Load and Deformation Distribution Along the Bolt During Assembly Process

Abstract

Bolt connection structure is widely utilized in modern large-scale equipment. Understanding the stress distribution along the bolt during assembly process is essential for improving its reliability. In this paper, assembly torque model is established to formulate the relationship between tightening torque and preload force, and a linear proportionality between them is obtained. The stress distribution during assembly process is modeled. Rotation angle and displacement distribution can be obtained from the proposed method. To validate our model, numerical simulation analysis is carried out to obtain the load and deformation distribution under different conditions. Comparison with the previous literature confirms the accuracy of the proposed model.

1 Introduction

Bolt connection structure is widely utilized in modern large-scale equipment, such as aviation spacecraft, wind turbine spindles, petrochemical pipeline docking and so on. In practical applications, fasteners are inevitably affected by external excitation, and the behavior of the contact surface is extremely intricate. Understanding the stress distribution in the bolt during the assembly process is essential for improving its reliability. However, due to the complexity of inter-thread contact and friction, accurately modeling the bolt assembly process is challenging.

Extensive studies have been done along the bolt contact problem. For the bolt head contact problem, Motosh developed a friction model that accurately establishes the relationship between tightening torque and axial force for threaded fasteners [1]. According to this model, the tightening torque is the sum of the thread contact torque and the bolt head friction torque. Nassar et al. modeled the stress distribution on a simplified bolt head surface and derived four distinct friction distribution, considering the effect of the tightening speed [2].

For the screw thread contact problem, Sopwith developed a widely accepted load distribution model [3]. In this model, the load between the threads is treated as a concentrated force acting in the middle of the screw thread cross-section. The strain in the bolt and nut threads is divided into three components: bending deflection in the threads of bolt and nut, axial recession due to radial compression in the threads of bolt and nut, axial recession due to radial contraction of bolt and expansion of nut caused by radial pressure between bolt and nut. Yamamoto further expanded the analysis of the deformation of the screw thread cross-section, categorizing the elastic deformation into five types: bending deformation δ₁, shearing deformation δ₂, inclination deformation of the thread root δ₃, shearing deformation of the thread root δ₄, and deformation due to radial expansion (nut) and radial shrinkage (bolt) δ₅ [4]. Besides, Nassar proposed five pressure distribution models for internal and external threads [5]. Stoeckley and Macke investigated the effect of tapered threads on load distribution [6].

In terms of experimental investigations into bolt stress distribution, Goodier employed an extensometer to measure both radial and axial deformation of nuts [7]. Goodier's experiments revealed that the load at the free end or in the middle of bolt is small, but becomes concentrated near the loaded end. It is important to note that both Goodier's experiments and earlier photo-elasticity studies have certain limitations, as their experimental models fail to accurately represent the 3D geometry of nuts and bolts. Hetenyi conducted 3D photo-elastic experiments using a series of nuts and identified stress concentration phenomenon [8]. In Hetenyi's experiments, the stress near the loaded face tends to decrease, which contrasts with Sopwith's model and Goodier's experiments showing a parabolic increase tread, as shown in Fig. 1. Brown and Hickson replicated Hetenyi's experiments using a new photo-elastic material called Fosterite [9]. Their results were consistent with Hetenyi's findings, showing a reduction in load distribution at the loaded face of the nut, as also observed in Kenny and Patterson's experiments [10]. This reduction may be attributed to the fact that the threads on the nut and bolt are not fully formed at the loaded face, resulting in lower thread stiffness and load capacity.

Fig. 1

Comparison of Sopwith’s results with those of Hetenyi and Goodier [11]

In addition, several other researchers have used FE software to investigate bolt stress [12]. Maruyama employed FE analysis and copper plating method to study the stress in nut-bolt joints [13]. Honglin Xu et al. studied the stress distribution of casing threaded joints according to the Yamamoto method, and established a 2D finite element model [14]. The theoretical results were found to be generally consistent with the FE simulation. Shikun Lu compared the stress distribution obtained from Yamamoto model with that from a 3D finite element model [15]. The findings revealed that the load and stress distribution results from both methods were essentially identical.

In previous theoretical models, the bolt axial stress distribution model was established, but the effect of the addition of tightening torque on the stress distribution during the actual assembly process was not considered. Similarly, in previous experiments and simulations, the axial force was added to simulate the preload force, without considering the intricacies of the assembly process. To address these limitations, this article aims to investigate the stress distribution of bolt fasteners during and after the assembly process, combined with Yamamoto thread deformation model.

The rest of this paper is organized as follows. In Sect. 2, the relationship between the tightening torque T and the preload force F is deduced. In Sect. 3, stress distribution along the bolt during the tightening process is studied. In Sect. 4, numerical simulations and experiments are carried out and relevant results are compared with the proposed model. Finally, the conclusions are summarized.

