Abstract
As the core component of a swivel bridge, the spherical hinge structure exhibits complex mechanical behavior under high pressure, making it challenging to describe accurately through theoretical analysis or on-site monitoring. In this context, this paper systematically examined the mechanical characteristics and key design parameters of a simplified spherical hinge structure using numerical simulations and formula derivation. Additionally, an indoor scaled test was conducted based on a representative spherical hinge structure to further investigate its mechanical properties. The findings indicated that contact pressure along the spherical hinge interface increased radially outward and followed a cubic distribution pattern. The support radius was identified as the most critical factor influencing the maximum stress value within the spherical hinge structure. In the aperture ratio range of 0.042 to 0.083, the upper and lower spherical hinges are unable to achieve coordinated deformation under high pressure, resulting in negative pressure at the edge of the pin hole. Furthermore, for spherical hinge structures with sliders, the presence of the slider introduced additional complexity to the stress state, leading to a coexistence of tension and compression on the surface of the lower spherical hinge. The highest stress occurred between the inner ring slider and the pin hole, an area that requires careful monitoring. The research provides valuable insights for the design of spherical hinges in swivel bridges and can guide the development of spherical hinge structures tailored to different bridge types in the future.
Keywords: Swivel bridge, Spherical hinge, Mechanical characteristics, Design parameter, Scaled model
Subject terms: Civil engineering, Computational methods
Introduction
In recent years, with advancements in bridge construction technology, the rotational construction method has become an increasingly prominent technique in the field1–3. At the heart of a swivel bridge, the spherical hinge structure plays a crucial role. It is primarily composed of an upper and lower spherical hinge, along with essential accessories such as the pin shaft and sliders4–6. Serving as the central component connecting the superstructure and lower bearing platform, the spherical hinge structure is integral to the central support system. Its main function is to evenly transfer the load of the superstructure to the lower bearing platform7,8. During construction, the rotation speed of the spherical hinge structure is precisely controlled to adjust the bridge’s rotational movement9–12.
The mechanical behavior of the spherical hinge structure under high-pressure is complex, and extensive research has been conducted by scholars both domestically and internationally13–16. Liu et al.17 proposed a critical overturning moment model for the concrete spherical hinge in the rotation process of bridges. The ground stress was calculated using the non-Hertz contact theory, and the model conclusions were verified through on-site monitoring. The results indicate that the non-Hertz method has superiority in predicting overturning resistance. Parsons et al.18 analyzed and solved the displacement of ball hinge under stress based on displacement method. Liu et al.19 used the Box-Behnken design method to design the response surface method (RSM) and derived an optimized design scheme for the spherical hinge structure. By comparing the structural contact stress and rotational traction force before and after optimization, the effectiveness and necessity of the optimization of the spherical hinge structure were verified. The results show that after optimization, the maximum contact stress and rotational traction force of the spherical hinge structure decreased by 13.86% and 8.42% respectively compared to before optimization. Li et al.20 obtained the optimal friction coefficient of the spherical hinge using the finite element analysis method, and evaluated the lubrication and friction performance of the spherical hinge structure through experiments conducted with self-developed tests. The finite element analysis results indicate that the optimal rotational friction coefficient of the spherical hinge structure is 0.031 to 1.131. The experimental results show that the friction coefficient of the graphene lubricating coating is 0.065, the wear scar diameter is 0.79 mm, and the maximum non-slip load is 426 N. Kim and Lee21 focused their research on spherical hinges, investigating the effects of frictional force and traction. The results of the study indicate that the results from strength theory are slightly lower than those from finite element calculations, as they did not consider the influence of displacement and eccentricity of the spherical hinge. Frictional resistance has a significant impact on the stress of spherical hinges. Che22 derived the formula for calculating the vertical compressive stress on the contact surface of a spherical hinge by using the concentrated force theory of half-space boundary action, and clarified the characteristics of stress distribution. Wei23 built a fine finite element model of the rotating structure through finite element analysis software, carried out static analysis on it, and determined the stress distribution on the contact surface of the spherical hinge.
Although extensive research has been conducted on the stress characteristics of spherical hinge structures24–27, a unified conclusion regarding the stress distribution patterns has yet to be reached. Current theoretical analyses are relatively simplistic and do not accurately capture the stress state of spherical joint structures. Therefore, this paper focuses on the swivel bridge over the Beijing-Harbin Railway on Xingfu Avenue in Changchun. Through a combination of numerical simulation and theoretical analysis, it characterizes the mechanical behavior of the spherical hinge structure and validates the effectiveness of both the numerical simulation and commonly used theoretical formulas using field data. Based on this validated simulation approach, the study then analyzes the impact of key design parameters on the stress state of the spherical hinge. Finally, a scaled model of the prevalent spherical hinge + slider structure is developed to further investigate the stress characteristics. The detailed process is illustrated in Fig. 1.
Fig. 1.
Flow chart of this paper.
Actual swivel bridge descriptions
Swivel bridge structure
The target bridge in this study is the Xingfu Road Swivel Bridge located in Changchun City, Jilin Province. This swivel bridge crosses the Beijing-Harbin Railway, one of China’s busiest lines. The bridge features a 50 m + 50 m prestressed concrete box girder with variable height in a T-shaped rigid frame configuration, measuring 100 m in total length and weighing 12,157.5 tons. During construction, the pile foundation, turntable, and piers are first completed. Brackets are then erected along the railway direction, followed by concrete pouring for the main girder. Afterward, the beams are rotated into their designated positions, and the turntable is sealed with concrete. Finally, the brackets are repositioned to construct the cast-in-place section of the side span. The bridge rotates at a designed speed of 0.02 rad/min over an 81.6°, taking approximately 71.2 min. A structural schematic of the actual swivel bridge is shown in Fig. 2.
