Fatigue strength analysis of bogie frames under random loads

Abstract

In this study, a method for analyzing the fatigue strength of a bogie frame under a random load was proposed. Based on the geometric features, a welded joint coordinate system was established to compute the stress components of each node in this coordinate system. With the influence of small amplitude cycles included, based on the corrected S–N curve and a method for calculating the equivalent constant amplitude stress, the node and comprehensive degree of utilization were calculated based on the anti-fatigue design grade of welded joints to evaluate the fatigue strength of a structure under a given lifespan. The FKM and International Institute of Welding methods were used to evaluate the fatigue strength of typical welded joints of a bogie frame. The characteristics of the node degree of utilization under different analytical methods were compared, and the results showed that when the time histories of the three stress components of the nodes had significant non-proportional features, the FKM method obtained conservative results. When the time histories of the three stress components of the nodes were synchronized, the criterion value specified by the International Institute of Welding method was the main factor affecting the distribution characteristics of the node degree of utilization. The analysis based on the International Institute of Welding method can effectively balance the lightweight design and reliability of the structure.

Introduction

The welding frame is an important bearing component of a bogie and its fatigue strength has attracted considerable attention. When a vehicle is running, the bogie frame is subjected to the car body vibration load, driving load, braking load, vibration load of the large mass suspension component, load of the suspension system components, and twist load caused by track irregularities. The stress on the bogie frame is randomly distributed.

In recent years, studies on the fatigue strengths of bogie frames with randomly distributed stresses have been conducted.^1–4 For the differences between the dynamic load characteristics of a vehicle running on straight and curved lines, Dietz et al. ¹ proposed a method for predicting the fatigue life of a railway bogie based on the frequency domain and time domain methods. Yang ² established a rigid-flexible coupled dynamics analysis model of the DF4D locomotive of the China Railway, introduced the low-order mode of the bogie frame, used the quasi-static superposition method to obtain the stress-time history and stress spectrum of the area to be inspected, and evaluated the fatigue strength of the bogie frame based on the Palmgren-Miner’s rule. Introducing the elasticity of the car body and bogie frame in a dynamic model of the vehicle system, Liu and Shao ³ calculated the characteristics of the dynamic load and dynamic stress of a vehicle system with the European Rail Research Institute track irregularities. Lu et al. ⁴ established a high-speed train dynamics analysis model to obtain the load and time history of each part of the bogie frame. The modal superposition method was used to study the influence of structural vibrations on the fatigue strength of the bogie frame based on the Palmgren-Miner linear cumulative damage criterion and the P–S–N curves of the materials. In the studies described above, which were all based on the time history of the uniaxial equivalent stress, the stress spectrum was generated using the corresponding counting method, the cumulative damage at the investigation point was calculated using the Palmgren-Miner’s rule, and the fatigue strength of the bogie frame was evaluated based on whether the cumulative damage was greater than 1.

According to recent studies,^5–10 stress characteristics have a significant impact on the fatigue strengths of welded structures under random loads. First, the variable amplitude stress cycle causes more damage to the structure than the constant amplitude cycle. Sonsino and Kueppers^5–8 suggested that a structure should be evaluated with a small amount of allowable accumulated damage to ensure that the fatigue strength is in a safe range. Second, in the multi-axial stress state, the stress component in the non-proportional stress state causes the principal stress direction to change over time, and the fatigue strength of the steel weld is severely weakened. Thus, a modified strength theory should be used to evaluate the fatigue strength of the steel weld.^5,9 Based on the notch stress method, Shen et al. ¹⁰ built a finite element model of a typical welded joint to explore the fatigue strengths of welded joints in multi-axial stress states using different strength criteria. In addition, the “FKM Guideline: Analytical Strength Assessment of Components” also proposed an engineering method for evaluating the structural fatigue strength in a non-proportional stress state.

For a bogie frame subjected to a random load, the stress in each area is in a significant multi-axial stress state, and the stress components are typically non-proportional. Hence, analyses based on currently available methods endanger the structural fatigue strengths. Considering the limitations of currently available engineering analysis methods and based on the results of fatigue tests of typical welded joints, the contribution of small amplitude cycles to structural fatigue damage was taken into account, and the influence of the stress characteristics under random loads was fully considered to study a method for evaluating the fatigue strength of a bogie frame, which is of great significance for ensuring its structural reliability.

