Parametric Analysis and Stiffness Investigation of Extended End-Plate Connection

Extended end-plate (EP) bolted connections are widely used in steel structures as moment-resisting connections. Most of these connections are semi-rigid or in other words flexible. The paper aims to study the behavior of such connections under the effect of column top-side cyclic loading using the finite element (FE) method. For semi-rigid connections, it is very vital to determine the moment-rotation relationship as well as the connection stiffness. These beam-column connections have been parametrically studied, the effect of joint type, shear forces, diameter of bolt, thickness of end-plate, and end-plate style were studied. Parametric studies show that the panel zone shear force is the key factor and has a significant effect on the connection stiffness. Finally, based on the component method, the stiffness of the bending component is improved, and the initial stiffness calculation model of the connection under column top-side cyclic loadings is established. The results show that the calculation model is in good agreement with the finite element analyses, and this proves that the calculation model proposed in this study could act as a reference method.


Introduction
Both the 1994 Northridge earthquake and the 1995 Hyogoken-Nanbu earthquake [1][2][3][4][5] resulted in widespread and unanticipated failures in steel frame beam-column welded joints. Plenty of investigations and researches indicate that the beam-column connection failure was caused by the brittle fracture of welds. The seismic behavior of semi-rigid connections, which exhibit better ductility than welded connections, has been widely studied. The semi-rigidity of the beam-column joint means that the relative rotation changes when the joint is subjected to certain moment, and the joint has certain rotational stiffness. The current design codes [6][7][8] of many countries require the moment-rotation (M-θ) relationship curve of the joint as the design basis.
Krishnamurthy, N. et al. [9] used flexible bolt connections for the first time in the United States, who developed a 3D FE analysis model and analyzed the influence of the end-plate thickness on the bolt connections; while Shi, G. et al. [10] compared the influence of dimensional parameters on the joint performance of eight flushes and extended end-plates and analyzed that the joint rotation mainly derives from the relative deformation between the end-plate and the column flange and the shearing deformation of the panel zone, respectively. Tao, W. et al. [11] used experiments and FE methods to analyze the space frame, considering the flexural resistance performance of the plane of component stiffness was improved, and a calculation model of joint initial rotation stiffness based on joint deformation was proposed, and verified with test data and parametric FE model.

Test Overview
In the experiment, six joint specimens of extended end-plate connections were designed, namely, the intermediate column joints (IC-EP1/2/3) and the edge column joint (EC-EP1/2/3), respectively. These joints were designed according to the standard for the design of steel structure [32], and the specimen size was derived from an actual high-rise building project. The length of the beam and the height of the column depend on the position of the frame anti-bending point. The details of the specimens are shown in Table 1. The variable parameters were the joint type, end-plate thickness, and bolt diameter. The beam was connected to the column by 10.9-grade friction high-strength bolts. Construction of high-strength bolts adopted the torque method to tighten. The initial tightening torque and final tightening torque of 10.9-grade M20 and M24 high-strength bolts are 280-446 N·m and 400-760 N·m, respectively. The corresponding pre-tightening (Fpre) of the two kinds of bolts is 155 kN and 225 kN. The beams and columns of the six test specimens all adopt hot-rolled I-shaped sections, the steel strength grade of all components of the joint is Q345B, and the basic configuration of the joints are shown in Figure 1. The end-plates and beams are connected by fully-penetrating butt welds, and all welds in the test specimen are first-grade welds. The friction surface of the connected member was prepared by sandblasting to obtain a friction coefficient of 0.44. The above two processes are completed in the factory standard workshop. A 1250 kN axial load was applied to the column top and remained constant by using a hydraulic jack. The ratio of the axial load was 0.3 for the columns.  The test setup is shown in Figure 2. The column was connected to the MTS hydraulic actuator through the loading plate. The beam tips are roller supported, the column bottom is hinged to the foundation, and the column top is connected to the reaction frame through directional support. Refer to the loading protocol of SAC joint venture (1997) (Figure 3), and the loading was controlled by story drift displacement in the whole process. The test terminated when joint failure occurred or the load device limit was reached. Table 1. Configuration details of joint specimens. The test setup is shown in Figure 2. The column was connected to the MTS hydraulic actuator through the loading plate. The beam tips are roller supported, the column bottom is hinged to the foundation, and the column top is connected to the reaction frame through directional support. Refer to the loading protocol of SAC joint venture (1997) (Figure 3), and the loading was controlled by story Materials 2020, 13, 5133 4 of 30 drift displacement in the whole process. The test terminated when joint failure occurred or the load device limit was reached.     The moment-rotation responses for all specimens are illustrated in Figure 4, and the test results are provided in Table 2. (The specimens IC-EP1/2 and EC-EP1/2 refer to previous research work [31]), and the specimen IC-EP3/EC-EP3 are newly added test data). Figure 4a,b shows that slight differences existed between the west connection and east connection. This conforms to the laws of mechanics, and both side connections with identical parameters are the same mechanical behaviors under symmetrical loading. However, compared with Figure 4c,d the initial rotational stiffness is quite different. This is because of the two forms of joint panel zones in different stress states and boundary conditions.  [31].
Materials 2020, 13    The moment-rotation responses for all specimens are illustrated in Figure 4, and the test results are provided in Table 2. (The specimens IC-EP1/2 and EC-EP1/2 refer to previous research work [31]), and the specimen IC-EP3/EC-EP3 are newly added test data). Figure 4a,b shows that slight differences existed between the west connection and east connection. This conforms to the laws of mechanics, and both side connections with identical parameters are the same mechanical behaviors under symmetrical loading. However, compared with Figure 4c,d the initial rotational stiffness is quite different. This is because of the two forms of joint panel zones in different stress states and boundary conditions. The moment-rotation responses for all specimens are illustrated in Figure 4, and the test results are provided in Table 2. (The specimens IC-EP1/2 and EC-EP1/2 refer to previous research work [31]), and the specimen IC-EP3/EC-EP3 are newly added test data. Figure 4a,b shows that slight differences existed between the west connection and east connection. This conforms to the laws of mechanics, and both side connections with identical parameters are the same mechanical behaviors under symmetrical loading. However, compared with Figure 4c,d the initial rotational stiffness is quite different. This is because of the two forms of joint panel zones in different stress states and boundary conditions.

Finite Element Modeling
The extended end-plate connection analysis model was established using Abaqus Standard ® [33] module. The nonlinear finite element (FE) method can save expensive cost and time of experimental work, and effectively avoid uncontrollable errors during test processes, which can intuitively reflect the stress distribution of each component in the FE simulations. On one side or both sides of column flanges, set extend end-plate connected to the beam as the object of this study, that is, intermediate column (IC) joint and edge column (EC) joint. The type and loading method of these research joints were evaluated. The symmetrical boundary conditions were worth considering, hence, the half-   Table note: K ji is the initial rotational stiffness of the joints; M y , M max , θ y , and θ u are defined by the key parameters of the connections; µ θ is the ductility coefficient of the test specimen; and µ θ = θ u /θ y .

Finite Element Modeling
The extended end-plate connection analysis model was established using Abaqus Standard ® [33] module. The nonlinear finite element (FE) method can save expensive cost and time of experimental work, and effectively avoid uncontrollable errors during test processes, which can intuitively reflect the stress distribution of each component in the FE simulations. On one side or both sides of column flanges, set extend end-plate connected to the beam as the object of this study, that is, intermediate column (IC) joint and edge column (EC) joint. The type and loading method of these research joints were evaluated. The symmetrical boundary conditions were worth considering, hence, the half-model (FE 1/2) and the quarter-model (FE 1/4) were established. These numerical models can reduce the occupation of storage space; meanwhile, an exploratory finite element analysis was conducted for the subsequent parameter analysis, expecting to find a model with moderate mesh density and acceptable Materials 2020, 13, 5133 6 of 30 accuracy in results. Additionally, the FE model that has been developed can effectively verify the six joint test data, paving the way for the content of the following section.

