Numerical modelling of RC strengthened columns under biaxial loading
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Abstract
During an earthquake, the reinforced concrete (RC) structures are subjected to deformations that may lead their structural elements to exceed the corresponding resistance limit state, forcing them to have nonlinear responses. The application of realistic numerical models that can represent the nonlinearity of each structural element requires full examination and calibration. Furthermore, simplified numerical approaches that can represent the seismic behaviour of original and strengthened RC elements are of full importance. For this, the experimental tests are useful to calibrate the numerical models, and thus to capture as well as possible the real response of the elements. The main goal of this work is to evaluate the efficiency of a simplified numerical approach to represent strengthened RC columns with steel and CFRP jacketing, subjected to biaxial horizontal loading. The numerical modelling efficiency will be evaluated by comparing the numerical results with the experimental ones in terms of sheardrift hysteretic behaviour, initial stiffness and stiffness degradation, maximum strength and energy dissipation. The results shows a good performance of the numerical models, mainly for the RC columns strengthened with CFRP jacketing technique.
Keywords
Columns Biaxial loading Numerical modelling Strengthening techniquesIntroduction
The available data regarding to RC columns subjected to biaxial cyclic bending and constant or variable axial load allows us to recognize that the biaxial bending effect is a very important topic for building structures in earthquake prone regions. Recently, Rodrigues et al. [11, 12, 13, 14] tested several number of RC fullscale column specimens under different horizontal loading patterns, always including the comparison of 2D test results with those of similar columns tested in (1D) bending. Rodrigues concluded that: (i) initial column stiffness is not much affected by 2D load path; (ii) maximum strength in one specific column direction for each 2D test is always lower than that of the corresponding 1D test; (iii) ultimate ductility is clearly reduced in columns under 2D loading paths; (iv) strength degradation is very reduced before ductility demands about 3, increasing thereafter; (v) 2D bending can introduce higher energy dissipation than 1D loading; (vi) viscous damping highly depends on biaxial load path. Recently some experimental tests were conducted To evaluate the efficiency of different strengthening technics to improve the RC columns behaviour [15, 16, 17, 18, 19, 20, 21, 22, 23, 24].
Some numerical modelling research works regarding to the use of simplified approaches to represent the RC columns cyclic behaviour can be found in the literature [25, 26]. Different modelling strategies are proposed by several authors regarding for example to lumped damaged models [27], fiber models [28], among others. However the number of studies about the numerical modelling of strengthened RC columns subjected to biaxial horizontal loading are scarce, and is of full importance to study numerical strategies that allow to analyse the effect of different strengthening technics application in the existing buildings response.
This study intends to evaluate numerical efficiency of a simplified approach to represent the biaxial cyclic behaviour of RC columns strengthened with CFRP and steel plates jacketing, through a distributed plasticity approach with the formulation based in forces coupled with a nonlinear variable confinement material model developed by Ferracuti and Savoia [29] to represent the reinforcement material effect. The numerical modelling efficiency was evaluated by comparing the numerical results with the experimental ones, in terms of: sheardrift hysteretic response, initial stiffness, stiffness degradation, maximum strength and energy dissipation.
Simplified modelling approach to represent strengthened RC columns behaviour
Overview of the experimental campaign
Specimen specifications, loading, material characteristics and strengthening technique
Group  Specimen  Geometry (cm × cm)  f _{ cm } (MPa)  f _{ yk } (MPa)  Ν (%)  Horizontal displacement path type  Strengthening technique 

