A New Single-Step Bolt Tightening Method for Multi-Bolt Flange Structures

Abstract

Single-step tightening is simple and fast compared to multi-step tightening, and it is therefore frequently used during the tightening of bolts in flange structures. However, single-step tightening to maintain uniformity is more difficult to achieve. To address this problem, existing methods such as the Elastic Interaction Coefficient Method (EICM) and Tetraparametric Assembly Method (TAM) have been investigated for load uniformity in single-step tightening. In order to improve the computational efficiency and accuracy of the traditional methods, a new single-step tightening method is proposed in this paper. This method can realize the design of the initial preload force simply by measuring the change in bolt load in a specific sequence. It is verified by numerical simulation that this method can realize the uniform distribution of bolt load. In addition, this paper will provide suggestions for the optimal tightening sequence for the single-step tightening method.

1. Introduction

Flange connections represent a standard method for joining mechanical systems. They are frequently employed in modern engineering equipment [1]. However, the elastic interaction between bolts influenced by the tightening sequence gives rise to bolt relaxation phenomena [2,3,4]. The above process hinders the successful implementation of bolt preload [5]. Consequently, the presence of irregular stress distributions, a reduction in fatigue strength [6], and unfavorable impacts on other mechanical properties of the joint system can be observed. These factors ultimately lead to the occurrence of safety accidents. Furthermore, the elastic interaction between bolts is subject to a number of influencing factors, including the geometry and materials of the components being joined, the magnitude of the load, the distance between bolts, and the configuration of the assembly [7,8,9]. As a result, it is a challenge to mitigate these interactions and achieve uniform load distribution.

Single-step tightening is more convenient and faster than multi-step tightening, so it is often used in flange connections. However, the elastic interaction between bolts is greater in single-step tightening compared to multi-step tightening, which leads to a more dispersed distribution of bolt loads in single-step tightening [10]. Therefore, unlike in multi-step tightening where the preload size for each bolt is the same when designing the tightening sequence, in single-step tightening [11], it is more important to design the preload size for each bolt under different tightening sequences to achieve a uniform distribution of bolts.

Among them, the Elastic Interaction Coefficient Method (EICM) is a method based on the principle of elastic interaction, which assumes that there is a linear relationship between the initial preload and residual preload when the tightening sequence is determined and expresses this relationship by means of an elastic coefficient matrix [12,13]. Once the target preload is set, the initial preload for each bolt can be obtained by solving the inverse of the elastic coefficient matrix and multiplying it with the target preload. Subsequent studies have validated the effectiveness of this method using finite element analysis (FEA) and physical experiments [14,15,16]. In addition, some studies have attempted to find the optimal tightening sequence by constructing a neural network agent model to achieve a more uniform preload distribution and further improve the effectiveness of the EICM [17,18]. These attempts have not significantly changed the consistency of the optimization results of EICM under different tightening sequences [15]. At the same time, the spring node model provides a theoretical basis for the EICM [19,20,21]. Even in the case of nonlinear materials such as flexible graphite gaskets, the EICM still ensures a uniform distribution of bolt loads [22]. However, the EICM requires measuring the impact on each already tightened bolt during the tightening process, making the computational cost high for multi-bolt flange structures [16].

To solve the problem of high computational cost of the EICM, a simplified method has been proposed—the Tetraparametric Assembly Method (TAM) [16]. The method neglects elastic interactions between distant bolts, thus greatly simplifying the elastic coefficient matrix. While the TAM simplifies the design process of the initial preload by requiring only a few parameters to be measured, its consistency in optimization results across different tightening sequences is not as robust as that of EICM. In addition, the TAM mainly focuses on the case of unilateral bolts, while it is inadequate for the treatment of bilateral bolts. Therefore, to address the limitations of the EICM and TAM, this paper presents a new method designed to simplify the EICM by considering bilateral bolts by simulating their operation in real situations. Compared with the EICM, this method only needs to measure the variation in individual bolt preload at a specific sequence, which significantly reduces the computational cost. Recommendations for the tightening sequence are also given to minimize the influence of the tightening sequence on the optimization results.

