Finite Element Analysis and Parametric Study of Panel Zones in H-Shaped Steel Beam–Column Joints

Abstract

This paper investigates the mechanical properties of a traditional welded rigid joint with a weakened panel zone under seismic load. The created finite element model is calibrated by the high-strength steel joint test, carried out by the team in the early stage, and the effectiveness of the finite element method was verified. The finite element software ABAQUS is used to investigate the influence of different joint web thicknesses on the mechanical properties of middle column joints under a low-cyclic-loading test. Supported by a validated numerical model, the ductility, energy dissipation, and other properties of different thicknesses of panel zone column webs are carefully analyzed. The results indicate that the thickness of the web plate in the panel zone significantly affects the location of the joint plastic hinge. The ultimate loading capacity of the joints increased significantly with an increase in the thickness of the webs in the panel zones. Compared with the joint with a weakened panel zone, the hysteresis curve of the strengthened joint is fuller; meanwhile, it cannot alleviate the stress concentration at the weld holes of the web. When the thickness of the joint domain web is too weak, excessive deformation in the joint domain will lead to a decrease in the bearing capacity of the joint, causing damage. The stiffness degradation coefficient of the web-thickened specimen was found to be dominated and controlled by the stiffness of the beam; however, with an increase in the thickness of the web, the stiffness degradation coefficient remained basically unchanged. Finally, a recommendation for weakened beam–column interior joints based on the steel frame panel zone is made, which will lay a foundation for the simulation and analysis of the seismic performance of this structure.

1. Introduction

Over the past decades, the seismic codes for buildings have been continuously updated, reiterated, and improved worldwide to achieve buildings that can withstand severe ground motion without collapsing [1]. One of the basic principles in modern code provisions is to encourage the use of ductile building configurations, structural systems, and materials. A structure possesses ductility if it can withstand large inelastic strains without significant loss of strength, allowing it to avoid instability and collapse. Since the 1960s, engineers have considered welding steel frame buildings as one of the most ductile systems in building construction standards [2,3,4]. Steel frame buildings are generally unaffected by structural damage caused by earthquakes; it is assumed that, if damage were to occur, then it would be limited to the ductile yielding of the components and connections. According to this model, earthquake-induced collapse should be impossible. As a result of this idea, many large industrial, commercial, and institutional buildings employing welded-steel framing systems have been constructed, particularly in the western United States [5,6].

However, the Northridge earthquake of 17 January, 1994, in the United States challenged this ductile structure model [7]. The failure state of the steel structure in the Northridge earthquake was mainly caused by crack propagation in the butt weld at the flange area of the beam end. The steel frame structure was considered to have superior seismic performance according to assessment of its performance during the 1995 Kobe earthquake in Japan; however, it was found that the failure of the beam–column joints caused significant damage. The typical failure of traditional welded bending beam–column joints has caused many scholars to examine the cause of the fault and design potential alternatives and solutions for the joint connection structure; they have been trying to solve the problem of the brittle failure that occurs in steel beam–column joints in terms of joint connection form and design [8,9,10,11,12,13]. Subsequently, several studies and finite element analyses on the mechanical properties of panel zones have been carried out to assess the seismic performance of this structural system. Some essential guiding conclusions have been provided in the literature; these provide a basis for engineering practice and theoretical research. Peter et al. [14] experimentally investigated seven beam–column joints, obtaining the load capacity of the panel zones. The design equations of the beam–column joints were fitted and improved in accordance with the curve data of the test results. Fielding et al. [15] analyzed the role of high shear stress and high axial force state on the yielding of rigidly connected frame joints by theory, considering that the design method of the AISC formula was too conservative. Li et al. [16] proposed a beam–column hinge joint with a replaceable energy dissipation component and compared the effects of the long-span ratio of steel beams on the seismic performance of the joints. Krawinkler et al. [17] summarized the effects of lateral loads on bending frames through experimental studies and theoretical analyses. According to the experimental results, the panel zones have good energy dissipation properties and ductility with stable hysteretic performance. Three years later, these results were consulted in formulating the joint design method that was provided in the AISC code at that time. Krawinkler et al. [18] discussed the effect of high shear stresses at joints on the frame’s strength, stiffness, and energy dissipation capacity. The study results revealed that the joints were highly ductile and withstood large repeated inelastic strains without any loss of strength. Bai et al. [19] investigated the seismic performance of outer-loop-reinforced steel beam–column connections under bi-directional cyclic loading and analyzed the hysteretic response of moments versus rotations. The hysteretic loops of all the specimens exhibited stable ductile properties, indicating a good energy dissipation capacity. El-Tawil et al. [20] numerically investigated the effect of panel zone yielding on the fracture of bolted welded steel joints. The finite element simulations of this panel zone illustrated that the existing AICS code for calculating panel zone yielding capacity is not applicable to edge joints. Reza Amani et al. [21] also numerically studied the double-web columns to estimate localized flange buckling. Based on the two methods, some formulas for calculating the local flange bending of double-web columns were proposed, and their validity and accuracy were verified. Sina Sarfarazi et al. [22] performed a finite element analysis of 432 beam–column joints. They proposed a mathematical model to consider the effect of axial forces on the behavior of the panel zones in cruciform columns. The finite element model verified the accuracy of the mathematical model. Deylami et al. [23] developed a finite element model of six beam–flange weakening joints using ANSYS 2003. They analyzed the parameters, such as the thickness of the panel zones, to investigate the interaction between the panel zones and the flange weakening beams. It was indicated that the suitable thickness of the panel zones can effectively participate in the structure’s energy dissipation and avoid the premature destruction of the beams. Ye et al. [24] proposed an innovative hybrid beam–column joint with a robust connecting system and a weak structural component; the experimental results demonstrated that the new hybrid beam–column joint could reduce the stress concentration effect at the joints. Lin et al. [25] connected square steel tubular columns to H-shaped steel beams with T-shaped steel and unilateral bolts for cyclic loading tests and numerically investigated them. The numerical experiments indicated that the seismic capacity of the joint can be improved by increasing the size of the T-shaped beam. Hou et al. [26] carried out comparative seismic performance tests and numerical simulation analyses of bare and internal frames; the results showed that the loading capacity of the structure is significantly improved with an increase in the thickness of the column.

