Research on the stiffness of the improved bolt-column (IBC) joint subjected to eccentric and combined loads

Abstract

The Improved Bolt-Column (IBC) joint is a kind of novel semi-rigid connections in spatial structures. By modifying the cone section of the bolt-column joint, the bending capacity of the joint is enhanced. However, in-depth research on its mechanical behavior under the combined action of bending moment and axial force remains limited. This study first investigates the failure modes and mechanical properties of IBC joints under eccentric loading. Moment-rotation curves and axial load-axial displacement curves under different plate thicknesses and eccentricities are obtained, and the stress distribution of the joints is analyzed using ABAQUS package. The results indicate that increases in both plate thickness and eccentricity result in greater initial stiffness and ultimate moment capacity of the joints. The critical eccentricity is identified as e = 200 mm; beyond this value, the mechanical behavior of the joints under eccentric loading becomes similar to that under pure bending. Furthermore, as the plate thickness increases, the failure mode gradually shifts from yielding of the conical section to yielding and fracture of the bolts. By analyzing the ultimate moments of four groups of joints with different plate thicknesses under eccentricities ranging from 10 to 200 mm, a predictive formula between eccentricity and ultimate moment is established. When the joints are subjected to combined axial force and bending moment, an increase in axial tension significantly reduces both the initial bending stiffness and the ultimate moment capacity. In contrast, an increase in axial compression also leads to a decrease in the ultimate moment capacity; however, the initial bending stiffness first decreases and then increases. A parametric analysis is conducted on the bending capacity of the joints under different magnitudes of axial force, and a formula is developed to quantify the influence of axial force on the bending capacity.

Keywords

IBC joint eccentric load combined bending and axial force finite element analysis predictive formula mechanical behavior

Introduction

The investigation of the mechanical performance of IBC connections is of significant scientific and practical importance, as the strength, stiffness, and durability of these connections directly influence the behavior of the entire space lattice structure. Moreover, current researches on their mechanical properties were primarily conducted under single-load conditions, such as strong-axis bending, weak-axis bending, axial force, or shear force. Xiao et al.¹ improved upon the traditional bolted column connection by adding a web plate, two flange plates, and two additional bolts, thereby proposing a novel semi-rigid connection known as the IBC connection. Its mechanical performance was analyzed under individual load conditions, validating its superior performance in bending, tension, and shear compared to the traditional bolted column connection.

Single-layer latticed shells, due to their large span and complex load distribution, represent a critical subject for researching semi-rigid connections. Fujimoto et al.² conducted buckling experiments and a numerical analysis on a single-layer latticed dome. Most connections in spatial structures are semi-rigid, such as the MERO joint system,³ bolted-ball joints,^4,5 the T.U.U.-S ball joints,⁶ S14, D14, and D06 joints,⁷ and space truss connectors.⁸ These lattice structures are widely used in large public buildings, including stadiums and exhibition halls, where their structural behavior is vital to ensuring stability. Xiao et al.⁹ examined how the stiffness, dimensions, and imperfections of IBC joints affect the shell’s load-bearing capacity. López et al.^10,11 and Kato et al.¹² conducted related researches demonstrating that the rigidity of joints is vital to single-layer latticed shells’ performance. The structural integrity and safety of these systems are directly influenced by the mechanical behavior of semi-rigid connections.

In reality, joints in latticed shells are not subjected to idealized single-load conditions, such as pure bending or pure axial force. Instead, they are primarily under the combined action of bending moments and axial forces. Ma et al.^13,14 investigated the mechanical performance of bolted column connections under various in-plane combined loads, including moment-shear, moment-compression, and moment-tension interactions. Zhang et al.¹⁵ systematically investigated the effects of axial compressive loads on self-piercing riveted joints through integrated experimental and numerical simulation approaches. Dai et al.¹⁶ conducted experimental and numerical analyses on shear anchor (AAH) joints under eccentric loading, deriving the correlation curve between axial force and bending moment capacity. Their studies also included an analysis of how the axial force-to-bending moment ratio influences the initial rotational stiffness. Zhang et al.¹⁷ investigated the mechanical behavior of double-ring joints under pure bending, combined bending-shear, and eccentric loading conditions. Based on the full-section plastic criterion, the researchers developed a connection between the ultimate moment ratio and axial load eccentricity. Additionally, using an exponential function for nonlinear fitting, Zhang et al.¹⁷ established a relationship of the initial rotational stiffness ratio to eccentric axial loading. Separately, Wang et al.^18,19 examined the influence of sheet material, thickness, and edge distance on the mechanical properties and failure mechanisms of self-piercing riveted joints under quasi-static loading conditions. Han et al.²⁰ highlighted that joint systems in latticed shells are subjected to the combined action of axial force and bending moment in practical engineering, and investigated the mechanical performance of joints under eccentric loading through experiment and numerical simulation. Furthermore, the eccentric bearing capacity and failure mechanism of prefabricated hub joints were derived, and these results with behaviors under pure axial compression and pure bending conditions were also systematically compared. In a related study, Fan et al.²¹ demonstrated that axial compression will increase the initial bending stiffness of joint. The investigation further revealed that compressive axial load produces differential effects on the bending capacity across joint types and accelerate the yielding phase in moment-rotation responses. These results verify that axial force substantially affects both the load-bearing capability and rigidity of prefabricated connections. Xiong et al.²² investigated the semi-rigid behavior of aluminum alloy wedge connections under combined bending moments and axial forces. Therefore, when studying the mechanical properties of new joint types, the combined effect of bending moment and axial force must be fully considered. Wang et al.²³ investigated the mechanical behavior of bolt-column joint under combined bending moment and axial force.

