Optimization of Swivel Spherical Hinge Structure Design Based on the Response Surface Method

Abstract

The accurate analysis of key components of a spherical hinge structure directly affects bridge quality and safety during construction. Considering the key components of a spherical joint structure as the research object, a refined calculation model for the spherical joint is established to examine its stress using finite element analysis. The influence of design parameters on the mechanical characteristics of the spherical hinge structure is systematically analyzed. The response surface method (RSM), devised using a Box–Behnken design, is used to optimize the design of the spherical hinge structure parameters. A response surface model is established to derive the scheme of the optimized spherical hinge structure design. Moreover, by comparing the structural contact stress and rotational traction force before and after optimization, the effectiveness and necessity of the spherical hinge structure optimization are verified. The result comparison shows that the maximum contact stress and rotational traction force in the spherical hinge structure after optimization are reduced by 13.86% and 8.42%, respectively, compared with those before optimization. The relative error between the calculated and predicted values is approximately 3%, indicating that the RSM is feasible for optimizing key components of the spherical hinge structure. Its optimization effect is evident. Based on the identified optimal parameters of the spherical hinge structure, a range of recommended design parameters for the key structure of the rotating spherical hinge at different load carrying capacities is established using the interpolation method, which provides a valuable reference for engineering practice.

1. Introduction

A spherical hinge is a key force-bearing component of a swivel bridge, which plays a vital role in the swivel of the bridge during construction. The spherical hinge stress state is affected by material selection and structural size. Moreover, it is closely related to the starting traction torque and stability of the swivel structure [1,2,3,4]. The swivel spherical hinge is typically composed of an upper rotary table, upper spherical hinge, lower spherical hinge, lower cushion cap, central pin shaft, and other structures, which bear the vertical load from the upper bridge structure. Therefore, to ensure the safety of a swivel bridge during construction, optimizing the design of the spherical hinge structure and selecting the best spherical hinge optimization parameters are necessary.

In recent years, many studies have been conducted on the design of spherical hinge structures. Che et al. [5] proposed an optimization algorithm for a simplified calculation, relative to the current specifications, based on the structure and stress characteristics of the spherical hinge. Using a comparison and analysis with existing measured engineering data, the spherical hinge design was ensured to be reasonable and reliable. Mo et al. [6] performed a stress analysis based on the spherical hinge parameter values of actual engineering projects and determined the bearing and curvature radii for spherical hinge structures under different tonnages. Huang et al. [7] proposed a simplified contact mechanics model based on the contact mechanics theory considering stress in the polytetrafluoroethylene slider in the contact process. Optimization of the simplified model significantly improved the calculation efficiency of the model. Tian et al. [8] used node coupling technology and finite element analysis to conduct numerical simulations of the entire construction process for a spherical hinge structure to ensure the safety and reliability of its construction. Feng et al. [1] presented the superstructure rotation method (SRM), which can optimize bridge construction and reduce the impact on traffic, safety, and the overall budget. The construction technology of the project was presented, including the installation process for the traction system, the design and precise control method for the traction system, and the optimized design method for the tie rod tension by considering seven different loading conditions during the construction process. Liu et al. [9] proposed a critical overturning moment model for concrete ball hinges to ensure safety during the rotation of swing bridges. Non-Hertzian contact theory was used to calculate the surface stresses, and the model was validated with field monitoring. The results showed that the non-Hertzian method is superior in predicting the overturning resistance. Xiao et al. [10] studied the seismic safety of single-tower, cable-stayed bridges during the construction of the turnings. A finite element model for the construction phase was developed based on the actual project using ANSYS finite element analysis software (version 2020). The analysis showed that the locating pin at the center of the ball end hinge was subjected to excessive shear forces under 6- and 7-degree seismic excitation, which may lead to damage or destruction of the ball joint.

The response surface method (RSM) is an optimization method that combines test design and modeling. A mathematical model considering design variables and response values is developed using regression analysis. It has the advantages of fewer test times, shorter cycles, and high accuracy. Moreover, it considers the interaction among various factors, and it has been applied in several fields [11,12]. Aziz et al. [13] used a response surface model and experimental design, combined with finite element analysis, to optimize the design of a surface wave band gap and then verified the effectiveness of response surface model optimization. Zhan et al. [14] used the cable force optimization method combining the RSM and particle swarm optimization algorithm to establish the cable force optimization objective function. The method significantly improved uniformity in the cable force and bending moment of the main beam in the entire bridge. Datta et al. [15] established an insertion height model using experimental design and the RSM. They considered heating temperature and insertion time as independent variables and demonstrated that heating temperature had the greatest impact on the insertion height. Kadir O et al. [16] used RSM to evaluate a 94-year-old RC arch bridge and found that the bridge deck needs to be strengthened and refurbished to support larger vehicle loads. Kumar A et al. [17] demonstrated the efficiency of the RSM in damage identification for a six-story shear building and found that the RS model generated from the first three translational modal frequencies and corresponding first two mode shapes could detect and localize damage with proper quantification. Kang et al. [18] proposed a KELM-based RSM for parameter inverse analysis of concrete dams to efficiently identify material parameters and minimize computation time. Juan et al. [19] proposed using the RSM to simplify building energy performance models and demonstrated its effectiveness using a case study on a single-family house in three different climates. Mujahid et al. [20] used the RSM to assess the optimal replacement rate of waste foundry sand as a substitute for fine aggregate in concrete mixtures for improved environmental sustainability. Priyanka et al. [21] optimized the properties of eco-friendly, self-cleaning concrete containing marble dust and stearic acid using the RSM and found that a partial replacement rate of 30% was optimal for the self-cleaning property. Alina et al. [22] used the RSM to optimize the extraction yield of biologically active compounds from crushed Merlot grapes using ultrasound treatment before maceration. Sushrut S. et al. [23] optimized fuel flow rates in a DF engine using the RSM for improved brake thermal efficiency and reduced exhaust emissions. Allah et al. [24] used the RSM to optimize the electrochemical disinfection of canal water using stainless steel electrodes and found that interelectrode spacing is the most significant factor affecting disinfection efficiency. Mathad R. et al. [25] used the RSM to optimize the performance of a diesel engine using biodiesel fuels while reducing emissions, and they found that the combination of 2B 3G, IVM at 90 degrees, and an NG of six grooves resulted in the highest brake thermal efficiency while lowering emissions of smoke, CO, and HC.

