Finite element modelling of bond–slip at anchorages of reinforced concrete members subjected to bending
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Abstract
The bond mechanism plays a decisive role when anchorage failure in a reinforced concrete (RC) member occurs. A refined model of bond–slip is therefore needed to analyse RC members by analytical or finite element (FE) modelling where the anchorage of reinforcement is critical, such as at a lapped-splice or in any situation where the strength of a bar is required to be developed to achieve its ultimate strength. This paper presents results of FE modelling of some experimental RC specimens with anchorage of deformed steel reinforcing bars in tension, including the cases of end development and lapped splices of bars in specimens subjected to bending. The FE modelling of the anchorage length specimens with bond–slip laws calibrated from experimental results was better able to predict the load–deflection behavior observed during experimental tests. A comparison of the results obtained using the calibrated bond–slip laws with those using conventional Fédération Internationale du Béton bond–slip laws is made and discussion is provided on the effects of the anchorage lengths and bar diameters on the bond–slip relationships.
Keywords
Bond–slip Reinforced concrete Anchorage length Finite element modelling1 Introduction and background
The assumption of perfect bond between the concrete and reinforcing bar is often a sufficient assumption when modelling the ultimate behavior of a large reinforced concrete (RC) structure, but it is not of course appropriate when anchorage failure occurs. The bond mechanism affects the collapse load behavior of a structural member or parts of a structural member where anchorage of reinforcement is critical. The bond stress–slip law specified in the widely accepted Fédération Internationale du Béton (FIB) model code [1] is principally based on results of pull-out tests on reinforcement anchored in concrete for a very short length. However, the conditions favorable for the development of somewhat uniform bond stress in a short anchorage length of a RC test specimen are not representative of the conditions of bond stress in the longer anchorage lengths of practical RC members. Local bond stress–slip relationship may become highly complex due to bond deterioration with splitting cracks in RC members [2]. Highly fluctuating bond stress conditions may arise within the anchorage lengths of bars in RC flexural members with the presence of cracks crossing the anchorage length [3].
Significant research efforts have been put into more realistic assessment and modelling of bond–slip in relatively long anchorage lengths in RC test specimens with conditions resembling those in practical RC members such as those in RC beam-column joints (e.g., [4, 5]) or in RC members with predominant flexural conditions (e.g., [3, 6, 7, 8, 9, 10]) with monotonic as well as cyclic load applications. Analytical modelling of bond–slip laws was undertaken by Mazumder [11] to assess the bond stress–slip relations along critical anchorage lengths of deformed steel bars in tension in some full-scale RC specimens subjected to bending. The analytical modelling results was compared with the test results [3, 7, 8] from experimental RC beam and slab specimens, referred to herein as anchorage length specimens, which included typical cases of end development (with development length l_{d}) and lapped splices (of length l_{s}) of deformed bars in tension. The anchorage length specimens were experimentally tested by the authors to determine average ultimate bond stress (τ_{avg,exp}) calculated from the measured maximum load (P_{max}) at anchorage failure. The analytical modelling procedure proposed by Yankelevsky [12] was adapted by Mazumder [11] with a modified solution procedure to determine peak local bond stress (τ_{y}) and other constituent parameters of a representative bond–slip law for each of the selected anchorage length specimens. It was found that calibration of a conventional bond–slip model was required to reproduce bond zone conditions and local bond stresses within the anchorage of a selected specimen so that the calculated average ultimate bond stress (τ_{avg,cal}) from the analytical model is in good agreement with that (τ_{avg,exp}) observed during the experiment of the specimen.
The load–deflection behavior of the RC test specimens was also observed during the experiment up to the point of anchorage failure. This paper mainly aims at presenting results of finite element (FE) modelling [11] of some selected anchorage length specimens where the calibrated bond–slip laws were shown to simulate satisfactorily the collapse load behavior of the anchorage length specimens. A comparison of the results obtained using the calibrated bond–slip laws with those using the conventional FIB bond–slip law is also made in this paper and discussion is provided on the effects of the anchorage lengths and bar diameters on the bond–slip relationships.