The new method proposed in this paper is dedicated to reducing the computational complexity and improving the efficiency while giving suggestions for the tightening sequence. Specifically, Section 1 demonstrates the EICM using a spring-node model and discusses the effect of the tightening sequence on the load distribution. Further, in Section 2, the simplification approach to the EICM is presented, and a new single-step tightening method is proposed. In order to validate the proposed method, an FE model for numerical simulation is presented in Section 3. Based on FEA, the optimization effects of the TAM and the new method under three different tightening sequences are further compared and analyzed in Section 5, and finally the recommended tightening sequence for single-step tightening is given.

2. The Influence of Tightening Sequence on Elastic Interaction

2.1. Validation of EICM Using the Spring-Node Model

Figure 1 illustrates the tightening process. Despite the nonlinear behavior observed in the contact between the flange and gasket [23], the elastic interaction can be equated to a spring if the deformation of both the bolt and the connector is within the range of the elastic deformation. Therefore, the process of bolt tightening can be described by a spring-node model [19,20,21]. Elastic deformation is required so that the deformation of each bolt is consistent with the deformation of its associated node. As a result, a set of equilibrium equations describing the elastic deformation behavior is generated during bolt tightening.

Tightening clockwise produces the following system of equilibrium equations for elastic deformation as we tighten sequentially from the first bolt to the n-th bolt:

$$\left[\begin{array}{ccccc}{K}_{1,1}^{-1}+K_{b1}^{-1}& K_{2,1}^{-1}& K_{1,3}^{-1}& \cdots & K_{1,n-1}^{-1}\\ K_{\mathrm{2,1}}^{-1}& K_{\mathrm{2,2}}^{-1}+K_{b2}^{-1}& K_{\mathrm{2,3}}^{-1}& \cdots & K_{2,n-1}^{-1}\\ K_{\mathrm{3,1}}^{-1}& K_{\mathrm{3,2}}^{-1}& K_{\mathrm{3,3}}^{-1}+K_{b3}^{-1}& \cdots & K_{3,n-1}^{-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ K_{n-\mathrm{1,1}}^{-1}& K_{n-\mathrm{1,2}}^{-1}& K_{n-\mathrm{1,3}}^{-1}& \cdots & K_{n-1,n-1}^{-1}+K_{b\left(n-1\right)}^{-1}\end{array}\right]\left\{\begin{array}{c}\Delta S_{1,n}\\ \Delta S_{2,n}\\ \Delta S_{3,n}\\ ⋮\\ \Delta S_{n-1,n}\end{array}\right\}=-\left\{\begin{array}{c}{K}_{1,n}^{-1}{S}_n\\ K_{2,n}^{-1}{S}_n\\ K_{3,n}^{-1}{S}_n\\ ⋮\\ K_{n-1,n}^{-1}{S}_n\end{array}\right\}$$

(1)

where $\Delta S_{i,n}$ is the value of the change in preload force of the i-th bolt after preloading the n-th bolt; $S_n$ is the preload force applied to the n-th bolt; $S_{i,j}$ is the elastic interaction stiffness between the i-th and j-th bolts; $S_{i,i}$ is the axial stiffness of the i-th bolt hole, and $S_{bi}$ is the i-th bolt stiffness. The $\Delta S_{i,n}$ is equal to

