Development of a numerical model to simulate the behavior of plate shear connectors applied to slender cross-section concrete-filled steel tube

Abstract The cross-section of concrete-filled steel tube (CFST) columns is classified according to the local slenderness of the steel tube as slender, non-compact, or compact. There are analytical models to determine the axial force of locally slender CFST columns, however studies on load transfer from beams connected to these columns are scarce. This paper presents the process of the development and validation of a numerical model to simulate the behavior of Crestbond connectors applied to CFST columns with slender steel tubes, based on the results of experiments carried out. The numerical model presented can be used for the analysis and design of the load transfer in slender columns using connectors known as composite dowels.


INTRODUCTION
The competitiveness and applicability of concrete-filled steel tubes (CFST) columns are associated with the interaction between the steel tube and the concrete core and the difficulty to connect them to the beams. In this way, when the shear stress exceeds the natural bond strength between the steel tube and the concrete core, it is necessary to apply mechanical devices such as shear connectors. Among the devices standardized by ABNT NBR 8800:2008 [1], there are stud bolts and by ABNT NBR 16239:2013 [2], there are bolted shear connectors. Several studies on the load transfer through these connectors have already been carried out, including De Nardin and El Debs [3] and Santos et al. [4] The behavior of the concrete-filled steel tubes (CFST) columns is also influenced by the local slenderness of the composite cross section (λ), since it is a function of the ratio between the diameter (or side, for rectangular sections) and the thickness of the tube [5]- [7]. The American standard AISC 360-16 [8] classifies CFSTs according to the local slenderness of the cross section. According to this standard, the cross section can be classified as being slender, noncompact or compact, providing the design recommendations for determining their axial strength. However, the load transfer from the beams connected to these columns still has few studies and solutions.
De Nardin and El Debs [3] verified the contribution of stud bolts, in the shear strength in the steel-concrete interface in CFST columns submitted to push-out tests. Starossek et al. [5] and Cardoso [9] experimentally evaluated the rupture mode of CFST columns subjected to an axial load while using stud bolts as shear connectors, proving the efficiency of bolted shear connectors for such application. Cardoso [9] experimental results later served as basis for studies by Santos [10] and Prates [11]. Santos [10] proposed a new analytical model to describe the behavior of bolted shear connectors when used as shear connectors in CFST. Prates [11] studied the behavior of bolted shear connectors, when used as shear connectors in columns composed of cold formed steel tubes filled with concrete, a simple and innovative solution for this type of connection, eliminating welding. The load transfer in the CFST by bolted shear connectors was also the study object of Ribeiro and Sarmanho [12] and Xavier et al. [13] Younes et al. [14] and Tao et al. [15], who evaluated the load transferred using bolted connectors in CFST columns subjected to axial and cyclical loads, respectively. Although the way in which force is applied is different, Younes et al. [14] as for Tao et al. [15] also corroborated the efficiency of this type of tool as shear connector in CFST columns.
The Crestbond connector initially studied for application in beams [16], began to be tested as a load transfer device in CSFT columns. The Crestbond dowels allow compatibility with the reinforcement rebars of the CSFT columns, and its steel plate can be designed to integrate with a beam-column connection (Figure 1).  [17] Different studies carried out by researchers at the School of Engineering at Universidade Federal de Minas Gerais (UFMG) prove the efficiency of the Crestbond connector to be used as a shear load transfer device in compact section CFST columns [17]- [19] After the efficiency of the Crestbond connector in application in CFST columns was proven, both the behavior of the Crestbond connector and the composite cross section started to be analyzed in specific ways. In this context, the present study aimed to develop and validate a numerical model to analyze the shear load transfer in locally slender composite CFST columns with the Crestbond connector.

