Fracture Threshold Analysis and Parameter Matching of Cut-Out Induced Bolts for Subway Couplers

Abstract

The overload protection device is crucial in ensuring the orderly absorption of kinetic energy by the coupler buffer device. This paper studies an overload protection bolt with a cut-out zone. In the bolt impact experiment, a premature fracture of 10.9-grade M24 bolts was observed. Based on the analysis of the results, it was concluded that this phenomenon was caused by the mismatch between the mechanical properties of the bolts and the dynamic performance of the coupler. Building on this test, a numerical simulation model was established and subsequently validated. The width and depth of the inducing structure were selected as the research objects. Using the Latin Hypercube method, 78 sets of cut-out zone structure parameters were generated, and numerical simulations were performed on the cut-out induced bolts. The simulation results indicate that the peak force generated by the coupler collision leads to necking in the cut-out induced bolts, which consequently weakens their mechanical properties to some extent. Therefore, it is necessary to consider a strength margin when designing cut-out induced bolts. Based on the simulation results, a surrogate model was constructed, and the optimal bolt cut-out zone was obtained through optimization: a width of 17.74 mm and a depth of 1.37 mm. The surrogate model predicted a fracture force of 1894.13 kN for the bolts. An impact test was conducted to verify the performance of the optimized cut-out induced bolts. The experimental results showed that the cut-out induced bolts broke after the crush tube completed its kinetic energy absorption, with a fracture force of 1828.44 kN, which was a 3.59% difference from the predicted value of the surrogate model. After optimization, the fracture force of the cut-out induced bolts increased from 1147.5 kN to 1828.44 kN (a 59.34% improvement), while the fracture time extended from 20.9 ms to 69 ms, fully meeting the design requirements of the overload protection device.

1. Introduction

The orderly and controllable energy absorption of the coupler buffer device can effectively reduce casualties in train collision accidents. The role of the overload protection device is to promptly disconnect the coupler from the vehicle body, allowing subsequent anti-climbing energy absorption devices to come into contact and absorb energy. In the collision accidents, Figure 1 indicates that the impact kinetic energy is absorbed in the following order: shock absorbers, crushed tube, anti-climbing energy absorbers, and the train body [1,2,3,4]. The purpose of the overload protection device is to disconnect the coupler (which includes the shock absorber and crushed tube) from the train body after the crushed tube has completed its energy absorption, allowing the coupler to freely retract. Otherwise, the long, slender structure of the coupler can act like a lever, lifting the train and potentially causing climbing between vehicles or derailment [5,6,7]. Additionally, if the coupler cannot retract freely, the high-strength coupler can impede the contact of the train’s anti-climbing energy absorption device, interfering with the normal operation of the collision energy management system [8,9].

Currently, the forms of overload protection devices on couplers mainly include drum sleeve type, shear break bolt type, tensile break bolt type, and aluminum alloy plate fracture type [9,10]. The drum sleeve-type overload protection device requires the installation of guide rails during the installation process and needs special tools to apply pre-tightening force. Improper installation can lead to large errors in pre-tightening force, making it inconvenient for practical use [9]. Yao et al. [9] studied the aluminum alloy plate fracture-type overload protection device by adding a cut-out zone and optimizing the circular cross-section, successfully reducing the breaking force value, and found that the method of optimizing the circular cross-section was more effective than adding a cut-out zone. Guan et al. [10] investigated the effects of impact kinetic energy, bolt preload, and transition end structure on the aluminum alloy plate fracture-type overload protection device. Although the aluminum alloy plate fracture-type overload protection device has the advantages of a simple structure and convenient installation, there is currently no set of general design guidelines, nor a general relationship between the transition section design and the breaking force value, that is not conducive to engineering applications. The shear break bolt type and tensile break bolt type are currently the most commonly used forms of overload protection devices. In this study, the tensile break bolt-type overload protection device was chosen as the research object.

The research methods for bolts mainly include experimental methods and numerical simulation methods. Eom et al. [11] studied the tensile properties of bolted end plate connections for structural angle sections using experimental methods. The results indicated that the primary failure mode of the structure was thread failure in the bolts. Additionally, Gong [12], Teh [13], and Wan [14] all studied the mechanical behavior of bolts in various engineering structures under tensile conditions using experimental methods. Guzas [15] conducted tensile tests on bolts using four simulation models for modeling, including a detailed model, simplified model with end linkages, simplified model with shell head and embedded end, with a hole, and simplified model with shell head and embedded end, without a hole. The results indicated that the simulation data most closely matched the experimental data when the diameter of the bolt shank was equal to the major diameter of the bolt and the diameter of the threaded region was equal to the minor diameter of the thread. Mi et al. [16] proposed a novel triangular headed one-sided bolt and studied the performance of T-stubs using triangular headed one-sided bolts through numerical and experimental methods. They also provided an analysis of the factors influencing the strength of T-stubs using triangular headed one-sided bolts. Liang et al. [17] studied the load-bearing capacity and fracture behavior of superalloy bolts under various loading conditions. By comparing the ultimate loads of bolts under combined tensile-bending loads in experiments and simulations, they validated the effectiveness of the finite element model. Guo et al. [18] investigated the influence of different arrangements of Q690 high-strength bolts on the failure modes and ultimate load-bearing capacity of connection structures through static tensile tests and numerical simulations. Lemma [19], Liu [20], and Ahmed [21] all used a combination of experimental and simulation methods to study specific bolted connection structures.