2 Analytical modelling of bond–slip relations in RC members
Parameters for defining mean bond stress–slip relationships of deformed bars (according to [1])
Parameters (1) | Pull-out | Splitting | ||||
---|---|---|---|---|---|---|
ε_{s} < ε_{s,y} | ε_{s} < ε_{s,y} | |||||
Good bond condition (2) | All other bond condition (3) | Good bond condition | All other bond conditions | |||
Unconfined (4) | Stirrups (5) | Unconfined (6) | Stirrups (7) | |||
τ_{y} | 2.5√f′_{c} | 1.25√f′_{c} | 7.0 (f′_{c}/20)^{0.25} | 8.0 (f′_{c}/20)^{0.25} | 7.0 (f′_{c}/20)^{0.25} | 8.0 (f′_{c}/20)^{0.25} |
s_{1} | 1.0 mm | 1.8 mm | s(τ_{y}) | s(τ_{y}) | s(τ_{y}) | s(τ_{y}) |
s_{2} | 2.0 mm | 3.6 mm | s_{1} | s_{1} | s_{1} | s_{1} |
s_{3} | c_{clear}^{a} | c_{clear}^{a} | 1.2s_{1} | 0.5c_{clear}^{a} | 1.2s_{1} | 0.5c_{clear}^{a} |
α | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 | 0.4 |
τ_{f} | 0.40τ_{y} | 0.40τ_{y} | 0 | 0.4τ_{y} | 0 | 0.40τ_{y} |
2.1 Analytical modelling and solutions for calibrating bond–slip of anchorage length specimens
Calibrated bond–slip laws for selected specimens and results of analytical modelling
Specimens (bond–slip laws) (1) | Constitutive parameters of bond–slip laws (confined/unconfined, good bond condition) ε_{s} < ε_{s,y} (2) | Bond zone length (x_{y}, x_{2}) (mm) (3) | Average bond stress within anchorage τ_{avg} (MPa) (4) | Structural and material parameters (5) | |||||
---|---|---|---|---|---|---|---|---|---|
τ_{y} (MPa) | s_{1} (mm) | s_{2} (mm) | s_{3} (mm) | α | τ_{f} (MPa) | ||||
DL-1, l_{d} = 10d_{b} (calibrated) | 8.80 | 0.25 | 0.25 | 3 | 0.4 | 0 | x_{y} = 119.7 | τ_{avg,exp} = 7.69 τ_{avg,cal} = 7.72 | d_{b} = 16 mm; c = 25 mm; c_{max} = 60 mm; f_{c} = 38.5 MPa; f_{sy} = 546 MPa |
DL-3, l_{d} = 20d_{b} (calibrated) | 6.20 | 0.8 | 0.8 | 10 (≈ r_{b})^{a} | 0.4 | 0 | x_{y} = 145.1 | τ_{avg,exp} = 5.98 τ_{avg,cal} = 5.98 | |
DL-1 and 3 (FIB bond–slip law) | 10.19 | 0.35 | 0.35 | 0.42 | 0.4 | 0 | x_{y} = 107.2 (DL-1) | τ_{avg,exp} = 7.69 τ_{avg,cal} = 7.51 | |
DL-6, l_{d} = 10d_{b} (calibrated) | 12.80 | 0.40 | 0.40 | 4 | 0.4 | 0 | x_{y} = 80.9 | τ_{avg,exp} = 11.92 τ_{avg,cal} = 11.90 | d_{b} = 12 mm; c = 25 mm; c_{max} = 60 mm; fʹ_{c} = 38.5 MPa; f_{sy} = 561 MPa |
DL-7, l_{d} = 15d_{b} (calibrated) | 10.40 | 0.6 | 0.6 | 6 | 0.4 | 0 | x_{y} = 132.3 | τ_{avg,exp} = 9.41 τ_{avg,cal} = 9.35 | |
DL-8, l_{d} = 20d_{b} (calibrated) | 9.00 | 0.7 | 0.7 | 7 (≈ r_{b})^{a} | 0.4 | 0 | x_{y} = 234.8 | τ_{avg,exp} = 6.72 τ_{avg,cal} = 6.72 | |
DL-6 to 8 (FIB bond–slip law) | 11.86 | 0.51 | 0.51 | 0.61 | 0.4 | 0 | No real solution | – | |
BL-2, d_{b} = 20 mm, l_{s} = 400 mm; unconfined (calibrated) | 4.85 | 0.5 | 0.5 | 1 | 0.4 | 0 | x_{y} = 100 | τ_{avg,exp} = 3.87 τ_{avg,cal} = 3.84 | c = 25 mm; c_{max} = 70 mm; A_{tr}/s = 1.57; K_{tr} = 3.9%; f′_{c} = 43.0 MPa (for BL-2 and BL-5) and 36.1 MPa for BL-9; f_{sy} = 534 MPa |
BL-5, d_{b} = 20 mm, l_{s} = 300 mm; confined (calibrated) | 7.00 | 0.5 | 0.6 | 0.8 | 0.4 | 2.0 | x_{y} = 95.9 x_{2} = 206.2 (3 bond zones) | τ_{avg,exp} = 4.51 τ_{avg,cal} = 4.50 | |
BL-9, d_{b} = 20 mm, l_{s} = 400 mm; confined (calibrated) | 5.35 | 0.5 | 0.5 | 5.0 | 0.4 | 0 | x_{y} = 220.5 | τ_{avg,exp} = 4.89 τ_{avg,cal} = 4.89 | |
BL-2, 5, 9 (FIB bond–slip law) | τ_{y(split 1,2)} are 4.84 MPa for BL-2, 7.03 MPa for BL-5 and 5.34 MPa for BL-9 |
The experimentally measured end slip of the debonded bars at P_{max} was used to decide reasonable initial values of the slips s_{1} or s_{2} in Fig. 2 and the post-peak end slip of the bars at 0.4P_{max} gave an indication of the spread of the tail end (s_{3}) of a bond–slip law chosen for the analytical modelling procedure. Since more than one set of real solutions of unknowns for a specimen could be determined for different bond–slip laws, further analyses were performed using them to calculate the local bond stress at locations of the ribs of the reinforcement within the anchorage length (l_{d} or l_{s}) of each specimen. The average of the calculated local bond stresses (τ_{avg,cal}) at P_{max} was compared for consistency with the experimentally determined average ultimate bond stress (τ_{avg,exp}) for that specimen.