1  PC12N10S  30 × 50  8.4  573.7  0.24  Diamond  CFRP jacketing 
2  PC12N11S  Steel jacketing  
1  PC12N12S  CFRP jacketing  
2  PC12N15S  14.8  575.6  0.14  Diagonal −45^{o}  Steel jacketing  
2  PC12N16S  Diamond  Steel jacketing  
1  PC12N17S  Diagonal −45^{o}  CFRP jacketing  
1  PC12N18S  Diamond  CFRP jacketing 
Numerical modelling strategy description
Fiber discretization was adopted to represent the behaviour at section level, where each fiber was associated with the corresponding uniaxial stress–strain law. The sectional moment–curvature state of the beam and column elements is then obtained through the integration of the nonlinear uniaxial stress–strain response of the individual fibers into which the section has been subdivided (Fig. 4c). Through the integration of the nonlinear response, the stress–strain uniaxial relationship in each individual fiber of the control section are submitted, results in the stress–strain state of the element. The nonlinearity of each column was obtained directly from the nonlinear behaviour of the fibers, which were is so accurate as much as the accuracy of the fibers. In this study the number of 600 fibers was assumed. Taking into account that the Seismostruct [30] does not have the possibility to take into account with the bondslip effect, and regarding to the considerations made by Bousias [32] in his parametric study to evaluate the bondsplit consideration in the numerical modelling with simplified approaches, the bondsplit effect was neglected.
Some numerical modelling approaches of the confinement effect provided by reinforcement materials can be found in the literature, mainly regarding to micromodelling strategies. For the present study, a simplified numerical approach proposed by Ferracuti and Savoia [29] was adopted, which the main goal is through a uniaxial variable confinement model determines the concrete strength obtained with the jacketing material (CFRP and steel plates in the present study). This nonlinear uniaxial variable confinement model follows the constitutive cyclic rules proposed by Mander et al. [33] for the compression and the Yankelevsky for the tension [34]. The confinement effects provided by the jacketing material are represented through the use of the rules proposed by Spoelstra and Monti [30, 35]. This numerical model approach allow to represent the interaction between the containment devices (jacketing material) due to the concrete lateral deformation, through and iterative incremental approach. The relationship between the axial and lateral stresses is implicitly derived through equilibrium between the confined concrete (dilation) and the containment device.
It should be noted that the elements have the same control sections characteristics throughout the entire length. What does not happen in reality because this reinforced control sections are only located in the seismic zone interventions, 0.50 m from the base.
However, the fact of the control sections located above the seismic reinforcement is different, because they are in linear elastic behavior which does not significantly affect the final response of the element. Additionally two original RC columns were modelled to calibrate the numerical strengthened models, despite as can be observed in Table 1 the material properties are slightly different for each RC columns group. In the next subsections are summarized the values adopted in the modelling process of the RC columns and the most important considerations.
RC columns strengthened with CFRP plates—Group 1
Homogenous confined concrete material parameters adopted for the numerical models—Group 1
Group  Samples  Compressive strength f _{ c } (kPa)  Elasticity modulus E _{ CFRP } (GPa)  Strain at peak strength ε _{ c } (‰)  ρ_{CFRP} 

1  PC12N10S  8400  240  4.5  0.0018 
PC12N12S  
PC12N17S  15,950  
PC12N18S 
Parameters adopted for the reinforcement material adopted in the numerical models—Group 1
F_{y} (kPa)  E_{s} (GPa)  μ  Ro  a1  a2  a3  a4 

550,000  200  0.015  19.2  18.6  0.05  0.01  2 
RC columns strengthened with steel plates jacketing—Group 2
Homogenous confined concrete material parameters adopted for the numerical models—Group 2
Group  Samples  Compressive strength f _{ c } (kPa)  Elasticity modulus E _{ steel } (GPa)  Strain at peak strength ε _{ c } (‰)  Ρ_{steel} 

2  PC12N11S  8400  210  4.5  0.0013 
PC12N15S  15,950  
PC12N16S 
Parameters adopted for the reinforcement material adopted in the numerical models—Group 2
F_{y} (kPa)  E_{s} (GPa)  μ  Ro  a1  a2  a3  a4 

550,000  200  0.015  19.2  18.6  0.05  0.01  2 
Evaluation of the numerical modelling efficiency
The numerical results will be presented along the present section for of all the specimens and were compared with the experimental ones in terms of sheardrift hysteretic behaviour, sheardrift envelopes, maximum strength, initial stiffness, secant stiffness degradation, tangent stiffness degradation and energy dissipation and are presented in the next subsections.
Sheardrift hysteretic curves results

Generally, it can be stated that the numerical responses were similar to the experimental results. Some problems were observed in some numerical models to capture the experimental response for large displacements, mainly in terms of the strength degradation. Similar problems are reported by other authors in the numerical modelling, with simplified approaches, of original RC columns subjected to biaxial loading [38]. Nevertheless, it is possible to observe already some strength degradation in the numerical results;

Regarding to the results of the Group 1 (Fig. 8) it can be observed a better numerical results for the columns subjected to diagonal—45º loading pattern when compared with the results for the diamond loading pattern. The same was also verified for the Group 2 RC columns subjected to the same loading pattern, as illustrated in Fig. 9;

