1 INTRODUCTION
Beam-column connection in Reinforced Concrete (RC) frame is one of the most critical parts of the structure which has important role in seismic performance of these buildings. When a beam-column connection of a moment resisting frame is subjected to lateral forces, beam-column joint is prone to joint shear failure due to high shear stress which appears in the joint panel as a result of opposite sign moments on both sides of the joint core. This type of failure is not favorable because it has undesirable effects on seismic performance of RC buildings, especially moment resisting frames. The joint shear failure is a brittle type of failure which can adversely affect ductility of the RC frames. Also, early occurrence of this failure causes the beams not to reach their ultimate flexural capacity. Finally, the exposure of severe damage in the joint region may lead to total collapse of the building when the structure is in the seismically active regions and designed according to non-seismic design guidelines before the application of modern seismic design codes (ACI 318-14, ACI 352R-02, ASCE 41-13 and NZS3101). Therefore, this type of failure has undesirable effects on global seismic performance of RC structures, especially moment resisting frames.
Considerable amount of experimental and analytical studies have been carried out to investigate the seismic performance of RC beam-column joints subjected to lateral earthquake loading ((^{Elsouri and Harajli, 2013}), (^{Lee et al., 2009}), (^{Masi et al., 2013}) and (^{Sasmal et al., 2013})) or development of retrofitting techniques for vulnerable existing RC beam column joints ((^{Eslami and Ronagh, 2014}), (^{Esmaeeli et al., 2014}), (^{Vecchio et al., 2016}), (^{Shwan and Abdul Razak, 2016}), (^{Esmaeeli et al., 2015}) and (^{Campione et al, 2015})). Some of these researches were concentrated on shear resistance and behavior of RC beam-column joints. As one of the first studies in this field (^{Hanson and Connor, 1967}) suggested a quantitative definition of joint shear. Based on their definition, joint shear was determined from a free-body diagram, at mid-height of the joint panel. (^{Pauley et al, 1978}) introduced qualitative analytical shear resistance mechanisms of RC beam-column joints including a concrete strut and a truss. The first mechanism is a diagonal compression concrete strut that transfers the compression forces from the beam and column actions without the contribution of shear reinforcement. The second mechanism is a stress mechanism that transfers bond forces from the longitudinal bars utilizing horizontal and vertical joint shear reinforcement and concrete struts, therefore the shear resistance in the joint region corresponding to concrete truss mechanism is attributed to the bond capacity between reinforcement and concrete. These basic studies were followed by experimental researches on RC joints aimed at evaluation of their shear behavior ((^{De Risi et al., 2016}), (^{Kotsovou and Mouzakis, 2012}) and (^{Hakuto et al., 2000})). The tests were done on different joint types such as exterior, interior, knee and T joints in different scales. In these experimental studies the effects of different parameters such as geometrical properties of the joint panel, concrete compressive strength, transverse joint reinforcement, beam longitudinal reinforcement, existence of RC slab, column axial load and etc. were investigated on the joint shear strength and load-displacement curves of the RC joints. (^{Clyde et al, 2000}) and (^{Pantelides et al, 2002}) carried out seismic tests on some half-scale exterior beam-column joint specimens in order to determine performance levels according to FEMA273 guidelines. (^{Park and Mosalam, 2012}) tested four full-scale exterior beam-column joints with orthogonal transverse beams and floor slabs under variable column axial loads. They investigated the behavior of unreinforced exterior joints subjected to cyclic loading. Their investigation was focused on the effect of some crucial parameters, such as joint aspect ratio and beam longitudinal reinforcement ratio on the shear strength and deformability of exterior connection joints comparing their test results with the ASCE 41 seismic design recommendations.