2 Assembly Torque Model

The purpose of this section is to establish the relationship between tightening torque T and preload force F. A brief analysis shows that during the bolt assembly process, as depicted in Fig. 2, the tightening torque T is equal to the sum of the bolt head contact torque T_t and the thread contact torque T_b. In the subsequent discussion, T_t and T_b are analyzed separately.

Fig. 2

Assembly torque analysis

2.1 Bolt Head Contact Torque T_t

Before analyzing the bolt head contact torque, it is necessary to determine the stress distribution on the contact surface. The following assumptions are made:

• i. The bolt head and the support surface are considered as rigid bodies.

• ii. The contact stress between the bolt head and the support surface is uniformly distributed.

On the contact surface, the moment generated by friction can be expressed as

$$ T_{t} = \iint {\mu_{t} \sigma r}{\text{d}}A, $$

where normal stress $\sigma = \, F/A_{h}$, $A_{h}$ is the contact area of the bolt head, $\mu_{t}$ is the friction coefficient of the bolt head contact surface. Then the above equation can be formulated as

$$ T_{t} = \mu_{t} Fr_{t} , $$

(1)

where $r_{t} = \iint {r{\text{d}}A/A_{h} }$.

2.2 Thread Contact Torque T_b

According to Yamamoto method, the bolt thread can be simplified as a circular helix with radius a (a is effective radius of the thread), on which the thread contact force is assumed to act. The helix equation can be expressed as

$$ x = a\cos \theta , \, y = b\cos \theta , \, z = b\theta , $$

where θ denotes the angular coordinate of a point on the helix, b = p/(2π), p is the pitch of screw.

The global and local coordinate systems are established on the helix. As shown in Fig. 3, e (x, y, z) is the global coordinate system with the origin at the bottom center of the bolt, z-axis along the axial direction of the bolt, and x-axis pointing to the start of the helix. For any point O_l on the helix, there are three local coordinates. The direction of e₁ (x₁, y₁, z₁) can be obtained by rotating e counterclockwise about z-axis by θ_l (θ_l is the coordinate of the helix angle corresponding to point O_l). e₂ (x₂, y₂, z₂) is obtained by rotating e₁ counterclockwise about x₁-axis by α (α is thread lead angle). e₃ (x₃, y₃, z₃) is obtained by rotating e₂ counterclockwise about y₂-axis by β (β is half of thread included angle).

Fig. 3

Coordinate systems on the helix

The contact forces acting on the thread surface are assumed to consist of a normal force f and a friction force μf, where μ is the friction coefficient between the threads. In e₃ coordinate system, the thread contact force can be expressed as (0, μf, −f). According to the coordinate transformation relationship, in e₁ coordinate system, the projections of the contact forces on the coordinate axes are

$$ f_{z1} = \left( {\mu \sin \alpha - \cos \alpha \cos \beta } \right)f{ = } - \left( {\sqrt {\mu^{2} + \cos^{2} \beta } \cos \left( {\alpha + \xi } \right)} \right)f, $$

$$ f_{y1} = \left( {\mu \cos \alpha + \sin \alpha \cos \beta } \right)f = \left( {\sqrt {\mu^{2} + \cos^{2} \beta } \sin \left( {\alpha + \xi } \right)} \right)f, $$

where $\xi = \mu /\cos \beta$.

The thread contact torque and the preload force satisfy

$$ T_{b} = \int\limits_{0}^{s} {f_{y1} } a{\text{d}}s, \, F = \int\limits_{0}^{s} {f_{z1} } {\text{d}}s, $$

where s is the arc length coordinate of a point on the helix. Thus, we can obtain the relationship between thread contact torque T_b and preload force F.

$$ T_{b} = Fa\tan (\alpha + \xi ) $$

(2)

Then the tightening torque can be expressed as

$$ T = T_{b} + T_{t} = k_{FT} F, $$

(3)

where $k_{FT} = \mu_{t} r_{t} + a{\text{tan}}(\alpha + \xi )$.

3 Stress Distribution Along the Bolt During Assembly

In Lu's study, it is evident that the nut stress is very small compared to the bolt stress [15]. Consequently, when solely considering stress distribution along the bolt, the overall compressive deformation of the nut can be disregarded.

3.1 Thread Contact Stiffness

As illustrated in Fig. 4, according to Yamamoto model [4], the thread contact deformation is categorized into five types [5]: bending deformation δ₁, shearing deformation δ₂, inclination deformation of the thread root δ₃, shearing deformation of the thread root δ₄, and deformation due to radial expansion (nut) and radial shrinkage (bolt) δ₅.

