Experimental study on mechanical performance of frame connected by improved end-plate joint

Abstract

An improved end-plate joint has been applied to connect H-beams and H-columns in the weak-axis direction. However, there is still a lack of research on the impact of this joint on the strong-axis mechanical behavior of frames. To address this gap, monotonic loading tests were conducted on two steel frames: one braced and the other unbraced, both connected with improved end-plate joints. The mechanical performance of the frames in the strong-axis direction was analyzed based on test phenomena, load-displacement curves, and deformations of the frame and joint core area. Finite element models were established for braced and unbraced frames to investigate the effects of parameters such as cross-brace presence, brace connecting plate presence, stiffener rib configuration, and beam-column joint type on the strong-axis mechanical performance of the frames. A simplified calculation model and stiffness equation for the frames are proposed. The results show that, compared to steel frames with traditional end-plate joints, the initial stiffness, yield load, and ultimate bearing capacity of those with improved end-plate joints are moderately increased. When compared to welded frames, the yield load and ultimate bearing capacity are comparable, while the initial stiffness decreases by 13%. Installing braces significantly enhances the stiffness and bearing capacity of the frame, shifting the failure mode from end-plate bending failure to net section failure of the tension brace.

Keywords

steel frame extended end-plate beam-column joint mechanical performance stiffness equation

Introduction

Currently, end-plate joints are widely used in the beam-to-column connections for light steel industrial structures and some multi-storey steel frame residential buildings. Compared to other fully bolted joints, end-plate joints offer lower manufacturing costs and greater efficiency in onsite construction under equivalent connection conditions. A well-designed end-plate joint is generally considered semi-rigid, providing adequate bearing capacity to meet design requirements while enhancing joint ductility. These joints also improve the distribution of bending moments and reduce beam deformation in frames (Yorgun and Bayramoğlu, 2001). Over the years, extensive research has been conducted on the mechanical behaviour, design methods, and computational models of beam-column end-plate joints in steel structures.

Regarding the research on end-plate joint connecting frame beam to steel tube column or H-shaped steel column in the strong-axis direction (strong-axis end-plate joint), (Shi et al., 2008) developed numerical models for end-plate joints, comparing results with tests. Mohamadi-Shoore and Mofid (2011) modeled the moment-rotation curve of extended end-plate joints using an exponential fit, validated by tests. Sofias et al. (2014) tested full-scale extended end-plate joints with weakened flanges, confirming good seismic performance. Wang and Wang (2016) investigated blind bolted end-plate joints in square steel tube-H beam connections, evaluating tensile properties. Thai and Uy (2016, 2017) and Wang et al. (2018) studied CFST column-H beam joints, analyzing rotational stiffness and flexural capacity. Waqas et al. (2019) found concrete slabs enhanced joint stiffness in composite structures. Boudia et al. (2020) performed parametric FEA on stiffened extended end-plate joints. Zhang et al. (2020) tested nut-free high-strength bolted joints, while (Sun et al., 2021) proposed a flexural capacity formula for T-bolted joints. New stiffener designs were explored by (Fan et al., 2022; Solhmirzaei et al., 2021) derived bearing capacity formulas for flush/extended end-plate joints with long bolts. Cai et al. (2022) studied TOB-bolted joints’ seismic behavior, and (Tran and Kim, 2022) used ANN to model moment-rotation relationships. Liu et al. (2023) designed curved end-plate joints for circular steel tubes, highlighting prying force effects, while (Zhao et al., 2023) proposed similarity error compensation for scaled models. Collectively, these studies confirm the excellent mechanical performance of strong-axis end-plate joints. Currently, strong-axis end-plate joints have relatively mature simplified calculation models and design methods.

In structures, it is common for the weak-axis direction of frame column to also need to be connected to frame beam. For steel tube column, the same or similar end-plate joint can be used in both main-axis directions. However, for H-shaped steel column, if end-plate joints are to be used in the weak-axis direction, improvements to the stiffener design or the addition of new connecting elements are necessary. In response to this issue, researchers have studied the mechanical performance and design methods of the end-plate joint connecting frame beam to H-shaped steel column in the weak-axis direction (weak-axis end-plate joint). Cabrero and Bayo (2007a, 2007b) welded flange plates with bolt holes to H-column flanges parallel to the web, connecting them to beam end-plates, and derived initial stiffness formulas for both directions via experiments. Kim et al. (Kim et al., 2008) directly attached end-plates to column webs with transverse stiffeners at joint ends, conducting experimental studies. Spavier and El Debs (2022) concrete-filled H-sections and used long bolts in the weak-axis to connect columns to end-plates, performing monotonic tests. Nie et al. (2024) installed a U-shaped component on frame columns (flanges connected to column flanges, web parallel to column web), forming end-plate joints via high-strength bolt connections between beam end-plates and the U-component bottom plate, with experimental and FEA validation. The research results above indicate that compared to strong-axis end-plate joints, weak-axis end-plate joints exhibit lower load-bearing capacity and ductility, making it challenging to achieve widespread application.