A series of researchers have collected the test results in literature and have developed empirical and analytical relations to predict the RC joint shear capacity ((^{Pauletta et al., 2015}), (^{Muhsen and Umemura, 2011}), (^{Park and Mosalam, 2012}), (^{Lima et al., 2012}), (^{Wong and Kuang, 2014}), (^{Wang et al., 2012}), (^{Jeon et al., 2014}) and (^{Kassem, 2016})). For example (^{Kim and Lafave, 2007}) collected experimental results of semi-static cyclic tests carried out by different researchers on various types of RC beam-column connections (exterior, interior and corner joints). All specimens had a minimum amount of joint confinement and the main objective of the study was to investigate the most influential parameters affecting the joint shear load-displacement behavior. The parameters investigated were material property, joint panel geometry, confining reinforcement, reinforcement bond condition and the column axial force. A design approach for RC beam-column joint was presented by (^{Kotsovou and Mouzakis, 2012}). The basic assumption of their model was that the load transferred from the linear elements to the joint is mainly resisted by a diagonal strut mechanism. In a state of the art article, (^{Lima et al, 2012}) reviewed some of the analytical and empirical models introduced in the last two decades for predicting shear strength of RC beam-column joints. Some simplified models are also developed for simulation of the joint shear behavior in nonlinear static and dynamic analysis of RC frames ((^{Sharma et al., 2011}), (^{Favvata et al., 2008}) and (^{Shayanfar et al., 2016})). For instance, model developed by (^{Shayanfar et al, 2016}) is one of the recent models for this purpose. They have introduced a frame model to simulate the RC joints in nonlinear analysis of RC frames.
A series of numerical studies have been done on RC joints using FEM analysis. For example (^{Niroomandi et al, 2014}) performed numerical investigation of affecting parameters on the shear failure of non-ductile exterior joints. According to their numerical results, two crucial parameters influencing the joint shear behavior were joint aspect ratio and beam longitudinal reinforcement ratio.
Although a majority of numerical studies have been carried out on RC beam-column connections ((^{Parvin and Granata, 2000}), (^{Mostofinejad and Talaeitaba, 2006}), (^{Niroomandi et al, 2010}), (^{Mahini and Ronagh, 2011}), (^{Masi et al., 2013}) and (^{Haach et al., 2008})) by FEM softwares such as ANSYS, ABAQUS and DIANA, these studies have concentrated on simulating flexural behavior of beams and columns adjacent to the joint region and are not focused on joint shear behavior of RC connections, while in many cases the shear strength and behavior of joints control the overall response of RC beam-column connections subjected to seismic actions. It should be noted that capturing shear dominant failure mode in discontinuity region of connection is a very complicated numerical simulation problem.
The suggested modeling technique in this paper has been conducted by means of the commercial FEA program ABAQUS and calibrated by modeling and analyzing experimentally tested exterior and interior beam-column connections in which the governing failure mode during simulated seismic actions on the specimens was the joint shear failure type. The comparison between numerical and experimental results, indicates the ability of the proposed method in simulating the governing joint shear behavior even at post peak phase.
2 FINITE ELEMENT MODELING
Nonlinear finite element analysis of RC beam-column connections with joint shear failure as the governing failure mode is performed using ABAQUS/Standard. Geometrical and material nonlinearities are considered in order to properly simulate the behavior of connections and the joint shear failure mode under seismic loading. In pre-processing stage, geometry and boundary conditions, element types, material properties and nonlinear analysis solution are defined. Finite element simulation of the complex behavior of concrete as a non-homogeneous and anisotropic material is a challenge in the finite element analysis of reinforced concrete structures and their components. Among constitutive models defining concrete nonlinear behavior as a quasi-brittle material available in ABAQUS, such as smeared and brittle cracking models, the Concrete Damage Plasticity (CDP) is selected and introduced to the numerical model. The main and essential elements of any model based on classical plasticity theory, which are the "yield criteria", "flow rule" and "hardening rule" are all effectively considered in damage plasticity model. Numerical modeling of RC joint shear behavior calibrated by experimental results of other researchers is the main strategy of this study. To verify the model, two RC beam-column connections with exterior and interior joint configurations are selected and simulated. The numerical modeling is first calibrated by experimental results of a quasi-static cyclic test on a half scale exterior beam-column connection which is previously done by (^{Clyde et al, 2000}) and then applied to an interior full scale RC beam-column connection which is tested by (^{Hakuto et al, 2000}) under simulated cyclic load. In both cases the ultimate failure of specimens were attributed to the joint shear failure.
2.1 Elements
3D 8-noded hexahedral (brick) elements having 3 degrees of freedom in each node (translations in X, Y and Z directions) are utilized for modeling concrete elements with reduced integration (C3D8R) to prevent the shear locking effect. In order to model reinforcements, 2-noded truss elements (T3D2) having 3 degrees of freedom in each node (translations in X, Y and Z directions of global coordinates system) are used. The embedded method with perfect bond between reinforcement and surrounding concrete is adopted to properly simulate the reinforcement-concrete bonding interaction. It is notable that the effects usually associated with reinforcement-concrete interface, such as bond slip and dowel action are modeled indirectly by defining "tension stiffening" into the reinforced concrete model to approximately simulate load transfer across cracks through the rebar ((^{ABAQUS user’s manual, 2014})).