Material Models
The stress-strain relationship of the steel can be the simplified trilinear model and considers the plastic hardening of the material. The Von Mises yield criterion is adopted to determine whether the steel reaches the yield point in the multi-axial stress state. When the equivalent stress of the steel exceeds the yield stress, the steel will undergo plastic deformation. A bilinear kinematic hardening model was applied to the high-strength bolt constitutive, which is very suitable for high-strength steel. The material parameters of the FE model correspond to every actual tensile coupon test result to better verify the mechanical properties of the joint. Poisson's ratio was assumed to be 0.3. See Table 3 for detailed information about the material properties of steel and bolts. The stress-strain relationship of the steel and bolt can be simplified as Figure 5 with the material constitutive curve presented by Bahaari, M. and Sherbourne, A.N. [34].
the occupation of storage space; meanwhile, an exploratory finite element analysis was conducted for the subsequent parameter analysis, expecting to find a model with moderate mesh density and acceptable accuracy in results. Additionally, the FE model that has been developed can effectively verify the six joint test data, paving the way for the content of the following section.

Material Models
The stress-strain relationship of the steel can be the simplified trilinear model and considers the plastic hardening of the material. The Von Mises yield criterion is adopted to determine whether the steel reaches the yield point in the multi-axial stress state. When the equivalent stress of the steel exceeds the yield stress, the steel will undergo plastic deformation. A bilinear kinematic hardening model was applied to the high-strength bolt constitutive, which is very suitable for high-strength steel. The material parameters of the FE model correspond to every actual tensile coupon test result to better verify the mechanical properties of the joint. Poisson's ratio was assumed to be 0.3. See Table  3 for detailed information about the material properties of steel and bolts. The stress-strain relationship of the steel and bolt can be simplified as Figure 5 with the material constitutive curve presented by Bahaari, M. and Sherbourne, A.N. [34] (a) (b)

Finite Element Modeling
All parts were modeled using the 8-node linear brick incompatible mode element (C3D8R), which reduced integration and used hourglass control. The model is divided into five parts, namely, beams, columns, end-plates, bolts, and web stiffeners. Tie contact was used for the welding relationship between the end-plates and steel beam and does not consider other weld modeling. The general mesh size for the entire model was moderate mesh density, and there were at least three layers in the thickness direction. All components were controlled by a structured mesh. The abovementioned IC joint of the full model (FE all), the half-model (FE 1/2), and the quarter-model (FE 1/4) corresponding the total number of elements is about 104,000, 52,000, and 26,000 elements, respectively, and normal hard contact to simulate the extrusion phenomenon between bolts and plate. Tangential penalty function was used to simulate friction between the end-plate and the column flange (the friction coefficient of 0.44). The shapes and mesh division diagrams of the components are shown in Figures 6 and 7.

Finite Element Modeling
All parts were modeled using the 8-node linear brick incompatible mode element (C3D8R), which reduced integration and used hourglass control. The model is divided into five parts, namely, beams, columns, end-plates, bolts, and web stiffeners. Tie contact was used for the welding relationship between the end-plates and steel beam and does not consider other weld modeling. The general mesh size for the entire model was moderate mesh density, and there were at least three layers in the thickness direction. All components were controlled by a structured mesh. The above-mentioned IC joint of the full model (FE all), the half-model (FE 1/2), and the quarter-model (FE 1/4) corresponding the total number of elements is about 104,000, 52,000, and 26,000 elements, respectively, and normal hard contact to simulate the extrusion phenomenon between bolts and plate. Tangential penalty function was used to simulate friction between the end-plate and the column flange (the friction coefficient of 0.44). The shapes and mesh division diagrams of the components are shown in Figures 6 and 7.

Boundary Conditions and Symmetry
Considering the symmetry boundary conditions and the loading direction, to divide an appropriate element mesh, the accuracy of the finite element analysis result can be guaranteed. Due to the difference of boundary conditions, the IC joint adopts half-model (FE 1/2) and quarter-model (FE 1/4), and the EC joint only adopts half-model (FE 1/2) for simulation calculation. The above symmetrical models combined themselves into a full-model (FE all), for specific values refer to Table  4, which were compared with the experimental results and aim to explore a kind of analysis model. The model not only meets the requirements of analysis accuracy but also saves calculation costs as much as possible, and does an exploratory study for the parameterized analysis in the subsequent sections. The boundary conditions of the model are consistent with the experimental settings of the specimen, the symmetrical model only deforms in the XOZ plane, so the initial setting limits the UY translation direction and RX and RZ rotation. The top of the column is directional support, which corresponds to a restricted rotation in the RY direction; the bottom of the column is hinged support, this restricts UX and UZ translation; and the beam tip is roller support, which only restricts UZ translation.  Figure 7a,b

Boundary Conditions and Symmetry
Considering the symmetry boundary conditions and the loading direction, to divide an appropriate element mesh, the accuracy of the finite element analysis result can be guaranteed. Due to the difference of boundary conditions, the IC joint adopts half-model (FE 1/2) and quarter-model (FE 1/4), and the EC joint only adopts half-model (FE 1/2) for simulation calculation. The above symmetrical models combined themselves into a full-model (FE all), for specific values refer to Table  4, which were compared with the experimental results and aim to explore a kind of analysis model. The model not only meets the requirements of analysis accuracy but also saves calculation costs as much as possible, and does an exploratory study for the parameterized analysis in the subsequent sections. The boundary conditions of the model are consistent with the experimental settings of the specimen, the symmetrical model only deforms in the XOZ plane, so the initial setting limits the UY translation direction and RX and RZ rotation. The top of the column is directional support, which corresponds to a restricted rotation in the RY direction; the bottom of the column is hinged support, this restricts UX and UZ translation; and the beam tip is roller support, which only restricts UZ translation.

Boundary Conditions and Symmetry
Considering the symmetry boundary conditions and the loading direction, to divide an appropriate element mesh, the accuracy of the finite element analysis result can be guaranteed. Due to the difference of boundary conditions, the IC joint adopts half-model (FE 1/2) and quarter-model (FE 1/4), and the EC joint only adopts half-model (FE 1/2) for simulation calculation. The above symmetrical models combined themselves into a full-model (FE all), for specific values refer to Table 4, which were compared with the experimental results and aim to explore a kind of analysis model. The model not only meets the requirements of analysis accuracy but also saves calculation costs as much as possible, and does an exploratory study for the parameterized analysis in the subsequent sections. The boundary conditions of the model are consistent with the experimental settings of the specimen, the symmetrical model only deforms in the XOZ plane, so the initial setting limits the UY translation direction and RX and RZ rotation. The top of the column is directional support, which corresponds to a restricted rotation in the RY direction; the bottom of the column is hinged support, this restricts UX and UZ translation; and the beam tip is roller support, which only restricts UZ translation.  Figure 7a,b

Loading Type
In order to simulate the loading effect and boundary constraint of the beam tip and the column top-side, coupling constraints are applied to the specified area to eliminate unrealistic stress and strain concentration. Each model applies three types of loads, the first was the constant load that was applied at the middle of the bolt shank to simulate the bolt pretension force, the second is to maintain an axial compression ratio of 0.3 and apply a constant pressure value on the column-top, while the third was the cyclic load that the displacement was applied in the form of small steps at the column top-side to generate the moment on the connection. The cyclic displacement history of FE is similar to the experimental loading protocol for increasing amplitude, and both were incrementally imposed.

Finite Element Analysis and Test Result Comparison
These FE models were established and solved by using Abaqus software (Version 6.14, SIMULIA, RI, USA). As shown in Figure 8, the FE and test hysteresis curves for all specimens show satisfactory agreement. As the load-displacement increases in the later stage, local differences begin to appear between the two. This is because the constitutive relationship of the steel used for the FE is that the elastoplasticity of kinematic hardening is different from the constitutive curve of the actual material in the test, and the initial geometric defects and testing errors make FE and testing slightly different. However, the error range is within the controllable area, and the finite element analysis and test hysteresis curves of all models show satisfactory consistency. Load asymmetry

Loading Type
In order to simulate the loading effect and boundary constraint of the beam tip and the column top-side, coupling constraints are applied to the specified area to eliminate unrealistic stress and strain concentration. Each model applies three types of loads, the first was the constant load that was applied at the middle of the bolt shank to simulate the bolt pretension force, the second is to maintain an axial compression ratio of 0.3 and apply a constant pressure value on the column-top, while the third was the cyclic load that the displacement was applied in the form of small steps at the column top-side to generate the moment on the connection. The cyclic displacement history of FE is similar to the experimental loading protocol for increasing amplitude, and both were incrementally imposed.