Research [26,27,28] indicates that the bearing radius, curvature radius, pin radius, swivel weight, and other relevant parameters of the swivel ball joint considerably affect the structural design of the rotation system. They also control the calculation and determination of the rotation starting torque and the structural design of the entire traction system. However, presently, designing the structural dimensions of spherical hinges is mainly based on engineering experience. The structural calculation model for spherical hinges is relatively simple, and no universally recognized design standard has been established. Therefore, the formulation of a reliable structural analysis theory for spherical hinges that optimizes and enhances the existing design as well as improves the rationality and economy of the design to satisfy the requirements of engineering accuracy is urgent. In order to further investigate the relationship between the design parameters, such as the radius of curvature, pin radius, bearing radius, and the contact stress and rotational traction in the spherical hinge structure, and to find the best parameters for rotating the key parts of the spherical hinge to reduce energy loss in the spherical hinge during rotation. The finite element model for the spherical hinge structure with different curvature radii, pin radii, and support radii is established according to engineering requirements, and the response surface model is obtained by fitting the finite element analysis results to the bearing radii, curvature radii, and pin radii of the ball hinge structure. Using a significance analysis of the coefficients in the response surface model and an analysis of the 3D response surface diagram, the best optimization parameters are obtained to guide engineering practice.

In this study, static characteristic analysis and the structural optimization design for the key components of a spherical hinge structure are investigated. First, by establishing a refined finite element spherical joint model, a stress analysis of the key components in the spherical joint structure is performed. Subsequently, the influence of the curvature, pin, bearing radii, and other design parameters of the spherical joint structure on the mechanical characteristics of the spherical joint structure is systematically analyzed. Finally, the factors influencing the bearing radius, curvature radius, and pin radius of the spherical hinge structure are theoretically evaluated using the response surface experimental design method. Furthermore, the optimization parameters of the rotary spherical hinge structure are determined. The feasibility and effectiveness of the response surface optimization method are verified using an assessment of the structural performance of the optimized scheme.

2. Establishment of a Finite Element Model for the Spherical Joint Structure

The swivel system is a key part of swivel construction and includes a lower turntable, upper turntable, and swivel traction system. The main components of the lower turntable are the base and framework, lower ball joint, central positioning pin shaft, slide way, and jack reaction seat. The main components of the upper rotary table are the upper spherical hinge, supporting feet, and sand box. The main components of the traction system are the traction reaction seat and traction rope [29,30]. In this study, a swivel structure capable of bearing a 30,000 t weight is set as the research object. Finite element analysis software is used to establish a finite element analysis model for the key parts of the spherical hinge structure, as shown in Figure 1. The analysis is conducted on the key parts of the spherical hinge structure, including the bearing, curvature, and pin.

To improve the analysis efficiency of the finite element software, 30,000 t was directly applied to the surface of the upper turntable in the form of a uniformly distributed load instead of the main beam component of a building [31]. Under boundary conditions, all six degrees of freedom on the bottom surface of the lower turntable are constrained. For the upper turntable, the upper and lower spherical hinges release the degrees of freedom (U2) along the gravity direction. The contact relationship between the surfaces on each part of the spherical hinge structure is set. The contact attribute in the normal direction is set as hard contact to prevent penetration during analysis. The direct tangent direction of the contact surface is set as Coulomb friction; the friction coefficient is set as 0.06. The grid of each component is divided according to its importance, and the key parts of the spherical joint are divided into dense grid elements.

3. Analysis of the Key Components of Spherical Hinge Structures

The structural analysis and calculation for the spherical hinge structure of the rotating system indicate that the compressive stress level of the upper and lower rotary table contact surfaces of the spherical hinge is closely related to several parameters, such as the bearing, curvature, and pin radii. These parameters determine not only the structural design of the rotating system but also the rotational starting torque and structural design for the entire traction system [32]. The magnitude of the contact stress is an important indicator of whether the spherical joint is safe during rotation. The contact stress is also a control parameter for the design of the spherical joint structure and structural dimensions of the rotating traction system [33]. In this study, the influence of the bearing radius, curvature radius, pin radius, and other design parameters on the mechanical characteristics of spherical joints is analyzed.