EXPERIMENTAL PROGRAM
Two shear tests were carried out in order to experimentally evaluate the behavior of the Crestbond shear connector when used as load transfer device in slender section of the CFST columns. The execution of the tests was defined by adapting the standard shear test procedures (push-test) prescribed in Annex B of the European Standard EN 1994-1-1:2004 [20] As in Cardoso [18] experiments, the displacement of the CFST was restricted, while the steel tube was kept free to slide in relation to the concrete core. In addition, a release agent was applied to the internal surface of the steel tubes to minimize friction at the material interface, causing the force in the section to be transmitted, mainly, through the Crestbond CR56b-R12 shear connectors with the geometric pattern proposed by Verissimo [16] The relative displacement between the steel tube and the concrete core was measured using displacement transducers (DTs) fixed vertically in strategic positions to capture linear displacements between the tube and the concrete. From the force versus vertical relative displacement curves obtained experimentally, it was possible to develop the numerical models of the present study.

Compact cross section CFST
In the study by Cardoso [18], the CFST models with Crestbond shear connectors had compact tubular cross sections (local slenderness limited to 34 ≤ λ ≤ 40) and had the force applied to the steel tube through a set of end-plates ( Figure 2). The prototypes were 100 cm high with a gap of 5 cm between the top of the concrete core and the plate for applying force to the test device. The configurations of the shear test can be seen in Figure 2.

Slender cross section CFST
The authors experimentally evaluated the behavior of the Crestbond connector applied to slender section CFST (local slenderness equal to 153) and, due to the great local slenderness of the steel tubes, a new shear test methodology was developed [21] In this test program, the force was introduced directly into a Crestbond connector.
The CFSTs were 75 cm high and were locked by end-plates rigid enough to prevent rotation of the model that had a Crestbond with two steel dowels (CR2D). The configurations of the shear test can be seen in Figure 3. The configuration of the tested models can be seen in Figure 4 and their geometrical properties are shown in Table 1. The present study has the purpose to develop a finite element model able to simulate the behavior of Crestbond connectors applied in thin-walled CFST columns. Among the studies previously presented Cardoso (2018) was the principal reference by this work, because it analyzed CFST columns with compact cross-section using the same connectors.
The numerical results of Cardoso [18] were good from a numerical study calibrated based on an experimental analysis realized with more than 20 models. In this sense, the present study aims to be a continuation of the work started by Cardoso [18], but now to evaluate the slender thin-walled CFST columns tested in the same laboratory and developed by the same research group.
So, the experimental models studied by Cardoso [18] are present in Table 1 with the experimental models of the present work, because these models together formed the basis of the numerical model developed. *λ represents the thinness index of the cross section of the composite column | ** This series features only one model | S1 -Section 219.1 x 6.4 | S2 -Section 355.6 x 9.5 | S3 -Section 320 x 250 x 8.2 | CR -Crestbond Connector | Da -number of steel dowels | Dc -number of concrete dowels | F -Concrete frontal strength | SF -Without concrete frontal strength | ConA -Self-compacting concrete | E -Stirrup | | The S1-02CR-E models; S2-02CR-E and S3-02CR-E had internal reinforcement From the data obtained during the tests, the curves Force (kN) x Relative Displacement (mm) shown in Figure 5 and Table 2 were obtained.

Generalities
The numerical analysis was developed using the finite element program ABAQUS -version 6.14. The numerical model was elaborated with the same parameters adopted in the study by Cardoso [18] The final shape of the numerical models can be seen in Figure 6.

Finite Element Mesh
The mesh elements adopted for the modeling of the steel tube, concrete and connector were of the C3D8 type (Continuum, hexahedral and linear), a type of solid element that has eight nodes (only at the vertices) with three degrees of freedom per node (translations in the three main directions X, Y and Z). This element is shown in Figure 7. Concrete core CR2D Figure 7. C3D8 Mesh Element [19] The sizes of the mesh elements were defined through a study of mesh sensitivity and refinement and its discretization are illustrated in Figure 8. In Crestbond and in the region limited by around the shear connectors, mesh elements with a length of 8 mm (II) were used, which was increased as the elements moved away from the region of concentration of efforts. For the concrete and tube located in the upper region of the connector, mesh elements with a length varying between 10 mm and 15 mm (I) were applied. For the parts located in the region below the connector, the mesh was divided into elements with a length varying between 10 mm and 20 mm (III). Finally, mesh elements with a width of 10 mm were chosen for the direction corresponding to the model's transversal axis (IV). The mesh distribution methods adopted were structured mesh and sweep mesh. Due to the complexity of the model, divisions were made to favor the allocation of the finite element mesh and are shown in Figure 9. Based on the studies carried out by Santos [21] and Cardoso [18], the contact between concrete, steel profile and shear connector was simulated considering a static friction coefficient of 0.5 for the contact between the connector and the concrete, 1.0 between the connector and tube, and 0.17 between the tube and the concrete core. It was considered that in the test the friction between the steel tube and the concrete was minimized with paint and the release agent inside the tube.
The contact between concrete, steel tube and shear connectors, was simulated through face-to-face interactions and it was necessary to define separately in each contacts pairs which was the stiffer element. The rigid type of contact was defined in all the interactions which is the one that admits the minimum penetration between the surfaces of the elements. The contact pairs on the surfaces were defined as: Concrete-to-Tube ( Figure 10a); Concrete-to-Connector ( Figure 10b); Tube-Connector (Figure 10c). Connector-to-Tube.