The ultimate goal of studying the mechanical performance of end-plate joints is to apply these joints to the connection between beam and column in frame structures. Currently, some researchers have investigated the overall performance of steel frames connected using end-plate joints. Cassiano et al. (2017) performed parametric FEA on progressive collapse resistance of flush end-plate framed structures. Wang et al. (2018a, 2018b) experimentally studied progressive collapse of end-plate connected frames. Hou et al. (2023) tested and modeled steel frames with SCBBs in end-plate joints, proposing a flexural capacity prediction model. Zhou et al. (2023) conducted cyclic tests on 3D end-plate framed structures for seismic evaluation. Zheng et al. (2023) experimentally compared extended end-plate connected stainless steel frames with conventional steel frames. Ning et al. (2024) cyclic-tested two stainless steel frames: one end-plate connected, the other welded. The research results above indicate that using end-plate joints in the strong-axis direction of structural frame columns is feasible, and the frame’s progressive collapse resistance and seismic performance meet the code requirements.

From existing research, it can be observed that the application and study are primarily focused on strong-axis end-plate joints. Research on weak-axis end-plate joints is relatively limited, and these studies also indicate that if traditional end-plate joints are adopted in the weak-axis direction, the mechanical performance of these joints are significantly reduced compared to the same joints in strong-axis. Morever, these joints are prone to failure, imposing significant limitations on their practical application.

To improve the mechanical performance of weak-axis end-plate joints and make the joints in both directions as equi-strong as possible, this paper proposes an improved end-plate joint by introducing welded cross-shaped stiffeners and connecting plates on the column. Following sufficient research on the mechanical properties of improved end-plate joints, the joints are used in the beam-column connection in the strong-axis direction of the frame. Two single-span frames were fabricated: one with flexible cross-brace (braced frame) and another without (unbraced frame). The mechanical performance in the strong-axis direction of the brace and unbraced frames was analyzed based on test phenomena, horizontal load-frame lateral displacement curve, and deformations of frame and joint core area. Furthermore, finite element models (FEMs) were developed for more frames to study the impact of parameters such as presence of brace, stiffener rib form, and joint type on the mechanical performance in the strong-axis direction. Finally, simplified calculation models for the brace and unbraced frame were established. The conclusions drawn in this study can serve as a reference for the application of improved end-plate joints in steel frame structures.

Experiment design

Joint design

The construction of the improved end-plate joint proposed in this paper is illustrated in Figure 1 (a)–(c). The improvement of the joint involves replacing the horizontal stiffening ribs in the traditional end-plate joint with cross-shaped stiffening ribs and welding a connecting plate onto the stiffening ribs. The advantage of the improved joint is that it can be used to connect the frame beam to the H-shaped steel column and achieve similar mechanical performance in both the strong-axis and weak-axis directions by properly designing the thicknesses of stiffeners and connecting plates.

Figure 1.

Schematic diagram of the improved end-plate joint. (Note. 1 - connecting plate; 2 - cross-shaped stiffening rib; 3 - weak-axis frame beam; 4 - strong-axis frame beam).

Test specimens design

In this paper, the improved end-plate joints are used to connect the frame beam to the H-shaped steel frame column in the strong-axis direction. A total of two frames were designed, differing by the presence of flexible cross-bracing, as shown in Figure 2. The cross-section of the frame column is H200 × 200 × 6 × 10, with a length of 2.70 m, and the center height of the joint region is 2.50 m. The cross-section of the frame beam is H280 × 150 × 6 × 10, with a length of 3.00 m. The cross-section of the brace is C100 × 50 × 3 × 3, with a length of 3.55 m. The centerline of the brace is coincident with the center point of the joint region and the bottom center point of the column. The two braces are positioned on opposite sides of the joint plate without direct contact. Since the centroid of the brace section is not located in the symmetric plane of the frame, both braces function as eccentrically loaded components. Beam-column joints are assembled using M16 grade 8.8 friction-type high-strength bolts with a preload of 80 kN. The braces are attached to the node plates with M22 grade 10.9 high-strength bolts, each subjected to a preload of 190 kN.

Figure 2.

Machining drawing of the test specimen (units: mm).

Material mechanical properties test

Specimens for material tests were cut from the flanges of the fabricated columns, beams, and braces, with the elastic modulus (E), yield strength (f_y), ultimate tensile strength (f_u), and elongation after fracture (δ₅) of different component steels listed in Table 1.

Table 1.

Material mechanical properties.

Specimen location	Specimen thickness t/mm	Specimen width b/mm	Elastic modulus E/GPa	Yield strength f_y/MPa	Ultimate tensile strength f_u/MPa	Elongation after fracture δ₅/%
Column flange	10	15	208	398	500	33
Beam flange	10	15	206	402	503	33
Brace	3	15	217	442	533	29

Sensor layout

The main sensors utilized in this test are force sensors, displacement meters and strain gauges. The layout of displacement meters on the specimens is shown in Figure 3, where D0 to D5 are designated for monitoring the deformation of the frame columns, D6 for monitoring the mid-span deflection of the frame beams, D7 and D8 for monitoring the vertical deformation at the top of the columns, D9 and D10 for monitoring the out-of-plane deformation at the top of the columns. The average of the readings of D0 and D1 is defined as the total lateral displacement of the frame.

Figure 3.

Displacement meters layout diagram.

Test setup and loading scheme

The experiment was conducted on the 10,000 kN multi-functional structural testing system at Tongji University’s Structural Testing Laboratory, with the loading device shown in Figure 4.