2.2 Analysis Approach
Finite Element Analysis (FEA) of the connection joint specimens is performed in a nonlinear static analysis format and the analysis procedure considers both material and geometric nonlinearities. In a nonlinear analysis, the total specified loads acting on a finite element body will be divided into a number of load increments. At the end of each increment the structure is in approximate equilibrium and the stiffness matrix of structure will be modified in order to reflect nonlinear changes in structure's stiffness.
ABAQUS/Standard uses the Newton-Raphson method to obtain solutions for nonlinear problems. Newton-Raphson equilibrium iterations provide convergence at the end of each load increment within tolerance limits for all degrees of freedom in the model. Residual load vector, which is the difference between the internal forces (the loads corresponding to the element stresses) and the external applied loads are analyzed again by Newton-Raphson approach. Subsequently, the program carries out a linear solution using residual loads and considering initial stiffness of structure, in order to check the convergence criteria. If the residual load vector is less than the current tolerance value, the external and internal forces are in equilibrium (this tolerance value is set to 0.5% of an average force in the structure, averaged over time). If convergence criteria are not satisfied, the stiffness matrix is updated, the residual load vector is re-calculated and a new solution is achieved. ABAQUS/Standard also checks that the displacement correction is small relative to the total incremental displacement and it is regularized in such a way that both convergence checks (loads and displacements) must be satisfied before a solution considered to be converged for that load increment.
2.3 Concrete Damage Plasticity Model
Concrete damage plasticity is used as the governing concrete material plasticity model over the whole geometry of the specimens. The model is a plasticity-based model which is developed using concepts of continuum damage mechanics and the application of scalar damaged elasticity in combination with isotropic tensile and compressive plasticity to properly represent the inelastic behavior of concrete ((^{Lubliner et al., 1989})). The main two failure mechanisms of the concrete material are tensile cracking and compressive crushing according to fundamental assumptions of damage plasticity model. The evolution of the yield (or failure) surface and the degradation of elastic stiffness in damage plasticity model are controlled by two hardening variables which are tensile and compressive equivalent plastic strains (
Where
Here
Typical values of the ratio (_{
σb
}
_{0} /_{
σc
}
_{0}) for concrete based on experimental results are reported in the range from 1.10 to 1.16, yielding values of α between 0.08 and 0.12 ((^{Lubliner et al, 1989})). Finally, the parameter y determines the shape of the yield surface in the plasticity model and enters the yield function only for stress states of triaxial compression, when
The coefficient _{ Kc } is the ratio of the hydrostatic effective stress in tensile meridian to that on the compressive meridian when the maximum principal stress is negative. This coefficient defines the shape of yield surface in the deviatory plane. The shape of deviatory plane was first assumed to be circular as in the classic Drucker-Prager strength hypothesis (_{ Kc } = 1). The CDP model suggests to assume default value of _{ Kc } = 2/3 based on triaxial stress test results. Figure 1 shows a plane stress cross section of the yield surface in principal stress space together with the yield surface cross section on the deviatory plane corresponding to two values of _{ Kc } representing different strength criteria.
The flow rule in concrete damage plasticity is introduced to the model as an essential element of plasticity theory utilizing non-associated flow potential functionG(σ) The CDP model uses Drucker-Prager hyperbolic function as non-associated flow potential function according to Eq.5.
The equation involves three material parameters. The first one is dilation angle ψ which is a concrete performance characterizing parameter when it is subjected to triaxial compound stress state. The next parameter ∈ is eccentricity which adjusts the shape of hyperbola in the plastic potential flow function. The eccentricity parameter is a small positive value and can be estimated as the ratio of concrete tensile strength to its compressive strength. The default value of eccentricity parameter is considered to be equal to 0.1. Finally, the last parameter of plastic flow potential function is the concrete initial uniaxial tensile strength _{ σt } _{0}.
Damage in CDP model is associated with the failure mechanisms consist of concrete cracking and crushing, therefore the occurrence of damage leads to reduction in elastic stiffness. To introduce damage to the model Eq.6 is applied.
Where E _{0}is the initial (undamaged) elastic stiffness of the material and the operator (:) denotes the product of the related tensors. Based on the scalar-damage theory, the stiffness degradation is isotropic and defined by a degradation variable d. Damage is defined in both tensile and compressive states _{ dt } and _{ dc } as functions of plastic strains. Damage parameter can take values in the range from zero (corresponding to undamaged material) to one (corresponding to fully damaged material).