In order to meet the requirements of structural fatigue strength evaluation under multi-axial stress state, this study first constructed the weld coordinate system based on the geometric characteristics of the finite element model, and obtained the node’s stress components in the weld coordinate system. Using the linear superposition method, the time history of each stress component of the node is obtained, and the stress spectrum is generated. Aiming at the defects of FKM method, the calculation method of constant amplitude cyclic equivalent stress is proposed. The structural fatigue strength is evaluated based on the modified strength theory under non-proportional stress conditions. Based on the above method, the fatigue strength of the metro bogie frame under random load is analyzed, and the analysis results obtained by different methods are compared.

Node stress component in the weld coordinate system

Fatigue strength analysis of welded joints is based on the nominal stress method. According to the International Institute of Welding (IIW), the German Mechanical Engineering Society and the European Committee for Standardization, the fatigue strength of welds is directly determined by the normal stress in the direction of the welds $({\sigma }_{x\prime })$ , the normal stress perpendicular to the direction of the welds $({\sigma }_{y\prime })$ , and the shear stress in the plane formed by the above two directions $({\tau }_{x\prime y\prime })$ .^9,11,12 Frame welds have various forms. If the plane mentioned above is used as the X′O′Y′ plane and a three-dimensional (3D) weld coordinate system is established with the normal direction of the plane as the Z′ axis, there are angles between each coordinate axis of the coordinate system and that of the global coordinate system for most nodes, and finite element analysis results cannot directly output the joint stress component in the weld coordinate system. When evaluating the structural fatigue strength, it is necessary to compute the direction vector of each coordinate axis based on the geometrical features of the nodes, establish the weld coordinate system, and obtain the stress component of each node in the coordinate system.

Welds of bogie frames can generally be divided into two types: straight and curved. The coordinate system of both types of welds can be established by examining the vector relationship between nodes, adjacent nodes, and external reference nodes.

Coordinate system of straight welds

Figure 1 shows the straight welds. If a node is defined as node A, its neighbor node is node B, and the reference node C is in the weld plane. Then, the direction vector of $\mathit{O}\prime X-\prime$ along the X′ axis in the weld coordinate system is defined as follows

$$\mathit{O}\prime \mathit{X}\prime =(x_B-x_A)\mathit{i}+(y_B-y_A)\mathit{j}+(z_B-z_A)\mathit{k}$$

(1)

where $x_i$ , $y_i$ , and $z_i(i=A,B,C)$ are the coordinate values of the node in the global coordinate system.

Figure 1.

Straight welds.

The direction vector of the Y′ axis in the weld coordinate system, $\mathit{O}\prime Y-\prime$ , resides in a plane composed of vectors $\mathit{O}\prime X-\prime$ and $\mathit{CA}$ , and is perpendicular to vector $\mathit{O}\prime X-\prime$ . Thus, vector $\mathit{O}\prime Y-\prime$ is computed as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathit{O}\prime \mathit{Y}-\prime =(a-\frac{1}{\mathrm{ad}+\mathrm{be}+\mathrm{cf}})\mathit{i}+(b-\frac{1}{\mathrm{ad}+\mathrm{be}+\mathrm{cf}})\mathit{j}}\\ \multicolumn{1}{c}{+(c-\frac{1}{\mathrm{ad}+\mathrm{be}+\mathrm{cf}})\mathit{k}}\end{array}$$

(2)

where a i , b j , and c k are the components of the unit vector of vector $\mathit{O}\prime X-\prime$ on the three coordinate axes of the global coordinate system, and d i , e j , and f k are the components of the unit vector of vector CA on the three coordinate axes of the global coordinate system.

The direction vector of the Z′ axis of the weld coordinate system, $\mathit{O}\prime Z-\prime$ , is the vector product of vectors $\mathit{O}\prime X-\prime$ and $\mathit{O}\prime Y-\prime$ whose direction is determined according to the right-hand rule based on the directions of these two vectors. After converting the three vectors that represent the directions of each coordinate axis in the weld coordinate systems into unit vectors, the direction cosine of each coordinate axis in the overall coordinate system was obtained.