Figure 4.

Test setup.

During the loading: Initially, a vertical force of 1400 kN is applied at a loading rate of 100 kN/min in the vertical direction. After loading, the axial compression ratio of each column is 0.35. After applying the vertical force, the load is held for 5 minutes before proceeding with horizontal loading. Horizontal loading is controlled by displacement, using monotonic step loading. Each load step is 5 mm. The average reading of D0 and D1 is considered for each step. Before yielding, the loading rate is 1 mm/min, and after yielding, it is 3 mm/min.

Test results and analysis

Test phenomenon

Unbraced frame

During the loading process where the total lateral displacement of the frame increased from 0 to 35 mm, as the horizontal actuator contracted, the frame underwent gradual lateral displacement without any notable anomalies. In the loading stage from 35 to 40 mm, the end-plate on the tension side of the joint region was observed to separate from the frame column flange, accompanied by slight bending deformation of the end-plate, as shown in Figure 5(a). Subsequently, the bending deformation of the end-plate continued to increase. Loading was stopped when the force sensor on the horizontal actuator indicated that the load on the specimen had dropped to 85% of the maximum load. At this stage, the end-plate on the tension side of the joint region was found to be completely separated from the frame column flange, with a maximum gap of approximately 10 mm, as shown in Figure 5(b). Upon disassembly, the high-strength bolts in the joint showed no signs of failure, as depicted in Figure 5(c). However, obvious plastic deformation was clearly visible on the end-plate, as shown in Figure 5(d). Therefore, these observations confirmed that the frame’s failure mode was end-plate failure.

Figure 5.

Test Phenomena of unbraced frame.

Braced frame

During the loading stage from 10 to 15 mm, noticeable out-of-plane buckling of the compression brace was observed, as shown in Figure 6(a). As the loading progressed from 20 to 25 mm, a noise was heard from the tension brace end. After loading, bolt slippage was evident, as shown in Figure 6(b). At this stage, the frictional connection had failed, and the bolts began to bear load through contact with the hole wall. When the total lateral displacement reached 35 mm, the bolt holes expanded significantly, resulting in substantial plastic deformation and flange twisting, as shown in Figure 6(c). By the time the total lateral displacement reached 55 mm, the brace fractured at the net section, and severe twisting of the end-plate flange was observed, as shown in Figure 6(d). Loading was terminated when the load dropped to 85% of the peak value, as indicated by the force sensor. At this stage, the end-plate exhibited negligible deformation. Upon specimen disassembly, no bolt failure was detected.

Figure 6.

Test Phenomena of braced frame.

Load-displacement curve

The horizontal load-total lateral displacement curves for the unbraced frame and the braced frame are shown in Figure 7.

Figure 7.

Load-displacement curves of the two frames.

The maximum loads (

F_{\max}

) and the ultimate displacements (

∆_{u}

) of the two frames were obtained from the load-displacement curves. The initial lateral stiffness (

K_{0}

) of the frame was determined by calculating the slope of the tangent at the origin on the curves. The yield displacement (

∆_{y}

) and yield load (

F_{y}

) of the frame were obtained through the energy equivalence method (Feng et al., 2017). The displacement ductility factor was determined by calculating the ratio of

∆_{u}

∆_{y}

. The mechanical performance of the two frames is listed in Table 2.

Table 2.

Mechanical properties of the two frames.

Frame type	$K_{0}$ /kN·mm^-1	$F_{y}$ /kN	$∆_{y}$ /mm	$F_{\max}$ /kN	$∆_{u}$ /mm	μ
Brace	5.12	130.60	34.56	143.70	109.79	3.18
Unbraced	12.52	292.10	29.65	327.80	63.82	2.15

From the table, it can be observed that the displacement ductility factors for both the unbraced frame and the braced frame are greater than 2.0. Due to the inclusion of flexible cross braces, the braced frame exhibited a 144% increase in initial stiffness, a 123% increase in yield load, a 128% increase in maximum load, and a 32% decrease in displacement ductility factor. Combining the test phenomena, it can be concluded that the inclusion of flexible cross braces effectively enhances the initial stiffness and ultimate load capacity of the frame. Braces can serve as the first line of defense for the structure, preventing other components and joints from failing.

Analysis of brace failure mechanisms

For the eccentric compression brace, due to the combined effects of axial compression and bending moments, and considering the brace’s high slenderness ratio resulting in a very low stability coefficient, the brace undergoes bending instability around the weak axis early. This leads to a sharp increase in strain on the brace flanges (S1 and S5). By analyzing the test data and observations, it is determined that the horizontal load on the frame at the point of instability is approximately 22.4 kN, with an average strain of about 55 με on the brace section due to axial forces. Considering the material properties of the specimen, the axial compression force is about 6.5 kN. Using the formula for the stability bearing capacity of compression members given in the Chinese code (GB 50017–2017, 2017), the stability bearing capacity of this brace is calculated to be approximately 3.6 kN. The experimental value exceeds the design value specified by the code, consistent with practical conditions.

