# Experimental Study of Constant-Amplitude Fatigue Performance ofM39 High-Strength Bolts in Grid Structures with Bolt-Sphere Joints

Authors: Zichun Zhou; Honggang Lei; Xu Yang; Bin Qiu
Journal: Mobile Information Systems
 (2022)

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## Abstract

Grid structures with bolt-sphere joints are widely used in industrial plants. With the installation of suspension cranes, high-strength bolts are subjected to repeated alternating loads and cause fatigue problems. Due to their dispersive nature, the degree of difficulty and cost of fatigue tests are extremely high. Due to the lack of a recognized fatigue design method, engineering designers have encountered great difficulties, and the promotion and application of such methods are severely restricted. In this paper, we successfully implemented a constant-amplitude fatigue test of M39 high-strength bolt specimens on an American MTS Landmark 370.50 fatigue testing machine and obtained the correspondingS-N curve from the statistical analysis of the test data. Using the nominal stress amplitude Δσ and the hot point stress amplitude Δσk as the design parameters, we established a corresponding constant-amplitude fatigue design method. We performed a microscopic analysis of the fracture surface using a TESCAN Mira3 LMH scanning electron microscope and revealed the fatigue failure mechanism of the high-strength bolts.


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## Body

## 1. Introduction

Grid structures with bolt-sphere joints are space structures with a high degree of statical indeterminacy [1]. Due to their suitable stress ranges, factory prefabrication, on-site assembly, and broad scope of applicability, they are widely used for industrial building construction. To meet production needs, the bolt-sphere grid structures in industrial buildings are usually installed with suspension cranes for lifting and transporting equipment. The installation of suspension cranes takes full advantage of the statically indeterminate structure of the grid to quickly spread the concentrated force, and the grid can also be flexibly laid out to change the industrial flow processes. Grid structures have been used for a broad range of applications in construction, aerospace, metallurgy, machinery, and light industries [2].In the reciprocating loading process of the suspension crane on the bolt-sphere grid structure, fatigue will occur at the bolt-sphere joint of the grid and result in fatigue failure [3]. Fatigue failure is a type of brittle failure that usually occurs suddenly, and the failure is complete. The goal of studying fatigue is to establish a fatigue calculation method or a fatigue life estimation method. Although the fatigue problem of the grid structure under the action of a suspended crane has gained attention, to date, there has not been an explicit stipulation on how to carry out the fatigue design. In China’s national standard grid structure design and construction procedures (JGJ7-91) [4], article 1.0.4 stipulates that “The fatigue strength and construction of grid structures directly subjected to medium or heavy class loads of suspension cranes should be determined through special tests.” This is an evasive statement. In the classification of components and joints that require fatigue calculations in Appendix E of design code for steel structures [5], connecting structures related to grid structures were not included. With no standards to rely on, designers can only act based on experience. This can not only put engineering design at risk but has also greatly restricted the large-scale application of suspension cranes in the grid structures of industrial plants. In this regard, the experimental and theoretical research on the fatigue of bolt-sphere joints in grid structures has important scientific value and practical engineering significance.Nam et al. investigated a novel method to construct a unified stress-life (S-N) curve that was proposed for the three different bolts with different sizes and lengths subjected to various external axial loading conditions and preloads, ranging from 44% to 88% of the ultimate tensile stress [6]. Maljaars and Euler drew on a large number of fatigue tests to assess bolts and bolted connections, which revealed the possibility to improve current design specifications [7]. Bartsch and Feldmann analyzed the fatigue test data and concluded that bolts should be designed with possibly large diameters and be placed as close as possible to the tension flanges, while end plates should be dimensioned thick enough, at least as thick as the bolt diameter [8]. Ding et al. investigated the reliability of the high-strength bolts’ fake tightened-up phenomena of the bolted spherical joints and developed a group of calculation formulas for predicting the tensile strength of bolted spherical joints with different bolt screwing depths [9]. Yuan et al. studied the bearing performance of the bolt-sphere joint portion of the bolted spherical node with stochastic pitting corrosion damage [10].The key to the fatigue problem of the grid structure is at the joints. Researchers have performed theoretical and experimental studies on the fatigue performances of the welded hollow sphere joints and the bolt-sphere joints commonly used in grid structures. These results are summarized as follows. In 1994, researchers carried out uniaxial tensile constant-amplitude fatigue tests on twelveM24 and three M33 high-strength bolts, and they obtained 15 effective fracture points and the corresponding S-N curves [11]. In addition, microscopic analysis of the fracture surface was performed using a PSEM-500X scanning electron microscope and EDA X-711 spectrometer. In 1995, researchers carried out uniaxial tensile constant-amplitude fatigue tests on four M14, eight M24, and eight M33 high-strength bolts and obtained 20 effective fracture points [2]. They obtained the corresponding S-N curves and established an empirical formula using the stress amplitude as the design parameter. Additionally, they found that the tensile fatigue strength of the high-strength bolts was related to the stress ratio ρ.In 2016 and 2017, researchers conducted constant amplitude fatigue tests onM20 and M27 high-strength bolts and fitted the corresponding S-N curves to the data [12, 13]. A preliminary constant amplitude fatigue calculation method was established based on a design expression of the allowed stress amplitude. In 2019, researchers conducted a variable-amplitude fatigue test on M30 high-strength bolts and revealed the variable-amplitude fatigue performance of the M30 high-strength bolts [14].The above-cited studies on fatigue problems and the research methods adopted, without exception, were fatigue tests. Based on this research trend, fatigue tests will remain the most reliable research tool for investigating fatigue performance for the foreseeable future. This article presents the first systematic investigation of the constant-amplitude fatigue performances ofM39 high-strength bolts in grid structures with bolt-sphere joints, aimed at establishing a fatigue calculation method.