Visco-plastic regularization of the concrete constitutive equations is defined in the numerical model using a generalization of Duvaut-Lions regularization. As a result, the plastic strain tensor and consequently the stiffness degradation variable were modified. Visco-plastic strain rate tensor (
Here _{ εpl } is the evaluated plastic strain and the visco-plastic stiffness degradation variable _{ dv } is defined according to Eq.8:
Where d is the evaluated degradation variable. The stress-strain relation based on viscoplastic model is defined according to Eq.9:
2.4 Material Model
2.4.1 Concrete Material Modeling
Uniaxial stress-strain behavior of concrete is simulated utilizing Hognestad type parabola ((^{Hognestad, 1951})). The uniaxial stress-strain behavior of concrete can be categorized into three main domains. The first one represents the linear-elastic branch which continues to reach the stress level of _{ σc } _{0} that is taken as _{ σc } _{0} = 0.4 _{ f’c } . The second stage shows the hardening part of the concrete uniaxial compressive stress-strain behavior which describes the ascending branch of the stress-strain relationship reaching to the peak load at the corresponding strain level ε _{0} = 2_{ f’c/Ec } The last part of concrete uniaxial compressive stress-strain relationship attributes to the post-peak softening behavior and therefore represents the initiation and progression of compressive damage in the concrete material until the ultimate compressive strain_{ εu }Figure 2 shows the compressive stress-strain behavior of concrete which is introduced to the presented numerical model.
The uniaxial stress-strain behavior of concrete in tension consists of two parts. As shown in Figure 3, the first part is a linear elastic behavior up to concrete tensile strength_{ σt } _{0} The second phase initiates together with crack occurrence and its propagation in concrete material under tension demonstrating a descending branch in the uniaxial tensile stress-strain diagram. The behavior in this phase is modeled by a softening procedure which can be modeled using linear, bilinear or nonlinear stress-strain relationships ((^{Belarbi and Thomas, 1994})). According to analysis assumptions in this study the linear behavior is applied to the model. Figure 3 demonstrates details of tensile softening assumptions in the presented model.
The ultimate tensile strength of concrete is estimated by Eq.10 ((^{Genikomsou and Polak, 2015}) and (^{Wang and Vecchio, 2006})).
Damage is defined both for uniaxial tension and compression during softening procedure in concrete damage plasticity model. Damage in compression occurs just after reaching to the maximum uniaxial compressive strength corresponding to strain level ε _{0} The degradation of elastic stiffness in softening regime is characterized by two damage variables, _{ dt } and _{ dc } corresponding to tensile and compressive damage, respectively, which are assumed to be functions of the plastic strains. Tensile and compressive damage in concrete damage plasticity model in the presented numerical model is assumed to be according to equations and diagrams of Figure 4.
2.4.2 Steel Reinforcement Modeling
The uniaxial tensile stress-strain behavior of reinforcement was assumed to be elastic with conventional Young’s modulus and Poisson’s ratio. The plastic behavior is also modeled including yield stress and corresponding plastic strain. Properties of plastic phase is defined to the model using bilinear behavior.
Most finite element studies of RC structures do not consider bond-slip of reinforcing steel and the inherent interaction between reinforcement and concrete in RC members. The post-failure behavior for direct straining is modeled with tension stiffening in CDP model, which helps to define the strain-softening behavior for cracked concrete, therefore in order to consider concrete-reinforcement interaction such as bond slip, an indirect approach which defines "tension stiffening" into the reinforced concrete finite element model is applied as mentioned before in this study. Two different approaches to introduce tension stiffening to the numerical model are the application of post-failure stress-strain relation and the definition and application of fracture energy cracking criterion. The presented numerical model utilizes the first method (post-failure stress-strain relation) to appropriately introduce the tension stiffening behavior of reinforced concrete, the method in which post-failure properties of reinforced concrete is defined by giving the post-failure stress as a function of cracking strain (
The elastic strain of undamaged material (_{ εelot } ) is defined according to Eq.12:
When unloading data are available the tension softening data can be provided in terms of tensile damage-cracking strain relationship. A linear tension stiffening data based on uniaxial tensile damage definition in CDP model is introduced to the numerical model of the beam-column connections according to the diagram of Figure 5.
