1 Introduction

The total distance covered by the Chinese high-speed railway network was approximately 29000 km at the end of 2018, and it is expected to reach 38000 km by 2025. Safety and durability have become paramount. Among the safety-critical structures, the high-speed welded bogie frame is known to play a crucial role in transferring the static load acting on the vehicle to the hollow axles, in addition to controlling the wheelsets on straight and curved tracks and supporting the wheels, axles, brakes, and suspensions [1,2,3]. In general, the welded bogie frame is designed to operate for over 25 years with regular inspection according to the maintenance procedures.

The welding process may not only lead to a strength/geometry mismatch, but may also induce various defects such as inclusions, pores, cracks, and undercuts, thereby resulting in serious stress concentration under fatigue load [4,5,6]. Nevertheless, it is widely reported that some failures usually originate from extrinsic defects probably due to abnormal impact and vibration, and not from welded joints [7,8,9]. To tackle such problems, ample research has been conducted on materials and structures by performing elaborate simulations and experiments using standard procedures [2, 10, 11]. The locations with high stresses on a railway bogie can be accurately determined within the framework of the nominal stress approach by incorporating multibody system dynamics (MBSD) and/or finite element method (FEM) [8, 12, 13].

To keep higher safety and reliability of bogie frames for a light rail vehicle, three fatigue assessment procedures in terms of equivalent stress ranges have been carefully examined using highly loaded welded joints, such as the endurance limit approach with the maximum loading amplitude, cumulative damage approach with the loading spectrum of fatigue test program, and cumulative damage approach with the vertical force range spectrum from MBS simulation. It has been confirmed that the MBS-based loading spectrum provides more realistic loads, which act on the bogie frame, for fatigue assessment, depending on the actual railway conditions [3, 14,15,16]. However, it is well-known that the fatigue assessment based on the endurance limit approach provides relatively conservative results for the allowable stress and accumulation damage; nevertheless, such an approach serves as a quick and standard procedure for the safe life analysis of railway safety-critical components [17,18,19]. Moreover, most studies have employed the simplified bogie frame model composed of hybrid solid and plate elements for the residual life prediction in the presence of defects [4, 20, 21]. Novel assessment methods using the master S–N curve, hot spot stress, and notch stress have been proposed to consider the uneven stress distribution due to geometry discontinuity or strain gradient [14, 22,23,24]. Recently, a fracture mechanics-based damage tolerance design approach has been successfully applied to aircraft structures, high-speed railway axles, disc brakes, and nuclear pressure pipes for remaining life and resulting inspection interval predictions [25,26,27,28,29].

The brake unit of a bogie system is one of the vital structures significantly affecting the operation safety and ride comfort of high-speed railway vehicles. Furthermore, frequent deceleration and acceleration occur during operation because of routine stopping and starting. This might induce a high stress zone around the brake unit bracket, particularly while braking in an emergency.

This study conducts life prediction and durability assessment for the brake unit bracket of a bogie frame using a damage cumulative approach and an advanced damage tolerance design method. The mechanical properties, probabilistic fatigue S–N curves, and fatigue crack growth (FCG) rate of a weathering steel were measured as important input parameters for the solid finite element model of the bogie frame. The standard NASGRO equation was selected as the fitted FCG rate model with stress ratios of − 1 and 0. To obtain the actual fatigue performance and lifetime, an eight-stage loading spectrum acting on the Wuhan–Guangzhou line was employed.

2 Experimental Procedures

2.1 Materials

The widely used weathering steel material consisting of equiaxial ferrite and pearlite was used for the bogie frame model under normalised and rolled states [30]. The mechanical properties obtained from the monotonic tensile tests with three effective specimens are listed as follows: 409 MPa for the 0.2% offset proof yield strength Rp0.2, 567 MPa for the ultimate tensile strength Rm, 26% for the fracture elongation, 0.3 for the Poisson’s ratio υ, and 206 GPa for the Young’s modulus E. The main chemical compositions of the parent metal (PM) in terms of the maximum percentage weight include 0.19% C, 0.035% S and P, 0.55% Si, 0.70% Ni, 0.35% Mo, and 0.17% Zr. The compositions of other alloying elements, such as Mn, Cr, and Cu, vary in the ranges of 0.45%–1.60%, 0.35%–0.85%, and 0.20%–0.60%, respectively. The chemical compositions and basic properties of such a material with good toughness are comparable to those of hot-rolled SMA490BW.