## 2. Materials and Methods

### 2.1. Design of Fatigue Specimen

The fatigue test procedures for grid structures with bolt-sphere joints were created according to the specifications in the technical specification for space frame structures (JGJ7-2010) [15], the bolted spherical nodes of steel grid structures (JGT10-2009) [16], and the high-strength bolts for joints of space grid structures (GBT 16939-2016) [17].M39 high-strength bolts: the M39 thread specifications were as follows: the nominal length L = 128 mm, the performance class was 9.8 s, the effective cross-sectional area was 976 mm2, the ultimate tensile load was 878–1074 kN, and the material was 40 Cr. Since the wedge grip on the fatigue testing machine can accommodate a maximum diameter of 55 mm but the nut diameter of the M39 high-strength bolt was 60 mm, it was necessary to modify the nut of the M39 high-strength bolt and cut it down to a diameter less than 55 mm.Bolt-sphere BS200: the outer diameter of the No. 45 steel bolt-sphere was 200 mm. The bolt-sphere was machined from a blank forged steel sphere. The material met the specifications of the high-quality carbon structural steel (GB/T 699) [18]. Three pairs of threaded holes were machined on each sphere. When one thread failed, the sphere could be rotated to use the threads in another direction for fatigue testing. Thus, a bolt-sphere could be efficiently used three times.


### 2.2. Test Program

#### 2.2.1. Static Failure Test

To understand the mechanical properties of the high-strength bolts used in the tests and determine the magnitude of the applied load in the subsequent constant-amplitude fatigue tests, the high-strength bolts were first subjected to a tensile test before the fatigue test. Randomly select threeM39 high-strength bolts, prepare test samples according to relevant provisions of ISO 6892-1:2009 [19], as shown in Figure 1, and conduct static performance tensile tests on high-strength bolt materials with a WAW-1000 kN microcomputer-controlled electro-hydraulic servo universal testing machine. See Table 1 for the mechanical property test results of the obtained materials and the stress-strain curve. All mechanical property indexes of the high-strength bolts meet the requirements.Figure 1
. The s force—displacement curve of theM39 high-strength bolt.Table 1
Static test results of high-strength bolts.
Bolt numberσ0.2σ0.2¯σbσb¯δ¯ (%)Ψ¯ (%)M39-a783.6781.6964.4969.711.244M39-b757.9951.1M39-c803.4993.6σ0.2—Conditional yield strength (MPa); σ0.2¯—average value of conditional yield strength (MPa); σb—ultimate tensile strength (MPa); σb¯—average value of ultimate tensile strength (MPa); δ¯—rate of extension; Ψ¯—rate of contraction.


#### 2.2.2. Constant-Amplitude Fatigue Test

In order to obtain a completeS-N curve, we will set several stress ranges, each of which contains more than 2 specimens. The stress ratio is set to 0.1 and 0.6, respectively, and the values of the maximum stress and the minimum stress in this range can be calculated according to formulas (1) and (2).(1)R=σminσmax,(2)Δσ=σmax−σmin.According to the loading capacity of the fatigue testing machine, the loading system shown in Table2 is formulated.Table 2
Results of constant-amplitude fatigue tests for M39 high-strength bolts.
SpecimenMaximum stressσmax (MPa)Minimum stressσmin (MPa)Stress range Δσ (MPa)Fatigue lifeN (×104)Stress ratioRM39-3138538.5346.52.970.1M39-3238538.5346.52.720.1M39-3331531.5283.53.460.1M39-3431531.5283.54.350.1M39-37280282528.420.1M39-38280282526.110.1M39-392102118913.500.1M39-402102118915.170.1M39-3517517.5157.534.450.1M39-3617517.5157.524.390.1M39-138523115420.620.6M39-238523115420.130.6M39-438523115419.020.6M39-535021014048.310.6M39-635021014041.890.6M39-735021014028.690.6M39-931518912643.510.6M39-1031518912657.240.6M39-1131518912639.520.6M39-1328016811238.410.6M39-1428016811229.840.6M39-1628016811240.020.6M39-2628016811239.520.6M39-2728016811249.790.6M39-172451479842.300.6M39-1824514798102.700.6M39-192451479848.360.6M39-282451479861.790.6M39-292451479847.080.6M39-2121012684159.150.6M39-2221012684143.960.6M39-2321012684272.370.6M39-2421012684125.490.6M39-252101268485.700.6All the constant-amplitude fatigue tests of the high-strength bolts were completed on the MTS Landmark 370.50 hydraulic servo fatigue testing machine. The maximum applied load of the testing machine was 500 kN, the minimum load was 15 kN, and the frequency of the loading cycles could reach 60 Hz. Under a hydraulic cycle, the actuator at the bottom of the fatigue testing machine produced a vertical displacement and applied an axial tension to the bolt held in the fixture. The applied stress load was exactly the same as the load experienced by the bolt in actual engineering applications, as shown in Figure2.Figure 2
. Fatigue test machine and loading device.Using the test setup described above, fatigue tests were conducted on 34 high-strength bolts under constant-amplitude alternating loads. The stress ratioρ of this test was set to the range of 0.1 to 0.6, and the loads were applied in a specified order.Figure 3
. TheS-N curve of the M39 high-strength bolt (R = 0.1 and R = 0.6).