3 VERIFICATION OF THE FINITE ELEMENT MODELING
An exterior half-scale beam-column connection namely test#2 conducted by (^{Clyde et al, 2000}) and a full scale interior beam-column connection namely unit-01 tested by (^{Hakuto et al, 2000}) are chosen to validate the numerical model. Both specimens were designed to have joint shear failure as their governing failure mode. To enforce shear failure in the joint, the beam longitudinal reinforcement of exterior beam-column connection was increased to prevent yielding and early degradation of the beam. To represent non-ductile detail of the joint, no transverse reinforcement was considered within the joint panel in both cases (exterior and interior beam-column connections). It is worth mentioning that the interior beam-column connection tested by (^{Hakuto et al, 2000}) is identical to an interior beam- column connection of an existing RC building constructed in New Zealand before mid-1960s.
3.1 Boundary Conditions, Loading, Dimensions and Details
Geometry and reinforcement details of the exterior and interior joints are shown in Figure 6. A schematic representation of the loading apparatus regarding both test connections is shown in Figure 7. According to boundary conditions applied to the exterior beam-column connection specimen, column was simply supported at both ends and the compressive axial load was applied to the column’s end using a hydraulic cylinder. The column axial compressive load is applied to the top of column corresponding to 10% of the compressive strength of concrete in the first step. The lateral load was applied in second step to the beam tip through a loading collar in a quasi-static cyclic manner. The interior beam-column connection had a different boundary condition and loading procedure. The pinned connection at the bottom of the column restrained its lateral displacement, but the column was allowed to rotate and elongate. The ends of the beams were connected by pin-ended steel members to the reaction floor, so that the ends of the beams were free to rotate and to translate horizontally but restrained vertically. No axial load was applied to the column during the test since according to (^{Hakuto et al, 2000}) the column axial pressure may contribute to the load bearing capacity of the joint making an unfavorable condition. Loading was applied to the interior beam-column connection in one step only (lateral load). Cyclic horizontal loading was applied to the top end of the columns of the interior test unit using a double acting hydraulic jack.
3.2 Material Properties of the Test Specimens
The measured uniaxial concrete compressive strength of the test specimens, the yield strength and the ultimate tensile strength of reinforcement used in the tests, are reported in Table 1.
Connection specimen | Concrete compressive strength f^{ ' } _{ c } (MPa) | Reinforcement type | Bar size | Yield strength F_{ y } (MPa) | Ultimate tensile strength F_{ u } (MPa) |
---|---|---|---|---|---|
Exterior connection (Test #2) | 46.2 | Beam longitudinal | #9(28.65mm) | 454.4 | 746 |
Column longitudinal | #7(22.23mm) | 469.5 | 741.9 | ||
Stirrups | #3(9.53mm) | 427.5 | 654.3 | ||
Interior connection (Test 01) | 41 | Beam and Column longitudinal | D24(24mm) | 325 | Not reported |
Stirrups | R6(19mm) | 339 | Not reported |
4 FINITE ELEMENT ANALYSIS
4.1 Elements
A uniform mesh size of 40 mm is chosen for the concrete elements over the whole geometry in both exterior and interior connection specimens as shown in Figure 8. The same size for reinforcement mesh is also adopted for steel bars. With this configuration, the exterior RC connection specimen has 11144 elements and 13414 nodes and the interior connection has 17414 elements and 21088 nodes. Details regarding element types are presented in Table 2.
4.2 Model Geometry and Boundary Conditions
Restraints were defined at both top and bottom surfaces of the specimen's column in the exterior test specimen according to boundary conditions addressed in the test setup. Restraints are also defined in the interior connection specimen according to the test setup boundary conditions both on beam and column tip surfaces. Details regarding to the geometry and boundary conditions of the RC exterior and interior beam-column connections which are applied to the finite element models are illustrated in Figure 9.
Loading is introduced to the model in two separate steps in exterior beam-column connection. The column compressive axial load is applied to the column top surface in the first step which remained constant during the analysis procedure.
The second step corresponds to the monotonic lateral loading of the specimen by applying the lateral displacement at the beam's end surface. However for the interior connection, only one step of load that is related to the lateral loading of the specimen is applied at column's top surface.
4.2 Material Parameters and Calibration of the Model
The numerical model is calibrated on the experimental test results of exterior connection specimen as a reference experiment. The calibrated numerical model is then utilized to conduct nonlinear finite element analysis on the interior connection specimen (Test01).