The eccentric tension brace does not occur stability issues. The data indicates that the strain on the mid-section of the brace is relatively uniform. However, during the final stages of loading, the net section rupture at the bolt hole causes axial rebound in the brace, resulting in a slight decrease in strain towards the end of the loading process. Before rupture, the maximum average strain on the mid-section is approximately 1940 με. Considering the material properties of the specimen, the axial tension force is about 437.2 kN, with approximately 291.8 kN in the horizontal direction. At this point, the horizontal load of the frame is approximately 327.5 kN.

Finite element analysis

Finite element models

Using ABAQUS 2020 for finite element analysis (FEA) of the test, in addition to the two tested specimens, five additional specimens were designed based on key parameters, including the presence of cross-bracing, brace plates, stiffener configurations, and beam-column joint types. The numbers and differences of each specimen are listed in Table 3. In the table, ‘N’ represents the unbraced frame, ‘B’ represents the braced frame, and the design of specimen N-4-F is specifically to assess the influence of brace plates on the mechanical performance of the frame.

Table 3.

Specimen number and difference.

Specimen number	Research method	Are there braces?	Are there brace plates?	Stiffening rib form	Beam-column joint form
N-1-T	TEST	No	No	Cross-shaped	End-plate joint
N-1-F	FEA	No	No	Cross-shaped	End-plate joint
N-2-F	FEA	No	No	Horizontal	End-plate joint
N-3-F	FEA	No	No	Cross-shaped	Welded-joint
N-4-F	FEA	No	Yes	Cross-shaped	End-plate joint
B-1-T	TEST	Yes	Yes	Cross-shaped	End-plate joint
B-1-F	FEA	Yes	Yes	Cross-shaped	End-plate joint
B-2-F	FEA	Yes	Yes	Horizontal	End-plate joint
B-3-F	FEA	Yes	Yes	Cross-shaped	Welded-joint

In the finite element models, all components are created using solid elements with the element type C3D8R. In the FEMs, all components are created using the three-dimensional hexahedral element with reduced integration (C3D8R). In the FEMs, the initial global mesh size was set to 150 mm, but the calculation results were insufficiently accurate. Subsequently, when the global mesh size was reduced to 50 mm, the test results were basically consistent with the FEA results. Therefore, this size was adopted as the mesh size for all models. Additionally, in all models, the plates were each divided into four meshes along the thickness direction.

Based on material test results, the constitutive model for component utilizes a three-linear segment model with a descent section. Parameters such as the elastic modulus (E), yield strength (f_y), ultimate tensile strength (f_u), etc., are determined based on the material tests for different specimens. For high-strength bolts, a two-linear segment model without a strengthening section is used, grade 8.8 bolt yield strength (fy) is 640 MPa, and grade 10.9 bolt yield strength is 900 MPa, respectively. Within the model, surface-to-surface contact is used for interactions between all components. In the contact properties, the normal direction is set as “hard contact” and the tangential direction as “penalty function,” with a friction coefficient of 0.3.

In the FEMs, distribution beams, column base anchorage, horizontal connection-2 and hinge pin present from the test setup were excluded. Instead, complete fixed constraints are applied at the bottom of the frame columns and a uniformly distributed load with a total of 700 kN is applied on the top surface of each frame column. Model of horizontal connection-1 is established to consider its impact on the beam-column joints. Additionally, horizontal displacement loads were applied at the center of the pin holes, as shown as in Figure 8. It has been verified that the above simplification of boundary conditions has no effect on the calculation results.

Figure 8.

Boundary conditions of FEMs.

Finite element analysis results

In the finite element models, the load-displacement curves for each specimen are shown in Figure 9, where “o” denotes the yield point and “Δ” denotes the ultimate load point of the curve. The data in the labels represent the displacement and force of each point, respectively.

Figure 9.

Load-displacement curves of the frames.

From these figures, it can be observed:

(1) The load-displacement curves obtained from the test and FEA of the two frames match, validating the accuracy and reliability of the finite element models;

(2) Whether or not cross-brace is installed, the initial stiffness and ultimate bearing capacity of the frames using traditional end-plate joints and improved end-plate joints are similar. However, the frame using traditional end-plate joints exhibits better ductility;

(3) In the frames using welded joints, the initial stiffness and ultimate bearing capacity are improved due to enhanced constraints between beams and columns. However, ductility is slightly reduced;

(4) A comparison of the curves N-1-FEA and N-4-FEA shows that the presence of brace plates significantly enhances the initial stiffness and ultimate bearing capacity of the frame. The main reason is that the installation of brace plates increases the section height of the beam and column and reduces the calculated length of the column to some extent.

In the finite element model, the failure modes of frames N-1-F and B-1-F are illustrated in Figure 10. The failure mode of the unbraced frame exhibits severe bending deformation at the end-plate, while the braced frame exhibits failure due to net section rupture at the tension brace, consistent with the experimental results.

Figure 10.

Failure modes of FEM.