The experimental anchorage length slab specimens with end development of bars had dimensions of 2000 mm × 600 mm × 200 mm and the beam specimens with lapped splice bars (BL-2, 5, 9) had dimensions 2.3 m long, 250 mm wide and 300 mm high. The anchorage lengths of the different specimens are given in column 1 of Table 2 and structural and material parameters of the specimens are provided in column 5 of the same table. It may be observed that the FIB bond–slip laws for anchorage length specimens (e.g., for DL-1 and DL-3 in Table 2) with identical structural and material parameters for a bar are the same regardless of the variations of the anchorage lengths. The calibrated bond–slip laws for the various specimens, however, are found to vary significantly. While the FIB bond–slip laws showed minor variations between the anchorage specimens with different bar sizes (e.g., for d_{b} = 16 mm and d_{b} = 12 mm), the calibrated bond–slip laws are found to be significantly different. This indicates that variations of anchorage lengths and bar diameters significantly affect the bond–slip laws for anchored bars in practical RC members.
3 Finite element modelling of the anchorage length specimens
The efficacy of a FIB bond–slip law and of a calibrated bond–slip law for a specimen was further examined using FE modelling [1] to simulate the behavior of the laboratory test specimens, including load–deformation characteristics, failure load, crack development and progression. Three dimensional (3D) FE modelling was undertaken in order to compare the effects of using the analytically calibrated bond–slip laws and the specified FIB bond–slip laws for modelling the RC specimens. The FE modelling software Atena-v.4.2.7 [20] was used. The effects of varying the bond–slip laws and other associated model parameters on the load versus vertical deflection (Δ) behavior of some selected anchorage length specimens were assessed. The observed load–deflection behavior of an anchorage length specimen was simulated by implementing chosen bond–slip laws in a 3D FE model of each of the selected test specimens.
3.1 Materials and methods for FE modelling of anchorage length specimens
The dimensions, geometry, material properties, constitutive material models and the loading arrangement in the FE model of an anchorage length specimen were implemented in compliance with those of the practical test specimen. The material model chosen for the concrete was a fracture-plastic constitutive model [20]. The concrete material model combines constitutive models for (tensile) fracturing and compressive (plastic) behavior. The fracture model for concrete employs Rankine failure criterion and exponential softening and it can be implemented with either a rotating or a fixed crack model. The input material properties for the concrete and reinforcement of the model were the same as measured at the time of testing the specimens (column 5 of Table 2). The fracture energy of concrete (G_{F}) was estimated according to the equation given by Vos [21]. The loading and support conditions in the laboratory tests were simulated in the FE analyses.
Reinforcement bars were embedded within concrete macro-element by inputting its geometric locations in the model in global coordinates. The pre-processing routine of the Atena program decomposed reinforcing bars into individual truss elements embedded into solid elements of concrete. The steel stress–strain properties of the embedded truss elements are typically linked to concrete elements through fictitious interface of bond-link elements [22, 23, 24, 25]. The interface is automatically introduced on the material level in most FE modelling programs or packages. The 3D FE modelling system of Atena essentially includes three types of finite elements: concrete continuum elements (3D), bar truss elements (constant strain) and automatically introduced bond-link elements (constant slip). The nodal displacement formulation for the embedded bar elements in Atena is extended to include a bond element with its new degree of freedom s representing bond slip, which is the difference between concrete and bar displacements on the element boundary. The slip degree of freedom (s) is accordingly introduced into the expression for stress evaluation of a bar element. During the FE model simulation with Atena, the inner iterative process for reinforcement continues until achieving acceptably low out-of-balance forces that are searched according to the slip of the linked bond element [22].