$$\Delta S_{i,n}=-{\displaystyle \frac{\left|\begin{array}{ccccc}{K}_{\mathrm{1,1}}^{-1}+K_{b1}^{-1}& \cdots & K_{1,n}^{-1}& \cdots & K_{1,n-1}^{-1}\\ ⋮& \ddots & ⋮& \cdots & ⋮\\ K_{i,1}^{-1}& \cdots & K_{i,n}^{-1}& \cdots & K_{i,n-1}^{-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ K_{n-\mathrm{1,1}}^{-1}& \cdots & K_{n-1,n}^{-1}& \cdots & K_{n-1,n-1}^{-1}+K_{b\left(n-1\right)}^{-1}\end{array}\right|}{\left|\begin{array}{ccccc}{K}_{\mathrm{1,1}}^{-1}+K_{b1}^{-1}& \cdots & K_{1,i}^{-1}& \cdots & K_{1,n-1}^{-1}\\ ⋮& \ddots & ⋮& \cdots & ⋮\\ K_{i,1}^{-1}& \cdots & K_{i,i}^{-1}+K_{bi}^{-1}& \cdots & K_{i,n-1}^{-1}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ K_{n-\mathrm{1,1}}^{-1}& \cdots & K_{n-1,i}^{-1}& \cdots & K_{n-1,n-1}^{-1}+K_{b\left(n-1\right)}^{-1}\end{array}\right|}}\cdot S_n$$

(2)

where the stiffness is determined by the system itself, and the elements contained within the determinant are determined by the tightening sequence. Therefore, as the tightening sequence is determined, $\Delta S_{i,n}$ and $F_n$ are linearly related and can be expressed as follows:

$$\Delta F_{i,n}=A_{i,n}\cdot S_n$$

(3)

where $A_{i,n}$ is the elastic interaction coefficient between the i-th bolt and the nth bolt. Assuming a total of m bolts in the flange system, the initial and residual preloads of the bolts can be expressed as the following relationship [12,13]:

$$\left\{\begin{array}{c}\begin{array}{c}{S}_{f1}\\ S_{f2}\end{array}\\ ⋮\\ S_{fn-1}\\ S_{fn}\end{array}\right\}=\left[\begin{array}{ccccc}{A}_{\mathrm{1,1}}& A_{\mathrm{1,2}}& \cdots & A_{1,n-1}& A_{1,n}\\ 0& A_{\mathrm{2,2}}& \cdots & A_{2,n-1}& A_{2,n}\\ ⋮& \cdots & \ddots & ⋮& ⋮\\ 0& 0& \cdots & A_{n-1,n-1}& A_{n-1,n}\\ 0& 0& \cdots & 0& A_{n,n}\end{array}\right]\left\{\begin{array}{c}\begin{array}{c}{S}_1\\ S_2\end{array}\\ ⋮\\ S_{n-1}\\ S_n\end{array}\right\}$$

(4)

where Equation (4) can also be expressed in the following form:

$${\mathit{S}}_f=\mathit{A}\cdot {\mathit{S}}_i$$

(5)

where ${\mathit{S}}_f$, ${\mathit{S}}_i$, and $\mathit{A}$ represent the residual load, the initial load, and the elastic coefficient matrix, respectively. Once the elastic coefficient matrix $\mathit{A}$ has been determined, optimization for bolt tightening can be achieved based on the target residual preload. The initial preload required to achieve the target residual preload can be obtained by inverting elastic coefficient matrix $\mathit{A}$:

$${\mathit{S}}_i={\mathit{A}}^{-1}\cdot {\mathit{S}}_f$$

(6)

The method described above is known as the EICM. Although the EICM can achieve the target preload distribution, it is necessary to measure the change in load of the remaining bolts as each bolt is tightened and recalculate the corresponding elastic coefficient matrix when the tightening sequence is changed.

2.2. The Impact of Tightening Sequence on Load Distribution

Figure 2 illustrates the division of the flange into two and five zones. Divide the flange into two zones. Zone I bolts the 1st to the 30-th and zone II bolts the 31-st to the 60-th. Tighten the first bolt of each zone in turn, i.e., 1→31, then tighten the second bolt of each zone, i.e., 2→32, and so on for all bolts. If the flange is divided into five zones, tighten them in a similar manner.