Boundary Conditions
To reproduce the test conditions, the following boundary conditions were adopted: (I) restriction of the concrete core to vertical displacement to simulate the rigid support of the test; (II) rotation and horizontal displacement restrictions to represent the symmetry of the real model and (III) springs with stiffness (k) calculated in order to simulate the stiffness of the locking plates of the experimental model. The boundary conditions are shown in Figure 11.

Displacement Application and Data Acquisition
To simulate the monotonic loading with displacement control of the experimental test and to avoid possible convergence problems, it was decided to introduce displacement increments and obtain the corresponding reaction forces to get the force versus displacement curve of the model.
For this purpose, a reference node was created, called as Reference Point (RP) connected to the Crestbond connector plate, as shown in Figure 12. In this way, as each increment was applied, the connector moved and, because of the rigid connection with the other materials of the model, transferred the displacement in the section. In addition, it is important to note that numerical data acquisition was also carried out through the RP. For the nonlinear analysis of the models, the Dynamic Implicit analysis method was used with the quasi-static option. This method was used by Santos [10] and Cardoso [18] and it shows good convergence and satisfactory results, proving to be efficient for the analysis of shear connectors applied in CFST columns.

Concrete Constitutive Model
The Concrete Damaged Plasticity model (CDP) was used for the simulate the concrete core constitutive model and it is available in the ABAQUS library. This model is suitable for modeling fragile materials and has been widely applied for numerical modeling involving confined concrete.
The input parameters required for this model are the dilatation angle (ψ); the ratio between the compressive yield stress in the biaxial and uniaxial state (σb0⁄σc); the ratio between the second invariant stress of the tension meridian and the second invariant stress of the compression meridian (K c ); viscosity parameter (μ vis ); and the eccentricity (ϵ). Based on the studies by Aguiar [22] and Cardoso [18], in this work adopted: ψ = 36 °, σ b0 ⁄σ c = 1.16, K c = 2/3, μ vis = 0.00005 and ϵ = 0.1.
To represent the behavior of the confined concrete subjected to compression, it was adopted the stress versus strain relationship, proposed by Pavlović et al. [19] illustrated in Figure 13. Pavlović et al. [19] proposed a continuous sinusoidal curve that starts at the point D, presented by the European standard EN 1992-1-1:2004, and continues until the strain ε cu = 0.01, defined by the following equation: The stresses at the D and E points are defined as f cuD = f cu1 = σ c (ε cu1 ); f cuE = α/f cm . Strains at the D and E points are defined as ε cuD = ε cu1 ; ε cuE = 0.03, respectively. The remaining parameters are defined as: α = 20, α tD = 0.5, α tE = 0.10 e β = f cm / f cu1 .
To reproduce the behavior of concrete under tension, the same curve used in Cardoso numerical study was adopted in this work [18] This author used the stress curve (σ t ) versus crack opening (W c ) proposed by Tahmasebinia et al. [23], as in Figure 14.
The damage variables d c (damage to uniaxial compression) and d t (damage to uniaxial tension), were calculated, where = 1 − e = 1 − [23], and inserted in the ABAQUS program according to the plastic deformations equivalent to tension and compression, respectively.