#### 2.2.3. Experimental Procedure

(1)
Using the MTS Landmark 370.50 fatigue tester (Figure3), constant-amplitude fatigue tests were performed on two high-strength bolts at a time. The load was applied according to the desired stress amplitude on the bolt.(2)
The test stopped automatically, and the MTS Landmark 370.50 fatigue testing machine was shut down when the bolt fractured and failed under fatigue.(3)
The number of final cycles at which the bolt broke was recorded, the fractured surface was photographed, and the test information was analyzed (Figure3).(4)
The failed bolt was replaced with a new bolt, and the next round of constant-amplitude fatigue loading was conducted with a new stress amplitude. The undamaged bolt then became the test specimen for the variable-amplitude fatigue tests, and so forth, until all the high-strength bolts were tested.


## 2.1. Design of Fatigue Specimen

The fatigue test procedures for grid structures with bolt-sphere joints were created according to the specifications in the technical specification for space frame structures (JGJ7-2010) [15], the bolted spherical nodes of steel grid structures (JGT10-2009) [16], and the high-strength bolts for joints of space grid structures (GBT 16939-2016) [17].M39 high-strength bolts: the M39 thread specifications were as follows: the nominal length L = 128 mm, the performance class was 9.8 s, the effective cross-sectional area was 976 mm2, the ultimate tensile load was 878–1074 kN, and the material was 40 Cr. Since the wedge grip on the fatigue testing machine can accommodate a maximum diameter of 55 mm but the nut diameter of the M39 high-strength bolt was 60 mm, it was necessary to modify the nut of the M39 high-strength bolt and cut it down to a diameter less than 55 mm.Bolt-sphere BS200: the outer diameter of the No. 45 steel bolt-sphere was 200 mm. The bolt-sphere was machined from a blank forged steel sphere. The material met the specifications of the high-quality carbon structural steel (GB/T 699) [18]. Three pairs of threaded holes were machined on each sphere. When one thread failed, the sphere could be rotated to use the threads in another direction for fatigue testing. Thus, a bolt-sphere could be efficiently used three times.


## 2.2. Test Program

### 2.2.1. Static Failure Test

To understand the mechanical properties of the high-strength bolts used in the tests and determine the magnitude of the applied load in the subsequent constant-amplitude fatigue tests, the high-strength bolts were first subjected to a tensile test before the fatigue test. Randomly select threeM39 high-strength bolts, prepare test samples according to relevant provisions of ISO 6892-1:2009 [19], as shown in Figure 1, and conduct static performance tensile tests on high-strength bolt materials with a WAW-1000 kN microcomputer-controlled electro-hydraulic servo universal testing machine. See Table 1 for the mechanical property test results of the obtained materials and the stress-strain curve. All mechanical property indexes of the high-strength bolts meet the requirements.Figure 1
. The s force—displacement curve of theM39 high-strength bolt.Table 1
Static test results of high-strength bolts.
Bolt numberσ0.2σ0.2¯σbσb¯δ¯ (%)Ψ¯ (%)M39-a783.6781.6964.4969.711.244M39-b757.9951.1M39-c803.4993.6σ0.2—Conditional yield strength (MPa); σ0.2¯—average value of conditional yield strength (MPa); σb—ultimate tensile strength (MPa); σb¯—average value of ultimate tensile strength (MPa); δ¯—rate of extension; Ψ¯—rate of contraction.