Hence, this study investigates the mechanical performance of IBC joint under the combined action of bending moment and axial force. Finite element simulations are conducted using ABAQUS package, considering the following variables such as joint plate thickness, loading condition, and load eccentricity, to reveal the failure mechanism of the joint under eccentric load. By varying the eccentricity, the load-displacement curves for the IBC joints under different combinations of moment and axial force are obtained, and a parametric analysis of eccentricity versus ultimate bending capacity is performed. Additionally, the influence of different axial force levels on the initial bending stiffness and moment capacity of the IBC joints under a constant bending moment is also examined.

Establishment of the finite element model for the IBC joint

Establishment of the joint model

As depicted in Figure 1, the IBC joint is an assembly consisting of a hollow column node, high-strength bolts, washers, and a conical head component. This joint is applicable for connecting H-section steel beams and box girders in practical structures. The conical head is fabricated from five steel plates, that is a front plate, an end plate, a web plate, and two flange plates. A key construction feature is that the conical heads are pre-welded to the ends of the member before arriving on site. Thereby, the connection to the column node is facilitated during assembly by installing four high-strength bolts, eliminating the need for any field welding. Figure 2 specifies the detailed dimensions of the connection, where both the flange and web share a common thickness t. The relevant material properties are listed in Table 1. The applicable standard is GB/T 700-2021, which primarily specifies the designations, technical requirements, test methods, and other relevant provisions for carbon structural steels. In addition, based on GB 50017-2017, the classification limits for joint stiffness are defined. A joint is classified as semi-rigid when its initial rotational stiffness $K_{i}$ satisfies $\frac{0.5 E I_{b}}{L_{b}} \leq K_{i} \leq \frac{30 E I_{b}}{L_{b}}$ .

Figure 1.

The IBC joint: (a) components of IBC joint, (b) a single IBC joint unit, and (c) IBC Joint assembly.¹

Figure 2.

Dimensions of IBC joint (mm): (a) top view, (b) 1-1 section, (c) 2-2 section, and (d) member section.

Table 1.

Material parameters of components.¹

Components	Material	Yielding strength	Ultimate strength	Elasticity modulus	Poisson’s Ratio $μ$
Components	Material	f_y (N/mm²)	f_u (N/mm²)	E (N/ ${mm}^{2}$ )	Poisson’s Ratio $μ$
Flanges and web	Q235 steel	304	465	207,848	0.33
Front and end plates		235	235	210,000	0.33
Beam		269	407	205,569	0.33
Bolts and gaskets	40Cr steel	975	1171	213,302	0.33
Column node	45Mn steel	355	355	210,000	0.29

A finite element (FE) model of a single column and its corresponding joint is developed in ABAQUS, accounting for both geometric and material nonlinearities. The analysis utilized a single-bar model, justified by geometric symmetry and the joint loading conditions. In parametric studies, the use of a simplified “single-strut model” can significantly improve computational efficiency, avoid the complex contact and boundary conditions associated with a full assembly, and facilitate variable control. Feng et al.²⁴ investigated the central ring-sleeve joint using a single-strut model with a fixed support. Yang et al.,²⁵ in their analysis of the mechanical behavior of bolted joints in single-layer reticulated shells, proposed a simplified mechanical model based on the component method and derived formulas for the initial rotational stiffness and yield moment of the joint. These studies demonstrate that simplified models can effectively capture the key mechanical characteristics of joints with acceptable accuracy. As depicted in Figure 3, at the column joint’s cross-section, a fixed support is implemented, restricting all translational degrees of freedom (X, Y, and Z, respectively). Tie contact interactions are defined for both the bolt-to-joint interfaces and the welded regions, encompassing the connections among the conical head’s five plates and that linking the endplate to the beam. Conversely, a surface-to-surface contact definition is applied to the bolt and front plate interfaces, utilizing a “Hard” contact model for normal behavior and a Coulomb friction model (with a coefficient of 0.3¹) for tangential behavior. According to GB 50017-2017, the four high-strength bolts in the model have a diameter of 24 mm, and the preload for each bolt is set to 225 kN accordingly. This preload is applied in ABAQUS using the “Bolt Load” function, which simulates the initial tensile stress generated during bolt installation. This approach is essential for accurately reproducing the initial contact state between the components at the joint and the subsequent load transfer path. The eccentric loading applied to the model is characterized by an eccentricity value e, representing the perpendicular distance from the reference point (RP) to the central axis.

Figure 3.

Load application and boundary constraints of the IBC joints.