Note that the semi-automated metal active gas (MAG) welding system was used to manufacture the bogie frames of China Railway high-speed (CRH) railway vehicles, which have a speed range of 200–500 km/h. However, the present study focused on the failure effect of the PM plate (thickness of 10 mm) with a high strain region, which was identified from a geometry-optimised bogie frame structure. The intrinsic strength/geometry mismatch due to the welding process has been effectively avoided by performing a detailed system dynamic simulation and a full-scale benchmark test. Therefore, the assessment was based on the fact that the crack was initiated far away from the welded zone.

2.2 Experiments

A standard monotonic tensile test was performed at a constant strain rate using an MTS 809 testing machine. This test served as a reference for the fatigue experiments. The strain rates were 0.005 and 2 mm/min before and after the yield point, respectively.

Identical hourglass-shaped plate specimens as shown in Figure 1 (gauge size of 70 × 10 mm2) were prepared for both low cycle fatigue (LCF, R = − 1 and strain rate of 0.002/s) and high cycle fatigue (HCF, R = − 1 and alternating sinusoidal mode) tests. An MTS 810 servo-hydraulic testing machine and an INSTRON 8872 high-frequency fatigue machine were selected for the LCF and HCF tests, respectively. During the HCF testing, the tests were manually terminated when the lifetime exceeded 107 cycles, thus indicating an unbroken specimen.

Figure 1
figure 1

Geometry and dimensions in millimeters of specimens for: a LCF tests; b HCF tests

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Based on the probabilistic endurance curves, the HCF results were used to conduct a safe life assessment test on the braking bracket using a conventional nominal stress method. However, to ensure the safety and reliability of the bracket in service, the cyclic stress–strain responses of the high strain region with a potential defect were measured to predict the FCG resistance. Under a remote reversed loading, there exists a reversed plastic zone around the crack where the fatigue damage occurs. This shows that the LCF behaviour controls the fatigue cracking.

It has been widely argued that different geometries of the middle-crack tension (MT) and compact tension (CT) specimens can lead to considerable discrepancy in the FCG rate, even though the same ASTM E 647 standard is used under the same material and loading conditions [31, 32]. Therefore, for a reliable remaining life prediction based on the damage tolerance approach, the experimental long crack FCG rate and threshold stress intensity factor (SIF) range (∆Kth) were determined using standard MT specimens for R = − 1 and 0.

3 Modelling

3.1 Constraint and Loading Conditions

Figure 2 shows the structure of the modelled bogie frame, along with the applied braking forces and the corresponding constraint conditions. The bogie frame selected in this study is a box-type welded frame, consisting of two side beams and two cross beams, which form an H-shaped frame.

Figure 2
figure 2

Bogie frame structure with braking forces and boundary conditions

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To ensure the computational accuracy of the finite element analysis, most parts of the bogie frame were meshed using the structural eight-node solid brick element (linear C3D8 type in ABAQUS). Nevertheless, a few tetrahedron (C3D4 type) and pentahedron (C3D6 type) elements were used in the smooth transition parts for welding lines and the arc transition parts within the bogie frame, respectively. The element size was approximately 5 mm, which corresponded to a total of 2557391 elements and 1938972 nodes for the bogie frame model. The weathering steel, which is used for the current bogie frame, was modelled as a linearly elastic material.

As shown in Figure 2, the locations (1) to (4) represent the axle boxes subjected to displacement constraints in the x, y, and z directions and rotation ones about the x and y directions. To avoid any over-constraint or under-constraint, a virtual axle model is employed, as shown in Figure 3. The primary suspension between the frame and the axle box is in the form of steel and rubber springs, represented by red lines, respectively. The stiffness values in the three directions are applied to the steel and rubber springs. The red lines represent the rigid beam elements, which are used to simulate the axle and axle box.