### 2.2.2. Constant-Amplitude Fatigue Test

In order to obtain a completeS-N curve, we will set several stress ranges, each of which contains more than 2 specimens. The stress ratio is set to 0.1 and 0.6, respectively, and the values of the maximum stress and the minimum stress in this range can be calculated according to formulas (1) and (2).(1)R=σminσmax,(2)Δσ=σmax−σmin.According to the loading capacity of the fatigue testing machine, the loading system shown in Table2 is formulated.Table 2
Results of constant-amplitude fatigue tests for M39 high-strength bolts.
SpecimenMaximum stressσmax (MPa)Minimum stressσmin (MPa)Stress range Δσ (MPa)Fatigue lifeN (×104)Stress ratioRM39-3138538.5346.52.970.1M39-3238538.5346.52.720.1M39-3331531.5283.53.460.1M39-3431531.5283.54.350.1M39-37280282528.420.1M39-38280282526.110.1M39-392102118913.500.1M39-402102118915.170.1M39-3517517.5157.534.450.1M39-3617517.5157.524.390.1M39-138523115420.620.6M39-238523115420.130.6M39-438523115419.020.6M39-535021014048.310.6M39-635021014041.890.6M39-735021014028.690.6M39-931518912643.510.6M39-1031518912657.240.6M39-1131518912639.520.6M39-1328016811238.410.6M39-1428016811229.840.6M39-1628016811240.020.6M39-2628016811239.520.6M39-2728016811249.790.6M39-172451479842.300.6M39-1824514798102.700.6M39-192451479848.360.6M39-282451479861.790.6M39-292451479847.080.6M39-2121012684159.150.6M39-2221012684143.960.6M39-2321012684272.370.6M39-2421012684125.490.6M39-252101268485.700.6All the constant-amplitude fatigue tests of the high-strength bolts were completed on the MTS Landmark 370.50 hydraulic servo fatigue testing machine. The maximum applied load of the testing machine was 500 kN, the minimum load was 15 kN, and the frequency of the loading cycles could reach 60 Hz. Under a hydraulic cycle, the actuator at the bottom of the fatigue testing machine produced a vertical displacement and applied an axial tension to the bolt held in the fixture. The applied stress load was exactly the same as the load experienced by the bolt in actual engineering applications, as shown in Figure2.Figure 2
. Fatigue test machine and loading device.Using the test setup described above, fatigue tests were conducted on 34 high-strength bolts under constant-amplitude alternating loads. The stress ratioρ of this test was set to the range of 0.1 to 0.6, and the loads were applied in a specified order.Figure 3
. TheS-N curve of the M39 high-strength bolt (R = 0.1 and R = 0.6).


### 2.2.3. Experimental Procedure

(1)
Using the MTS Landmark 370.50 fatigue tester (Figure3), constant-amplitude fatigue tests were performed on two high-strength bolts at a time. The load was applied according to the desired stress amplitude on the bolt.(2)
The test stopped automatically, and the MTS Landmark 370.50 fatigue testing machine was shut down when the bolt fractured and failed under fatigue.(3)
The number of final cycles at which the bolt broke was recorded, the fractured surface was photographed, and the test information was analyzed (Figure3).(4)
The failed bolt was replaced with a new bolt, and the next round of constant-amplitude fatigue loading was conducted with a new stress amplitude. The undamaged bolt then became the test specimen for the variable-amplitude fatigue tests, and so forth, until all the high-strength bolts were tested.


## 2.2.1. Static Failure Test

To understand the mechanical properties of the high-strength bolts used in the tests and determine the magnitude of the applied load in the subsequent constant-amplitude fatigue tests, the high-strength bolts were first subjected to a tensile test before the fatigue test. Randomly select threeM39 high-strength bolts, prepare test samples according to relevant provisions of ISO 6892-1:2009 [19], as shown in Figure 1, and conduct static performance tensile tests on high-strength bolt materials with a WAW-1000 kN microcomputer-controlled electro-hydraulic servo universal testing machine. See Table 1 for the mechanical property test results of the obtained materials and the stress-strain curve. All mechanical property indexes of the high-strength bolts meet the requirements.Figure 1
. The s force—displacement curve of theM39 high-strength bolt.Table 1
Static test results of high-strength bolts.
Bolt numberσ0.2σ0.2¯σbσb¯δ¯ (%)Ψ¯ (%)M39-a783.6781.6964.4969.711.244M39-b757.9951.1M39-c803.4993.6σ0.2—Conditional yield strength (MPa); σ0.2¯—average value of conditional yield strength (MPa); σb—ultimate tensile strength (MPa); σb¯—average value of ultimate tensile strength (MPa); δ¯—rate of extension; Ψ¯—rate of contraction.