As discussed above, it is noted that the failure mode of the panel zone and the overall steel frame structure under earthquake action requires research attention. To systematically study the seismic and mechanical performance of steel joints with a weakened panel zone, this paper first goes through a previous set of high-strength steel joint numerical model calibrations to verify the finite element model and the analytical method [27]. The effects of web thickness of different panel zones on the mechanical properties of mid-column joints under low-cycle reciprocating loads were investigated. By reducing the web thickness of the column or adding welded reinforcement plates to change the web thickness of the panel zone, 11 groups of finite element models (Q345-T8~Q345-T24) were established. The following factors were carefully analyzed: the influence of the thickness change of the panel zone on the stress development; the deformation law; the bending moment–shear ratio hysteresis performance of the panel zone and the equivalent story shear force–drift ratio; the skeleton curve; the energy dissipation of the steel joint. Based on the design stage of the joint, a design recommendation is made: researchers and developers should fully consider the energy dissipation capacity of the panel zone after yield. This recommendation provided a basis for the engineering design.

2. Finite Element Validation

2.1. The Design of Joint

In the early stage, our team conducted a seismic performance test on welded high-strength steel joints [28]. A set of high-strength steel specimens and ordinary steel specimens were designed. The beam and column members were made from Q235 ordinary steel and Q690 high-strength steel, respectively; their specific dimensions are shown in Figure 1. And 10.9 grade M27 high-strength bolts connect the replaceable joint specimen and the loading aid.

2.2. Test Loading Program

The overall loading device is shown in Figure 2. The bottom and top of the square steel tube column are hinged with the steel frame to ensure that part of the specimen can rotate during the loading process. A 60-ton range jack is installed on the upper part of the beam end and the beam end is subjected to reciprocating loading. The top of the vertical jack is connected to the top beam of the steel frame. In the test, the specimens were loaded using full displacement control. The vertical reciprocating load was applied to the beam end of the specimen using a jack, and the drift ratio was used as the control parameter, defined as the ratio of the applied displacement at the loading end to the beam length (3500 mm). The loading system of the specimen OJ and the specimen HJ-1 is loaded at 0.25%, 0.5%, 1.0%, 2.0%, 3.0%, 4.0%, and 5.0% of the drift ratio; each stage is loaded for two cycles. When the bearing capacity drops below 85% of the peak bearing capacity or the specimen breaks, the loading stops. If the specimen is not damaged after loading, then the amplitude of the inter-layer displacement angle is increased by 1% until the specimen reaches failure. The loading system is shown in Figure 3 [29,30].

2.3. Establishment of Finite Element Model

2.3.1. Performance Parameters of Steel

The steel constitutive relation of the specimen in the finite element model is based on the test results of the material specimen. Equations (1)–(3) convert these into a true stress–strain relationship and the material constitutive form required for the finite element model. The steel adopts a trilinear model considering the strengthening stage and the descending section, as shown in Figure 4.

$${\sigma }_t={\sigma }_{nom}(1+{\epsilon }_{nom})$$

(1)

$${\epsilon }_t={ln}^{(1+{\epsilon }_{nom})}$$

(2)

$${\epsilon }_p=\left|{\epsilon }_t\right|-\frac{\left|{\sigma }_t\right|}{E}$$

(3)

where $E$ denotes elastic modulus of material; ${\sigma }_t$ denotes real stress of the material; ${\sigma }_{nom}$ denotes nominal stress of the material; ${\epsilon }_t$ denotes real strain of the material; ${\epsilon }_{nom}$ denotes nominal strain of material; ${\epsilon }_p$ denotes plastic strain of materials.

