**3. Case Study**

In this section, the effectiveness of the proposed method is verified by a bolt pretightening test and simulation analysis.

#### *3.1. Modal Test*

As shown in Figure 5, the study object is a bolted joint structure which is composed of two lap plates and four bolts. The size of the lap plate is measured at about 200 mm long × 80 mm wide × 8 mm deep and the connecting length is 80 mm. The lap plate is made of aluminum alloy. Bolts and nuts are made from low carbon steel of which the nominal diameter is M10. The material parameters of the above two components are shown in Table 2.

**Figure 5.** Bolted lap structure (mm).

**Table 2.** The material parameters of components.


The test sample bolted structure was made by machining operation. A torque wrench was used to control the pre-tightening torque of a single bolt in the group. In this way, the modal test is carried out by hammering under different pre-tightening torque as shown in Figure 6.

**Figure 6.** Modal test of the connection structure.

The piezoelectric accelerometers used in this modal test, had a mass of 20 g, charge sensitivity of 6.05 pC/ms2, and frequency range from 0.5 Hz to 5 kHz. The CL–YD–303 hammer had a reference sensitivity of 3.99 pC/N. CRAS V7.0, a vibration and dynamic signal acquisition and analysis system developed by Nanjing AnZheng Software Engineering Company, was used as the analytical instrument, and MaCras was used as the modal analysis software. To simulate free boundary conditions, a spring rope suspension was applied to counteract gravity; this method is effective in avoiding imported errors which affect the dynamic characteristics of the structure. The suspension plane is orthogonal to the test direction to avoid the effect of suspension conditions on the test results. The accelerometer is arranged at the end of the structure to avoid the vibration type node. Thirteen test points are arranged along the length direction of the lapping structure. The sampling frequency is set to 5000 Hz. The arrangement of measuring points is shown in Table 3.

**Table 3.** Distance of measuring points from origin (point *O*).


The pre-tightening torque *TN* = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 25, 28 N·m are applied to the bolt through a torque wrench. The first four order modal frequencies under different pre-tightening forces are obtained through multiple modal tests, which are all shown in Table 4. The experimental modal data are plotted and the relationship between the test frequency and the pre-tightening torque is obtained, as shown in Figure 7.


**Table 4.** Experimental modal frequency under different pre-tightening torque.

As can be seen from Figure 7, with the increase of pre-tightening torque the first four order modal frequencies also increase at the same time. The modal frequencies of the first, the third and the fourth order gradually increase with the increase of the pre-tightening torque, while the second modal frequencies tend to be stable when the pre-tightening torque comes to 14 N·m.

**Figure 7.** The relationship between the experimental frequency and the pre-tightening torque.

The damping ratio of the bolted connection structure is related to many factors, such as material performance, medium condition, surface roughness, processing method, pretightening force state and loading frequency, the influence of which is mostly reflected in nonlinear relations. In addition, the measurement errors will also have an impact on the damping ratio. When measuring the frequency response functions, the measured result of the damping ratio is larger than the actual result due to the effect of accelerometer weighting functions. The first four order damping ratios under different pre-tightening forces were obtained, and are all shown in Table 5. Furthermore, the relationship between the experimental damping ratio and the pre-tightening torque was obtained as is shown in Figure 8.


**Table 5.** Experimental damping ratio under different pre-tightening torque.

**Figure 8.** Experimental damping ratio depends on the pre-tightening torque.

As shown in Figure 8, the relationship between the damping ratio of the connection structure and the pre-tightening torque is generally non-linear, and there is no rule to be found. Meanwhile, the relationship between the first four order experimental modal shapes and the pre-tightening torque is shown in Figure 9.

As can be seen from Figure 9, the modal shapes are generally smooth. On the one hand, the connection stiffness is enough to provide good linearity to the structure; on the other hand, it is impossible to measure the detailed characteristics of the modal shapes in the connecting part due to the limited test precision. At the same time, it can be found that the mechanical properties of the bolt connection tend to be stable with the increase of the pre-tightening force.

**Figure 9.** The first four order experimental modal shapes under different pre-tightening torque.

#### *3.2. Modelling and Parameter Identification*

When thin layer elements are used to model bolted structures, the role of bolts on the structure is replaced by thin layer elements established on the contact surface. A schematic diagram of the method is shown in Figure 10.

**Figure 10.** Parameterization of contact interface.

However, the pressure distribution on the bolt joint surface is not uniform. The pressure gradually decreases along the radial direction, as shown in Figure 11. The pressure distribution of the joint surface is shown in Equations (19)–(21).

$$P\_{\text{max}} = \frac{3F\_b}{\pi (r\_m^2 + r\_m r\_i + r\_i^2)}\tag{19}$$

$$P(r) = -\frac{P\_{\text{max}}}{r\_m - r\_i}(r - r\_m) \tag{20}$$

$$r\_m = r\_b + h \cdot \tan a \tag{21}$$

where *Fb* is the bolt pre-tightening force, *ri* is the radius of the bolt hole, *rm* is the maximum contact radius, *rb* is the radius of the bolt head, and *h* is the thickness of the connected piece.

**Figure 11.** Schematic diagram of pressure distribution in bolted joint.

We assuming that the normal dynamic stiffness *Kn* and tangential dynamic stiffness *K<sup>τ</sup>* of the joint surface per unit area under uniform pressure are shown in Equations (22) and (23)

$$K\_n = \mathfrak{a}\_n \cdot P\left(r\right)^{\beta\_n} \tag{22}$$

$$K\_{\pi} = \alpha\_{\pi} \cdot P(r)^{\beta\_{\pi}} \tag{23}$$

where *P*(*r*) is the pressure of the joint surface, *α<sup>n</sup>* and *β<sup>n</sup>* are normal characteristic parameters of the joint, *ατ* and *βτ* are tangential characteristic parameters of the joint.