## 2.2.2. Constant-Amplitude Fatigue Test

In order to obtain a completeS-N curve, we will set several stress ranges, each of which contains more than 2 specimens. The stress ratio is set to 0.1 and 0.6, respectively, and the values of the maximum stress and the minimum stress in this range can be calculated according to formulas (1) and (2).(1)R=σminσmax,(2)Δσ=σmax−σmin.According to the loading capacity of the fatigue testing machine, the loading system shown in Table2 is formulated.Table 2
Results of constant-amplitude fatigue tests for M39 high-strength bolts.
SpecimenMaximum stressσmax (MPa)Minimum stressσmin (MPa)Stress range Δσ (MPa)Fatigue lifeN (×104)Stress ratioRM39-3138538.5346.52.970.1M39-3238538.5346.52.720.1M39-3331531.5283.53.460.1M39-3431531.5283.54.350.1M39-37280282528.420.1M39-38280282526.110.1M39-392102118913.500.1M39-402102118915.170.1M39-3517517.5157.534.450.1M39-3617517.5157.524.390.1M39-138523115420.620.6M39-238523115420.130.6M39-438523115419.020.6M39-535021014048.310.6M39-635021014041.890.6M39-735021014028.690.6M39-931518912643.510.6M39-1031518912657.240.6M39-1131518912639.520.6M39-1328016811238.410.6M39-1428016811229.840.6M39-1628016811240.020.6M39-2628016811239.520.6M39-2728016811249.790.6M39-172451479842.300.6M39-1824514798102.700.6M39-192451479848.360.6M39-282451479861.790.6M39-292451479847.080.6M39-2121012684159.150.6M39-2221012684143.960.6M39-2321012684272.370.6M39-2421012684125.490.6M39-252101268485.700.6All the constant-amplitude fatigue tests of the high-strength bolts were completed on the MTS Landmark 370.50 hydraulic servo fatigue testing machine. The maximum applied load of the testing machine was 500 kN, the minimum load was 15 kN, and the frequency of the loading cycles could reach 60 Hz. Under a hydraulic cycle, the actuator at the bottom of the fatigue testing machine produced a vertical displacement and applied an axial tension to the bolt held in the fixture. The applied stress load was exactly the same as the load experienced by the bolt in actual engineering applications, as shown in Figure2.Figure 2
. Fatigue test machine and loading device.Using the test setup described above, fatigue tests were conducted on 34 high-strength bolts under constant-amplitude alternating loads. The stress ratioρ of this test was set to the range of 0.1 to 0.6, and the loads were applied in a specified order.Figure 3
. TheS-N curve of the M39 high-strength bolt (R = 0.1 and R = 0.6).


## 2.2.3. Experimental Procedure

(1)
Using the MTS Landmark 370.50 fatigue tester (Figure3), constant-amplitude fatigue tests were performed on two high-strength bolts at a time. The load was applied according to the desired stress amplitude on the bolt.(2)
The test stopped automatically, and the MTS Landmark 370.50 fatigue testing machine was shut down when the bolt fractured and failed under fatigue.(3)
The number of final cycles at which the bolt broke was recorded, the fractured surface was photographed, and the test information was analyzed (Figure3).(4)
The failed bolt was replaced with a new bolt, and the next round of constant-amplitude fatigue loading was conducted with a new stress amplitude. The undamaged bolt then became the test specimen for the variable-amplitude fatigue tests, and so forth, until all the high-strength bolts were tested.


## 3. Results and Discussion

### 3.1. Analysis of Fatigue Test Results

At present, the most commonly used fatigue curve expression is the log-log expression, given as follows:(3)lgΔσ=A+BlgN±2s,where s is the statistical deviation of the test points.Using the MTS Landmark 370.50 fatigue tester, we carried out constant-amplitude fatigue tests on 34M39 high-strength bolts. Ten of the specimens had a stress ratio of σ = 0.1, 24 of the specimens had a stress ratio of σ = 0.6, and only one specimen, M39-23, did not fail after 2 × 106 cycles. Thus, there were 33 effective data points available for the regression analysis.Perform regression analysis on 10 constant amplitude fatigue test data with a stress ratio of 0.1 to obtain the correspondingS-N curve (Figure 3). The corresponding equation is as follows. Correlation coefficient R2 = 0.958, [Δ σ]2 × 106 = 76.97 MPa.(4)lgΔσ=−0.2924lgN+3.7153±0.1109.The regression analysis is conducted on 23 constant amplitude fatigue test data with a stress ratio of 0.6 to obtain the corresponding S-N curve (Figure3). The corresponding equation is as follows. Correlation coefficient R2 = 0.697, [Δ σ]2 × 106 = 42.48 MPa.(5)lgΔσ=−0.3229lgN+3.9568±0.2319.It can be concluded that the fatigue test data obtained under different stress ratios have a good correlation through the regression analysis, but the correlation between the constant amplitude fatigue test data of theM39 high-strength bolts with a stress ratio of R = 0.1 is better than that of M39 high-strength bolts with a stress ratio of R = 0.6, and the fatigue strength is higher.Now, the 10 constant amplitude fatigue test data points ofM39 high-strength bolts with a stress ratio of R = 0.1 and the 23 constant amplitude fatigue test data points of M39 high-strength bolts with a stress ratio of R = 0.6 are analyzed together, and the constant amplitude fatigue S-N curve of the M39 high-strength bolt connection is shown in Figure 4. The correlation coefficient R2 = 0.914, [Δ σ]2 × 106 = 40.82 MPa. The power S-N curve of M39 high-strength bolts with the stress amplitude Δ σ as the parameter is shown in Figure 5.(6)lgΔσ=−0.3229lgN+3.9568±0.1065.Figure 4
. TheS-N curve of the M39 high-strength bolt (33data).Figure 5
. The powerS-N curve of the M39 high-strength bolts with the stress amplitude Δσ as the parameter.It can be seen from Figures4 and 5 that when the regression analysis is performed with the stress amplitude Δσ as the parameter, the correlation of the regular fatigue test data is good, indicating that the stress comparison has no effect on the fatigue intensity.


