Evolution characteristics and performance evaluation of extended end-plate joints with single and double side connections under cyclic loading

Abstract

Experimental studies were conducted on four extended end-plate joints subjected to cyclic loading at the column top, investigating the evolving patterns of the joints' mechanical performance. The paper provides a detailed analysis and discussion of the test joints' failure modes, ductility, stiffness degradation, and energy dissipation capacity. The Mann-Kendall (M − K) trend analysis tool was applied to the mechanical response curves, identifying key performance evolution points (evolution initiation point P and overall yield point Q). The trends in bolt forces, deformations, and strains at critical joints were effectively validated, revealing the transition of the energy system from quantitative to qualitative changes and the component's failure process from stability to instability. Additionally, based on the experimental joints' hysteresis curves and energy dissipation capacity, a theoretical hysteresis model was established to predict the joint's hysteresis curve and cumulative dissipated energy accurately. According to EC3 requirements, joints were classified as partially rigid connections. The experimental results of the initial rotational stiffness and plastic moment were further used to evaluate the calculated values in existing standards EN 1993-1-8, ANSI/AISC 358-16, and GB 51017-2017. The results indicate that extended end-plate connections possess sufficient strength, joint rotational stiffness, ductility, and energy dissipation capacity, making them suitable for seismic moment frames.

1. Introduction

In the context of the extensive background of industrialization and modularization in construction, the joints of plate connections are gaining widespread application in multi-story steel frame structures due to their advantages of simple assembly, time-efficient construction, and the ability to achieve good performance at relatively low costs [1,2]. As typical semi-rigid (flexible) connections, plate connections differ from commonly used pin connections and rigid connections [3], exhibiting nonlinear mechanics and non-continuous connections [4]. Accurately assessing the instability characteristics of plate connection joints under complex loads such as earthquakes and wind loads is crucial for the safety and reliability of structures. The risk of failure at critical locations [5,6] in plate connections can lead to catastrophic collapses of buildings [7,8]. This risk can be attributed to the sequential evolution of performance [9] and the cumulative progression of damage, representing the gradual transformation of the energy system in plate joints from quantitative to qualitative changes, continually triggering failure risks [10].

Numerous scholars have extensively researched the structural performance of plate connections. Early studies involved monotonic static tests [2,11] and elastic-plastic theoretical mechanical analyses [12], decomposing plate connections into T-shaped elements and applying yield line theory to calculate their load-bearing capacity and deformation [13]. With the widespread adoption of computer technology, finite element methods and numerical calculations [14] have been employed. Additionally, cyclic loading tests have been conducted to study their hysteresis behavior [15], focusing on their hysteretic performance [16]. Currently, with the rise of big data technology, databases of beam-column connection joints [17] are gradually being established. From the structural performance of demountable precast concrete column-column connections [18] to innovative hybrid beam-column connections [19], from innovations in seismic damping technology [20] to fracture toughness issues in the field of high-strength steel structures [21], and finally the behavior of high-strength steel built-in H-section columns [22], which provide new ideas and methods for solving the problems and structural design of main connections and explore the potential of seismic damping in infrastructural potential in infrastructure seismic mitigation, these studies provide an important reference for optimizing the design and improving the structural performance and also reflect the continuous concern and efforts for safe and reliable building structures. The hysteresis performance of plate connections has always been a major theme in seismic research on steel structures. Scholars have paid special attention to their structural performance to ensure design safety. Guo et al. [23] compared the hysteresis behavior, stiffness, and strength of reinforced and unreinforced extended plate connections through experiments, indicating that ribbed connections have higher load-bearing capacity and energy dissipation, enhancing connection stiffness, which significantly influences their cyclic behavior. Chen [24] and Barigozzi [25] evaluated the mechanical properties of unreinforced extended plate connection joints through monotonic loading tests, suggesting that semi-rigid design methods better match actual performance. Shi et al. [15] conducted monotonic and cyclic loading tests on eight extended plate connection joints, introducing ribbed reinforcements to the plate and column flange. The study showed that both column flange and plate reinforcements can improve joint stiffness and load-bearing capacity. Wang et al. [26] studied the flexural behavior of weak-axis and strong-axis plate connections, asserting that these connections remain typical semi-rigid connections. The above studies indicate that extended plate connections can be used in steel frames in high-seismic regions because they possess sufficient strength, stiffness, ductility, and energy dissipation capacity. This prevents bolt fracture-induced brittle failure by allowing other more ductile components to locally failure [27].

The current research methods for structural failure of steel joints are mainly based on static elastoplastic analysis of structural performance indicators [28], finite element modeling analysis (rod model, solid model, fine model, multiscale model, spring rod model) [29], and incremental dynamic analysis (IDA) [30], to establish the correspondence between the joint failure state and the distinctive values of the load-displacement curve, seismic capacity, seismic susceptibility, and other parameters. However, the failure evolution analysis of the overall joint structure is either too cumbersome or too simple. The actual structure is three-dimensional and spatial effects are not considered, and the existing analysis methods present some invalidity in the face of the multiple directions of external effects, the uncertainty of susceptibility and the randomness of the influencing factors [31], which cannot reproduce the complex working behavior and damage mechanisms of the joint structure. Moreover, due to the lack of theoretical concepts and the incompetence of technical methods, the large amount of strain data from the test that is not easily accessible is not sufficiently valued, which leads to the failure of this part of the data to give deep analysis [32,33], where new research tools have been needed to reveal the unseen effects of changes in the structural response on the evolution of the joint failure.

The literature review above reveals that, to date, most experiments have only analyzed and compared data-level mechanical responses, without defining their evolving characteristics or conducting accurate performance evaluations. The inherent mechanics and underlying patterns have not been sufficiently revealed. Moreover, the critical component of the panel zone has not been fully considered. Previous experiments usually applied loads to the beam end to simulate unidirectional bending moments for single-sided connections or symmetric bending moments for double-sided connections. Both loading methods overlook the case of asymmetric bending moments in double-sided connection joints, where two opposing bending moments are most unfavorable for the panel zone. Considering the sequential and uncertain nature of seismic loads, there exists a coupled superposition effect of moments generated at the joint connection [8,34]. Thus, studying the mechanical performance of double-sided connection joints with opposing bending moments is necessary. This paper adopts the displacement application method at the column top, which, according to the principles of mechanics, produces opposing bending moments on both sides of double-sided connection joints. This approach more realistically reflects the stressing state of the panel zone, facilitating the consideration and analysis of critical mechanical components of the joint. Compared to the beam-end loading method, the column-top loading method enhances the means to improve the experimental analysis of joint force performance. The two methods differ in the proportion of rotational deformation and the semi-rigidity of joints, hence a comprehensive study of the structural performance of joints with double-sided plate connections is still needed.

The structural behavior and failure evolution of four extended plate joints, including two double-sided (DS) connection joints and two single-sided (SS) connection joints, were investigated by using the structural stressing state methodology. The innovative application of the M − K criterion identifies the characteristic points of the performance evolution and verifies the reliability of the results with the traditional three methods of yield point calculations, and then analyzes the working state (hysteretic performance, bolt force development, joint deformation and strain, degradation of stiffness and accumulation of energy dissipation) and other aspects of verification of failure in the structural system from quantitative to qualitative changes, reflecting the gradual evolution of material failure and joint mutation damage characteristics, and revealing the mechanical response mechanism and evolution of degradation law behind it at a deep level. A theoretical hysteresis model based on the experimental results was developed to predict the joint's hysteresis curve and cumulative dissipated energy. Furthermore, the accuracy of existing design standards, namely the American guide ANSI/AISC 358 [35], European regulations EN1993-1-8 [12], and Chinese GB 50017 [36], was evaluated based on the critical structural performance characteristics of joints (moment bearing capacity and initial rotational stiffness).

2. Experimental program

Four joints with extended end-plate were designed and fabricated in this study concerning various design standards and guidelines, with special attention paid to key points such as load-carrying capacity, stability, ductility, and design of connecting bolts of the joints. During the test, cyclic loading is applied to simulate the effects of the earthquake and vertical gravity loads to study the force response and performance of the joints under different working conditions, which mainly include the initial rotational stiffness of the joints, bending capacity, energy dissipation capacity, ductility, and damage modes. In order to reasonably collect the above data, a variety of instruments and sensors are used for data collection and monitoring, and the selection of these experimental programs and instruments can ensure the reliability and accuracy of the test. The experimental study also involves the testing and analysis of material properties, including the mechanical property testing of common steel and high-strength bolts, which provides the basis and foundation for the subsequent test loading and theoretical calculation. Overall, by comprehensively utilizing the design standards and guidelines of different countries, deeply analyzing the design requirements and checking steps of the joints, and combining the material property tests and detailed test setups, this study aims to comprehensively assess the performance of the steel joints, and to provide reliable guidance and reference for the engineering practice.

2.1. Design basis

Based on the steel structure design standards and guidelines of various countries [12,[35], [36], [37], [38], [39], [40]], to meet several design criteria such as “strong column weak beam,” corresponding design requirements are specified for the yielding capacity, stability, strength, and ductility of the panel zone. The components of the extended end-plate joint should adhere to the following provisions.

• (1)“Strong column weak beam” design

This calculation ensures that the steel beam enters the elastic-plastic state before the steel column.

$$W_c\left(f_{yc}-N_p/A_c\right)⩾1.1{\eta }_y\left(W_b{f}_{yb}+V_{b}s\right)$$ (1)

where N_p is the axial force of the column under combined loads, s is the distance from the plastic hinge position to the column flange face, A_c and W_c are the column cross-sectional area and modulus of section, f_yb and f_yc are the design strengths of the beam and column, η_y is the steel strengthening factor, generally taken as η_y = 1.15, V_b is the design shear force at the plastic hinge section of the beam, calculated according to Eq. (2).

$$V_b=V_G+\frac{2W_b{f}_{yb}}{L-h_{bw}-\frac{2}{3}{h}_c}$$ (2)

where V_G is the shear force under representative gravity loads, L is the span of the beam, h_bw is the height of the beam web, and h_c is the height of the column section.

• (2)Yielding capacity design of the panel zone

$$\left(M_{b1}+M_{b2}\right)/V_{pl}\le \left(4/3\right)f_{yv}/{\gamma }_{\text{RE}}$$ (3)

where f_yv is the design shear strength of the steel, f_yv = f_y/√3, f_y is the design strength of the steel, M_b1 and M_b2 are the full-section plastic bending capacities of the beams on both sides, V_pl is the effective volume of the panel zone, Y_re is the seismic adjustment factor for the panel zone load-bearing capacity, generally taken as 0.75.