2.3.2. Load, Boundary Conditions, and Meshing

The boundary conditions of the finite element model and the applied loads are shown in Figure 5a. The load applied in the model includes the bolt pretension force and the displacement load at the beam end. The replaceable specimen is connected with a 10.9-grade M27 high-strength friction bolt in front of the reusable beam–column member. The bolt pre-tightening force is P = 290 KN. To satisfy the rotation requirements in the test, only the end of the column is allowed to rotate around the X-axis. The mesh division of the model element is shown in Figure 5b. In order to meet the rotation requirements of the hinged support in the test, the boundary conditions are set to ${ U}_1=U_2=U_3=0,{UR}_2={UR}_3=0$, and only the end of the column is allowed to rotate around the X axis. The lower side of the loading end of the beam needs out-of-plane constraints. In the test, the out-of-plane deformation of the loading end is prevented from causing the loading problem. The boundary conditions are set to ${ U}_1=0$; only the X-axis displacement of the loading end is limited.

2.4. Comparison between Test and Finite Element Analysis Results

2.4.1. Stress Cloud

Figure 6 shows the stress cloud of the finite element modeling results of the OJ specimen compared with the test results. The steel beam showed slight out-of-face deformation when the specimen was loaded to 4% of the drift ratio. As the displacement of the beam end continuously increased, the steel beam exhibited apparent out-of-face buckling, as shown in Figure 6c. When the displacement loading reached 6% of the drift ratio, the lower flange of the steel beam closest to the core area cracked. The stress concentration phenomenon of the lower flange of the steel beam can be observed. Local buckling occurred at both the upper and lower flanges, as depicted in Figure 6 and Figure 7. Figure 7 illustrates the comparison between the stress cloud diagram of the finite element modeling results and the test results of high-strength steel specimen HJ-1. When the displacement of the beam end was loaded to 5% of the drift ratio, obvious cracks appeared on the upper flange of the beam end, and then the upper flange of the steel beam began to crack. Then, the specimen is monotonically loaded to a 7% drift ratio. There was evident stress concentration on the upper flange of the steel beam, and plastic deformation occurred in the steel beam’s reserved welding hole, as shown in Figure 7b.

2.4.2. Hysteresis and Skeleton Curve

As plotted in Figure 8, at the initial loading stage, the specimen OJ test is consistent with the finite element modeling results. When the drift ratio reaches 3%, the strength and stiffness of the steel degrade rapidly during the test loading process, and the overall stiffness of the finite element model is too large. Therefore, the bearing capacity obtained through the finite element analysis in the positive loading direction is larger than the experimental value, and the negative loading direction agrees with the test results. The skeleton curve is shown in Figure 9; the hysteresis loop obtained through the finite element analysis is fuller. In the initial loading stage, the finite element simulation results are in good agreement with the test results. As the displacement of the beam end increases, the decrease in slope of the stiffness in the finite element model is less than the test value, and the hysteresis loop of the finite element modeling results is fuller than the test results. Comparing the hysteresis and skeleton curves of the test and the finite element modeling results, it can be observed that there is high consistency between the finite element modeling results and the test results.

2.4.3. Stiffness Degradation

Figure 10 compares the two specimens’ stiffness degradation between the test results and the finite element modeling results. At the beginning of the loading stage, the stiffness of specimen OJ is not degraded; after the drift ratio reaches 1.0%, the stiffness gradually starts to degrade with displacement of the beam end, and the overall stiffness results of the finite element simulation are slightly higher than the test results. The stiffness of the specimen HJ-1 did not degenerate before the drift ratio reached 2%. When the drift ratio reaches 3%, the cumulative damage of the specimen leads to a rapid decrease in stiffness and bearing capacity. The slope of the stiffness degradation curve of the OJ specimen shows a decreasing trend with the increase in displacement. The decrease in stiffness is slowed down, while the stiffness degradation curve of the HJ-1 specimen shows a continuous and significant downward trend. This can be explained by the fact that the ordinary steel joint has better ductility after reaching the ultimate bearing capacity than the high-strength steel joint, allowing more significant joint deformation; therefore, the trends of the stiffness degradation curves of the two specimens are different.

3. Study on the Mechanical Performance of Steel Joints with a Weakened Panel Zone

3.1. Introduction of FEM Model

3.1.1. Design of Specimen

The specimens in this FEM simulation had a 3.3 m column height and 7.2 m beam-span. Considering the different thicknesses of column web in the joint core area, eleven rigid joint specimens were designed. The specimen number is composed of steel type and the effective thickness of the panel zone, which are given in

Table 1. List of design joints specimens.

Specimen	Column Cross-Section (mm)	Beam Cross-Section (mm)	${\mathit{\alpha }}_{\mathit{s}\mathit{c}\mathit{w}\mathit{b}}$	${\mathit{t}}_{\mathit{p}\mathit{z}}$	${\mathit{\alpha }}_{\mathit{p}\mathit{z}}$	${\mathit{\phi }}_{\mathit{p}\mathit{z}}$
B345-T8	400 × 200 × 8 × 16	330 × 150 × 10 × 14	1.60	8	0.60	0.92
B345-T9	400 × 200 × 9 × 16		1.64	9	0.68	0.82
B345-T10	400 × 200 × 10 × 16		1.67	10	0.75	0.74
B345-T11	400 × 200 × 11 × 16		1.71	11	0.83	0.67
B345-T12	400 × 200 × 12 × 16		1.74	12	0.90	0.62
B345-T13	400 × 200 × 13 × 16		1.78	13	0.98	0.57
B345-T14	400 × 200 × 14 × 16		1.82	14	1.05	0.53
B345-T18	400 × 200 × 14 × 16	330 × 150 × 10 × 14	1.82	18	1.35	0.41
B345-T20			1.82	20	1.50	0.37
B345-T22			1.82	22	1.65	0.33
B345-T24			1.82	24	1.80	0.31

Note: ${\alpha }_{scwb}$ denotes the coefficient of strong column and weak beam; $t_{pz}$ denotes the web thickness of panel zone; ${\alpha }_{pz}$ denotes the coefficient of strength of panel zone; ${\phi }_{pz}$ denotes the coefficient of the balance of panel zone.