Figure 3
figure 3

Virtual axle model simulating the primary suspension, rubber spring, axle, axle box, and the connecting relationship between them

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The braking forces, shown in Figure 2, are located at the mass centre points of the brake units, which are installed on the brake unit brackets. The braking forces acting on the brake unit brackets at the two ends of the bogie frame are equal in magnitude but opposite in direction. The eight-stage loading spectrum data (see Figure 4) obtained from the loading tests [33] on the Wuhan–Guangzhou line are used for the brake disc assessment. The load spectrum represents a distance of 76700 km of the CRH electrical multi-unit (EMU) at a speed of 350 km/h. The block loads 2–8 occupy a small proportion of the spectrum, i.e., 4.3%, whereas the block load 1, which takes over 95% of the spectrum, corresponds to a very low load of approximately 3.32 kN.

Figure 4
figure 4

Eight-stage braking load spectrum obtained from the Wuhan–Guangzhou line of the CRH EMU

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The first principal stress distribution in the S11 direction of the bogie frame under peak load 1 is plotted to represent the loading capacity, as shown in Figure 5.

Figure 5
figure 5

Stress distribution on the bogie frame under load 1

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The maximum stress occurs clearly in the arc region of the brake unit bracket, where a relatively high stress concentration is observed under braking load. It is reasonable to assume that the crack initiates from the arc region. Hence, to ensure the accurate assessment to the safety and durability of the bracket, the meshes are refined locally with different crack depths so that a more accurate analysis of the FCG behaviour can be performed.

3.2 Brake Unit Bracket with a Defect

Ample theoretical and experimental studies have shown that a fatigue crack generally takes an elliptical or a semi-elliptical shape as observed from the fracture face. In the BS 7910 code, the single planar surface and embedded cracks or defects are generally characterised as standard elliptical or semi-elliptical shapes, and the maximum distances along the length and width directions are selected as the long and short axes, respectively. As shown in Figure 6, a semi-elliptical crack with an aspect ratio (a/c) of 0.6 is introduced in the arc part where the maximum stress occurs. The minimum elemental edge along the crack front is one tenth of the crack depth a.

Figure 6
figure 6

Bogie frame configuration, finite element meshes, and brake unit bracket considering a crack: a bogie frame, b brake unite bracket, c configuration of semi-elliptical crack

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Figure 7 shows the principal stress distribution in the S11 direction with a 1 mm crack under load 3 (shown in Figure 4). A clear stress concentration is observed at the crack front, with a maximum stress of approximately 169.1 MPa. Once a crack is initiated in the stress concentration region, the stress at the crack front tends to be significantly higher than that in the case without a crack or any other defects. As a result, the calculated SIF range at the crack front increases rapidly. In this case, all the load levels, shown in Figure 4, contribute to the subsequent crack growth. This analysis shows that after introducing a crack in the model, the local stress state visibly deteriorates under the same external load condition. However, the local stress state under this condition is more realistic and accurate.

Figure 7
figure 7

Result of brake unit bracket with a 1 mm crack under braking load 3

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To further reflect the influence of a defect, a crack with a 2-mm depth was induced in the maximum stress region (see Figures 5 and 7) of the brake unit bracket. The eight-stage braking load spectrum was applied to the bogie to evaluate the stress status. Figure 8 shows the first principal stress values for the structure with and without a defect.

Figure 8
figure 8

Maximum stress of brake unit bracket under eight-stage braking load spectrum

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It is clear that a defect can significantly amplify the stress in the arc region. This indicates that a crack may deteriorate the stress distribution of the bracket dramatically and lead to a sudden fatigue accident. Such a stress amplification phenomenon (see Figure 8) shows that the conventional nominal stress method is clearly unsuitable to evaluate damaged structures.

3.3 Fatigue Damage

3.3.1 Low Cycle Fatigue

It is generally believed that a cyclic plastic zone is locally consolidated before the crack penetrates the characteristic length [31, 32]. Such a global deformation phenomenon can be related with the local damage behaviour ahead of a growing crack, thus derives a crack growth model even under an elastic loading condition.