Validation of IBC joint

To validate the finite element (FE) model developed in this study, an IBC joint model with a web and flange thickness of t = 6 mm is established. To gain the moment-rotation response and identify the failure mechanism, a pure bending moment is applied at the joint’s end. A comparative analysis is conducted against data from Xiao et al.⁹ As shown in Figure 4, the moment-rotation curve of the proposed model shows negligible discrepancy with that of the reference model. The main failure mode involved the development of a plastic hinge within the conical head region, which aligns well with findings documented in prior literature. From the stress contour under the ultimate moment shown in Figure 5, it can be observed that the failure mechanism of the model in this study is generally consistent with that reported in the reference literature under the same condition (pure bending). As indicated in the figure, under pure bending, significant yielding occurs at the two flanges of the joint. The middle plate experiences relatively low stress near the neutral axis, while more pronounced yielding is observed at both sides. Additionally, the bolts on the tension side do not exhibit obvious yielding when the plate thickness is relatively small. Consequently, the numerical model developed here reliably captures the IBC joint’s structural response.

Figure 4.

Validation of the finite element model.

Figure 5.

Finite element model validation based on stress contours.

Mesh sensitivity analysis

To perform a mesh sensitivity analysis and ensure that the simulation results are not affected by discretization errors, the influence of different mesh densities on the joint performance was investigated for the front plate and bolts in a joint with a plate thickness of 6 mm. Three mesh densities were selected for the front plate: 5, 6, and 7 mm; and three mesh densities for the bolts: 2.5, 3.5, and 4.5 mm. All other attributes of the joint, such as material properties, contact definitions, and boundary conditions, were kept unchanged. As shown in Figure 6, the comparison of the moment-rotation curves under different mesh densities reveals that the overall trends and data remain generally consistent, indicating that the various mesh densities can effectively capture the fundamental mechanical behavior of the joint. Based on these results, a mesh density of 6 mm was adopted for the front plate and 3.5 mm for the bolts in this study.

Figure 6.

Moment-rotation curves under different mesh densities: (a) mesh sensitivity analysis of high-strength bolts and (b) mesh sensitivity analysis of fore plate.

Analysis of the mechanical performance of IBC joints under eccentric force

To study the mechanical behavior of the IBC joint under eccentric loads, a series of 84 finite element (FE) models are developed. The study considers key parameters including the flange and web thickness t (i.e. 6, 8, 10, 12, 14, and 16 mm, respectively), and the load eccentricity e (i.e. 10, 20, 30, 40, 50, and 60 mm, as well as the pure bending case). The loading configuration is illustrated in Figure 7. Such loading conditions are common in practical engineering.

Figure 7.

Eccentric loading: (a) eccentric tension and (b) eccentric compression.

Eccentric tension

Figure 8 presents the moment-rotation curves of the joint under eccentric tension for different plate thicknesses and eccentricities, and Figure 9 shows the corresponding load-displacement curves. The curve corresponding to e = ∞ represents the moment-rotation relationship for the case of pure bending. In this study, the bending moment is generated by an eccentric load and is calculated as the product of the axial load and the eccentricity. The bending moment induces flexural deformation in the component, with its direction consistent with that of the eccentricity. The rotation refers to the angular displacement of the end section of the joint (the section between the back plate and the component) relative to its original position under the applied moment. The force refers to the axial load applied to the joint. The displacement refers to the linear displacement of the end section of the joint (the section between the back plate and the component) relative to its initial position under the axial load.

The numerically simulated curves exhibit a three-stage nonlinear behavior:

Elastic stage: This phase is characterized by an essentially linear relationship between moment and rotation (load and displacement). The slope of the curve corresponds to the joint’s initial stiffness. Throughout this stage, the joint experiences no plastic deformation, with all components—including the bolts—remaining in the elastic state. Consequently, any deformation incurred is fully recoverable once the load is removed.

Yielding stage: The yielding stage is initiated when the bending moment or load attains a critical value, leading to a nonlinear response characterized by a gradual reduction in slope. This deviation from linearity signifies the commencement of yielding and the development of discernible plastic deformation in localized parts of the joint or material. Despite this, the structural member maintains a degree of its overall bearing capacity.

Failure stage: In this stage, the slope of the curve progressively decreases until it plateaus and approaches zero, indicating that the moment or load has reached or is approaching its maximum value—the ultimate moment or ultimate load. Significant plastic deformation develops in the joint, and its load-carrying capacity peaks. This may be accompanied by local instability and buckling, leading to large deformations or fracture. After the ultimate capacity is attained, the curve may exhibit a descending trend. This post-peak descent represents a loss of load-carrying capacity, typically associated with the failure of the joint. The steepness of this descent is indicative of the component’s ductility. A sharp drop suggests poor ductility and brittle failure (e.g. bolt local buckling), whereas a gradual decline signifies good energy dissipation capacity and ductile behavior.

Figure 8.

Moment-rotation curves of the IBC joint under eccentric tension: (a) t = 6 mm, (b) t = 8 mm, (c) t = 10 mm, (d) t = 12 mm, (e) t = 14 mm, and (f) t = 16 mm.

Figure 9.