Fig. 4

Thread contact deformation

The formulas for these five kinds of deformations are as follows.

$$ \delta_{1} = \frac{{3f\sin \beta \left( {1 - \nu^{2} } \right)}}{4E}\left\{ {\left[ {1 - \left( {2 - \frac{{d_{m} }}{d}} \right)^{2} + 2\ln \left( {\frac{d}{{d_{m} }}} \right)} \right]\cot^{3} \beta - 4\left( \frac{h}{d} \right)^{2} \tan \beta } \right\}, $$

$$ \delta_{2} = \frac{{6f\sin \beta \cot \beta \left( {1 + \nu } \right)}}{5E}\ln \left( {\frac{d}{{d_{m} }}} \right), \, $$

$$ \delta_{3} = \frac{{12f\cos \beta \left( {1 - \nu^{2} } \right)h}}{{\pi Ed^{2} }}\left( {h - \frac{{d_{m} }}{2}\tan \beta } \right), $$

$$ \delta_{4} = \frac{{2f\cos \beta \left( {1 - \nu^{2} } \right)}}{\pi E}\left[ {\frac{p}{d}\ln \left( {\frac{{p + {d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}}}{{p - {d \mathord{\left/ {\vphantom {d 2}} \right. \kern-0pt} 2}}}} \right) + \frac{1}{2}\ln \left( {4\frac{{p^{2} }}{{d^{2} }} - 1} \right)} \right], $$

$$ \delta_{5s} = \frac{{fd_{p} \left( {1 - \nu_{s} } \right)\cos \beta \tan^{2} \beta }}{{2pE_{s} }}, \, \delta_{5m} = \frac{{fd_{p} \cos \beta \tan^{2} \beta }}{{2pE_{m} }}\left( {\frac{{d_{0}^{2} + d_{p}^{2} }}{{d_{0}^{2} - d_{p}^{2} }} + \nu_{m} } \right), $$

where E is Young's modulus, υ is Poisson's ratio, h is the thread pitch line height, d is the thread root width, d_m is the thread width at the pitch line, d_p is the effective diameter of the thread, d₀ is the outer diameter of the nut, and p is the pitch of thread. The subscripts m and s denote the nut and bolt, respectively.

Consequently, the thread contact stiffness of the bolt and nut can be expressed as follows.

$$ k_{i} = f{\text{cos}}\beta /\delta_{i} , $$

where $\delta_{i} = \delta_{1i} + \delta_{2i} + \delta_{3i} + \delta_{4i} + \delta_{5i},$ i represents s and m, respectively.

3.2 Tensile and Torsional Equilibrium Equations

After calculating the thread contact stiffness, we can proceed to establish the tensile and torsional equilibrium equations for the bolt.

Referring to Fig. 5, the length of the connected part is l_m1, the length of the nut is l_m2, the length of the free section of the bolt is l_m3, and the origin of the global coordinate system is located at the bottom center of the nut. The axial coordinates for the material points on the bolt and nut are denoted as z_s and z_m, respectively.

Fig. 5

Element force analysis

The tensile and torsional equilibrium equations of the bolt can be obtained from the element force analysis.

$$ {\text{d}}F + f_{z1} {\text{d}}s = 0,{\text{ d}}T - f_{y1} a{\text{d}}s = \, 0. $$

Combining the coordinate transformation equations and the relationship between stress and strain,

$$ F = EA\frac{{{\text{d}}u}}{{{\text{d}}z_{s} }},\quad T = GI_{p} \frac{{{\text{d}}\varphi }}{{{\text{d}}z_{s} }},\quad {\text{d}}s = \frac{{{\text{d}}z}}{{{\text{sin}}\alpha }} = 0, $$

the above equilibrium equations can be expressed as follows.

$$ \begin{gathered} EA\frac{{{\text{d}}^{{2}} u}}{{{\text{d}}z_{s}^{2} }} + \left( {\frac{\mu }{{{\text{cos}}\beta }} - {\text{cot}}\alpha } \right)k_{s} \delta_{s} = 0, \, \hfill \\ GI_{p} \frac{{{\text{d}}^{{2}} \varphi }}{{{\text{d}}z_{s}^{2} }} - \left( {\mu \frac{{{\text{cot}}\alpha }}{{{\text{cos}}\beta }} + 1} \right)ak_{s} \delta_{s} = 0, \hfill \\ \end{gathered} $$

where A denotes the bolt cross sectional area, I_p denotes the second polar moment of area of the bolt cross-section, G stands for the shear modulus. u and φ represent the displacement and rotation angle of the bolt, respectively.