### 3.2. Analysis of Fatigue and Fracture

In this experiment, the failure mode of theM39 high-strength bolts was consistently fractured at the first thread below the surface of the bolt-sphere on the shank of the bolt damage, as shown in Figure 1. To study the fatigue failure mechanism, the surfaces of the fatigue fracture on the high-strength bolt were examined systematically under an EVOMA15 scanning electron microscope (Zeiss, Germany). In this experiment, the failure mode of the M39 high-strength bolts was consistently fractured at the first thread below the surface of the bolt-sphere on the shank of the bolt damage. A representative fracture failure, designated as M39-10, was chosen for macroscopic and microscopic analysis.In the macroscopic photograph in Figure6(a), the failure was brittle, in nature and there were no evident signs of plastic deformation near the fracture. There were three characteristic zones: a fatigue source zone, a fatigue crack steady-propagation zone, and an instantaneous failure zone. The fatigue source zone was the earliest failure that occurred for the M39-10 high-strength bolts. In this region, the crack growth rate was slow, and the repeated opening and closing of the cracks caused friction on the mating surfaces of the fracture. The fatigue source zone was therefore macroscopically flat and smooth. There were multiple fatigue source zones at the notch around the circumference of the bolt fracture. Microscopically, the thread of a high-strength bolt had a notch where there was large residual stress. Being constructed from industrial materials, high-strength bolts often contain slag, voids, inclusions, indentations, bruises, and regions of inhomogeneous and discontinuous microstructures and chemical compositions. Fatigue cracks often occurred int these positions. Figure 6(b) shows the stress concentration regions where fatigue cracks occurred at the surface notch.Figure 6
Fracture morphology of a bolt.The second stage of fatigue failure was steady crack propagation, with the path of propagation perpendicular to the principal stress direction. On the macrophotographs of this region, the macroscopic characteristics of the fracture surface were relatively smooth and occupied more than half of the fracture. The fatigue bands could be clearly observed on the microphotograph. Figure6(c) shows the fatigue bands that appeared at this stage. The key characteristic is that they were basically parallel to each other, and the direction of the band was perpendicular to the local crack propagation direction. The bands produced by brittle fatigue failure were uneven and irregular, but overall, the region could still be recognized.The rough and protruding areas in the lower parts of Figure6(d)show the instantaneous failure zone. This was a region formed by the unstable propagation after the fatigue crack had grown to a critical size. The microscopic morphology of this region mostly manifested as static-load transient characteristics, with more dimples, as shown in Figure 6.


### 3.3. Establishing a Calculation Method for Constant-Amplitude Fatigue

Based on the results of the constant-amplitude fatigue tests of high-strength bolts described above, we established the following calculation method for the constant-amplitude fatigue of the high-strength bolts in grid structures with bolt-sphere joints using the nominal stress amplitude Δσ as the design parameter.


#### 3.3.1. Nominal Stress Amplitude Method

This method uses Δσ as the design parameter and the following equations:(7)Δσ≤Δσ,Δσ=CN1/β,where Δσ is the nominal stress amplitude (MPa) at the joint of the high-strength bolt, N is the number of cycles, and C and β are parameters. Article 6.2.1 of the standard GB50017-2003 design specification for steel structures stipulates that the converted stress amplitude of the non-welded part is Δσ = σmax—0.7 σmin. In this study, it was concluded that 0.7 should be changed to 1.0, or Δσ = σmax—σmin. σmax and σmin are the maximum and minimum stresses (MPa) at the joint of the high-strength bolt, respectively, and [Δσ] is the allowable stress amplitude (MPa) of the high-strength bolt joint.TakingN = 2 × 106 as the base period, it follows from (7) that the allowable stress amplitude of M39 high-strength bolts should be [Δσ]2×106 = 40.82 MPa. The values of C and β were 2.23 × 1010 and 2.5131, respectively, based on (7).


#### 3.3.2. Allowable Hot-Spot Stress Amplitude Method

This article uses the hot stress amplitude Δσh as the design parameter, which can better reflect the real stress state of the destructive position, which is more scientific.In this method, the hot-spot stress amplitude Δσh was used as a design parameter, and both sides of (8) were multiplied by the fatigue notch coefficient Kf, which was in the range of 4.35–4.89. For safety reasons, the minimum value of 4.35 was taken on the right-hand side of the equation. Using N = 2 × 106 as the base period, the allowable hot-spot stress amplitude of the M39 high-strength bolts was [Δσh] = 177.57 MPa(8)Δσ_h=K_fΔσ≤Δσ_h.


