Bolt Pull-Out Failure Analysis and Structural Optimisation for Heavy-Duty Rod End Bearings via a Combined Numerical-Analytical Method

Abstract

Rod-end spherical bearings are widely used in heavy machinery, wind power, and transportation. Their bolted connections directly determine structural safety but are prone to pull-out failure under maximum articulation angle and heavy load. This study employs finite element (FE) simulation to elucidate the failure mechanism and, combined with Timoshenko beam theory, systematically analyses the effects of end cap parameters (size, height, modulus) on bolt head lateral force and bending moment. Results show that two-piece end caps induce abnormal contact and severe stress concentration under combined lateral and axial loads. A spigot design with optimised bolt number and contact geometry is proposed, reducing the additional bending moment from $1.882\times {10}^4$ N·mm to $2.193\times {10}^3$ N·mm and lateral load from $8.236\times {10}^5$ N to $7.092\times {10}^4$ N (over 88% reduction), bringing stress within a safe range. Although the numerical analysis was not directly verified experimentally, the experimental confirmation of the design’s functionality supports the optimisation. This study clarifies the pull-out mechanism and provides insight for anti-pull-out connections under high lateral forces.

1. Introduction

In modern machinery and precision equipment, reliable connection and efficient power transmission between moving mechanisms are core factors ensuring overall performance. Rod-end spherical bearings, featuring excellent shock resistance, high load-bearing capacity, and multi-axis rotation characteristics, are widely used in fields such as aviation, construction machinery, and wind power generation [1,2,3,4,5,6]. Most existing studies have focused on the performance and service-life issues of the bearing body under dynamic cyclic loads (e.g., wear and fatigue) [7,8,9,10,11], while research specifically targeting the special working condition of extreme static heavy loads remains relatively scarce. The deep-water deployment system developed in-house for this study also encounters this operational challenge: under low-speed, high-load static conditions, the rod-end spherical bearing must adjust the direction of the pulley to transmit the load from the steel wire rope. During the test, the rod-end spherical bearing did not exhibit the expected ball head wear; instead, typical failures occurred, including end cap and bolt fractures, as well as ball rod scratching (see Figure 1). Therefore, the strength of the overall bolted connection under this working condition has become the bottleneck restricting the reliability of the system.

It is noteworthy that the rod-end spherical bearing adopts a radial spherical joint structure without rolling elements, whose overall load-bearing capacity is highly dependent on the bolted connection between the spherical inner ring and the external mating ring. Bolted connections are widely used in various equipment types due to their simple structure and ease of assembly and disassembly [12,13,14]. However, their mechanical properties are significantly influenced by factors such as contact nonlinearity and uneven load distribution [15,16]. Therefore, under complex loading conditions, the precise characterisation and prediction of their mechanical behaviour remain a critical challenge in structural engineering. Generally, bolted connections are categorised as either tension-type or shear-type based on their resistance mechanism: the former primarily resists tensile forces acting along the bolt axis, while the latter withstands lateral forces perpendicular to it [17]. For flange-bolted connections, Liu et al. [18] employed finite element analysis to establish the relationship between system tightness, additional loads, and gasket residual stress. Their work confirmed that the additional bending moment generates significant bending stress in the bolts, thereby impairing the connection’s tightness. Huang et al. [19] explored the origin of prying force in flange connections, proposing a modified formula and revealing that prying force contributes to both bolt axial stress and an additional bending moment, which causes uneven stress distribution. This further highlights the additional bending moment as a decisive factor in the performance of flange bolted connections. In addition, regarding bolt loosening induced by lateral forces, Hu et al. [20] developed a finite element model incorporating the thread lead angle. Their work systematically revealed the evolution of contact conditions and residual preload during the loosening process, identifying load amplitude as the dominant factor influencing bolt loosening. Huang et al. [21] created a three-dimensional FE model of bolted angle steel connections in transmission towers, examining the effects of preload, bolt hole geometry, and lap surface area on the slip curve. Yamagishi et al. [22] proposed a slip displacement estimation formula for bolts of various dimensions and tightening forces, validated experimentally. Amir et al. [23] theoretically developed analytical models for bending moment and stress in concentric bolted connections under lateral forces, showing that preload relaxation causes bolt bending stresses, thus impairing connection stability. The aforementioned studies have examined the effects of lateral forces on bolt connection performance from multiple perspectives, whilst also considering bolt bending stresses arising from additional bending moments. Nevertheless, it is important to note that the bending stresses at the bolt head not only stem from these additional moments but are also significantly influenced by the lateral forces directly acting on the bolt head. Regarding connection structures with a spigot, previous studies have either examined their dynamic behaviour [24] or analysed the parameter sensitivity of their bending stiffness [25]. Nevertheless, there has been insufficient investigation into how the spigot and bolts interact under static heavy loads, especially concerning the lateral force distribution at the bolt head and its coupling with the additional bending moment. The rod-end spherical bearing under investigation, when subjected to oblique loading, exhibits a composite failure mode characterised by lateral-bending moment coupling. Bolted connection failure in this scenario completely compromises the bearing structure’s integrity. Therefore, understanding the failure mechanism requires a detailed examination of the critical bolted connection’s behaviour under static heavy loads, focusing specifically on the bolt head’s stress state—including its spigot configuration—under combined lateral and bending moments.

The FE method has been widely adopted and extensively developed in bolted joint research. Qian et al. [26] developed a three-dimensional high-fidelity FE model featuring a tapered thread for an M15 × 1.5 bolt to simulate preload forces. Comparison with experimental torque-tension data revealed that the FE simulations agreed more closely with experimental results than traditional theoretical calculations, thereby more accurately reflecting the bolt preloading process. Zhao et al. [27] utilised solid FE analysis to study the dynamic response of bolted joints in a pump-turbine head cover and stay ring, uncovering the characteristics of stress distribution and plastic deformation. Belardi et al. [28] introduced a computationally efficient modelling strategy for bolted connections, representing the bolt shank with beam elements and incorporating two spring elements to capture friction and clearance effects. This method achieved significant computational savings over solid models without compromising accuracy. Hence, the selection of an appropriate bolt modelling approach is dictated by the specific research aims and required precision. Current FE modelling strategies for bolted connections comprise four main types: solid bolt models, coupled bolt models, spider bolt models, and boltless models. Coupled and spider models, which utilise beam or line elements for bolt representation, offer superior computational efficiency relative to solid models; however, they are unable to deliver the level of detailed stress analysis achievable with solid elements [29,30]. Solid bolt models may be further classified into two types: detailed models incorporating thread geometry and smooth-shaft models without threads. The former offer higher fidelity in representing local bolt features but at a significantly increased computational cost. Since the present study aims to elucidate the effects of lateral forces and additional bending moments on bolts—with interest in the overall stress response rather than local thread stresses—the smooth-shaft solid model is chosen for the subsequent simulations.