Coordinate system of curved welds

It is currently difficult to accurately predict the curvatures of non-circular curved welds. Since the distances between the nodes in the finite element analysis model are much smaller than the radii of the curves of the welds, it is sufficient to replace the actual shape of the welds with the arc formed by the inspection point and the two adjacent points and meet most engineering needs. The coordinate system of the curved welds is shown in Figure 2, where node D is the inspection point, nodes E and F are two adjacent nodes, and H and J are the midpoints of line segments DE and DF, respectively. When establishing the weld coordinate system, the external reference point of the welds, that is, the coordinates of the center G of the arc, should be obtained first.

Figure 2.

Curved welds.

The curved weld joints are shown in Figure 2, and vector JG can be calculated as follows

$$\mathit{JG}=\lambda {\mathit{e}}_{\mathit{JG}}$$

(3)

where e_JG is the unit vector whose direction is consistent with that of vector JG , which is determined by the vector product of vector v and vector FD ; vector v is the unit vector determined by the vector product of vector FD and vector DE ; and $\lambda$ is a coefficient that can be obtained as follows

$$\lambda =-\frac{\mathit{HJ}·\mathit{DE}}{{\mathit{e}}_{\mathit{JG}}·\mathit{DE}}$$

(4)

After the unit vector e _JG and the coefficient $\lambda$ are obtained, the coordinates of the center G can be calculated as follows

$$x_G=x_J+\lambda r$$

(5)

$$y_G=y_J+\lambda s$$

(6)

$$z_G=z_J+\lambda t$$

(7)

where r i , s j , and t k are the components of the vector e _JG on the three coordinate axes of the global coordinate system.

Based on the coordinates of the center G and the inspection point D, the direction vector of the Y′ axis of the curve weld coordinate system, $\mathit{O}\prime Y-\prime$ , can be determined. Since the direction vector of the X′ axis, $\mathit{O}\prime X-\prime$ , lies in the plane formed by vectors $\mathit{O}\prime Y-\prime$ and DF , and is perpendicular to vector $\mathit{O}\prime \mathit{Y}-\prime$ , it can be calculated using equation (2), and the direction vector $\mathit{O}\prime \mathit{Z}-\prime$ of the Z′ axis can subsequently be obtained.

Node stress components in the weld coordinate system

The stress components of the nodes in the global coordinate system include the normal stress components, that is, ${\sigma }_x$ , ${\sigma }_y$ , and ${\sigma }_z$ , and the shear stress components, that is, ${\tau }_{\mathrm{xy}}$ , ${\tau }_{\mathrm{yz}}$ , and ${\tau }_{\mathrm{xz}}$ . The conversion equation of the stress components in different coordinate systems is as follows ¹³

$$\begin{array}{c}\multicolumn{1}{c}{\left(\begin{array}{ccc}\begin{array}{c}{\sigma }_{x\prime }\hfill \\ \multicolumn{1}{c}{{\tau }_{x\prime y\prime }}\\ \multicolumn{1}{c}{{\tau }_{x\prime z\prime }}\end{array}\hfill & \begin{array}{c}{\tau }_{x\prime y\prime }\hfill \\ \multicolumn{1}{c}{{\sigma }_{y\prime }}\\ \multicolumn{1}{c}{{\tau }_{y\prime z\prime }}\end{array}\hfill & \begin{array}{c}{\tau }_{x\prime z\prime }\hfill \\ \multicolumn{1}{c}{{\tau }_{y\prime z\prime }}\\ \multicolumn{1}{c}{{\sigma }_{z\prime }}\end{array}\hfill \end{array}\right)=\left(\begin{array}{ccc}\begin{array}{c}{l}_1\hfill \\ \multicolumn{1}{c}{l_2}\\ \multicolumn{1}{c}{l_3}\end{array}\hfill & \begin{array}{c}{m}_1\hfill \\ \multicolumn{1}{c}{m_2}\\ \multicolumn{1}{c}{m_3}\end{array}\hfill & \begin{array}{c}{n}_1\hfill \\ \multicolumn{1}{c}{n_2}\\ \multicolumn{1}{c}{n_3}\end{array}\hfill \end{array}\right)}\\ \multicolumn{1}{c}{\left(\begin{array}{ccc}{\sigma }_x\hfill & {\tau }_{\mathrm{xy}}\hfill & {\tau }_{\mathrm{xz}}\hfill \\ \multicolumn{1}{c}{{\tau }_{\mathrm{xy}}}& {\sigma }_y\hfill & {\tau }_{\mathrm{yz}}\hfill \\ \multicolumn{1}{c}{{\tau }_{\mathrm{xz}}}& {\tau }_{\mathrm{yz}}\hfill & {\sigma }_z\hfill \end{array}\right)\left(\begin{array}{ccc}{l}_1\hfill & l_2\hfill & l_3\hfill \\ \multicolumn{1}{c}{m_1}& m_2\hfill & m_3\hfill \\ \multicolumn{1}{c}{n_1}& n_2\hfill & n_3\hfill \end{array}\right)}\end{array}$$