Fig. 2.
Swivel bridge over the Beijing-Harbin Railway on Xingfu Avenue.
Spherical hinge structure
The spherical hinge structure consists of two discs with matching curvature, each measuring 3900 mm in diameter and 40 mm in thickness. Made of Q345 steel, the spherical hinge is divided into upper and lower sections, forming the core of the rotation system in horizontal rotation construction. The upper spherical hinge connects to the upper turntable, which has a diameter of 1080 mm and a height of 80 mm, while the lower spherical hinge is embedded in the lower bearing. A structural schematic of the spherical hinge is shown in Fig. 3.
Fig. 3.
Spherical hinge structure diagram.
Test and method
Finite element numerical simulation
According to the actual size of the target bridge, the “Optistruct” module of “HyperWorks 2017” simulation software is used to model and analyze the spherical hinge structure. To ensure calculation accuracy, the simulation model dimensions match the design specifications. Additionally, in line with the engineering requirements and analysis objectives, certain auxiliary components, such as positioning pins, sliders, and feet, are excluded from the model.
Spherical hinge model establishment
Following the design dimensions, this study established four components for the spherical hinge: the lower bearing platform, lower spherical hinge, upper spherical hinge, and upper turntable. Four established component models are shown in Fig. 4. Additionally, to facilitate solution convergence and maintain calculation accuracy, finer meshing is used near the pin in the spherical hinge, while coarser meshing is applied in areas farther from the pin. The model includes Penta6 and Hex8 solid elements, totaling 177,988 elements.
Fig. 4.
Spherical hinge structure model.
Material properties definition
Based on engineering design requirements, Q345 steel is chosen as the material for the upper and lower spherical hinge, while C50 concrete is used for the upper turntable and lower bearing platform. Detailed material parameters are provided in Table 1.
Table 1.
Material parameters of spherical hinge model Q345 steel.
| Material | Poisson’s ratio | Elastic modulus (MPa) | Density (KN/m3) | Coefficient of friction |
|---|---|---|---|---|
| Q345 | 0.3 | 2.06E + 05 | 7850 | 0.06 |
| C50 | 0.2 | 34,500 | 2200 | N/A |
Contact behavior settings
As a critical component of the horizontal swivel bridge, the spherical hinge requires accurate simulation of the contact state between the upper and lower spherical hinges. The quality and accuracy of this contact simulation are crucial for the success of the overall structural numerical analysis. Based on previous research, this study uses penalty friction for tangential contact behavior, a method commonly used in finite element software. For normal behavior, hard contact is specified, with a friction coefficient of 0.06.
Load and boundary constraints
The ideal model assumes that the spherical hinge structure only carries the vertical load transmitted from the upper beam, with no unbalanced moments. In accordance with engineering design requirements, the vertical load from the upper beam is converted into a pressure load of 1.33 MPa, applied to the upper turntable. Additionally, a node set is created at the bottom node of the lower bearing platform to constrain its displacement and rotational degrees of freedom. The lower bearing platform only supports the load transmitted from the upper structure.
Moreover, this study examined the impact of four key design parameters of the spherical hinge structure on its stress state: the curvature radius, support radius, pin hole radius, and pin hole length. The dimensions and explanations of these design parameters for the target bridge are shown in Fig. 5.
Fig. 5.
Schematic of key design parameters.
Additionally, a control variable method is used to design multiple sets of analysis schemes to explore the influence of four key design parameters on mechanical properties of spherical hinge structure, as outlined in Table 2.
Table 2.
Four key design parameters and actual dimensions.
| Influencing factors | Value range(mm) | Actual size(mm) | |||||
|---|---|---|---|---|---|---|---|
| Curvature radius | 4000 | 6000 | 8000 | 10,000 | 12,000 | ∞ | 10,000 |
| Support radius | 450 | 1200 | 1950 | 2200 | 2700 | 3450 | 1950 |
| Pin hole radius | 0 | 300 | 450 | 600 | / | / | 150 |
| Upper spherical hinge pin hole length | 510 | 610 | 810 | 910 | / | / | 710 |
| Lower spherical hinge pin hole length | 525 | 625 | 825 | 925 | / | / | 725 |
Theoretical formula derivation
In simplified mechanics theory of spherical hinge, the frictional force along the tangent direction of the hinge disk primarily affects the frictional torque and rotational traction. This frictional force is largely determined by the friction coefficient and contact pressure, making the contact pressure at the spherical hinge interface a critical factor in analyzing the hinge disk’s force distribution. The contact pressure at this interface aligns with the normal stress generated by the radial forces on the spherical hinge interface. This study derives and analyzes the contact pressure of the spherical hinge interface using an engineering simplification algorithm, the normal distribution force on the boundary of a half-space body, and Hertz contact theory, with the derivation’s accuracy verified through the aforementioned finite element analysis.
Engineering simplification algorithm
Assuming that the contact surface of the spherical hinge is a flat disk, the radius is R, and the upper transfer load is F, the contact pressure of the spherical hinge interface can be calculated by formula (1), which is also a calculation method generally accepted in engineering.
 |
1 |
Normal distributed force on the boundary of a half-space body
Assuming that the uniformly distributed normal load q acts on the circular area with radius a as an example, the stress at any point on the spherical hinge structure can be solved by superposition method. For the stress at any point on the Z axis, the load area can be divided into differential rings, as shown in Fig. 6.