For dividing the flange into two zones, Equation (1) when tightening to the second zone can be expressed as follows:

$$\left[\begin{array}{cccccc}{K}_{\mathrm{11,11}}^{-1}+K_{b11}^{-1}& \cdots & K_{\mathrm{11,1}n}^{-1}& K_{\mathrm{11,21}}^{-1}& \cdots & K_{\mathrm{11,2}(n-1)}^{-1}\\ ⋮& \ddots & ⋮& \cdots & ⋮& ⋮\\ K_{1n,11}^{-1}& \cdots & K_{1n,11}^{-1}+K_{b1n}^{-1}& K_{1n,21}^{-1}& \cdots & K_{1n,2(n-1)}^{-1}\\ K_{\mathrm{21,11}}^{-1}& \cdots & K_{21,n1}^{-1}& K_{\mathrm{21,21}}^{-1}+K_{b21}^{-1}& \cdots & K_{\mathrm{21,2}(n-1)}^{-1}\\ ⋮& ⋮& ⋮& \cdots & \ddots & ⋮\\ K_{2(n-1),11}^{-1}& \cdots & K_{2(n-1),1n}^{-1}& K_{2(n-1),21}^{-1}& \cdots & K_{2(n-1),2(n-1)}^{-1}+K_{b2\left(n-1\right)}^{-1}\end{array}\right]\left\{\begin{array}{c}{A}_{\mathrm{11,2}n}\\ ⋮\\ A_{1n,2n}\\ A_{\mathrm{21,2}n}\\ ⋮\\ A_{2(n-1),2n}\end{array}\right\}=-\left\{\begin{array}{c}{K}_{\mathrm{11,2}n}^{-1}\\ ⋮\\ K_{1n,2n}^{-1}\\ K_{\mathrm{21,2}n}^{-1}\\ ⋮\\ K_{2(n-1),2n}^{-1}\end{array}\right\}$$

(7)

where $A_{ij,nm}$ is the stiffness between the j-th bolt in the i-th zone and the m-th bolt in the n-th zone. The elastic interaction between the bolts decreases with increasing distance [19]. Therefore, it is possible to consider only the stiffness of neighboring bolts. The system of equations combining the symmetry of the flange structure can be simplified to the following form:

$$\left[\begin{array}{ccccccccc}{K}_h^{-1}+K_b^{-1}& K^{-1}& & & & & & & \\ & K^{-1}& K_h^{-1}+K_b^{-1}& K^{-1}& & & & & \\ & & & \ddots & & & & & \\ & & & K^{-1}& K_h^{-1}+K_b^{-1}& & & & \\ & & & & & K_h^{-1}+K_b^{-1}& K^{-1}& & \\ & & & & & & & \ddots & \\ & & & & & & & K^{-1}& K_h^{-1}+K_b^{-1}\end{array}\right]\left\{\begin{array}{c}{A}_{\mathrm{11,2}n}\\ A_{\mathrm{12,2}n}\\ ⋮\\ A_{1n,2n}\\ A_{\mathrm{21,2}n}\\ ⋮\\ A_{2(n-1),2n}\end{array}\right\}=-\left\{\begin{array}{c}0\\ 0\\ ⋮\\ 0\\ 0\\ ⋮\\ K^{-1}\end{array}\right\}$$

(8)

where $K$ is the elastic interaction stiffness of neighboring bolts; $K_h$ is the axial stiffness of the bolt hole and $K_b$ is the bolt stiffness. Clearly, the elastic interaction coefficients beyond the coefficient between the neighboring bolts $A_{2(n-1),2n}$ are zero. $A_{2(n-1),2n}$ is constant.

A similar approach can be used to reach the same conclusion if the flange structure is divided into a different number of zones. Consequently, the resulting residual preloads of the bolts in the same zone k are, respectively,

$$\left\{\begin{array}{c}{S}_{fkf}=S_{kf}+A\cdot S_{k(f+1)}+A\cdot S_{(k-1)e}\\ S_{fkc}=S_{kc}+A\cdot S_{k(c+1)}\\ S_{fkl}=S_{kl}\end{array}\right.$$

(9)

where $S_{fkf}$ is the residual preload of the first tightened bolt in zone k; $S_{kf}$ is the initial preload of the first tightened bolt in zone k; $A$ is the elastic interaction coefficient between the neighboring bolts; $S_{k(f+1)}$and $S_{(k-1)e}$ are the initial preloads applied to bolts adjacent to the first tightened bolt of zone k; $S_{fkc}$ is the residual preload of bolts in zone k except for the first and last tightened bolts; $S_{kc}$ is the initial preload of bolts in zone k except for the first and last tightened bolts; $S_{k(c+1)}$ is the initial preload applied to bolt adjacent to the bolts in zone k except for the first and last tightened bolts; $S_{fkl}$ is the residual preload of the last tightened bolt in zone k; $S_{kl}$ is the initial preload of the first tightened bolt in zone k.

