# An Analytical Model for Rotation Stiffness and Deformation of an Antiloosening Nut under Locking Force

Authors: X. J. Jiang; J. Hong; G. Q. Shao; L. B. Zhu; Y. S. Zhu
Journal: International Journal of Rotating Machinery
 (2014)

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## Abstract

Screw fasteners are undoubtedly one of the most important machine elements due to their outstanding characteristic to provide a high clamping force just with a simplified design. However, the loosen vibration is their inherent and inevitable fault. The friction locking approach is one of the basic locking fastener categories by enhancing the bearing load on the contact surface of thread by applying a locking force on an antiloosening nut. This locking force may cause more severe deformation in the nut. The contact stress distribution on the nut would be changed and that can cause the variation of the friction torque for the bolt joint. However, there exists no established design calculation procedure that accounts for the rotation deformation and its stiffness of the antiloosening nut under the locking force. The main objective of the work is to develop an analytical solution to the rotation deformation problem encountered in the antiloosening nut. The proposed model is supported by comparison with numerical finite element analysis of different sizes of joint elements and different applied forces.


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## Body

## 1. Introduction

Screw fasteners are undoubtedly one of the most important machine elements. Their design simplicity to provide a high clamping force is the outstanding feature of screw fasteners. However, they have an inherent and inevitable fault: they all loosen eventually by vibration [1–3].Some researches intended to make clear of the phenomenon of self-loosening of bolt based on the vibration theory [4–9]. However, the popularly known antiloosening fasteners do not seem to possess much resistance to loosening by the test results [10, 11] for the effectiveness of screw threads, spring washers, nylon inserted nuts, double nuts, and eccentric nuts of few sizes to resist loosening. Sase et al. [11] in the year 1998 introduced and evaluated the step lock bolt (SLB) with regard to its antiloosening performance using a displacement based loosening device. They found the presence of desirable antiloosening characteristics of SLB.The friction locking approach is one of three basic locking fastener categories established by major companies [12, 13]. The mechanism of friction locking is to enhance the friction force by increasing the bearing load or the friction coefficients of contact surfaces of nut or bolt head and thread. One of the methods to enhance the friction force is to increase the bearing load of the contact surface of thread by applying one force on the nut in outside direction perpendicular to the bearing surface of nut. However, the bearing force and the bending deformation of nut will be influenced by this applied force. The bending deformation of the nut is determined by its bending rigidity that may never be discussed by other researches. In the present study, an analytical model was proposed to evaluate the bending rigidity of the nut under preload. The effects of the different forces produced by the small screw and applied on the nut were investigated on the bearing force and the bending deformation of the nut. The theoretical calculation procedure that considers this effect is presented in detail hereafter.