• (3)Stability design of the panel zone

The stability of the panel zone is required according to the thickness of the column web plate in the “Steel Structure Design Code” [36] and the “Seismic Design Code for Buildings” [41]. The stability coefficient of the panel zone is:

$${\lambda }_{cw}=\left(h_{cw}+h_{bw}\right)/t_{cw}\le 90$$ (4)

The specific meanings of various parameters and their corresponding positions are shown in Fig. 1. Based on Table 1 and Fig. 1, the stability coefficients λ_cw of the designed joints are calculated according to the above equation, and all four joints have λ_cw of 27.4, meeting the specified conditions.

Fig. 1.

Design information: (a) end-plate connection parameters; (b) joint parameter information.

Table 1.

Configuration details of joint specimens. (Bolt grade: 10.9).

Joints type	Specimen	Beam (mm)	Column (mm)	End-plate thickness (mm)	Bolt Diameter (mm)	Initial Tightening Torque (N·m)	Final Tightening Torque (N·m)	Bolt pretension force Fpre (kN)
Double-side connection joint	DS1	300 × 200 × 8 × 12	300 × 300 × 10 × 15	16	20	280	446	155
	DS2			16	24	400	760	225
Single-side connection	SS3			16	20	280	446	155
	SS4			20	20	280	446	155

Also, to comply with EN1993-1-8 [12] and ensure sufficient rotational capacity, the slenderness ratio (h_bw/t_cw) of the column web plate should satisfy Eq. (5), and the thicknesses t_ep and t_cf of the column flange or end-plate should satisfy Eq. (6).

$$\frac{h_{bw}}{t_{cw}}\le 69\epsilon$$ (5)

$$t_{ep}or{t}_{cw}\le 0.36d_b\sqrt{\frac{f_{ub}}{f_y}}$$ (6)

where h_bw has the specific meaning mentioned above (refer to Fig. 1 for specific location, and repetitive joint dimension parameters are not repeated, the same below), ε is the out-of-plane deformation limit of the column web plate, d_b is the nominal bolt diameter, and f_ub is the ultimate tensile strength of the bolt.

• (4)Ductility design

Ductility design aims to ensure that the joint has sufficient deformation capacity, energy dissipation capacity, and rotational capacity. The failure modes of end-plate connections mainly include [42]: 1) bending failure of the end-plate and column flange; 2) yielding in the panel zone before bolts and end-plates (shear failure); 3) yielding of the end-plate or column flange before the bolts. To avoid other forms of brittle failure, the ductility design of the joint needs to be verified.

• 1)Bending failure of the end-plate and panel zone column flange

Under seismic action, to ensure that the failure mode of the end-plate and column flange is bending failure rather than shear failure around the bolts, the following conditions are required for the end-plate and column flange:

$$N_{ut}=\frac{b_{ep}{t}^2{f}_y}{4e_f}+\frac{\left(e_f+e_w\right)t^2{f}_y}{2e_v}\le N_v=\eta \pi d_{b}t{f}_{yv}$$ (7)

where b_ep is the width of the end-plate, t is the thickness of the end-plate or column flange, d_b is the bolt diameter, η is the ratio of the outer diameter of the bolt head or nut contact area to the nominal diameter of the bolt, generally taken as 1.7; e_f and e_w are the distances from the center of the bolts at both ends to the beam flange and web surface, respectively.

• 2)Panel zone yielding before bolts and end-plate

The bearing capacity of the end-plate connection should be controlled by the panel zone, i.e., the design value M_ep,b of the bending capacity corresponding to the end-plate and bolts, and the design value M_s of the panel zone bending capacity should satisfy the following Eq. (8):

$$M_{ep,b}=\frac{m{N}_{t1}\sum y_i^2}{y_1}\ge M_s=\frac{4}{3}{f}_0{V}_p=\frac{4}{3}\frac{f_y}{\sqrt{3}}{V}_p$$ (8)

where N_t1 is the bearing capacity design value of the bolt and end-plate segment at the first row of bolts, $N_{t1}=\min \left\{N_{u,ep},N_{tu,b}\right\}$, N_u,ep is the bearing capacity design value of the end-plate segment at the first row of bolts, $N_{u,ep}=\frac{b_{ep}{t}_{ep}^2{f}_y}{6e_f}+\frac{\left(e_f+e_v\right)t_{ep}^2{f}_y}{3e_v}$, N_tu,b is the design tensile bearing capacity of a single bolt, $N_{tu,b}=0.8P$, y_i is the lever arm length, which can be referred to Ref. [42].

• 3)Yielding of the end-plate or column flange before the bolts

In ductility design, the ultimate bearing capacity N_u,ep of the end-plate should not exceed the ultimate tensile bearing capacity N_tu,b of the bolt, i.e.,

$$N_{tu,b}\ge N_{u,ep}$$ (9)

where: $N_{tu,b}=A_{eb}{f}_{ub}$, A_eb is the effective area of the bolt, f_ub is the minimum value of the ultimate tensile strength of the bolt.

• (5)Bolt Connection Design

• 1)Bolt arrangement: Refer to the relevant methods in the “Steel Structure Design Manual” [43]. The bolt arrangement for the end-plate connection should be symmetric. Two rows of bolts are preferred, and one row of bolts is set on the inner and outer sides of the beam flange, forming an effective T-shaped component under load. Depending on the actual requirements, multiple rows of bolts can also be added on the inner side of the flange.
• 2)Bolt selection: Preferably, high-strength friction-type bolts with diameters usually ranging from 12 to 30 mm are used. In this study, M20 and M24 bolts were used.
• 3)Bolt preload: F_pre can be determined by the following Eq. (10).

$$F_{\text{pre}}=0.7A_e^b{f}_u^b/{\gamma }_M$$ (10)

where Ab e and f b u are as in Eq. (9), and the local safety factor γ_M is 1.1.

• (6)Horizontal Stiffening Rib Design for Column Web Plate

According to the provisions of the “Steel Structure Design Code” (GB50017-2017) [36] for the design of horizontal stiffening ribs on the column web plate, the widths bs and thicknesses t_s of the stiffening ribs should satisfy the following Eqs. (11), (12):

$$b_{\mathrm{s}}>h_{cw}/30+40$$ (11)

$$t_{\mathrm{s}}>b_{\mathrm{s}}/15$$ (12)

where t_s and b_s are the thickness and width of the stiffening ribs, respectively. In addition, according to the relevant provisions of the “Design Code for Steel Structures of High-rise Civil Buildings” (JGJ99-98) [44]: for structures with seismic fortification, the thickness of horizontal stiffening ribs should be 0.5 to 2 times the thickness of the beam flange; the centerline of the horizontal stiffening rib should align with the centerline of the beam flange.

2.2. Joint configuration

According to the requirements outlined above, four basic specimens were designed, and the details of the specimens can be found in Fig. 2. The four extended end-plate joints are made of I-shaped section steel, including two specimens with double-side (DS) connected extended end-plate joints and the other two with single-side (SS) connected extended end-plate joints. The corresponding specimen numbers are DS1/2 and SS3/4, where DS and SS represent whether the beam is connected to the column on both sides or one side only. The dimensions of the steel beam and column sections are 300 × 200 × 8 × 12 mm and 300 × 300 × 10 × 15 mm, respectively, and the end-plate dimensions are 500 × 250 × 16/20 mm. The studied parameters include end-plate thickness, bolt diameter, and single or double-sided connections. The thickness of the column web stiffening rib is equal to the beam flange thickness (12 mm). Additionally, to eliminate local yielding issues at the loading point, stiffening ribs (12 mm) are placed at the beam end and column top, as illustrated in Fig. 2a.

Fig. 2.

Geometric details of joints (all dimensions in mm): (a) end-plate connection parameters; (b) details of end-plate (full scale); (c) details of end-plate (1/2 scale).

These specimens are derived from typical multi-story steel frame joints, where the flexural point of the steel frame is generally at l/2 under the combined action of horizontal seismic forces and vertical gravity loads. The cantilever beam length was chosen to be 1500 mm, and the column height was set at 2100 mm. Except for the high-strength bolts, all other steel materials are Q345B steel. The bolts used are high-strength friction-type bolts (Grade 10.9). All bolts were tightened using the calibrated wrench method, following the construction sequence of initial tightening and final tightening. Torque wrenches were calibrated for torque on the pre-tensioned bolts. Table 1 lists the initial and final tightening torques along with the bolt pre-tension force (F_pre), calculated based on the bolt diameter specified in the code. The factor γ_M is set to 1.1 to account for safety, as recommended to prevent relaxation of pre-tensioned force [45]. The friction surfaces of the end-plate and column flange are pre-treated with a sandblasting process, with a coefficient of friction of 0.45 and a gap of 2 mm between bolt holes and bolts.

The geometric dimensions and bolt-hole arrangement for the extended end-plate are shown in Fig. 2b and c. The end-plate and beam are joined by full-penetration butt welds, and all welds in the specimen are considered Class 1 welds using E50 electrodes. The welding work is completed in the factory, and the design and fabrication of the specimens comply with relevant standards and regulations. Stiffening ribs are placed at the panel zone and are aligned with the beam flange on three sides, and fillet welding is performed. According to the 8-degree fortification requirement, the axial compression ratio μ_N of the column is set to 0.3. The axial force F_c (F_c = 0.3f_ycA_c = 1211 kN) is kept constant throughout the entire test, where f_yc is the nominal yield strength of Q345B steel and A_c is the column cross-sectional area. The axial force is applied before the displacement loading based on seismic design standards.

2.3. Materials properties

Standard tensile material tests were conducted on plates of the same batch, same specifications, and the same grade used for processing beams and columns [35]. Tensile specimens were prepared according to the room temperature tensile test standard GB/T228.1–2010 [46]. Samples were taken from different locations, including beam flanges, column flanges, beam webs, column webs, end-plates, and bolts. Three replicate samples were prepared for each part, resulting in a total of 15 samples. Six samples were obtained for different bolt diameters (M20 and M24). Samples of ordinary steel plates were obtained directly from hot-rolled I-beams using wire-cutting technology, while round samples for bolts were machined from threaded rods using CNC milling. Fig. 3a and b shows the processing drawings and samples before and after the test, respectively, while Fig. 3c and d shows the plate and bolt samples before and after the test. A universal testing machine with a 300 kN capacity was used for testing (Fig. 3e). Tensile specimens were tested until fracture. For each specimen, two strain gauges and one built-in extensometer were used to monitor and record the development of strain, and stress values were determined by dividing the recorded force by the original cross-sectional area. The loading rate during the test complied with standard requirements.