Comparison of mechanical performance of different test specimens

According to the FEA results, the initial stiffness, yield load, yield displacement, ultimate bearing capacity, and displacement ductility coefficient of each specimen have been obtained and organized in Table 4, and the comparison of various mechanical performance indicators is shown in Figure 11. From the table and figures, the following conclusions can be drawn:

(1) By comparing N-1-F with N-2-F and B-1-F with B-2-F, it is found that compared to traditional horizontal stiffeners, cross-shaped stiffeners can slightly increase the initial stiffness, yield load, and ultimate bearing capacity in the main axis direction of the frame. In the unbraced frame, the initial stiffness increases by 1.73%, the yield load by 2.64%, and the ultimate bearing capacity by 3.22%. In the braced frame, the initial stiffness increases by 6.04%, the yield load by 3.12%, and the ultimate bearing capacity by 3.60%;

(2) By comparing N-1-F with N-3-F and B-1-F with B-3-F, it can be observed that the form of the beam-column joint has little impact on the yield load and ultimate bearing capacity of the frame. Frames using end-plate joints exhibit slightly higher ductility, but the joint form significantly affects the initial stiffness of the frame. The initial lateral stiffness of frames using end-plate joints is approximately 83% of that of frames using welded joints;

(3) By comparing N-1-F, N-4-F, and B-1-F, it can be found that the contributions of the frame itself, brace plates, and cross-bracing to the initial stiffness of the frame are 5.30 kN/mm, 1.68 kN/mm, and 6.00 kN/mm, respectively. The contribution of brace plates to the stiffness of the braced frame is 15%. Regarding the ultimate bearing capacity of the frame, the contributions are 145.52 kN, 24.45 kN, and 157.83 kN, respectively. Brace plates contribute 8% to the ultimate bearing capacity of the braced frame. The presence of cross-bracing and brace plate increases the initial stiffness of the frame by 145% and the ultimate bearing capacity by 125%, aligning with the test results.

Table 4.

Mechanical performance of each specimen.

Number	$K_{0}$ /kN·mm^-1	$F_{y}$ /kN	$∆_{y}$ /mm	$F_{\max}$ /kN	$∆_{u}$ /mm	μ
N-1-T	5.12	130.60	34.56	143.70	109.79	3.18
N-1-F	5.30	130.40	32.51	145.42	116.05	3.57
N-2-F	5.21	127.05	32.06	140.89	130.93	4.08
N-3-F	6.38	134.89	31.38	148.62	104.72	3.34
N-4-F	6.98	152.94	28.02	169.97	104.38	3.72
B-1-T	12.52	292.10	29.65	327.80	63.82	2.15
B-1-F	12.98	306.60	29.05	337.40	65.87	2.27
B-2-F	12.24	297.32	28.97	325.67	68.90	2.37
B-3-F	15.70	309.11	24.02	337.80	53.33	2.22

Figure 11.

Comparison of mechanical performance.

Simplified calculation model of the frame using improved end-plate joint

Simplified methods of the frames

The frames studied in this paper can be simplified as follows:

(1) Consider the joint core area as a rigid body, neglecting the impact of shear deformation on the overall deformation of the frame, while also disregarding any axial deformations of all components during horizontal loading processes.

(2) Based on the component method, the extended end-plate joint is equivalent to a spring with rotational stiffness, and the stiffness of the spring will be derived in the next section.

(3) The influence of the rigid joint core area on the initial stiffness and load-bearing capacity of the frame is manifested by reducing the length of the column. In the simplified model, the length of the column is the vertical distance from the bottom of the beam to the top of the stiffener at the base of the column.

(4) In the simplified model, beam, column, and brace components are represented by line elements. Compression brace is prone to instability, so the simplified model only considers the effect of tension brace, and it is approximately assumed that braces only bear axial tension.

After simplification, the computational model of the frame is shown in Figure 12. The figures indicate element numbers, some element parameters, node numbers, degrees of freedom present in the frame, global and local coordinate systems, and other relevant information. The local coordinate systems of beam elements are aligned with the global coordinate system.

Figure 12.

Simplified calculation model of the frame.

Stiffness equation of the simplified frame

Element stiffness equation

Eurocode 3 (EN 1993-1-8, 2005) provides a method for calculating the initial stiffness of end-plate joints. The initial rotational stiffness of the joint is a combination of the stiffness of various components. For a traditional extended end-plate joint, the components include the column web in shear, the column tension flange, the column compression flange, tension bolts, compression bolts, and the end plate in bending.

For the improved end-plate joint studied in this paper, with cross-shaped stiffeners and connecting plates, only the stiffness of the tension bolts (bt) and end plate in bending (epb) need to be considered, assuming that the stiffness of other components is infinitely large. The computational model of the joint is shown in Figure 13.

Figure 13.

Stiffness calculation model of the joint.

Calculate the stiffness of the end-plate according to equation (1).

k_{epb} = \frac{192 E I_{ep}}{d^{3}}

(1)

where I_eq is the moment of inertia of the entire cross-section of the end-plate about the 1-1 axis, d is the center distance between the two tension bolts vertically, taken as 90 mm, as shown in Figure 13.

Calculate the stiffness of a single tension bolt according to equation (2).

k_{bt} = \frac{E A_{0}}{η L}

(2)

where A₀ is the effective cross-section of the bolt. To prevent loosening of the bolt, spring washers are used. An adjustment factor η is introduced to consider the influence of the spring washers on the axial stiffness of the bolt. In this paper, η is taken as 2.0.

According to Eurocode 3 (EN 1993-1-8, 2005), the equivalent lever arm (Z_eq) for the joint is 277.5 mm. Calculate the equivalent stiffness of the two rows of tension bolts using equation (3).

k_{bt}^{'} = \frac{2 \sum_{r} k_{bt} h_{r}}{Z_{eq}} = \frac{4 Z}{Z_{eq}} k_{bt}

(3)

where Z is the center distance between the flanges of the frame beam, taken as 270 mm, as shown in Figure 18.

