Calculation of the effective bearing contact radius for precision tightening of bolted joints

Abstract

Torque control is usually the only method for tightening bolts in some precision assembly applications. However, the scatter of the torque–tension relationship may significantly decrease the accuracy of the preload, which conflicts with the high requirement for mechanical accuracy in such precision assemblies. An important, but often ignored, factor affecting the accuracy of the torque–tension relationship is the effective bearing contact radius. In this article, a three-dimensional finite element model of a typical bolted joint was developed to obtain the actual bearing pressure distribution, based on which the effective bearing contact radius can be further calculated. Then, a parametrical study was conducted to systematically investigate the effects of various geometrical, material, and frictional factors on the effective bearing contact radius. Based on the numerical results, a comprehensive and quantitative evaluation of the relative accuracy of each traditional method of calculating the effective bearing contact radius was made. In particular, it was found that the effective bearing contact radius, calculated based on the assumption of uniform bearing pressure distribution, was always relatively accurate regardless of the geometrical, material, and frictional conditions considered. This study will be helpful in increasing the accuracy of preload, thus ensuring mechanical accuracy and quality for precision assemblies.

Keywords

Bolted joints torque control effective bearing contact radius precision assembly finite element analysis

Introduction

Bolted joints are used widely in many applications due to their ease of assembly and disassembly. The safety, reliability, and quality of bolted joints are determined largely by the magnitude and stability of the clamp load or preload. To achieve a desired preload, various tightening methods have been developed, such as torque control, torque–angle control, stretch control, and yield control.¹ Among these methods, the torque control method is the most commonly used because it can be implemented readily without using complex or expensive tools. In several precision assembly applications, such as some small-batch precision optical systems,² the torque control method is usually the only applicable tightening method for bolted joints because of the restrictions on operating space and the limitations of assembly cost. In order to ensure the preload accuracy in precision assembly applications, the allowed percentage error of the torque–tension relationship might be as small as several percents. For example, Li et al.² numerically calculated the effects of preload on the deformation of a 400-mm class reflecting mirror which is a key part in a high-power solid laser facility. They found that the peak-to-valley (PV) value of the wavefront can change 2 nm when the preload changes only 1 N. For precision optical systems, the PV value is typically required to be smaller than 63 nm (i.e. $λ / 10$ , $λ = 0.6328 μ m$ ). Thus, even several percent errors of the preload (e.g. when the preload is several hundred Newton) may cause a considerable change on the PV value. Therefore, there is a need to improve the accuracy of the torque–tension relationship to ensure preload accuracy for precision assemblies using a torque control tightening method.

Some analytical expressions of torque–tension relationship have been developed by Motosh³ and Nassar and Yang⁴

T = F (μ_{b} r_{b} + r_{t} \frac{\tan α + μ_{t} \sec β}{1 - μ_{t} \sec β \tan α})

(1)

T = F (μ_{b} r_{b} + \frac{p}{2 π} + \frac{μ_{t} r_{t}}{\cos β})

(2)

T = F (μ_{b} r_{b} + r_{t} \frac{\tan α + μ_{t} \cos α \sqrt{1 + \tan^{2} α + \tan^{2} β}}{1 - μ_{t} \sin α \sqrt{1 + \tan^{2} α + \tan^{2} β}})

(3)

where T is the input torque, F is the preload, $μ_{b}$ is the friction coefficient between the turning head/nut and its bearing surface, $μ_{t}$ is the friction coefficient between threads, $α$ is the helix angle, $β$ is the half of the thread profile angle, $r_{b}$ is the effective bearing contact radius, $r_{t}$ is the effective thread contact radius, and p is the pitch of the threads.

It can be seen from equations (1) to (3) that the torque–tension relationship of bolted joints is determined by geometric factors ( $α$ , $β$ , and p), friction coefficients ( $μ_{b}$ and $μ_{t}$ ), and effective contact radii ( $r_{b}$ and $r_{t}$ ). Since the geometrical factors are fixed for a specific type of bolted joint, the accuracy of the torque–tension relationship depends mostly on the accuracy of the friction coefficients and the effective contact radii. In practice, friction coefficients are affected by various factors, such as material class,⁵ surface roughness,⁶ tightening speed,⁷ and the number of tightening cycles.⁸ Thus, friction coefficients may scatter considerably for the same type of bolted joint.⁶ However, using proper surface coatings⁹ or lubricants,^7,10 the scatter of friction coefficients could be reduced significantly. For example, Nassar and Zaki⁹ demonstrated experimentally that the 1-σ scatter of friction coefficients could be controlled to less than 5% using a zinc/aluminum coating composition. Thus, the effects of friction coefficients on the accuracy of the torque–tension relationship can be controlled technically to satisfy the requirements of precision assembly applications.