Tube and Connector Steel Constitutive Model
To simulate the steel behavior of the connectors and the tubes, an elastoplastic model with hardening was adopted based on the multilinear theoretical diagram in Figure 15. This model was used by Cardoso [18] The part of the graph that represents the smooth unloading (G-H in Figure 15) was adopted as a numerical solution to complete the analysis in a regime of large deformations.

Generalities
The same characteristics were adopted as in Item 2.2 with the exception for the fictitious springs that were added to the numerical model to represent the locking plate of the test device, as illustrated in Figure 16a. The numerical results were validated by comparing them to the force versus relative displacement curve obtained experimentally, evaluating the initial stiffness, the maximum force value obtained in the analyzes and the deformed configuration of the models. Figure 17 shows the comparison between the numerical and experimental curves of the analyzed models. By comparing the curves, it is possible to observe that the numerical results were close to the experimental ones. The slope of the two curves of model E practically overlapped, where the numerical model ensured the representation of a connection with an initial stiffness close to that obtained experimentally. In addition, it is observed that in C Series Model numerical curve practically coincided with the maximum force and E Series Model remained below the experimental curve, demonstrating a more conservative numerical model than the experimental one, as in Tables 3 and 4.  (1) Defined at numerical model.    Figure 20 shows images of the numerical and experimental C series model after testing. The numerical model was able to reproduce where the concrete damage was more pronounced like the observed in the experimental analysis. For that, the results for the variable DamageT, which represents the degradation of the concrete tensile stiffness, are evaluated in the numerical model. The red region indicates a complete no stiffness, which culminated in the separation of the concrete core by tensile stresses.

Model Geometry Transition
To simplify the numerical model so that the simulation time and the computational cost of the analyzes were lower, some sensitivity tests were necessary to validate the transition of the model geometry, including the removal of the springs and the evaluation of the model behavior with two symmetries.

Sensitivity Test of the Locking Plate Stiffness
To enable the analysis of 1/4 of the numerical model, it was necessary to remove the springs that simulated the locking plates in the test. In addition to simplifying the model, considering zero stiffness for springs allows simulating situations more similar that occur in reality, in which there is less restriction for the connector. For that, the boundary conditions were modified, releasing the horizontal translation of the plate of the connector and restricting the regions corresponding to the location of the locking plates, as shown in Figure 16.
From the results obtained, it was possible to observe that the removal of the spring led to a more conservative model, favoring safety, shown in Figure 21 and Tables 5 and 6.

Sensitivity Test of Transition to Double Symmetry
As concluded the evaluations that proved the possibility of removing the springs that simulated the locking plates and after observing that the curves remained more conservative (below the experimental curve), a sensitivity test was performed for the double symmetry, allowing the modeling only 1/4 of the model, illustrated in Figure 22. The model reduction to 1/4 provided more conservative results (Figure 23 and Tables 7 and 8). Furthermore, there was a reduction in processing time per model with 1/4 configuration without springs of about 10 hours, in relation to the time required for the simulation of 1/2 models with spring. Thus, it was adopted the numerical 1/4 model without springs to be used in parametric studies and analytical models of shear load transfer in slender composite sections columns through Crestbond connectors.

CONCLUSIONS
This work presents the development of a numerical model to simulate the behavior of a Crestbond shear connector applied in CFST columns with slender composite section. Although analytical models for determining the axial strength of CFST already exist, studies on the load transfer from the beams connected to these columns are scarce. Two experiments were carried out, expanding the datasets used for the development of the presented numerical model and demonstrating the applicability of this new solution.
From the calibrated numerical model, sensitivity tests were performed to reduce the processing time, which resulted in a representative and conservative numerical model. The numerical results were validated from the comparison with the relative force/displacement curves obtained experimentally, evaluating the initial stiffness, the maximum force value obtained in the analyzes and the deformed configuration of the models. The final model uses only 1/4 of the real models, due to its double symmetry, and can be used for study and design of the shear load transfer in CFST columns with slender composite section using shear connectors, which is a fundamental contribution to the development of new research and further development of a analytical model.