Fig. 3.

Stress-strain curves of material coupons: (a) the results of tested steel plate material properties; (b) the results of tested bolt material properties; (c) steel plate samples; (d) Bolt standard samples; (e) material property testing device.

The obtained stress-strain (σ-ε) curves for the materials are plotted in Fig. 3a and b. The average measured material properties for ordinary steel and high-strength bolts are listed in Table 2. E₀ is the initial Young's modulus, E₀ is the elastic modulus; σ_y and σ_0.2 are the yield strengths of ordinary steel and high-strength bolts, respectively [47]; σ_u is the ultimate tensile strength; ε_y (ε_0.2) and ε_u are the corresponding yield strain and ultimate strain; ε_f is the elongation at fracture. The ratio of ultimate tensile stress to yield strength (σ_u/σ_y or σ_0.2) for ordinary steel and high-strength bolts is observed to be 1.5 and 1.2, respectively, indicating that ordinary steel has more significant strain hardening capacity. The ratio of ultimate tensile strain to yield strain (ε_u/ε_y or ε_0.2) for different thicknesses of ordinary steel exhibits higher ductility compared to high-strength bolts. These test results provide a basis for the material parameters of the test loading system and subsequent capacity prediction.

Table 2.

Material properties of specimens.

Sample	t or d (mm)	E0 (GPa)	σy (σ0.2) (MPa)	σu (MPa)	εy (ε0.2) (%)	εu (%)	εf (%)	σu/σy (σu/σ0.2)	εu/εy (εu/ε0.2)
End-plate	16	207.2	336.2	485.6	0.162	18.0	19.2	1.4	111.1
Beam flange	12	206.9	337.4	523.2	0.163	19.7	21.5	1.6	120.9
Beam web	8	198.6	371.7	534.7	0.187	20.0	21.1	1.4	107.0
Column flange	15	201.8	365.4	535.5	0.181	25.0	26.9	1.5	138.1
Column web	10	207.6	375.9	533.4	0.181	22.7	24.6	1.4	125.4
M20 bolt	20	234.7	924.5	1126.5	0.394	19.2	23.5	1.2	48.7
M24 bolt	24	218.8	926.3	1111.8	0.423	17.8	22.8	1.2	42.1

Note: t or d is the plate thickness and bolt diameter; E₀ is the elastic modulus; σ_y (σ_0.2) and σ_u are the yield strength and tensile strength; ε_y (ε_0.2) and ε_u are the corresponding yield strain and tensile strain; ε_f is the elongation after break rate.

2.4. Setup and loading

This experiment applies cyclic loading to the top side of the column to allow the force mechanism to transition from inter-story displacement loading to the connection bending moment between the beam and column. Significant shear deformation occurs in the intermediate panel zone, enabling a comprehensive consideration of the stressing state changes in the panel zone and the entire joint under cyclic loading. The test apparatus used is illustrated in Fig. 4a and b, including a schematic of boundary conditions, a 3D reconstruction, and on-site photos. The 3D reconstruction displays important components of the test apparatus in different colors for detailed description, including a rigid crossbeam, rigid connectors, a hydraulic servo MTS actuator, lateral supports, hydraulic jacks, and hinge supports. The differences in the DS and SS joint test setups lie in the constraint of the single or double-sided beam connection and the boundary conditions of the panel zone. In the DS joint, both sides of the beam end bear anti-symmetric loads to simulate the bending moment borne by the joint in a moment-resistant frame during an earthquake, making this setup more consistent with the actual load-bearing components in engineering practice. In Fig. 4a, the east and west sides of the DS joint beam end are labeled “East” and “West.” The boundary conditions of the test joint are set as follows: a hinge boundary condition is provided at both the bottom of the column and the beam end, and the column top is connected to the sliding support through an anchor rod. Lateral restraint devices are used to prevent out-of-plane torsional instability, and the column axial force is applied by a 600 kN capacity hydraulic jack from the top of the column, maintaining a constant force throughout the entire cyclic loading process. For the DS joint, the column axial force is divided between the east and west sides, while for the SS joint, all axial forces are applied on one side. The column top connection is on the MTS loading system, applying cyclic reciprocating displacements. The built-in load and displacement sensors record the applied force and corresponding displacement values.

Fig. 4.

Test setup for beam-to-column joint specimens: (a) Double-side connection joints; (b) Single-side connection joints; (c) Loading protocol.

To check if the components are in good contact and the test equipment is functioning properly, a pre-load is applied at a 0.1 % inter-story drift rate before the test loading. The formal loading history includes two stages: in the first stage, the axial force at the column top is calculated based on the axial compression ratio limit of the prototype frame, applied through a jack and kept constant. In the second stage, the loading protocol adopts the loading procedure proposed by SAC (1997) [48] for cyclic testing, with the loading amplitudes shown in Fig. 4c. Quasi-static cyclic loading is applied according to the loading protocol, controlling the entire test process through inter-story displacement. In the initial 6 cycles, the inter-story displacement ratios are 0.375 %, 0.5 %, and 0.75 %; before yielding the specimen, 4 cycles are performed with a 1.0 % inter-story displacement ratio. Subsequently, 2 cycles are performed with a 1.5 % inter-story displacement ratio; thereafter, the inter-story displacement increment is 1.0 %, and 2 cycles are performed at each stage until complete failure. The loading speed during the elastic stage of the specimen is 5 mm/min, and the loading speed increases to 12 mm/min after entering the plastic stage.

2.5. Instruments

Fig. 5 illustrates the detailed instrumentation scheme for the beam-column connection specimen, employing a combination of linear variable differential transformers (LVDTs), strain gauges, and force sensor elements. As depicted in Fig. 5a, this setup aims to investigate the local yielding and plastic characteristics of key components. The column component utilizes a total of 14 uniaxial strain gauges (c₁-c₁₄) and one triaxial strain gauge (cw_1-3) to monitor the strain development in the column flanges and the panel zone. For the beam component and end-plate, six uniaxial strain gauges (b₁-b₆) and eighteen uniaxial strain gauges (ep₁-ep₁₅) are employed, respectively. As shown in Fig. 5b, four LVDTs (D₁-D₄) are used to record the bending deformation of the two end-plates in the DS joint, while the SS joint only requires two LVDTs on one side. Two additional LVDTs (P₁–P₂) are diagonally placed in each column web area to monitor shear deformation. Simultaneously, on the east and west sides of the beam end and column top, one horizontal LVDT each (L₆-L₇, L₁) is arranged for subsequent verification of the joint connection rotation calculation, with specific placement details shown in Fig. 4a. At the bottom of the column, one horizontal LVDT (D₆) is set to verify that there is no displacement at the column base. Furthermore, four calibrated hollow cylindrical force sensors (B₁–B₄) are used for the two rows of bolts on the end-plate to monitor the applied pre-tension force and the further development of bolt force during the entire loading process. All data signals are collected through the DH-3816 N static strain measurement system to capture strain and displacement at each measurement point. The test loads and displacements at the beam end and column top loading points are automatically recorded by the built-in force and displacement sensors of the hydraulic servo actuator. The force at the beam end support is recorded by a force sensor placed below. It is worth noting that, since the SS joint only has a beam connection on one side, the test setup and measurement point arrangement for the SS joint are the same as the DS joint, except for the unused east side.

Fig. 5.

Detailed instrumentation plan for beam-to-column joint specimens: (a) Layout of strain gauges for beam, column and end-plate parts; (b) Bolt dynamometer and LVDTs.

3. Test results

3.1. Failure modes

Under cyclic loading, all specimens experienced low-cycle fatigue failure. The observed failure modes in the joints are illustrated in Fig. 6 and summarized in Table 3, along with the reported cycle numbers and failure characteristics. For DS1 and DS2 specimens, the DS joints underwent shear buckling in the panel zone and moderate bending deformation between the end-plate and column flanges. With increasing cycles, the shear deformation in the panel zone of DS1 became more pronounced, extending to severe cracking at the weld between the west end-plate and the lower flange of the beam (Fig. 6a). In DS2, as the bending deformation of the end-plate increased on the tension side of the east beam, corresponding to bolt slip and failure on the compression side of the west beam, the end-plate disconnected from the column flange (Fig. 6b). DS joints exhibited sufficient ductility before limiting joint performance due to severe weld cracking or bolt failure.

Fig. 6.

Failure modes of tested joints: (a) DS1; (b) DS2; (c) SS3; (d) SS4; (e) S–ISC–1; (f) S-ESC-1.

Table 3.

Failure modes of tested joints.

Specimen	Cycles	Limit interlayer drift ratio θu (rad)	Key failure characteristics
DS1	33	0.054	Considerable shear deformation in the panel zone, moderate bending deformation of the end-plate and column flange, and fractures located near fillet welds for west side beam-to-end-plate
DS2	32	0.051	Considerable bending deformation in the end-plate and part of the threaded slip teeth broke causing the end-plate to disconnect from the column flange
SS3	34	0.061	The presence of shear deformation in the panel zone and moderate bending deformation in the end-plate and column flange
SS4	35	0.063

For SS3 and SS4 specimens, the panel zone of the SS joints experienced over 50 % less shear deformation compared to DS joints, with moderate bending deformation between the end-plate and column flanges (Fig. 6c and d). This indicates different failure mechanisms compared to DS joints. The boundary constraints of the panel zone in DS joints with beams connected on both sides differ from SS joints. The failure in both DS joints was initiated by pronounced shear deformation in the panel zone, causing cracking of the weld connecting the end-plate and beam flange or bolt slip, while the failure in SS joints was due to excessive bending deformation of the end-plate and column flange.

It is noteworthy that during the initial stages, the joints experienced primarily elastic deformation, characterized by axial tensile deformation. The degradation of specimen strength and stiffness was not significant. As the loading amplitude increased, a certain level of non-elastic deformation (shear deformation in the panel zone) appeared in the initial plastic phase. Accumulative damage to the specimen was mainly attributed to the yielding of the panel zone and the development of end-plate cracking. The failure of the specimen was an evolving process, gradually progressing from inception to advanced damage, with damage caused by cyclic loading growing steadily. After reaching the ultimate load, damage growth accelerated. With a higher number of cycles, the accumulated damage eventually led to specimen failure.

Due to connections on both sides, DS joints exhibited a relatively uniform failure mode, resulting in overall better load-bearing and deformation capacities. SS joints, on the other hand, might be more prone to localized damage on the connected side, leading to an overall reduction in joint performance. All tested joints exhibited distinct ductile features, including significant bending deformation of the end-plate and column flange, as well as substantial shear deformation in the DS joint column webs. The failure occurred due to significant deformations in the later stages of loading, causing weld cracking, bolt fracture, and bending.