.

3.1.2. Establishment of Finite Element Model

Based on the information provided in Table 1, the general finite element method (FEM) software ABAQUS 2021 was utilized to establish the FEM models. The model mesh is divided by the C3D8R solid element.

Table 2. The meshing of each component for the joint models.

Part Name	Element Type	Delineation Technique	Element Number
Beam	$\mathrm{C}3\mathrm{D}8\mathrm{R}$	Structured, sweep	13,780
Column	$\mathrm{C}3\mathrm{D}8\mathrm{R}$	Structured	6536
Rib stiffener	$\mathrm{C}3\mathrm{D}8\mathrm{R}$	Structured	90
Stiffened sheet	$\mathrm{C}3\mathrm{D}8\mathrm{R}$	Structured	375

and Figure 11 shows the number of meshing elements of each component in the specimen. To simplify the calculation and effectively reproduce the stress state of the panel zone, the following elements adopt tie fixed constraints: the weld connections between the steel plates, such as the butt weld between the upper and lower flanges of the steel beam and the side of the steel column flange; the butt weld between the transverse stiffener of the joint region and the inner side of the column flange; the fillet weld connection between the web of the steel beam and the column flange; the fillet weld connection between the transverse stiffener of the joint region and the column web. The boundary conditions was shown in Figure 12.

3.1.3. Steel Constitutive

The constitutive relation of the steel is a trilinear model, considering the strengthening section and the descending section, as shown in Figure 13. According to the material test by Wang [31], the corresponding data were obtained; these are given in

Table 3. The mechanical property parameters of steel.

$\mathit{E}$	$\mathit{\mu }$	${\mathit{\sigma }}_{\mathit{y}}$	${\mathit{\epsilon }}_{\mathit{y}}$	${\mathit{\sigma }}_{\mathit{u}}$	${\mathit{\epsilon }}_{\mathit{u}}$	${\mathit{\sigma }}_{\mathit{s}\mathit{t}}$	${\mathit{\epsilon }}_{\mathit{s}\mathit{t}}$
2.05 × 105 MPa	0.3	369 N/mm2	0.179%	514 N/mm2	4.46%	416 N/mm2	7.34%

Note:$E$ denotes the modulus of elasticity; $\mu$ denotes the poisson ratio; ${\sigma }_y$ denotes the yield strength; ${\epsilon }_y$ denotes yield strain; ${\sigma }_u$ denotes the ultimate strength; ${\epsilon }_u$ denotes the ultimate strain; ${\sigma }_{st}$ denotes the destructive strength; ${\epsilon }_{st}$ denotes the destructive strain.

.

3.1.4. Loading System

The model was loaded through full displacement control with variable amplitude; the specific displacement loading amplitude is depicted in Figure 14.

3.2. Finite Element Analysis Results

3.2.1. Instantaneous Stress Cloud

Considering that some specimens have similar stress development and failure modes during the loading process, the following three specimens were subjected to stress analysis: B345-T8, B345-T14, and B345-T22. The stress, strain law, and loading phenomenon of the rest of the specimens during the loading process are recorded in

Table 4. Loading phenomena.

Specimen	${\mathit{P}}_{\mathit{y}}$	${\mathit{\epsilon }}_{\mathit{e}}$	${\mathit{P}}_{\mathit{u}}$	${\mathit{\sigma }}_{\mathit{b}}$(MPa)	${\mathit{\sigma }}_{\mathit{j}}$(MPa)	${\mathit{\sigma }}_{\mathit{b}}$$/{\mathit{\sigma }}_{\mathit{j}}$	Plastic Hinge
B345-T8	38 mm of the first circle panel zone	1.365 × 103	94.96	490.9	509.6	0.96	panel zone
B345-T9	38 mm of the first circle panel zone	0.941 × 103	107.41	489.7	510.4	0.96	panel zone
B345-T10	38 mm of the first circle panel zone	0.448 × 103	118.51	502.2	505.9	0.99	panel zone
B345-T11	28.5 mm of the first circle right beam weld toe	0.774 × 104	129.70	509.6	510.0	0.99	panel zone/steel beam
B345-T12	28.5 mm of the first circle right beam weld toe	1.430 × 104	140.91	508.9	507.5	1.00	panel zone/steel beam
B345-T13	28.5 mm of the first circle button flange of right beam	1.378 × 104	151.22	510.9	503.4	1.02	panel zone/steel beam
B345-T14	38 mm of the second circle right beam weld toe	1.358 × 104	170.84	508.7	495.6	1.03	steel beam
B345-T18	38 mm of the second circle upper flange of left beam	2.95 × 103	198.98	512.9	475.3	1.07	steel beam
B345-T20	38 mm of the second circle upper flange of left beam	1.85 × 103	210.69	513.2	436.7	1.18	steel beam
B345-T22	38 mm of the second circle button flange of right beam	1.34 × 103	222.60	509.2	394.6	1.29	steel beam
B345-T24	38 mm of the second circle button flange of right beam	8.65 × 104	247.88	511.6	387.36	1.32	steel beam