Figure 9 shows a mixed softening–hardening behaviour, which is in contrast to the monotonic tensile curve. The softening behaviour gradually changes to a hardening one with the increase in the cyclic strain amplitude. The cyclic yield stress (σcy) is the stress at 0.2% strain on the cyclic stress–strain curve. This value can be determined by drawing a line parallel to the modulus of elasticity through the 0.2% strain offset at the zero stress point. The stress at which the line intersects the cyclic stress–strain curve is taken as the cyclic yield stress and is used in Eq. (1).

Figure 9
figure 9

Monotonic tensile and cyclic stress–strain curves of the weathering steel

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The cyclic stress–strain relationship, in Figure 9, shows a deformation behaviour, particularly when a crack is initiated. This study ignores the cyclic hardening phenomenon while establishing the FCG rate model. Figure 10 shows the strain-life curve of the weathering steel including the elastic, plastic, and total strains. The elastic ε ea and plastic strain amplitudes ε pa were separated from the total one εa. According to the Coffin–Manson relation and Basquin equation, the LCF life can be calculated using the following equation:

$$\varepsilon_{a} = \varepsilon_{a}^{e} + \varepsilon_{a}^{p} = \frac{{\sigma^{\prime}_{f} }}{E}(2N_{f} )^{b} + \varepsilon^{\prime}{}_{f}(2N_{f} )^{c} ,$$
(1)

where E, σ′f, b, ε′f, and c are the Young’s modulus, fatigue strength coefficient, fatigue strength index, fatigue toughness coefficient, and fatigue toughness index, respectively. The fatigue ductility properties of the weathering steel are b = − 0.109, c = − 0.593, σ′f = 1298.57 MPa, and ε′f = 0.310.

Figure 10
figure 10

Strain-life curve of the weathering steel

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For the present weathering steel, the transition life (NT), which is the number of cycles when the elastic strain range is equal to the plastic one, is found to be 3735 cycles, as demonstrated in Figure 10. For a fatigue life less than NT, the plastic component prevails over the elastic one, whereas for a higher cycle number, the contrary is the case.

3.3.2 High Cycle Fatigue

As shown in Figure 11, the probabilistic fatigue S-N curves and fatigue limits are concluded from the HCF experimental data. The fatigue strength limit is approximately 225 MPa (at the knee point when ND = 3.62×106 cycles) with a survival probability of 50%. The material parameters C and m in Eq. (2) for the S–N curve are the basic data required for the fatigue life prediction using the cumulative damage approach.

$$C = N_{i} \sigma_{{{\text{a}}i}}^{m} .$$
(2)
Figure 11
figure 11

Fatigue S–N curves with survival probabilities of 5%, 50%, and 95% for weathering steel

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The probabilistic fatigue S–N curve with a survival probability of 95% and a confidence level of 95% is selected for the component design. This curve ensures the fatigue life at a specified stress level with the aforementioned reliability [34].

4 Results and Discussions

4.1 Fatigue Limit Diagram Analysis

The conventional and most frequently used method for the fatigue assessment of bogie frames is the fatigue limit diagram analysis along with the nominal stresses. It is a convenient and relatively conservative approach to quickly evaluate the fatigue design of frame structures. Table 1 lists the maximum and minimum principal stresses at several points on the brake unit bracket after the FE calculations under the eight-stage braking load spectrum. According to the tensile test results, given in Section 2.1 and Figure 9, the 0.2% offset proof yield strength Rp0.2 and the ultimate tensile strength Rm are 409 MPa and 567 MPa, respectively. The FKM Guideline [35] has suggested a safety factor of 1.35 corresponding to the yield strength of steel structures. As a result, the permissible stress at the yield point of the weathering steel is approximately 303 MPa.