Based on the above analysis, this study investigates the failure mechanism of integral bolted connections in rod-end spherical bearings under extreme static loads, emphasising the role of the bolt head in connection strength under combined lateral forces and additional bending moments. The research encompasses three main components: firstly, combining experimental failure observations with FE simulation to identify the failure mode and underlying cause of the bearing under extreme oblique tension; secondly, examining the stress distribution and failure mechanism of the bolt head under coupled loading, and evaluating the influence of key structural parameters on connection behaviour through a comparative analysis of Timoshenko beam theory and FE simulation; and finally, proposing design enhancements for reinforcement. VDI 2230-1 (2015), while providing coefficients for the ultimate sliding load in static friction for bolted connections, does not delve into the underlying mechanisms of its influence [31]. Consequently, this study not only offers specific guidance for failure-resistant design of rod-end spherical bearings under extreme conditions but also provides a mechanical analysis method for annular bolted connections under combined loading that can be extended to similar structures such as flanges and pressure vessels. This offers theoretical foundations and engineering references for their safety assessment and optimised design under extreme loading conditions.

2. Finite Element Modelling and Failure Analysis of Rod-End Spherical Bearings

2.1. Finite Element Modelling of Rod-End Spherical Bearings

To accurately analyse the actual performance of the rod-end spherical bearing, this study adopted its real structure as the prototype and established an FE model using the Ansys Workbench 2022 R1. The model was designed to replicate the actual extreme tension-lifting conditions, and it comprehensively considered key factors including external equivalent loads, bolt preload, structurally critical dimensions, and material constitutive relationships. Based on detailed two-dimensional design drawings, a three-dimensional (3D) assembly model of the rod-end spherical bearing was constructed in SolidWorks 2022. To improve computational efficiency while ensuring the accuracy of subsequent FE analysis, the 3D model was subjected to reasonable geometric simplification: key features affecting the mechanical response were retained, while non-essential details such as threaded holes and unnecessary fillets were removed. The simplified model, shown in Figure 2, consists of bolts, end caps, a base, and a ball rod. Its spherical joint can provide three degrees of freedom for rotation, which meets the attitude adjustment requirements of the connected pulley mechanism.

The material selection was guided by structural functionality requirements. The ball rod was fabricated from 40CrMo alloy structural steel, while the end caps and base were made of 40CrMnMo high-strength medium-carbon quenched and tempered steel. Both materials offer tensile strength, toughness, and fatigue resistance. Grade 8.8 M24 standard bolts were selected to ensure the reliability of the bolted connection under axial loading conditions. The main physical and mechanical properties of each material are presented in

Table 1. Materials and corresponding physical and mechanical properties.

Materials	Elastic Modulus (GPa)	Poisson’s Ratio	Yield Strength (MPa)	Tensile Strength (MPa)	Density (kg/m3)
40CrMo	210	0.3	980	1080	7850
40CrMnMo	210	0.3	785	980	7850
Grade 8.8 bolt	206	0.3	640	800	7850

.

Since the mechanical structure exhibited damage during testing and had reached the yielding stage, a bilinear isotropic hardening elastoplastic model was utilised to represent the material’s plastic response. The model features a linear elastic part and a plastic hardening part, the slope of which defines the tangent modulus. The tangent modulus for this study was computed using the plastic hardening stage formula proposed by Cha et al. [32], as follows:

$$E_s^P=0.005E_s$$

(1)

where $E_s^P$ is the tangent modulus and $E_s$ is the elastic modulus of the material.

Regular meshes enhance computational efficiency while ensuring the calculation accuracy of sensitive areas. In this simulation, a hexahedral mesh was adopted, as this mesh type ensures the stability of numerical calculations and the accuracy of the solution method [33], while enabling precise control over the mesh aspect ratio, Jacobian coefficient, and warping factor. The analysis utilised Solid185, an eight-node hexahedral element with three translational degrees of freedom at each node. The element’s ability to degenerate into pentagonal and tetrahedral forms facilitates mesh generation for complex geometries. It accounts for material nonlinearities, including plasticity and hyperelasticity, as well as geometric nonlinearities such as large deformation and large strain [34]. To validate mesh convergence, a local mesh independence analysis was first conducted on a simplified bolt connection model, as illustrated in Figure 3a. By refining the mesh on the bolt shank cross-section, comparisons of stress, sectional stress, and element/node counts were made across different mesh densities, with results presented in

Table 2. Mesh independence of the elastoplastic single-bolt connection model.

Element Size	2.32 mm	0.5 mm	1 mm
Number of nodes	15,231	70,571	222,300
Number of elements	12,616	62,676	202,210
Maximum Stress under 15,000 N (MPa)	623.67	662.17	634.3
Maximum Stress under 30,000 N (MPa)	648.02	648.49	647.08

. Balancing computational accuracy with available resources, a final element size of 2.32 mm was adopted, with local refinement applied to the bolt shank surface. Subsequently, HyperMesh 2022 was employed to mesh the rod-end spherical bearing, with stringent control over mesh quality: maximum aspect ratio of 4.06, maximum skewness of 0.59266, 0.1% of elements with Jacobian ratio below 0.7, and a minimum Jacobian ratio of 0.66. The overall model comprised 563,832 elements and 615,399 nodes.