Thus, when the same initial preload is applied, the average residual preload of the bolt for different numbers of zones is

$${\stackrel{-}{S}}_f={\displaystyle \frac{{\sum }^n{\sum }^{m-2}{S}_{fkc}}{n(m-2)}}+{\displaystyle \frac{{\sum }^n(S_{fkf}+S_{fkl})}{2n}}=S_{fkc}$$

(10)

where $m$ is the number of bolts in zone k; $n$ is the number of zones. Consequently, the average residual preloads of different tightening sequences are equal.

The tightness of the connection is influenced by the uniformity of the residual preload of the bolts [11]. The uniformity of preload is measure by the standard variance (SD). When the same initial preload is

$$\sigma =\sqrt{{\displaystyle \frac{{\sum }^n{\sum }^{m-2}{\left(S_{fkc}-{\stackrel{-}{S}}_f\right)}^2+{\sum }^n(S_{fkf}-{\stackrel{-}{S}}_f{)}^2+{\sum }^n(S_{fkl}-{\stackrel{-}{S}}_f{)}^2}{m\times n}}}=\sqrt{{\displaystyle \frac{2}{m}}}·A·{\stackrel{-}{S}}_f$$

(11)

Because the number of bolts $m\times n$ in the flange structure is fixed, the number of bolts in zone k is inversely proportional to the number of zones. Hence, as the number of zones increases, the degree of preload dispersion increases.

3. New Methodology for Bolt Tightening Sequence Optimization

3.1. Tetraparametric Assembly Method

To reduce the computational cost of the EICM, the TAM was developed [16]. As shown in Figure 3, the closer the relative distance between bolts, the greater the elastic interaction [19]. Therefore, the TAM focuses on the elastic interactions between adjacent bolts or bolts separated by one bolt. Elastic interactions between bolts that are further apart are not considered, and the elastic interaction coefficients are zero. By measuring the possibilities of all elastic interaction coefficients under simplified conditions, appropriate elastic interaction coefficients are input into the elastic coefficient matrix based on different tightening sequences. It is worth noting that since the TAM is a simplification for the ECIM, the parameters of the TAM and EICM are of the same nature. Therefore, for the same structure, the magnitude of the load does not affect the parameters of the TAM. This selective consideration effectively reduces the computational cost of the EICM. Finite element analysis and physical experiments for different tightening sequences indicate that the TAM can achieve a uniform distribution of loads effectively.

3.2. A Novel Single-Step Bolt Tightening Method for Multi-Bolt Flange Structures

Like the TAM, the proposed method also aims to reduce the EICM by reasonably simplifying the elastic interaction coefficients. Unlike the TAM, the simplification in the proposed method is achieved by simulating the actual flange bolt tightening process through a local bolt tightening. This simplification approach is more in line with the actual situation of flange bolt tightening. The schematic diagram of the proposed methodology is shown in Figure 4. After processing the parameters obtained for a specific tightening sequence, the optimization of the bolt load can be achieved given only the tightening sequence and the target residual preload.

The range of elastic interactions considered in the proposed method is the eight bolts adjacent to each other on both sides. Therefore, considering the flange assembly with twelve bolts illustrated in Figure 5, the preset tightening sequence is 1→2→3→4→5→9→10→11→12.