3.1. Effect of the Bearing Radius on the Mechanical Characteristics of Spherical Hinge Structures

The bearing radius is a key parameter in the structural design of spherical joints. Generally, a bearing radius analysis is performed by simplifying the contact surface of the upper and lower turntables into a plane. The average value of the vertical compressive stress is obtained, and the bearing radius of the spherical joint is determined by controlling the level of the compressive stress so that the allowable stress of the material is not exceeded. To examine the mechanical characteristics of spherical hinge structures with different bearing radii, four types of spherical hinge structures with bearing radii of ZCR₁ = 2400 mm, ZCR₂ = 3000 mm, ZCR₃ = 3600 mm, and ZCR₄ = 4200 mm are selected. A finite element analysis is performed to calculate the stress state of the spherical hinge structures. Contact pressure nephograms for the spherical joint interfaces with four bearing radii are shown in Figure 2.

As shown in Figure 2, the maximum contact stresses in the spherical joints considering the four radii (ZCR₁ = 2400 mm, ZCR₂ = 3000 mm, ZCR₃ = 3600 mm, and ZCR₄ = 4200 mm) are 89.05, 81.91, 74.03, and 66.08 MPa, respectively. Let the maximum contact stress value of 89.05 MPa with ZCR₁ = 2400 mm be the standard. The stress differences with respect to stresses corresponding to ZCR₂, ZCR₃, and ZCR₄ are 7.14, 15.02, and 22.97 MPa, respectively; with a decrease in the bearing radius, the contact stress decreases by 8.02%, 16.87%, and 25.79%, respectively. As shown in Figure 3, the contact stress in the spherical joint with four bearing radii gradually increases within 2500 mm from the center of the spherical joint. The contact stress reaches a maximum within 3000–3500 mm from the center of the spherical joint. Then, the contact stress gradually decreases with increasing distance from the center of the spherical joint.

3.2. Effect of the Curvature Radius on the Mechanical Characteristics of Spherical Hinge Structures

To discuss the mechanical characteristics of spherical hinge structures, four spherical hinge structures with curvature radii of QLR₁ = 8000 mm, QLR₂ = 10,000 mm, QLR₃ = 12,000 mm, and QLR₄ = $\infty$ mm are selected. A finite element analysis is performed to calculate the stress state of the spherical hinge structures. Contact pressure nephograms for the spherical joint interfaces with the four curvature radii are shown in Figure 4.

As shown in Figure 4, the maximum contact stresses in the spherical joints considering the four curvature radii (QLR₁ = 8000 mm, QLR₂ = 10,000 mm, QLR₃ = 12,000 mm, and QLR₄ = $\infty$ mm) are 63.99, 66.08, 71.67, and 79.32 MPa, respectively. Let the maximum contact stress value of 79.32 MPa corresponding QLR₄ = $\infty$ mm be the standard. The contact stress differences considering the stresses corresponding to QLR₁, QLR₂, and QLR₃ are 15.33, 13.24, and 7.65 MPa, respectively; as the curvature radius increases, the contact stress decreases by 19.33%, 16.69%, and 9.64%, respectively. As shown in Figure 5, the rising trend in the contact stress in the spherical joint with four bearing radii is relatively gentle within 2500 mm from the center of the spherical joint; the rising curves are virtually coincident. Within 3000–3500 mm from the center of the spherical joint, the rising rate of the contact stress with QLR₄ = $\infty$ increases significantly. When the maximum value is reached, the contact stress in the spherical joint gradually decreases with the increasing distance from the center of the spherical joint.

3.3. Effect of the Pin Radius on the Mechanical Characteristics of Spherical Hinge Structures

To discuss the mechanical characteristics of spherical hinge structures with different pin radii, four spherical hinge structures with curvature radii of XZR₁ = 0 mm, XZR₂ = 135 mm, XZR₃ = 270 mm, and XZR₄ = 400 mm are selected. A finite element analysis is performed to calculate the stress state of the spherical hinge structures. Contact pressure nephograms for the spherical joint interfaces with the four pin radii are shown in Figure 6.

The maximum contact stresses in the spherical joints with four pin shaft radii (XZR₁ = 0 mm, XZR₂ = 135 mm, XZR₃ = 270 mm, and XZR₄ = 400 mm) are 74.73, 76.78, 78.72, and 79.34 MPa, respectively, as shown in Figure 6. Let the maximum contact stress value of 79.34 MPa corresponding to XZR₄ = 400 mm be the standard. The contact stress differences in the spherical joints considering XZR₁, XZR₂, and XZR₃ are 4.61, 2.56, and 0.62 MPa, respectively; with a decreasing bearing radius, the contact stress also decreases by 5.81%, 3.23%, and 0.78%, respectively. Compared with the other three pin shaft radii, Figure 7 shows that XZR₄ = 400 mm yields the highest contact stress at the center distance of the spherical joint of approximately 2500 mm. After reaching the maximum, the contact stress rapidly decreases with increasing distance from the center.