Finite Element Analysis and Test Result Comparison
These FE models were established and solved by using Abaqus software (Version 6.14, SIMULIA, RI, USA). As shown in Figure 8, the FE and test hysteresis curves for all specimens show satisfactory agreement. As the load-displacement increases in the later stage, local differences begin to appear between the two. This is because the constitutive relationship of the steel used for the FE is that the elastoplasticity of kinematic hardening is different from the constitutive curve of the actual material in the test, and the initial geometric defects and testing errors make FE and testing slightly different. However, the error range is within the controllable area, and the finite element analysis and test hysteresis curves of all models show satisfactory consistency. As presented in Figure 9c,d, the IC-EP1 model shows that the failure mode was the welding between the beam flange and end-plate reached the ultimate stress, and then reaches the bolt yield stress, which also corresponds to the bolt breaking phenomenon in the test (in Table 2). The IC-EP2 model had serious local buckling in the column web (Figure 9a), and the failure mode was excessive shear deformation of the panel zone in the test (in Table 2). Both the numerical model and the experimental failure phenomenon of the EC-EP1/2/3 joint show large bending deformation of the end-plate. (in Figure 9 b,e,f and Table 2). As presented in Figure 9c,d, the IC-EP1 model shows that the failure mode was the welding between the beam flange and end-plate reached the ultimate stress, and then reaches the bolt yield stress, which also corresponds to the bolt breaking phenomenon in the test (in Table 2). The IC-EP2 model had serious local buckling in the column web (Figure 9a), and the failure mode was excessive shear deformation of the panel zone in the test (in Table 2). Both the numerical model and the experimental failure phenomenon of the EC-EP1/2/3 joint show large bending deformation of the end-plate. (in Figure 9b,e,f and Table 2).
As shown in Figure 10, when the joint rotation reaches 0.05 rad, it shows that all the key components of the joint have buckled, which was reflected when the joint's ultimate moment begins to decline and the secant stiffness degrades to a very low level in the moment-rotation (M-θ) skeleton curve. The M-θ skeleton curves attained by numerical calculations in comparison with the moment-rotation skeleton Materials 2020, 13, 5133 9 of 30 curves obtained from the tests. The results show that there was good agreement between the test results and the FE simulation in the elastic stage. Additionally, in the stage after yielding, there was some disparity due to the simplification of the material constitutive behavior of the materials and test errors. Materials 2020, 13, x FOR PEER REVIEW 9 of 31 As shown in Figure 10, when the joint rotation reaches 0.05 rad, it shows that all the key components of the joint have buckled, which was reflected when the joint's ultimate moment begins to decline and the secant stiffness degrades to a very low level in the moment-rotation (M-θ) skeleton curve. The M-θ skeleton curves attained by numerical calculations in comparison with the momentrotation skeleton curves obtained from the tests. The results show that there was good agreement between the test results and the FE simulation in the elastic stage. Additionally, in the stage after yielding, there was some disparity due to the simplification of the material constitutive behavior of the materials and test errors. Table 5 lists the comparison between the finite element analysis and the test results. The ratio of initial rotational stiffness was defined as the finite element data divided by the test result. The mean and standard deviation of this ratio are 1.003 and 0.064, respectively. Therefore, it can be concluded that in terms of accuracy, calculation time, and storage space, the finite element analysis of the IC joint and the EC joint using the quarter-model (FE 1/4) and the half-model (FE 1/2) can meet subsequent parameter research and stiffness analysis requirements.  Table 5 lists the comparison between the finite element analysis and the test results. The ratio of initial rotational stiffness was defined as the finite element data divided by the test result. The mean and standard deviation of this ratio are 1.003 and 0.064, respectively. Therefore, it can be concluded that in terms of accuracy, calculation time, and storage space, the finite element analysis of the IC joint and the EC joint using the quarter-model (FE 1/4) and the half-model (FE 1/2) can meet subsequent parameter research and stiffness analysis requirements.

Parametric Research
Based on the above finite element verification and experimental research, a parameter study was conducted to develop three-dimensional FE models with variable parameters to simulate the IC joints and EC joints of the extended end-plates connection. Both material and geometry nonlinearities were considered in the analysis. This study aims to understand the connection behavior of such joints under cyclic loading on the column top-side and to determine the valid parameters on ultimate moment, rotation capacity and initial rotational stiffness.

Model Description
A total number of 144 3D-FE models were created for the connection of the extended end-plate beam to column joints to study their behavior under cyclic loading. These models and the investigated parameters are summarized in Figure 11. The model contains IC joints and EC joints with the same size information, and the number of models each accounts for half; The beam of FE model uses the same section (H300 × 200 × 8 × 12 mm), The specific parameters: two groups of column profile (H300 × 300 × 10 × 15 mm and H300 × 250 × 8 × 12 mm), the beam length (800, 1500 mm), the thickness of the end-plates (12, 16 and 20 mm), the diameter of the bolts (16, 20 and 24 mm) and the end-plate styles S1 and S2, Refer to Figure 12 for specific configuration information. The type of these bolts is 10.9 high-strength bolts, and the corresponding pretensions are 58 kN, 155 kN and 225 kN, respectively. The model gives joint connection size information according to the label to create different models, and the model label number can be used to extract data for each model. Because the half mode (FE 1/2) and quarter mode (FE 1/4) of the EC joints and the IC joints are given in the above section of the FE verification, the accuracy can meet the analysis accuracy requirements. Hence, in accordance with the above-mentioned rules, the symmetric model was adopted for parametric analysis. In Table 6, model labels 001-072 are the IC joints, and model labels 073-144 are the EC joints, except for setting the difference between the single side and double side connection, both account for half and other parameters are the same.
As shown in Figure 13, the detailed mesh and boundary conditions are given. The material model, element type, boundary conditions, and load type are consistent with the previous modeling content of the finite element verification section. The model calculation results are summarized in Table 6.