## 2. Analytical Model

The three basic elements that form an antiloosening nut discussed in present research are the upper nut, the lower nut, and the small screw, as shown in Figure1. The lower nut is supported by clamped parts. The small screw is passed through the bolt hole in the upper nut. As the small screw is tightened, the extra large bearing force will be applied on the contact surface of nut threads that can enhance the antiloosening ability of bolted joint. Therefore, the tightening force produced by the small screw can be considered a locking force on the antiloosening nut.Picture of an antiloosening nut.
(a)
Overall view picture
(b)
Section view pictureThe model used to simulate the joint with the antiloosening nut is shown in Figure2. For the case involving clamped parts, bolt and nut are included. As with bolt force Fb, when a locking force fb is applied to the small screw, different rotation displacements and distortions at the nut flank are generated for the abovementioned joint components.Figure 2
Bolted joint model under locking force.Before conducting the elastic interaction analyses, it is worth noting that the theoretical reference should be used to treat the different joint elements.(1)
The nut flank rotation is treated using the theory of beam on an elastic foundation without nonlinear distortion (Figure3). The rotation angle of the nut flank is considered to be the rotation angle in the point of effective bearing friction radius rb of the nut bearing surface.(2)
The bolt part connected to the nut is represented by a compression spring of linear elastic stiffnesskn. The connected point of the spring to the nut is at the center point of the height tn of the nut.(3)
The connected point of bolt and nut is simplified without the threads.The elastic spring Kn is given by
(1)Kn=Etnrd,
where rd is the radius of the bolt axis, tn is the height of the nut, and E is Young’s modulus.Figure 3
Nut flank rotation analysis.The rotational stiffness of the nut flank, which is formed by the nut flank and the connected part of bolt to the nut, is characterized by two parametersknM and kbf depending on the dimensions of the joint elements. kbf is the bending rigidity of the nut flank due to a bending moment produced by the nut bearing load and can be calculated by the beam theory. knM is the rotation rigidity due to the bending moment effect produced by the nut bearing load on the connected part of bolt to the nut. The expression of the rigidity Kbf is
(2)Kbf=2EInrb-rd,
where rb is the effective bearing friction radius. In is moment of inertia. KnM can be equivalently replaced by the compression rigidity of Kn as follows:
(3)KnM=Kn(tn2)2=Etn34rd.Therefore, the rotational stiffness of the nut flank can be written as(4)K=KnMKbfKnM+Kbf=Etn36rb-2rd,
where rb can be calculated by
(5)rb=∫rdroprdr∫rdropdr≈∫rdropr2dr∫rdroprdr.The equilibrium consideration in the joint is obtained for both the bolt initial preloading statei and final small screw tightening state f, such that
(6)Pni=Fb,
and, from the small screw tightening equilibrium,
(7)Pnf+fb=Fb,
where Pn is the bearing force of nut.The initial and final rotation deformations of the nut flank are given by the following equations:(8)θni=MniK=2Fb(3rb2-4rbrd+rd2)Etn3,θnf=MnfKθnf=(2Fb(3rb2-4rbrd+rd2)kkk+2fb(3rfbrb-rfbrd-3rb2+rbrd))×(Etn3)-1,
where
(9)Mni=Pni(rb-rd),Mnf=Pnf(rb-rd)+fb(rfb-rd)Mnf=Pni(rb-rd)+fb(rfb-rb)
in which rfb is the radius of the small screw loading point. It is worth noting that as fb=0, θni=θnf.The effective bearing friction radiusrb can be obtained from the calculation results from Nassar’s theory [14]. In that research, four scenarios of underhead load distribution are discussed (i.e., uniform underhead load, linearly decreasing load, exponentially decreasing load, and sinusoidal underhead load). A uniform pressure distribution simulates the use of a strong steel bolt and nut to clamp onto a much weaker clamping surface, such as plastic surfaces or soft washers. Other pressure distributions considered would simulate contact stress concentration at the edge of the fastener hole or represent the zero pressure conditions at minimum or maximum radii of the contact area. Nassar’s theoretical equations [14] can be listed below, where rb-t means the theoretical calculation result of rb.Uniform underhead load:(10)rb-t=0.656(γ2.9-1)rdγ1.9-1.Linearly decreasing load:(11)rb-t=0.95rd(γ2-1)γ1.9-1(6γ3.9-23.4γ+17.411.3γ3-33.93γ+22.62).Exponentially decreasing load:(12)rb-t=0.95(γ2-1)rd0.1γ1.9-1(∫rdγrdγ1.9e-a(r-rd)dr∫rdγrdγe-a(r-rd)dr),
where a=ln⁡(pmax⁡/pmin⁡)/rd(γ-1).Sinusoidal underhead load:(13)rb-t=0.95(γ2-1)rd0.1γ1.9-1×(∫rdγrdγ1.9sin[π(r-rd)/rd(γ-1)]dr∫rdγrdγsin[π(r-rd)/rd(γ-1)]dr),
where γ is the value of the maximum to minimum contact radii ratio.


## 3. Numerical FE Model

To validate the analytical model that estimates the rotation rigidity and deformation of the nut flank, a two-dimensional axisymmetric numerical FE model was constructed. A 2D 4-node element PLANE42 with two degrees of freedom per node in ANSYS [15] was used to model the bolt joint, as shown in Figure 4. Young’s modulus is 195 GPa and Poisson’s ratio is 0.3. In the definition of contact pairs for finite element analysis, the contact surface of clamped part is assigned to be the target element TARGE169, and the contact surface of nut is the contact element CONTA172.Figure 4
2D finite element model.The antisymmetry DOF constraint alongx-axis is applied on the bolt axis. The bottom of the clamped part is also restrained in displacement in y direction. The right side of the clamped part is also restrained in displacement in x direction. The loading is applied in two steps. The first step consists of applying an initial nut-up. This is achieved by imposing to the bolt end-plane nodes, an equivalent axial node force to produce the target initial bolt-up stress of 28.8 MPa. A locking force fb produced by the small screw that depends on the case studied is applied in the second step. Comparisons between the analytical method and finite element method results were conducted on different geometries of joint elements and different applied forces.