4. Optimization Process for Spherical Joint Structures Based on the RSM

4.1. Principles and Characteristics of the RSM

The RSM is a process parameter optimization method that combines mathematical methods and statistical analysis [34]. When a nonlinear relationship exists between the response variables and influencing factors of the optimization object, the quadratic term cannot be estimated using a conventional experimental design; in this case, the RSM becomes the best tool for optimizing process parameters. In general, the form of the real response function is unknown. The RSM is used to fit the second-order mathematical relationship model between the response output and influencing factors as an approximation of the real function relationship. The relationship is analyzed and optimized to determine the level of a group of factors for optimizing response variables.

Generally, RSMs are implemented in three phases: parameter screening, region search, and optimization processing [35,36].

(1)

Parameter screening

In the initial phase of the experiment, multiple influencing factors are introduced. However, their importance cannot be determined because of cost and efficiency constraints. Hence, factors that are unimportant must be eliminated using a screening test before a response surface study is conducted. By screening a few important factors, the number of subsequent tests is reduced considerably, thus ensuring efficiency.

(2)

Region search

When the test area is far from the optimal region, the response variable is approximately linear relative to the influencing factors. A designed first-order test is conducted, and a first-order mathematical model is fitted to obtain the following:

$$y={\beta }_0+{\displaystyle \sum _{i=1}^k{\beta }_i{x}_i+\epsilon },$$

(1)

When the curvature test indicates that the fitted function has a significant curvature effect, the second-order model must be fitted. At this point, the test region can be considered close to or within the optimal region. The third stage, i.e., optimization processing, can then be started.

(3)

Optimization processing

When the experimental region is close to or within the optimal region, the response variable is nonlinearly related to the influencing factor. A second-order design is implemented, and a second-order model is fitted:

$$y={\beta }_0+{\displaystyle \sum _{i=1}^k{\beta }_i{x}_i+{\displaystyle \sum _{i=1}^k{\beta }_i{}_i}}{x}_i{}^2+{\displaystyle \sum _{\begin{array}{l}i=1\\ i<j\end{array}}^k{\beta }_{ij}{x}_i{x}_j+\epsilon },$$

(2)

where $\beta$ is the unknown coefficient; $k$ is the number of design variables; $y$ is the predicted response value; ${\beta }_0$ is the offset term coefficient; ${\beta }_i$ is the linear offset term coefficient; ${\beta }_{ii}$ is the second-order offset term coefficient; ${\beta }_{ij}$ is the interaction coefficient; and $\epsilon$ is the total error (including random, modeling, and systematic errors).

The RSM has the following application conditions [37]: (1) significant factors are identified; (2) the number of significance factors is small; (3) the randomness error is small; (4) the linear model is inadequately fitted; and (5) the model is used for the optimization of parameters. Because the RSM uses experimental design, data analysis, and model fitting, it has the advantages of mathematical rigor and logic, and the method considers the interaction among controllable factors. Consequently, the model established using the method has high reliability.

4.2. Response Surface Model Construction

Various response surface experimental design methods are available [38]. The most commonly used techniques are the Box–Behnken design (BBD) and central composite design (CCD). The BBD is a common experimental design method for response surface optimization. It is suitable for optimizing experiments involving two–five factors. The BBD considers three horizontal influencing factors coded with −1, 0, and 1. The design table considers 0 as the value for the central point; +1 and −1 are the high and low values corresponding to the cubic point, respectively. The distribution of the experimental points for the BBD is shown in Figure 8 (three factors are considered an example).

The CCD experimental design method is composed of a $2^k$ full factorial design or a partial factorial design. It has the $2^{k-p}$ addition of axial and central points to the cubic points in the two-level design. For $2^k$ axial points, the following are added: ($\pm a, 0, 0, \cdots , 0$), ($0, \pm a, 0, \cdots , 0$), and ($0, 0, 0, \cdots , \pm a$). The $n_c$ central point is ($0, 0, 0, \cdots , 0$). Figure 9 shows the CCD for $k$ = 2.

Because no axial point exists, the BBD requires fewer tests and is more economical. The optimal process-level value obtained using optimization does not exceed the maximum value range, which is particularly applicable to tests with special requisites or safety requirements [39]. In this study, the BBD method is selected to optimize the key parts of the spherical hinge structure. Because of the action of a vertical load, the spherical hinge structure is in a compressive state over an extended period. Its internal stress value determines the area of the structure prone to damage. Accordingly, this study considers the maximum stress value of the spherical hinge structure as one of the response values. The smaller the maximum stress value, the safer the state of the spherical hinge structure. In addition to the maximum stress value, the traction required by a spherical joint to overcome friction during rotation is an important evaluation factor. The smaller the traction, the lower the energy consumption required during construction. Accordingly, in this study, the contact stress in a spherical joint is calculated by numerical simulation. The rotational traction force calculated using formula is considered the response of the test design.