The effects of distant bolts 6, 7, and 8 on bolt 1 are neglected with consideration limited to the impact of bolts closer to bolt 1:

$$\left\{\begin{array}{cc}{A}_{\mathrm{1,1}}=1& j=1\\ A_{1,j}={\displaystyle \frac{S_{1,j}-S_{1,j-1}}{S_j}},& 2\le j\le 5\cup 9\le j\le 12\\ A_{1,j}=0,& j=\mathrm{6,7},8\end{array}\right.$$

(12)

where $S_{1,j}$ and $S_{1,j-1}$ are the preload on the 1st bolt when the j-th bolt and (j − 1)-th bolt are tightened, respectively. The coefficient $A_{1,j}$ is the influence of tightening the j-th bolt on the 1st bolt. This gives a complete set of elastic interaction matrix coefficients affecting the 1st bolt. Both $A_{1,2}$ and $A_{1,12}$ denote the coefficients of bolts adjacent to the 1st bolt. Notably, the coefficient $A_{1,12}$ depicts a more complex coupling condition compared to $A_{1,2}$. So, these two coefficients represent predetermined values at different coupling complexities. As shown in Equation (1), in a flange structure with consistent bolt spacing, the elastic interaction matrix coefficients are uniform for each bolt when the bolt spacing and coupling are identical. Therefore, a recursive strategy can logically start with the elastic interaction coefficient of bolt 1:

$$\left\{\begin{array}{cc}{A}_{i,1}=A_{i-\mathrm{1,12}},& 1<i\le 12\\ A_{i,j}=A_{i-1,j-1},& 1<i,j\le 12\end{array}\right.$$

(13)

The elastic interaction coefficients for each bolt can be determined by Equation (8), which are then used to build the matrix ${\mathit{A}}^{\prime }$. Matrix ${\mathit{A}}^{\prime }$ can be represented as follows:

$${\mathit{A}}^{\prime }=\left[\begin{array}{cccc}{A}_{\mathrm{1,1}}& A_{\mathrm{1,2}}& \cdots & A_{\mathrm{1,12}}\\ A_{\mathrm{2,1}}& A_{\mathrm{2,2}}& \cdots & A_{\mathrm{2,12}}\\ ⋮& ⋮& \ddots & ⋮\\ A_{\mathrm{12,1}}& A_{\mathrm{12,2}}& \cdots & A_{\mathrm{12,12}}\end{array}\right]$$

(14)

The matrix ${\mathit{A}}^{\prime }$ contains the simplified elastic interaction coefficients between all bolts. With the target residual preload ${\mathit{S}}_f$ and the specific tightening sequence 1→7→2→8→ 3→9→4→10→5→11→6→12, the focus is on solving the initial preload ${\mathit{S}}_i$. As described in Equation (4), when the bolt tightening sequence is the same as the ordering of the ranks in the matrix, the matrix is an upper triangular matrix. Using this property, the elastic interaction coefficients in the specific tightening sequence can be retained, and the transformation matrix $\mathit{E}$ can be obtained by performing row operations on the identity matrix in sequence according to the specific tightening sequence. Matrix ${\mathit{A}}^{\prime }$ suffers simultaneous row and column alterations using $\mathit{E}$. Hence, by preserving solely the upper triangular constituent of the correction matrix, the matrix $\mathit{A}$ is obtained:

$$\mathit{A}=\left\{\begin{array}{c}\mathit{E}\cdot {\mathit{A}}^{\prime }\cdot {\mathit{E}}^T,i\ge j\\ 0 , i<j\end{array}\right.$$

(15)

$$\mathit{A}=\left[\begin{array}{ccccc}{A}_{\mathrm{1,1}}& A_{\mathrm{1,7}}& \cdots & A_{\mathrm{1,6}}& A_{\mathrm{1,12}}\\ 0& A_{\mathrm{7,7}}& \cdots & A_{\mathrm{7,6}}& A_{\mathrm{7,12}}\\ 0& 0& \ddots & ⋮& ⋮\\ 0& 0& \cdots & A_{\mathrm{6,6}}& A_{\mathrm{6,12}}\\ 0& 0& \cdots & 0& A_{\mathrm{12,12}}\end{array}\right]$$

(16)

Once matrix $\mathit{A}$ is constructed, the initial preload can be calculated by Equation (6).