Load-displacement curves of the IBC joint under eccentric tension: (a) t = 6 mm, (b) t = 8 mm, (c) t = 10 mm, (d) t = 12 mm, (e) t = 14 mm, and (f) t = 16 mm.

It can be observed that the flange and web thickness significantly influences the joint’s load-carrying capacity. As shown in Figure 8, a greater thickness corresponds to a higher initial stiffness and a larger ultimate moment. The initial stiffness represents the joint’s resistance to rotational deformation under low load levels. This observed trend can be explained by the following factors: (Ⅰ) An increase in the moment of inertia, which is a key geometric parameter governing the joint’s bending stiffness. It is defined as the cross-section’s ability to resist bending deformation; (Ⅱ) enhanced material resistance to deformation. With greater plate thickness, the material exhibits a significant improvement in its capacity to withstand tensile and compressive stresses, thereby reducing its susceptibility to local deformation; (Ⅲ) mitigation of local buckling. When the plate thickness is small, local buckling is likely to occur in regions with concentrated loads, such as the joint interface. This phenomenon, characterized by localized instability, compromises the overall bending stiffness of the joint. Furthermore, the underlying mechanisms for the improvement in ultimate moment with increased plate thickness are multifaceted, including: (Ⅰ) Optimization of load-bearing distribution. An increase in plate thickness promotes a more uniform stress distribution within the joint, thereby mitigating stress concentration effects; (Ⅱ) enhanced plastic deformation capacity. With greater thickness, the joint can sustain a higher bending moment without immediate failure. The key advantage of thicker plates lies in their ability to provide increased plastic reserve capacity; (Ⅲ) reinforcement of the bolted connection zone. Thicker flanges and webs expand the effective bearing area for the high-strength bolts, thereby reducing the likelihood of material cracking or fracture caused by excessive stress in the bolts; (Ⅳ) The significant improvement in shear stiffness afforded by thicker plates results in enhanced stability of the joint when subjected to high bending moments. It is also observed that, at a constant plate thickness, increasing the load eccentricity enhances both the joint’s initial stiffness and its ultimate moment capacity. Examination of the curves shows that decreased eccentricity hastens the shift to plastic deformation, as the structural behavior becomes governed primarily by the marked axial tension under such loading. Meanwhile, the incremental gain in the ultimate load gradually decreases with each equal step of increased eccentricity. To illustrate, for an IBC joint featuring 6 mm thick flange and web plates, increasing the eccentricity from 10 to 20 mm raises the ultimate moment by 4.6 kN m, while an increase from 50 to 60 mm yields only a 3.2 kN·m improvement. This nonlinear scaling behavior stems from the fact that higher eccentricities modify the structural force path, accentuating stress concentration effects—notably in vulnerable areas such as bolt groups and slender plate segments—which in turn constrains the extent of achievable gains in ultimate bearing capacity.

As shown in Figure 9, an increase in the thickness of the flanges and web leads to a higher ultimate load and a significant improvement in the axial tensile stiffness of the joint. Based on the curves, the progression of the plastic region within the joint is gradual. Furthermore, the displacement can reach around 30 mm before significant load reduction occurs. This behavior occurs because, under eccentric tension, the bolts yield first as the load increases, resulting in a decrease in the slope of the curve. At this stage, although the system stiffness declines, the flanges and web remain within their elastic range, partially compensating for the loss of load-bearing capacity. Additionally, as the eccentricity increases, the maximum axial force that the joint can sustain decreases—meaning the ultimate load becomes smaller—while the deformation displacement of the joint increases.

The stress contour plots of the cone component and the high-strength bolts are selected at the loading stage corresponding to the ultimate moment of the joint. As is shown in Figure 10, the following observations regarding different flange and web thicknesses and eccentricities can be made: (Ⅰ) For flange and web thicknesses ranging from 6 to 10 mm, joint failure is primarily caused by the yielding of the flange and web plates. The relatively small plate thickness corresponds to inferior initial tensile/bending stiffness and ultimate tensile/bending strength. In this case, the high-strength bolts experience only localized yielding at the root without fracturing. As the plate thickness increases, a stress concentration zone gradually forms at the connection between the web and the front plate. At thicknesses of 12 and 14 mm, a significant stress concentration remains at the web-to-front plate junction, but the stress on the flange decreases noticeably. Concurrently, the yielded regions of the high-strength bolts expand, leading to widespread yielding and eventual fracture. Thus, the joint failure transitions to a combined mode involving buckling of the conical head section and fracture of the bolts. At a plate thickness of 16 mm, the stress in the conical head section is substantially reduced. Only small areas of the flange and web experience high stress, with minor buckling occurring. In contrast, the bolts exhibit a much larger yielded area than in previous cases, resulting in their fracture. Consequently, the joint failure at this thickness is dominated by bolt fracture, while the conical head itself remains intact; (Ⅱ) under small eccentricity, the joint behavior under eccentric tension closely resembles concentric axial tension. This results in uniformly high stress levels across the conical head, particularly near the junctions of the flanges, web, and front plate, leading to widespread yielding. As the eccentricity increases, the stress concentration zone within the conical head shifts toward the tension side (i.e. the side in the direction of eccentricity), while stress on the compression side decreases correspondingly. A similar stress redistribution occurs in the bolts: stress increases significantly in the bolts on the tension side but decreases only marginally on the compression side. However, this stress variation is much less pronounced in the bolts compared to the conical head section. (Ⅲ) From the stress contour plots, it can be observed that the joint undergoes plastic deformation prior to failure, forming a plastic hinge with limited rotational capacity. Local buckling occurs in the web, flange, and bolts. Based on the provisions regarding ductility in GB 50017-2017, it can be concluded that the IBC joint exhibits good ductility and is not susceptible to brittle failure.