To accurately simulate the actual load conditions of the FE model, the following boundary conditions were imposed based on the system’s working principles and operating conditions:

1. Rod load application: The load acting on the ball rod is derived from the pulley and roller to which it is connected, as shown in Figure 4. During system operation, the steel wire rope is routed around the pulley and roller, and the tension difference between the two sides and frictional effects generate a resultant force and frictional torque. These loads were ultimately applied to the connection interface of the ball rod in an equivalent manner. The calculation formula is given as follows:

$$\left\{\begin{array}{l}{F}_{3x}=F_{1x}+F_{2x}\\ F_{3y}=F_{1y}+F_{2y}\\ M_{rod}=M_{pulley}+M_{roller}-F_{1x}{L}_{1y}-F_{2x}{L}_{2y}+F_{1y}{L}_{1x}+F_{2y}{L}_{2x}\end{array}\right.$$

(2)

Where $F_{1x}$, $F_{1y}$, and $M_{pulley}$ denote the force components and frictional torque from the middle pulley; $F_{2x}$, $F_{2y}$, and $M_{roller}$ correspond to those from the right roller; and $F_{3x}$, $F_{3y}$, and $M_{rod}$ represent the force components and torque from the rod-end spherical bearing. The parameters $L_{1x}$ and $L_{1y}$ are the x- and y-distances from the pulley centre to the ball rod load application point, while $L_{2x}$ and $L_{2y}$ are the corresponding x- and y-distances from the roller centre.

The pulley mechanism is located 200 m horizontally from the shore winch at a water depth of 500 m. It is subjected to combined net buoyancy loads of 3 t (from a small buoy) and 10 t (from a floating body), while accounting for the sag effect of the intermediate wire rope (with a unit weight of 4.32 kg/m). The loads acting on the pulley centre are: force $1.0963\times {10}^5$ N in the x-direction, force $2.2124\times {10}^5$ N in the y-direction, and bending moment $1.3414\times {10}^6$ N·mm. The loads acting on the roller centre are: force $3.7184\times {10}^4$ N in the x-direction, force $-5.4348\times {10}^4$ N in the y-direction, and bending moment $-2.66974\times {10}^5$ N·mm. According to formula (2), the equivalent loads at point 3 (the ball rod position) in the diagram are: force $1.4682\times {10}^5$ N in the x-direction, force $1.6689\times {10}^5$ N in the y-direction, and bending moment $7.719\times {10}^7$ N·mm.

2. Frictional contact: All potential sliding interfaces are modelled as frictional contacts. The normal contact stiffness factor of 1 is specified, with automatic update during iterations and default penetration tolerance. The contact pairs include: the ball rod spherical surface and end cap spherical surface; the interface between the two end caps; the end cap–base interface; and the bolt–end cap interfaces. According to mechanical design guidelines, the friction coefficient for steel-on-steel contact is 0.15 under dry conditions and 0.1–0.12 under lubricated conditions [35]. As the bearing operates submerged in a lubricated environment, a friction coefficient of 0.1 is assigned to all frictional contacts.

3. Bolted connection: The contact areas between the bolt heads and the upper surfaces of the end caps, as well as between the bolt shanks and the smooth bolt holes in the end caps, were defined as frictional contact, while the connection between the threaded portions of the bolts and the threaded holes of the base was set as bonded contact.

4. Bolt preload: The preload of the bolts was set in accordance with the VDI 2230 standard to prevent bolt loosening. For the Grade 8.8 M24 alloy steel bolts adopted in this study, the applied preload value was 1.68 × 10⁵ N [31].

5. Fixed support: A fixed support was applied to the bottom surface of the base, constraining all its degrees of freedom to simulate the actual installation conditions.

2.2. Failure Modes and Structural Stress Analysis

FEA reveals the distribution of equivalent stress and equivalent plastic strain throughout all components. According to the yield criterion, yielding occurs when the equivalent stress exceeds the material’s yield strength or when the equivalent plastic strain becomes non-zero, resulting in permanent plastic deformation. If the equivalent stress continues to increase to the ultimate strength, fracture will eventually occur. As depicted in Figure 5a and Figure 6a, the FE results reveal that stress concentration and yield zones in the ball rod are concentrated within its contact interface with the compressed-side end cap. The equivalent stress here exceeds the material’s yield strength, indicating incipient cracking. This numerical observation is consistent with the compressive damage location identified in the experiment. Notably, additional pronounced stress concentration appears at the stress-bearing corner of the rod-end neck, where equivalent plastic strain is also localised. As shown in Figure 5b,c and Figure 6b,c, the compressed-side end cap displays markedly larger yield zones and higher yield stresses than the prying-side end cap. Yielding on the compressed side is localised at two key sites: the ball rod–end cap spherical contact, where equivalent stress exceeds the yield strength, and the bolt hole-bolt interface. These two high-stress concentrations are principally responsible for the increased fracture susceptibility of this end cap. In contrast, the prying-side end cap exhibits relatively small yield zones only around the bolt holes and at the spherical surface. As illustrated in Figure 5d and Figure 6d, localised yielding is observed in the base at the ball rod contact interface and around the compressed-side bolt holes. Figure 5e and Figure 6e demonstrate that although the prying-side bolts (Nos. 3–8) display larger yield zones than the compressed-side bolts (Nos. 1, 2, 9–12), the peak equivalent plastic strain occurs on the compressed side. Furthermore, contact pressure analysis at the end cap-to-bolt interface (Figure 7) shows higher pressures on the compressed-side bolts, identifying this locally excessive contact stress as a significant contributor to their elevated fracture susceptibility.

2.3. Load Transfer and Stress Distribution in the Bolt Group

2.3.1. Mechanical Analysis of the Bolt Group

From the stress contour diagram of the bolts shown in Figure 5e, it can be observed that the bolts were subjected to both axial tension and bending moment. This bending moment originated from the additional bending moment and lateral force borne by the bolt heads. Therefore, it was necessary to clearly determine the magnitude of the lateral force and additional bending moment at the head of each bolt. Analysis of the results presented in Figure 8 indicated that the load distribution within the bolt group was highly directional. The axial tensile force (y-direction) and the additional bending moment about the x-axis were the predominant components, while the lateral forces (x- and z-directions), the bending moment about the z-axis, and the torque about the y-axis were minor. When the rod-end spherical bearing was loaded, one end cap was pried upward while the other was pressed downward, with each end cap subjected to an overturning moment. Therefore, the prying-side end cap bolts (Nos. 3–8) carried higher overall axial loads than the compressed-side end cap bolts (Nos. 1, 10–12). However, the axial forces in the compressed-side bolts (Nos. 2 and 9) were greater than those in the prying-side bolts (Nos. 4–7). Within the same end cap, due to the torsional effect induced by the overturning moment and the longer lever arm, the bolts farthest from the end cap centre—such as the compressed-side bolts (Nos. 2 and 9) and the prying-side bolts (Nos. 3 and 8)—exhibited higher axial forces than the other bolts on the same side.