There are many factors that affect the axial impact resistance performance of bolts, in addition to basic material and diameter; studies have shown that the impact resistance performance of bolts is also related to the bolt preload and the length of the connecting threads [22,23,24]. The tensile failure of bolts mainly includes two forms: thread failure and bolt shank failure [16,25]. When the bolt experiences thread failure, its breaking force value will be much lower than the force required to break the smooth shank of the bolt, which is contradictory to the need for a controllable and stable breaking force value in overload protection devices. In the study of the dynamic mechanical properties of thin-walled structures, controlling deformation through inducing structures has been proven to be reliable [26,27,28]. Based on this design concept, cut-out induced bolts have emerged. These bolts weaken the strength of the cut-out zone by setting a cut-out zone on the smooth shank of the bolt, allowing tensile failure to occur at this location and providing better control over the breaking force value. Yao et al. [29] studied the effect of preload on the cut-out induced bolt. The results showed that excessive preload force could cause deformation of the cut-out zone, thereby affecting the breaking force value of the bolt. It can be seen that reasonable cut-out zone structural parameters are an effective means to improve the stability of the fracture mode of the cut-out induced bolts and the reliability of the overload protection function.

From the analysis of the current research status on the overload protection device, it can be found that the matching relationship between the fracture threshold of the overload protection device and the working force of the coupler energy-absorbing device is still unclear. When designing the fracture threshold of the overload protection device, designers lack clear specifications and usually set the fracture threshold of the overload protection system slightly higher than the working force of the coupler energy-absorbing device. The overload protection device designed in this way tends to exhibit premature failure during impact tests.

Therefore, this paper adopts a method combining simulation and experimental verification to investigate the influence of structural parameters of the cut-out induced bolt’s cut-out zone on the fracture threshold, reveal the mechanism behind the premature fracture phenomenon of the overload protection device, and clarify the matching relationship between the fracture threshold of the tensile break bolt-type overload protection device and the working force of the coupler energy-absorbing device.

Based on the aforementioned research background, this paper begins with a cut-out induced bolt impact test and investigates the reasons for premature fracture of the cut-out induced bolts during the test. A finite element simulation model aligned with the test was established. The two key structural parameters of the bolt cut-out zone, width and thickness, were extracted as the research objects. Numerical simulation methods were used to conduct a parametric study of these two parameters, and machine learning methods were employed to form a surrogate model. Finally, the surrogate model was combined with an optimization algorithm to optimize the parameters of the cut-out zone structure of the cut-out induced bolts, and experiments were conducted for verification.

2. Experiment and Simulation Analysis

2.1. Impact Experiment

2.1.1. Cut-Out Induced Bolt

Figure 2 shows the tensile break bolt-type overload protection on the coupler. Its specification is M24, and the material grade is 10.9. It can be observed from the figure that a cut-out zone is set on the smooth shank of the bolt. This structure can be described by two structural parameters: width and depth. By varying the width and depth of the cut-out zone, the cross-sectional shape of the bolt can be altered, thereby enabling control over the bolt’s impact resistance strength. The currently used bolt cut-out zone has a width of 10.4 mm and a depth of 2.14 mm.

As shown in Figure 3, the cut-out induced bolts of the subway train coupler are installed between the coupler yoke and the buffer front plate, used to connect the buffer and the coupler yoke. When a collision occurs, the impact force is transmitted through the coupler head, expansion tube, and buffer to the bolts. Due to the constraint of the coupler yoke, a pair of tensile forces is generated on the bolts. When the tensile force exceeds the strength of the bolts, the bolts break, the connection between the coupler and the coupler yoke is invalid, and the coupler is no longer constrained, thus completing the function of overload protection. It is particularly important to note that the fracture threshold of the overload protection device is collectively determined by all four cut-out induced bolts. Should a significant sequential fracture of the cut-out induced bolts occur during impact, the fracture threshold of the overload protection device would substantially decrease. Therefore, the four cut-out induced bolts should maintain maximum consistency in both structure and material properties to ensure simultaneous fracture during impact events. During installation of the cut-out induced bolts, a torque wrench (corresponding tightening torque: 1.92 N⋅m) was used to apply a 400 N preload force to each bolt.

2.1.2. Experimental Setup

Figure 4 shows the collision test bench at Central South University used for this experiment. The test bench includes a fixed end, a moving end, and data acquisition equipment. The main part of the moving end is the test trolley, which is adjusted by placing weights on it to regulate the trolley’s mass. The tested coupler is placed at the center of the front of the bench, and experimental overload protection devices consisting of nine energy-absorbing tubes are installed on both sides. Unlike the coupler’s overload protection device, the experimental overload protection device is designed to protect the experimental equipment in case of accidents. The main part of the fixed end is the rigid wall, where three load cells are installed at the corresponding positions of the tested coupler and the experimental overload protection devices. The load cells mainly consist of coupling devices and force sensors. The force sensor components used in the test included two models, SFC-500t and SFC-200t, manufactured by the Beijing Great Wall Institute of Measurement and Testing Technology (Beijing, China). Their maximum measurement ranges were 0–5000 kN and 0–2000 kN, respectively, with measurement errors not exceeding 0.3%. The data acquisition equipment included a high-speed camera and speed sensor. The high-speed camera model used was MEMRECAM HX-6E, manufactured by the NAC Image Technology (Tokyo, Japan), with a maximum resolution of 2560 × 1920 pixels. It offered six shooting speed options: 100 fps, 300 fps, 500 fps, 1000 fps, 3000 fps, and 5000 fps, with a speed accuracy of ±0.01%. In the test, the shooting speed was uniformly set at 3000 fps. The speed sensor (Model TD400C) was manufactured by the SIMENS (Munich, German), with a measurement error not exceeding 0.5%. The high-speed camera was set up above and to the left of the track, while the speed sensor was located at the center of the track to record the speed of the bench before it impacted the rigid wall. Both were triggered by a trigger on the side of the track. According to the energy absorption parameters of the tested coupler, the mass of the test trolley for this experiment was 29 t.