Figure 10.

Von mises stress distribution in joints with different plate thicknesses under eccentric tensile loading: (a) t = 6 mm, (b) t = 8 mm, (c) t = 10 mm, (d) t = 12 mm, (e) t = 14 mm, and (f) t = 16 mm.

Quantification of the three stages of the curve

The three-stage division is illustrated in Figure 11, where $K_{i}$ is the initial rotational stiffness, representing the slope of the elastic segment, and $K_{j}$ is the yield stiffness. According to the component method in Eurocode 3, the yield of the joint depends on whether the flange and web or the high-strength bolts yield first. As indicated by the cloud maps, when the plate thickness is small, the flange and web yield first; when the plate thickness is large, the high-strength bolts yield first. This phenomenon also applies to the failure stage.

Figure 11.

Three failure stages of IBC joint.

In the study by Li et al.,²⁶ it was shown that before joint failure, the ultimate moment of the joint, denoted as M_u, is determined, which is equal to 10% of $K_{i}$ . The yield point is quantified using the bilinear model and the energy equivalence method as specified in GB50017. As shown in Figure 12, assuming the existence of a yield moment $M_{j} = a M_{u}$ (where a is the yield coefficient and $M_{u}$ is the ultimate moment), the value of a can be calculated using the energy equivalence method, ensuring that the area enclosed by the moment-rotation curve equals that enclosed by the bilinear model. Additionally, $K_{i}$ represents the initial rotational stiffness, and $θ_{u}$ represents the rotation corresponding to the ultimate moment. For joints with varying plate thicknesses, the yield stiffness can be quantified using this method.

Figure 12.

Bilinear model.

Based on the finite element model, the bilinear model characteristic parameters (a, $θ_{u}$ , $K_{i}$ , and $M_{u}$ ) of the IBC joint were obtained and are listed in Table 2.

Table 2.

Bilinear model characteristics of IBC joint.

t (mm)	e (mm)	a	θ_u (rad)	K_i (kN m/rad)	M_u (kN m)
12	10	0.940	0.004602	3233.9	6.08
12	20	0.893	0.008364	3443.2	11.85
12	30	0.886	0.013212	3699.4	17.04
12	40	0.881	0.016243	3884.5	21.66
12	50	0.907	0.022477	4034.2	25.86
12	60	0.912	0.027588	4158.5	29.71

Nonlinear fitting was performed using equation (1) for the yield coefficient (a), ultimate rotation ( $θ_{u}$ ), initial bending stiffness ( $K_{i}$ ), and ultimate bending moment ( $M_{u}$ ). The independent variable in the fitting formula was selected as the eccentricity e of the eccentric load applied to the joint. Model data from the joint with a plate thickness of 12 mm were used for the fitting, with the resulting coefficients listed in Table 3. The fitted curves were then compared with the original model data. As shown in Figure 13, the data points generally lie on the ideal fitting line, indicating that the fitted formula can accurately predict the key parameters of the bilinear model.

y = c_{1} e^{4} + c_{2} e^{3} + c_{3} e^{2} + c_{4} e + c_{5}

(1)

Table 3.

Table of fitted curve coefficients.

y	c₁	c₂	c₃	c₄	c₅
a	3.75e-8	−4.30e-6	1.74e-4	−3.97e-3	9.69e-1
θ_u	−1.00e-9	1.25e-7	−3.54e-6	4.50e-4	5.00e-4
K_i	1.09e-4	−1.95e-2	1.15e+0	−1.14e+0	3.14e+3
M_u	0.00e+0	1.67e-5	−2.82e-3	6.27e-1	−1.87e-1

Figure 13.

Comparison between actual values and fitted curves.

Figure 14 presents a comparison of the moment-rotation curves obtained from the finite element model and the fitted bilinear model. It can be observed that the bilinear model accurately captures both the stiffness and flexural capacity of the joint, demonstrating strong agreement with the numerical results.

Figure 14.

Comparison of the moment-rotation relationship between bilinear and actual models.

Based on the bilinear model, the yield stiffness can be quantified, which more clearly illustrates the yield stage of the joint.