The plane integration method is applied to derive the swivel traction force, $F$. The plane integration method calculation model assumes that the bottom surface of the bearing foot is in full contact with the slideway and the upper and lower parts of the grinding center, ignoring the contact gap caused by the arrangement of the sliding pads [40,41,42]. All contact surfaces can be regarded as the integration of countless parts of micro-rings, as shown in Figure 10. Some of the rings can be simplified into small rectangular infinitesimal elements.

The friction moment at the grinding center is deduced from the integral as $M_1$:

$$M_1=p{\mu }_{1}0{\displaystyle \underset{0}{\overset{R_0}{\int }}{\displaystyle \underset{0}{\overset{2\pi }{\int }}{r}^{2}d\theta dr=\frac{2\pi R_0^{3}p{\mu }_1}{3}}},$$

(3)

The friction moment, $M_2$, at a single support foot is

$$M_2=p{\mu }_2{\displaystyle \underset{R_1}{\overset{R_2}{\int }}{\displaystyle \underset{0}{\overset{\theta }{\int }}{r}^{2}d\theta dr=\frac{R_2^3-R_1^3}{3}p{\mu }_2\theta }},$$

(4)

According to the torque balance, $M_1+M_2=Fl$.

The final traction force, $F$, is

$$F=\frac{2\pi R_0^3{\mu }_1+\left(R_2^3-R_1^3\right)\theta {\mu }_2}{3l}p,$$

(5)

where $p$ is the pressure per unit area of the support system; ${\mu }_1$ is the friction coefficient at the pin; ${\mu }_2$ is the friction coefficient at the foot; $R_0$ is the pin radius; $R_1$ is the radius of the circle where the inner edge of the foot is located; $R_2$ is the radius of the circle where the outer edge of the foot is located; $l$ is the traction force arm; and $F$ is the traction force.

When multiple methods for experimental design are available, the economy of the experimental design, efficiency, and applicability of structural optimization analysis must be comprehensively considered. In this study, the BBD experimental method, which has high fitting accuracy in the Design–Expert optimization analysis software, is used for the experimental design and analysis. Each selected design variable has three levels in the BBD experimental design, indicated using −1, 0, and +1. In this study, the support, curvature, and pin radii are used as the three design variables. Moreover, a BBD combination test is conducted. The test factors and level codes are summarized in

Table 1. Test factors and level design.

Experimental Factors	Unit	Coding Level and Value
		−1	0	1
Bearing radius	mm	2400	3000	3600
Curvature radius	mm	8000	10,000	12,000
Pin radius	mm	0	200	400

.

4.3. Test and Analysis of the Response Surface Model

Simulation tests were conducted according to the BBD to establish refined finite element models of ball hinges. A numerical simulation analysis was performed to obtain the contact stress and rotational traction of the ball hinge structure. Each influencing factor was considered an independent variable and contact stress and rotational traction were considered evaluation indices [43,44] (

Table 2. BBD test results.

Number	Bearing Radius (mm)	Curvature Radius (mm)	Pin Radius (mm)	Contact Stress (MPa)	Rotational Traction (kN)
1	2400	12,000	200	98.6	1733.3
2	3000	10,000	200	64.8	2192.5
3	3600	12,000	200	57.2	2386.7
4	2400	10,000	0	85.8	1768.7
5	3000	10,000	200	66.3	2169.8
6	3000	8000	400	60.8	2003.8
7	2400	8000	200	81.9	1712.4
8	3000	8000	0	59.4	2201.7
9	3000	12,000	0	71.2	2240.5
10	3000	12,000	400	78.5	1998.4
11	3000	10,000	200	66.9	2157.9
12	2400	10,000	400	91.3	1694.2
13	3000	10,000	200	65.3	2142.7
14	3600	10,000	0	53.9	2428.6
15	3000	10,000	200	67.4	2176.2
16	3600	8000	200	51.6	2316.8
17	3600	10,000	400	55.8	2287.9

).

The data samples listed in Table 2 are the contact stress and rotational traction force calculated using the models based on quadratic polynomial regression. The models, formulated using the Design Export 13 software, are as follows:

$$\begin{array}{ll}{Y}_1=& 66.14-17.39A+6.48B+2.01C-2.78AB-0.9AC+\\ & 1.48BC+5.2A^2+0.98B^2+0.355C^2\end{array}$$

$$\begin{array}{ll}{Y}_2=& 2167.82+313.93A+15.53B-81.9C+12.25AB-16.55AC-\\ & 11.05BC-98.39A^2-32.14B^2-24.58C^2\end{array}$$

where $Y_1$ is the contact stress; $Y_2$ is the rotational traction; $A$ is the bearing radius; $B$ is the curvature radius; and $C$ is the pin radius.

With the derived mathematical models for the contact stress and rotational traction force, a further reliability analysis was conducted to evaluate their validity. The details are summarized in

Table 3. Reliability test of response regression models.

Model	Standard Deviation	Average Deviation	$\mathit{C}.\mathit{V}.\mathit{\%}$	${\mathit{R}}^2$	${\mathit{R}}^2{}_{\mathit{a}\mathit{d}\mathit{j}}$	S/N
$Y_1$	1.40	69.22	2.02	0.9853	0.9794	44.48
$Y_2$	34.11	2094.83	1.63	0.9836	0.9725	30.26

.