2.1.3. Results Analysis

According to the data transmitted back by the speed sensor, the speed of the test bench when it impacted the rigid wall was 7.504 m/s. The high-speed photography camera recorded the displacement–time image of the coupler collision process, as shown in Figure 5. Combining the high-speed photography images with the force–time curve, T = 0 ms when the couplers were in contact according to the high-speed photography, at which point the contact force between the couplers was zero. When T = 8.2 ms, the expansion tubes began to crush, and the contact force between the couplers fluctuated around 1200 kN. When T = 21.2 ms, the contact force between the couplers was about to rapidly drop to zero, so it was judged that the bolts break at this time. When T = 24 ms, the broken bolts can be clearly observed flying out from the high-speed photography images. When T = 40 ms, the couplers completely retracted and separated from the rigid wall, and the overload protection devices on both sides were in contact with the rigid wall.

The force sensor on the rigid wall recorded the corresponding force–time data, which were filtered to obtain the data shown in Figure 6. From the force–time data, it can be observed that the data from the force sensor are zero between T = 2.9 ms and T = 6.9 ms. This may be due to the presence of a certain gap between the internal components of the coupler buffer and the buffer housing. During the loading process, this gap needs to be consumed first, leading to pressure unloading.

The appearance of the bolts after the test is shown in Figure 7. It can be observed from the figure that the bolts experienced axial tensile necking at the cut-out zone, followed by a process of breaking at the necking surface, with a relatively flat fracture surface.

Figure 8 shows the force–displacement curve from the experiment, where the red segment represents the force–displacement characteristics of the expansion tube. It can be observed that the displacement of the expansion tube in the experiment is approximately 92.03 mm, which is much less than the designed effective displacement of 245 mm. At this stage, the cut-out induced bolts exhibit a fracture force of 1147.5 kN and a fracture duration of 20.9 ms. This is due to the insufficient strength of the cut-out induced bolts, causing the bolts to break prematurely during the collision, resulting in dangerous conditions in the experiment.

However, it is also important to note that the strength of the cut-out induced bolt should not be too high. A too-strong bolt may fail to promptly disconnect the coupler from the vehicle body, causing the vehicle body to experience excessive impact force, which can lead to undesirable deformation of the vehicle body. Therefore, it is necessary to optimize the cut-out zone of the cut-out induced bolt to ensure that the bolt does not break during the normal energy absorption of the coupler but can quickly break after the coupler completes its energy absorption work.

2.2. Simulation Analysis

2.2.1. Finite Element Model

The numerical simulations were performed employing a finite element model constructed in LS-DYNA. As shown in Figure 9, the structure of the train coupler mainly included the coupler head, expansion tube, buffer, cut-out induced bolts, coupler yoke, and coupler seat. To simplify the calculation of the model, beam elements were used to simulate the dynamic characteristics of the expansion tube and buffer, and *MAT_GENERAL_NONLINEAR_6DOF_DISCRETE_BEAM (MAT_119) was selected as the material model for the beam elements. The coupler head and coupler seat underwent less deformation during collisions and were not the focus of the study, so shell elements were used for simulation, and *MAT_RIGID (MAT_020) was selected as the material model for the shell elements.

Bolts, coupler yoke, and buffer front plate were simulated using solid elements, as shown in Figure 10. The bolts used *MAT_PIECEWISE_LINEAR_PLASTICITY (MAT_024) as their material model. The buffer front plate and coupler yoke underwent less deformation during collisions and could be considered rigid bodies in the calculation process, so *MAT_RIGID (MAT_020) was selected as the material model. The connection between the buffer front plate and the buffer beam element was achieved through *CONSTRAINED_EXTRA_NODE.

As shown in Figure 11, the finite element simulation model for the test conditions established contact relationships between the coupler heads, crushed tube, buffers, and coupler yoke using *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE, and the friction coefficient was set to 0.2, referencing relevant literature [30]. Additionally, to ensure that the impact of kinetic energy remains unchanged during the collision, corresponding mass points are placed on the test trolley, and the total model mass is adjusted to match the test mass.

The finite element model consists of 157,728 nodes and 140,369 elements. With the exception of the overload protection device (composed of pentahedral and hexahedral elements) and the mechanical models of the expansion tube and buffer (beam elements), the model primarily employs a combination of triangular and quadrilateral shell elements. The finite element model features a global average mesh size of 10 mm, with local refinement to 1–3 mm at the bolt cut-out zone locations (corresponding to the structural dimensions of the cut-out zone).

2.2.2. Material Parameter

• Buffer and expansion tube

Since the complete coupler force–displacement characteristics were not obtained in the full-scale vehicle test, ideal coupler force–displacement characteristics were used as a substitute in the simulation modeling of the expansion tube. Figure 12 exhibits the force–displacement curves of the expansion tube and buffer. According to the design energy absorption parameters of the coupler, the effective stroke of the expansion tube is 245 mm, with an average crush force of 1200 kN. The quasi-static compression force–displacement characteristics of the EFG3 rubber friction buffer were obtained through quasi-static compression experiments [31].

2.

Bolts and Other Components of Coupler

The bolt material is 40Cr and the stress–strain relationship of the 40Cr material obtained by Zhang [29] on the universal fatigue testing machine is shown in Figure 13. The blue curve represents the engineering stress–strain relationship of the 40Cr material, which is mainly based on the assumption that the cross-sectional area of the bolt does not change during the tensile process, and is obtained through Equations (1)–(3).

$$\left\{\begin{array}{c}\sigma ={\displaystyle \frac{F}{A}}\\ \epsilon ={\displaystyle \frac{l-l_0}{l_0}}\end{array}\right.$$

(1)

$${\sigma }_t=\sigma (1+\epsilon )$$

(2)

$${\epsilon }_t=\mathrm{l}\mathrm{n}(1+\epsilon )$$

(3)

In the equations, $\sigma$ represents the engineering stress; $\epsilon$ represents the engineering strain; $F$ represents the tensile load; $A$ represents the original cross-sectional area of the sample standard distance section; $l_0$ represents the original length of the sample standard distance section; $l$ represents the length of the sample standard distance section after extension; ${\sigma }_t$ represents the true stress; ${\epsilon }_t$ represents the true strain.