Eccentric compression

Figures 15 and 16 respectively display the joint’s moment-rotation and load-displacement relationships under eccentric compressive loading. Under this loading condition, the joint exhibits a lower maximum axial force and smaller deformation displacement compared to eccentric tension. However, a consistent trend is observed that as the eccentricity increases, the ultimate load decreases while the deformation displacement increases. From the moment-rotation curve, it can be observed that the joint exhibits a relatively steep slope during the plastic stage, with its bearing capacity beginning to decline after reaching the ultimate limit. This behavior occurs because, under eccentric compression, the conical head section yields first. Figure 17 illustrates the stress distribution at the joint under eccentric compression. It can be observed that for smaller plate thicknesses, joint failure is primarily caused by the extensive yielding of the flange and web on the compression side of the conical head, where large-area high stress drives the section to converge toward a plastic limit. For larger plate thicknesses, the yielded area in the conical head reduces; instead, localized yielding in the bolts expands from the center toward the root regions. Consequently, the joint failure under these conditions results from combined localized yielding in both the conical head and the bolts. Furthermore, a larger eccentricity enlarges the stress concentration zone at the junction of the compression-side flange and web, and also increases the localized yielding area in the bolts on the tension side.

Figure 15.

Moment-rotation curves of the IBC joint under eccentric compression: (a) t = 6 mm, (b) t = 8 mm, (c) t = 10 mm, (d) t = 12 mm, (e) t = 14 mm, and (f) t = 16 mm.

Figure 16.

Load-displacement curves of the IBC joint under eccentric compression: (a) t = 6 mm, (b) t = 8 mm, (c) t = 10 mm, (d) t = 12 mm, (e) t = 14 mm, and (f) t = 16 mm.

Figure 17.

Von mises stress distribution in joints subjected to eccentric compression: (a) t = 6 mm, (b) t = 8 mm, (c) t = 10 mm, (d) t = 12 mm, (e) t = 14 mm, and (f) t = 16 mm.

Mechanical performance of IBC joints under eccentric force

Four additional joint models with eccentricities of e = 100, 200, 600, and 1000 mm, respectively, are established for the joint with a plate thickness of t = 10 mm to extend the parametric study. Figure 18 illustrates the variation in initial bending stiffness and plastic moment resistance for this joint. In the provided illustration, the notation $S_{j; ini}^{e} / M_{j; ini}^{0}$ corresponds to the ratio between the initial flexural rigidity subjected to eccentric compression and that in pure bending. Correspondingly, $M_{\sup}^{e} / M_{\sup}^{0}$ signifies the ratio of the plastic moment capacity under eccentric compression to its counterpart under pure bending. $e_{t}$ represents the eccentricity of the joint under eccentric tension, and $e_{p}$ represents the eccentricity of the joint under eccentric compression. The plotted data demonstrate that eccentric tensile and compressive loads exert comparable influence on plastic moment resistance, whereas their impact on initial bending stiffness is markedly distinct. Curve examination further identifies a two-phase mechanical response.

Stage Ⅰ (e < 200 mm): in this stage, the initial bending stiffness and moment capacity of the joint under eccentric loading differ significantly from those under pure bending, indicating that the eccentric force has a notable influence on the mechanical performance of the joint. Under tensile loading with eccentricity, the joint’s initial flexural rigidity decreases compared to the pure bending condition. Conversely, when subjected to eccentric compression, its initial stiffness exceeds that observed in pure bending. In both eccentric tension and compression, the moment capacity is lower than that under pure bending. However, as the eccentricity increases from 20 to 200 mm, the plastic moment resistance of the joint shows a clear increasing trend.

Stage Ⅱ (e ⩾ 200 mm): During this phase, the discrepancies in initial flexural stiffness and moment capacity between eccentrically loaded and purely bent joints substantially decrease, eventually approaching convergence. Consequently, an eccentricity of e = 200 mm is established as the critical threshold. Beyond this limit, the joint’s structural response closely aligns with its behavior under pure bending. The critical eccentricity of e = 200 mm is only applicable to the joint dimensions and materials used in this study.

In the small eccentricity stage, axial compression dominates, the bending moment is relatively small, and the joint deformation is primarily characterized by compression, resulting in a high rotational stiffness that decreases gradually. In the moderate eccentricity stage, the joint enters a critical contact state where partial separation occurs at the contact interface. The dominant action gradually shifts from axial compression to bending moment. During this stage, the abrupt change in contact conditions leads to the peak variation in rotational stiffness. In the large eccentricity stage, the bending moment dominates, the contact state becomes stable, and the variation in rotational stiffness tends to level off.

Figure 18.

Variations in initial stiffness and moment capacity of the joint under eccentric force.

Figure 19 shows the relative values of initial flexural rigidity and plastic moment resistance between eccentrically tensile and compressive conditions. The plotted trends reveal that for e < 200 mm, both properties are markedly reduced under tension relative to compression. Conversely, when e ⩾ 200 mm, the distinction in mechanical behavior between tensile and compressive loading diminishes progressively, with the two responses converging as eccentricity grows.

Figure 19.

Ratios of initial stiffness and moment capacity of the joint under eccentric tension and compression.

