### 3.1 Recession Curve of Clamping Force

The recession curves of the bolt clamping force under composite excitation are continuous and decrease smoothly. In general, the curve can be divided into three stages: the material loosening period, the structural loosening period, and the fatigue fracture period [30]. During the material loosening period, the plastic deformation of the bolt material rapidly reduces the bolt clamping force; during the structural loosening period, the relative movement between the internal and external thread contact surfaces of the bolt causes the nut to rotate, which slowly reduces the bolt clamping force; finally, during the fatigue fracture period, the bolt cracks and expands, resulting in an instantaneous fracture of the bolt, and the clamping force reduces to zero. Notably, an explicit and unified dividing method to determine the dividing points of each stage of the recession curve of the bolt clamping force has not been established in previous studies, and the determinant condition of bolt loosening and fatigue has also remained undefined.

Since the load applied to the bolt is sinusoidal, the clamping force of the bolt is fluctuating, and the recession curve of bolt clamping force is a load broadband. In order to obtain the dividing point between the structural loosening period and the fatigue fracture period, we propose to use the tangent point between the mean curve of the clamping force broadband and the tangent line with a specific angle as the dividing point. Based on a statistical analysis of the bolt clamping force recession curves obtained herein, we found that when the tangent angle changes from 0° to 90°, 45° is the best angle, so the tangent point of the the mean curve of the clamping force broadband with a 45° tangent can serve as the dividing point between the structural loosening period and the fatigue fracture period (denoted by *N*_{2_k}, *k* = 0.3, 0.4, 0.5, and 0.6). This tangent point is unique. The bolt clearly demonstrates structural loosening before the tangent point, and fatigue fracture after the tangent point, which proves the correctness of the method. Moreover, Jiang et al. [9] believed that for high-strength bolts with hard material, the clamping force was reduced to about 90 % of the initial preload when the bolt entered the structural loosening period, but this definition was conservative. Jiang et al. [31] took the moment when the clamping force was reduced to about 70% of the initial preload as the standard for full loosening of high-strength bolts, which was dangerous. We find that for high-strength bolts, when the clamping force of the bolt was reduced to 80% of the initial preload, the bolt fully entered the structural loosening period; therefore, the clamping force can be used to determine whether there is significant bolt loosening. If the bolt clamping force reduces to 80% of the initial preload during the structural loosening period (denoted by *N*_{L_k}), i.e., *N*_{L_k} <*N*_{2_k}, the bolt undergoes loosening failure; if the bolt clamping force does not reduce to 80% of the initial preload until the fatigue fracture period (denoted by *N*_{F_k}), i.e., *N*_{F_k} > *N*_{2_k}, the bolt undergoes fatigue failure.

The recession curves of the bolt clamping force with *R*=0.15 mm/kN and *R* = 0.06 mm/kN were obtained as shown in Figure 3 (0.6–4 means that the transverse displacement amplitude is 0.6 mm and the axial load amplitude is 4 kN, and 0.6–10 is also defined in the same way). As shown, under different initial preloads, when *R*=0.15 mm/kN, all the recession curves satisfy the condition *N*_{L_k} < *N*_{2_k}, i.e., the bolts undergo loosening failure; when *R*=0.06 mm/kN, all the recession curves satisfy the condition *N*_{F_k} > *N*_{2_k}, i.e., the bolts undergo fatigue failure. Furthermore, the failure life of the bolt first increases and then decreases with the increase in the initial preload at the same *R*, and the failure life of the bolt differs with the same transverse load but different axial loads. Therefore, the failure life of a bolt is determined by the load amplitude and the preload, and the optimal preload of 8.8 grade M8 × 1.25 × 70 high-strength bolts for the longest life is *F*_{0.5}.

### 3.2 Influence Law of Load Ratio

To further study the influence of *R* on the competitive bolt failure mode according to the criterion established in Figure 3, the failure modes of the of all the bolt specimens listed in Table 1 were distinguished, and the failure mode of each bolt specimen was obtained. To intuitively visualize the failure modes of the bolts, the *R*–significant degree of failure curves under different preloads were plotted, as shown in Figure 4(a) (as the bolts undergo loosening failure when *R* > 0.15 mm/kN, they are not shown in the figure); *S*_{L} represents the significant degree of bolt loosening failure and *S*_{F} represents the significant degree of bolt fatigue failure. *S*_{L} and *S*_{F} were calculated using the following equations:

$${S}_{\mathrm{L}}=100\mathrm{\%}\frac{{N}_{{2}_{k}}-{N}_{{\mathrm{L}}_{k}}}{{N}_{{2}_{k}}},$$

(2)

$${S}_{\mathrm{F}}=100\mathrm{\%}\frac{{N}_{{2}_{k}}-{N}_{{\mathrm{F}}_{k}}}{{N}_{{2}_{k}}}.$$

(3)

Based on Figure 4(a), it can be concluded that when *R* ≤ 0.06 mm/kN, the bolts undergo fatigue failure, and when *R* ≥ 0.08 mm/kN, the bolts undergo loosening failure. Therefore, there is a clear competitive failure relationship between the two failure modes under composite excitation, and a critical value of *R* exists for bolt loosening failure and fatigue failure. When the ratio of the transverse load to the axial load is higher than the critical value of *R*, bolt loosening failure occurs, and when it is less than the critical value of *R*, bolt fatigue failure occurs. This implies that the critical *R* value is an inherent property of the bolt that is related to the bolt material, size, and assembly method, but is unrelated to the magnitude of the load. In addition, due to the large preload when *k*=0.6, the bolt is prone to local plastic deformation, which makes the data dispersion of loosening failure relatively large and makes the curve fluctuate significantly. Furthermore, Figure 4(a) shows that when the numerical value of *R* is around 0.075 mm/kN, the bolt reaches a critical state of loosening and fatigue, and *S*_{L} and *S*_{F} tend toward zero. Therefore, the critical *R* value of loosening and fatigue failure of 8.8 grade M8 × 1.25 × 70 high-strength bolts under composite excitation is 0.075 mm/kN. In addition, the *R*–significant degree of failure curves under different preloads have the same failure regularity, which proves that the failure mode of the bolt is only determined by *R* and is unaffected by the initial preload. Thus, the failure mode of bolts subjected to composite excitation can be directly predicted based on the critical *R* value.