However, in the actual tensile process, the cross-sectional area of the specimen will decrease. To further obtain the true stress–strain relationship of the material, it is necessary to make corresponding corrections to the engineering stress–strain curve. Zhang assumed that after the specimen undergoes necking, the cross-section at the necking position appears as a circular arc shape as shown in the figure, and then it is possible to obtain the following true stress–strain correction formulas (Equations (4)–(7)) based on the principle of volume conservation during the tensile process.

$${\sigma }_t={\displaystyle \frac{F}{A}}={\displaystyle \frac{F}{\pi D_i^2}}$$

(4)

$${\epsilon }_t=\ln \left({\displaystyle \frac{A}{A_i}}\right)=\mathrm{l}\mathrm{n}({\displaystyle \frac{D}{D_i}})$$

(5)

$$L_0={\displaystyle \frac{∆L_i(2D_0^2+D_i^2)}{D_0^2-D_i^2}}$$

(6)

$$D_i=\sqrt{{\displaystyle \frac{3D_0^2{L}_0}{L_0+∆L_i}}-2D_0^2}$$

(7)

In the equations, $A$ represents the original cross-sectional area, $A_i$ represents the cross-sectional area at time $i$, $D$ represents the original diameter of the necking section, and $D_i$ represents the minimum diameter of the necking section at each time $i$ after necking occurs. In the equations, $L_0$ represents the length before necking, $L_i$ represents the length after necking, $D_0$ represents the diameter before necking, and $D_i$ represents the outer arc radius after necking. $L_0$ can be obtained by substituting the minimum diameter of the necking section at fracture and the final deformation into Equation (6). $D_0$ can be obtained before necking occurs based on the principle of volume conservation. By substituting the minimum cross-sectional diameter $D_i$ of the necking section into Equations (4) and (5) at any time, the true stress–strain values of the necking section can be obtained.

The stress–strain curve of 40Cr obtained from Figure 13 is input into the LCSS parameters of the bolt material model (MAT_024). Other components of the coupler, such as the coupler seat and coupler yoke, are also made of steel and have similar basic physical parameters such as density and Young’s modulus. Therefore, the same parameters as the bolt are used, as shown in

Table 1. Basic mechanical property parameters of bolts and other components of coupler [29].

Density (g/cm3)	Young’s Modulus (GPa)	Poisson Ratio	Yield Stress (MPa)
7.82	206.08	0.3	835.27

[29].

2.2.3. Validation

After the simulation calculation, the calculated force–time curve was compared with the force–time curve collected from the test. As can be seen from the curves in Figure 14, the time history trend of the coupler contact force is roughly similar. Additionally, as described in Section 2.1.3, due to installation errors and other reasons, there is a certain amount of free travel between the buffer and the internal components of the buffer. Therefore, during the model calibration process, a certain amount of free travel was also given to the buffer beam’s force–displacement curve to synchronize the contact force time history changes between the test and simulation.

In the simulation, the coupler heads did not fully couple at the beginning of the calculation, leaving a certain gap. Therefore, the moment when the contact force starts to rise was set as T = 0 ms, as shown in Figure 14. When T = 1.19 ms, the initial peak force F = 2032 kN was reached; when T = 2.35 ms, the contact force dropped to zero, entering the free travel stage; when T = 6.9 ms, the free travel ended, and the contact force began to rise again; when T = 9.76 ms, the expansion tube entered the stable crushing stage, with an average crushing force F_mean = 1200 kN; when T = 20.4 ms, the bolt broke, and the contact force started to decrease.

As shown in Figure 15, the deformation of the bolts in the simulation and test is as follows: when T = 0 ms, the coupler heads are fully coupled, about to generate contact force, and the bolts are in the state of being preloaded with no deformation; when T = 10 ms, under the simulation calculation, the expansion tube enters the stable crushing stage, and the bolts at the cut-out zone begin to show slight deformation; when T = 20 ms, under continuous force, the bolts at the cut-out zone continue to deform, the cross-section area decreases, and a more obvious necking phenomenon can be observed, with the bolts at the brink of breaking; at T = 22 ms, the bolts have broken, and the two parts fly apart. In summary, the bolt breaking time in the test is T = 21.2 ms, and the bolt breaking time obtained from the simulation calculation is T = 20.4 ms, with a difference of approximately 3.77%, proving the model’s validity.

3. Effect of Cut-Out Zone Structural Parameters

3.1. Design of Experiments (DOE)

Methods to enhance the mechanical performance of bolts primarily include three approaches: improvement of material mechanical properties, increase in preload force [29], and structural optimization. In addition to traditional metal heat treatment methods for enhancing bolt mechanical performance [32], recent studies have successfully demonstrated the application of composite materials in bolts [33]. Therefore, the mechanical performance improvement approaches using composite materials [34,35] also represent effective alternatives. While material property improvements exhibit certain uncertainties and preload force increases offer limited performance enhancements, this study therefore focuses on optimizing the notch groove structure. Before exploring the effects of the depth and width of the cut-out zone on the fracture threshold of bolts, it is first necessary to test a system of four bolts in a coupler yoke. Therefore, the corresponding experimental design needs to be carried out. The experimental design scheme employs the Latin Hypercube method to generate samples. The reason for using the Latin Hypercube method to discretize sample points is that it not only ensures that the sampling values are evenly distributed throughout the sample space but also prevents the sampling values from being overly clustered. A total of 78 sets of structural parameters were generated, and the resulting test groups are shown in Appendix A 

Table A1. The data of DOE.