2.3.2. Stress Analysis of Bolts

Under the combined action of axial force and bending moment, the maximum normal stress on the bolt cross-section is determined by the superposition of the stress components induced by axial load and bending [36]. When the material response remains within the linear elastic range, the maximum sectional stress can be expressed as:

$${\sigma }_{\max }={\displaystyle \frac{M}{W}}+{\displaystyle \frac{T}{A}}$$

(3)

where ${\sigma }_{\max }$ denotes the maximum normal stress on the bolt cross-section, $M$ is the additional bending moment acting on the bolt, $W$ represents the section modulus in bending, $T$ corresponds to the axial force in the bolt, and $A$ is the cross-sectional area of the bolt.

When clamping connecting members, the bolt head constraint behaves as a vertical member under combined axial force, lateral force, and bending moment. Accordingly, this study adopts a simplified mechanical model for quantitative stress analysis, representing the bolt as a cantilever beam fixed at its base and loaded at its free end with axial force, lateral force, and a bending moment. The strength calculation formula for the fixed-end cross-section is given by:

$${\sigma }_{\max }={\displaystyle \frac{\sqrt{{\left(F_{Hx}L\pm M_z\right)}^2+{\left(F_{Hz}L\pm M_x\right)}^2}}{W}}+{\displaystyle \frac{T}{A}}$$

(4)

where $F_{Hx}$ and $F_{Hz}$ denote the lateral forces applied to the bolt head in the x- and z-directions, respectively, while $M_x$ and $M_z$ represent the additional bending moments about the z- and x-axes at the bolt head. The sign (positive or negative) is determined by whether the sense of the moment induced by the lateral force aligns with the direction of the corresponding additional bending moment, in accordance with the defined coordinate system. A matching sense is taken as positive, while an opposite sense is taken as negative.

To validate the theoretical model described above, a comparative finite element analysis was conducted based on the simplified linear-elastic mechanical model, as illustrated in Figure 9. In this model, a bolt preload of $1.6084\times {10}^5$ N was applied, and a lateral force of $30000$ N was introduced on the vertical plane of the end cap. The FE results indicated a lateral force of $1.4013\times {10}^4$ N and a bending moment of $4.8544\times {10}^5$ N·mm at the bolt head. To evaluate stress distribution, multiple cross-sections were selected along the bolt axis from the base joint surface to the bolt head, and their maximum equivalent stresses were extracted. A comparison between finite element results and theoretical solutions revealed that stress values in the middle smooth shank section of the bolt agreed well with theoretical predictions. Over the range of 10–66 mm, the maximum deviation of 6.64% was observed at the 10 mm location. However, near the base joint surface and the bolt head, stress singularities arising from local geometric discontinuities resulted in significant deviations between FE results and theoretical values. At the smooth shank surface, with an element size of 2.32 mm, the FE results showed a maximum deviation of 71.73% from the theoretical values. Hence, this model is appropriate for analysing nominal stress distributions within the connection region but is not intended to capture localised peak stresses. It is important to note that the theory is valid only for bolt stress solutions within the linear elastic range; once plastic deformation occurs, the model ceases to represent the actual bolt stress state. Nonetheless, the elastic solution serves as the fundamental criterion for assessing bolt yielding. For the geometric and loading parameters considered in this study, the model offers reliable engineering guidance for evaluating whether the material is susceptible to yield failure.

To investigate the stress distribution in the bolts of the original rod-end spherical bearing assembly, and considering the symmetry of both the structure and applied loads, bolts Nos. 1, 2, and 12 on the compressed-side end cap and bolts Nos. 3, 4, and 5 on the prying-side end cap were selected for stress analysis, as shown in Figure 10. Near the joint interface, the theoretical maximum stresses in the bolt shank cross-section differ notably from the FE results. This is attributable to two causes: firstly, the linear-elastic basis of the theoretical model, compared with the elastoplastic constitutive model used in FE analysis, which leads to overestimation by theory in yielded regions; secondly, stress concentration due to geometric discontinuities and high interface contact pressure, which elevates the FE values above the theoretical predictions.

The bending moment along the bolt shank displays a bimodal distribution, with maxima at the bolt–end cap and end cap–base interfaces. As illustrated in Figure 11a, the moment at the bolt head is primarily an additional bending moment, responsible for the peak bending stress. However, substantial bending stress also develops at the joint interface, influenced jointly by lateral forces and the additional bending moment. The maximum bolt bending moment arises from the superposition of lateral forces and additional bending moments, with the direction and magnitude of these forces being the primary controlling parameters. As shown in Figure 11b, the primary factors affecting bolt strength are contact pressure, bending stress, and axial force. Among these, contact pressure exerts the most significant influence, while the effect of bending stress is also considerable, comparable in magnitude to that of axial force. Among all bolts, Bolt No. 12 experiences the greatest stress concentration at the bolt–end cap interface. To enhance connection reliability, the following design measures are recommended: firstly, optimising the structural interface design to minimise contact pressure; secondly, reducing lateral forces and additional bending moments through structural modifications, thereby lowering the bending stress on the bolts.

In summary, FE analysis performed in ANSYS identified several critical risk zones in the rod-end bearing assembly of the adaptive pulley mechanism. These include the contact interface between the ball rod and the end cap, the bolted connection zones (including the bolt head and the joint interface), and the region surrounding the end cap bolt holes. These zones are prone to crack initiation and represent structural weak points. Further analysis reveals that bolt fracture is primarily driven by two mechanisms: first, the abnormal contact stress induced by relative sliding between the end cap and the bolt shank; second, the lateral force and additional bending moment applied to the bolt head. Therefore, to improve structural reliability, design optimisation should focus on two aspects: first, suppressing end-cap sliding through improved fitting methods and controlled bolt preloading; second, identifying and reducing the factors that induce excessive lateral force and additional bending moment on the bolt head.