Note: $P_u$ denotes the ultimate bearing capacity; $P_y$ denotes the position of the first yield point; ${\epsilon }_e$ denotes the equivalent plastic strain; ${\sigma }_b$ denotes when the displacement of the beam end is 228 mm at the stress of the beam end; ${\sigma }_j$ denotes when the displacement of the beam end is 228 mm at the stress of the panel zone.

.

According to Figure 15, Figure 16 and Figure 17, before the displacement of the beam end reaches 38 mm, specimens B345-T8, B345-T14, and B345-T22 are in the elastic stage. The equivalent stress of the panel zone and the steel beam is linear with the loading displacement of the beam end. The maximum equivalent stress of specimen B345-T8 appears at the center point of the joint domain, while specimens B345-T14 and B345-T22 appear in the beam–column flange docking area. When the plastic strain (PEEQ > 0) is presented at and near the center of the panel zone, the loading displacements of the beam ends of specimens B345-T8, B345-T14, and B345-T22 are 38 mm, 38 mm, and 76 mm, respectively. After the elastic–plastic stage, as the displacement of the beam end continues to increase, the panel zone of specimen B345-T8 shows obvious bulging deformation along the diagonal direction and specimen B345-T14 has a small out-of-plane deformation. The steel beam web within the height range of the column beam has obvious out-of-plane deformation, and the lower flange of the right beam and the upper flange of the left beam have local buckling. For specimen B345-T22, whose steel beam web within the height range from the column beam has obvious out-of-plane deformation, the lower flange of the right beam and the upper flange of the left beam have local buckling.

When the B345-T8 specimen enters the failure stage, the deformation of the panel zone expands, but the steel beam does not buckle, indicating that, when the panel zone is weak, it almost bears all the deformation of the joint. Nevertheless, when the B345-T14 specimen enters the failure stage, the steel beam produces a plastic hinge, the stress in the joint area falls back, and the bearing capacity of the joint decreases to 89.6% of the ultimate bearing capacity. For specimen B345-T22, when the beam end displacement reaches 228 mm, the drift ratio is 6%, the failure stage is entered, and the bearing capacity at the time of failure is only 60.81% of the ultimate bearing capacity.

3.2.2. Equivalent Interstory Shear Force–Drift Ratio Hysteresis Curve and Skeleton Curve

In the finite element post-processing module, the components of the drift ratio and the shear ratio of the panel zone are obtained. The displacements of the left beam, the right beam, the top of the column, and the bottom of the column are measured, as shown in Figure 18. Through the four sets of data, the drift ratio of the joint during the loading process can be calculated according to Equation (4). Combined with the force analysis diagram of joints in Figure 18 and Figure 19, the equivalent layer shear force–drift ratio hysteresis curves of 11 groups of joint specimens can be calculated, as shown in Figure 20.

$${\theta }_d=\frac{{\delta }_1-{\delta }_2}{L}-\frac{{\delta }_3-{\delta }_4}{H}$$

(4)

where ${\delta }_1$denotes the displacement of left end of beam; ${\delta }_2$denotes the displacement of right end of beam; ${\delta }_3$denotes the displacement of up end of column; ${\delta }_4$ denotes the displacement of bottom end of column; $L$ denotes the width of beam; $H$denotes the height of column.

As depicted in Figure 20, the hysteresis curves of five groups of specimens, B345-T14, B345-T18, B345-T20, B345-T22, and B345-T24, present the phenomenon of saturation. This indicates that the thicker the panel zone, the more significant the energy dissipation capacity of the joint will be. With the increase in the panel zone’s thickness, the joint specimen’s ultimate bearing capacity is significantly improved. The maximum equivalent story shear forces of the B345-T8, B345-T14, and B345-T24 specimens during loading are 94.96, 170, and 247.88, respectively. In the hysteresis curve of the specimen with a thickness of 8~14 mm, it can be found that the hysteresis curve of the joint shows a certain yield characteristic after the drift ratio exceeds 3.7%; this is in contrast to the hysteresis curve of specimens with a thickness of 18~24 mm.

In order to more intuitively compare the influence of the weakening or strengthening of the panel zone on the joint, the equivalent story shear force–drift ratio hysteresis curves of three specimens B345-T10, B345-T14, and B345-T24 were selected to draw the same diagram, as shown in Figure 21. It can be seen from the diagram that, with an increase in the effective thickness of the joint area of the specimen, the ultimate bearing capacity of the specimen improves to a certain extent. Compared with the panel-zone-reinforced specimen, the hysteresis curve of the panel-zone-weakened specimen shows a pinch phenomenon, indicating that the energy dissipation capacity is general. A possible reason for this is that the panel zone is too weak, which leads to excessive deformation in the panel zone and a decrease in the bearing capacity of the joint.