Comparing the numerical results of the Group 2 it can be observed that the pinching effect is better adjusted to the experimental hysteretic results for the Group 1 specimens, mainly in the weak direction. In terms of energy dissipation, it was observed that the Group 2 that the numerical models dissipated higher energy than the observed in the experimental results.
Sheardrift envelopes
In terms of capture of the RC columns initial stiffness, it is observed that the numerical models have better results for the weak direction when compared with the other direction.
Initial stiffness ratio
Comparing the results for each Group, it is observed that the results are better for the Group 1, illustrated in Fig. 12a, where was found slight differences between the numerical and experimental results, 5–60 % for the strong direction and 0–25 % for weak direction. For Group 2, plotted in Fig. 12a, it is observed differences about 45–50 % for the stronger direction and between 0–30 % for the weak direction.
Maximum strength ratio
Better results were obtained for Group 1, when compared with the Group 2 results, as reported by the slight difference between 5–15 % obtained for the RC columns strengthened with CFRP jacketing and the 5–40 % observed for the RC strengthened columns with steel plates jacketing.
Stiffness degradation

The stiffness degradation was satisfactory captured by the numerical models, however it is observed that the results are better in the weak direction;

The numerical models obtained better stiffness degradation representation for larger drift values, mainly after 1.5 % of drift;

Comparing the results between the RC columns of Group 1, it is possible to observe that the numerical models are better for the diagonal 45º load path. The same was not observed for the RC columns of Group 2.
Stiffness normalized degradation
As observed for the evolution of the stiffness degradation the results shows to be better for the Group 1 RC columns. Again, comparing the stiffness normalized degradation between the RC columns of Group 1, it is possible to observe that the numerical models are better for the diagonal 45º load path. The same was again not observed for the RC columns of Group 2.
Energy dissipation
Regarding to the results obtained for RC columns of Group 1, it is observed slow differences for the samples subjected to the diamond load pattern, with differences between 5–45 %. The results of the Group 2 shows to be better than for Group 1, and this can be observed through the difference between the 5–50 % and the 15–25 % of difference observed respectively. It was also observed that the diagonal 45º load path presents better results than the diamond load path.
It can be observed also a good performance of the numerical models for lower drift values (0–3 % drift), and justified by the insufficient capacity to represent the strength degradation for higher drift values, the results of the final energy dissipated is higher than the observed in the experimental ones.
Total energy dissipation
Conclusions
The main objective of the present study was to evaluate the efficiency of a simplified approach to represent RC strengthened columns with two types of strengthening techniques (CFRP and steel plates jacketing) subjected to biaxial horizontal loading and constant axial load. The modelling strategy was based on the distributed plasticity element with forcebased formulation, with particular attention to the modelling process of the strengthening material and the respective effect on the RC columns response. This numerical model approach allow to represent the interaction between the containment devices (jacketing material) due to the concrete lateral deformation, through and iterative incremental approach. The relationship between the axial and lateral stresses is implicitly derived through equilibrium between the confined concrete (dilation) and the containment device.
The numerical modelling efficiency was evaluated by comparing the numerical results with the experimental ones. Globally it was observed that the numerical models represented satisfactory the original RC columns for both of the Groups. Relatively to the RC columns of Group 1 it was observed that all of them were well represented, especially the RC column subjected to the diagonal 45º horizontal load path. The same was observed for the group 2 where the best representation of the experimental behavior was obtained for the column under the same horizontal load path.
In terms of initial stiffness and maximum strength it was observed that the numerical models reproduced accurately the experimental response with better results with slight differences around 5–15 % when compared with the differences observed for Group 2 around 5–40 %. Regarding to the stiffness degradation the results shows to be better for the Group 1 RC columns. Again, comparing the stiffness degradation between the RC columns of Group 1, it is possible to observe that the numerical models are better for the diagonal 45º load path. The same was again not observed for the RC columns of Group 2. It was verified that the original RC columns numerical columns obtained higher energy dissipation values in the experimental test when compared with the numerical results. However for the strengthened columns the same wasn’t verified, with higher values for the numerical models, as consequently of the difficulty of the maximum strength degradation. All numerical models showed limitations in the representation of the pinching effect at the dischargerecharge phase.
The global findings of the present research work allow to conclude that this modelling strategy for strengthened columns can help the designers to evaluate the benefits in the structural response by introducing some jacketing reinforcements in RC buildings.
Notes
Acknowledgments
This paper reports research developed under financial support provided by “FCT—Fundação para a Ciência e Tecnologia”, Portugal, through the research project PTDC/ECM/102221/2008. HR participated in the design of the study and carried out the nonlinear analysis. AF worked on the results obtained and helped to draft the manuscript. HV and AA conceived of the study, and participated in its design and. All authors read and approved the final manuscript.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no competing interests.
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