3. Mechanism of Lateral Force and Additional Bending Moment at the Bolt Head

3.1. Theoretical Modelling of the Single-Bolt Connection

To systematically analyse the key mechanisms and factors influencing the lateral force and additional bending moment in the bolted connection, a simplified single-bolt analysis model was established, as illustrated in Figure 12a. In flange connections, lateral shear forces are typically balanced by frictional forces generated through bolt preload. To prevent bolt overload caused by increased preload, Yang et al. [37] introduced a spigot structure to carry the primary lateral force. Regarding bolt modelling, Yang et al. [38] treated bolts as cantilever beams, establishing a mechanical model for bending moments and axial forces. However, this equivalent beam model did not account for the influence of the spigot structure on the bolt load path. Therefore, this paper establishes a mechanical analysis model for bolts applicable to both spigot and spigotless connections. Building upon equivalent beam theory, the model incorporates the coordinated deformation relationship between the end caps and the bolts. It determines the distribution of external lateral forces between the end caps and the bolts, while analysing the influence of parameters such as end cap dimensions on the lateral forces and additional bending moments acting on bolts within spigot connections.

The single-bolt analysis model was simplified as follows. As shown in Figure 12b, the influence of the base was neglected, and the entire system was treated as fixed at the base support. The bolt and the end cap were each idealised as a cantilever beam with fixed ends, thereby forming a statically indeterminate structure. To facilitate the mechanical solution, the contact interaction at the joint interface between them was equivalently represented by a pair of interacting forces and moments. Consequently, the loads on the two components can be decoupled: the bolt primarily bears the lateral force and additional bending moment transmitted from the end cap, while the end cap is subjected to a concentrated force and a bending moment at its free end from the bolt, as well as the uniformly distributed loads along its span.

Classical beam theory rests on the assumption of small deflections, where beam displacement is negligible relative to its length. Geometric nonlinearity is omitted, and cross-sectional dimensions are considered constant throughout deformation. Euler–Bernoulli beam theory further invokes the plane cross-section hypothesis, postulating that sections normal to the neutral axis remain planar and normal after deformation. Shear effects are neglected, confining the theory to slender beams [39]. By contrast, Timoshenko beam theory considers lateral shear deformation and section rotation. Shear force is non-zero, and plane sections are no longer constrained to remain perpendicular to the neutral axis after deformation [40]. This renders the theory suitable for analysing short beams, composite beams, and beam structures under high-frequency excitation. Consequently, given the relatively large section height-to-span ratio of the end cap and bolt structure examined herein, shear deformation cannot be neglected. Timoshenko beam theory is therefore employed to determine the rotation angle and deflection under applied loads. At the bolt head interface, a distributed load acts on the end cap’s bearing surface. Assuming the bolt behaves as a cantilever beam, the formulae for calculating its rotation angle, deflection, and slope are as follows:

$$\left\{\begin{array}{l}\phi {\left(x\right)}_L={\displaystyle \frac{\left(2F_{2}Lx-F_2{x}^2+2M_{2}x\right)}{2E_L{I}_L}}\\ {{\omega }^{\prime }}_L\left(x\right)={\displaystyle \frac{\left(2F_{2}Lx-F_2{x}^2+2M_{2}x\right)}{2E_L{I}_L}}+{\displaystyle \frac{F_2}{k_L{G}_L{A}_L}}\\ \omega {\left(x\right)}_L={\displaystyle \frac{\left(3F_{2}L{x}^2-F_2{x}^3+3M_2{x}^2\right)}{6E_L{I}_L}}+{\displaystyle \frac{F_{2}x}{k_L{G}_L{A}_L}}\end{array}\right.$$

(5)

where $F_2$ and $M_2$ denote the lateral force and additional bending moment, respectively, acting on the bolt head; $E_L$, $I_L$, $L$, and $A_L$ represent the elastic modulus, second moment of area, length, and cross-sectional area of the bolt shank; and $k_L$ and $G_L$ correspond to the shear correction factor and shear modulus of the bolt material.

Under a uniformly distributed load applied to the end cap, idealised as a cantilever beam, the expressions for its rotation angle, deflection, and slope are given by:

$$\left\{\begin{array}{l}\phi {\left(x\right)}_D={\displaystyle \frac{\left(6F_{1}Lx-3F_1{x}^2+6M_{1}x+q\left(3L^{2}x+x^3-3x^{2}L\right)\right)}{6E_D{I}_D}}\\ {\omega }^{\prime }{\left(x\right)}_D={\displaystyle \frac{\left(6F_{1}Lx-3F_1{x}^2+6M_{1}x+q\left(3L^{2}x+x^3-3x^{2}L\right)\right)}{6E_D{I}_D}}+{\displaystyle \frac{F_1}{k_D{G}_D{A}_D}}+{\displaystyle \frac{q\left(L-x\right)}{k_D{G}_D{A}_D}}\\ \omega {\left(x\right)}_D={\displaystyle \frac{\left(12F_{1}L{x}^2-4F_1{x}^3+12M_1{x}^2+q\left(6L^2{x}^2+x^4-4x^{3}L\right)\right)}{24E_D{I}_D}}+{\displaystyle \frac{F_{1}x}{k_D{G}_D{A}_D}}+{\displaystyle \frac{q\left(2Lx-x^2\right)}{2k_D{G}_D{A}_D}}\end{array}\right.$$

(6)

where $F_1$ and $M_1$ denote the force and moment, respectively, acting on top of the end cap; $E_D$, $I_D$, and $A_D$ represent the elastic modulus, second moment of area, and cross-sectional area of the end cap, respectively; $k_D$ and $G_D$ correspond to the shear correction factor and shear modulus of the end cap; and $q$ is the uniformly distributed load acting on one side of the end cap.