3.2. Hysteresis curves

From the M-θ, M_p-θ_s, and M-θ_ep hysteresis curves shown in Fig. 7 for all specimens, two groups can be identified. The first group includes specimens DS1 and DS2, whose hysteresis curves initially follow nearly straight-line cycles during early loading with almost no residual deformation during unloading, indicating that all parts of the joint were in an elastic working stage. As the load increased, the hysteresis curves deviated from a straight line and became loopy, with increasing residual deformation during unloading and more apparent stiffness degradation, signifying the gradual transition of the joint into an elastoplastic working stage. During displacement-controlled loading, although the phenomenon of stiffness degradation was more pronounced than before, horizontal loads could continue to increase, exhibiting evident elastoplastic strengthening characteristics. Their connection rotation θ_ep developed stably, and about 2/3 of the proportion of the total rotation θ was contributed by shear rotation θ_s. These joints demonstrated good energy dissipation behavior, and a notable feature was that strength hardly decreased during cyclic loading.

Fig. 7.

Moment–rotation curves for four specimens: (a) Moment–shearing rotation (M_p-θ_s) curves; (b) Moment–connection rotation (M-θ_ep) curves; (c) Moment–connection rotation (M-θ) curves.

The second group includes specimens SS3 and SS4, their shear rotation θ_s developed relatively steadily, but due to different plane boundary conditions than DS joints, the connection stiffness significantly decreased. The development of connection rotation θ_ep was insufficient. Although the proportion of connection rotation θ_ep to total rotation θ was small, the residual deformation of the joint increased significantly, and the rates of stiffness and strength degradation accelerated. The hysteresis loops began to pinch, reducing the loop area and exhibiting a bow or inverted S shape.

Generally, DS joints showed more stable hysteresis behavior compared to SS joints. Their hysteresis curves appeared full, while SS joints exhibited pinching. The differences in hysteresis performance and related energy dissipation between DS and SS joints can be attributed to the differences in failure modes exhibited by the two types of joints. The shear failure in the plane of DS joints exhibited high toughness, indicating good seismic performance. For the tested SS joints, failure mechanisms, such as bolt bending between the column flange and end-plate, plastic cracking near the beam-end-plate weld, end-plateend-plateand lever action development in the thin end-plate, might cause pinching of the M–θ hysteresis curve.

3.3. Skeleton curves

The envelope curve, defined as the skeleton curve connecting the peak points of the first cycle hysteresis curves under each load level, reflects the strength and deformation capacity of the structure more clearly for all tested joints. As shown in Fig. 8, the skeleton curves of the joints have long, gentle descending sections, indicating good ductility and energy dissipation capabilities. The skeleton curves for DS joints connected on the east and west sides are essentially coincident, suggesting similar initial stiffness and strength degradation in different directions of the connection. For cyclic loading of the specimens, the trends of the skeleton curves in Fig. 8 are roughly similar before reaching the maximum moment. The moment-rotation relationship can be considered linear before rotation of 0.005 rad; in the inelastic stage, as the inter-story drift ratio increases, the moment-rotation curve turns towards the x-axis, and the bearing capacity decreases with the increase in cyclic loading. When the rotation reaches 0.05 rad for all six specimens, at which point the joint can still withstand a moment above 0.5M_max, according to the failure mode of the end-extended end-plate joint under cyclic loading, the joint still exhibits robustness.end-plate.

Fig. 8.

Skeleton curves of tested joints: (a) DS1; (b) DS2; (c) SS3/4.

3.4. Joint ductility and rotational capacity

Joint ductility is defined as the ability to undergo large deformations without a significant decrease in strength. The ductility coefficient μ is defined as:

$$\mu ={\theta }_u/{\theta }_{\mathrm{y}}$$ (13)

The values of the ductility coefficient for the specimens are shown in Table 4. The ductility coefficient for the SS joint (SS3) is 1.3, while for DS1, a joint with end-plates connected on both sides, the ductility coefficient increases to 2.25. When the bolt diameter is increased for SS4, the ductility coefficient decreases to 1.7. DS joints exhibit larger values of the ductility coefficient compared to SS joints. The range of ductility coefficient μ for all tested specimens is 1.2–5, indicating good ductility.

Table 4.

Joint ductility and rotation capacity.

Specimen	Connection	Mmax (kN m)	Kj (kN·m/mrad)	μ	ΔMmax/Mmax,DS1 (%)	ΔKj/Kj,DS1 (%)	Δμ/μDS1 (%)
DS1	West	241.7	8.22	1.9	–	–	–
	East	236.5	8.71	2.7	−2.2	6.0	42.1
DS2	West	236.7	7.87	3.1	−2.1	−4.3	63.2
	East	217.6	7.95	3.2	−10.0	−3.3	68.4
SS3	/	229.2	5.76	1.3	−5.2	−29.9	−31.6
SS4	/	210.2	5.76	1.6	−13.0	−29.9	−15.8

Note: M_max, K_j and μ correspond to the average values of peak bending moment M_max, initial rotational stiffness K_j and ductility coefficient μ, respectively, and Δ is the difference between a certain specimen and the DS1 specimen west-side connection.

Regarding rotational capacity, Table 4 lists the peak moment M_max, initial rotational stiffness K_j, and ductility coefficient μ for different-side connections of all specimens, reflecting the rotational capacity of the joints at different stages. The elastic rotation of most end-plate-connected specimens during the test was about 0.012 rad. Under cyclic displacement, the rotational stiffness degraded significantly, and the bearing capacity decreased accordingly. However, the plastic deformation capacity of the specimen joints was good, with rotations at the ultimate state (θ_u) greater than 0.03 rad, as required by the U.S. FEMA [49]. For DS2, the maximum rotation reached 0.07 rad. According to the test results, suitably designed end-plate connections typically exhibit satisfactory hysteresis performance. Specifically, the appropriate selection of the strength and stiffness of the end-plate relative to the bolt group can reduce pinching,end-plateend-plate and improve the stability of hysteresis loops, rotational capacity, and overall ductility [50].

4. Hysteresis performance evolution analysis

The key characteristics of the skeleton curves of the specimens were studied to identify the performance evolution points of the joints. Previous research relied on determining the yield point to assess the elastic-plastic point, ductility, and failure performance of the joints at different stages. Alternatively, by analyzing the changing trend of the mechanical response curve of the joints, trend analysis tools (Mann-Kendall test) were employed to determine critical performance evolution points. This provided a more detailed and in-depth analysis of the evolution of stressing states at different stages, reflecting well in other mechanical response indicators (bending deformation, shear deformation, bending strain, shear strain, joint stiffness, bolt force, and joint energy dissipation).

4.1. Confirmation of performance evolution points

Three traditional methods were commonly used to determine the values of rotation and bearing capacity at the yield point in the early stages. The bearing capacity control method was adopted here to determine the yield points of the specimens, including the graphical method, equal energy method, and R. Park method [51,52]. The average values of the results obtained from these three methods were used as the criteria for determining the yield points. Finally, the limit point was identified by the load decreasing to 85 % of its maximum value. The three methods for determining the yield point are illustrated in Fig. 9. Taking DS1 as an example, parameters and ductility coefficients for both positive and negative sides of the connection skeleton curves of the DS1 joint were calculated using the three methods, and the results are summarized in Table 5.

Fig. 9.

Method for determination of structural performance points: (a) geometric graphic method; (b) energy equivalence principle method; (c) the R.Park method.

Table 5.

Three methods about the value of the rotation and load at the yield point for typical DS1 specimens.

Specimen	Load directions	Method	Yield point		Ductility (μ)	Specimen	Load directions	Method	Yield point		Ductility (μ)
			θy (0.01 rad)	My (kN m)					θy (0.01 rad)	My (kN m)
DS1 (West)	Positive	Ⅰ	3.39	175.05	1.7	DS1 (East)	Positive	Ⅰ	2.07	144.89	1.2
		Ⅱ	2.07	189.40	2.3			Ⅱ	1.96	188.81	2.6
		Ⅲ	1.91	175.76	2.5			Ⅲ	2.17	186.90	2.3
		Mean	2.46	−154.80	2.2			Mean	2.07	−205.29	2.0
	Negative	Ⅰ	−3.39	−200.71	1.7		Negative	Ⅰ	−1.80	−213.19	1.4
		Ⅱ	−3.19	−201.05	1.8			Ⅱ	−1.92	−200.87	2.6
		Ⅲ	−3.59	175.05	1.6			Ⅲ	−1.69	144.89	2.9
		Mean	−3.39	189.40	1.7			Mean	−1.80	188.81	2.3

Graphical Method (Fig. 9a): Tangent to the initial segment was drawn at point OA. A line was drawn from the peak point to intersect the horizontal line passing through the peak point at point A. The perpendicular line AB was drawn to intersect the curve at point B, and line OB was extended to intersect the horizontal line at point C. The displacement of point C was considered as the yield displacement, and the corresponding point on the curve was the yield point Q.

Equal Energy Method (Fig. 9b): An ideal elastic-plastic two-segment line with equal envelope area replaced the original curve. The displacement of the inflection point of the two-segment line was considered as the yield displacement, and the corresponding point on the curve was the yield point Q. This method required numerical integration and couldn't be obtained graphically.

R. Park Method (Fig. 9c): The point A was set α times the peak moment on the curve. A line was drawn from the origin and point A to intersect the horizontal line passing through the peak point at point B. The displacement of point B was considered as the yield displacement, and the corresponding point on the curve was the yield point Q.

All test results for the specimens are summarized in Table 6, presenting average values obtained from the three methods. Due to the cyclic loading regime, structural performance values were both positive and negative. The calculated results represented the moment and rotation corresponding to the yield point, peak point, and ultimate point. The initial rotational stiffness k_j corresponds to the positive or negative cut-off rotation stiffness of the first displacement cycle and K_j is the initial rotational stiffness for each side connection. μ is the ductility coefficient. As shown in Table 5, the initial rotational stiffness and ductility of DS joints were significantly greater than those of SS joints, while the difference in bending strength between the two was relatively small. Additionally, there was an improvement in rotational capacity between them, attributed to the larger bending deformation components in SS joints allowing for a quicker initial drop in rotational stiffness, making the joint more prone to rotation.

Table 6.

Test results of tested joint specimens.