	Width (mm)	Depth (mm)		Width (mm)	Depth (mm)
1	5.00	1.50	40	6.88	2.11
2	2.41	0.56	41	9.76	2.94
3	6.38	2.20	42	12.49	1.05
4	3.78	0.19	43	17.67	0.48
5	5.34	2.11	44	9.13	0.94
6	6.53	0.13	45	3.65	0.60
7	10.78	0.88	46	3.17	1.43
8	1.40	1.91	47	8.96	0.79
9	16.04	2.41	48	1.10	0.29
10	17.01	0.01	49	13.01	0.36
11	12.28	2.84	50	10.02	0.74
12	11.94	2.26	51	8.51	0.56
13	7.87	1.32	52	7.71	2.09
14	15.79	1.06	53	15.76	1.89
15	8.70	1.13	54	10.76	2.52
16	3.36	0.69	55	5.45	2.00
17	14.69	1.40	56	16.64	1.87
18	9.14	1.43	57	4.93	0.12
19	7.38	2.65	58	5.23	1.30
20	10.44	2.14	59	2.95	0.38
21	2.85	2.04	60	13.27	2.75
22	4.34	0.93	61	4.35	2.35
23	16.78	2.52	62	0.24	0.88
24	17.74	1.51	63	1.90	2.30
25	12.95	0.75	64	15.56	2.46
26	1.86	2.97	65	6.11	2.90
27	5.59	2.77	66	16.69	2.22
28	8.48	1.73	67	7.30	1.73
29	15.03	1.91	68	12.06	1.70
30	0.13	1.77	69	1.42	1.15
31	0.75	0.37	70	2.46	1.58
32	4.71	0.62	71	6.43	2.58
33	13.88	1.17	72	11.45	1.54
34	14.27	0.44	73	0.55	0.19
35	13.01	2.69	74	14.26	1.11
36	11.22	0.26	75	14.89	0.01
37	11.07	1.24	76	17.23	2.81
38	13.53	2.69	77	0.99	2.53
39	14.53	1.39	78	17.74	1.42

. The distribution of sample points is illustrated in Figure 16. Based on the structural parameters provided in Appendix A Table A1, the finite elements in the cut-out zone of the four cut-out induced bolts (grouped as one set) were modified in the finite element model. The preload force of each cut-out induced bolt was maintained consistently at 400 N.

3.2. Analysis of Bolt Failure Modes

By extracting the force–time curves from the simulation results and combining them with the element failure times, the force–displacement curves were filtered to obtain the fracture time and fracture force for each set of bolts. The results are shown in Appendix A 

Table A2. The result of numerical simulation.

	Fracture Time (ms)	Fracture Force (kN)		Fracture Time (ms)	Fracture Force (kN)
1	53.15	2044.39	40	52.95	1762.44
2	53.40	2068.11	41	11.95	1234.98
3	52.95	1731.50	42	53.25	2035.52
4	53.45	2072.15	43	53.30	2057.84
5	53.05	1879.79	44	53.25	2046.64
6	53.40	2070.02	45	53.35	2067.34
7	53.25	2043.87	46	53.20	2057.26
8	53.25	2071.29	47	53.25	2054.46
9	12.35	1370.98	48	53.45	2069.72
10	53.45	2081.01	49	53.35	2064.07
11	12.10	1285.53	50	53.25	2052.19
12	12.30	1327.45	51	53.30	2061.17
13	53.20	2032.12	52	52.95	1738.84
14	53.25	2013.06	53	12.30	1267.01
15	53.20	2042.36	54	12.20	1336.78
16	53.35	2068.02	55	53.10	1930.31
17	53.15	1896.53	56	12.35	1271.25
18	53.20	2003.83	57	53.45	2070.19
19	12.10	1307.97	58	53.20	2047.02
20	17.90	1182.55	59	53.40	2069.73
21	53.15	2044.65	60	12.25	1335.87
22	53.25	2060.60	61	52.95	1800.22
23	12.30	1345.84	62	53.45	2071.38
24	53.15	1803.23	63	53.15	2052.59
25	53.25	2052.84	64	12.35	1371.14
26	53.05	1902.01	65	11.75	1133.62
27	12.05	1296.80	66	12.30	1323.97
28	53.10	1918.21	67	53.15	1952.27
29	12.40	1276.54	68	53.10	1815.22
30	53.40	2071.84	69	53.35	2069.51
31	53.45	2070.99	70	53.25	2062.79
32	53.35	2066.22	71	12.20	1355.25
33	53.20	2000.67	72	53.15	1901.54
34	53.30	2056.69	73	53.45	2074.48
35	12.30	1357.96	74	53.20	2012.81
36	53.35	2070.69	75	53.50	2087.22
37	53.25	2023.61	76	12.10	1245.23
38	12.00	1246.76	77	53.05	1742.35
39	53.15	1905.54	78	53.10	1825.82

. The simulation results can be categorized into three main types: bolts breaking before the expansion tube starts working; bolts breaking during the operation of the expansion tube; and bolts breaking after the expansion tube has completed its work.

Taking the case of bolts breaking during the operation of the expansion tube (the 20th set of bolt conditions in the DOE) as an example, as shown in Figure 17, the simulation calculation provides a time series graph of the total bolts area, the coupler contact force, and the total cross-sectional force of the bolts.

From the graph, it can be observed that during the collision process, the change in the cross-sectional area of the bolts undergoes a total of five stages. The first stage is from the beginning of the collision to the moment when the cross-sectional force of the bolts reaches the initial peak force. During this stage, the maximum stress generated in the bolts does not reach the yield stress of the material, and the cross-sectional area of the bolts remains essentially unchanged.

Comparing the magenta line and the light blue line in the graph, it can be seen that due to the presence of mass points and shock absorbers, there is a noticeable lag in the transmission of the initial peak force from the coupler to the bolts cross-section. Therefore, the change in the cross-sectional area of the bolts does not follow the change in the coupler contact force.