In practice, the effective thread contact radius, $r_{t}$ , is generally approximated by the mean thread radius. It has been demonstrated that this approximation is sufficiently accurate. Nassar et al.¹¹ compared the effective thread contact radius with the mean thread radius and found that the percentage difference between them was generally less than 1% regardless of the assumed pressure distribution. However, the effects of the effective bearing contact radius, $r_{b}$ , on the accuracy of the torque–tension relationship have usually been overlooked. It can be seen from equations (1) to (3) that the effective bearing contact radius appears in the term $F μ_{b} r_{b}$ which is the bearing friction torque component. It is the torque needed to overcome the friction between the turning head/nut and its bearing surface. Previous studies have demonstrated that this component contributes about 50% to the torque–tension relationship.¹² Therefore, the effective bearing contact radius can have a significant effect on the accuracy of torque–tension relationship.

The effective bearing contact radius is defined as follows¹³

r_{b} = \frac{\int_{r_{b - \min}}^{r_{b - \max}} P_{b} (r) r^{2} dr}{\int_{r_{b - \min}}^{r_{b - \max}} P_{b} (r) rdr}

(4)

where $P_{b} (r)$ is the bearing pressure distribution, $r_{b - \min}$ is the minimum radius on the bearing surface, $r_{b - \max}$ is the maximum radius on the bearing surface, and $r$ is a variable that represents the distance to the axis. It can be seen that the value of effective bearing contact radius depends on the bearing pressure distribution. However, there is no precise analytical expression of the bearing pressure distribution, thus the effective bearing contact radius cannot be directly calculated using equation (4). In practice, the effective bearing contact radius has been approximately calculated using different methods

r_{b} = \frac{r_{b - \max} + r_{b - \min}}{2}

(5)

r_{b} = \frac{r_{b - \max} + D / 2}{2}

(6)

r_{b} = \frac{2}{3} r_{b - \min} \frac{γ^{2} + γ + 1}{1 + γ}

(7)

r_{b} = \frac{1}{2} r_{b - \min} \frac{γ^{4} - 4 γ + 3}{γ^{3} - 3 γ + 2}

(8)

where $γ$ is the ratio of $r_{b - \max}$ to $r_{b - \min}$ , and D is the nominal diameter of the bolt. The detailed explanation of equations (5)–(8) can be found in Appendix 1. Equations (5)–(8) only provide approximate evaluations of the effective bearing contact radius. If too much error was introduced in these calculations, the error of the predicted torque–tension relationship would be relatively high. This would negatively impact the accuracy of the preload of bolted joints, thus decreasing mechanical accuracy and quality, especially for precision assemblies. However, to the best of our knowledge, there is no reported study that has systematically analyzed the accuracy of the various methods for calculating the effective bearing contact radius. This will be the focus of this study.

Methodology

It is not trivial to accurately determine the value of $r_{b}$ through experiments. Even though coatings and lubricants can be used, the scatter of friction coefficients cannot be completely avoided, thus can cause measurement error on the effective bearing contact radius. One potential way to accurately calculate the bearing pressure distribution and effective bearing contact radius is to use finite element analysis (FEA). In this study, a three-dimensional (3D) finite element model of a typical bolted joint structure is built, and the torque control strategy by directly applying a moment to the nut is simulated.^14–16 Then, the effective bearing contact radius is calculated based on the FEA results of bearing pressure distribution

r_{b} = \frac{\sum_{1}^{n} A_{i} r_{i} P_{i}}{\sum_{1}^{n} A_{i} P_{i}}

(9)

where $A_{i}$ is the area of the ith element on the nut bearing surface, $r_{i}$ is the distance from the center of the ith element to the axis, $P_{i}$ is the average pressure on the ith element, and n is the total number of elements on the nut bearing surface. Equation (9) is a direct discretization of equation (4), thus its error will be negligible if the element number, n, is high enough. Usingequation (9) as the reference, the relative accuracy of various methods for approximating the effective bearing contact radius can be analyzed and compared.