(8)

where ${\sigma }_{x\prime }$ , ${\sigma }_{y\prime }$ , ${\sigma }_{z\prime }$ , ${\tau }_{x\prime y\prime }$ , ${\tau }_{y\prime z\prime }$ , and ${\tau }_{x\prime z\prime }$ are the six stress components of the nodes in the weld coordinate system; $l_1$ , $m_1$ , and $n_1$ are the direction cosines of the X′ axis of the weld coordinate system in the global coordinate system; $l_2$ , $m_2$ , and $n_2$ are the direction cosines of the Y′ axis of the weld coordinate system in the global coordinate system; and $l_3$ , $m_3$ , and $n_3$ are the direction cosines of the Z′ axis of the weld coordinate system in the global coordinate system.

Fatigue strength evaluation method for welded joints under random load

Time history of the stress components

When the vehicle is running on an irregular track, the bogie frame is subjected to various loads that vary with time. At time $t$ , the stress components of a node under various loads can be calculated as follows

$$\begin{array}{c}\multicolumn{1}{c}{{({\sigma }_{i\prime j\prime ,1,t},{\sigma }_{i\prime j\prime ,2,t},\dots ,{\sigma }_{i\prime j\prime ,n,t})}^T=\mathrm{diag}(F_{1,t},F_{2,t},\dots ,F_{n,t})}\\ \multicolumn{1}{c}{\times ({\sigma }_{i\prime j\prime ,1},{\sigma }_{i\prime j\prime ,2},\dots ,{\sigma }_{i\prime j\prime ,n}{)}^T}\end{array}$$

(9)

where ${\sigma }_{i\prime j\prime ,k,t}(i\prime =x\prime ,y\prime ,j\prime =x\prime ,y\prime ,k=1,2,\dots ,n)$ is the stress component of the node under the kth load at time $t$ , and ${\sigma }_{i\prime j\prime ,k}$ is the stress component of the nodes in the welded joint coordinate system under the unit load of the kth load.

The bogie frame exhibited a linear stress–strain relationship under various loads. At time $t$ , the stress component of the node in the weld coordinate system, ${\sigma }_{i\prime j\prime ,t}(i\prime =x\prime ,y\prime ,j\prime =x\prime ,y\prime )$ , can be calculated as follows

$${\sigma }_{i\prime j\prime ,t}=\sum _{k=1}^n{\sigma }_{i\prime j\prime ,k,t}$$

(10)

Influence of average stress

The rain-flow counting method was used to process the time history of each stress component to gain a 3D stress spectrum composed of the mean value, amplitude value, and the number of cycles of each stress component. As recommended by the FKM Guideline, ${\sigma }_{\mathrm{ae}}$ , the symmetric cyclic equivalent stress of each two-dimensional (2D) stress spectrum block, can be calculated as follows

$${\sigma }_{\mathrm{ae}}=\{\begin{array}{cc}{\sigma }_a·(1-M_{\sigma })/K_E,\hfill & R>1\hfill \\ \multicolumn{1}{c}{{\sigma }_a·(1+M_{\sigma }·{\sigma }_m/{\sigma }_a)/K_E,}& -\infty \le R\le 0\hfill \\ \multicolumn{1}{c}{{\sigma }_a·\frac{(1+M_{\sigma })(3+M_{\sigma }·{\sigma }_m/{\sigma }_a)}{(3+M_{\sigma })·K_E},}& 0<R<0.5\hfill \\ \multicolumn{1}{c}{{\sigma }_a·\frac{3{(1+M_{\sigma })}^2}{(3+M_{\sigma })·K_E},}& R\ge 0.5\hfill \end{array}$$

(11)

where ${\sigma }_a$ is the cyclic stress amplitude of the spectrum block; ${\sigma }_m$ is the mean cyclic stress of the spectrum block; $R$ is the cyclic stress ratio, which for normal stress is $R=({\sigma }_m-{\sigma }_a)/({\sigma }_m+{\sigma }_a)$ and for shear stress is $R=(\left|{\sigma }_m\right|-{\sigma }_a)/(\left|{\sigma }_m\right|+{\sigma }_a)$ ; and $M_{\sigma }$ and $K_E$ are the average stress sensitivity and the residual stress coefficients, respectively, whose values were determined following the FKM Guideline.