The second stage is the initial peak force phase. During this stage, the cross-sectional force of the bolts begins to rise sharply, and the material undergoes permanent plastic deformation as it cannot withstand the corresponding stress, causing the area of each bolt to suddenly decrease. However, the bolts do not break at this point.

The third stage is the transition from the initial peak force to the average crush force. After passing the initial peak force, the force on the bolts’ cross-section decreases, the deformation of the bolts stops, and the cross-sectional area remains unchanged.

The fourth stage is the average crush force phase. In this stage, due to the improper design of the cut-out zone, the structure of the bolts after being impacted by the initial peak force can no longer withstand the load of the average crush force. During this process, the cross-sectional area of the bolts continues to decrease until it eventually cannot withstand the corresponding load and breaks at a certain moment.

The fifth stage is the post-fracture phase. In this stage, since the bolts have already broken, they can no longer withstand the corresponding load, so the cross-sectional area remains at the size it was at the moment of fracture. Additionally, the unloading of the cross-sectional force of the bolts, after passing through the shock absorber, also experiences a corresponding lag in being transmitted to the coupler contact surface.

Taking the case of bolts breaking before the expansion tube starts working (the 29th set of bolt conditions in the DOE) as an example, as shown in Figure 18, the first and second stages are the same as before. However, in the third stage, after the cross-sectional force of the bolts reaches the initial peak force and starts to decline before returning to the platform force, there is still a reduction in the cross-sectional area of a certain bolt within the bolt group. Before the coupler contact force and the cross-sectional force of the bolts return to the platform force, the minimum cross-sectional area of the bolts is insufficient to withstand the load, causing a certain bolt to break, leading to the failure of the entire bolts group and prematurely entering the fifth stage, the post-fracture phase.

Taking the case of bolts breaking after the expansion tube has completed its work (the 57th set of bolt conditions in the DOE) as an example, as shown in Figure 19, the first and second stages remain unchanged.

During the transition from the initial peak force to the stable crushing phase of the expansion tube (the third stage) and the stable crushing phase of the expansion tube (the fourth stage), the cross-sectional area of the bolts will experience minor changes due to the peak force generated when the expansion tube starts. After that, it remains stable until the expansion tube enters the stiffening phase. Unlike the first scenario (where bolts break during the stable crushing phase of the expansion tube), in this case, the fifth stage is the coupler stiffening phase. During this stage, both the coupler contact force and the cross-sectional force of the bolts continue to rise, and the cross-sectional area of the bolts decreases until they break. The sixth stage is equivalent to the fifth stage in the first scenario, which is the post-fracture phase.

Evidently, due to the existence of inertial effects, the force on the bolts during the collision does not change synchronously with the coupler contact force. Inertial effects act as a filter and a delay throughout the process. Therefore, during the first stage, when the coupler contact force reaches the first initial peak force, the cross-sectional force of the bolts does not significantly increase but is delayed by 4–5 milliseconds, and the increase is much less than the amplitude of the initial peak force. Secondly, before entering the stable crushing phase, the bolts’ cross-section will be subjected to peak force impacts from both the shock absorber and the expansion tube (corresponding to the peak forces in stages two and three). Under the impact of the peak force, the bolts will deform, weakening their resistance to tensile forces.

3.3. The Establishment of the Surrogate Model

Four methods, Least Squares Regression (LSR), Moving Least Squares Method (MLSM), HyperKriging (HK), and Radial Basis Function (RBF), were used to fit the fracture force with respect to the two independent variables (groove depth and groove width). The LSR method minimizes the sum of the squares of the differences between the fitted values and the original values, with its objective shown in Equation (8). The MLSM is based on the same principle as the Least Squares method, but the weighting coefficients for the sampling points are functions of the distance between the sample points and the evaluation points. The HK method generates an interpolation model based on the existing sampling points. The RBF method uses the linear superposition of several basis functions (including linear functions, cubic functions, thin-plate spline functions, Gaussian functions, etc.) for fitting. Both the RBF method and the HK method typically pass through the original data more accurately.

$$\min E={\sum }_{i=1}^n{\left(f_i^{predicted}-f_i\right)}^2$$

(8)

In the equation, $n$ represents the number of calculations, $f_i$ represents the original value, and $f_i^{predicted}$ represents the fitted value.

To characterize the prediction accuracy of the fitted model, the coefficient of determination (R²) and the relative average absolute error (RAAE) are commonly used, with their calculation formulas shown in Equations (9) and (10):

$$R^2=1-{\displaystyle \frac{\sum _{i=1}^n{\left[y_i-{\widehat{y}}_i\right]}^2}{\sum _{i=1}^n{\left[y_i-\overline{y}\right]}^2}}$$

(9)

$$RAAE={\displaystyle \frac{\frac{1}{n}\sum _{i=1}^n\left[abs\left(y_i-{\widehat{y}}_i\right)\right]}{\sqrt{\left(\frac{1}{n}\sum _{i=1}^n{\left[y_i-\overline{y}\right]}^2\right)}}}$$

(10)

where $y_i$ represents the actual data value, ${\widehat{y}}_i$ represents the model prediction value, $\overline{y}$ represents the average of the actual data and $n$ represents the number of data points. The closer $R^2$ is to 1 and the lower RAAE is, the better the fitted model fits the original data.

As shown in

Table 2. Certainty coefficients of different fitting methods.