Considering the structural continuity and deformation coordination, at the bolt head-to-end cap contact interface, when $x=L$, the deflection and rotation of the end cap and bolt are equal when the single-bolt connection is subjected to lateral loading, which can be expressed as:

$$\left\{\begin{array}{l}\phi {\left(L\right)}_L=\phi {\left(L\right)}_D\\ \omega {\left(L\right)}_L=\omega {\left(L\right)}_D\end{array}\right.$$

(7)

At the bolt head-to-end cap interface, the lateral forces on each part form an action-reaction pair. The bending moment sustained by the end cap consists of the bolt’s reaction moment plus an additional bending moment induced by the combination of bolt preload and end cap cross-sectional eccentricity. Therefore, the lateral force relationship between the bolt head and end cap, and the moment relationship, are expressed as:

$$\left\{\begin{array}{l}{F}_1=-F_2\\ M_1=-M_2+M_e\\ M_e=F_{Pre}{e}_0\end{array}\right.$$

(8)

where

$$M_e=F_{Pre}{e}_0$$

(9)

where $M_{\mathrm{e}}$ denotes the bending moment induced by the bolt preload with respect to the centroid of the end cap cross-section; $F_{Pre}$ represents the clamping force exerted by the bolt on the end cap; and $e_0$ denotes the eccentricity distance between the centroid of the bolt cross-section and that of the end cap cross-section.

By combining Equations (7) and (8), the lateral force acting on the bolt head under a uniform distributed load is derived as follows:

$$F_2={\displaystyle \frac{a}{b}}q$$

(10)

where $a$ and $b$ denote, respectively:

$$\left\{\begin{array}{l}a={\displaystyle \frac{L^3}{12\left(E_D{I}_D\right)}}+{\displaystyle \frac{L}{k_D{G}_D{A}_D}}\\ b=\left({\displaystyle \frac{L^2}{6\left(E_L{I}_L\right)}}+{\displaystyle \frac{L^2}{6\left(E_D{I}_D\right)}}+{\displaystyle \frac{2}{k_D{G}_D{A}_D}}+{\displaystyle \frac{2}{k_L{G}_L{A}_L}}\right)\end{array}\right.$$

(11)

For spigotless structures, the interface between the end cap and the base can only resist lateral force up to the maximum static friction. Consequently, the lateral force relationship differs from that in spigot structures. While lateral forces in spigot structures are still calculated using Equation (10), the lateral force on the bolt head in spigotless connections is given by:

$$F_2=\left\{\begin{array}{l}{\displaystyle \frac{a}{b}}q,\left|qL+F_1\right|<\mu F_{pre}\\ qL-\mu F_{pre},\left|qL+F_1\right|>\mu F_{pre}\end{array}\right.$$

(12)

Incorporating the lateral forces obtained above into the deformation compatibility Equation (7) gives the additional bending moments for both spigot and spigotless bolted connections, which are both expressed as:

$$M_2={\displaystyle \frac{\left(-3F_{2}L\cdot E_D{I}_D+\left(q{L}^2-3F_{2}L\right)\left(E_L{I}_L\right)+6M_e\cdot E_L{I}_L\right)}{6\left(E_L{I}_L+E_D{I}_D\right)}}$$

(13)

When a concentrated force is applied to the end cap at position $x=L$, the lateral forces and additional bending moments acting on the bolt heads for both spigot and spigotless configurations are, respectively:

1. For the spigot structure, the lateral force on the bolt head is:

$$F_2={\displaystyle \frac{c}{d}}{F}_3$$

(14)

where $c$ and $d$ denote, respectively:

$$\left\{\begin{array}{l}c={\displaystyle \frac{L^2}{12E_D{I}_D}}+{\displaystyle \frac{1}{k_D{G}_D{A}_D}}\\ d={\displaystyle \frac{L^2}{12E_D{I}_D}}+{\displaystyle \frac{L^2}{12\left(E_L{I}_L\right)}}+{\displaystyle \frac{1}{k_D{G}_D{A}_D}}+{\displaystyle \frac{1}{k_L{G}_L{A}_L}}\end{array}\right.$$

(15)

2. For the spigotless structure, the lateral force on the bolt head is:

$$F_2=\left\{\begin{array}{l}{\displaystyle \frac{c}{d}}{F}_3,\left|F_3+F_1\right|<\mu F_{pre}\\ F_3-\mu F_{yu},\left|F_3+F_1\right|>\mu F_{pre}\end{array}\right.$$

(16)

3. The additional bending moment for both structures is:

$$M_2={\displaystyle \frac{\left(-F_{2}L\cdot E_D{I}_D+\left(F_3-F_2\right)L\cdot E_L{I}_L+M_e\cdot E_L{I}_L\right)}{2\left(E_L{I}_L+E_D{I}_D\right)}}$$

(17)

Based on the beam model, the lateral force and additional bending moment at the bolt head depend on the bending stiffness of the bolt, the bending stiffness of the end cap, the bolt shank height, and the bolt preload. Among these parameters, the bending stiffness is determined by the material’s elastic modulus and the cross-sectional geometry (i.e., shape and dimensions).

3.2. Analysis of Factors Affecting Lateral Force and Additional Bending Moment at the Bolt Head

The end cap has dimensions of 60 mm × 60 mm × 70 mm, with the centre of its circular hole offset by 4 mm from the centre of the square. The base measures 60 mm × 60 mm × 100 mm. The bolt has a shank length of 100 mm and a diameter of 23.67 mm. The friction coefficient at the joint interface is 0.1, and the bolt preload is 1.68 × 10⁵ N. A uniform load was applied to the side of the end cap closest to the bolt. By varying the magnitude of this uniform load, the variations in the lateral force and additional bending moment at the bolt head were analysed for single-bolt connections both with and without a spigot structure.

As shown in Figure 13a,b, the comparison between the FE and theoretical model results indicates that both the lateral force and the additional bending moment at the bolt head increase with the applied external load. In contrast to the spigotless connection, the spigot structure significantly reduces the lateral force transferred to the bolt head, and this reduction becomes more pronounced as the load increases. In the theoretical model with sufficient bolt preload, the beam-model predictions are generally lower than the corresponding FE results for the spigot design; this deviation gradually increases with increasing load, while the proportion of the external load carried as lateral force by the bolt head remains below 10%. In the spigotless connection, once the lateral force at the contact interface exceeds the maximum static friction force, the remaining load is entirely transferred to the bolt head, causing a sharp increase in both its lateral force and additional bending moment. In summary, the bolted connection incorporating a spigot structure achieves a more favourable load distribution, thereby resulting in higher load-carrying capacity and improved structural reliability.