The fatigue strength of the weld of the bogie frame is affected by the stress cycle characteristics and the residual stress state, and a relatively high residual stress will weaken the fatigue strength of the structure. Therefore, for complex structures in a state of global residual stress, Hobbacher ⁹ suggested that the influence of the average stress should be ignored, and the cyclic stress spectrum should be directly determined based on the stress amplitude and the number of cycles of each 2D stress spectrum block.

Node degree of utilization component under variable amplitude load

For the longitudinal load-carrying fillet weld, Figure 3 shows the different S–N curves given by the IIW Recommendation and FKM Guideline, the parameters of which are listed in Table 1. The FKM Guideline uses an S–N curve with a fatigue limit cut-off line. ¹⁴ When the number of stress cycles $N\ge 5\times {10}^6$ , the S–N curve of a welded joint is a horizontal straight line, and the stress cycle below the fatigue limit will not affect the structural fatigue strength. According to recent research results, for structures subjected to variable amplitude stress cycles, small stress cycles will cause fatigue damage. Therefore, while analyzing the fatigue strengths of structures subjected to variable amplitude stress cycles, the number of stress cycles at knee point of the S–N curve should be defined as $N_D={10}^7$ based on Hobbacher’s ⁹ suggestion, and the curve should be extended with the exponent $k_2=2·k_1-1$ when $N\ge N_D$ .

Figure 3.

S–N curves recommended in different studies. (a) Normal stress S–N curve and (b) shear stress S–N curve.

Table 1.

Characteristic parameters of the S–N curve.

Types of stress	Reference	Number of stress cyclesat the inflection point, ND	Curve index,k1 (N < ND)	Curve index,k2 (N ≥ ND)
Normal stress	IIW Recommendation	1 × 107	3	5
	FKM Guideline	5 × 106	3	–
Shear stress	IIW Recommendation	1 × 108	5	9
	FKM Guideline	1 × 107	5	–

IIW: International Institute of Welding.

For simplicity, the FKM Guideline did not consider the influence of the fatigue limits of the welded joints when calculating the equivalent constant amplitude stress. When the equivalent constant amplitude stress ${\sigma }_e$ is greater than the maximum stress amplitude ${\sigma }_{\mathrm{ae},1}$ , ${\sigma }_e\mathrm{is}\mathrm{equal}\mathrm{to}{\sigma }_{\mathrm{ae},1}$ . However, this method has the following three drawbacks. First, when calculating the cyclic equivalent constant amplitude stress, the S–N curve was extended using the exponent $k_1$ . Although the influence of the small amplitude cycle of the stress amplitude ${\sigma }_{\mathrm{ae},i}<{\sigma }_D$ ( ${\sigma }_D$ is the cyclic stress amplitude corresponding to the inflection point of the S–N curve) was taken into consideration, the relatively small index $k_1$ caused the influence to be greater than it was in the actual system. As a result, the structural fatigue strength analysis tends to be conservative. Second, for a small stress amplitude cycle, and even if the calculated equivalent constant amplitude stress is less than the fatigue limit ${\sigma }_D$ of the welded joints, the degree of utilization is still greater than 0, as stipulated by the FKM Guideline. However, since FKM Guideline uses anS–N curve with a fatigue limit cut-off line, from the fatigue damage theory, under the action of these stress cycles whose amplitude is less than the fatigue limit, the structure has an infinite fatigue life with a damage value of 0. This contradicts the conclusion that “material utilization is greater than 0.” Third, according to the test results, Sonsino ⁸ suggested that the allowable accumulated damage of the welded joints under a variable amplitude load should be 0.5. According to the method proposed by the FKM Guideline, the obtained range of the allowable accumulated damage is $0.5\le D_m\le 1$ , which makes the structural fatigue strength result being risky. Therefore, when evaluating the fatigue strength of structures subjected to variable amplitude stress loads using the S–N curve recommended by the FKM Guideline, the constant amplitude equivalent stress calculation method and the allowable accumulated damage $D_m$ should be corrected.