Table 1 Analyzed stresses for the load spectrum shown in Figure 4
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Moreover, the allowable fatigue stress is 155 MPa. The mean stress σm = (σmax + σmin)/2 and stress amplitude σa = (σmax − σmin)/2 were then calculated. The values are indicated in the Goodman diagram shown in Figure 12. The data points under the load levels 1–5 are below the PM line, whereas the data points under the load levels 6–8 are above the line. Therefore, although only a small proportion of the load spectrum is occupied by the load levels 6–8 (i.e., only approximately 0.000907%), the fatigue strength of the brake unit bracket could not meet the requirements of the fatigue design. However, as discussed above, although the fatigue limit diagram analysis is convenient, it can be over-conservative.

Figure 12
figure 12

Goodman diagram for brake unit bracket under the eight-stage braking load spectrum

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4.2 Cumulative Damage Approach

The structural design of a brake unit bracket using the fatigue limit diagram approach is rather conservative. To predict the fatigue life and determine the validity of the braking load conditions more accurately, the cumulative fatigue damage approach was applied. Based on the eight-stage braking load spectrum, the cumulative damage D was subsequently calculated using the linear damage accumulation Palmgren–Miner rule as shown in Eq. (3):

$$D = \sum\limits_{i = 1}^{k} {\frac{{n_{i} }}{{N_{i} }}} .$$
(3)

A significant proportion of the eight-stage load spectrum is occupied by cycles at relatively lower stress levels. Therefore, the contribution of these cycles to the fatigue damage must be considered. To reasonably consider the damage caused by all the stress levels, the actual cumulative damage Dact of the eight-stage load spectrum was derived based on the Miner linear fatigue cumulative damage rule and the S-N curve obtained from the HCF tests.

$$D_{\text{act}} = \sum\limits_{i = 1}^{8} {\frac{{n_{i} }}{{N_{i} }}} = \sum\limits_{i = 1}^{8} {\frac{{n_{i} \sigma_{{{\text{a}}i}}^{m} }}{C}} .$$
(4)

After obtaining Dact by Eq. (4) and corresponding distance Lact, which was introduced in another study [33], the total operation distance L of the brake unit bracket can be calculated as follows:

$$\frac{{D_{\text{cri}} }}{{L_{\text{cri}} }} = \frac{{D_{\text{act}} }}{{L_{\text{act}} }},$$
(5)

where Dcri is the total critical damage with a recommended value of 0.3 according to the FKM Guideline [35] for non-welded components of steel materials.

The calculated results show that Dact = 3.44 × 10−6 and Lcri = 6.68 × 109 km. If the operating mileage of the CRH is set to be 600000 km per year, the brake unit bracket of the bogie frame can be safely used for 11136.5 years, which is more than adequate. It is generally assumed that the total lifetime of railway bogie frames is approximately 25 years. Therefore, the brake unit bracket satisfies the service requirement.

However, if a crack with a depth of 2 mm (which is equal to the minimum detectable crack size using current detection techniques [8, 17, 29, 32, 36]) already exists in the arc part of the brake unit bracket, the damage value will increase to Dact = 1.69 × 10−2, thus reducing the total distance to Lcri = 1.36 × 106 km. In this case, the total service life of the brake unit bracket is 2.27 years, which still meets the safety requirement of the maintenance policy (grade 2) for the Chinese high-speed railway. Therefore, the safety of the brake unit bracket can be guaranteed before the critical crack is detected.

It is worth noting that the proportion of loads 5–8 in the load spectrum is lower than 0.1%; however, the damage due to these loads accounts for as much as 98.27% of the total damage. This interesting result shows that although the frequency of high loads is low, it significantly affects the damage accumulation.

4.3 Damage Tolerant Approach

The shortcoming of the conventional design is that it considers the material as a homogeneous continuum without defects. In fact, initial defects are almost inevitable in such structures, thereby in turn making the actual strength lower than the nominal value. In contrast, the fracture mechanics-based approach considers that the structure is defective, which forms the basis for investigating the fracture failure mechanism and carrying out the safety evaluation.