## 4. Results and Discussion

Figure5 shows the result of the stress distribution along y-axis. The stress concentration can be found at the connecting corner of bolt and nut that produces the stress of about 260 MPa. The rotation of the nut flank can be observed obviously. The results of nut flank rotation after applied loads obtained from the proposed analytical approach are compared to those of FE model for the different applied loads and different sizes of joint elements.Figure 5
y-axis stress and deformation of joint under preload force.Figure6 shows the variation of the average values of the contact pressure of bolt and nut with the change of the locking force from the small screw. It can be seen that the average value decreases with the increase in the locking force. A difference between the analytical model and FEA that may be owing to the stress concentration at the detached point of the contact area of the bolt and nut is also shown for two different bolt force cases. Figure 7 depicts the rotation deformation of the nut flank after the locking force is applied. From Figure 7, it can be found that the effective bearing friction radius rb obtained from the calculation results from Nassar’s theory for different underhead load distribution cannot effectively calculate the rotation deformation of the nut flank for different bolt force Fb besides the exponentially decreasing load case. The results given by the uniform distribution and the sinusoidal distribution are very close. On the other hand, the results given by the linear distribution and the exponentially decreasing distribution are also close. However, the exponentially decreasing distribution gives much better results than other distributions as Pmax⁡/Pmin⁡=5. Hereafter, the proportional coefficient Pmax⁡/Pmin⁡=5 for the exponentially decreasing load distribution is considered to calculate the rotation stiffness and deformation for the different sizes of joint elements at Fb=200 kN and fb=3500 N. It should be noted that γ ratio is 1.4 for all cases.Average value of the underhead contact pressure under different locking force.
(a)
F
b
=
200 kN
(b)
F
b
=
400 kNRotation deformation of nut flank.
(a)
F
b
=
200 kN
(b)
F
b
=
400 kNFigure8 shows the comparison of the results for the different rD case by the analytical model and FEA.Rotation stiffness and deformation of nut flank with variation ofrD.
(a)
Rotation deformation
(b)
Rotation stiffnessThe analytical and FEA results are in a good agreement with a maximum difference of 0.9% between the two methods. The increase inrD will generate a smaller rotation stiffness of the nut flank according to (2), which contributes to a larger value of the rotation deformation of the nut flank. From Figure 9, a good agreement between this stiffness and these deformations for different clearance dn between the bolt and the clamped part is obtained. The decrease in the rotation stiffness k as dn increase will be enhanced by (4). Therefore, as Fb and fb are fixed, the bending moment Mnf can be shown to increase as dh increases, which leads to the glowing of the rotation deformation of the nut flank. The results of the calculated stiffness and deformations of the nut flank with the analytical model as shown in Figure 10 are in good agreement with those obtained by the FEM for the case of different height tn of the nut. The difference between the two methods is less than 20%. The difference between the results is due to the linear simplification made on the nut flank mechanical behavior. According to (4), the increase in the heighttn will raise the rotational stiffness of the nut flank. The rotation deformation will decrease in this case.Rotation stiffness and deformation of nut flank with variation ofdn.
(a)
Rotation deformation
(b)
Rotation stiffnessRotation stiffness and deformation of nut flank under different locking force with variation oftn.
(a)
Rotation deformation
(b)
Rotation stiffnessIn order to analyze the effect of the role position of the locking forcefb, the different rfb case is discussed in this paper. Good agreement results for the different rfb case by the analytical model and by the FEA (Figure 11) are shown. In this case, the rotation stiffness will not be changed because of the same geometry size of the joint elements. The built-up of rfb will increase the bending moment Mhf, so the rotation deformation is increased with the growth of rfb.Figure 11
Rotation deformation of nut flank under different locking force with variation ofrfb.


## 5. Conclusions

An analytical method has been developed to analyze a bolt joint with an antiloosening nut. The developed model based on the beam theory has shown rationality solution in the rotation stiffness and deformations of the nut flank. The analyses are verified against the accurate 2D FEM on different sizes of joint elements and different applied forces. The analytical results of the rotation stiffness and deformations compare quite well with those of FEM. It is confident that the analytical model has the ability to predict the deformation and the contact load of the antiloosening nut.



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*Source: 410813-2014-09-29.xml*