In Table 3, $C.V.$ is the coefficient of variation; $R^2$ is the coefficient of determination; $R^2{}_{adj}$ is the corrected coefficient of determination; and S/N is the signal-to-noise ratio.

Table 3 indicates that the signal-to-noise ratios of the contact stress and rotational traction force are 44.48 and 30.26, respectively. Both ratios exceed 4, indicating that the models can be used to achieve an optimal design. The coefficient of determination represents the degree of fit for each model: the closer the coefficient of determination is to 1, the higher the fitting degree of a model. The coefficients of determination for the contact stress and rotational traction force models are 0.9853 and 0.9836, respectively. These values indicate that the models are well-fitted, the predicted values have a good correlation with measured values, and the prediction error is small. The coefficients of variation for the models are less than 5%, indicating that the detection accuracy of the models is sufficient. The coefficients of determination for the contact stress and rotational traction response models are 0.9754 and 0.9725, respectively. These coefficients indicate that the two models effectively predict and interpret 97.54% of the contact stress response values and 97.25% of the rotational traction response values, respectively. Hence, the models can optimize the spherical joint structure using the BBD method.

As summarized in

Table 4. Variance analysis and significance test of the quadratic polynomial model for contact stress.

Source of Variance	Quadratic Sum	Freedom	Mean Square	$\mathit{f}$ Value	p-Value	Significance
Model	2951.93	9	327.99	167.65	<0.0001	Quite significant
A	2418.6	1	2418.6	1236.28	<0.0001	Quite significant
B	335.41	1	335.41	171.44	<0.0001	Quite significant
C	32.4	1	32.4	16.56	0.0048	Significant
AB	30.8	1	30.8	15.74	0.0054	Significant
AC	23.24	1	23.24	13.66	0.0023	Significant
BC	8.7	1	8.7	4.45	0.0729	Not significant
A2	114.07	1	114.07	58.31	0.0001	Significant
B2	4.04	1	4.04	2.07	0.1937	Not significant
C2	0.5306	1	0.5306	0.2712	0.6186	Not significant
Residual	13.69	7	1.96	-	-	-
Lack of fit	9	3	3	2.56	0.1931	Not significant
Pure error	4.69	4	1.17	-	-	-
Sum	2965.62	16	-	-	-	-

and

Table 5. Variance analysis and significance test of the quadratic polynomial model for rotational traction force.

Source of Variance	Quadratic Sum	Freedom	Mean Square	$\mathit{f}$ Value	$\mathit{p}$ Value	Significance
Model	8.97 × 105	9	99,676.88	85.68	<0.0001	Quite significant
A	7.88 × 105	1	7.88 × 105	677.68	<0.0001	Quite significant
B	1928.21	1	1928.21	1.66	0.2389	Not significant
C	53,660.88	1	53,660.88	46.13	0.0003	Significant
AB	38,600.25	1	18,600.25	16.51	0.0034	Significant
AC	11,095.61	1	31,095.61	39.41	0.0009	Significant
BC	488.41	1	488.41	0.4198	0.5377	Not significant
A2	40,756.25	1	40,756.25	35.03	0.0006	Significant
B2	4348.03	1	4348.03	3.74	0.0945	Not significant
C2	2544.94	1	2544.94	2.19	0.1827	Not significant
Residual	8143.6	7	1163.37	-	-	-
Lack of fit	6730.93	3	2243.64	6.35	0.053	Not significant
Pure error	1412.67	4	353.17	-	-	-
Sum	9.05 × 105	16	-	-	-	-

, the $f$ values of the contact stress and rotational traction force models are 167.65 and 85.68, respectively. The p-values of the models are considerably less than 0.05 (p < 0.0001), indicating that the regression models have a good fit. The significance of the influence of each variable on the response value of the regression model passes the p-value test: the smaller the p-value, the higher the significance of the influence on the response value. When p < 0.01, the effect is particularly significant, and when p < 0.05, the effect is significant. From the perspective of the degree of influence, the $f$ value of each factor can reflect its influence on the response: the greater the $f$ value, the greater the influence on the response value.

As summarized in Table 4, in the regression equation, the primary items are A (p < 0.0001), B (p < 0.0001), and C (p < 0.0001); the secondary item is A2 (p < 0.0001). These items have significant effects on the maximum stress response value, indicating that the bearing, curvature, and pin radii have a distinct influence on the contact stress in the spherical hinge structure. In terms of the degree of influence, the bearing, curvature, and pin radii are 1236.28, 171.44, and 16.56, respectively. Therefore, the order for the degree of influence of the factors on the contact stress in the spherical hinge structure is bearing radius (A) > curvature radius (B) > pin radius (C). In the maximum stress regression model coefficient, the quadratic term, AB, has a value of 15.74. This indicates that the interaction between the support and curvature radii has a considerable impact on the contact stress in the spherical joint structure [45,46,47].