Specimen	Connection	Load directions	Yield point		Peak point		Ultimate point		Initial stiffness		Ductility (μ)
			θy (0.01 rad)	My (kN m)	θmax (0.01 rad)	Mmax (kN m)	θu (0.01 rad)	Mu (kN m)	Kj
									(kN m/mrad)
DS1	West	Positive	2.46	180.07	4.9	231.2	4.9	231.2	8.14	8.22	2.0
		Negative	−3.39	−185.52	−5.9	−252.1	−5.9	−252.1	8.29		1.7
	East	Positive	2.07	173.53	5.1	227.7	5.1	227.7	8.18	8.71	2.5
		Negative	−1.80	−206.45	−5.0	−245.2	−5.0	−245.2	9.25		2.8
DS2	West	Positive	1.72	182.50	5.1	232.5	6.8	214.1	8.09	7.87	4.0
		Negative	−2.45	−193.63	−5.3	−240.8	−5.3	−240.8	7.66		2.2
	East	Positive	1.98	176.50	6.4	209.2	6.4	209.2	7.52	7.95	3.2
		Negative	−1.94	−183.32	−5.1	−225.9	−6.1	−223.7	8.39		3.1
SS3	/	Positive	3.6	184.76	4.1	217.8	5.5	142.5	5.53	5.76	1.4
		Negative	−3.5	−207.90	−4.0	−240.6	−5.5	−156.9	5.99		1.2
SS4	/	Positive	3.5	183.35	3.4	203.4	5.7	181.0	5.41	5.76	1.6
		Negative	−3.5	−190.36	4.5	217.0	−5.7	−176.5	5.17		1.6

From the perspective of the ratio θ_u/θ_y, which compares the rotational capacity under plastic conditions to that under elastic conditions, it is advisable to use end-plate connections with moderately thick large bolts to enhance seismic performance. According to the provisions of GB50011-2010 ″Code for Seismic Design of Buildings” [41], for multi-story steel structures, the limit for elastic inter-story displacement rotation [θ_e] is set at 1/250 rad, and the limit for elastic-plastic inter-story drift ratio [θ_p] is set at 1/50 rad. As shown in Table 6, the yield rotation θ_y is approximately (2.6–9) [θ_e], and the plastic ultimate rotation θ_u is approximately (7.9–3.4) [θ_p]. The analysis results concerning the elastic displacement angle limit and elastic-plastic displacement angle limit for steel structures indicate that such joints exhibit good ductility, meeting the requirements of seismic design.

The Mann-Kendall (M − K) test tool was introduced to define the trend changes of working performance parameters in the moment sequence M_k (as shown in Fig. 11a). The M − K tool effectively distinguished whether a natural process was in a state of random fluctuation or had a deterministic trend [53,54]. Using the joint model at different moment levels, when the moment level reaches a certain threshold, the corresponding mechanical response (such as the joint rotation in Fig. 11a) rapidly increases and even undergoes a sudden change. The stressing state then exhibits characteristics of transitioning from quantity to quality, indicating a potential for failure from that moment of abrupt change. To identify the starting point of failure and analyze the entire failure process, the M − K tool process is as follows [55,56].

• (1)At different moment levels, for the θ_k sequence (θ_k = {θ₁, θ₂, …, θ_n}) or other mechanical responses (in terms of parameter sequences), construct a rank sequence r_p to represent the sample cumulative count of θ_p>θ_q (1<q ≤ p), and define a parameter S_k as follows:

$$S_k=\sum _{p=1}^k{r}_p$$ (14)

where r_p is:

$$r_p=\{\begin{array}{cc}+1& {\theta }_p>{\theta }_q\left(1⩽q⩽p\right)\\ 0& \text{otherwise}\end{array},\left(q=1,2,3,...,p\right)$$ (15)

where S_k is the cumulative count of the sample: r_p is used to judge the growth of the θ_j sequence, and "+1″ indicates that the p-th value in the θ_j sequence is greater than the q-th value.

• (2)The θ_k sequence at different moment levels conforms to the characteristics of being random and independent. Define a new statistical variable (UF):

$$U{F}_k=\frac{\left(S_k-E\left(S_k\right)\right)}{\sqrt{V\left(S_k\right)}}$$ (16)

where UF_k is the standard normal distribution statistic, and E(S_k) and V(S_k) are the mean and variance of the parameter Sk.

• (3)When UF_k = 0, at a given significance level α, the θ_j sequence conforms to |UF_k |≥U_α, indicating a significant trend change in this sequence. The formulas for calculating E(S_k) and V(S_k) are as follows:

$$E\left(S_k\right)=k\left(k-1\right)/4$$ (17)

$$V\left(S_k\right)=k\left(k-1\right)\left(2k+5\right)/72$$ (18)

• (4)Extract UF_k statistical data and plot the UF_k-M_j curve.
• (5)Reverse the θ_k sequence to obtain the γ_k sequence, then repeat (1)–(4) to form the UB_k-M_j curve. The relationship between the θ_k sequence and the γ_k sequence is as follows:

$$\theta \left(i\right)=\gamma \left(n-i+1\right),k=1,2,...,n$$ (19)

• (6)Determine the intersection points of the UF_k and UB_k curves as the first performance characteristic point P on the θ_k-M_k curve. where the UF_k and UB_k curves are the statistical sequences in the Mann-Kendall (M − K) test, where UF is the standard normal distribution statistic in the θ_k ordinal case, and UB is the standard normal distribution statistic in the θ_k inverted case.
• (7)Using the θ_k sequence after the first performance characteristic point P, repeat the above process to obtain the intersection points of the UF_k and UB_k curves, defining them as the second performance characteristic point Q on the θ_k-M_k curve, considered as its performance evolution characteristic point.

Fig. 11.

Detection of three stages for θ−M curves with M − K trend tool: (a) θ−M and M−K statistical curves of DS1 specimen; (b) evolution analysis of θ−M curves for different specimens.

Therefore, using the above method as a tool for defining failure evolution, one can determine the performance evolution characteristic points of the joint model as criteria for essential changes in stressing states. The comparison results between the M − K method and the traditional three methods for calculating moments are listed in Table 7, and individual comparisons can be seen in Fig. 10. Due to material and testing errors, there is some dispersion in the calculated moments for each connection in the four specimens. The average ratio of the traditional method calculated value (M_y) to the M − K trend tool calculated value (Q) is 0.977, with a standard deviation of 0.113 and an average absolute error of 0.116. Although some specimens exceed 15 %, the overall relative error fluctuates within the range of −15 %–15 %, indicating a good correlation between the M − K tool calculated values and the traditional three methods. It can identify the characteristic point of trend change in the whole curve and take Q as the joint yield point, it studies the trend and growth of the parameter from the data series θ_k, and searches for the mutation point from yielding at the material level to failure at the component or structural level, which has a better correlation with the traditional three methods and proves the rationality of this aspect. However, further specific discussions are required regarding the performance evolution patterns of the specimens.

Table 7.

Calculation results of characteristic moments of traditional method and M − K method.

Specimen	Connection	Load directions	P	Q	My,I	My,II	My,III	Ratio
			(kN·m)	(kN·m)	(kN·m)	(kN·m)	(kN·m)	Q/My,I	Q/My,II	Q/My,III
DS1	West	Positive	157.79	178.57	175.05	189.40	175.76	1.02	0.94	1.02
		Negative	146.89	170.49	154.80	200.71	201.05	1.10	0.85	0.85
	East	Positive	131.73	161.72	144.89	188.81	186.90	1.12	0.86	0.87
		Negative	110.44	155.56	205.29	213.19	200.87	0.76	0.73	0.77
DS2	West	Positive	179.09	202.34	168.24	195.16	184.10	1.20	1.04	1.10
		Negative	183.25	204.77	198.38	195.61	186.90	1.03	1.05	1.10
	East	Positive	165.17	184.90	183.69	177.05	168.76	1.01	1.04	1.10
		Negative	168.04	193.73	173.60	188.89	187.48	1.12	1.03	1.03
SS3	/	Positive	156.93	183.26	190.37	180.74	183.18	0.96	1.01	1.00
		Negative	141.45	185.33	205.98	202.78	214.93	0.90	0.91	0.86
SS4	/	Positive	179.71	190.44	176.49	186.33	187.25	1.08	1.02	1.02
		Negative	131.22	170.70	182.19	194.04	194.84	0.94	0.88	0.88
Mean	–	–	–	–	–	–	–	1.02	0.95	0.97
St.dev	–	–	–	–	–	–	–	0.12	0.10	0.11

Fig. 10.

Comparison between the traditional method and M − K method for calculating characteristic moments.

4.2. Verification of performance evolution states

To fully analyze the working behavior and evolution characteristics of the end-plate connection joints, three stages of the joint model loading process were delineated based on the θ_k-M_k curve of the typical DS1 specimen's east-side connection. The Mann-Kendall tool was used to detect the two performance characteristic points (initiation point P of failure evolution and overall yield point Q of the joint model) in the θ_k-M_k curve.

In Fig. 11a, the two performance characteristic points (P, Q) of the DS1 specimen were obtained by calculating the intersection points of the UF and UB curves. The working behavior of the joint model underwent a qualitative change before and after the performance characteristic points. The three stages corresponding to these points were: (1) The stage from 0 to 157.79 kN m, where the θ_k-M_k curve of the DS1 specimen changed stably and increased rapidly, indicating a stable structural stressing state, and the entire joint was basically in a linear elastic working stage. (2) The stage from 157.79 to 178.57 kN m, where the θ_k-M_k curve exhibited a nonlinear form, and the growth rate significantly slowed. The joint model began to undergo failure evolution as some components yielded and entered the plastic stage. However, the entire joint remained in the elastic-plastic transition stage, accumulating the energy necessary for subsequent unstable failure. The performance characteristic point P was defined as the initiation point of failure evolution. (3) The final stage from 178.57 to 227.7 kN m, where the plastic development of the main force components of the joint model resulted in a rapid development of the θ_k values with the increasing M_k, indicating the joint model's behavior gradually entering an unstable stage. The joint began to undergo the evolution process of continuous energy accumulation and progressive structural failure. Here, the second performance characteristic point Q could be characterized as the overall yield point of the joint model, signifying the critical point where the stressing state of the specimen became unstable. In this stage, the joint was no longer safe, and any random factor could lead to excessive displacement and complete failure with the continuous increase of M_j. This analysis identified an initiation point P of failure evolution before the traditionally defined yield point, providing new insight into the initiation of structural failure. This approach also opened up a new perspective for the analysis of structural failure evolution, shifting from the transformation of quantity to quality in the development of mechanical responses, and emphasized the importance of understanding structural failure as an accumulation and progressive evolution process, rather than focusing solely on the random final failure point. This highlighted the need to establish a better design intuition based on redundancy and the intrinsic behavior of the structure.