Figure 22 illustrates the equivalent story shear force–drift ratio skeleton curves of 11 groups of joint specimens. From these, it can be observed that an increase in the thickness of the panel zone is correlated with a significant increase in the ultimate bearing capacity of the B345-T8, B345-T9, and B345-T10 specimens. When the drift ratio reaches 3%, the bearing capacity decreases; with an increase in the subsequent drift ratio, the bearing capacity tends to stabilize. For specimens B345-T18, B345-T20, B345-T22, and B345-T24 whose panel zone was strengthened, during the forward loading stage, when the drift ratio reached 4%, the bearing capacity gradually declined. In contrast, during the reverse loading stage, the ultimate bearing capacity decreases significantly when the drift ratio reaches 3%, and the bearing capacity is only 70.8% of the ultimate bearing capacity.

3.2.3. The Moment–Panel Zone Shear Ratio Curves of Joint Specimens

According to the panel zone shear ratio calculation Equation (5) and Figure 19, the panel zone bending moment–shear ratio (M_pz–$\theta$_pz) curve can be calculated.

$${\theta }_{pz}=\frac{{\delta }_5-{\delta }_6}{2}\frac{\sqrt{h_{pz.0}^2+b_{pz.0}^2}}{h_{pz.0}{b}_{pz.0}}$$

(5)

where ${\delta }_5$ and ${\delta }_6$denote the deformation of panel zone in two diagonal directions; $h_{pz.0}$ denotes the the width of the panel zone; $b_{pz.0}$ denotes the the height of the panel zone.

It can be seen in Figure 23 that there is a significant difference between the influences of the web-weakened and -strengthened options on the rotating angle and energy dissipation of panel zones. The hysteresis curve of the bending moment–panel zone shear ratio of the B345-T14 specimen is compared and analyzed with other specimens. From the data of six specimens with weakened panel zones, it can be observed that, with a decrease in the effective thickness of the panel zone, a pike shape can be seen in the hysteresis curve, while the hysteresis loop area is larger. With the reduction in the effective thickness of the panel zone web, the proportion of the panel zone participating in energy dissipation gradually increases, and the shear rotation angle of the panel zone increases. The maximum shear rotation angle of specimen B345-T8 can reach 0.58, while that of specimen B345-T14 is only 0.27. The results show that the weaker the panel zone is, the stronger the ability to develop a plastic rotation angle will be. For the hysteresis curve of the strengthened specimen in the panel zone, the bending moment of the panel zone increases; additionally, the shear deformation angle begins to decrease to 0.038, and the envelope area of the hysteresis loop decreases with the increase in the effective thickness of the panel zone. This indicates that the ratio of the energy dissipation of the panel zone to the overall energy dissipation of the joint decreases. From the above results, it can be concluded that the slight weakening of the thickness of the panel zone can better make full use of the plastic deformation and energy dissipation capacity, while the strengthening of the panel zone may lead to the failure of the panel zone to develop a sufficient plastic rotation angle.

3.2.4. Misses Equivalent Stress Distribution in the Panel Zone

As plotted in Figure 24, two stress paths, BC and DE, were selected to study the stress distribution and development in the panel zone. Considering the large number of specimens, three groups (B345-T8, B345-T14, and B345-T24) were selected for this subsection to compare the equivalent stresses on stress path 1.

Figure 25 shows the Misses equivalent stress distribution curves of the B345-T8, B345-T14, and B345-T24 specimens along stress path 1. In the elastic stage, the stress distribution of the three groups of specimens in the diagonal direction is relatively uniform. In the yield stage, for specimen 345-T8, the maximum stress of the center point of the panel zone reaches 384.03 MPa, while the stress bands of the B345-T14 and B345-T24 specimens are formed along the diagonal, and the maximum stresses can reach 492.72 MPa and 378.30 MPa, respectively. After the specimen enters the failure stage, the deformation of the joint continues to increase, and the stress of the panel zone of the three groups of specimens decreases to different degrees. The latter two groups of specimens with plastic hinges at the beam end are the most obvious.

From the comparison of the three groups of specimens in the elasticity, yield, and failure stages, it can be observed that the overall effective stress of the B345-T24 specimens with the panel-zone-strengthened is at a lower level. This can be explained by the fact that the steel beam bears the primary deformation of the joint and the strengthening plate participates in sharing the deformation of the panel zone and energy dissipation, thus reducing the stress in the panel zone. Conversely, the specimens with the weakening of column webs, B345-T8, show different degrees of decrement in the panel zone stresses, showing a fall in stress after the plastic hinge is formed; this is because its panel zone bears the primary deformation among the whole joint.

3.2.5. Ultimate Load and Ductility Coefficient

The ductility coefficient of the joints is calculated according to Equation (6). The yield load, yield displacement, ultimate load, ultimate displacement, and ductility coefficient of the B345-T8, B345-T11, B345-T14, B345-T20, and B345-T24 specimens are listed in

Table 5. Displacement ductility factors.