As summarized in Table 5, the primary items in the regression equation are A (p < 0.0001) and C (p = 0.0003), and the secondary items are A2 (p = 0.0006), AB (p = 0.0018), and AC (p = 0.0062). These items have significant effects on the response values of the rotating traction force. In terms of the degree of influence, the bearing and pin radii are 677.68 and 46.13, respectively. These values indicate that the influence of the bearing radius on the rotational traction force in the spherical hinge structure is greater than that of the pin radius. In the regression model coefficient for rotational traction force, the values of the quadratic terms, AB and AC, are 16.51 and 39.41, respectively, indicating that the interaction between the bearing, curvature, and pin radii is significant.

Based on the foregoing analysis, a response surface model is plotted considering the interaction between the two output variables of contact stress and rotational traction force in a spherical joint structure with bearing radius, curvature radius, and pin radius, as shown in Figure 11, Figure 12 and Figure 13.

The response surface diagrams for design variables on the influence of the force and deformation in the spherical hinge structure are shown in Figure 11, Figure 12 and Figure 13. These diagrams are used to evaluate the interaction effect of any two variables on the force and deformation in the spherical hinge structure. These figures indicate that for the contact stress response index of the spherical hinge structure, the interaction between the bearing, curvature, and pin radii is significant. By contrast, the interaction between the curvature and pin radii is weak. These interactions are characterized by the steeper surface slopes of the response surface graphs in Figure 11 and Figure 12 compared to the graph in Figure 13. As shown in Figure 11 and Figure 12, the contact stress in the spherical hinge structure decreases rapidly as the bearing radius increases. By contrast, the contact stress in the spherical hinge structure gradually decreases as the curvature and pin radii decrease. Therefore, the reduction rate of the contact stress in the spherical hinge structure is primarily determined by the bearing radius.

For the response index of the rotational traction force in the spherical hinge structure, the following are shown in Figure 14, Figure 15 and Figure 16: the interaction between the bearing and pin radii is most significant; the interaction between the bearing and curvature radii is more significant; and the interaction between the curvature and pin radii is not significant. These interactions are characterized by the fact that the surface slopes of the curved surfaces in Figure 14 and Figure 15 are steeper than those in Figure 16. As shown in Figure 14 and Figure 15, the rotational traction force in the spherical hinge structure rapidly increases with the bearing radius. When the bearing radius is small, the influence of the pin radius on the rotational traction force is greater than that of the curvature radius. The shape of the contour map reflects the intensity of the interaction between the two factors. The elliptical shape indicates that the interaction between the two factors is significant, whereas a circular shape indicates the opposite. As shown in Figure 14 and Figure 15, the contour map is approximately elliptical, indicating that the interaction between the bearing and pin radii is significant for the response index of the rotational traction force in the spherical hinge structure. By contrast, the interaction between the pin and curvature radii is insignificant as indicated by the approximately circular contour line.

5. Analysis and Discussion of Optimization Results for a Swivel Spherical Hinge

5.1. Optimization Results for a Swivel Spherical Hinge

Using the RSM analysis, the three optimization parameters (bearing, curvature, and pin radii) are found to have different degrees of influence on the contact stress and rotational traction force in the spherical joint structure. The optimal design parameters of the spherical hinge structure are calculated using the function of the optimization module in the design software. To verify the reliability of the optimization results, the structural parameters of the spherical joint before and after the optimization are analyzed and calculated using the finite element method, as summarized in

Table 6. Comparison of results before and after optimization.

Name	Unit	Before Optimization	After Optimization	Predicted Value
Bearing radius	mm	2750	3130.2	-
Curvature radius	mm	10,000	8936.6	-
Pin radius	mm	135	176.5	-
Contact stress	MPa	79.2	68.3	66.2
Rotational traction force	kN	1989.5	1821.9	1758.5

.

5.2. Analysis and Discussion of Optimization Results

After implementing the RSM, the optimized predicted and simulation values were compared. The contact stress and rotational traction force of the ball-hinge structure parameters are 68.3 and 1821.9 kN, respectively. Deviations are observed in the predicted values; however, these differences are not considerable. The errors of the simulation values for contact stress and rotational traction force are 3.07% and 3.48%, respectively, and the relative error is approximately 3%. Therefore, the regression model fits the parameters with a high degree of confidence and can accurately optimize the ball-hinge structure parameters. After optimization, the rotational traction force and maximum stress decreased by 13.86% and 8.42%, respectively, compared with those before optimization.

Based on the literature review and the previous discussion, it was found that the three optimization parameters of bearing radius, curvature radius, and pin radius have different degrees of influence on the contact stress and rotational traction in the ball-hinged structure. The smaller the value, the more balanced the ball-hinged structure will be, the smoother the rotation process will be, and the lower the cost will be. In this paper, the sensitivity of different radii of curvature, pin radii, and bearing radii hinge design parameters on the ball hinge structure is explored using a single-factor and multi-factor, respectively, and it is determined that the bearing radius of the rotating ball hinge under the influence of a single factor is the most significant influencing factor on the maximum contact stress value of the ball hinge structure. In addition, the interaction between the bearing radius and radius of curvature of the rotating ball hinge under the influence of a multi-factor is the most significant influencing factor on the maximum contact stress value of the ball hinge structure. The interaction between the radius of support and the radius of curvature is the most significant factor influencing the maximum contact stress value of the ball hinge structure. On this basis, the Box–Behnken design method is used to determine the optimal parameters of the rotating ball hinge structure for solid projects and the recommended range of rotating ball hinge design parameters for different rotating tonnage, which can provide guidance and suggestions for field construction,