Finally, calculate the series spring stiffness according to equation (4).

k_{eq} = \frac{1}{\frac{1}{k_{bt}^{'}} + \frac{1}{k_{epb}}}

(4)

Calculate the initial rotational stiffness of the joint according to equation (5).

S = Z^{2} k_{eq}

(5)

For ease of expression,, define the stiffness ratio between the spring and the beam as μ₁ and the stiffness ratio between the spring and the column as μ₂. Calculate according to equation (6).

μ_{1} = \frac{S l}{E I_{b}}, μ_{2} = \frac{S h}{E I_{c}}

(6)

The element stiffness equation for the beam is given by equation (7).

(\begin{array}{l} M_{2} \\ M_{3} \end{array}) = S (\begin{array}{l} \frac{3}{μ_{1} + 6} + \frac{1}{μ_{1} + 2} \frac{3}{μ_{1} + 6} - \frac{1}{μ_{1} + 2} \\ \frac{3}{μ_{1} + 6} - \frac{1}{μ_{1} + 2} \frac{3}{μ_{1} + 6} + \frac{1}{μ_{1} + 2} \end{array}) (\begin{array}{l} θ_{2} \\ θ_{3} \end{array})

(7)

The element stiffness equation for the column is given by equation (8).

(\begin{array}{l} F_{1} \\ M_{1} \end{array}) = \frac{S}{h^{2}} (\begin{array}{l} \frac{12}{μ_{2} + 4} \frac{6 h}{μ_{2} + 4} \\ \frac{6 h}{μ_{2} + 4} \frac{4 h^{2}}{μ_{2} + 4} \end{array}) (\begin{array}{l} μ_{1} \\ θ_{1} \end{array})

(8)

The stiffness matrix for the brace has only one element, and it requires coordinate transformation. The stiffness equation for the brace is given by equation (9).

F_{1} = \frac{E A_{t}}{l_{t}} \cos^{2} α u_{1}

(9)

where A_t is the cross-sectional area of the tension brace, l_t is the length of the tension brace, α is the angle between the brace and the horizontal direction, as shown in Figure 12. For ease of expression, let

k_{t} = \frac{E A_{t}}{l_{t}} \cos^{2} α

Frame stiffness equation

By combining the stiffness matrices of the four elements, the stiffness matrix of the entire frame is obtained. The stiffness equation for the frame is given by equation (10), where γ is introduced to account for the contribution of the brace plates to the frame stiffness. According to the analysis in section 4.3, for braced frame, γ is taken as 1.15, and for unbraced frame, γ is taken as 1.00. In the unbraced frame, let k_t = 0 to neglect the stiffness of the brace. In the equation, the sum of F₁ and F₄ represents the horizontal load acting on the frame.

(\begin{array}{l} F_{1} \\ M_{1} \\ \begin{array}{l} M_{2} \\ \begin{array}{l} M_{3} \\ M_{4} \\ F_{4} \end{array} \end{array} \end{array}) = γ S (\begin{array}{c} \begin{array}{c} \frac{12}{(μ_{2} + 4) h^{2}} \\ \frac{6}{(μ_{2} + 4) h} \\ \begin{array}{c} 0 \\ 0 \\ \begin{array}{c} 0 \\ 0 \end{array} \end{array} \end{array} & \begin{array}{c} \begin{array}{c} \begin{array}{c} \frac{6}{(μ_{2} + 4) h} \\ \frac{4}{μ_{2} + 4} \end{array} \\ 0 \\ 0 \end{array} \\ 0 \\ 0 \end{array} & \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} 0 \\ 0 \end{array} \\ \frac{3}{μ_{1} + 6} + \frac{1}{μ_{1} + 2} \\ \frac{3}{μ_{1} + 6} - \frac{1}{μ_{1} + 2} \end{array} \\ 0 \\ 0 \end{array} & \begin{array}{c} \begin{array}{c} \begin{array}{c} 0 \\ 0 \end{array} \\ \frac{3}{μ_{1} + 6} - \frac{1}{μ_{1} + 2} \\ \frac{3}{μ_{1} + 6} + \frac{1}{μ_{1} + 2} \end{array} \\ 0 \\ 0 \end{array} & \begin{array}{c} \begin{array}{c} \begin{array}{c} \begin{array}{c} 0 \\ 0 \end{array} \\ 0 \\ 0 \end{array} \\ \frac{4}{μ_{2} + 4} \\ \frac{6}{(μ_{2} + 4) h} \end{array} & \begin{array}{c} \begin{array}{c} \begin{array}{c} 0 \\ 0 \end{array} \\ 0 \\ 0 \end{array} \\ \frac{6}{(μ_{2} + 4) h} \\ \frac{12}{(μ_{2} + 4) h^{2}} + \frac{k_{t}}{S} \end{array} \end{array} \end{array} \end{array}) (\begin{array}{l} μ_{1} \\ θ_{1} \\ \begin{array}{l} θ_{2} \\ θ_{3} \\ \begin{array}{l} θ_{4} \\ μ_{4} \end{array} \end{array} \end{array})

(10)

Error of the simplified calculation model

Obtain displacements and rotations of each freedom degree within the elastic range from the finite element model. Then, theoretically calculate (TC) the internal forces of the frame using equation (10). Compare the theoretical internal force results (F_c) with test results (F_t) and FEA results (F_m) to assess the error of the simplified model. The comparison results are shown in Figure 14 and Table 5, where e₁ represents the error between theoretical calculation and test results, and e₂ represents the error between theoretical calculation and FEA results. The calculation method for the errors is defined in equation (11).

e_{1 (2)} = \frac{F_{3} - F_{1 (2)}}{F_{3}} \times 100 %

(11)

Figure 14.