Failure Modes
The FE analysis shows the different typical failure modes in Figure 14. According to the location of the failure, it can be divided into six representative failure modes, namely bolt failure(BF), column flange failure(CFF), column panel zone buckling(CPZB), end-plate failure(EPF), weld between column flange and column stiffener(WCF-CS), and weld between end-plate and beam flange(WEP-BF). As shown in Figure 14a, for connections with medium thickness end-plate and bolt diameters (89-ECBC1D20t16L8S1), the failure only occurs in the weld between the end-plates and the beam flange (WEP-BF) when it reached the ultimate stress. It can be observed that as shown in Figure 14b,c, the connections with large bolt diameter (31-ICBC1D24t16L15S1), their failure model is panel zone buckling occurs at the column web (CPZB), because of the IC joint bear enormous moment, this results in immense shear stress on the column web. By referring to Table 6, in the model (91-ECBC1D20t16L15S1), due to the EC joint only on one side connection, the deformation of the column web area is slight, yet the connection deformation is obvious. Thus the failure occurs in the end-plate (EPF). Simultaneously, for models with all thin end-plates (12 mm), the failure is almost due to excessive bending deformation of the end-plates (EPF). For the thick end-plate and crude bolt diameter models (095-ECBC1D20t20L15S1), the failure occurs at the column flange (CFF), because the connection part is strong and the column flange is relatively weak. For larger column profiles (H300 × 300 × 10 × 15 mm), the stiffener weld (WCF-CS) between the column web and the column flange has failed (77-ECBC1D16t16L8S1). In the model (11-ICBC1D16t20L15S1), the bolt reaches its ultimate stress and fails (BF). As shown in Figures 15 and 16, the ultimate moment (M max ) of the joint is always less than the plastic flexural resistance (M bp ) of the beam or column. Therefore, the plastic hinge did not occur in the beam section, and the column section is not buckled. Failure always occurs in the connection component or panel zone. with the same size information, and the number of models each accounts for half; The beam of FE model uses the same section (H300 × 200 × 8 × 12 mm), The specific parameters: two groups of column profile (H300 × 300 × 10 × 15 mm and H300 × 250 × 8 × 12 mm), the beam length (800, 1500 mm), the thickness of the end-plates (12, 16 and 20 mm), the diameter of the bolts (16, 20 and 24 mm) and the end-plate styles S1 and S2, Refer to Figure 12 for specific configuration information. The type of these bolts is 10.9 high-strength bolts, and the corresponding pretensions are 58 kN, 155 kN and 225 kN, respectively. The model gives joint connection size information according to the label to create different models, and the model label number can be used to extract data for each model. Because the half mode (FE 1/2) and quarter mode (FE 1/4) of the EC joints and the IC joints are given in the above section of the FE verification, the accuracy can meet the analysis accuracy requirements. Hence, in accordance with the above-mentioned rules, the symmetric model was adopted for parametric analysis. In Table 6, model labels 001-072 are the IC joints, and model labels 073-144 are the EC joints, except for setting the difference between the single side and double side connection, both account for half and other parameters are the same. As shown in Figure 13, the detailed mesh and boundary conditions are given. The material model, element type, boundary conditions, and load type are consistent with the previous modeling content of the finite element verification section. The model calculation results are summarized in Table 6  with the same size information, and the number of models each accounts for half; The beam of FE model uses the same section (H300 × 200 × 8 × 12 mm), The specific parameters: two groups of column profile (H300 × 300 × 10 × 15 mm and H300 × 250 × 8 × 12 mm), the beam length (800, 1500 mm), the thickness of the end-plates (12, 16 and 20 mm), the diameter of the bolts (16, 20 and 24 mm) and the end-plate styles S1 and S2, Refer to Figure 12 for specific configuration information. The type of these bolts is 10.9 high-strength bolts, and the corresponding pretensions are 58 kN, 155 kN and 225 kN, respectively. The model gives joint connection size information according to the label to create different models, and the model label number can be used to extract data for each model. Because the half mode (FE 1/2) and quarter mode (FE 1/4) of the EC joints and the IC joints are given in the above section of the FE verification, the accuracy can meet the analysis accuracy requirements. Hence, in accordance with the above-mentioned rules, the symmetric model was adopted for parametric analysis. In Table 6, model labels 001-072 are the IC joints, and model labels 073-144 are the EC joints, except for setting the difference between the single side and double side connection, both account for half and other parameters are the same. As shown in Figure 13, the detailed mesh and boundary conditions are given. The material model, element type, boundary conditions, and load type are consistent with the previous modeling content of the finite element verification section. The model calculation results are summarized in Table 6

Effect of Shear and Column Size
The column top-side loading method is extremely sensitive to the shear effect of the panel zone, and the shear deformation of IC joints is more obvious than that of EC joints. Adopting different beam lengths to change the moments of the connections affects the shear force in the panel zone. Figure 15a,b shows two groups of curves: group (I) represents the results of the large column section (H300 × 300 × 10 × 15 mm) and group (II) represents the results of the small column section (H300 × 250 × 8 × 12 mm). In each group, the following relationships are shown: (1) beam length variation, (2) different end-plate styles, and (3) column section. For large column sections (num 017, 019, 018, and 020 in Figure 15a), the higher the shear value, the larger the shear angular rotation, and the larger the connection stiffness as well. On the other hand, for small column sections (125, 127, 126, and 128 in Figure 15b), for a lower shear force, the ultimate flexural resistance capacity is increased by 20% as a whole, but the relative rotation of the limit is reduced. Figure 15 shows that at the same value of rotation of the large column section or the small column section, the connection limit rotation is almost the same, but in considering the stiffness degradation, the IC joint with more considerable shear value has the highest stiffness at the connection hardening starting point, then the connection stiffness at the ultimate shear rotation has the minimum value ( Figure 16).

Failure Modes
The FE analysis shows the different typical failure modes in Figure 14. According to the location of the failure, it can be divided into six representative failure modes, namely bolt failure(BF), column flange failure(CFF), column panel zone buckling(CPZB), end-plate failure(EPF), weld between column flange and column stiffener(WCF-CS), and weld between end-plate and beam flange(WEP-BF). As shown in Figure 14a, for connections with medium thickness end-plate and bolt diameters (89-ECBC1D20t16L8S1), the failure only occurs in the weld between the end-plates and the beam flange (WEP-BF) when it reached the ultimate stress. It can be observed that as shown in Figure 14b,c, the connections with large bolt diameter (31-ICBC1D24t16L15S1), their failure model is panel zone buckling occurs at the column web (CPZB), because of the IC joint bear enormous moment, this results in immense shear stress on the column web. By referring to Table 6, in the model (91-ECBC1D20t16L15S1), due to the EC joint only on one side connection, the deformation of the column web area is slight, yet the connection deformation is obvious. Thus the failure occurs in the end-plate (EPF). Simultaneously, for models with all thin end-plates (12 mm), the failure is almost due to excessive bending deformation of the end-plates (EPF). For the thick end-plate and crude bolt diameter models (095-ECBC1D20t20L15S1), the failure occurs at the column flange (CFF), because the connection part is strong and the column flange is relatively weak. For larger column profiles (H300 × 300 × 10 × 15 mm), the stiffener weld (WCF-CS) between the column web and the column flange has failed (77-ECBC1D16t16L8S1). In the model (11-ICBC1D16t20L15S1), the bolt reaches its ultimate stress and fails (BF). As shown in Figures 15 and 16, the ultimate moment (Mmax) of the joint is always less than the plastic flexural resistance (Mbp) of the beam or column. Therefore, the plastic hinge did not occur in the beam section, and the column section is not buckled. Failure always occurs in the connection component or panel zone.

Effect of Shear and Column Size
The column top-side loading method is extremely sensitive to the shear effect of the panel zone, and the shear deformation of IC joints is more obvious than that of EC joints. Adopting different beam lengths to change the moments of the connections affects the shear force in the panel zone.

Effect of End-Plate Thickness
The effect of end-plate thickness on the moment-rotation curve of the connection is shown in Figure 16, which indicates that the connection flexural resistance capacity is increased with an increase in the thickness of end-plates, but the ultimate rotation of the connection is decreased with an increase in the thickness of end-plates, resulting in low connection ductility. Furthermore, the initial stiffness of the connection increases with an increase in the end-plate thickness. These figures show that, in most cases, increasing the thickness of end-plate causes the increase in the stiffness of the joint to vary between 4% and 15%, while the thick end-plate has the effect of slowing the degradation of the stiffness of joints. Additionally, the joints with a medium end-plate thickness (16 mm) have the most significant rotation capability. However, for the thin end-plate thickness (12 mm), its failure mode is EPF or WEP-BF. This is because the end-plate is so thin that the connection does not exert the comprehensive performance of the joint and fails prematurely. Figure 17 shows the effect of changing the end-plate thickness on the ultimate flexural resistance of the connection. Under the condition of the same bolt diameter and beam length, the column section (large column: H300 × 300 × 10 × 15 mm; small column: H300 × 250 × 8 × 12 mm), IC joint, and EC joint are used as variables. For the small size column (H300 × 250 × 8 × 12 mm) of the connection, when the end-plate thickness is increased from 12 mm to 20 mm, the ultimate load of the IC joints and EC joints are increased by about 20%. Meanwhile, when the end-plate with a thickness of 16 mm is used, the ultimate load of the IC joint increases by about 9.7% compared to the EC joint, because the failure of the connection occurred in the column flange. For large-sized column (H300 × 300 × 10 × 15 mm) joints, when the thickness of the end-plate is increased from 12 to 20 mm, the ultimate flexural resistance of the IC joint and EC joint increases by 19% and 15%, respectively. As displayed in Figure 16, the endplate thickness has a significant effect on the initial joint stiffness, but it only has a partial effect on the stiffness degradation of the entire loading process. The thinner the end-plate, the faster the stiffness degradation, and vice versa.

