15(3), Bai, Y. L. (2014). Behavior of modeling of RC columns fined with large-rupture-strain FRP composites. Ph. D thesis, The Hong Kong Polytechnic University. Chen, Z.H. and Hu X.M. (2009). Design of Steel-Concrete Composite Structures. Beijing, China Architecture & Building Press. (in Chinese). Daniel, I.M., and Ishai, O. (2006). Engineering Mechanics of Composite Materials (2nd ed.). New York:Oxford University Press. Dhakal, R.P. and Maekawa, K. (2002). Reinforcement stability and fracture of cover crete in reinforced 471

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