Specimen	${\mathsf{\Delta }}_{\mathit{y}}\left(\mathbf{m}\mathbf{m}\right)$	${\mathit{P}}_{\mathit{y}}\left(\mathbf{k}\mathbf{N}\right)$	${\mathsf{\Delta }}_{\mathit{u}}\left(\mathbf{m}\mathbf{m}\right)$	${\mathit{P}}_{\mathit{u}}\left(\mathbf{k}\mathbf{N}\right)$	$\mathit{\mu }$
B345-T8	47.5	59.72	228	99.53	4.80
B345-T11	55.48	72.09	228	126.82	4.11
B345-T14	63.84	97.80	191.52	162.99	3.02
B345-T20	83.98	114.57	199.12	190.95	2.37
B345-T24.	88.54	135.74	199.12	226.23	2.25

Note: ${\mathsf{\Delta }}_y$ denotes the yield displacement; $P_y$ denotes the yield load; ${\mathsf{\Delta }}_u$ denotes the ultimate displacement; $P_u$ denotes the ultimate load; $\mu$ denotes the ductility coefficient.

.

$$\mu =\frac{{\delta }_u}{{\delta }_y}$$

(6)

where ${\delta }_u$ denotes the ultimate displacement; ${\delta }_y$ denotes the yield displacement; $\mu$ denotes the ductility coefficient.

It can be seen from Table 5 that with an increase in web thickness, the ductility coefficient of the joint decreases first and then tends to stabilize at about 2.25. When the thickness of the panel zone is thin, the time of the joint in the elastic stage is shorter at the initial loading stage, so the joint enters the plastic deformation stage earlier. The deformation of the panel zone almost occupies most of the deformation of the joint; the plastic deformation is large, and the bearing capacity does not decrease significantly. Therefore, the ductility of the specimen is better. With the increase in the thickness of the column web, the elastic deformation capacity of the joint increases, and the plastic deformation of the joint decreases after yielding, so its ductility begins to fall with the increase in the thickness of the panel zone. Due to the high strength of the panel zone, the stress concentration and yield phenomenon transfer to the beam end, the accumulation of flange buckling and web buckling plastic deformation at the beam end leads to the plastic hinge at the beam end, and the bearing capacity of the joint depends on the stiffness of the beam. Therefore, after the thickness of the joint domain reaches 22 mm, the ductility coefficient of the joint tends to be stable.

3.2.6. Stiffness Degradation

The stiffness degradation coefficient is defined as the absolute of the degraded stiffness at all levels of displacement loading to the cutline stiffness at the initial elastic loading stage, and the results are plotted in Figure 26.

It can be found from Figure 26 that the joints with weakened panel zones reach yield in the panel zones first. With the increase in displacement, the stress and deformation of the panel zones increase rapidly, forming a plastic hinge in the panel zones. While the joints with welded steel plates reach yield in the beam end section first, with an increase in displacement, the plastic stresses are shifted to the flanges and the welded holes, forming a plastic hinge at the beam end. The stiffness degradation coefficient curve of the panel-zone-strengthened specimen is higher than that of the panel-zone-weakened specimen. In addition, the panel-zone-strengthened specimen’s stiffness degradation curve is gentler.

3.2.7. Equivalent Viscous Damping Coefficient

The energy dissipation capacity of the joint can be expressed by the equivalent viscous damping coefficient, which is calculated based on the equivalent principle of damping force. The damping ratio of the whole specimen under a certain displacement state in the whole loading system can represent the energy dissipation capacity of the specimen. It can be calculated according to the ratio of the maximum hysteresis loop ABC of the hysteresis curve in Figure 27 to the area surrounded by the abscissa S₁ (blue shade area) and the area of the triangle OBD S₂ (yellow shade area). Combining Equation (7) and Figure 27, the equivalent viscous damping coefficients for 11 groups of specimens B345-T8~B345-T24 were plotted in Figure 28.

$${\mathsf{\xi }}_{\mathrm{e}}=\frac{1}{2\mathsf{\pi }}\times \frac{{\mathrm{S}}_1}{{\mathrm{S}}_2}=\frac{1}{2\mathsf{\pi }}\times \frac{{\mathrm{S}}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{O}\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{O}}}{{\mathrm{S}}_{\mathsf{\Delta }\mathrm{O}\mathrm{B}\mathrm{D}}}$$

(7)

Figure 28 shows the equivalent viscous damping coefficient of the specimens; here, it can be seen that the panel-zone-weakened joints and the B345-T14 specimens are more than 0.3; this indicates that the seven groups of specimens exhibit good energy dissipation capacity, especially those with weakened panel zones. The equivalent viscous damping coefficient of the joints shows a trend of rising and then decreasing with the increase in the thickness of the panel zone; this can be explained by the fact that the excessively weak panel zones have large deformation. Therefore, the plastic hinge is formed prematurely, and the bearing capacity of the joint decreases rapidly. The panel zone’s energy dissipation capacity cannot be fully used. The equivalent viscous damping coefficients of specimens B345-T10, B345-T11, and B345-T12 are 0.398, 0.451, and 0.407, respectively. The remaining four specimens include B345-T18, B345-T20, B345-T22, and B345-T24. The effective thickness of the joint domain is increased by adding welded steel plates in the panel zone. The deformation capacity of the reinforced specimen during the loading process is mainly controlled by the deformation of the steel beam. When the plastic hinge is formed at the beam end, the bearing capacity of the joint decreases rapidly. The ductility of the joint is poor, which demonstrates that, through appropriately reducing the thickness of the panel zone, the steel beams in the joint and the panel zone share the energy dissipation of the entire structural deformation; thus, the energy dissipation of panel zone was significantly improved.