From Fig. 11b, it is evident that performance characteristic points P and Q for all four joints were described using the Mann-Kendall tool. The numerical values for each specimen are listed in table. All four curves exhibited distinct characteristics of performance discontinuity, reflecting changes in stressing states brought about by moment levels (M_k) during each loading cycle. SS joints, in particular, showed a significantly higher initiation point P of failure evolution compared to DS joints. This was attributed to the large deformation in the shear domain of SS joints leading to an early failure evolution. For both types of joints, once the moment level M_j exceeded the Q value, the specimen immediately entered an extremely unstable state. This occurred as local buckling or component fracture immediately occurred, suggesting that the specimens did not fully utilize their nonlinear deformation capabilities.

4.3. Development of bolt forces

During the entire experimental loading process, the time-history curves of bolt tension in the specimens were measured using four calibrated annular force sensors. These curves were employed to monitor the development of bolt forces in each joint. As shown in Fig. 12, the shape of the time-history curves is generally consistent with the experimental loading protocol, indicating the effectiveness of the measured bolt tension data obtained through this testing method. In the early stages of loading, the bolt tension values are minimal, attributed to the pre-tensioning of high-strength bolts before formal loading. During the initial loading, external loads applied in the experiment induce tensile stress in the bolts, initially offsetting the compressive stress from pre-tensioning. A comparison between the bolt force time-history curves for the DS joint and the SS joint in Fig. 12, b reveals distinct patterns. The SS joint's curve exhibits a stepped shape (with periodic abrupt increases in later stages), while the DS joint's curve shows a sinusoidal pattern (with a pronounced smooth transition in the middle). This indicates that the bolt forces in the DS joint are more evenly distributed throughout the entire loading process. Therefore, in testing and design, particular attention should be given to potential bolt brittle fractures in the SS joint due to uncoordinated loading from the single-sided connection.

Fig. 12.

Time-history of bolt force development for Single or double side connection joint: (a) DS1; (b) SS3.

Fig. 13 illustrates the development of bolt forces in the four specimens. In Fig. 13a, the moment (M_j) - bolt force (F_B) curve for the DS1 specimen demonstrates that, during the elastic phase, the bolt tension is primarily linearly distributed, with the tensile strain induced by the bending moment at the connection point always maintaining a linear distribution. This occurs with the neutral axis at the centroid of all bolts. In the later stages of loading, bolts in the tensile zone not only endure axial forces but also significant bending moments, causing a change in the trend of bolt force growth towards the Y-axis. The bolt forces in the first row of bolts (B₁, B₂) in the DS joint are generally greater than those in the second row of bolts (B3, B4) initially. This is since the upper-row bolts are farther away from the rotation center of the plate connection. Both rows of bolts continue to share the load in the later stages. In contrast, Fig. 13c depicts the SS joint, where, as the first-row bolts (B₁, B₂) enter the yielding state in the later stages, the tensile force suddenly transfers to the second-row bolts (B₃, B₄) at the connection point, causing a rapid increase in their tension. This results in a swift development of forces in the second-row bolts. As the bending moment increases and the stressing state transitions from axial force to pure bending force, it eventually leads to bolt shear failure or bending of the screw.

Fig. 13.

Bolt force development curves: (a) DS1; (b) DS2; (c) SS3; (d) SS4.

The development of bolt forces in the four joints shown in Fig. 13 also reflects, to varying degrees, the failure evolution of stressing states near the initiation point (P) and the overall yielding point (Q). In the initial 0-P stage, the bolt forces are generally at a lower level, increasing significantly after the performance characteristic point, particularly after the overall yielding point Q. This is closely related to the occurrence of pry force after the plate yields. The bolt forces in the DS joint grow relatively uniformly throughout the entire loading process, while the SS joint is more prone to sudden increases in the later stages. Hence, in testing and design, greater attention should be paid to unexpected increases in bolt forces in the SS joint leading to bolt brittle fractures.

4.4. Development of joint deformation and strain

Fig. 14 illustrates two typical development changes in joint deformation, namely, the evolution of shear deformation in the panel zone and the separation of the end-plate from the column flange. As shown in Fig. 14a, the shear deformation in the panel zone of the SS joint gradually increases with the applied load, but the magnitude of the increase is limited. In contrast, after the Q value, the growth rate of shear deformation in the panel zone of the DS joint accelerates. In the early stages, shear deformation primarily occurs in the panel zone, while the deformation at the connection remains relatively small. As shown in Fig. 14b, the bending deformation at the connection of the SS joint is consistently greater than that of the DS joint. From the graph, it is evident that, at the point of failure, the shear deformation in the panel zone of the DS joint is approximately twice that of the SS joint, and the bending deformation of the second joint is 5–10 times that of the first joint.

Fig. 14.

Deformations of the panel zone and separation of the end-plate from the column flange: (a) shear deformation of the panel zone; (b) gap due to bending deformation between end-plate and column flange.

Fig. 15 displays the development of strains in the panel zone and at the plate connection for the DS and SS joints. The strain development in critical regions of DS and SS joints is generally similar, with strains at various key positions gradually increasing with the applied load. Fig. 15a shows that the bending strain at the plate connection of the SS joint is slightly higher than that of the DS joint, attributed to the dominant bending deformation in the SS joint and the asymmetric loading due to the single-sided connection. When the bending moment at the connection (M) reaches the overall yielding point Q, the weld at the connection between the end-plate and the beam flange in the DS joint cracks, resulting in a reduction in strain due to the failure of the strain gauge. Additionally, Fig. 15b demonstrates that the main surface strain in the panel zone of the DS joint is significantly higher than that in the SS joint, exceeding the maximum range of the strain gauge before failure. This is related to the more pronounced shear deformation in the panel zone of the DS joint as shown in Fig. 6, which corresponds to the failure mode.

Fig. 15.

Development of strains in the panel zone and end-plate connections for beam-column joints: (a) Strain development of end-plate connections; (b) Principal strain development of the panel zone.

4.5. Degradation of stiffness

The joint secant stiffness K_j for the j-th load level, defined by Eq. (20), is commonly used to assess the degradation of the stiffness of the tested beam-column connections.

$$K_j=\sum _{i=1}^n{M}_{j.i}/\sum _{i=1}^n{\theta }_{j,i}$$ (20)

Here, M_j,i and θ_j,i represent the peak moment and the corresponding rotation, respectively, in the i-th cycle (i = 1,2,3) under the j-th load level, and n is the number of load cycles at the j-th load level. As shown in Fig. 16, the degradation curves of the secant stiffness for all four joints illustrate the change in secant stiffness with rotation, depicting both positive and negative directions of the hysteresis loop secant stiffness K_j. In Fig. 16a, it is apparent that K_j undergoes a rapid initial increase, followed by a noticeable decrease with increasing load level. During the elastic phase, the joint secant stiffness is not maximum at the starting point (origin) but should be when the elastic phase of some components in the joint ends and enters the plastic or strain-hardening phase. Fig. 16b and c reveal that the degradation of the secant stiffness K_j for the DS joint is more severe than that for the SS joint. All DS joints have larger secant stiffness values than SS joints. The secant stiffness of the joints exhibits a trend of initially increasing and then decreasing. After the strain-hardening phase of some components in the joint, the secant stiffness rapidly deteriorates, with the DS joint experiencing a faster degradation rate than the SS joint, resembling exponential decay. When any component in the experiment reaches the ultimate stress or strain, the secant stiffness of the joint tends to remain constant.

Fig. 16.

Stiffness degradation of tested joint specimens: (a) Division of joint stiffness degradation; (b) Double-side connection joints; (c) Single-side connection joints.

4.6. Accumulation of energy dissipation capacity

The evaluation of the cyclic energy dissipation capacity of the joints employs three key parameters: total dissipated energy (W_exp), equivalent viscous damping ratio (ξ_eq), and energy dissipation capacity (E_ec). As shown in Fig. 17, the dissipated energy (W) describes the cumulative dissipated energy in each loading stage, where W is the area under the M-θ hysteresis curve for a specific cyclic time. The evaluation of the cyclic energy dissipation capacity of the joints employs three key parameters: total dissipated energy (W_exp), equivalent viscous damping ratio (ξ_eq), and energy dissipation capacity (E_ec). A larger E_ec indicates a stronger energy dissipation capacity. S_ABC and S_CDA represent the upper and lower halves of the hysteresis curve, respectively, while S_OBE and S_ODF denote the corresponding triangular areas (Fig. 17a). The results of the energy dissipation indices for each specimen are listed in Table 8, with W_exp/W_exp,SS3 comparing each specimen to the control specimen SS3 for analysis. W_pre represents the predicted dissipated energy according to subsequent theoretical hysteresis models, as discussed in Section 5.1.

Fig. 17.

Energy dissipated in each step and parameter definitions: (a) Definition of the moment versus rotation hysteresis loop; (b) DS1; (c) DS2; (d) SS3/4.

Table 8.

Energy dissipation indices of joint specimens.

Specimen	Wexp (kJ)		Eec		ξeq		Wexp/Wexp,SS3		Wpre	Wexp/Wpre
	West	East	West	East	West	East	West	East
DS1	195.6	160.5	1.709	1.713	0.272	0.273	2.53	2.07	222.2	0.88
DS2	209.7	215.5	1.642	1.712	0.261	0.273	2.71	2.78	207.2	1.01
SS3	–	77.4	–	0.976	–	0.155	–	1.00	86.4	0.90
SS4	–	98.2	–	1.018	–	0.162	–	1.27	94.8	1.04
Average	–	–	–	–	–	–	–	–	–	0.96
St. dev	–	–	–	–	–	–	–	–	–	0.08

As depicted in Fig. 17b–d, the dissipated energy (W) for each hysteresis loop increases with the number of loading cycles. Notably, the energy dissipation during the elastic phase is less than 5 % of the total dissipated energy. Additionally, the energy dissipation capacity of the first hysteresis loop in each loading stage is slightly higher than that of subsequent loops. Analyzing Fig. 17b and c, the change in energy dissipation with the number of load steps for each specimen describes the energy dissipation on both sides of the connection in the DS joint. Clearly, with increasing displacement amplitude, the energy dissipation capacity gradually increases. Due to the larger amplitude before failure, the DS joint exhibits higher energy dissipation capacity. Comparing Fig. 17b–d, it is evident that the energy dissipation capacity of the DS joint is roughly twice that of the SS joint.