When considering the influence of a small amplitude cycle on the structural fatigue strength for structures subjected to variable amplitude loads, the method proposed by Hobbacher ⁹ was used to extend the S–N curve with the exponent $k_2=2·k_1-1$ when $N\ge N_D$ . The S–N curve was constructed based on the allowable accumulated damage $D_m$ under a variable amplitude load to calculate the equivalent constant amplitude stress. When computing the damage from the cycle according to the Palmgren-Miner rule, we obtained the corrected equivalent constant amplitude stress, ${\sigma }_e$ , as follows ⁹

$${\sigma }_e=\{\begin{array}{c}{\left[\frac{\sum (n_i·{\sigma }_{\mathrm{ae},i}^{k_1})+{\sigma }_D^{k_1-k_2}·\sum (n_j·{\sigma }_{\mathrm{ae},j}^{k_2})}{D_m·N_D}\right]}^{1/k_1},{\sigma }_e\ge {\sigma }_D\hfill \\ \multicolumn{1}{c}{{\left[\frac{{\sigma }_D^{k_2-k_1}·\sum (n_i·{\sigma }_{\mathrm{ae},i}^{k_1})+\sum (n_j·{\sigma }_{\mathrm{ae},j}^{k_2})}{D_m·N_D}\right]}^{1/k_2},{\sigma }_e<{\sigma }_D}\end{array}$$

(12)

where $N_D$ is the number of stress cycles corresponding to the inflection point of the S–N curve; ${\sigma }_D$ is the allowable stress amplitude of the cycle at the inflection point, where ${\sigma }_D=0.5·\mathrm{FAT}·(N_C/N_D{)}^{1/k_1}·f_d$ ; $\mathrm{FAT}$ denotes the anti-fatigue design grade of the welds; $N_C$ is the number of cycles corresponding to the anti-fatigue design grade of the welds, where $N_C=2\times {10}^6$ ; f_d is the thickness coefficient, such that $f_d=1$ when $d<25\mathrm{mm}$ and $f_d=1$ when $d\ge 25\mathrm{mm}$ , where $f_d=(25/d{)}^m$ and d is the thickness of the material used to manufacture the welds; m is the index determined by the form of the welds, and its values are tabulated in the book by Hobbacher; ⁹ ${\sigma }_{\mathrm{ae},i}$ is the cyclic stress amplitude with an amplitude greater than ${\sigma }_D$ , where $n_i$ is the corresponding number of cycles; ${\sigma }_{\mathrm{ae},j}$ is the cyclic stress amplitude with an amplitude less than ${\sigma }_D$ , where $n_j$ is the corresponding number of cycles; $k_1$ is the index of the S–N curve when the number of cycles $N<N_D$ ; $k_2$ is the index of the S–N curve when the number of cycles $N\ge N_D$ ; and $D_m$ is the allowable accumulated damage. For steel welds, Hobbacher ⁹ suggested that $D_m=0.5$ .

The nodes’ degree of utilization of each stress component, $a_{i\prime j\prime }(i\prime =x\prime ,y\prime ,j\prime =x\prime ,y\prime )$ , was calculated as follows

$$a_{i\prime j\prime }=\frac{{\sigma }_{e,i\prime j\prime }·j_F}{{\sigma }_{D,i\prime j\prime }}$$

(13)

where $j_F$ is the safety factor, the value of which was obtained by the IIW Recommendation or FKM Guideline.

Fatigue strength under non-proportional load

In the non-proportional stress state, the direction of the principal stress of the weld joints changes with time, which deteriorates the fatigue strength of the structure. For the purpose of engineering simplifications, the IIW Recommendation and FKM Guideline proposed methods for evaluating structural fatigue strengths suitable for non-proportional stress cycles.