3.3. Recommendations

The current research on panel zones mainly focuses a consideration of the effect of panel zone deformation in design, aiming to elucidate approaches for fully utilizing the energy dissipation capacity of panel zones after yielding in seismic designs. In this paper, the hysteresis characteristics, bearing capacity, ductility, energy dissipation, and mechanical properties, such as the panel zone deformation in finite element models with different panel zone web thicknesses, are carefully discussed. Our findings indicate that the panel zone equilibrium design criterion proposed by FEMA-355D [32] is not fully applicable when the equilibrium design coefficient of the panel zone is between 0.7 and 0.9. In cases of severe buckling deformation in the panel zone of the specimen, the joint is buckled in advance. Hence, the steel beams fail to sufficiently utilize their deformation capacity. It is noted that the weakening of the panel zone can lead to full utilization of the energy dissipation capacity of panel zones and the deformation capacity. However, too weak a web thickness of the panel zone will cause severe buckling deformation, which is unfavorable for flange welds. On this basis, it is proposed that the equilibrium design coefficients of the panel zone should meet the requirements of Equations (8) and (9):

$${\phi }_{pz}=\frac{\sum W_{pb}{f}_{yz}\times \frac{L}{L-h_c}\frac{H-h_b}{H}}{0.55f_{y.pz}{h}_c{t}_{pz}{h}_b}$$

(8)

$$0.6\le {\phi }_{pz}\le 0.7$$

(9)

where ${\phi }_{pz}$ denotes the balance design coefficient of the panel zone; $f_{yz}$ denotes the tensile yield strength of panel zone; $L$ denotes the span of the beam; $H$ denotes the height of the column; $h_c$ denotes the width of the panel zone; $h_b$ denotes the height of the panel zone; $f_{y.pz}$ denotes the shear yield strength of panel zone; $t_{pz}$ denotes the thickness of panel zone web; $W_{pb}$ denotes plastic section modulus of beam.

When the panel zone equilibrium coefficient ranges from 0.6 to 0.7, the specimen yields and deforms in both the panel zone and the beam end cross-section during the loading process. This scenario makes full use of the energy dissipation capacity of the panel zone after yielding and reduces the demand on the beam plastic rotation angle.

4. Conclusions

In this paper, a numerical model is calibrated for a high-strength steel joint test to verify the accuracy of the finite element model as well as the analysis method. Then, a parametric study of panel zones with different thicknesses (8 mm, 9 mm, 10 mm, 11 mm, 12 mm, 13 mm, 14 mm, 18 mm, 20 mm, 22 mm, and 24 mm) of H-shaped beams and columns were numerically analyzed. Based on the obtained results, the following conclusion can be drawn:

(1)

For the existing test specimens, the validity of the finite element model has been verified. The two finite element models can reasonably reproduce the test results and verify the validity of the finite element model, which provides the finite element model and analysis basis for the subsequent H-shaped steel beam–column joints and parametric analyses.

(2)

The effective thickness of the web of the panel zone has a significant effect on the position of the plastic hinge of the joint. The B345-T8 (8 mm), B345-T9 (9 mm), and B345-T10 (10 mm) specimens with weakened webs form plastic hinges in the panel zone, while the B345-T18 (18 mm), B345-T20 (20 mm), B345-T22 (22 mm), and B345-T24 (24 mm) specimens with thickened webs form plastic hinges at the beam end.

(3)

With the increase in the web thickness of the panel zone, the ultimate bearing capacity of the joint is significantly improved. However, the specimen with excessive weakening of the web thickness of the joint domain has a significant out-of-plane deformation due to its weak panel zone, which causes it to be damaged in advance; therefore, the hysteresis curve of the joint is pinched. Compared with the specimens with weakened webs in the panel zone, the hysteretic curves of the strengthened joints are fuller. However, in the process of increasing the displacement of the beam end, the joint cannot protect the weld seam in the butt joint area of the beam–column flange, nor can it effectively alleviate the stress concentration at the web weld hole.

(4)

Through the analysis of the ductility coefficient, the stiffness degradation, and the energy dissipation capacity, the ductility and energy dissipation capacity of the partially weakened specimens were found to be better than those of the strengthened specimens. However, when the web thickness of the panel zone is too thin, the out-of-plane buckling deformation of the panel zone will cause a significant decrease in the bearing capacity of the joint, resulting in premature failure of the specimen.

(5)

The stiffness degradation coefficient of the web-thickened specimen is dominated and controlled by the stiffness of the beam. However, with an increase in the web thickness, the stiffness degradation coefficient remains basically unchanged with an increase in the displacement.

(6)

Based on the FEMA-355D equilibrium design criterion, when the optimal equilibrium design coefficient is in the range of 0.6~0.7, both the panel zone and the steel beam participate in the energy dissipation of the joint. Compared with ordinary joints, the ductility and energy dissipation of the joints are improved.