Fig. 18a demonstrates that the dissipation capacity on both sides of the DS joint is roughly equal, while the DS joint exhibits greater energy dissipation capacity than the SS joint. This confirms that the involvement of the panel zone in deformation can achieve better ductility and energy dissipation [57]. Both the cumulative dissipated energy (W_total) and energy dissipation capacity (E_ec) gradually increase with each loading stage. In the 0.05 rad relative rotation of the joints, the dissipated energy for all four full-scale specimens exceeds 60 kJ, and the cumulative dissipated energy for the DS joint ranges from 160 kJ to 220 kJ at failure, demonstrating an ideal energy dissipation effect for this type of joint before failure.

Fig. 18.

Cumulatively dissipated energy and development curves of the energy dissipation coefficient: (a) Cumulatively dissipated energy of tested joint specimens; (b) Double-side connection joint and single-side connection joint.

The energy dissipation coefficient E_ec is obtained based on the characteristics of each cycle and is further used to calculate the equivalent viscous damping ratio ξ_eq, as shown in Fig. 18b. At a 0.05 rad relative rotation of the joints, all specimen values for E_ec are greater than 1, indicating excellent energy dissipation capacity in these specimens. The smaller value for the SS joint is due to the pinch effect.

5. Prediction and evaluation of test performance

5.1. Comparison with empirical hysteresis models

Considering the more stable hysteresis curve of the DS joint compared to the SS joint, a bilinear kinematic hardening hysteresis model (shown in Fig. 19a) is selected to approximate the DS joint's hysteresis curve [15]. For the SS joint, a trilinear hysteresis model (see Fig. 19b) is employed to approximate the joint's hysteresis curve and can further be used to compare with the obtained test curves.

Fig. 19.

Moment-rotation (M-θ) hysteresis model: (a) Follow-up enhancement model; (b) Trilinear hysteresis model; (c) Plastic hardening factor r_pl; (d) Unloading stiffness factor r_ul.

The initial stiffness K_j is obtained by linear fitting of the elastic range of the hysteresis curve of the test joint. Plastic hardening stiffness K_pl and unloading stiffness K_ul are determined by introducing two reduction coefficients r_pl and r_ul, respectively. These coefficients are obtained by linear fitting of partial curves in the plastic hardening range and unloading range of the M-θ hysteresis curve, as shown in Fig. 19c and d. The average values of plastic hardening coefficient r_pl and unloading stiffness coefficient r_ul for the DS joint are 0.069 and 0.981, respectively. The corresponding values for the SS joint are 0.245 and 0.810.

Fig. 20 shows the comparison between the empirical hysteresis model-based analysis curves and the test curves for two typical DS joints and SS joints (specimens DS1 and SS3). It can be observed that the predicted curves are quite consistent with the test curves until significant stiffness and strength degradation occurs due to crack initiation and propagation. However, the slight contraction effect is not captured. Additionally, Table 8 summarizes the predicted cumulative dissipated energy values, and the average ratio of test energy dissipation W_exp/W_pre is 0.96 with a standard deviation of 0.08. It can be concluded that, although the degradation patterns of joint stiffness and strength are not considered, the empirical hysteresis model, determined by coefficients from the test curves, can adequately reproduce the hysteresis response. Both the bilinear kinematic hardening model and the trilinear hysteresis model can well describe the M-θ hysteresis characteristics of DS and SS joints, with calculated results in good agreement with experimental results, meeting the relevant requirements for seismic design and analysis of steel frames.

Fig. 20.

Comparison of moment-rotation (M-θ) hysteresis curves of different specimens: (a) DS1; (b) SS3.

5.2. Classification of connection joints

The classification of full-scale or reduced-scale DS and SS joints as semi-rigid or partially strength-connected joints is determined based on the rotational stiffness and strength of the M-θ skeleton curves. The classification method from European standard EN1993-1-8 [12] is referenced.

• 1)Rigid Connection: K_j ≥ 8 EI_b/L_b (for non-sway frames) or K_j ≥ 25 EI_b/L_b (for sway frames); Pinned Connection: K_j ≤ 0.5 EI_b/L_b; Semi-rigid Connection: 0.5 EI_b/L_b < K_j < 8 EI_b/L_b (for non-sway steel frames) or 0.5 EI_b/L_b < K_j < 25EI_b/L_b (for sway frames). Here, K_j is the initial rotational stiffness of the test specimen, EI_b/L_b is the flexural stiffness of the connecting beam, E is the elastic modulus of steel, I_b is the second moment of area of the beam's cross-sectional area, and L_b is the span of the beam (distance from the column center to the loading point).
• 2)Strength Classification: M_d ≥ M_bp, fully strength-connected; when M_d ≤ 0.25 M_bp, pinned connection; when0.25 M_bp < M_d < M_bp, partially strength-connected.

The plastic moment M_bp is 261.3 kN m used in the experiment is 261.3 kN m. In Fig. 21, the peak moment values for the four specimens are 93.8 %, 89.6 %, 90.5 %, and 86.4 % of M_bp. Table 6 shows that the initial rotational stiffness of all specimens during the test falls within the range of 0.5EI_b/L_b -8EI_b/L_b (EI_b/L_b = 15.8 kN m/mrad). According to the rotational stiffness and strength classification, the extended end-plate joints used in cyclic loading are classified as semi-rigid joints; all specimens, when failure occurs, have moments between 0.86 M_bp and 0.94 M_bp, classified as partially strength-connected.end-plate.

Fig. 21.

Classification of tested joints: (a) Braced frame; (b) None braced frame.

5.3. Evaluation of existing design standards

This study evaluates the relevant design formulas for extended end-plate connection joints. Calculation formulas for determining the plastic resistance moment and initial rotational stiffness in seismic applications are provided in the American standard ANSI/AISC 358end-plate [35], European standard EN1993-1-8 [12], and Chinese standard GB 50017 [36]. This study compares and analyzes the predicted values with the obtained test results, considering the design methods for determining the degree of extended end-plate connections for DS and SS joints based on the component method in EN 1993-1-8 [12] and GB 50017 [36].

• (1)American AISC Guide

According to the AISC design guide, the moment-carrying capacity of the four extended end-plate connection joints without stiffeners is determined. Using the required bolt diameter d_b,req, required end-plate thickness t_p,req and column flange thickness t_cf from the connection properties, the moment strength (M_f) is obtained through Eqs. (21), (22), (23), (24):

$$d_{b,req}=\sqrt{\frac{2{M_f}_{，1}}{\pi {\phi }_n{F}_{nt}\left(h_0+h_1\right)}}$$ (21)

$$t_{ep,req}=\sqrt{\frac{1.11M_{f，2}}{{\phi }_d{f}_{yep}{Y}_p}}$$ (22)

$$t_{cf}=\sqrt{\frac{1.11M_{f,3}}{{\phi }_d{f}_{ycw}{Y}_c}}$$ (23)

$$M_f=\min \left(M_{f,1};M_{f,2};M_{f,3}\right)$$ (24)

where M_f,1 is the calculated moment at the connection, F_nt is the nominal tensile load of the bolt, ℎ₀ is the distance from the centerline of the compressed flange to the outermost tension side bolt, ℎ₁ is the distance from the centerline of the compressed flange to the innermost tension side bolt, f_yep and f_ycw are the minimum yield strength of the end-plate material and column web material, respectively, Y_p and Y_c are the yield line mechanism parameters for the end-plate and column web (here the joint connection conforms to the behavior of thick end-plates, and the levering effect of additional bolt forces is not considered separately), with two resistance coefficients Ф_n = 0.9 and Ф_d = 1. To better match the real situation, the nominal yield strength determined in the material property tests is used for comparison with experimental results. end-plateIt is worth noting that Eqs. (21), (22), (23), (24) only consider the bending capacity of the end-plate and column flange and the tensile capacity of the bolt, without considering the shear capacity of the joint zone. Therefore, AISC guidance may have limitations in situations where joint zone failure occurs in DS joints.

• (2)European EC3 Standard

EN1993-1-8 [12] adopts the component method, considering the strength of each joint component. In the end-plate-bolt connection, the tension area is idealized through an equivalent T-shaped joint. Unlike the AISC guide, the European standard does not distinguish between thick and thin end-plates. Three failure modes of T-shaped components are defined: complete flange yielding, bolt failure plus flange yielding, and bolt failure. For stiffness calculation, the joint is assumed to be a mechanical model composed of rigid links and multiple springs. Explicit design equations are provided for strength and stiffness.end-plate The initial rotational stiffness and plastic resistance moment can be determined by Eqs. (25), (26), (27), (28), It is worth mentioning that the approximate value of the transformation parameter β is taken as 2 for joints connected by two-sided beams under antisymmetric loading, whereas for joints connected by one-sided beams, the value is set to 1.

$$K_{j,ini}=\frac{E_0{z}^2}{\mu \sum _i\frac{1}{k_i}},\mu =1$$ (25)

$$M_{j,Rd}=\sum _r{h}_r{F}_{T,Rd}$$ (26)

$$F_{T,Rd}=\min \left(F_{T,1Rd};F_{T,2Rd};F_{T,3Rd}\right)$$ (27)

$$F_{T,1Rd}=\frac{f_{yep}{l}_{eff,1}{t}_{ep}^2}{m};F_{T,2Rd}=\frac{0.5f_{yep}{l}_{eff,1}{t}_{ep}^2+1.44n{f}_{ub}{A}_{eb}}{m+n};F_{T,3Rd}=1.44f_{ub}{A}_{eb}$$ (28)

where μ = 1, k_i is the stiffness coefficient of the i-th basic component, z is the lever arm, F_T,Rd is the effective design tensile strength of the r-th row of bolts, ℎ_j is the distance from the j-th row of bolts to the compressed center, assuming the compressed center is located in the middle thickness of the beam's compressed flange. Here, t_ep is the thickness of the end-plate, m is the distance to the center of the bolt angle weld on the rolled section, n = min(e_x,1.25 m), e_x is the vertical edge distance of the outer hole, l_eff is the calculated length of the end-plate; f_ub is the nominal ultimate strength of the bolt, A_eb is the bolt cross-sectional area.