Fig. 6.
A half-space body is subject to a normal distribution of forces at the boundary.
According to previous studies, the stress distribution of spherical hinge structure is shown in formula (2)28.
 |
2 |
Hertz contact theory
Hertzian theory is a theory that describes the contact and pressure distribution of elastomers, where when two elastic bodies come into contact, a pressure is created in the contact area. The distribution of these pressures is usually not uniform, but has a certain shape, as shown in Fig. 7. Hertzian theory generally applies to contact between a ball and a plane, or between two spherical objects, and contact between a cylinder and a plane.
Fig. 7.
Contact diagram of two elastomers.
Some relevant studies have applied Hertz theory to the spherical hinge structure, and its formula is shown in (3).
 |
3 |
Where, qmax is the pressure at the center of the contact area and also the maximum compressive stress, ρsinψ is the distance between any point inside the contact area and the center of the contact area, and a is the radius of the prototype contact area.
Laboratory scaled model test
Assembly of spherical hinge model
According to the structural model of the indoor rotating spherical hinge, the steel structure of the spherical hinge is designed, the main structure of the spherical hinge is processed, and the material is consistent with the actual engineering, as shown in Fig. 8.
Fig. 8.
Indoor rotary ball hinge structure.
To ensure the precision of model processing and reduce the number of model hoisting times, a reasonable pouring and assembly sequence is adopted to construct the indoor rotating spherical hinge structure. The specific assembly process is divided into four steps. The assembled spherical hinge structure is shown in Fig. 9.
Fig. 9.
The assembled spherical hinge structure model.
-
Step 1
Connect the lower ball hinge to the support through 4 positioning bolts, and level the horizontal direction. Then the foam plate is laid, the lower ball hinge surface is waterproofed, and the lower ball hinge and the support structure are inverted.
-
Step 2
Install positioning screws, traction screws, and stiffening transverse partitions according to the positions of the reserved holes in the pier column template. In addition, high-strength screws are used on both sides of the template, and steel wire binding is used between the positioning screws.
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Step 3
The pier formwork is fitted with the upper ball hinge, and asbestos is laid on the joint surface, and solid glue is applied for waterproof treatment, and then concrete is poured. The loading pier column is divided into two pouring molding, the first pouring to drown the circumferential floor of the upper ball hinge, and then the pressure sensor is laid after the vibration, coating and health preservation have a certain strength. Second pouring to the top of the loading pier column formwork.
-
Step 4
After all the concrete health care is completed, the dimensional accuracy is measured, and the support and lower ball hinge are screwed with the positioning steel plate after meeting the design accuracy requirements. Then install the positioning pin, apply lubricating oil, install the upper ball hinge and load pier column. Finally, it is lifted to the loading table to fix the positioning plate and the loading table.
Measuring point arrangement of spherical hinge structure model
In this paper, high precision foil resistance strain gauge is selected. Considering that the temperature of ordinary soldering iron will be particularly high, which will lead to defects in the strain gauge during welding, the purchased models are solderless with terminals. The strain gauge selected in this paper is BX120-5AA, the base material of the strain gauge is phenolic aldehyde epoxy, the wire grid material is imported Constantan, and the applicable temperature is -30℃~60℃. In order to obtain the strain data of the indoor rotating spherical hinge structure model, the strain gauge was pasted according to the method shown in Fig. 10.
Fig. 10.
Measurement point distribution of lower ball hinge surface.
Spherical hinge loading test
In this paper, a multi-functional hydraulic servo loading system is adopted to carry out hierarchical loading, which can realize monotone loading, low-cycle reciprocating loading, pseudo-dynamic loading, hybrid analog loading and other loading forms, as shown in Fig. 11. The multi-functional hydraulic servo loading system is used to load the rotating spherical hinge structure model in 5 stages, loading up to 122KN, followed by a set of limit loading up to 150KN. The loading test adopts force control mode and the loading speed is 15KN/min.
Fig. 11.
Spherical hinge model loading test.
Results and discussion
Numerical simulation calculation results
The displacement cloud diagram of the spherical hinge structure is shown in Fig. 12.
Fig. 12.
Displacement cloud diagram of the spherical hinge structure.
As can be seen from Fig. 12, the upper turntable is the part with the largest average deformation, and the largest deformation occurs at the edge of the upper turntable, which is also consistent with the actual situation. The maximum deformation solved by OptiStruct finite element analysis algorithm is 30.61 mm, which indicates that the displacement solved by OptiStruct finite element analysis algorithm has high similarity.
The interface contact pressure cloud diagram of the spherical hinge structure is shown in Fig. 13.
Fig. 13.
The interface contact pressure cloud diagram of the spherical hinge structure.
From Fig. 13, it can be observed that the distribution of contact pressure increases radially from the inside to the outside, which is significantly different from the ideal uniform distribution in engineering design. In this paper, the contact pressure of the interface between the upper and lower spherical hinges is extracted, and the relationship between the contact pressure of the spherical hinge interface and the distance from the center of the spherical hinge is obtained, as shown in Fig. 14.
Fig. 14.
Contact pressure distribution of the spherical hinge interface.
From Fig. 14, it can be seen that the contact pressure ranges from − 0.413 to 25.11 MPa. Through fitting the contact pressure of the interface between the spherical hinge and the distance from the center of the hinge, it is found that the interface contact pressure has a significant correlation with the distance from the center of the spherical hinge, exhibiting a cubic function distribution.