3.3 Supplementary Deformation Conditions

During the process of bolt tightening, the material points along the bolt and nut satisfy the following deformation criteria:

• i. Matching material points occupy the same position along the z-axis.

  $$ z_{s} + u - \delta_{s} {\text{cos}}\alpha = z_{m} + \delta_{m} {\text{cos}}\alpha $$

  (4)

• ii. Assuming no deformation of the reference helix, the corresponding angles of the matching material points are the same.

  $$ \theta_{m} (z_{m} ) = \theta_{s} = \theta_{s0} (z_{s} ) - \varphi (z_{s} ), $$

  (5)

  where

  $$ \theta_{m} = {{2\pi z_{m} } \mathord{\left/ {\vphantom {{2\pi z_{m} } p}} \right. \kern-0pt} p},\quad \theta_{s0} = {{2\pi z_{s} } \mathord{\left/ {\vphantom {{2\pi z_{s} } p}} \right. \kern-0pt} p}. $$

3.4 Boundary Conditions

At the boundary z_m = 0, the tension and torque are both zero.

$$ EA\frac{{{\text{d}}u}}{{{\text{d}}z_{s} }}|_{{z_{m} = 0}} = 0,\quad GI_{p} \frac{{{\text{d}}\varphi }}{{{\text{d}}z_{s} }}|_{{z_{m} = 0}} = 0 $$

(6)

At the boundary z_m = l_m2, the displacement and torque satisfy

$$ u{\mkern 1mu} |_{{z_{m} = l_{m2} }} = - \frac{{Fl_{m1} }}{EA},\quad GI_{p} \frac{{{\text{d}}\varphi }}{{{\text{d}}z_{s} }}|_{{z_{m} = l_{m2} }} = T_{b} $$

(7)

Combining the complementary conditions with the boundary conditions, the displacement and angle of the bolt in the contact section can be obtained. Afterwards, the deformation information of the bolt in the non-contact section can be solved according to the thread contact torque, and then the deformation of the whole bolt can be deduced.

4 Numerical Simulation Analysis

4.1 Load Percentage Analysis

After establishing the theoretical model, the following numerical simulation analysis is carried out. The loading on each thread turn is analyzed and compared with the existing literature. In order to validate the accuracy of this model and also to compare the results with the existing literature, M10 and M20 bolts are used in the simulation. The different bolt parameters are listed in Table 1. The cross-section of M10 bolt head is a circle with a diameter of 16 mm, and the cross-section of M20 bolt head is a square hexagon with a side length of 17 mm.

Table 1 Formatting sections, subsections and subsubsections

M10 bolts with only five turns of thread are used to study the load distribution on each turn of thread. Comparison of the results of this model with those of other studies [12] is shown in Fig. 6. A satisfactory agreement between different calculation methods can be found. The first three turns of threads near the loaded face account for almost 80% of the total preload. The last turn of the thread carries little force.

Fig. 6

Load distribution on each turn of threads

In order to investigate the load percentage of the bolt with more turns of threads, M20 bolts with 20 turns of effective thread are use. The load proportion analysis of the first five turns of threads is shown in Fig. 7a. It can be seen that the axial force on the first five turns of threads constitutes approximately 70% of the total preload, and this proportion gradually increases as the coefficient of friction μ increases. This phenomenon suggests that as the thread friction coefficient increases, the load concentration becomes more pronounced. Then the loads on the first five turns of threads are represented by the histogram in Fig. 7b, which are consistent with Lu's results [15].

Fig. 7

The load on the first five turns of threads

4.2 Rotation Angle and Displacement Distribution

With this model, if the tightening torque applied to the bolt is known, the deformation information of the whole bolt can be obtained. In order to perform simple calculation, assuming an applied torque of 1.4 N m for the M10 bolt, the coefficient of friction at the contact interface is 0.1. The preload force can be calculated as 1 kN. The values of l_m1, l_m2, l_m3 are 0.2 mm, 0.3 mm, 0.2 mm, respectively. Then the rotation angle and displacement distribution are depicted in Fig. 8.

Fig. 8

The rotation angle and displacement distribution

Therefore, based on the tightening torque of the bolt, we can obtain information about the angle of the bolt head and the overall displacement of the free section of the bolt. This provides theoretical support for the use of corner control to add bolt preload in engineering design.

5 Conclusions

In this paper, the theoretical modeling and numerical simulation of the axial force and torque distribution during bolt assembly are performed. Assembly torque model is established. The linear relationship between tightening torque T and preload force F is formulated. Stress distribution along the bolt during assembly process is modeled. The tensile and torsional differential equations are developed by element force analysis of the bolt. Combined with the supplementary deformation conditions and boundary conditions, the force and displacement distribution can be deduced. Numerical simulation analysis is carried out. The loading on each thread turn is analyzed and compared with the existing literature. A satisfactory agreement between different calculation methods can be found. The study of load and deformation distribution of the bolt is of guiding significance for the assembly process in practical engineering, and at the same time provides initial conditions for the study of vibration dynamics.