• (3)Chinese GB Standard

GB 50017 [36] provides formulas for calculating the moment-carrying capacity and initial rotational stiffness. Specifically, the ultimate carrying capacity of the joint is determined based on the shear strength of the column web plate, neglecting the influence of axial force and bending deformation, as shown in Eq. (29). This indicates that this expression can only be applied to two DS joints (DS1 and DS2) experiencing zone deformation. The rotational stiffness of the joint is composed of the shear spring stiffness (R₁) of the column web plate and the bending spring stiffness (R₂) of the connection, which can be calculated by Eqs. (30), (31), (32). The bending rotational stiffness (R₂) of the connection is determined by the bending stiffness of the end-plate, with a reduction factor of 1.1.

$$M\le \frac{4}{3\sqrt{3}}{f}_{\mathrm{y}}{h}_{bw}{h}_{cw}{t}_{\text{cw}}$$ (29)

$$R=\frac{R_1{R}_2}{R_1+R_2}$$ (30)

$$R_1=G{h}_{bw}{h}_{cw}{t}_{\text{cw}}$$ (31)

$$R_2=\frac{6E_0{I}_{ep}{h}_1^2}{1.1e_f^3}$$ (32)

where M is the sum of the moments at both ends of the beam, f_y is the yield strength of the column web plate, h_bw, h_cw, and t_cw are the height, width, and thickness of the column web plate, respectively, G and E₀ are the shear modulus and elastic modulus of the material, respectively, I_ep is the second moment of area of the end-plate, ℎ₁ is the distance from the center of the beam's compressed flange, e_f is the distance from the center of the outer tension-side bolt row to the beam's flange.

The predicted initial rotational stiffness values and moment resistance are compared with experimental values, and the calculated strength and stiffness are listed in Table 9. According to EN 1993-1-8 and AISC 358-16, the design predictions are based on experimentally determined material properties and nominal seam dimensions. The average values of K_j,EC3/K_j,exp and M_Rd,EC3/M_Rd,exp are 0.98 and 0.99, respectively, with corresponding standard deviations of 0.15 and 0.10. The average ratio of strength, M_Rd,AISC/M_Rd,exp, is 0.94 with a standard deviation of 0.33. This suggests that the predictions for initial rotational stiffness and plastic moment resistance are slightly conservative, which can be explained by the lack of explicit consideration of the reinforcing effect of stiffeners on the column web plate in the experiment [58]. This effect has a mutual influence on shear deformations in the joint zone.

Table 9.

Comparison of experimental results with predicted results from codified methods.

Specimen	Connection	Kj,EC3 (kN·m/mrad)	Kj,GB (kN·m/mrad)	$\frac{{K_{j,}}_{EC3}}{{K_{j,}}_{\exp }}$	$\frac{{K_{j,}}_{\text{GB}}}{{K_{j,}}_{\exp }}$	MRd,AISC (kN·m)	MRd,EC (kN·m)	MRd,GB (kN·m)	$\frac{M_{Rd,AISC}}{{M_{Rd,}}_{\exp }}$	$\frac{M_{Rd,EC3}}{{M_{Rd,}}_{\exp }}$	$\frac{M_{Rd,GB}}{{M_{Rd,}}_{\exp }}$
DS1	West	7.40	13.07	0.90	1.59	150.92	221.70	148.43	0.62	0.92	0.61
	East	7.40	13.07	0.85	1.50	150.92	216.93	148.43	0.64	0.92	0.63
DS2	West	7.48	13.14	0.95	1.67	150.92	255.56	148.43	0.66	1.12	0.65
	East	7.47	13.14	0.94	1.65	150.92	243.20	148.43	0.70	1.12	0.69
SS3	/	6.57	13.13	1.14	2.28	150.92	141.23	148.43	1.01	0.94	0.99
SS4	/	7.26	17.97	1.26	3.12	150.92	168.63	148.43	0.84	0.94	0.83
Mean	–	–	–	0.98	1.70	–	–	–	0.75	0.99	0.73
St.dev	–	–	–	0.15	0.65	–	–	–	0.15	0.10	0.15

For the predictions according to the Chinese standard GB 50017, the average ratio of K_j,GB/K_j,exp is 1.70, with a corresponding standard deviation of 0.65. This significant overestimation is due to the neglect of the mechanical properties of two key components (bolt tensile action and end-plate bending) in the calculation. In reality, the stiffness of the joint results from the combined contributions of various components. GB 50017 only calculates shear stiffness and bending stiffness, while neglecting the shear resistance capability of the joint zone. The average ratio of M_Rd,GB/M_Rd,exp is 0.73, with a high standard deviation of 0.15, indicating that the prediction of moment resistance is overly conservative. This can be explained by the lack of consideration for the strengthening effect of stiffeners and the impact of the other three critical components: bending components (end-plate and column flange), and tensile components (bolts).end-plate This suggests the need for improvements to the prediction of joint strength and stiffness according to Chinese standard GB 50017 to better meet engineering design requirements.

6. Conclusion

This study evaluates the hysteresis performance and evolution characteristics of four beam-column joints with extended end-plates connection, and analyzes in detail the operational performance and failure evolution of various mechanical responses (hysteresis performance, connecting moment capacity, strain development, bending and shear deformation development, bolting and prying forces, degradation of stiffness, and energy dissipation). It is further used to validate the empirical hysteresis model and to evaluate the design equations used to calculate such joints in existing design standards. The following conclusions can be drawn.

• (1)The tested center-column joints exhibit stable hysteresis curves and good ductility, and show higher energy dissipation capacity than the side-column joints, thanks to the full participation of the panel zone in the shear deformation and energy dissipation during loading.
• (2)The M − K trend tool is feasible in identifying the joint yield point, obtains a small difference between the overall yield point Q and the values calculated by the three traditional methods, and has an advantage in identifying the performance evolution characteristics and the specimen performance pattern, which can provide a reliable mathematical tool for the structural performance evaluation in the identification of the trend in the whole process.
• (3)By analyzing the development trend and degradation response of the working state parameters (bolt force, deformation, strain, stiffness degradation, and energy dissipation) of the joint model, the working state curves of the strain changes before and after the characteristic loads are graphically shown, and the leaping characteristics of the working state of the joint model are identified; and the progressive evolutionary failure of the material and the sudden structural damage under the “destructive” loading are revealed, which illustrates that the level of the stressing state of each component determines the basic evolutionary behavior of the structure.
• (4)The side column joints at the early stage of loading are prone to brittle bolt fracture due to the uncoordinated stresses on one side of the connection, which leads to uneven distribution of bolt stresses. In contrast, the center column joint exhibits more uniform bolt force distribution and greater energy dissipation capacity.
• (5)The empirical hysteresis model can adequately reproduce the joint hysteresis response, and the bilinear follower strengthening model and the trilinear hysteresis model can better describe the M-θ hysteresis characteristics of the center column joints and the side column joints, respectively, which can meet the requirements for the design and analysis of steel frames.
• (6)The relevant design formulas in the three beam-column joint standards, ANSI/AISC 358, EN 1993-1-8, and GB 51022, were evaluated to calculate the respective moment-carrying capacities and initial rotational stiffnesses, and the overall conservative prediction criteria were obtained for all three. end-plate

CRediT authorship contribution statement

Liang Luo: Writing – original draft, Software, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Hang Sun: Writing – review & editing, Resources, Funding acquisition, Conceptualization. Shengcan Lu: Validation, Supervision, Software, Investigation. Xi Li: Visualization, Supervision, Resources, Investigation. Huan Yuan: Writing – review & editing, Visualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Nomenclature

Symbols Content

A_c

Column cross-sectional area

A_eb

Effective area of bolt

b_ep

Width of end-plate

b_s

Width of stiffening rib

d_b

Bolt diameter

E

Modulus of elasticity of steel

E(S_k)

Average value of parameter Sk

E_ec

Energy dissipation factor

e_f

Distance from the center of the bolt hole to the flange of the beam

e_w

Distance from the center of the bolt hole to the surface of the web

e_x

Vertical side distance of the outer hole

F_nt

Nominal resistance of the bolt

F_pre

Preloading force of the bolt

F_T,Rd

Effective design tensile strength of bolts of the rth row of bolts

f_ub

Ultimate tensile strength of the bolt

f_y

Design strength of steel

f_yb

Design strength of beam

f_yc

Design strength of column

f_ycw

Minimum yield strength of column flange

f_yep

End-plate

f_yv

Design value of steel shear strength

G

Shear modulus

ℎ₁

Distance from centerline of compressed flange to inner row of bolts on tensile side

h_bw

Height of beam web

h_c

Height of column section

h_cw

Height of column web

ℎ_j

Distance from the j-th row of bolts to the center of compression

I_b

Second moment of area of beam

I_ep

Second moment of area of end-plate

k_i

Stiffness coefficient of the i-th basic component

K_j

Initial rotational stiffness

K_pl

Plastic hardening stiffness

K_ul

Unloading stiffness

L

Span of the beam

L_b

Span of beam (distance from center of column to loading point)

l_eff

Calculated length of end-plate

m

Distance from the center of the fillet weld of a rolled section bolt

M_b1&M_b2

Plastic bending capacity of full section of beam on both sides

M_bp

Plastic bending moment of beam section

M_ep,b

Design value of bending capacity of panel zone

M_f,1

Calculated moment at the joint

N_{b t}

Design value of tensile capacity of single bolt

N_p

Axial force of column under combined load

N_t1

Design value of bearing capacity of bolts and end-plate segments at the first row of bolts

N_tu,b

Ultimate tensile capacity of bolts

N_u,ep

Design value of bearing capacity of end-plate section at first row of bolts

P

First performance characteristic point

Q

Second performance characteristic point

r_p

Determination of the growth of the sequence of θ_j

r_pl

Plastic hardening stiffness reduction factor

r_ul

Plastic hardening stiffness reduction factor

s

Distance between plastic hinge position and column flange surface

S_k

Cumulative number of samples

t_cf

Thickness of column flange

t_cw

Thickness of column web

t_ep

Thickness of the end-plate

t_s

Thickness of stiffening rib

V(S_k)

Variance of the parameter S_k

V_b

Design shear in beam with plastic hinge section

V_G

Shear force at representative values of gravity loads

V_pl

Effective volume of panel zone

W_c

Column section modulus

W_exp

Total dissipated energy

W_pre

Predicted value of dissipated energy for theoretical hysteresis model

Y_c

Yield line mechanism parameters for column flange

y_i

Force arm length

Y_p

Yield line mechanism parameter of end-plate

Y_re

Seismic adjustment factor for panel zone bearing capacity, generally 0.75

γ_M

Bolt local safety factor

ε

Column web plate surface deformation limitation amount

η

Ratio of bolt head outer diameter to bolt diameter

η_y

Strengthening factor of steel, generally n_y = 1.15

θ_u

Rotation in the limit state

θ_y

Rotation at the moment of yielding

λ_cw

Stability coefficient of panel zone

μ

Ductility coefficient

ξ_eq

Equivalent viscous damping factor

Ф_n&Ф_d

Resistance coefficient