The deformation of the upper and lower spherical hinges is magnified by 2000 times, as shown in Fig. 15.
Fig. 15.
Deformation diagram of the upper-lower spherical hinge magnified by 2000 times.
Under high pressure, the upper and lower spherical hinges undergo distortion, with minimal deformation around the pinhole, presenting a “W” shape with smaller deformations at the center and edges and larger deformations in the middle, which differs from the ideal arc surface. The vertical displacement of the upper spherical hinge at the pin hole to be 0.4218 mm, and the vertical displacement of the lower spherical hinge to be 0.4295 mm. The vertical displacement of the lower spherical hinge is greater than that of the upper spherical hinge, which also explains the occurrence of negative contact pressure at the edge of the pin hole.
The vertical stress cloud diagram of the lower spherical hinge is shown in Fig. 16.
Fig. 16.
The vertical stress cloud diagram of the lower spherical hinge.
From Fig. 16, it can be observed that the vertical stress of the lower spherical hinge structure increases radially from the center, with the maximum vertical stress occurring at the circumference, at around 20 MPa, meeting the compressive requirements of high-strength steel.
Given that the actual structure is simplified in the numerical simulation, its effectiveness requires validation through field data. This study performs numerical simulations based on three completed swivel bridges: the Dongfeng Road Interchange Swivel Bridge in Siping City, the Ziqi Road Interchange Swivel Bridge in Siping City, and the Huixin Road Railway Interchange Swivel Bridge in Dehui City, as is shown in Fig. 17.
Fig. 17.
Numerical simulation of actual rotating bridge.
The effectiveness of the simulation is verified by comparing the results with test data from these sites. Relevant data are presented in Table 3.
Table 3.
Validation of the Numerical Simulation Method.
| Engineering | Measuring point | Measured value (MPa) | Numerical simulation value (MPa) | Numerical simulation error (%) |
|---|---|---|---|---|
| Dongfeng Road | 1 | 17.72 | 18.16 | 2.48 |
| 2 | 19.11 | 20.31 | 6.28 | |
| 3 | 19.54 | 20.54 | 5.12 | |
| Ziqi Road | 1 | 13.38 | 13.67 | 2.17 |
| 2 | 13.79 | 14.18 | 2.83 | |
| Huixin Road | 1 | 1.43 | 1.55 | 8.39 |
| 2 | 1.39 | 1.52 | 9.35 | |
| 3 | 1.96 | 1.82 | -7.14 | |
| 4 | 0.98 | 1.05 | 7.14 | |
| 5 | 1.37 | 1.47 | 7.30 |
As can be seen from Table 3, the calculation results of numerical simulation have high accuracy, and the error between calculation and measurement is between 2.17% and 9.35%. In addition, it should be noted that the reason for the small measured pressure value of the Huixin road bridge is that its upper structure is not yet completed and its load is small.
Theoretical formula analysis results
Contact pressure calculation results on spherical hinge interface based on four theoretical analysis method in Table 4.
Table 4.
Contact pressure calculation results on spherical hinge interface.
| Algorithm type | Distance from the center of the spherical hinge (mm) | ||||||
|---|---|---|---|---|---|---|---|
| 150 | 450 | 750 | 1050 | 1350 | 1650 | 1950 | |
| Simplified algorithm | 10.18 | 10.18 | 10.18 | 10.18 | 10.18 | 10.18 | 10.18 |
| The half space body is subject to uniform force | 2.27 | 2.30 | 2.36 | 2.46 | 2.59 | 2.76 | 2.97 |
| Hertz theory | 27.62 | 27.62 | 27.62 | 27.62 | 27.62 | 27.62 | 27.62 |
| Numerical simulation | -0.156 | 0.441 | 2.169 | 5.024 | 8.778 | 12.45 | 24.98 |
According to Table 4, there are significant differences between the interface contact pressures obtained by the simplified algorithm, the algorithm considering uniform loading on the half-space body, and Hertz theory compared to numerical simulations. The interface contact pressure obtained by the algorithm considering uniform loading on the half-space body increases radially from the inside out along the radius, which is consistent with the distribution form obtained from numerical simulations. However, the interface contact pressure calculated by this algorithm is much smaller than the simulated values. This is because in the model of the half-space body subjected to uniform loading, the half-space body is assumed to be constrained by a plane in one direction and infinite elastic in other directions, with radial stress mainly dependent on depth. However, actual spherical hinge structures have boundaries. The maximum interface contact pressure calculated by Hertz theory for the spherical hinge is 27.62 MPa, and it remains almost unchanged within a finite support cord length, with an error of 10.57% compared to the maximum pressure value obtained from numerical simulations. This is because, according to the previous formula derivation, the maximum pressure stress occurs at the center of the contact surface in Hertz theory. Since the curvature radii of the upper and lower spherical hinges are very close and the length of the support cord (mn) used for calculating the contact pressure at different positions of the spherical hinge is much smaller than the radius (a), it leads to the pressure values within the range of the spherical hinge interface being equal to the maximum pressure stress. In summary, the simplified algorithm and the algorithm considering uniform loading on the half-space body differ significantly from the finite element results. The simplified algorithm tends to be conservative, and its use in engineering may lead to deviation in results. The algorithm considering uniform loading on the half-space body can effectively describe the distribution form of interface contact pressure in spherical hinge structures, while Hertz theory can simply predict the maximum pressure stress on the interface of spherical hinge structures.