The *R*–life curves under different preloads are shown in Figure 4(b). As shown, the failure life of the bolt first increases significantly, and then gradually decreases with the increase in *R* under composite excitation (except when the axial load is zero). When the transverse load is constant, the axial load gradually decreases with the increase in *R*. Conversely, as the axial load increases, the failure life of the bolt first increases slowly and then decreases rapidly. This indicates that a small axial load restrains bolt failure; when the axial load exceeds a certain value, the large composite stress of the bolt under composite excitation causes a large plastic deformation in the thread, which reduces the fatigue life of the bolt. The force diagrams of the bolt thread surface under composite excitation are shown in Figure 5 (*F*_{k} is the initial preload of the bolt, *F*_{T} is the external transverse excitation, *F*_{SA} is the axial excitation of the bolt, *α* is the helix angle, *μ*_{1} is the static friction coefficient, and *μ*_{2} is the sliding friction coefficient). According to Junker’s oblique block model [3], when the bolt is not subjected to an external load, although the block has a tendency to slide downward, the inclined plane and the block remain in a static state, i.e., the bolt thread is in a self-locking state, because the downward force *F*_{k}sin*α* along the inclined plane is less than the static friction force *μ*_{1}*F*_{k}cos*α*. When the bolt is only subjected to an external transverse excitation *F*_{T}, relative sliding motion occurs between the inclined plane and the block, and the friction between them changes from static friction to sliding friction. As the sliding friction force *μ*_{2}*F*_{k}cos*α* is less than *μ*_{1}*F*_{k}cos*α*, the self-locking state of the thread is broken and the block begins to slide downward, i.e., the internal and external threads of the bolt move relative to each other, and the bolt becomes loose. When the bolt is affected by composite excitation, the axial load of the bolt increases and becomes the resultant force of *F*_{k} and *F*_{SA}, the downward force along the inclined plane becomes (*F*_{k}+*F*_{SA})sin*α*, and the static friction force becomes *μ*_{1}(*F*_{k}+*F*_{SA})cos*α*. The transverse excitation *F*_{T} leads to transverse sliding between the thread contact surfaces; however, owing to the increase in the positive pressure on the inclined plane and the block contact surfaces, the sliding friction force *μ*_{2}(*F*_{k}+*F*_{SA})cos*α* is higher than (*F*_{k}+*F*_{SA})sin*α* within a certain range. Consequently, no relative movement occurs between the thread contact surfaces, and the bolt does not become loose. Therefore, to a certain extent, small axial loads can restrain bolt failure; however, when the axial load increases, the large composite tensile and shear stress causes a large plastic deformation in the thread surfaces, and the large axial load accelerates bolt failure, thereby reducing the fatigue life of the bolt.

### 3.3 Load–life Curves

The load–life curves of the bolts subjected to composite excitation were plotted as shown in Figure 6(a) (the ordinate is a logarithmic coordinate). As shown, when the transverse load is constant, the axial load–life curves do not decrease monotonously. At small axial loads, the failure life of some bolts is longer than that under pure transverse loads. Therefore, axial loads do not accelerate the failure of the bolt. The above analyses prove that small axial loads can restrain bolt failure. As shown in Figure 6(b), at different initial preloads, the significant degree of the restraining effect of small axial loads on bolt failure varies. When the preloads are small (*k*=0.3, 0.4, 0.5), except for some singular data points, the axial loads generally accelerate the bolt failure, but when the preloads are large (*k*=0.6), the small axial loads restrain the bolt failure. This is because under composite excitation, the downward force along the thread is (*F*_{k}+*F*_{SA})sin*α*, the transverse static friction force is *μ*_{1}(*F*_{k}+*F*_{A})cos*α*, and the transverse sliding friction force is *μ*_{2}(*F*_{k}+*F*_{A})cos*α.* When *F*_{k} decreases, the positive pressure on the thread and the transverse static friction force decrease. If *F*_{T} is higher than *μ*_{2}(*F*_{k}+*F*_{A})cos*α*, transverse relative movement occurs between the thread surfaces, and the upward friction along the thread surface changes from *μ*_{1}(*F*_{k}+*F*_{A})cos*α* to *μ*_{2}(*F*_{k}+*F*_{A})cos*α*. As *μ*_{2}(*F*_{k}+*F*_{A})cos*α* cannot offset the downward force (*F*_{k}+*F*_{SA})sin*α* along the thread surface, the bolt becomes loose and the failure process of the bolt accelerates. Thus, when the preload is large, the axial load can restrain the bolt failure to some extent; when the preload is small, the clamping load tends to zero, the thread contact surfaces experience a downward relative movement, and the axial load can no longer restrain the bolt failure. In addition, Figure 6(a) also indicates that the average failure life of a bolt is affected by both the external load and the initial preload, and the load–life curve with *k*=0.5 is the highest. This further proves that the failure life of a bolt is determined by the load amplitude and the preload, and the optimal preload for 8.8 grade M8×1.25×70 high-strength bolts is *F*_{0.5}.