The parameters in the EICM, the TAM and the proposed method are expressed in different symbols, but the physical meaning and the method of operation are similar. The number of bolts to be tightened and the number of parameters to be measured in the three methods for an O-ring with n (n ≥ 9) bolts are given in

Table 1. Comparison of different methods of calculating costs.

	EICM	TAM	The Proposed Method
the number of bolts	n	5	9
the number of parameters	n(n − 1)/2	4	8

.

4. Experimental Validation

To validate the effectiveness of the proposed method, a finite element model is created in ABAQUS 2022. As demonstrated in Figure 6, the model is the area where the spacecraft combustion chamber and nozzle are connected. The model has 60 M16 bolts with all materials defined as steel (E = 210 GPa, ν = 0.3, ${\sigma }_s$ = 355 MPa). The friction coefficient μ is 0.2. An initial preload of 0.1 N is applied to the untightened bolts, whereas for tightened bolts, a fixed length is maintained post-preload application. The C3D8R element type is adopted with all meshes consisting of hexahedral elements. The final assembly has 6,884,250 nodes and 5,530,800 elements.

To validate that the FE model effectively predicts load change in bolts during joint assembly, a test bench with identical physical properties to the FE model is developed, as shown in Figure 7. We compared FEA and experimental errors for three tightening sequences: clockwise, diagonal and star pattern tightening [24,25]. In the case of clockwise sequential tightening, the bolts are tightened in the sequence 1→2→3→⋯→60. The diagonal pattern tightening sequence and the five-points star pattern tightening sequence are shown in Figure 2a and Figure 2b, respectively. The diagonal tightening sequence divides the bolts into two zones, zone I bolts 1 to 30 and zone II bolts 31 to 60. Tighten the first bolt of each zone in turn, i.e., 1→31; then, tighten the second bolt of each zone, i.e., 2→32, and so on for all bolts.

A torque of 147 N⋅m is applied to the bolt using a torque wrench. The changes of bolt length in the loading process are measured by an ultrasonic rangefinder with a precision of 0.001 mm. It should be noted that the torque wrench has an application torque accuracy of ±3%. The sleeve indicated in Figure 7 guarantees the consistent positioning of the ultrasonic transducer above the bolt throughout measurement. Combined with the bolt size and material, the calculated bolt stiffness is 237,233 N/mm [26]. So, according to Hooke’s law, the preload can be calculated from the change in bolt length. Given the linear relationship between both torque and preload [27], an applied torque of 147 N⋅m is equivalent to applying an initial preload of 54,652 N. The amount of preload ensures that the resulting stress does not exceed 80% of the yield strength.

The experimental steps are as follows:

• Measure the initial length of each bolt.
• Tighten the 1st bolt and fix the probe on the 1st bolt, and then tighten the other bolts in sequence to obtain the change in preload force of the 1st bolt.
• After all the bolts are tightened, measure the change in the length of each bolt to obtain the residual preload force.

The FEA results are compared with the experimental results. Figure 8 illustrates the average residual preload and maximum relative error of all bolts, and Figure 9 shows the variation in the 1st bolt for different tightening sequences. The FEA results and experimental results exhibit close agreement.

The discrepancies between the experimental outcomes and the FEA results can be attributed to four primary sources of error: the precision of the torque applied via the wrench, the accuracy of the ultrasonic measurement techniques, the discrepancies between the FEA parameters and the actual frictional coefficients, and the thread damage incurred due to the repeated disassembly of the bolts [28].

The experimental results for the change in preload of the 1st bolt are closer to the FEA results than the residual preload of the individual bolts. The improvement in accuracy is related to the measurement method of the 1st bolt. The ultrasonic sensor is fixed to the head of the 1st bolt, thereby reducing the possibility of error that can arise from fluctuations in environmental variables.

In conclusion, these results demonstrate that the FE model can accurately obtain the preload force of the bolts.