The interface contact pressure obtained from numerical simulations is relatively close to the actual situation, and it follows a cubic distribution with respect to the distance from the center of the spherical hinge. In order to further obtain the traction force between the rotating body and the spherical hinge, it is assumed that the contact between the upper and lower spherical hinges is a surface contact. Under the action of upper loads, the contact pressure at each node on the interface of the spherical hinge is denoted as p0. At a horizontal radius r from the vertical line passing through the center of the sphere on the interface of the spherical hinge, a small area dl × ds is taken, where dl represents the small length of the arc in the horizontal direction at the small area, and ds represents the small length in the tangent direction to the curvature radius R on the spherical hinge surface at the small area. Then, the total frictional torque is calculated as shown in formula (4).
 |
 |
 |
4 |
Where µ is the friction coefficient of the spherical hinge interface, with a value of 0.06 as adopted in this study. P0 represents the contact pressure at each node on the spherical hinge interface, and its fitted relationship with the distance to the axis is denoted as 
. R stands for the curvature radius of the spherical hinge interface, which is 10,000 mm in this project. γ is the angle between the curvature radius of the spherical hinge passing through the axis boundary and the vertical centerline passing through the sphere center, with a value of γ = arcsin0.015 in this project. β is the angle between the curvature radius of the spherical hinge passing through the support radius and the vertical centerline passing through the sphere center, with a value of β = arcsin0.195 in this project.
The formula for calculating the simplified traction force of the rotating body in the engineering context is as shown in formula (5).
 |
5 |
Where R is the radius of the spherical hinge plane, R = 1950 mm; W is the total weight of the rotating body, W = 121575KN; D is the diameter of the turntable, D = 10,800 mm; µ is the friction coefficient of the spherical hinge, µ = 0.06.
This paper calculates the simplified traction force of the rotating body as 878KN by substituting relevant data, while the finite element-derived traction force is 1093KN. The relative error between the two is 24.49%. It is observed that the simplified algorithm tends to yield conservative results, potentially leading to deviation when applied in engineering. For this project, two sets of four ZLD200 hydraulic, synchronous, automatic continuous traction systems are installed on both sides of the turntable. The maximum traction force on one side can reach 800t, approximately 8000KN, meeting the theoretical requirement for traction force.
Laboratory scaled model test results
Through the loading test on the spherical hinge structure model of the rotating body, the real-time stress data of the lower spherical hinge was obtained and plotted, as shown in Fig. 18.
Fig. 18.
The stress values of each measuring point of the ball hinge surface under different loads.
As shown in Fig. 18, the data acquired at various measurement points have both positive and negative values, indicating that the lower spherical hinge undergoes a complex force state combining tension and compression under loading. This is jointly caused by the squeezing between sliders and the extension of the spherical hinge under loading. The stress on the lower spherical hinge increases continuously with the increase of load, indicating a certain correlation between stress on the lower spherical hinge and the load. Among them, the measurement point with the greatest stress variation is QM22, with a stress of 2.709 MPa under a 150KN load, located in the region between inner sliders and the pin pre-boring hole.
The stress at the position where the hinge surface of the lower ball is in contact with the slide block is large, especially in the area where the side of the slide block is in contact with stress concentration. The stress around the reserved hole of the pin is small, which also shows that the pin mainly plays the role of positioning in the whole structure. The regional stress between the outer ring slider and the outer edge of the lower ball hinge surface is small, and the influence of external load is small, which belongs to the non-significant region of the measuring point. By extracting the compressive stress data of the lower ball hinge surface, it can be found that the lower ball hinge is not in an ideal state of complete compression, except for the area in contact with the slider, the whole is in a state of tension, which may be due to the local deformation of the lower ball hinge after the load is transferred to the lower ball hinge through various dispersed sliders.
Under the action of load, the overall surface shape of the lower ball hinge structure is distorted, and the local subsidence occurs, which leads to the coexistence of tension and compression stress on the lower ball hinge surface. The maximum contact pressure appears on the edge of the slider installation reserve, and all point to the inner position of the ball hinge axis, and the maximum contact stress of the outer ring is always greater than the contact pressure of the inner ring. This is because under the action of self-weight and external load, the sliding block is subjected to the shear force pointing to the central axis, and the farther the outer ring sliding block is from the central axis, the greater the shear stress pointing to the central axis. In conclusion, the stress distribution on the lower spherical hinge is more complex compared to other positions, closely related to the direction of patches, and shows minor variations.
Influence analysis of design parameters on spherical hinge
Influence analysis of curvature radius on spherical hinge
To explore the stress distribution of the spherical hinge structure under different curvature radii, this paper selected spherical hinge structures with curvature radii of 4000 mm, 6000 mm, 8000 mm, 10,000 mm, 12,000 mm, and infinite. Following the calculation method described earlier, the stress states of the spherical hinge structures were computed. The interface contact pressure contour maps for the six curvature radii of spherical hinges are shown in Fig. 19.
Fig. 19.
Interface contact pressure contour maps of spherical hinges with different curvature radii.
From Fig. 19, it can be observed that the distribution of contact pressure at the interface of spherical hinges with six different curvature radii is basically the same, increasing radially from the inside out. The maximum contact pressure occurs at the circumference of the spherical hinge interface and increases continuously with the curvature radius. When the curvature radius is 4000 mm, the maximum contact pressure at the spherical hinge interface is 23.53 MPa, while for an infinite curvature radius (i.e., a flat surface), the maximum contact pressure at the spherical hinge interface is 27.24 MPa, which is 15.77% higher. Within the range of curvature radii from 4000 to 8000 mm, the contact pressure at the spherical hinge interface is positive, with the minimum contact pressure occurring at the edge of the pin hole and decreasing gradually with the increase in curvature radius. When the curvature radius of the spherical hinge exceeds 1000 mm, negative pressure appears at the edge of the pin hole and increases with the curvature radius. This may be due to the lower spherical hinge being embedded in the lower bearing pedestal and constrained by the surrounding concrete, resulting in the upper and lower spherical hinges being unable to achieve synchronous coordinated deformation. The closer the spherical hinge interface is to a plane, the more noticeable the bending deformation of the lower spherical hinge at the edge of the pin hole, leading to negative pressure at the edge of the pin hole. This paper extracts the contact pressure data at the interface of spherical hinges with six curvature radii, as shown in Fig. 20.