Specifically, a typical bolted joint structure is modeled in this study. Figure 1 shows two-dimensional (2D) sketches of the bolt, nut, and joint structures, where A=6.4mm, B=15mm, L=40mm, D = 10 mm, S=15.7mm, E=18.1 mm, M = 8 mm, and H₁ = H₂ = 10.5 mm. A finite element mesh was built with the commercial preprocessor HyperMesh 12.0^®. Here, the mesh of the bolted joint is generated based on a strategy consistent with Fukuoka et al.’s¹⁷ method which accurately takes into account the helical geometry and generates an orderly 3D hexahedral mesh for the bolted joint.

Figure 1.

2D sketches of bolt, nut, and bolted joint: (a) bolt (and its bearing surface), (b) nut (and its bearing surface), and (c) bolted joint.

The mesh of the entire bolted joint is shown in Figure 2. It can be seen that the bolt head and nut are simplified to cylinders for convenience of meshing. The FEA was implemented using the commercial software ANSYS 14.5^®. The mesh type is SOLID185. Young’s modulus is 200 GPa, and Poisson’s ratio is 0.3. There are four contact interfaces in this model. They are the interfaces between the mating threads, nut and upper plate, upper and lower plates, and bolt head and lower plate. A friction coefficient value of 0.15 is assigned to all sliding interfaces for the basic model. To simulate the torque control method, the target tightening torque, 80 N m, is applied directly to the upper surface of the nut using MPC184 elements, as shown in Figure 3. Other boundary conditions include the following: (1) the bolt head surface is constrained in all directions and (2) the outer surfaces of the clamped plates are constrained in the lateral direction but free to move along the bolt axis. In addition, a grid refinement study was conducted to determine the appropriate mesh density of the model.

Figure 2.

Mesh of the entire bolted joint.

Figure 3.

Applying tightening torque using MPC184 elements.

Results and discussions

The bearing effective contact radius is obtained based on the bearing pressure distribution. One example of the bearing pressure distribution is shown in Figure 4. In this section, the values of $r_{b}$ calculated using equations (5)–(8) are compared with those obtained with the FEA model (equation (9)) under various geometrical, material, and frictional conditions. The specific conditions considered in this study include the value of hole clearance $(D_{0} - D)$ , the ratio of the maximum bearing radius $(r_{b - \max})$ to the minimum bearing radius $(r_{b - \min})$ , the stiffness ratio of fasteners to clamped plates, and friction coefficient.

Figure 4.

Bearing pressure distribution (torque is 80 Nm, D is 10 mm, D₀ is 11.2 mm, Young’s modulus is 200 GPa, and friction coefficient is 0.15).

Effect of hole clearance

To evaluate quantitatively the effect of hole clearance on the accuracy of equations (5)–(8), FEA models with three different values of hole clearance (0.225, 0.600, and 0.975 mm) were built by changing the value of D₀, and the bearing pressure distribution and $r_{b}$ were obtained. The bearing pressure distributions under different hole clearances obtained from FEA are shown in Figure 5. The values of $r_{b}$ calculated using equations (5)–(8) and their percent errors compared with the FEA results are also summarized in Table 1. It can be seen that both equations (5) and (7) give relatively good approximations of $r_{b}$ . The percentage errors of these two equations are smaller than 1.7% in this set for comparison. Equation (6) only gives a good approximation when the hole clearance is very small. It can also be seen that the accuracy of equation (8) is relatively low over the entire range of hole clearance (with a percent error generally >4%).

Figure 5.

Bearing pressure distribution under different hole clearances.

Table 1.

Error analysis under different hole clearances.

Hole clearance (mm)	FEA (mm)	Equation (5) (mm)	Error (%)	Equation (6) (mm)	Error (%)	Equation (7) (mm)	Error (%)	Equation (8) (mm)	Error (%)
0.225	6.5179	6.5375	0.3	6.425	−1.43	6.6253	1.65	6.1628	−5.45
0.600	6.7187	6.725	0.09	6.425	−4.37	6.7877	1.03	6.3943	−4.83
0.975	6.9043	6.9125	0.12	6.425	−6.94	6.9549	0.73	6.6296	−3.98

FEA: finite element analysis.