Concrete material parameters used in the presented analysis consists of concrete Young’s modulus of elasticity (_{ E0 } ), Poisson’s ratio (ν) and concrete compressive and tensile strengths. The poison’s ratio value for concrete material is considered to be equal to 0.2. The concrete damage plasticity input parameters which were discussed in section 2.3 are considered in the plasticity model as presented in Table.3.
Plasticity Parameters | Notation | Parameter's Value considered in the model |
---|---|---|
Dilation angle | ψ | 35º (calibrated) |
Shape factor | K_{c} | 0.667 (default value) |
Stress ratio | σ_{b0}/σ_{c0} | 1.16 (default value) |
Eccentricity | ∈ | 0.1 (default value) |
The uniaxial tensile stress-strain behavior of reinforcement was assumed to be elastic with Young’s modulus (_{ Es } = 2 × 10^{5} MPa) and Poisson’s ratio (_{ vs } = 0.3). The plastic behavior is also modeled including yield stress and corresponding plastic strain. Properties of plastic phase is defined to the model using bilinear behavior. Figure 10 illustrates the typical stress-strain relationship of reinforcement introduced to the numerical model. Typical reinforcement properties are also presented in Table 4 for both exterior and interior beam-column connections.
Connection specimen | Reinforcement type | Yield stress | Yield strain | Ultimate strength | Ultimate strain |
---|---|---|---|---|---|
F_{ y } (MPa) | ε_{ y } | F_{ u } (MPa) | ε_{ u } | ||
Exterior connection (Test#2) | Beam Longitudional bar | 454.4 | 00.00227 | 746 | 0.09 |
Column longitudinal bar | 469.5 | 0.00235 | 741.9 | 0.09 | |
Stirrups | 427.5 | 0.00214 | 654.3 | 0.11 | |
Interior connection (Test01) | Beam & Column longitudinal bar | 325 | 0.001625 | 396.5 | 0.10 |
Stirrups | 339 | 0.001695 | 413.6 | 0.11 |
In order to investigate the role of parameters in constitutive equations of damage plasticity model and to calibrate the model, effects of different logical values of the dilation angle and the viscosity parameter and mesh sensitivity analysis of the model are presented.
The dilatancy of concrete material represents the occurrence of volume expansion when the material is subjected to triaxial stress state and the consequent inelastic strain. Internal actions on the connection joint specimens under simulated lateral loads induce high shear stress in the joint core, an example of which compound stress state leads to considerable dilatancy sensitiveness of the analytical model. A sensitivity analysis on dilation angle as a concrete material parameter is performed to investigate the parameter influence on lateral load-displacement response of the reference specimen. Figure 11 shows that by increasing the value of dilation angle, the displacement capacity and the ultimate failure load of the connection are increased subsequently.
A range between 31° to 42° of the dilation angle parameter is recommended for concrete material according to series of fundamental studies performed by different authors ((Lee and Fenves, 1988), (^{Wu et al., 2006}) and (^{Voyiadjis et al., 2009})). As shown in Figure 11 a value of 35° can reasonably capture the lateral load-deformation curve and the failure mode.
The influence of visco-plastic regularization on the CDP constitutive equations through the introduction of viscosity parameter was another source of calibration process of the finite element analysis in this study. Defining visco-plastic regularization to the numerical model was first proposed as a method to improve the rate-dependent plastic damage model that brings uniqueness to the incremental stress field in the constitutive equations. It was shown that while the appropriate choice of viscosity parameter and defining it to the model provides relaxation time for visco-plastic system of equations to overcome convergence difficulties in the softening regime, the influence of considering different values of viscosity parameter in monotonic loading cases do not impose considerable amount of changes in the behavior of concrete material under uniaxial tension and compression ((^{Lee, 1996})). It was shown that best results are obtained by small values of viscosity parameter compared to the pseudo time of analysis in the study performed by (^{Lee, 1996}). Series of guidelines and recommendations are suggested by various researchers in order to consider best values of viscosity parameter in the CDP model depending on the type of predominant internal actions which are active and the degree of nonlinearity that involved in different problems ((^{Wosatko et al, 2015}), (^{Genikomsou and Polak, 2015}), (^{Wei Ren et al, 2015})).
The influence of considering additional relaxation time for the visco-plastic system on the analysis results using different values of viscosity parameter is shown in Figure 12.