Effect of End-Plate Thickness
The effect of end-plate thickness on the moment-rotation curve of the connection is shown in Figure 16, which indicates that the connection flexural resistance capacity is increased with an increase in the thickness of end-plates, but the ultimate rotation of the connection is decreased with an increase in the thickness of end-plates, resulting in low connection ductility. Furthermore, the initial stiffness of the connection increases with an increase in the end-plate thickness. These figures show that, in most cases, increasing the thickness of end-plate causes the increase in the stiffness of the joint to vary between 4% and 15%, while the thick end-plate has the effect of slowing the degradation of the stiffness of joints. Additionally, the joints with a medium end-plate thickness (16 mm) have the most significant rotation capability. However, for the thin end-plate thickness (12 mm), its failure mode is EPF or WEP-BF. This is because the end-plate is so thin that the connection does not exert the comprehensive performance of the joint and fails prematurely. Figure 17 shows the effect of changing the end-plate thickness on the ultimate flexural resistance of the connection. Under the condition of the same bolt diameter and beam length, the column section (large column: H300 × 300 × 10 × 15 mm; small column: H300 × 250 × 8 × 12 mm), IC joint, and EC joint are used as variables. For the small size column (H300 × 250 × 8 × 12 mm) of the connection, when the end-plate thickness is increased from 12 mm to 20 mm, the ultimate load of the IC joints and EC joints are increased by about 20%. Meanwhile, when the end-plate with a thickness of 16 mm is used, the ultimate load of the IC joint increases by about 9.7% compared to the EC joint, because the failure of the connection occurred in the column flange. For large-sized column (H300 × 300 × 10 × 15 mm) joints, when the thickness of the end-plate is increased from 12 to 20 mm, the ultimate flexural resistance of the IC joint and EC joint increases by 19% and 15%, respectively. As displayed in Figure 16, the end-plate thickness has a significant effect on the initial joint stiffness, but it only has a partial effect on the stiffness degradation of the entire loading process. The thinner the end-plate, the faster the stiffness degradation, and vice versa.

Effect of Bolt Diameter
The influence of the bolt diameter on the ultimate flexural resistance of the connection is shown in Figure 18, which indicates that an increase in the bolt diameter can increase the ultimate flexural capacity of the connection, and the initial rotational rigidity will also increase (refer to Table 6). Under the conditions of the constant end-plate thickness (16 mm), beam length (1500 mm) and column section size (H300 × 300 × 10 × 15 mm, H300 × 250 × 8 × 12 mm), by changing the bolt diameter (16, 20, and 24 mm), the IC joints and EC joints are analyzed. For the larger column section group (H300 × 300 × 10 × 15 mm), regardless of whether it is an IC joint or an EC joint, when the bolt diameter increases from 16 to 24 mm, the ultimate load value increases by 32.1% and 25.6%, respectively. For the small column section group (H300 × 300 × 10 × 15 mm), the ultimate flexural resistance value only increases by about 10%. It is caused by the different weak components in the failure modes of the above two group models. On the other hand, when the bolt diameter is 20 mm, the IC joint and EC joint adopt the same column cross-section. Compared with both, the ultimate flexural resistance value of the IC joint is only slightly improved, which shows that the IC joint and EC joint are equivalent in ultimate flexural resistance capacity. By referring to Table 6, these connection joints with bolt diameter (16 mm and 20 mm) and medium thickness end-plate (16 mm and 20 mm) show excellent overall behavior of the connection in terms of the ultimate moment, joint stiffness, and ultimate rotation capacity.

Effect of Bolt Diameter
The influence of the bolt diameter on the ultimate flexural resistance of the connection is shown in Figure 18, which indicates that an increase in the bolt diameter can increase the ultimate flexural capacity of the connection, and the initial rotational rigidity will also increase (refer to Table 6). Under the conditions of the constant end-plate thickness (16 mm), beam length (1500 mm) and column section size (H300 × 300 × 10 × 15 mm, H300 × 250 × 8 × 12 mm), by changing the bolt diameter (16,20, and 24 mm), the IC joints and EC joints are analyzed. For the larger column section group (H300 × 300 × 10 × 15 mm), regardless of whether it is an IC joint or an EC joint, when the bolt diameter increases from 16 to 24 mm, the ultimate load value increases by 32.1% and 25.6%, respectively. For the small column section group (H300 × 300 × 10 × 15 mm), the ultimate flexural resistance value only increases by about 10%. It is caused by the different weak components in the failure modes of the above two group models. On the other hand, when the bolt diameter is 20 mm, the IC joint and EC joint adopt the same column cross-section. Compared with both, the ultimate flexural resistance value of the IC joint is only slightly improved, which shows that the IC joint and EC joint are equivalent in ultimate flexural resistance capacity. By referring to Table 6, these connection joints with bolt diameter (16 mm and 20 mm) and medium thickness end-plate (16 mm and 20 mm) show excellent overall behavior of the connection in terms of the ultimate moment, joint stiffness, and ultimate rotation capacity.

Effect of Bolt Diameter
The influence of the bolt diameter on the ultimate flexural resistance of the connection is shown in Figure 18, which indicates that an increase in the bolt diameter can increase the ultimate flexural capacity of the connection, and the initial rotational rigidity will also increase (refer to Table 6). Under the conditions of the constant end-plate thickness (16 mm), beam length (1500 mm) and column section size (H300 × 300 × 10 × 15 mm, H300 × 250 × 8 × 12 mm), by changing the bolt diameter (16,20, and 24 mm), the IC joints and EC joints are analyzed. For the larger column section group (H300 × 300 × 10 × 15 mm), regardless of whether it is an IC joint or an EC joint, when the bolt diameter increases from 16 to 24 mm, the ultimate load value increases by 32.1% and 25.6%, respectively. For the small column section group (H300 × 300 × 10 × 15 mm), the ultimate flexural resistance value only increases by about 10%. It is caused by the different weak components in the failure modes of the above two group models. On the other hand, when the bolt diameter is 20 mm, the IC joint and EC joint adopt the same column cross-section. Compared with both, the ultimate flexural resistance value of the IC joint is only slightly improved, which shows that the IC joint and EC joint are equivalent in ultimate flexural resistance capacity. By referring to Table 6, these connection joints with bolt diameter (16 mm and 20 mm) and medium thickness end-plate (16 mm and 20 mm) show excellent overall behavior of the connection in terms of the ultimate moment, joint stiffness, and ultimate rotation capacity.

Effect of End-Plate Section Size
Comparing the end-plates S1 and S2, the difference is the distance between each row of bolt holes and the end-plate size. The influence on the initial rotational stiffness of the joints is shown in Figure 19. Overall, these differences have an impact on the initial rotational stiffness of the joint. Because the bolt's hole distance in the end-plate S2 is smaller than end-plate S1, this restricts the bending deformation of the column flange and the end-plate, making this component contribute more to the stiffness of the joint, while the ultimate flexural resistance capacity is greatly affected by the failure mode, certain weak components may fail before the end-plate.
By referring to Table 6, it is found that under the same bolt diameter, end-plate thickness, and beam length, the ultimate flexural resistance of IC joints using the S1 type end-plate connection is higher than the S2 type end-plate connection. Figure 19 shows that the initial rotational stiffness of most S2 end-plate connections is slightly greater than that of S1 end-plates and the difference between them is within the range of 1-8%. For the EC joints of small-sized column sections, the S2 end-plate form is used to improve the joint stiffness, which is better than that of large-sized column sections.