5.3. Selection of Spherical Hinge Design Parameters

According to the response surface fitted model and the fitted curves for the influencing parameters, and with the great and small values of the strength index as the constraint and the rotation efficiency as the optimization target, the design parameters for the ball-hinged structure of this tonnage can be obtained in the following ranges: the curvature radius is about 8300 mm~15,000 mm, the bearing radius is about 1800 mm~4800 mm, and the pin radius is 0~300 mm. Using the design parameters of a 30,000 t spherical hinge structure as the standard, the recommended range of rotating spherical hinge design parameters for different rotating tonnage can be obtained using interpolation, as shown in

Table 7. Design parameter recommended values for the spherical hinge structure.

Rotational tonnage/t	15,000~18,000	18,000~21,000	21,000~24,000	24,000~27,000
Bearing radius/mm	1170~3020	1290~3540	1480~3890	1620~4160
Curvature radius/mm	4330~9680	5540~10,890	6230~12,850	7640~14,110
Rotational tonnage/t	27,000~30,000	30,000~33,000	33,000~36,000	36,000~39,000
Bearing radius/mm	1860~4780	2150~5630	2680~6010	3210~6820
Curvature radius/mm	8360~15,270	9880~16,950	11,320~18,470	13,590~20,360

(considering the small influence of the positioning pin on the overall structure and the convenience of processing and installation, the default radius of the pre-drilled hole for the positioning pin in this table is 150 mm).

6. Significance of Research

In the design process for a rotating bridge, the force analysis and design of a ball hinge is of great importance and directly affects the stability of the entire superstructure. At present, the design theory for the ball hinge in the bridge is conservative; no unified conclusion has been formed for the distribution pattern of the ball hinge forces. Meanwhile, the process for the theoretical solution is relatively simple, the forces on the ball hinge cannot be accurately described, and the control of the ball hinge size is mainly based on experience. In this study, using finite element analysis, a refined model for the rotating ball hinge is established and a force analysis is carried out. Subsequently, the influence of ball hinge structure design parameters such as the radius of curvature, pin radius, support radius, and pin depth on the force characteristics of the ball hinge structure is systematically analyzed. Finally, based on the Box–Behnken experimental design method, an optimal theoretical evaluation of the influencing factors of the support radius, radius of curvature, and pin radius of the ball-hinged structure was carried out, and the best-optimized parameters for the rotating ball-hinged structure in this project were determined.

7. Conclusions

In this study, static characteristic analysis and a structural optimization design for the key components of a swivel ball joint are conducted. The influence of the structural design parameters (such as bearing, curvature, and pin radii) on the mechanical characteristics of the spherical joint structure is systematically analyzed by establishing a refined finite element model for the rotary spherical joint. Subsequently, based on the BBD experimental method, the factors influencing the bearing, curvature, and pin radii of the spherical hinge structure are theoretically evaluated, the optimal optimization parameters are determined, and the structural parameters of the spherical hinge before and after optimization are analyzed. The specific conclusions are as follows:

(1)

The analysis of the key parts of the spherical joint structure, using the finite element method, showed that the contact stress in the spherical joint conforms to the distribution law, in which the contact stress first increases and then decreases with increasing distance from the center of the spherical joint. A change in the bearing radius and curvature radius of the spherical joint has a significant influence on the extreme value of the contact stress in the spherical joints. Conversely, a change in the pin radius has a negligible effect on the maximum contact stress in the spherical joint.

(2)

Using response surface analysis software, the response surface fitting models for the factors (bearing, curvature, and pin radii) influencing the contact stress and rotational traction force of the spherical joint structure are fabricated. The validity of the models is confirmed using reliability and significance analyses. Using a significance analysis of the coefficients of the models and a 3D response surface analysis, the bearing radius is found to be the most significant factor affecting the maximum contact stress and rotational traction force in spherical joint structures. The interactions between the support and curvature radii and between the bearing and pin radii significantly affect the maximum contact stress and rotational traction force in the spherical hinge structure, respectively.

(3)

Based on the BBD method, the optimal parameters for the key components of the spherical hinge structure are determined. The bearing, curvature, and pin radii of the spherical hinge structure are 3130.2, 8936.6, and 176.5 mm, respectively. By comparing and analyzing the calculated and predicted values of the spherical hinge structure optimization parameters, the relative error is found to be approximately 3%. This observation verifies the feasibility of the BBD method for optimizing the spherical hinge structure parameters. The maximum contact stress and rotational traction force in the spherical hinge structure after optimization are reduced by 13.86% and 8.42%, respectively, compared with the respective values before optimization.

(4)

Based on the identified optimal parameters of the ball joint structure, a range of recommended design parameters for the key structure of the rotating spherical hinge at different load carrying capacities was established using the interpolation method, which provides a valuable reference for engineering practice. The recommended values of the rotating spherical hinge design parameters provide a reference for optimizing the design of spherical hinges to minimize energy loss during rotation, improve overall performance and efficiency, and ensure successful implementation in practical applications.