Comparison of internal force obtained by three methods within the elastic range.

Table 5.

Comparison results.

Type of the frame	Disp./mm	F_t/kN	F_m/kN	F_c/kN	e₁/%	e₂/%
Unbraced frame	5	27.3	25.9	26.3	−3.80	1.38
	10	53.2	53.9	55.3	3.72	2.39
	15	73.7	79.7	81.7	9.78	2.44
	20	91.8	102.1	104.9	12.49	2.66
	25	108.1	117.9	123.8	12.68	4.80
Braced frame	5	62.1	64.3	65.8	5.66	2.28
	10	131.3	129.8	136.7	3.96	5.08
	15	192.9	187.3	200.5	3.77	6.54
	20	240.7	239.0	257.7	6.59	7.27
	25	273.9	283.3	310.1	11.67	8.65

From the graphs and tables, it is evident that in the initial loading stages, the simplified model can effectively calculate the mechanical performance of the frame with errors kept within 10%. However, as the load increases, the error of the simplified model gradually grows, and its mechanical performance is higher than test results and FEA results. The reason is that although only the elastic stage before yielding of the frame was considered for comparison, during this phase, some regions on the end-plate of the unbraced frame might have yielded, and the compression brace of the braced frame has become unstable, and there could also be other initial defects on the frame, leading to a decrease in both the load-bearing capacity and stiffness of the frame. The closer the load gets to the yield point, the more they decrease.

Conclusion

This paper conducted monotonic loading tests on the braced frame and the unbraced frames connected by improved end-plate joints. Subsequently, finite element analysis was performed on seven frames by varying parameters such as cross-brace presence, brace connecting plate presence, stiffener configurations, and beam-column joint types. The mechanical performance of the frame under monotonic loading was studied, and a simplified calculation model and stiffness equation for the frame were provided. Based on these studies, the following conclusions were drawn:

(1) The displacement ductility factors of both the unbraced and braced frames exceed 2.0. The failure mode of the former is severe end-plate bending deformation, while the latter is net section rupture of the tension brace. This indicates that braces act as the first line of defense, preventing premature failure of other components and joints.

(2) Compared to the unbraced frame, the braced frame exhibited a 144% increase in initial stiffness, a 123% increase in yield load, and a 128% increase in maximum load, demonstrating that installing flexible cross-brace effectively enhances the frame’s bearing capacity and stiffness.

(3) Compared to traditional end-plate joints, incorporating cross-shaped stiffeners and brace connecting plates improves the mechanical performance in the frame’s strong-axis direction. For the unbraced frame, this results in a 1.73% increase in initial stiffness, 2.64% in yield load, and 3.22% in ultimate bearing capacity. For the braced frame, the increases are 6.04%, 3.12%, and 3.60%, respectively. More importantly, cross-shaped stiffeners and connecting plates significantly reduce shear deformation in the joint core area, allowing it to be idealized as a rigid body.

(4) Compared to steel frames with welded joints, those with improved end-plate joints showed a 13% reduction in initial stiffness, while bearing capacity remained unaffected.

(5) Through simplifications and assumptions, steel frames with end-plate joints can be modeled as line element models with springs. The frame stiffness equation was theoretically derived, and comparisons with FEA results show that the simplified model’s calculation error for frame internal forces is within 10%.

(6) The above studies have demonstrated the feasibility of using improved end-plate joints to connect steel frames in the strong-axis direction. This paper will further investigate the improved end-plate joints for the weak-axis direction, aiming to truly promote the practical application of such joints.. In practical engineering, the improved end-plate joints enable fully bolted connections of steel frames, enhance the mechanical performance of the frame in the weak-axis direction, and improve the standardization level of components and construction efficiency of the structure, demonstrating great engineering application value.

Footnotes

ORCID iDs

Xiaonong Guo

Jinhui Luo

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Cabrero

Bayo

(2007a) The semi-rigid behaviour of three-dimensional steel beam-to-column joints subjected to proportional loading. Part I. Experimental evaluation. Journal of Constructional Steel Research 63: 1241–1253.

Cabrero

Bayo

(2007b) The semi-rigid behaviour of three-dimensional steel beam-to-column steel joints subjected to proportional loading. Part II: theoretical model and validation. Journal of Constructional Steel Research 63: 1254–1267.

Cai

Sun

Wang

, et al. (2022) Seismic behavior of TOBs bolted endplate connection to strengthened CFST column. Journal of Constructional Steel Research 197: 107493.

