## 1 Introduction  
As a kind of leading, large-span, spatial steel structures, grid structures have been universally applied in public and industrial buildings [1]. If the span of a structure is larger than 36 m, a traditional bridge crane cannot be adopted. However, grid structures with suspended cranes enjoy considerable advantages in hangars and industrial buildings because of the flexibility of the arrangement and the adaptability of the process flow; their application prospect is certain to broaden. Figure 1 illustrates the applications of grid structures with suspended cranes. As the burden, quantity, and operation frequency of suspended cranes increase, the fatigue problem of grid structures intensifies, so it has become a hot topic of research and a complex problem in academic and engineering fields [2]. In most cases, fatigue failure accidents occur suddenly with disastrous consequences. Thus, in-depth and systematic research on fatigue performance can reduce the potential risks and ensure the safety of grid structures. MERO Company, France, developed bolt-sphere joints in 1942 [1]. In the past 80 years, grid structures using bolt-sphere joints have been widely used globally. Under the load of suspended cranes, the fatigue of bolt-sphere joints primarily occurs at bolted connections. Figure 2 displays a standard bolt-sphere joint. [PERSON] et al. [3] investigated a new approach to plotting a uniform stress-fatigue life (S-N) curve for three bolts with different dimensions under different external, axial loading conditions and preloading, varying from 44% to 88% of the ultimate tensile stress. [PERSON] and [PERSON] [4] drew on many fatigue tests to assess bolts and bolted connections and explored the possibilities of improving current design specifications. [PERSON] and [PERSON] [5] analyzed fatigue test data and concluded that bolts need to be designed with a diameter as large as possible and placed near the tension flanges possibly close, while end plates must be at least as thick as the diameter of the bolts. [PERSON], [PERSON], [PERSON] et al. [6] examined the reliability of the fake tightened-up phenomena of high-strength bolts of bolted spherical joints and developed formulas for estimating the tensile strength of bolted spherical joints with various bolt screwing depths. In another work, [PERSON] [7] researched the bearing performance of bolt-sphere joints of bolted spherical joints with random, pitting corrosion damage. [PERSON] et al. [8] found that the appropriate thread rolling process can effectively reduce the tensile residual stress of bolts, and thus improve the fatigue life of bolts. [PERSON] [9] analyzed the failure mechanism of high-strength bolts, and concluded that damage to the zinc coating would reduce the fatigue performance of high-strength  
Figure 1: The applications of grid structures with suspended cranes. Figure 2: A schematic of a bolt–sphere joint. bolts. [PERSON] et al. [10] investigated the relationship between bolt loosening and fatigue failure, and the results could predict the fatigue life in specific circumstances. [PERSON] et al. [11] presented that the fatigue life of some alloys is affected more by material properties than by external loads under uncertain constant amplitude cyclic load. For research on the fatigue performance of bolt-sphere joints, scholars have conducted some theoretical and experimental studies. For example, [PERSON] and [PERSON] [12] performed 13 uniaxial, tensile fatigue tests on 40 Cr M14 high-strength bolts under constant amplitude and obtained the corresponding S-N curve. [PERSON] et al. [13] considered M24 and M33 high-strength bolts the research objects and conducted constant-amplitude fatigue tests on 12 members of the bolt-sphere-joined grid structure, 12 suspension-point specimens, and 2 bolt-sphere-joined truss specimens. Twenty-one fatigue failure points were determined in the tests, and the related S-N curve was developed. Furthermore, the checking formula was constructed when the stress amplitude was taken as the design parameter.