Effect of End-Plate Section Size
Comparing the end-plates S1 and S2, the difference is the distance between each row of bolt holes and the end-plate size. The influence on the initial rotational stiffness of the joints is shown in Figure 19. Overall, these differences have an impact on the initial rotational stiffness of the joint. Because the bolt's hole distance in the end-plate S2 is smaller than end-plate S1, this restricts the bending deformation of the column flange and the end-plate, making this component contribute more to the stiffness of the joint, while the ultimate flexural resistance capacity is greatly affected by the failure mode, certain weak components may fail before the end-plate.
By referring to Table 6, it is found that under the same bolt diameter, end-plate thickness, and beam length, the ultimate flexural resistance of IC joints using the S1 type end-plate connection is higher than the S2 type end-plate connection. Figure 19 shows that the initial rotational stiffness of most S2 end-plate connections is slightly greater than that of S1 end-plates and the difference between them is within the range of 1-8%. For the EC joints of small-sized column sections, the S2 end-plate form is used to improve the joint stiffness, which is better than that of large-sized column sections.  Figure 20 shows the deformation of the end-plate during the cyclic loading (last drift cycle). Firstly, due to the tension in the beam lower flange, the opening between the end-plate and the column flange appears at the beam lower flange(②,③). Then, as the load reverses, another opening between the end-plate and the column flange begins to appear at the beam web(④). By further load reversal, the tension is transferred to the upper beam flange, and the opening appears between the end-plate and the beam upper flange, when the tension increased in the upper flange and compression increased in the lower flange(⑤,⑥), the opening between the lower beam flange and the column flange begins to close(⑦).  Figure 20 shows the deformation of the end-plate during the cyclic loading (last drift cycle). Firstly, due to the tension in the beam lower flange, the opening between the end-plate and the column flange appears at the beam lower flange ( 2 , 3 ). Then, as the load reverses, another opening between the end-plate and the column flange begins to appear at the beam web ( 4 ). By further load reversal, the tension is transferred to the upper beam flange, and the opening appears between the end-plate and the beam upper flange, when the tension increased in the upper flange and compression increased in the lower flange ( 5 , 6 ), the opening between the lower beam flange and the column flange begins to close ( 7 ).

Differences between IC Joint and EC Joint
The difference between the IC joint and EC joint is whether to set up a single-sided connection or double-sided connection in the column flange, which results in the shear of the IC joint panel zone being twice the EC joint. At the ultimate loading capacity, the shear rotation of the IC joint and EC joint account for approximately 66% and 25% of the total rotation, respectively. Extracting the hysteresis curve from the cyclic load FE model shows that the characteristics of the IC joint are plumper, and their rotation span is slightly larger than the EC joint.
The size information of the 019-ICBC1D20t16L15S1 model (test specimen IC-EP1) and the 091-ECBC1D20t16L15S1 model (test specimen EC-EP1) are the same, the IC joint can enhance the joint stiffness remarkably compared to the EC joint, but the ultimate flexural resistance is basically the same. The above reason is that the strength is controlled by the failure of the weakest component. The two joint size's information is the same, and the weakest part is also unanimous, so the strength difference is slight; while the stiffness is determined by the stiffness contribution of each component. The panel zone boundary conditions of the IC joint are significantly different from the EC joint. The component stiffness of the IC joint panel zone is much greater than that of the EC joints. Under the column top-side loading method, the stiffness of the panel zone has a significant effect on the stiffness of the entire joint, so the stiffness of the two types of joints is obviously different. The difference between the IC joint and EC joint is whether to set up a single-sided connection or double-sided connection in the column flange, which results in the shear of the IC joint panel zone being twice the EC joint. At the ultimate loading capacity, the shear rotation of the IC joint and EC joint account for approximately 66% and 25% of the total rotation, respectively. Extracting the hysteresis curve from the cyclic load FE model shows that the characteristics of the IC joint are plumper, and their rotation span is slightly larger than the EC joint.
The size information of the 019-ICBC1D20t16L15S1 model (test specimen IC-EP1) and the 091-ECBC1D20t16L15S1 model (test specimen EC-EP1) are the same, the IC joint can enhance the joint stiffness remarkably compared to the EC joint, but the ultimate flexural resistance is basically the same. The above reason is that the strength is controlled by the failure of the weakest component. The two joint size's information is the same, and the weakest part is also unanimous, so the strength difference is slight; while the stiffness is determined by the stiffness contribution of each component. The panel zone boundary conditions of the IC joint are significantly different from the EC joint. The component stiffness of the IC joint panel zone is much greater than that of the EC joints. Under the column top-side loading method, the stiffness of the panel zone has a significant effect on the stiffness of the entire joint, so the stiffness of the two types of joints is obviously different.

Mechanical Model of the Initial Stiffness
The initial stiffness of the extended end-plate connection joint can be predicted according to the classical component method recommended by EC3 [7]. Meanwhile, based on the above parametric finite element research, the premise of the loading method on the top-side of the column is used to determine the components that contribute to the rotational stiffness of the end-plate connection. As exhibited in Figure 21, it can be seen that under the same directional moment, for the IC joint with end-plate connections on both sides, the shear deformation of the column web panel zone has a great influence on the initial rotational stiffness of the connection. The simplified spring model of the entire joint consists of eight springs simulating the deformation of two major parts. First, the connecting part is composed of three springs: the end-plate in bending (① epb), the column flange in bending (② cfb), and the bolt in tension (③ bt). Then, the column web part is composed of five springs, which simulate the shear (⑥ cws), tension (④ cwt) and compression deformation (⑦ cwc) of the column

Mechanical Model of the Initial Stiffness
The initial stiffness of the extended end-plate connection joint can be predicted according to the classical component method recommended by EC3 [7]. Meanwhile, based on the above parametric finite element research, the premise of the loading method on the top-side of the column is used to determine the components that contribute to the rotational stiffness of the end-plate connection. As exhibited in Figure 21, it can be seen that under the same directional moment, for the IC joint with end-plate connections on both sides, the shear deformation of the column web panel zone has a great influence on the initial rotational stiffness of the connection. The simplified spring model of the entire joint consists of eight springs simulating the deformation of two major parts. First, the connecting part is composed of three springs: the end-plate in bending ( 1 epb), the column flange in bending ( 2 cfb), and the bolt in tension ( 3 bt). Then, the column web part is composed of five springs, which simulate the shear ( 6 cws), tension ( 4 cwt) and compression deformation ( 7 cwc) of the column web, stiffener in tension ( 5 st) and in compression deformation ( 8 sc), respectively. The deformation of the entire joint is composed of these two parts, and the initial rotational stiffness of the connection can be expressed as: where K con and K cw are the stiffness of the connection and column web, respectively.
Materials 2020, 13, x FOR PEER REVIEW 21 of 31 where Kcon and Kcw are the stiffness of the connection and column web, respectively.

Stiffness Calculation of the End-Plate and Column Flange in Bending
For the calculation of the bending stiffness of the column flange and the end-plate, the deflection at the bolt hole of the rectangular plate is used to obtain the component stiffness. As displayed in Figure 22, for the No. I plate of T-stub parts, the AB side is considered to be a fixed boundary, because the stiffness outside the plane of the beam flange is considerable, which can provide sufficient restraint for the end-plate, the remaining three sides do not provide effective constraints in terms of stiffness contribution, and can then be simplified as free sides. According to the plates and shells theory, under this boundary condition, the deflection of the rectangular plate subjected to the concentrated load F/2 at the center can be expressed as ωm = αFab/2D, where ωm is the center deflection of the plate, α is the coefficient related to the length and constraints of the plate, D is the bending stiffness of the plate per unit width, D = Et 3 /12(1 − μ 2 ). For a rectangular cantilever plate with one fixed side and three free sides, the α coefficient reference [35] is 0.0465402. It can be concluded from the physical meaning that ωepb1 = F/2kepb1, where kepb1 = 1/δepb1 = D/(αab) = 1.79 × Etep 3 /((1 − μ 2 )ab), tep is the thickness of the end-plate; and the T-stub of No. II plate boundary conditions and bending stiffness are equivalent to No. I plate.
The bending stiffness of the end-plate is: ( ) The bending stiffness of the column flange is: where: E is the elastic modulus of steel, μ is the Poisson's ratio of the material; tcf and tep are the thickness of the column flange and end-plate respectively; acf, aep and bcf, bep are the calculated length and calculated height of the rectangular plate in the end-plate and column web, respectively.