Furthermore, the eccentricity significantly influences the failure mode of the joint. The stress distribution in the joint and bolts is shown in Figure 20, from which the following observations can be made: (Ⅰ) When e < 200 mm, the influence of axial force is more pronounced under smaller eccentricities, leading to local buckling in both flanges and the web. In contrast, when e ⩾ 200 mm, the bending moment plays a dominant role as the eccentricity increases, while the effect of axial force diminishes. In this case, local buckling occurs mainly in the tension-side flange and the region of the web adjacent to the tension- or compression-side flange, with only minimal buckling area on the opposite flange; (Ⅱ) under eccentric compression, when e ⩾ 200 mm, the neutral axis of the web gradually shifts toward the centerline of the web. However, when e < 200 mm, the neutral axis deviates toward the tension-side flange; (Ⅲ) under eccentric tension, when e < 200 mm, localized yielding occurs in bolts on both the tension and compression sides. However, when e ⩾ 200 mm, yielding concentrates primarily in the bolts located on the tensile side under eccentric tension, whereas under compressive eccentric loading, it remains confined solely to the root regions of these tension-side bolts.

Figure 20.

Von mises stress distribution in joints under different eccentricity ratios: (a) eccentric tension and (b) eccentric pressure.

In designing joints under eccentric tension or compression, a primary focus is on how the ultimate bending moment relates to load eccentricity. To address this, supplementary finite element models were created based on prior parametric studies. These models incorporated eccentricities of 100 and 200 mm for both tensile and compressive eccentric loading, covering plate thicknesses from 6 to 12 mm. Through this analysis, a dimensionless correlation between ultimate bending capacity and eccentricity was derived under both loading types, intended to uncover underlying behavioral mechanisms and parametric interactions.

The fitted curves shown in Figure 21 can be expressed by the following functions:

(Ⅰ) Eccentric tension:

0.92 e^{- 0.98 η_{d}^{0.85}} + η_{M}^{d} = 1

(2)

(Ⅱ) Eccentric compression:

0.62 e^{- 0.85 η_{d}^{0.65}} + η_{M}^{d} = 1

(3)

where d denotes the eccentricity, and $η_{M}^{d} = \frac{M_{\sup}^{d}}{M_{\sup}^{0}}$ $η_{d} = | \frac{d}{e_{c}} |$ . Both formulas yield $η_{M}^{d} \equiv 1$ within the domain of $η_{d} > 1$ . The theoretical curves in the figure were derived from ABAQUS simulations of joints with four different plate thicknesses, with eccentricities ranging from 10 to 200 mm for each thickness. A nonlinear fitting was performed to establish the relationship between eccentricity and ultimate moment. The horizontal axis represents the ratio of the applied eccentricity to the critical eccentricity, while the vertical axis represents the ratio of the ultimate moment under eccentric loading to that under pure bending. Based on these formulations, designers can predict the ultimate bending capacity of IBC joints under varying eccentricities.

Figure 21.

Relationship between moment capacity and eccentricity.

Mechanical performance of IBC joints under combined bending and axial force

In contrast to joints subjected to eccentric loading, this configuration maintains a constant axial force magnitude throughout the loading process. The axial force remains aligned with the beam centerline and directed toward the joint center. Figure 22 presents the corresponding moment-rotation relationship. Figure 23 illustrates the stress distribution within the conical head section of the joint, and Figure 24 summarizes the variations in three key mechanical parameters under different axial force levels, that is initial bending stiffness, yield moment, and plastic moment capacity. In the figures, $S_{j; i n i}^{N (t / p)} / S_{j; i n i}^{0}$ denotes the ratio of the joint’s initial bending stiffness under axial tension or compression to that under pure bending; $M_{\sup}^{N (t / p)} / M_{\sup}^{0}$ represents the ratio of the ultimate bending capacity under axial tension or compression to that under pure bending; and $M_{y}^{N (t / p)} / M_{y}^{0}$ indicates the ratio of the moment at yield under axial force to that under pure bending. $λ_{t} = N_{t} / N_{u}^{t}$ , $λ_{p} = N_{p} / N_{u}^{p}$ ; $N_{u}^{t} = 1220 k N$ , $N_{u}^{p} = 1330 k N$ , signifies the ultimate bearing capacity under pure axial tension or compression.

Figure 22.

Moment-rotation curves of the joint under different axial forces.

Figure 23.

Loading diagram.

Figure 24.

Variation patterns of joint stiffness and bending capacity under combined axial and flexural loading