## 3.1. Analysis of Fatigue Test Results

At present, the most commonly used fatigue curve expression is the log-log expression, given as follows:(3)lgΔσ=A+BlgN±2s,where s is the statistical deviation of the test points.Using the MTS Landmark 370.50 fatigue tester, we carried out constant-amplitude fatigue tests on 34M39 high-strength bolts. Ten of the specimens had a stress ratio of σ = 0.1, 24 of the specimens had a stress ratio of σ = 0.6, and only one specimen, M39-23, did not fail after 2 × 106 cycles. Thus, there were 33 effective data points available for the regression analysis.Perform regression analysis on 10 constant amplitude fatigue test data with a stress ratio of 0.1 to obtain the correspondingS-N curve (Figure 3). The corresponding equation is as follows. Correlation coefficient R2 = 0.958, [Δ σ]2 × 106 = 76.97 MPa.(4)lgΔσ=−0.2924lgN+3.7153±0.1109.The regression analysis is conducted on 23 constant amplitude fatigue test data with a stress ratio of 0.6 to obtain the corresponding S-N curve (Figure3). The corresponding equation is as follows. Correlation coefficient R2 = 0.697, [Δ σ]2 × 106 = 42.48 MPa.(5)lgΔσ=−0.3229lgN+3.9568±0.2319.It can be concluded that the fatigue test data obtained under different stress ratios have a good correlation through the regression analysis, but the correlation between the constant amplitude fatigue test data of theM39 high-strength bolts with a stress ratio of R = 0.1 is better than that of M39 high-strength bolts with a stress ratio of R = 0.6, and the fatigue strength is higher.Now, the 10 constant amplitude fatigue test data points ofM39 high-strength bolts with a stress ratio of R = 0.1 and the 23 constant amplitude fatigue test data points of M39 high-strength bolts with a stress ratio of R = 0.6 are analyzed together, and the constant amplitude fatigue S-N curve of the M39 high-strength bolt connection is shown in Figure 4. The correlation coefficient R2 = 0.914, [Δ σ]2 × 106 = 40.82 MPa. The power S-N curve of M39 high-strength bolts with the stress amplitude Δ σ as the parameter is shown in Figure 5.(6)lgΔσ=−0.3229lgN+3.9568±0.1065.Figure 4
. TheS-N curve of the M39 high-strength bolt (33data).Figure 5
. The powerS-N curve of the M39 high-strength bolts with the stress amplitude Δσ as the parameter.It can be seen from Figures4 and 5 that when the regression analysis is performed with the stress amplitude Δσ as the parameter, the correlation of the regular fatigue test data is good, indicating that the stress comparison has no effect on the fatigue intensity.


## 3.2. Analysis of Fatigue and Fracture

In this experiment, the failure mode of theM39 high-strength bolts was consistently fractured at the first thread below the surface of the bolt-sphere on the shank of the bolt damage, as shown in Figure 1. To study the fatigue failure mechanism, the surfaces of the fatigue fracture on the high-strength bolt were examined systematically under an EVOMA15 scanning electron microscope (Zeiss, Germany). In this experiment, the failure mode of the M39 high-strength bolts was consistently fractured at the first thread below the surface of the bolt-sphere on the shank of the bolt damage. A representative fracture failure, designated as M39-10, was chosen for macroscopic and microscopic analysis.In the macroscopic photograph in Figure6(a), the failure was brittle, in nature and there were no evident signs of plastic deformation near the fracture. There were three characteristic zones: a fatigue source zone, a fatigue crack steady-propagation zone, and an instantaneous failure zone. The fatigue source zone was the earliest failure that occurred for the M39-10 high-strength bolts. In this region, the crack growth rate was slow, and the repeated opening and closing of the cracks caused friction on the mating surfaces of the fracture. The fatigue source zone was therefore macroscopically flat and smooth. There were multiple fatigue source zones at the notch around the circumference of the bolt fracture. Microscopically, the thread of a high-strength bolt had a notch where there was large residual stress. Being constructed from industrial materials, high-strength bolts often contain slag, voids, inclusions, indentations, bruises, and regions of inhomogeneous and discontinuous microstructures and chemical compositions. Fatigue cracks often occurred int these positions. Figure 6(b) shows the stress concentration regions where fatigue cracks occurred at the surface notch.Figure 6
Fracture morphology of a bolt.The second stage of fatigue failure was steady crack propagation, with the path of propagation perpendicular to the principal stress direction. On the macrophotographs of this region, the macroscopic characteristics of the fracture surface were relatively smooth and occupied more than half of the fracture. The fatigue bands could be clearly observed on the microphotograph. Figure6(c) shows the fatigue bands that appeared at this stage. The key characteristic is that they were basically parallel to each other, and the direction of the band was perpendicular to the local crack propagation direction. The bands produced by brittle fatigue failure were uneven and irregular, but overall, the region could still be recognized.The rough and protruding areas in the lower parts of Figure6(d)show the instantaneous failure zone. This was a region formed by the unstable propagation after the fatigue crack had grown to a critical size. The microscopic morphology of this region mostly manifested as static-load transient characteristics, with more dimples, as shown in Figure 6.


## 3.3. Establishing a Calculation Method for Constant-Amplitude Fatigue

Based on the results of the constant-amplitude fatigue tests of high-strength bolts described above, we established the following calculation method for the constant-amplitude fatigue of the high-strength bolts in grid structures with bolt-sphere joints using the nominal stress amplitude Δσ as the design parameter.