Fig. 20.
Interface contact pressure of spherical hinges with different curvature radii.
From Fig. 20, it can be observed that the contact pressure at the interface of spherical hinges with six different curvature radii follows a similar trend with the distance from the center of the spherical hinge. The maximum and average contact pressures at the spherical hinge interface gradually increase with the increase in curvature radius. In other words, the closer the spherical hinge interface is to a plane, the greater the friction between the upper and lower spherical hinges. According to the formula, the traction forces required for the rotation of spherical hinge structures under six different curvature radii are calculated as 961KN (4000 mm), 1052KN (6000 mm), 1080KN (8000 mm), 1093KN (10000 mm), 1105KN (12000 mm), and 1143KN (infinite, i.e., flat). This implies that the larger the radius of the pin, the greater the traction force required for rotation. The spherical hinge structure with a curvature radius of 4000 mm can save 16% of traction power compared to a flat spherical hinge structure.
The Von Mises stress contour maps for spherical hinge structures with six curvature radii are shown in Fig. 21.
Fig. 21.
The Von Mises stress contour maps of the upper and lower spherical hinge structures.
From Fig. 21, it can be seen that the Von Mises stress distribution patterns of spherical hinge structures with six different curvature radii are essentially similar, all decreasing radially from the inside out, with the maximum Von Mises stress occurring at the edge of the pinhole and continuously increasing with the curvature radius. When the curvature radius is 4000 mm, the maximum Von Mises stress in the spherical hinge structure is 23.53 MPa, and when the curvature radius is infinite, i.e., flat, the maximum Von Mises stress at the spherical hinge interface is 27.24 MPa, which is relatively increased by 15.77%. The phenomenon of stress concentration at the edge of the hole can also be explained by the theory of elasticity, which considers stress concentration at the edge of the hole as a local phenomenon. Due to the presence of the hole, the stress and deformation states near the hole are completely different from those without a hole. Beyond several times the diameter of the hole, the stress is hardly affected by the hole, and the stress distribution and magnitude are almost the same as those without a hole. In summary, the curvature radius of the spherical hinge interface has a significant effect on the stress state of the spherical hinge structure. With the increase in the curvature radius of the spherical hinge interface, the phenomenon of stress concentration at the edge of the pinhole becomes more pronounced, and the average contact pressure and maximum contact force at the spherical hinge interface also increase accordingly. The traction force required for the rotation of the spherical hinge also gradually increases.
Influence analysis of support radius on spherical hinge
To explore the stress patterns of spherical hinge structures with different support radii, this study selected spherical hinge structures with six support radii of 450 mm, 1200 mm, 1950 mm, 2200 mm, 2700 mm, and 3450 mm, respectively. According to the calculation method described earlier, the stress states of the spherical hinge structures were determined. The contour maps of contact pressure at the spherical hinge interface for the six support radii are shown in Fig. 22.
Fig. 22.
Contour maps of contact pressure at the spherical hinge interface with different support radii.
From Fig. 22, it can be observed that the support radius has a significant impact on the contact pressure at the spherical hinge interface. As the support radius increases, the maximum contact pressure at the spherical hinge interface decreases accordingly. When the support radius is 450 mm, the maximum contact pressure at the spherical hinge interface is 545.4 MPa, exceeding the yield strength of Q345 steel. Therefore, the spherical hinge structure with a support radius of 450 mm cannot meet the actual load requirements. Within the range of support radii from 1950 to 2700 mm, negative pressure appears at the edge of the pinhole, indicating that the upper and lower spherical hinges cannot achieve coordinated deformation under the action of the load. Combining the previous discussion on the radius of the pinhole, this study defines the ratio of the pinhole radius to the support radius as the aperture ratio ( f ). By fitting the relevant data, it is found that when (0.042 < f < 0.083), the upper and lower spherical hinges cannot achieve coordinated deformation under the action of the load, and negative pressure is likely to appear at the edge of the pinhole. This paper extracts data on the contact pressure at the spherical hinge interface with different support radii, as shown in Fig. 23.
Fig. 23.
Contact pressure at the spherical hinge interface with different support diameters.
From Fig. 23, it can be observed that as the support radius increases, the curve of the contact pressure at the spherical hinge interface tends to flatten with the distance from the center of the spherical hinge. According to the formula for calculating traction force, the traction forces required for spherical hinge structures with six different support radii of 450 mm, 1200 mm, 1950 mm, 2200 mm, 2700 mm, and 3450 mm are calculated as 1117KN, 1351KN, 1092KN, 1122KN, 1035KN, and 1020KN respectively. It can be seen that the traction force required for the rotation of the spherical hinge structure does not show a clear pattern with increasing support radius. This may be due to the increase in contact area at the spherical hinge interface with increasing support radius, leading to a decrease in average contact pressure at the spherical hinge interface. With the interacting effects of contact pressure and contact area, the pattern of traction force variation with support radius is not evident.