Effect of the ratio of r_b–max to r_b–min

The ratio of the maximum bearing radius $(r_{b - \max})$ to the minimum bearing radius $(r_{b - \min})$ , $γ$ , is another important geometrical factor affecting the torque–tension relationship. For an ISO standard hexagonal bolt head and nut, $γ$ is approximately equal to 1.5. It is generally between 2 and 3 for most flanged heads and nuts.¹³ In this study, FEA models with three different values of $γ$ (1.57, 2.47, and 2.92) were built to evaluate the quantitative effect of $γ$ on $r_{b}$ . The bearing pressure distribution under different values of $γ$ obtained from FEA is shown in Figure 6. Values of $r_{b}$ under different values of $γ$ obtained from the FEA and calculated using equations (5)–(8) are summarized and compared in Table 2. It can be seen that the accuracy of equations (5)–(8) generally decreases with an increase in $γ$ . However, equation (7) can give relatively good approximations of $r_{b}$ over the entire range of $γ$ . Equation (5) only gives a good approximation of $r_{b}$ when $γ$ is relatively small (1.57). Equations (6) and (8) do not give good approximations of $r_{b}$ over the entire range of $γ$ .

Figure 6.

Bearing pressure distribution under different values of γ.

Table 2.

Error analysis under different ratios of r_b–max to r_b–min.

γ	FEA (mm)	Equation (5)(mm)	Error (%)	Equation (6)(mm)	Error (%)	Equation (7)(mm)	Error (%)	Equation (8)(mm)	Error (%)
1.57	6.7187	6.725	0.09	6.425	−4.37	6.7877	1.03	6.3943	−4.83
2.47	9.451	8.975	−5.04	8.675	−8.21	9.3981	−0.56	8.1725	−13.53
2.92	11.0927	10.1	−8.95	9.800	−11.65	10.7683	−2.92	9.1233	−17.75

FEA: finite element analysis.

Effect of the stiffness ratio of fasteners to clamped plates

To investigate the accuracy of equations (5)–(8) under various material combinations, finite element calculations were conducted with three representative stiffness ratios (the ratio of the elastic modulus of the fasteners to the clamped plates): 2, 1, and 0.5. Specifically, the elastic modulus of fasteners was fixed at 200 GPa, while the elastic modulus of the clamped plates was varied (100, 200, and 400 GPa). The bearing pressure distribution under different stiffness ratios obtained from FEA is shown in Figure 7. Values of $r_{b}$ under different stiffness ratios obtained from the FEA and calculated using equations (5)–(8) are summarized and compared in Table 3. It can be seen that both equations (5) and (7) give relatively good approximations of $r_{b}$ for different stiffness ratios. The percentage errors of these two equations are <2% in this data set. The accuracy of both equations (6) and (8) is relatively low over the entire range of stiffness ratio, with the percentage error generally >4%.

Figure 7.

Bearing pressure distribution under various stiffness ratios.

Table 3.

Error analysis under different ratios of fastener to joint stiffness.

Stiffness ratio	FEA (mm)	Equation (5)(mm)	Error (%)	Equation (6)(mm)	Error (%)	Equation (7)(mm)	Error (%)	Equation (8)(mm)	Error (%)
0.5	6.6642	6.725	0.91	6.425	−3.59	6.7877	1.85	6.3943	−4.05
1	6.7187	6.725	0.09	6.425	−4.37	6.7877	1.03	6.3943	−4.83
2	6.8097	6.725	−1.24	6.425	−5.65	6.7877	−0.32	6.3943	−6.10

FEA: finite element analysis.

Effect of friction coefficient

To investigate the accuracy of equations (5)–(8) under various friction coefficients, finite element calculations were conducted with three representative friction coefficient values: 0.05, 0.1, and 0.15. For simplification, $μ_{b}$ and $μ_{t}$ were set to be the same. The bearing pressure distribution under different friction coefficients obtained from FEA is shown in Figure 8. Values of $r_{b}$ under different friction coefficients obtained from the FEA and calculated using equations (5)–(8) are summarized and compared in Table 4. It can be seen that the value of $r_{b}$ obtained from FEA does not change much with the friction coefficient. In this set for comparison, both equations (5) and (7) still give relatively good approximations of $r_{b}$ regardless of the value of friction coefficient. The percentage errors of these two equations are <1.5% for this data set. Again, the accuracy of both equations (6) and (8) is still relatively low over the entire range of friction coefficient, with the percent error >4%.