From the presented data trends, key findings emerge: (Ⅰ) Both tensile and compressive axial loads lead to a pronounced reduction in plastic moment capacity. Under low values of $λ_{t}$ and $λ_{p}$ , the joint’s ultimate bending strength approximates its performance in pure bending. (Ⅱ) the influence of axial tension and compression on the initial bending stiffness of the joint exhibits distinct characteristics. Under axial tension, the initial bending stiffness decreases progressively with increasing tensile force and remains consistently lower than that under pure bending. In contrast, under axial compression, the initial bending stiffness initially increases with rising compressive force when $N_{p} \leq 532 kN$ ( $λ_{P} \leq 0.4$ ). However, beyond this threshold ( $N_{p} > 532 kN$ , $λ_{p} > 0.4$ ), the stiffness begins to decline. Notably, for $N_{P} \leq 665 kN$ ( $λ_{p} \leq 0.5$ ), the initial stiffness under combined compression and bending remains higher than that under pure bending. Once $N_{P} > 665 kN$ ( $λ_{p} > 0.5$ ), the stiffness falls below the pure bending value and subsequently decreases sharply with further increases in compressive force; (Ⅲ) axial tension and compression exhibit a generally consistent influence on the yield moment of the joint. As the axial force increases, the yield moment under combined axial-flexural loading decreases significantly. In all cases, the yield moment of the joint remains lower than that under pure bending conditions. Under varying levels of axial tension and compression, the bending stress distribution of the joint is illustrated in Figure 25. The following observations can be drawn: under relatively low axial tension, the yielding pattern in the conical head section shows little difference from that under pure bending. Localized yielding is primarily concentrated at the junction between the web and the tension-side flange. As the axial tension increases, the yielded region within the conical head begins to propagate from one side of the web toward its central area. Under axial compression, an increase in the compressive load causes the neutral axis of the web to shift gradually from the web centerline toward the compression-side flange. Axial tension and compression exert significantly different effects on bolt stress. Under tension, the stress level in the bolts on the tension side increases compared to that under pure bending. In contrast, under compression, the bolt stress level decreases.

Figure 25.

Stress distribution in joints subjected to varying axial forces: (a) tension-bending coupling and (b) compression-bending coupling.

A nonlinear regression is applied to the ratio between the ultimate bending moment under combined axial tension/compression and that in pure bending. The resulting fitted function, illustrated in Figure 26, allows for the prediction of the IBC joint’s bending capacity across a range of axial load scenarios. The relationship derived is given as follows

\frac{3}{2} λ_{(t / p)}^{3} - \frac{1}{4} λ_{(t / p)}^{2} + λ_{(t / p)} + λ_{M}^{N} = 1

(4)

where $λ_{(t / p)} = \frac{N_{(t / p)}}{N_{u}^{(t / p)}}$ , $λ_{M}^{N} = \frac{M_{\sup}^{N}}{M_{\sup}^{0}}$ . When the axial force is small, the bending capacity of the joint differs only marginally from that under pure bending. It can be considered that for $λ_{(t / p)} \leq 0.1$ , $λ_{M}^{N}$ may be approximated as 1.

Figure 26.

Correlation between moment capacity and axial force.

Conclusion

This study systematically investigates mechanical properties and failure modes of IBC joints with a total of 108 FE models, and establishes predictive formulas between ultimate moment and eccentricity/axial force. The investigated parameters included the thickness of the flanges and web in the conical head section, the eccentricity of the applied load, and the magnitude of the axial force under combined axial-flexural loading. The following principal conclusions can be drawn from the parametric investigation:

(Ⅰ) The initial stiffness and ultimate bending moment of the joint increase with greater flange and web thickness. Additionally, under constant plate thickness, larger eccentricities in eccentric loading lead to higher initial stiffness and ultimate moment. As plate thickness increases, the stress distribution transitions from extensive yielding in the conical head to significant yielding at the bolt roots, reaching the plastic limit. Correspondingly, the failure mode shifts from pronounced yielding of the conical head to bolt failure and fracture. Excessively thick plates cause high-strength bolts to yield and fracture before the conical head. Therefore, selecting an appropriate plate thickness is critical. Both insufficient and excessive thickness adversely affect the joint’s ultimate bending capacity.

(Ⅱ) The critical eccentricity for the joint under eccentric load is determined to be e = 200 mm. When e ⩽ 200 mm, the mechanical behavior of the joint is significantly influenced by the eccentric load. Specifically, under eccentric tension, both the initial bending stiffness and ultimate moment capacity are lower than those under pure bending. In contrast, under eccentric compression, the initial bending stiffness exceeds that of pure bending, while the ultimate moment capacity remains lower. When e > 200 mm, the influence of eccentricity on the joint’s mechanical performance becomes negligible, and its behavior closely approximates that under pure bending conditions.

(Ⅲ) Through parametric analysis of how ultimate bending moment correlates with eccentricity under both tensile and compressive eccentric loading, this research examines the influence of varying plate thicknesses. It subsequently develops formula for predicting the ultimate bending capacity of IBC joints under different eccentricities.

(Ⅳ) Under combined axial and flexural loading, higher tensile forces substantially reduce both the joint’s initial flexural rigidity and its peak moment capacity. For compressive axial loads, the ultimate bending strength similarly diminishes as compression rises. Conversely, the initial stiffness initially rises with compressive force before eventually decreasing. Specifically, when the axial force satisfies $N_{P} \leq 665 kN$ ( $λ_{p} \leq 0.5$ ), the initial stiffness under combined compression and bending remains higher than that under pure bending. A fitted formula is derived to predict the influence of different axial force levels on the bending capacity of the IBC joint under combined axial and moment loading.

Footnotes

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by National Natural Science Foundation of China (No.51408490), Natural Science Basic Research Program of Shaanxi (2022JM-234), and Training Programs of Innovation and Entrepreneurship for Undergraduates (XN2025008139), are gratefully appreciated.

ORCID iDs

Yuxing Fei

Huijun Li

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