In spigotless bolted connections, the friction coefficient significantly influences joint slippage. This study investigates its effect on bolt head load using coefficients of 0.05, 0.1, and 0.15. As shown in Figure 14, the friction coefficient determines the point of abrupt lateral force change on the bolt head. Higher values suppress this instability, enabling spigotless structures to approach the load capacity of spigot-type connections.

The effects of end cap cross-sectional dimensions, section height, and preload on the mechanical performance of a single-bolt spigot connection were examined. Under uniform distributed loads of 12,000 N and 22,400 N, the resulting behaviour is presented in Figure 15.

As shown in Figure 15a,b, the results from both the beam model and FE analysis indicate that increasing the cross-sectional dimensions of the end cap effectively reduces the lateral force and additional bending moment at the bolt head; however, beyond a certain size, this downward trend levels off. Figure 15c,d demonstrate that increasing the elastic modulus of the end cap leads to a slight decrease in the lateral force and additional bending moment in both the beam model and FE analysis, although the overall variation remains limited. According to Figure 15e,f, increasing the thickness of the end cap contributes to reducing the lateral force and additional bending moment at the bolt head, but the reduction gradually diminishes with further increases in thickness. As shown in Figure 15g,h, within the beam model, the lateral force at the bolt head remains constant regardless of preload variation. Conversely, in the finite element model, this lateral force decreases slightly with increasing preload, though the magnitude of change is minimal. This discrepancy likely arises because the beam model operates under the assumption that the cross-section remains planar throughout, whereas in the solid model, the cross-section ceases to be planar when subjected to the compressive effect of preload. Furthermore, both models demonstrate that the additional bending moment increases with preload. However, compared to the beam model, the additional bending moment in the finite element model is less sensitive to the bolt preload.

In conclusion, the incorporation of a spigot structure effectively prevents the sharp increase in lateral force and additional bending moment at the bolt head that occurs when the lateral load at the contact interface exceeds the maximum static friction force. To further improve the load-carrying capacity and structural reliability of the bolted connection, the cross-sectional dimensions and thickness of the end cap can be appropriately increased.

4. Structural Optimisation of Rod-End Spherical Bearings Based on Failure Mechanism

To optimise the structural performance of the rod-end spherical bearing, an integrated redesign of its two-piece end cap was conducted. The original split-type end cap was consolidated into a single integrated structure. This integrated design effectively resolves several issues arising from uneven load distribution in the original assembly, including excessive lateral force on a single end cap, combined axial and lateral forces on the connecting bolts, and high contact stress induced by sliding between the end cap and the bolt shank. Additionally, a spigot structure and other relevant optimised features were incorporated into the design. To verify the effectiveness of the improvement scheme, a comparative study was established, as illustrated in Figure 16. By comparing these models, the influence of different structural schemes on the mechanical performance and stress distribution of the rod-end spherical bearing can be systematically evaluated.

To quantitatively assess the structural performance improvement of the end caps, a comparative analysis of key mechanical indicators was conducted. This analysis included the maximum lateral force and additional bending moment at the connecting bolts, the peak stress in the bolt stress-concentration zones for different end-cap designs, and the maximum stresses at various cross-sections along the bolt shank. The corresponding results are presented in Figure 17 and Figure 18.

As shown in Figure 17, the integral spigot end cap structure exhibits a higher load-bearing capacity than the original integral structure. The maximum additional bending moment, lateral force, and axial force experienced by the most heavily loaded bolt in the improved end cap structure are the lowest among all comparison schemes. Compared to the original model, the maximum lateral force on a single bolt in the improved design decreased from $1.882\times {10}^4$ N to $2.193\times {10}^3$ N, representing a $88.35\%$ reduction. The additional bending moment decreased from $8.236\times {10}^5$ N·mm to $7.092\times {10}^4$ N·mm, a $91.39\%$ reduction. The axial force decreased from $2.232\times {10}^5$ N to $1.695\times {10}^5$ N, a $24.10\%$ reduction. Moreover, the enhanced load-bearing capacity of the improved model relative to the integral spigot structure stems primarily from the increased number of bolts (from 6 to 20), which reduces the load per bolt. Further analysis shows that non-monolithic end caps require a single cap to carry the full prying force or pressure arising from the ball rod’s relative horizontal inclination, exposing the bolts to combined high lateral forces, additional bending moments, and axial forces. Accordingly, non-monolithic end cap configurations should be avoided under severe combined lateral and axial loading conditions.

As shown in Figure 18, both theoretical calculations and finite element analysis results indicate that stress concentration is most pronounced in the bolts of the original model. Furthermore, a large volume of these bolts has entered the plastic strengthening stage. The substantial deviation between the theoretical values and finite element results can primarily be attributed to the combined effects of significant bolt yielding, stress concentration at the joint interface, and contact pressure between the bolt shank and the end cap bore. As illustrated in Figure 18b–d, the theoretical values for the central smooth shank section closely match FE results, as most stresses remain elastic. In the integral end cap, the bolt shanks exhibit uneven stress distribution, with a theoretical maximum of 584.63 MPa. For the flanged end cap, while some bolt shanks display relatively uniform stress and reduced bending, others retain uneven bending stress and significant inter-bolt variation, yielding a theoretical maximum of 624.66 MPa. The modified end cap markedly reduces stress disparities between bolts and promotes uniform stress distribution, with a theoretical maximum of 438.40 MPa.

In summary, the modified end cap not only effectively reduces the lateral forces and additional bending moments acting on the bolts but also achieves a more uniform stress distribution across all bolts. This significantly narrows the stress differential between bolts, with the maximum theoretical stress decreasing to 438.40 MPa, thereby substantially enhancing the load-bearing capacity of the bolted connection.