The Von Mises stress cloud diagrams for spherical hinge structures with different support radii are shown in Fig. 24.
Fig. 24.
Von Mises stress cloud diagrams for upper and lower spherical hinge structures.
From Fig. 24, it can be observed that within the range of aperture ratio ( f ) from 0.028 to 0.0625, stress concentration occurs along the edge of the reserved hole for the pin shaft, and as the support radius increases, the Von Mises stress values along the edge of the reserved hole for the pin shaft gradually decrease. When the aperture ratio is 0.228, the maximum Von Mises stress value of the spherical hinge structure is 367.3 MPa, which exceeds the material’s allowable stress of 345 MPa, indicating structural failure. The maximum Von Mises stress occurs at the outer edge of the spherical hinge structure, and there is no stress concentration phenomenon at the edge of the reserved hole for the pin shaft. When the aperture ratio is 0.022, the Von Mises stress value at the edge of the reserved hole for the pin shaft is 6.553 MPa, and the Von Mises stress value in the middle region of the hinge plate is 6.913 MPa. The stress concentration phenomenon at the edge of the reserved hole for the pin shaft is significantly weaker than that in the middle region of the hinge plate. It can be seen that stress concentration at the edge of the reserved hole for the pin shaft is only significant when the aperture ratio is within a certain range. This study fits the maximum Von Mises stress value of the spherical hinge structure with the support radius and calculates that when the support radius is less than 0.48 m, the spherical hinge structure fails and cannot meet the actual load. According to the engineering simplified algorithm, the minimum support radius for this project is 0.335 m, which is significantly smaller than the minimum support radius calculated by fitting. The shear design strength of steel is generally 0.58 times the tensile design strength, and this study takes 0.75 of the shear design strength of Q345 steel as the strength safety allowance value for the steel of the spherical hinge. According to the fitting equation of the maximum Von Mises stress value of the spherical hinge structure and the support radius, the support radius that satisfies the allowable strength of the spherical hinge structure is calculated to be 1.11 m, which is less than the actual design support radius of 1.95 m for the spherical hinge structure. For the load of this project, when the design support radius of the rotating spherical hinge structure is greater than 1.11 m, there will be excess support capacity. In summary, the support radius significantly affects the stress state of the spherical hinge structure. As the support radius increases, the maximum contact pressure and Von Mises stress value at the spherical hinge interface decrease accordingly. When the support radius is less than 0.48 m, the spherical hinge structure begins to fail and cannot meet the actual load.
Influence analysis of pin hole radius on spherical hinge
To discuss the stress forms of spherical hinge structures under different pin shaft radii, this paper selected spherical hinge structures with pin shaft reserve hole radii of 0 mm, 300 mm, 450 mm, and 600 mm, respectively. Following the calculation method described earlier, the stress states of the spherical hinge structures were calculated, as shown in Fig. 25.
Fig. 25.
Contact pressure cloud diagrams of spherical hinge interfaces with different pin shaft reserve hole radii.
From Fig. 25, it can be observed that the radius of the pin shaft reserve hole does not have a significant impact on the distribution pattern and magnitude of the contact pressure at the spherical hinge interface. The maximum contact pressures at the spherical hinge interface under five different pin shaft reserve hole radii are 24.88 MPa (0 mm), 25.11 MPa (150 mm), 24.99 MPa (300 mm), 25.22 MPa (450 mm), and 25.59 MPa (600 mm), with a maximum difference of less than 1 MPa. The minimum contact pressure at the spherical hinge interface occurs at the edge of the pin shaft reserve hole and gradually increases with the increase in pin shaft radius. When the pin shaft reserve hole radius is less than 300 mm, the contact pressure at the edge of the pin shaft reserve hole is negative, ranging from − 1.32 MPa to -0.417 MPa. When the pin shaft reserve hole radius is greater than 300 mm, the contact pressure at the edge of the pin shaft reserve hole is positive, ranging from 0.15 MPa to 2.18 MPa. This indicates that in the design of spherical hinge structures, the coordination deformation between the upper and lower spherical hinges can be ensured by adjusting the pin shaft radius. This paper extracts data on the contact pressure at the spherical hinge interface under five different pin shaft reserve hole radii, as shown in Fig. 26.
Fig. 26.
Contact pressure at the spherical hinge interface with different pin shaft reserve hole radii.
From Fig. 26, it can be observed that the distribution patterns of contact pressure at the spherical hinge interface for the five structural forms have a high degree of similarity and can be expressed by the same function. The increase in pin shaft radius does not result in the ideal scenario of reduced supporting area and a sharp increase in contact pressure. Calculations based on the derived formula from the previous section yield the traction forces required for the rotation of spherical hinge structures under five different pin shaft reserve hole radii, which are 1093KN (0 mm), 1093KN (150 mm), 1092KN (300 mm), 1089KN (450 mm), and 1081KN (600 mm). It can be seen that the radius of the pin shaft reserve hole does not have a significant impact on the traction force at the spherical hinge, but it slowly decreases with the increase in the pin shaft reserve hole radius.
The Von Mises stress cloud diagrams for the spherical hinge structures with five different pin shaft reserve hole radii are shown in Fig. 27.
Fig. 27.
Von Mises stress cloud diagrams of the upper and lower spherical hinge structures.