5. Numerical Examples

In order to verify the effectiveness of the proposed method, the numerical examples of three typical tightening sequences are optimized to achieve a uniform distribution of 54,652 N residual preload force by the proposed method and the TAM. Concurrently, a preload force of 54,652 N is uniformly applied to each bolt across all tightening sequences, establishing a benchmark for comparative optimization analysis. The bolt tightening sequence includes the three tightening sequences in Section 4: clockwise sequence tightening sequence, diagonal pattern tightening sequence and five-point star pattern tightening sequence. The specific results of the TAM parameters and the proposed methodological parameters are shown in

Table 2. The specific results of the TAM parameters.

α	β	γ	δ
−0.0499	−0.0503	0.0032	0.0039

and

Table 3. The specific results of the proposed methodological parameters.

A1,2	A1,3	A1,4	A1,5	A1,57	A1,58	A1,59	A1,60
−0.0499	0.0039	0.0007	0.0001	0.0003	0.0012	0.0007	−0.0499

, respectively.

Figure 10, Figure 11 and Figure 12 show the FE results for the three tightening sequences. In the radar diagram, the axis is the bolt serial number, and the concentric cycle is the contour of the residual preload. The mean relative error (MRE) is used to evaluate the closeness of optimization results from different methods to the target value. The uniformity of preload is measure by the SD. The comparison results are shown in

Table 4. Comparison results of the proposed method with TAM.

	Clockwise Sequential Tightening			Diagonal Pattern Tightening			Five-Points Star Pattern Tightening
	MRE	SD	Average Residual Preload	MRE	SD	Average Residual Preload	MRE	SD	Average Residual Preload
applying fixed preload	4.53%	466.07	52,174.6	4.60%	672.14	52,179.3	4.62%	1077.55	52,182.5
TAM	0.168%	32.73	54,743.9	0.136%	29.13	54,726.4	0.64%	876.82	54,718.2
the proposed method	0.037%	7.54	54,622.7	0.152%	31.69	54,584.2	0.132%	55.30	54,581.0

. The numerical examples manifest the efficacy of the proposed method in bolt tightening optimization.

Numerical simulation results indicate that compared to the fixed application of preload, the proposed method and TAM can achieve a uniform distribution of bolt loads with the error from the target load being within an acceptable range. There is no significant difference between the two methods when tightening in a clockwise sequence or a diagonal sequence. However, when using the five-point star tightening sequence, the preload distribution obtained by the proposed method is more uniform than that obtained by TAM. This phenomenon occurs because both methods simplify the elastic coefficient matrix, and the five-point star tightening sequence results in a more discrete distribution of loads. The proposed method more closely approximates the actual tightening conditions, thus providing a more accurate approximation of the elastic coefficient matrix for this sequence.

It is noteworthy that the conclusions drawn from the applying fixed preload are consistent with those derived in Section 2.1. As the number of partitions in the flange structure increases (the clockwise sequence can be considered as dividing the bolts into one zone, diagonal tightening and five-points star tightening can be considered as dividing the flange into two zones and five zones, respectively), the average residual preload of the bolt is approximated and the load distribution is more discrete when the same initial preload is applied. Combining the optimization results of the two methods, it can be observed that as the number of zones increases, the accuracy of the approximation of the elastic coefficient matrix decreases. Therefore, when using a single tightening method that simplifies the elastic coefficient matrix, it is best to use the clockwise tightening sequence. If considering the issue of bolt relaxation caused by coaxiality in actual production [29], diagonal tightening can also be adopted.

6. Conclusions

This paper studies the impact of tightening sequence on bolt load distribution using the spring-node model. It is observed that when only the elastic interaction between adjacent bolts is considered, the average residual preload of bolts is equal under different tightening sequences when the same initial preload is applied. The degree of load distribution dispersion increases as the complexity of the tightening sequence increases. The paper introduces a new single-step tightening method for flanges. This method can be achieved by tightening only local bolts, reducing computational costs compared to EICM. Finite element analysis results confirm the effectiveness of the proposed method in optimizing bolt tightening. When using such single-step tightening methods that simplify the elastic coefficient matrix, it is preferable to use either clockwise tightening or diagonal tightening.