4.3.1 Fatigue Crack Growth Rate

Recent studies have shown that the fatigue life predictions using the Miner cumulative damage rule can be over conservative even under the upper level of the service load spectrum. This is partially attributed to the intrinsic geometry discontinuity and extrinsic defects. Hence, to achieve a verifiable residual life, the assessment should be based on the framework of damage tolerance for a long crack. Figure 13 shows the obtained experimental FCG data, including the analytical solutions from the upper, lower, and optimal NASGRO models as described below [37]. As the residual life assessment is closely related to the value of threshold SIF range, it is important to determine this parameter properly. In present study, the threshold SIF range is experimentally obtained in accordance with the requirements described in ASTM E 647 standard using MT specimens. The threshold SIF ranges are 6.49 and 5.51 MPa∙m−1/2 when R = − 1 and 0, respectively.

$$\frac{\text{d}a}{\text{d}N} = C \cdot \Delta K_{\text{eff}}^{m} \left( {1 - \frac{{\Delta K_\text{th} }}{\Delta K}} \right)^{p} \left( {1 - \frac{{K_{\text{max} } }}{{K_\text{IC} }}} \right)^{ - q} ,$$
(6)

where ∆Kth, ∆Keff, and ∆Kmax are the threshold, effective, and maximum SIF ranges, respectively. C and m are the parameters in the Paris formula obtained from the FCG rate test, and KIC is the fracture toughness. p and q are artificially fitted parameters from the experimental data.

Figure 13
figure 13

Comparison of predicted FCG curves via upper, lower, and optimal NASGRO models and classical Paris equation with experimental data of weathering steel for: a R = − 1; b R = 0

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To consider the plasticity-induced crack closure effect, a crack opening function f can be introduced to modify ∆K and ∆Kth as follows [8, 17, 29, 32]:

$$\Delta K_{\text{eff}} = K_{ \text{max} } - K_{\text{op}} = \frac{1 - f}{1 - R}\Delta K,$$
(7)
$$\Delta K_{\text{th,eff}} = \frac{1 - f}{1 - R}\Delta K_{\text{th}} ,$$
(8)
$$f = \left\{ {\begin{array}{*{20}l} {\text{max} \left( {R,A_{0} + A_{1} R + A_{2} R^{2} + A_{3} R^{3} } \right),} \hfill & {R \ge 0,} \hfill \\ {A_{0} + A_{1} R,} \hfill & { - 2 \le R < 0,} \hfill \\ \end{array} } \right.$$
(9)
$$\begin{aligned} A_{0} & = \left( {0.825 - 0.34\alpha + 0.05\alpha^{2} } \right) \cdot \left[ {\cos \left( {{{\pi \sigma_{\text{max} } } \mathord{\left/ {\vphantom {{\pi \sigma_{\text{max} } } {2\sigma_\text{cy} }}} \right. \kern-0pt} {2\sigma_\text{cy} }}} \right)} \right]^{{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-0pt} \alpha }}} , \\ A_{1} & = \left( {0.415 - 0.071\alpha } \right) \cdot \left( {{{\sigma_{\text{max} } } \mathord{\left/ {\vphantom {{\sigma_{\text{max} } } {\sigma_{cy} }}} \right. \kern-0pt} {\sigma_\text{cy} }}} \right), \\ A_{2} & = 1 - A_{0} - A_{1} - A_{3} , \\ A_{3} & = 2A_{0} + A_{1} - 1, \\ \end{aligned}$$
(10)

where Kmax, σmax, σcy, and Kop are the maximum SIF, maximum nominal stress, cyclic yield stress obtained from the LCF test, and SIF required to fully open a crack, respectively. A0, A1, A2, and A3 are the coefficients of the Newman function f. α is the constraint parameter related to the stress status; here α = 3.0. The predicted curves based on the NASGRO model can be used to accurately estimate the FCG rate, particularly when the SIF range is close to the threshold region.

4.3.2 Residual Life Assessment

The fatigue performance of the brake unit bracket has been investigated using the classical nominal stress method. Over conservative lifetime predictions were obtained for the fatigue strength limit and permissible damage accumulation. To further enhance the durability, the brake unit bracket with existing defects was comprehensively investigated in terms of the fracture mechanics. Figure 14 shows the SIF range as a function of the crack depth under different loading conditions, wherein an initial semi-elliptical crack with a depth of 0.5 mm is included in the arc part of the brake unit bracket. This provides a qualitative evaluation of the fatigue property of the brake unit bracket.