Figure 8.

Bearing pressure distribution under different friction coefficients.

Table 4.

Error analysis under different friction coefficients.

Friction coefficient	FEA (mm)	Equation (5)(mm)	Error (%)	Equation (6)(mm)	Error (%)	Equation (7)(mm)	Error (%)	Equation (8)(mm)	Error (%)
0.05	6.7091	6.725	0.24	6.425	−4.23	6.7877	1.17	6.3943	−4.69
0.1	6.7193	6.725	0.08	6.425	−4.38	6.7877	1.02	6.3943	−4.84
0.15	6.7187	6.725	0.09	6.425	−4.37	6.7877	1.03	6.3943	−4.83

FEA: finite element analysis.

Validation of FEA results

The FEA model in this study accurately takes into account the helical geometry and has a high-quality 3D hexahedral mesh, with a seamless transition between the threaded section and the bolt shank. In addition, a careful mesh refinement study was done to ensure that the FEA results were grid-independent. Thus, it would be expected that the FEA results and corresponding conclusions obtained in this study are accurate. In this section, the accuracy of the FEA model is further validated by comparing the torque–tension relationship extracted directly from the FEA model with equation (3), which is the most accurate analytical torque–tension relationship to date. The results are shown in Figure 9. All geometric, material, and boundary conditions in the calculation are the same as the basic model, discussed in section “Methodology.” The $r_{b}$ value calculated using equation (7) is used as the input of equation (3).

Figure 9.

Comparison of the torque–tension relationship obtained from FEA and equation (3).

It can be seen that the torque–tension relationship from the FEA model is almost the same as that calculated based on equation (3). The relative error between these two sets of results is smaller than 0.5%. Given that these two sets of results are obtained using completely different strategies (numerical calculation and analytical derivation), the consistency of the results can be considered an evidence of the accuracy of both models (FEA model and analytical expression).

Conclusion

In this article, a 3D hexahedral finite element model of a typical bolted joint was built, based on which the tightening process of the bolt was simulated, and the actual bearing pressure distribution was calculated. Then, the effective bearing contact radius, based on the FEA results of bearing pressure distribution, was obtained. The relative accuracy of various methods of calculating the effective bearing contact radius was analyzed systematically and compared. The effects of the hole clearance, the ratio of the maximum to minimum bearing radius $(γ)$ , the stiffness ratio of fasteners to clamped plates, and friction coefficient were considered and discussed.

It was found that the effective bearing contact radius calculated based on the assumption of uniform bearing pressure distribution (equation (7)) was always relatively accurate for the conditions considered in this study. The relative error of equation (7) was generally less than 2%. The mean bearing radius (equation (5)) can accurately evaluate the effective bearing contact radius when $γ$ is relatively small. However, the accuracy of equation (5) decreases significantly as $γ$ increases. The percentage error of equations (6) and (8) varied significantly between 4% and 18%. Particularly, the accuracy of these two equations decreases significantly as $γ$ increases.

In conclusion, if the ratio of the maximum to minimum bearing radius is relatively small (less than 2, based on the results of this study), both equations (5) and (7) can satisfy the requirements for precision assembly applications. Otherwise, only equation (7) should be used in precision assembly applications. Considering that the allowed error of the torque–tension relationship might be as small as several percents for precision assembly applications, equations (6) and (8) are not suitable for these applications. The results and conclusions of this study can help to increase the accuracy of the torque–tension relationship for tightening bolted joints, thus ensuring mechanical accuracy and quality for precision assemblies.

Footnotes

Appendix 1

Academic Editor: Filippo Berto

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the National Natural Science Foundation of China (grant no. 51675050 and 51605030).

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Calculation of the effective bearing contact radius for precision tightening of bolted joints

Abstract

Keywords

Introduction

Methodology

Results and discussions

Effect of hole clearance

Effect of the ratio of rb–max to rb–min

Effect of the stiffness ratio of fasteners to clamped plates

Effect of friction coefficient

Validation of FEA results

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References

Effect of the ratio of r_b–max to r_b–min