Methods	R2	RAAE
LSR	0.8577	0.2411
MLSM	0.9849	0.0727
HK	0.9036	0.1235
RBF	0.9753	0.0840
LSR	0.8577	0.2411

, the MLSM has the highest coefficient of determination and the lowest relative average absolute error in fitting the fracture force, both of which are superior to the other three methods. Therefore, the MLSM is selected for fitting the fracture force. After the selection of the fitting method, the response diagrams between the depth and width of the cut-out zone and the fracture time and fracture force were drawn separately. In Figure 20, the response surface between the depth and width of the cut-out zone and the fracture force after fitting is shown. As evident from Figure 20, the fracture force of the cut-out induced bolts exhibits significant variation within specific regions, with values predominantly clustering at two distinct force levels: 1200 kN and 1800 kN. The depth of the cut-out zone emerges as the primary determinant influencing the dynamic mechanical performance of the cut-out induced bolts, as evidenced by the fracture force response surface showing pronounced variations along the depth dimension. In contrast, the width of the cut-out zone demonstrates comparatively minor influence on the dynamic mechanical properties, with the fracture force response surface displaying only marginal changes along the width dimension.

Based on the fracture time of the bolts group obtained from the simulation calculation results, it can be determined whether the bolt group has experienced premature breakage, and the data can be marked accordingly. The calculation results with a fracture time between 11 and 13 ms are marked as the first category, and the calculation results with a fracture time between 52 and 54 ms are marked as the second category.

Since there are too few samples of bolts breaking during the operation of the expansion tube (only the 20th group of conditions in the DOE), the data from the 20th set of conditions were marked as the first category. Combining the random forest algorithm to train the dataset, the structural parameter decision boundary for whether the bolts group experiences premature breakage can be obtained, as shown in Figure 21.

4. Fracture Threshold Optimization

4.1. Problem Definition

Due to the inability of the bolts in the test to meet the requirements of the overload protection device, it is necessary to optimize the structural parameters of the bolts. During the optimization process, the following conditions must be met: the fracture force of the bolts needs to have a corresponding matching relationship with the coupler parameters. An excessively large fracture force can lead to vehicle structure damage and excessive vehicle deceleration, while an excessively small fracture force can cause the bolts to fail either during or before the coupler’s operation. However, it is difficult to determine the breaking moment of the bolts solely from its fracture force, so the fracture time of the bolts needs to be considered in the definition of the objective. Based on the above conditions, the optimization objective can be defined as follows:

$$\left\{\begin{array}{c}\min F_{Fracture}\\ s.t.\begin{array}{c}{F}_{Fracture}\ge 1800\mathrm{k}\mathrm{N}\\ 52\mathrm{m}\mathrm{s}\le T_{Fracture}\le 54\mathrm{m}\mathrm{s}\end{array}\end{array}\right.$$

4.2. Algorithm and Process

The genetic algorithm was used as the optimization algorithm. Compared to gradient optimization algorithms, the optimal solution of the genetic algorithm may oscillate near local optima, while the genetic algorithm is more adept at global search and is more likely to find the global optimal solution. As shown in Figure 22, the genetic algorithm refers to all design points in each iteration as a population. When generating the initial population for the first time, random design points are first created as the initial generation population. The response and objective form the fitness of each design point, and most design points are eliminated based on their fitness ranking. In the remaining high-fitness design points, crossover operations that exchange design variables and mutation operations that cause random changes in some design variables are performed to generate the next-generation population for iteration. By continuously repeating the above operations, the fitness of the population increases, gradually approaching the global optimal solution.

Using the genetic algorithm with parameter settings as shown in

Table 3. GA parameters.

Maximum Iterations	Minimum Iterations	Population Size	Mutation Rate	Crossover Rate
50	25	60	0.01	0.5

, the optimal fracture force of the bolts for each generation of the population obtained after optimization is shown in Figure 23. Through optimization using the genetic algorithm, the final determination was made that the depth of the bolt cut-out zone should be 1.37 mm, and the width should be 17.74 mm.

4.3. Result Validation

4.3.1. Simulation Validation

In simulation validation, we reconstructed the cut-out induced bolts model based on the optimization results, ensuring that the rest of the model and boundary conditions remained unchanged, and then rebuilt the simulation model. Through computational simulation, we obtained the simulation results of the optimized bolts. Figure 24 shows the timing diagram of the coupler collision obtained from the simulation model calculation. The force–time curve obtained from the simulation calculation is also shown in Figure 25. The collision was considered to start when the coupler head was fully in contact, at which point T = 0 ms. When T = 10 ms, the buffer stroke was fully consumed, and the expansion tube was about to enter the stable crushing stage; when T = 49 ms, according to the force–time curve, it can be observed that the expansion tube was fully consumed at this point, and the coupler was about to enter the rigid phase, with the contact force beginning to rise; when T = 53.15 ms, based on the element failure situation reported in the simulation calculation, it can be judged that the bolts were pulled apart at this time; at T = 54 ms, the bolts can be observed flying out from the timing diagram.

The stress variation of each bolt is shown in Figure 26. When T = 0 ms, the coupler head was fully coupled, and due to the presence of preload, the maximum stress of the four bolts was concentrated in the cut-out zone, within the range of 516.84–519.25MPa. When T = 10 ms, the expansion tube was about to enter the stable crushing stage, and due to the influence of inertial effects, the bolt stress experienced a short rise and slight deformation, with the maximum stress of the four bolts ranging from 1002.92 to 1096.05MPa. When T = 30 ms, the expansion tube entered the stable crushing stage, with a relatively stable force value. The maximum stress was slightly lower than at T = 10 ms and remained stable. During this stage, the bolts did not undergo further deformation, with the maximum stress of the four bolts ranging from 807.65 to 877.45 MPa. When T = 53 ms, the expansion tube was fully consumed, and the entire coupler entered the rigid phase, with the force value rising. The bolt experienced a sudden increase in force, further deformation, and was about to break, with the maximum stress of the bolts ranging from 1050.79 to 1144.54 MPa. When T = 55 ms, the bolts broke, and the bolts stress was unloaded.