Figure 14
figure 14

FCG driving force curves at the deepest point of a growing crack for the brake unit bracket under load spectrum

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With the increase in the crack depth, the SIF range clearly increases. When the crack depth (a) is 0.5 mm under the loads 1–4, the calculated SIF range ∆K is lower than the threshold SIF range ∆Kth when R = − 1. In contrast, under the loads 5–8, ∆K is greater than ∆Kth.

Based on the experimental data, shown in Figure 13(a), Figure 15 shows the predicted FCG curves of the brake unit bracket subjected to actual fatigue load sequences considering a semi-elliptical crack with a 2.0 mm depth.

Figure 15
figure 15

Predicted life of brake unit bracket with an initial crack of 2 mm depth

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According to Figure 15, the lifetime of the brake unit bracket can be estimated from the initial crack depth of 2 mm to half the height of the brake unit bracket, i.e., 30 mm. The lifetime values obtained using the upper, optimal, and lower NASGRO models are 4.05 × 105 km, 5.54 × 105 km, and 1.69 × 106 km, respectively. It should be noted that the fatigue crack would propagate rapidly once a > 3.0 mm, which corresponds to approximately 5.54 × 105 km as per the optimal NASGRO model. To analyse the influence of the initial crack depth, the fatigue lives of the brake unit bracket with crack depths of 0.5 and 1 mm were also calculated. The equivalent service distances are approximately 4.24 × 107 km and 3.01 × 106 km, respectively, according to the optimal NASGRO model; these values are significantly greater than those when the initial crack depth is 2 mm. The fatigue life of the brake unit bracket could satisfy the service requirement of the bogie for all the initial crack depths considered. The initial crack depth significantly influences the fatigue life of the brake unit bracket, i.e., the life calculated with a 0.5 mm initial crack is 145.7 times that calculated with a 2-mm initial crack. Therefore, to obtain a more rational prediction of the fatigue life of the brake unit bracket, it is very important to detect the initial crack as carefully and accurately as possible.

However, it is worth noting that the results predicted using the three different NASGRO models were completely based on the artificially fitted parameters, and therefore, show a significant deviation. The equivalent service distance obtained using the lower NASGRO model is approximately four times that obtained using the upper NASGRO model. This is a disadvantage of the NASGRO model, which might consequently lead to significant error accumulation.

5 Conclusions

The fatigue life of a brake unit bracket under a measured braking load spectrum was predicted by integrating the standard nominal stress method and an advanced damage tolerance method. Prior to this, the static strength of a box-type welded frame, which is here modelled using high-accuracy solid elements rather than relatively simple plate and shell elements widely used, was first evaluated by adopting the finite element method. To perform the residual life assessment in the presence of a crack, the FCG rate was characterised using three different NASGRO models based on the data of FCG rate test. The following conclusions can be drawn from this study:

  1. (1)

    According to the finite element calculation under braking loads, a semi-elliptical crack with different depths was introduced in the stress concentrated area of the model, i.e., in the arc region of the brake unit bracket.

  2. (2)

    Based on the classical Miner rule, the damage is argued to be D = 0.3 to ensure a service life of over 30 years for the brake unit bracket under an eight-stage braking load spectrum. Even if a crack with a 2-mm depth exists, the brake unit bracket can be safely operated for more than 2.27 years.

  3. (3)

    The calculated residual fatigue life of the brake unit bracket with an initial crack having a 2-mm depth was found to be approximately 5.54 × 105 km based on the optimal NASGRO model. This can fully ensure the service safety and reliability of the bracket so that the crack can be detected in the subsequent maintenance process.

In future study, it is reasonably expected to introduce the multibody system dynamics to improve current static analysis. This is to take the influence of random loading spectrum into consideration especially when small-radius curved track exists. Besides, some defects, such as the lack of penetration and the lack of fusion during the welding, should be modelled according to non-destructive inspection or engineering experiences.