The relationship between the *E* and *G* of the gradient connecting layer and *K* is shown in Equations (24) and (25)

$$E(r) = \frac{K\_n h}{A} = \alpha\_n P\left(r\right)^{\beta\_n} h \tag{24}$$

$$G(r) = \frac{K\_{\tau}h}{A} = a\_{\tau}P\left(r\right)^{\oint\_{\tau}}h\tag{25}$$

where *A* is the area of the connecting layer and *h* is the thickness of the connecting layer. The influence area of the bolt connection is divided into three areas, as shown in Figure 12.

**Figure 12.** The influence area of the bolt connection, which is divided into three areas.

In order to facilitate calculation and ignore the influence of the screw hole, the gradient connection layer is divided into two areas and changed into a square in this paper. The contact pressure of each area is the average pressure of each layer.

The FEM of the connection structure as shown in Figure 13 is established by ignoring the influence of the bolt weight and screw hole. The lap plate is simulated with a solid element, and the interface is simulated with an isotropic thin-layer element. The contact stiffness close to the bolt is higher than that away from the bolt, and two different isotropic constitutive relations are used to simulate the contact surface and then used to identify the parameters. The parameters to be identified near the bolt area (the red area in Figure 13) are elastic modulus *E*<sup>1</sup> and shear modulus *G*1. For another area in the contact area, the parameters to be recognized away from the bolt are the elastic modulus *E*<sup>2</sup> and shear modulus *G*2. The ratio coefficient of the thin-layer element is Ratio = 10, and the density of the material is 0 t/mm3.

**Figure 13.** FEM of the connection structure.

It is important to note the setting of the initial value of the identification parameter. The parameters (*E*1, *G*1) of the thin layer element are of the same order of magnitude as the material parameters of the bolt where close to the bolt. However, the parameter selection (*E*2, *G*2) is an order of magnitude lower than the material parameters where away from the bolt. The initial value as well as the identification results are listed in Table 6. The iterative convergence curves of the identification parameters are as shown in Figure 14, when the pre-tightening torque is *TN* = 20 N·m.

**Table 6.** Initial value of the identified parameters.


(**a**) Iteration curve of *E*1, *G*<sup>1</sup> (**b**) Iteration curve of *E*2, *G*<sup>2</sup>

**Figure 14.** Iterative convergence curves of parameter selection.

Table 7 lists the modal frequency parameters before and after identification as well as the error between the above parameters and the experimental frequency. It can be seen from the identification results that the maximum updated error is no more than 2.5%. The results have comparatively higher identification accuracy.


**Table 7.** Comparison of modal frequency errors.

The elastic modulus *E* is greater than the shear modulus *G*, which reflects the characteristics of normal contact stiffness more than the tangential stiffness. The identification results (*E*1, *G*1) near the bolt area are far greater than that of the bolt area (*E*2, *G*2), which better reflects the actual situation. Therefore, the connection performance of the overlapping structure with multiple bolts can be well simulated by the of isotropic thin layer element.

Similarly, the material parameters of the thin layer element under different pretightening torque are obtained by the iterative solution method. Finally, the relationship between the identification parameters and the different pre-tightening torque is achieved, as shown in Figure 15.

**Figure 15.** The relation curves of identification parameters change with pre-tightening torque.

In order to verify the correctness of the method, we selected five points evenly in the interval of the fitting curve, obtained *E*1, *G*1, *E*<sup>2</sup> and *G*<sup>2</sup> through the parameter identification method, and put these values into the curve for verification. In Figure 16, the red dots are the verification points. It can be seen that the error between the red dots and the fitting curve is small, which proves the effectiveness of the method.

The curve obtained by this method is of high precision when a certain number of points and intervals are reasonably selected for fitting. The more points used, the more accurate the curve.

Through curve fitting, the relationship between the elastic modulus *E*<sup>1</sup> and *E*<sup>2</sup> and the shear modulus *G*<sup>1</sup> and *G*<sup>2</sup> under different pre-tightening torque was obtained. The resulting curves can guide the modeling of the same type of bolted joints. The specific operation is as follows: (*i*) The torque wrench can be used to determine the pre-tightening torque of the bolt. (*ii*) The parameter values of the thin layer element material *E*1, *G*1, *E*<sup>2</sup> and *G*<sup>2</sup> are drawn from the relation curves. Thus, an accurate dynamic model of the bolted joint structure could be established.

#### **4. Conclusions**

On the basis of the thin layer element with isotropic constitutive relation, this paper presents a method to simulate the stiffness of the joint surface of the bolt connection considering the variability of pre-tightening torque. The proposed method in this paper is verified by the pre-tightening test, and the following conclusions can be drawn:


**Author Contributions:** Conceptualization, D.J.; Funding acquisition, D.J.; Methodology, H.Q. and Z.C.; Resources, H.Q.; Software, Y.T.; Validation, Y.T. and Z.C.; Writing—original draft, Y.T.; Writing review and editing, H.Q., Z.C., D.Z. and D.J. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number No. 11602112.This research was funded by Natural Science Research Project of Higher Education in Jiangsu Province, grant number 20KJB460003. This research was funded by Qing Lan Project.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors are grateful for the support from the National Natural Science Foundation of China (No. 11602112), The Natural Science Foundation of the Jiangsu Higher Education Institutions of China (20KJB460003) and the Qing Lan Project of China.

**Conflicts of Interest:** The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### **References**