According to the diagram of figure 12, the differences in the responses appear only at the softening and to some extent at the hardening regions. For constant element mesh size, when the viscosity parameter was taken as a relative small value of 0.007985, the calculation procedure of the numerical results gave us the most accurate response in comparison to the experimental results. Also the concrete cracking pattern and the nature of joint shear failure were consistent with the test results.
In order to analyze mesh size sensitivity of the numerical model in this study, the finite element analysis results of the exterior connection specimen with 40 mm and 25 mm element mesh sizes are presented in terms of lateral load-displacement curve in Figure 13. Although effects associated with mesh size and strain localization in the presented numerical model are within the margins of error expected for most numerical simulations based on plasticity models, slight differences are observed in the peak lateral loads and displacements.
Slight mesh size dependency is available in most distributed plasticity models which consider strain softening phenomenon in the constitutive equations ((^{Genikomsou and Polak, 2015})). It should be investigated that the mesh size dependency does not considerably affect the overall response of the specimen in FEA. It is notable that best results are obtained when the coarse mesh (40 mm) is used which may be due to the fact that most damage processes, causing concrete cracking propagation usually involve length scales in the order of two to three dominant aggregate sizes of the base concrete material ((^{Bazant, 1986})).
Among the various methods to reduce the mesh size dependency due to the strain-softening localization in limited number of elements, one method is to introduce the characteristic internal crack length at the softening branch of the stress-strain relationship into the constitutive model. The other method is the introduction of concrete visco-plastic regularization to the numerical model. The second approach is used in this study but the model is still slightly mesh size dependent however the influence of mesh size dependency on pushover lateral load-displacement behavior of the studied beam-column connections is within the acceptance limits for numerical simulations.
5 FINITE ELEMENT ANALYSIS RESULTS
The FEA results of the RC beam-column connections subjected to lateral loading is presented in terms of force-displacement curves, ultimate loads and displacements and cracking pattern which are monitored at different key points of shear behavior of the connection joints. Comparison between force-displacement curves predicted by simulation and experimental results of exterior and interior connection specimens are presented in Figure 14 and Figure 15, respectively. Figure 14a, and Figure 15a, show the finite element analysis results of force-displacement curves which are compared with the envelope curve of the cyclic loading response shown in Figure 14b and Figure 15b which are obtained by experimental works of (^{Clyde et al, 2000}) and (^{Hakuto et al, 2000}) for exterior and interior connection test specimens respectively.
Peak lateral loads and displacements predicted by the numerical simulation and reported by experimental test results are presented in Table 5.
Connection specimen | Experimental results | Finite element analysis results | ||
---|---|---|---|---|
Peak lateral load (kN) | Displacement at peak lateral load (mm) | Peak lateral load (kN) | Displacement at peak lateral load (mm) | |
Exterior connection (Test#2) | 267 | 23.77 | 268.82 | 24.5 |
Interior connection (Test 01) | 89 | 60.5 | 87.38 | 58.97 |
The finite element analysis shows that the initial response is slightly stiffer than the test results. This may be due to effects of some assumed variables such as the choice of concrete tensile and compressive properties, or the uncertainties often involved with experimental efforts such as probable existence of material deficiencies and also the inherent differences which exist between response obtained by cyclic loading of the experiment and monotonic loading in presented analysis may also contribute in limited stiffer initial response of FEA comparing to testing.
As shown in Figure 14 and Figure 15, three key points of shear behavior that are specified on the FEA resultant lateral load-displacement diagrams for both exterior and interior connection specimens, reasonably match with reported experimental results by (^{Clyde et al, 2000}) and (^{Hakuto et al, 2000}), respectively. Point A displays measurable beam flexural and joint shear cracking in exterior specimen and first diagonal shear cracking of the joint core in the interior specimen. Point B displays the ultimate strength of both exterior and interior specimens and finally point C is related to the end of FEA procedure attributed to 25% drop of ultimate strength in exterior connection specimen test and 30% drop of ultimate strength in the interior connection specimen test.
Accuracy of the numerical model in capturing joint shear behavior is monitored through four output parameters which effectively define overall joint shear behavior including concrete tensile and compressive damage, reinforcement output stress and joint cracking pattern. The CDP model simulates the nonlinear behavior of concrete both in tension and compression appropriately. A significant and challenging aspect of damage plasticity model application is the appropriate definition of damage in the constitutive model. The value of maximum principal plastic strain is the main indicator of cracking initiation in concrete damage plasticity model. Cracks initiate when maximum principal plastic strain is positive and the orientation of cracks is considered to be perpendicular to the maximum principal plastic strains, therefore in order to visualize the direction of cracking, the maximum principal plastic strains output is investigated. Figure 16 and Figure 17 display concrete damage both in tension and compression together with reinforcement Von-Mises stress output and joint cracking pattern at three introduced key points of joint shear behavior related to exterior test#2 and interior test unit-01, respectively.