In addition to the incorporated spigot feature, the overall structure of the improved rod-end spherical bearing is illustrated in Figure 19. Its optimised design primarily manifests in five aspects: connection reinforcement, structural strengthening, contact optimisation, motion constraints, and material modification. Their respective functions are as follows:

(1)

Connection reinforcement: improving lateral load resistance

The core of this optimisation is load path reconfiguration through increasing the number of bolts from 12 to 20. This achieves two objectives: reducing individual bolt loads and increasing interface normal pressure to enhance static friction, allowing the friction interface to carry the majority of lateral forces. Consequently, the lateral forces on the new flange structure are significantly reduced, providing fundamental load-bearing capacity for the overall system under demanding conditions. The increased bolt count, however, introduces additional manufacturing complexity and component costs, requiring careful consideration of cost-performance trade-offs in engineering practice.

(2)

End cap structural reinforcement: enhancing stiffness

The core of this optimisation is geometric reconfiguration to improve the end cap’s shear resistance and suppress additional bending moments on the bolts. Countersunk holes increase the end cap thickness, enhancing shear capacity without extending the bolt shank’s shear length. Enlarged radial cross-sections simultaneously elevate bending stiffness. These combined measures mitigate end cap rotation under lateral forces, reducing bolt bending stresses and thus protecting the bolted connection.

(3)

Contact configuration optimisation: reducing peak contact stresses

The core of this measure is coordinated geometric scaling to improve contact performance. Simultaneous increases in the ball head radius, end cap socket radius, and base spherical radius enlarge the spherical contact area, optimising stress distribution and suppressing localised peak contact stresses. Furthermore, increasing the radius of the rod’s load-bearing section while appropriately shortening its length boosts the rod’s bending stiffness, mitigating stress concentration in this region.

(4)

Kinematic constraints and protection: mitigating edge contact risk

The core of this measure is a kinematic constraint designed to avert localised plastic failure. By enlarging the spherical bearing’s rated swing angle and introducing a supplementary rod-end pressure plate-to-end cap contact interface, the ball rod maintains surface contact with the end cap under extreme conditions, avoiding detrimental edge contact. This eliminates edge compression-induced stress concentration and plastic damage, thereby enhancing the structure’s resilience under severe loading.

(5)

Enhanced material performance: balancing load-bearing capacity and cost

This optimisation enhances the strength threshold of critical components at the material level. Upgrading key load-bearing parts—such as the ball rod, end caps, and pressure plates—to high-strength alloy steel 35MoCr (${\sigma }_{\mathrm{s}}=835$ MPa, ${\sigma }_{\mathrm{b}}=985$ MPa) directly improves the structure’s overall load-bearing capacity. Meanwhile, substituting the base material with 45 steel (${\sigma }_{\mathrm{s}}=355$ MPa, ${\sigma }_{\mathrm{b}}=600$ MPa) effectively controls manufacturing costs while meeting the base’s performance requirements.

In summary, the five enhancements each address distinct priorities: connection reinforcement and end cap strengthening enhance load-bearing capacity; contact optimisation improves stress distribution; kinematic constraints mitigate extreme-condition risks; and material modification elevates material performance utilisation. Among these, enhancements (1), (2), and (4) contribute significantly to overall rigidity and load-bearing capability, while enhancement (3) alleviates localised contact stresses between the rod and base. Enhancement (5) serves as a complementary engineering measure.

The equivalent stress distribution across key components within the equivalent plastic strain region after optimisation is illustrated in Figure 20 and Figure 21. The maximum stress at the base is 284.35 MPa, while that at the ball rod is 130.74 MPa. Both values remain below the yield strength of their respective materials, thereby satisfying the strength design requirements. Although minor yielding occurred in localised regions of the pressure plate and end cap, the modified structure has been successfully deployed in new buoyancy pull-down tests, achieving the required objectives without detachment or bolt failure. Simulation and experimental results indicate that, despite localised plastic strain and stress concentration, the current configuration retains sound engineering reliability. Subsequent studies may aim to refine the design by targeting stress concentration in these specific areas.

5. Conclusions

Based on experimental investigation and FE analysis of the faulty rod-end bearing, this study identified the critical regions of each component of the faulty rod-end spherical bearing, with particular focus on the strength of the end cap and bolted connection. By integrating FE simulation and Timoshenko beam theory, the influencing factors and variation patterns of the lateral force and additional bending moment at the bolt head were analysed. Subsequently, the structural design of the rod-end spherical bearing was optimised. The main conclusions are as follows:

• Load-carrying behaviour and failure modes of bolts: The lateral force and additional bending moment acting on the bolt head significantly increase the bending stress at the under-head fillet and the connection interface, thereby reducing the overall load-carrying capacity of the bolted joint. When the lateral force at the connection interface exceeds the maximum static friction of the joint, the load is transferred primarily to the bolt head, generating a larger additional bending moment. A further increase in lateral force causes the end cap to slip and press against the bolt, producing concentrated contact stress that accelerates connection failure.
• Influence of the spigot structure on load distribution: Incorporating a spigot structure effectively reduces the lateral force and additional bending moment transmitted to the bolt head. Furthermore, increasing the cross-sectional size and the height of the neutral plane in the connected components can further diminish the load acting on the bolt head. These findings provide a theoretical foundation for the design of bolt-connection structures with enhanced lateral load-carrying capacity.
• Integrated enhancement strategy for the rod-end spherical bearing design: Optimising the contact interface between the ball rod and the end cap of the rod-end spherical bearing reduces the risk of end-cap fracture and ball rod surface damage. Employing a larger ball head diameter helps prevent yielding of the base. Combining a keyway with a widened axial section of the end cap effectively suppresses excessive lateral force and additional bending moments. Utilising a counterbore design to increase the end-cap thickness enhances its shear capacity, while increasing the number of bolts improves load-distribution uniformity and reduces the load on individual bolts. Collectively, these measures provide a systematic theoretical reference and a practical technical approach for the design and application of rod-end spherical bearings under heavy-load conditions.

This study is primarily concerned with the functional application of rod-end spherical bearings, and therefore adopts simplifications including static loading, constant bearing articulation angles, and a fixed friction coefficient. These idealised assumptions introduce certain limitations. While the improved structure has been experimentally validated for practical use, systematic verification of the underlying theoretical model remains outstanding. Future work should include experimental validation of the static mechanical model to confirm its accuracy and applicability, as well as fatigue performance analysis under normal service loads to assess the structure’s long-term reliability.