We compared the fracture force obtained from the simulation model calculation with the predicted values from the fitting model. The surrogate model predicted the fracture force for the bolt group to be 1894.13 kN, while the simulation model calculated the fracture force for the bolt group to be 1855.59 kN, a difference of approximately 2.08%. The error was less than 5%, which proves the effectiveness of the fitting model from another perspective.

4.3.2. Experiment Validation

After completing the optimization of the cut-out zone, a verification experiment was conducted. To verify whether the optimized bolts could withstand the extreme collision conditions in real vehicles, the test plan was also adjusted, with the test setup as shown in Figure 27. The test was adjusted to a two-coupler collision energy absorption test. Test coupler one was installed on a rigid wall, and test coupler two was installed on a test bench. The buffer in test coupler one did not have an overload protection bolt installed, while the buffer in test coupler two had cut-out induced bolts installed between the buffer and the coupler yoke. The two couplers were installed with a height difference of 20 mm. In this test, the mass of the test bench was 32 t, the initial speed of the test bench recorded by the speed sensor was 6.90 m/s, and the total impact kinetic energy was 761.76 kJ.

The timing changes of the coupler collision were recorded by a high-speed camera as shown in Figure 28. The coupler was set to start contacting at T = 0 ms. When T = 26 ms, the buffers of both couplers were fully consumed, and the expansion tubes of the fixed-end and bench-end couplers began to act. When T = 72 ms, the expansion tube of the bench-end coupler stopped acting, and from this point on, only the expansion tube of the fixed-end coupler was consumed. When T = 95 ms, the expansion tube of the fixed-end coupler was fully consumed, and the expansion tube of the bench-end coupler started acting again. Unlike the previous test, in this test, the re-acting bench-end expansion tube successfully consumed the entire stroke. At T = 138 ms, all energy absorption devices of the two couplers were fully consumed, and the couplers entered the rigid phase. At T = 139 ms, the bench-end breakaway cut-out induced bolts broke, and the remaining kinetic energy was absorbed by the protective devices on both sides, causing the bench to rebound and the collision to end.

The force–time situation of the test is shown in Figure 29, with Roman numerals corresponding to key moments of the collision: (1) T = 0 ms, where the coupler heads are fully coupled, and the collision begins; (2) T = 26 ms, where the buffers of test coupler one and test coupler two are fully consumed, and the expansion tubes are about to start acting; (3) T = 72 ms, where the expansion tube of test coupler two stops crushing, and at this point, only the expansion tube of test coupler one is absorbing energy; (4) T = 95 ms, where the expansion tube of test coupler one is fully consumed, and the expansion tube of test coupler two is about to start acting again; (5) T = 138 ms, where all energy absorption devices of the couplers are fully consumed, and the couplers are about to enter the rigid phase, at which point the bolts need to be sheared to protect the vehicle body.

In this coupler coupling collision test, it is clear in the high-speed photography images that the breakaway bolts of the coupler broke shortly after the two expansion tubes had fully consumed their strokes.

Table 4. Comparison of the fracture force of cut-out induced bolts.

	Surrogate Model	Simulation	Experiment
Fracture Force (kN)	1894.13	1855.59	1828.44
Deviation from Experiment	3.59%	1.48%	/

shows that the fracture force was 1828.44 kN, differing from the fracture force obtained from the simulation calculation by 1.48%.

By comparing the two impact tests, it can be observed that the fracture force of the cut-out induced bolts increased from 1147.5 kN to 1828.44 kN, representing a 59.34% improvement in fracture threshold. Additionally, the fracture time of the cut-out induced bolts extended from 20.9 ms to 69 ms. After optimization, the cut-out induced bolts met the requirement for timely fracture in the coupler’s energy-absorbing components under overload conditions, thereby enabling the coupler’s energy absorption to satisfy the requirements specified in EN 15227 for C-II class vehicles at 25 km/h impact scenarios [36].

5. Conclusions

This paper investigates the phenomenon of premature breakage of tensile break bolt-type overload protection in the rigid wall impact test of a train coupler. Based on the cut-out zone of the bolts, the effects of the depth and width of the cut-out zone on the fracture time, fracture force, and fracture area were studied and analyzed. The following conclusions were drawn:

• The strength of the cut-out induced bolts need to match the collision dynamics behavior of the coupler. The bolts with excessive strength will prevent the timely disconnection of the coupler from the vehicle, leading to excessive impact acceleration and even climbing phenomena. The bolts with insufficient strength will cause the bolts to fail during the coupler’s operation, resulting in too low energy absorption by the coupler and preventing it from effectively absorbing kinetic energy.
• The simulation results indicate that during the coupler collision process, the peak force generated by the impact causes the cut-out induced bolts to neck, leading to a reduction in bolt strength. In the design of cut-out induced bolts, the influence of the peak impact force on the bolts should be considered, and a certain margin should be left in terms of strength.
• The experimental design for the cut-out zone structural parameters of the cut-out induced bolts was conducted using the Latin Hypercube method. The Moving Least Squares method was used to fit the fracture force from the simulation results. Additionally, the simulation results were categorized based on the working condition of the expansion tube at the time of bolt fracture, and the decision boundary for the cut-out zone structural parameters was determined using the Random Forest algorithm. Finally, a surrogate model was constructed based on the fitting and classification models, and combined with a genetic algorithm. The following optimal parameters for the cut-out zone of the overload protection bolts were obtained: a width of 17.74 mm and a depth of 1.37 mm. Through structural optimization of the cut-out zone, the cut-out induced bolts’ fracture force increased from 1147.5 kN to 1828.44 kN (a 59.34% improvement), with a fracture time extending from 20.9 ms to 69 ms. The structurally optimized cut-out induced bolts enable the coupler to meet the energy absorption requirements for C-II class vehicles specified in EN 15227.