Exterior beam-column connection specimen experienced beam tensile damage at point A as shown in Figure 16a. The concrete tensile damage extended to the vicinity of the joint and the beam longitudinal bars were placed under tension. No yielding of longitudinal beam bars and also no compressive damage in the joint core were observed at this stage. Measurable flexural cracks in the beam and limited shear cracks in the joint were the main source of slight stiffness change at point A which confirms reported experimental observations at this stage. Limited shear cracking of the joint rapidly spread to the whole joint at point B as shown in Figure 16b. Concrete tensile damage is developed to the joint region followed by yielding of the beam longitudinal reinforcement. The post peak behavior of the connection specimen begins at this point due to compressive damaging of the concrete diagonal strut.
Finally at the end of FEA procedure (point C) corresponding to 25% drop in ultimate strength, the failure of the connection specimen was occurred by crushing of the concrete diagonal strut in the joint region. Figure 16c shows the ultimate joint cracking pattern at point C. The ultimate failure of the beam-column connection specimen is attributed to the joint shear failure as in the experiments. Cracking pattern in the joint region obtained by FEA matches the cracking propagation pattern observed in the test. The diagonal cracks in the joint region are propagated to vicinity of the column.
Joint shear behavior described for the exterior beam-column connection specimen is also observed in the numerical output of the finite element analysis performed on the interior beam-column connection. Interior beam-column connection specimen experienced beam tensile damage at beam tension sides together with diagonal concrete tensile damage in the joint core at point A as shown in Figure 17a. Consequently, top of left side beam, bottom of right side beam and longitudinal bars of column on tension side were subjected to tension. No yielding of longitudinal beam bars and also no compressive damage in the joint core were observed similar to the exterior joint numerical output at this stage. Limited flexural cracks in the column and adjacent beams with diagonal tension cracking of the joint core were the main source of stiffness change at point A which is reported in experimental observations and obtained by the FEA outputs. Diagonal tension cracking of the joint rapidly spread to the vicinity of the column at point B as shown in Figure 17b. Concrete tensile damage is developed, and then is followed by yielding of the column and beam longitudinal reinforcement in the joint panel zone. The post peak behavior of the connection specimen also begins at this point due to the initiation of compressive damage in the concrete diagonal compressive strut.
Finally at the end of FEA procedure of the interior specimen (point C) corresponding to 30% drop in its ultimate strength, the failure of the connection specimen was occurred by crushing of the concrete diagonal strut in the joint region. As shown in Figure 17c the ultimate failure of the interior beam-column specimen is attributed to the joint shear failure as happened in the experiments. The numerical model shows good agreement between cracking pattern in the joint region obtained by FEA and the cracking propagation pattern observed in the test. Figure 18 presents a comparison between the cracking patterns of the joint region at ultimate strength (point B) for both exterior and interior specimens obtained by the FEA output and observed in the test results.
6 CONCLUSIONS
The study presented herein has endeavored to propose a suitable numerical model that describes the nonlinear shear behavior of reinforced concrete beam-column connections including poorly designed and detailed interior and exterior joints. Nonlinear finite element analysis of two RC beam-column connections with exterior and interior configurations is performed in order to capture the joint shear failure as the connection's governing failure mode when subjected to quasi static type lateral load. Concrete damage plasticity material model is applied to the numerical procedure as a distributed plasticity over the whole geometry of the specimens to appropriately simulate material nonlinearity. Two typical beam-column connections in reinforced concrete frame buildings built before the mid-1960s which were tested by other researchers under quasi-static cyclic loading were chosen as target model for this research. There was no transverse reinforcement considered in the joint panel zone.
The configuration of numerical model is implemented in finite element code ABAQUS. The finite element model is validated with experimental results. Numerical results presented in terms of joint shear capacity, deformations and cracking pattern at introduced key points of joint shear behavior conform appropriately to experimental results. The finite element analysis results confirms the capability of the developed finite element model in this study to predict RC beam-column joint shear behavior and can be further investigated and validated for different types of joints. The model can be used as an efficient and powerful numerical tool for further studies on different RC beam-column joints.