Stiffness Calculation of the End-Plate and Column Flange in Bending
For the calculation of the bending stiffness of the column flange and the end-plate, the deflection at the bolt hole of the rectangular plate is used to obtain the component stiffness. As displayed in Figure 22, for the No. I plate of T-stub parts, the AB side is considered to be a fixed boundary, because the stiffness outside the plane of the beam flange is considerable, which can provide sufficient restraint for the end-plate, the remaining three sides do not provide effective constraints in terms of stiffness contribution, and can then be simplified as free sides. According to the plates and shells theory, under this boundary condition, the deflection of the rectangular plate subjected to the concentrated load F/2 at the center can be expressed as ω m = αFab/2D, where ω m is the center deflection of the plate, α is the coefficient related to the length and constraints of the plate, D is the bending stiffness of the plate per unit width, D = Et 3 /12(1 − µ 2 ). For a rectangular cantilever plate with one fixed side and three free sides, the α coefficient reference [35]

Stiffness Calculation of the Bolt in Tension
For high-strength bolts, the pretension of the bolts significantly improves the initial stiffness of the connection. The coefficient γ is introduced in consideration of the effect of its pretension. Its value is referred to in [36], and generally γ = 10. The calculation formula of the tensile stiffness of two bolts in a single row is: where As is the effective area of the bolt, generally 80% of the nominal area of the bolt shank; Lb is the The bending stiffness of the end-plate is: Materials 2020, 13, 5133

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The bending stiffness of the column flange is: where: E is the elastic modulus of steel, µ is the Poisson's ratio of the material; t cf and t ep are the thickness of the column flange and end-plate respectively; a cf , a ep and b cf , b ep are the calculated length and calculated height of the rectangular plate in the end-plate and column web, respectively.

Stiffness Calculation of the Bolt in Tension
For high-strength bolts, the pretension of the bolts significantly improves the initial stiffness of the connection. The coefficient γ is introduced in consideration of the effect of its pretension. Its value is referred to in [36], and generally γ = 10. The calculation formula of the tensile stiffness of two bolts in a single row is: where A s is the effective area of the bolt, generally 80% of the nominal area of the bolt shank; L b is the calculated length of the bolt, L b = t ep + t cf + 2t wh + (t h + t n )/ 2, refer to Figure 23, where t ep is the thickness of the end-plate, t cf is the thickness of the column web, t wh is the thickness of the bolt washer, and t h and t n are the thickness of the bolt head and nut, respectively.

Stiffness Calculation of the Bolt in Tension
For high-strength bolts, the pretension of the bolts significantly improves the initial stiffness of the connection. The coefficient γ is introduced in consideration of the effect of its pretension. Its value is referred to in [36], and generally γ = 10. The calculation formula of the tensile stiffness of two bolts in a single row is: where As is the effective area of the bolt, generally 80% of the nominal area of the bolt shank; Lb is the calculated length of the bolt, Lb = tep + tcf + 2twh + (th + tn)/2, refer to Figure 23, where tep is the thickness of the end-plate, tcf is the thickness of the column web, twh is the thickness of the bolt washer, and th and tn are the thickness of the bolt head and nut, respectively.

Integrated Connection Stiffness
Integrate the stiffness of the connection joint of the extended end-plate, according to the beam end rotation conforms to the plane-section assumption and the superposition principle of displacement in the elastic stage, the rotation angle θ can be expressed as:

Integrated Connection Stiffness
Integrate the stiffness of the connection joint of the extended end-plate, according to the beam end rotation conforms to the plane-section assumption and the superposition principle of displacement in the elastic stage, the rotation angle θ can be expressed as: According to the physical meaning of the initial rotational stiffness of the joint, the stiffness of the connection part of the extend end-plate is: In the formula, M is the elastic ultimate moments at the connection, k con is the connection stiffness, k epb , k cfb , and k bt are the bending stiffness of the end-plate, column web, and the tensile stiffness of the bolt, respectively.

Tension and Compression Stiffness of the Column Web
According to the finite element analysis in the above section, it can be seen that the direction of force transmission, the axial force transmitted from the connection, causes the column web to be in a real state of compression and tension. The method of simplifying the column web to axial compression and axial tension plate can be used to calculate. Meanwhile, the influence of different load forms on the joint stiffness must also be considered. To calculate the initial rotational stiffness of the IC joints under the antisymmetric load and the EC joints under the asymmetric load.
The expression of the compression stiffness of the column web of the IC joint and EC joint is: where h cw is the calculated height of the column web ( Figure 24a); t cw is the thickness of the column web; b eff,c is the effective compression width of the column web, reference [37], if the column is hot-rolled steel, then b eff,c = t bf + 2h e,ep + 2t ep + 2(t cf + r c ); if the column is a welded steel section, b eff,c = t bf + 2h e,ep + 2t ep + 2(t cf + h e,c ), where h e,ep and h e,c are the effective heights of the weld of the end-plate and column web, r c is the root radius of the column flange weld, t bf is the thickness of the beam flange, and the ϕ value is 0.5.
According to the physical meaning of the initial rotational stiffness of the joint, the stiffness of the connection part of the extend end-plate is: In the formula, M is the elastic ultimate moments at the connection, kcon is the connection stiffness, kepb, kcfb, and kbt are the bending stiffness of the end-plate, column web, and the tensile stiffness of the bolt, respectively. According to the finite element analysis in the above section, it can be seen that the direction of force transmission, the axial force transmitted from the connection, causes the column web to be in a real state of compression and tension. The method of simplifying the column web to axial compression and axial tension plate can be used to calculate. Meanwhile, the influence of different load forms on the joint stiffness must also be considered. To calculate the initial rotational stiffness of the IC joints under the antisymmetric load and the EC joints under the asymmetric load.
The expression of the compression stiffness of the column web of the IC joint and EC joint is: where hcw is the calculated height of the column web ( Figure 24a); tcw is the thickness of the column web; beff,c is the effective compression width of the column web, reference [37], if the column is hotrolled steel, then beff,c = tbf + 2he,ep + 2tep + 2(tcf + rc); if the column is a welded steel section, beff,c = tbf + 2he,ep + 2tep + 2(tcf + he,c), where he,ep and he,c are the effective heights of the weld of the end-plate and column web, rc is the root radius of the column flange weld, tbf is the thickness of the beam flange, and the φ value is 0.5. The formula for calculating the tensile stiffness of the IC joint and EC joint is shown in Equations (8) and (9).
In the equation, IC and EC respectively represent the intermediate column joint and edge column joint; λ is to consider the influence coefficient of the bolt hole spacing, and the value λ = (p/w) 3 ; The formula for calculating the tensile stiffness of the IC joint and EC joint is shown in Equations (8) and (9).
In the equation, IC and EC respectively represent the intermediate column joint and edge column joint; λ is to consider the influence coefficient of the bolt hole spacing, and the value λ = (p/w) 3 ; and b eff,t is the effective tensile width of the column web. The calculation diagram is shown in Figure 24b. When m c is less than (p − d m )/2, then b eff,t is 2m c + d m ; when m c is greater than or equal to (p − d m )/2, then b eff,t is m c tan45 • + (d m + p)/2. Where m c is the distance from the center of the bolt hole to the welding foot of the column web, p and w are the vertical and horizontal distances between the centers of bolt holes, respectively. dm = 1.5 × d b , and d b is the nominal diameter of the bolt.