From Fig. 26, it can be observed that the edges and central positions of the pin shaft reserve holes are still the regions of stress concentration for the entire spherical hinge structure. When the spherical hinge is a continuous interface without gaps, the maximum Von Mises stress value for the spherical hinge structure is 35.37 MPa. With the appearance of pin shaft reserve holes, the maximum Von Mises stress value of the spherical hinge structure increases rapidly, gradually increasing with the radius of the pin shaft reserve hole. When the radius of the pin shaft reserve hole is 600 mm, the maximum Von Mises stress value for the spherical hinge structure is 50.88 MPa, which is 43.85% higher compared to the closed surface, requiring special attention to the stress concentration phenomenon at the edges of the spherical hinge holes in practical spherical hinge design. In summary, the radius of the pin shaft reserve hole has a significant impact on the stress state of the spherical hinge structure. As the curvature radius of the spherical hinge interface increases, the stress concentration phenomenon at the edges of the pin shaft reserve hole becomes more pronounced, and the distribution and magnitude of contact pressure at the spherical hinge interface are less affected by the radius of the pin shaft reserve hole.
Influence analysis of pin hole length on spherical hinge
To explore the stress forms of spherical hinge structures under different pin shaft lengths, this study selected eight spherical hinge structural forms, with the pin shaft reserve depths of the upper spherical hinge being 510 mm, 610 mm, 810 mm, and 910 mm, and the pin shaft reserve depths of the lower spherical hinge being 525 mm, 625 mm, 825 mm, and 925 mm, respectively. Following the calculation method described earlier, the stress states of the spherical hinge structures were computed. By extracting the contact pressure data of the spherical hinge interface models with different pin shaft reserve depths, as shown in Fig. 28.
Fig. 28.
The contact pressure of the spherical hinge interface with different pin shaft reserve depths.
According to Fig. 28, it can be seen that the pin shaft reserve depth does not affect the contact pressure at the spherical hinge interface or the distribution and magnitude of stress in the spherical hinge. Through the analysis of the Von Mises stress values of the upper turntable and lower base, it was found that stress concentration occurred at the circumferential edge of the upper spherical hinge pin shaft reserve hole top surface, which increased with the increase in the pin shaft reserve depth. When the pin shaft reserve depth of the upper spherical hinge was 910 mm, the Von Mises stress value at the circumferential edge of the pin shaft reserve hole top surface was 14.84 MPa, which was 16% higher compared to the corresponding stress value for a pin shaft reserve depth of 510 mm. Similarly, stress concentration also occurred at the circumferential edge of the lower spherical hinge pin shaft reserve hole bottom surface, which increased with the increase in the pin shaft reserve depth of the lower spherical hinge, as shown in Table 5. Therefore, it can be concluded that the pin shaft reserve depth has a relatively small impact on the contact pressure at the spherical hinge interface and the distribution and magnitude of stress in the spherical hinge structure.
Table 5.
Von Mises stress values at critical nodes under different pin shaft reserve depths.
| The upper spherical hinge pin shaft reserve depth (mm) | 510 | 610 | 710 | 810 | 910 |
|---|---|---|---|---|---|
| Von Mises stress value (MPa) | 12.79 | 13.24 | 13.69 | 14.47 | 14.84 |
| The lower spherical hinge pin shaft reserve depth (mm) | 525 | 625 | 725 | 825 | 925 |
| Von Mises stress value (MPa) | 3.18 | 4.18 | 4.23 | 4.54 | 5.21 |
Conclusion
This paper systematically examined the mechanical characteristics and key design parameters of a simplified spherical hinge structure using numerical simulations and formula derivation. Additionally, an indoor scaled test was conducted based on a representative spherical hinge structure to further investigate its mechanical properties. The specific conclusions are as follows:
The contact pressure at the spherical hinge interface increases radially from the center outward, following a cubic relationship with the distance from the center. Stress concentrations were observed along the edges of the pin shaft pre-drilled holes, with a recorded Von Mises stress of 44.72 MPa.
The curvature radius of the spherical hinge interface significantly influences the stress state of the hinge structure. As the curvature radius increases, stress concentration along the edges of the pin shaft pre-drilled holes becomes more pronounced. Both the average contact pressure and maximum contact force at the interface increase, along with the traction force required for rotation. The depth of the pre-drilled pin shaft holes, however, has only a minor effect on contact pressure and overall stress distribution.
The support radius also strongly impacts the stress state. As it increases, both the maximum contact pressure and Von Mises stress at the hinge interface decrease. When the support radius is less than 0.48 m, structural failure begins, as the structure can no longer support the load. Additionally, when the aperture ratio is between 0.042 and 0.083, the upper and lower spherical hinges struggle to deform uniformly under load, resulting in negative pressure along the edges of the pin shaft pre-drilled holes.
The addition of a slider complicates the stress state, creating a mix of tension and compression on the lower spherical hinge surface. As the load increases, stress in the lower hinge rises consistently, indicating a direct correlation between load and stress. The test point with the greatest stress variation, QM22, recorded a stress of 2.709 MPa under a load of 150 kN.
Acknowledgements
This research was funded by the Fundamental Research Funds for the Central Universities (Grant No. 2022QN1006), Jiangsu Province Double Innovation Doctor Program (Grant No. JSSCB20221491) and Jilin Scientific and Technological Development Program (Grant No. 20230203169SF).
Author contributions
W.G. (First Author): Conceptualization, Methodology, Software, Investigation, Formal Analysis, Writing - Original Draft.H.Y. : Data Curation, Writing - Original Draft.Y.L. : Resources, Supervision.W.T. : Investigation.W.C. (Corresponding Author): Conceptualization, Resources, Supervision, Writing - Review & Editing.
Data availability
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Associated Data
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Data Availability Statement
The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.