Cassiano

D’Aniello

Rebelo

(2017) Parametric finite element analyses on flush end-plate joints under column removal. Journal of Constructional Steel Research 137: 77–92.

de Souza Spavier

El Debs

ALHC

(2022) Experimental analysis of beam-to-column connection with partially encased concrete column with varying inertia axis. Journal of Constructional Steel Research 194: 107325.

European Committee for Standardization (CEN) . (2005). Eurocode 3: Design of Steel Structures—Part 1.8: Design of Joints (EN 1993-1-8). European Committee for Standardization (CEN).

Fan

Zhao

Hua

, et al. (2022) Seismic performance and analytical model of CFDST joint with endplates and long bolts. Structures 35: 483–499.

Feng

Qiang

(2017) Discussion and definition on yield points of materials, members and structures. Engineering Mechanics 34: 36–46, (in Chinese).

Hou

Wang

Chen

(2023) Cyclic behavior and mechanical model of a novel endplate connection to double-skin composite wall with slip-critical blind bolts. Journal of Building Engineering 63: 105453.

10.

Kim

(2008) Strength evaluation of beam–column connection in the weak axis of H-shaped column. Engineering Structures 30: 1699–1710.

11.

Liu

Chen

Jin

, et al. (2023) Prediction of yield strength and prying action of TOBs bolted arc-shaped endplate. Journal of Constructional Steel Research 208: 108039.

12.

Merad Boudia

Boumechra

Bouchair

, et al. (2020) Modeling of bolted endplate beam-to-column joints with various stiffeners. Journal of Constructional Steel Research 167: 105963.

13.

Ministry of Housing and Urban-Rural Development of the People’s Republic of China (MOHURD) (2017) Standard for Design of Steel Structures (GB 50017–2017). China Architecture and Building Press, (in Chinese).

14.

Mohamadi-Shoore

Mofid

(2011) New modeling for moment–rotation behavior of bolted endplate connections. Scientia Iranica 18: 827–834.

15.

Nie

Chen

, et al. (2024) An investigation on the seismic behavior of the end-plate connection between a steel beam and the weak-axis of an H-shaped column using a U-shaped connector. Buildings 14(4): 1087.

16.

Ning

Yang

, et al. (2024) Experimental study on seismic performance of stainless steel full-scale frames: global response. Journal of Constructional Steel Research 217: 108658.

17.

Shi

Wang

, et al. (2008) Numerical simulation of steel pretensioned bolted end-plate connections of different types and details. Engineering Structures 30: 2677–2686.

18.

Sofias

Kalfas

Pachoumis

(2014) Experimental and FEM analysis of reduced beam section moment endplate connections under cyclic loading. Engineering Structures 59: 320–329.

19.

Solhmirzaei

Tahamouli Roudsari

Hosseini Hashemi

(2021) A new detail for the panel zone of beam-to-wide flange column connections with endplate. Structures 34: 1108–1123.

20.

Sun

Liang

, et al. (2021) Experimental and theoretical modeling for predicting bending moment capacity of T-head square-neck one-side bolted endplate to tube column connection. Journal of Building Engineering 43: 103104.

21.

Thai

(2016) Rotational stiffness and moment resistance of bolted endplate joints with hollow or CFST columns. Journal of Constructional Steel Research 126: 139–152.

22.

Thai

Aslani

, et al. (2017) Behaviour of bolted endplate composite joints to square and circular CFST columns. Journal of Constructional Steel Research 131: 68–82.

23.

Tran

Kim

(2022) Revealing the nonlinear behavior of steel flush endplate connections using ANN-based hybrid models. Journal of Building Engineering 57: 104878.

24.

Wang

(2016) Yield and ultimate strengths determination of a blind bolted endplate connection to square hollow section column. Engineering Structures 111: 345–369.

25.

Wang

Zhang

, et al. (2018) Experimental investigation on seismic performance of endplate composite joints to CFST columns. Journal of Constructional Steel Research 145: 352–367.

26.

Wang

Chen

(2018a) Progressive collapse behaviour of endplate connections to cold-formed tubular column with novel slip-critical blind bolts. Thin-Walled Structures 131: 404–416.

27.

Wang

Chen

, et al. (2018b) Progressive collapse behaviour of extended endplate connection to square hollow column via blind hollo-Bolts. Thin-Walled Structures 131: 681–694.

28.

Waqas

Thai

(2019) Experimental and numerical behaviour of blind bolted flush endplate composite connections. Journal of Constructional Steel Research 153: 179–195.

29.

Yorgun

Bayramoğlu

(2001) Cyclic tests for welded-plate sections with end-plate connections. Journal of Constructional Steel Research 57: 1309–1320.

30.

Zhang

Zhao

Tian

, et al. (2020) Tensile behavior of the connection between nut-free high-strength bolt and endplate. Journal of Constructional Steel Research 174: 106301.

31.

Zheng

Zhang

, et al. (2023) Pseudo-static test on stainless steel frame with high-strength bolted extended endplate joint. Journal of Constructional Steel Research 207: 107980.

32.

Zhou

(2023a) Coupling error modeling and compensation of beam-column endplate connection system. Journal of Constructional Steel Research 208: 108005.

33.

Zhou

Zhao

, et al. (2023b) Experimental study of seismic performance of PEC column-steel beam 3D frame with endplate connection. Journal of Constructional Steel Research 202: 107765.