Shear Stiffness of Column Web
To calculate the deformation ∆ cws of the column web under shear, as shown in Figure 25, reference [38]. The column web can be assumed to be a short column that is only subjected to shear force V, which is generated by the moment of the connection at the upper and lower flanges of the beam. The deformation of the column web under the action of shear force is: where A sc is the shear area of the column web, as shown in the shaded part in Figure 25.
A sc = h c h cw − 2w cf t cf + (t cw + 2r c )t cf , and G is the shear modulus of steel. and beff,t is the effective tensile width of the column web. The calculation diagram is shown in Figure  24b. When mc is less than (p − dm)/2, then beff,t is 2mc + dm; when mc is greater than or equal to (p − dm)/2, then beff,t is mctan45° + (dm + p)/2. Where mc is the distance from the center of the bolt hole to the welding foot of the column web, p and w are the vertical and horizontal distances between the centers of bolt holes, respectively. dm = 1.5 × db, and db is the nominal diameter of the bolt.

Shear Stiffness of Column Web
To calculate the deformation δcws of the column web under shear, as shown in Figure 25, reference [38]. The column web can be assumed to be a short column that is only subjected to shear force V, which is generated by the moment of the connection at the upper and lower flanges of the beam. The deformation of the column web under the action of shear force is: where Asc is the shear area of the column web, as shown in the shaded part in Figure 25. Asc = hchcw − 2wcftcf + (tcw + 2rc)tcf, and G is the shear modulus of steel.
In the formula, when the connection belongs to the EC type, the column web is sheared on one side, and the β coefficient in the equation is 1; when the connection belongs to the IC joint type, the moments on both sides of the panel zone are equal and in the same direction, so the β coefficient in the equation is 2 if the moments on both sides of the panel zone are equal and in the opposite direction, so the β coefficient in the equation is 0, the bending moments offset each other, that is, no shear force. This paper does not involve such joints.

Tension and Compression Stiffness of Column Web Stiffener
The stiffeners of the column webs also adopted the similar axial tension and compression method at the column webs to calculate the stiffness. As shown in Figure 26, according to the theory of material mechanics, the tensile and compressive stiffness is calculated.
Tensile stiffness of column web stiffener: The shear stiffness of the web of the IC joint and EC joint is: EA sc h cw (11) k cws,EC = 1 4 × (1 + µ) EA sc h cw (12) In the formula, when the connection belongs to the EC type, the column web is sheared on one side, and the β coefficient in the equation is 1; when the connection belongs to the IC joint type, the moments on both sides of the panel zone are equal and in the same direction, so the β coefficient in the equation is 2 if the moments on both sides of the panel zone are equal and in the opposite direction, so the β coefficient in the equation is 0, the bending moments offset each other, that is, no shear force. This paper does not involve such joints.

Tension and Compression Stiffness of Column Web Stiffener
The stiffeners of the column webs also adopted the similar axial tension and compression method at the column webs to calculate the stiffness. As shown in Figure 26, according to the theory of material mechanics, the tensile and compressive stiffness is calculated.
Tensile stiffness of column web stiffener: Compression stiffness of column web stiffener: where t s is the thickness of the stiffener of the column web; b eff,st and b eff,sc are the effective widths of the stiffener under tension and compression, respectively. Considering that the bolt force diffuses at 45 • under tension.  (14) where ts is the thickness of the stiffener of the column web; beff,st and beff,sc are the effective widths of the stiffener under tension and compression, respectively. Considering that the bolt force diffuses at 45° under tension.
The rotation produced by the column web is: /ℎ The overall stiffness of the column web is: In the formula, h0 is the vertical distance between the center of the upper and lower flanges of the beam, h0 = hb -tbf, hb is the height of the beam, and tbf is the thickness of the beam flange. if (w ep -w bf )/2 ≥ t ep , then b eff,sc = min{w bf /2 + t cf + t ep ,w bf /2}, if (w ep -w bf )/2<t ep , then b eff,sc = min{w ep /2 + t cf ,b cf /2}. w ep , w bf , and w cf are the width of the connecting end-plate, the width of the beam flange, and the width of the column flange, respectively; b s is the width of the stiffener and t ep is the thickness of the end-plate. The above specific parameters are shown in Figure 26.

Integrated Column Web Stiffness
The shear, tension, and compression deformation of the column web and the tension and compression deformation of the stiffeners cause relative deformation between the panel zones, which in turn causes the beam ends to rotate. Under the action of the moment M, the five deformations of the column web are: The rotation produced by the column web is: The overall stiffness of the column web is: In the formula, h 0 is the vertical distance between the center of the upper and lower flanges of the beam, h 0 = h b -t bf , h b is the height of the beam, and t bf is the thickness of the beam flange.

Validation
First applied to the test specimens IC-EP1/2/3 and EC-EP1/2/3, to verify the improved component model for predicting the initial rotational stiffness, Table 7 summarizes the results of the component method model, tests, and the corresponding FE model. Compared with the experimental and finite element results, the theoretical predictions are consistent, and the error margin for individual specimens is 16%. To further validate the above-mentioned component model. The stiffness of the 144 finite element models of the above parameter analysis has been calculated. The model included parameters such as different column section sizes, end-plate thickness, bolt diameter, and end-plate size to verify the proposed component method. Table 8 compares the stiffness of FE analysis and component method calculations. Based on the comparative analysis of 144 FE models and component method calculations, ∆ represents the ratio of the difference between the two to FE, that is ∆ = |K T − K FE |/K FE . For the average of this ratio, the relative difference between the two is 8.99%. For dispersion, the standard deviation of this relative difference is equal to 6.88%. Due to the complexity and high cost of the test, the value calculated by the component method is still acceptable. The above two groups of verification data verify the validity of the initial rotational stiffness's theoretical calculation. So generally, the theoretical equation proposed in this article is of good applicability for most reasonably designed joints.

Conclusions
This study involved analyzing the performance difference between IC joints and EC joints of extended end-plate connection under the column top-side loading method. Firstly, based on the test data of six joints, a set of symmetrical models was developed to verify actual experimental results, and the failure modes and joint stiffness were analyzed and compared. The above work laid the foundation for parameter analysis; a total of 144 symmetrical 3D finite element models were created to study the influence of cyclic loads on the extended end plate beam-column connection with different parameters. Finally, a calculation equation for the initial rotation stiffness of the joints was presented based on the component method. The key conclusions of the study include the following: • According to the influence of different parameters on the mechanical behavior of the extended end-plate connection, the parametric FE models were established, and the effects on the flexural capacity, rotational stiffness and limit rotation of the joints are studied. Meanwhile, the joint typical failure model is divided into six types. It is also found that the joints also show nonlinear characteristics in the early stage, and some connected components have already yielded in the initial stage, and each component will not respond uniformly as a whole. The behavior of joints in the whole process is nonlinear.

•
In most studies in the past, the main focus has been on the study of the moment-rotation relationship of the connection under the beam tip loading method, but this article focuses on some force forms in actual engineering, using the column top-side loading method to study the parameter's comprehensive impact on joint performance. Additionally, the research on the panel zone shearing effect is focused, and it is concluded that the value of shearing force acting on the connection has a great influence on the mechanical behaviors of joints in this type of loading. Meanwhile, the shear rotation of the IC joint parameter model accounts for about 2/3 of the total connection rotation times, and the shear force in the panel zone is twice that of the EC joint. However, under the control of the failure mode of the weakest component, the ultimate flexural resistance capacity of the IC joint and EC joint remains consistent.

•
Increasing the diameter of bolts or the thickness of the end-plate in most cases enhances the behavior of the connection, in other words, increases both moment capacity and rotation capacity by a certain percentage. However, the yield rotation of the connection is decreased with an increase in the thickness of end-plates, resulting in low connection ductility. In spite of this, the use of medium-thickness end-plates with bolts of appropriate diameters can greatly improve the overall mechanical properties of joints.

•
According to the mechanical behavior of the joint components. The component model of the joint under the column top-side loading method is proposed, and the initial stiffness expression is established. The bending stiffness of the component was improved by adopting the large deflection calculation of the plate and shell theory, and considering the significant difference in the contribution of the shear stiffness component of the panel zone between the IC joint and EC joint. The expression is verified by experimental results and the 144-parameter FE model, which proves the reliability of the initial stiffness expression.