Load was applied by imposing small displacements. The analytically calibrated bond–slip law and the FIB bond–slip law were used as input parameters for the bond in two separate FE models of the specimens while keeping other model parameters the same. The load–deflection curves obtained from the FE models were compared with the respective load–deflection response that was observed in the laboratory.
4 Results and discussion
Since the FIB bond–slip laws are commonly used for modelling RC members where anchorage of reinforcement is critical, the discussion on analytical and FE modelling results is made using FIB bond–slip laws as a reference model. The qualitative and quantitative differences between the bond–slip law specified in the FIB model code or specifications [1] and the bond–slip laws analytically calibrated to accurately model the behavior of the anchorage conditions tested in the laboratory are discussed. However, the discussion in this section is principally focused on the FE model outputs and demonstrates the significant improvement obtained by using the calibrated bond–slip laws in the FE modelling.
4.1 Limitations of the conventional analytical bond–slip laws
The limitation of the conventional bond–slip laws for anchorage in RC members is that the effect of concrete deformation on bond stresses is neglected, as is commonly assumed when the steel strain is less than the yield strain, ε_{s} < ε_{s,y}. However, the concrete deformation is expected to vary depending on the variations of anchorage lengths or bar diameters. If there were no influence of concrete deformation on bond stresses, the average ultimate bond stresses would be the same for identical specimens (where the same materials are used in the same structural geometry) regardless of the variation of anchorage lengths. The bond–slip laws according to the FIB design model for the identical anchorage length specimens (e.g. for DL-1 and DL-3, in Table 2) are, in fact, the same regardless of the variations of the anchorage lengths or bar diameters whereas the constituent parameters of the calibrated bond–slip laws are found to be different for variable anchorage lengths or bar diameters in otherwise identical specimens. Therefore, the required changes of the constitutive parameters of the bond–slip laws for each specimen indirectly accounts for the effect of concrete deformation on bond stresses. Table 2 also shows that the constituent parameters of some of the calibrated bond–slip laws are significantly different from the FIB bond–slip laws for the respective specimens (e.g. the parameters for DL-3, DL-8 and DL-10).
4.2 Results of FE modelling of the anchorage length specimens
The comparison of the FE modelling results of load–deflection responses of specimens DL-3 and BL-2 shows that significant improvement is obtained by using the calibrated bond–slip laws compared to the FIB laws. Being reasonably representative of the true slips under different magnitudes of local bond stresses along the anchorage, the calibrated bond–slip models resulted in good agreement with the observed vertical deflection behavior of the specimens under loads. It was observed during the practical experiments that the splitting induced pull-out type bond failures progressed slowly for the specimens with the longer anchorages in contrast to the sudden splitting type bond failures that were typical of the specimens with the short anchorage lengths.
5 Conclusions
Analytical modelling of bond–slip laws was undertaken to calibrate local bond stress–slip relations along the anchorage lengths of deformed steel reinforcing bars in tension, including selected cases of end development and lapped splices of bars in some full-scale RC specimens subjected to bending. A viable analytical modelling procedure was used assuming linear approximation of a bond–slip law using constituent parameters reasonably based on experimental observations of each selected anchorage length specimen. It was found that recalibration of a conventional FIB bond–slip model was required to reproduce bond zone conditions and local bond stresses within the anchorage length of a specimen so that the calculated average ultimate bond stress from the analytical model solution is in good agreement with that observed during the laboratory testing.
The calibrated bond–slip laws and the FIB bond–slip relations were implemented in 3D FE models of selected anchorage length specimens. A comparison of the FE modelling results obtained using the calibrated bond–slip laws with those using the FIB bond–slip law is made in this paper. The FE modelling of the anchorage specimens with calibrated bond–slip laws was better able to predict the vertical deflection of a specimen at maximum load. The calibrated bond–slip laws for the anchorage specimens are also found to be significantly different from the FIB bond–slip laws for those specimens. Therefore, conventional bond–slip models need to be calibrated to account for the effects of the anchorage lengths and bar diameters on the bond–slip relationships.
Notes
Acknowledgements
The experimental work of the research project (Ph.D.) was undertaken with the financial support of the Australian Research Council (ARC) through an ARC Discovery grant to the second author. FE modelling and analyses with Atena software were carried out at the computer laboratory of the School of Civil and Environmental Engineering of UNSW, Sydney, Australia. These supports are gratefully acknowledged.
Funding
The experimental work of the research project was undertaken with the financial support of the Australian Research Council (ARC) through an ARC Discovery grant to the second author. The ARC Discovery (DP1096560) grant was given for the project entitled ‘Anchorage of reinforcement in concrete structures subjected to loading and environmental extremes’ for 3 years duration (2010–2012) of the project.
Compliance with ethical standards
Conflict of interest
On behalf of the authors, the corresponding author states that there is no conflict of interest.
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