### 3.3.1. Nominal Stress Amplitude Method

This method uses Δσ as the design parameter and the following equations:(7)Δσ≤Δσ,Δσ=CN1/β,where Δσ is the nominal stress amplitude (MPa) at the joint of the high-strength bolt, N is the number of cycles, and C and β are parameters. Article 6.2.1 of the standard GB50017-2003 design specification for steel structures stipulates that the converted stress amplitude of the non-welded part is Δσ = σmax—0.7 σmin. In this study, it was concluded that 0.7 should be changed to 1.0, or Δσ = σmax—σmin. σmax and σmin are the maximum and minimum stresses (MPa) at the joint of the high-strength bolt, respectively, and [Δσ] is the allowable stress amplitude (MPa) of the high-strength bolt joint.TakingN = 2 × 106 as the base period, it follows from (7) that the allowable stress amplitude of M39 high-strength bolts should be [Δσ]2×106 = 40.82 MPa. The values of C and β were 2.23 × 1010 and 2.5131, respectively, based on (7).


### 3.3.2. Allowable Hot-Spot Stress Amplitude Method

This article uses the hot stress amplitude Δσh as the design parameter, which can better reflect the real stress state of the destructive position, which is more scientific.In this method, the hot-spot stress amplitude Δσh was used as a design parameter, and both sides of (8) were multiplied by the fatigue notch coefficient Kf, which was in the range of 4.35–4.89. For safety reasons, the minimum value of 4.35 was taken on the right-hand side of the equation. Using N = 2 × 106 as the base period, the allowable hot-spot stress amplitude of the M39 high-strength bolts was [Δσh] = 177.57 MPa(8)Δσ_h=K_fΔσ≤Δσ_h.


## 3.3.1. Nominal Stress Amplitude Method

This method uses Δσ as the design parameter and the following equations:(7)Δσ≤Δσ,Δσ=CN1/β,where Δσ is the nominal stress amplitude (MPa) at the joint of the high-strength bolt, N is the number of cycles, and C and β are parameters. Article 6.2.1 of the standard GB50017-2003 design specification for steel structures stipulates that the converted stress amplitude of the non-welded part is Δσ = σmax—0.7 σmin. In this study, it was concluded that 0.7 should be changed to 1.0, or Δσ = σmax—σmin. σmax and σmin are the maximum and minimum stresses (MPa) at the joint of the high-strength bolt, respectively, and [Δσ] is the allowable stress amplitude (MPa) of the high-strength bolt joint.TakingN = 2 × 106 as the base period, it follows from (7) that the allowable stress amplitude of M39 high-strength bolts should be [Δσ]2×106 = 40.82 MPa. The values of C and β were 2.23 × 1010 and 2.5131, respectively, based on (7).


## 3.3.2. Allowable Hot-Spot Stress Amplitude Method

This article uses the hot stress amplitude Δσh as the design parameter, which can better reflect the real stress state of the destructive position, which is more scientific.In this method, the hot-spot stress amplitude Δσh was used as a design parameter, and both sides of (8) were multiplied by the fatigue notch coefficient Kf, which was in the range of 4.35–4.89. For safety reasons, the minimum value of 4.35 was taken on the right-hand side of the equation. Using N = 2 × 106 as the base period, the allowable hot-spot stress amplitude of the M39 high-strength bolts was [Δσh] = 177.57 MPa(8)Δσ_h=K_fΔσ≤Δσ_h.


## 4. Conclusions

By conducting constant-amplitude fatigue tests ofM39 high-strength bolts on an MTS fatigue testing machine, we obtained the S-N curve of the M39 high-strength bolts and established a design method for the constant-amplitude fatigue tests of the M39 high-strength bolts in grid structures with bolt-sphere joints. Taking N = 2 × 106 as the base period, we obtained the allowable stress amplitude of M39 high-strength bolts as [Δσ]2 × 106 = 40.82 MPa and the allowable hot-spot stress amplitude of M39 high-strength bolts as [Δσh] = 177.57 Mpa. In the calculation of the constant-amplitude fatigue, it is reasonable to take the stress amplitude Δσ as the design parameter. In calculating the fatigue strengths of high-strength bolts in grid structures with bolt-sphere joints, using the hot-spot stress amplitude as a design parameter yielded better agreement with the actual stress state of the high-strength bolts.By analyzing the constant-amplitude fatigue test data of theM39 high-strength bolts with two different stress ratios, we found a good correlation in the experimental data, indicating that the stress ratio had no effect on the fatigue strength. Observations of the fatigue source zone revealed that there may be one or more fatigue sources for the fracture of high-strength bolts. By observing the fatigue propagation zone, we found distinct fatigue bands in the fracture propagation zone of the high-strength bolts. Near the fatigue source zone, the fatigue bands were denser, and the fracture surface propagated slowly. Far from the fatigue source zone, the bands were far apart, and the fracture surface propagated faster.In summary, fatigue fracture is a very complex process, affected by a variety of internal and external factors. The internal cause of the failure fracture in this study mostly stress concentration at the thread of the high-strength bolt and the incomplete quenching during the manufacturing process. Under the action of a constant-amplitude alternating load, cracks were initiated and, under the tensile–tensile stress, the crack tip became blunted due to double slip, followed by further blunting by subsequent stretching action. Thus, the crack propagated forward by a certain distance, forming a steady propagation zone of the fatigue crack and resulting in a band. This was the earliest occurrence of fatigue failure.



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*Source: 4155732-2022-09-26.xml*