Based on the fatigue test results of steel welds in the non-proportional stress state, Sonsino ⁸ suggested that the criterion of $\mathrm{CV}=0.5$ should be used to correct the Gough–Pollard equation and that the evaluation should be carried out according to the criteria shown below in equation (14). The IIW standard also adopted this criterion to evaluate the fatigue strength of steel welds in the non-proportional stress state

$$a_{x\prime x\prime }^2+a_{y\prime y\prime }^2-a_{x\prime x\prime }·a_{y\prime y\prime }+a_{x\prime y\prime }^2\le \mathrm{CV}$$

(14)

If the evaluation method is unified with a node’s degree of utilization of less than 1, equation (15) must be rewritten as follows

$$\mathrm{DoU}=\sqrt{\frac{a_{x\prime x\prime }^2+a_{y\prime y\prime }^2-a_{x\prime x\prime }·a_{y\prime y\prime }+a_{x\prime y\prime }^2}{\mathrm{CV}}}\le 1$$

(15)

Due to the low elongation at the break of the welds, the FKM Guideline suggests that the node’s degree of utilization $\mathrm{Do}{U}_k$ should be calculated under a single load based on the maximum principal stress criterion as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{Do}{U}_k=\frac{1}{2}}\\ \multicolumn{1}{c}{[|a_{x\prime x\prime ,k}+a_{y\prime y\prime ,k}|+\sqrt{{(a_{x\prime x\prime ,k}-a_{y\prime y\prime ,k})}^2+4a_{x\prime y\prime ,k}^2}]}\end{array}$$

(16)

where $a_{i\prime j\prime ,k}(i\prime =x\prime ,y\prime ,j\prime =x\prime ,y\prime )$ is the node’s degree of utilization component under the kth load condition.

The FKM Guideline suggests superimposing the node’s degree of utilization in various load conditions to evaluate the fatigue strength as follows

$$\mathrm{DoU}=\sum _{k=1}^n\mathrm{Do}{U}_k\le 1$$

(17)

Fatigue strength of metro bogie under random load

Bogie frame characteristics under random load

To study the fatigue strength of the welded frame of metro bogies under random loads, a multi-body dynamics model of the vehicle system was established. When the vehicle passed an S-shaped curve with a radius of 450 m and a high-level track irregularity, the time histories of the vertical load between the wheel and rail, the vertical and lateral loads of the air spring, the lateral stop and damper loads, and the twist load experienced by the bogie frame were calculated according to the method proposed by Zhu. ¹⁵

For the metro bogie frame as shown in Figure 4, it mainly bears the vertical and lateral loads of the air spring, the bump stop load, the damper load and the torsional load. Figures 5–9 show the time history of each load. As the vehicle passes the curve under the action of the unbalanced centrifugal force load, the air springs on the left and right sides of the same bogie will be loaded or unloaded, respectively, with the vertical load of the air springs taking on an opposite variation tendency. The lateral loads of the air springs on both sides have the same variation tendency, while the lateral stop load has a significant pulsation cycle characteristic. Mainly affected by the irregular tracks, the torsional load and damper load borne by the bogie frame are randomly distributed.

Figure 4.

Metro bogie frame and its loads.

Figure 5.

Time history of the vertical load on the central spring.

Figure 6.

Time history of the lateral stop load.

Figure 7.

Time history of the lateral load on the air spring.

Figure 8.

Time history of the damper load.

Figure 9.

Time history of the torsional load.

Regular hexahedral elements were used for the finite element model (Figure 10). Since the analysis is based on the nominal stress method, it is not necessary to simulate the exact shape of the welded joint when building the finite element model. According to the recommendations of DVS, the stress at a certain distance, which is 1–1.5 sheet thickness from the weld toe, is used as the nominal stress of the joint. ¹⁶ A stress test was carried out on a bogie frame based on the vertical, lateral, and torsional loads specified in EN 13749 “Railway applications-Wheelsets and bogies-Method of specifying the structural requirements of bogie frames.”Table 2 shows the simulation and test results of two measurement points at the welded joint between the transom and inner web of the side frame, as well as two measurement points on the lower cover plate of the side frame. For the single-axis gauges or three axes rosettes, the stresses in the longitudinal direction of the strain gauges or the von Mises stresses are used for comparison. By comparing the results, it was found that for the above-mentioned four measurement points, the results obtained by the finite element model were larger than the test results under the corresponding working conditions, with a maximum relative error of 24.12%. The model could effectively predict the stress distribution of the structure under various loads. Based on this model, the distribution of stress on the bogie frame under a unit load for various loads was obtained and converted into the welded joint coordinate system.

Figure 10.

Finite element model of the bogie frame.

Table 2.

Finite element results and that of the stress test.

Loads
	Finite element results (MPa)	Stresstestresults(MPa)	Relativeerror (%)	Finite element results (MPa)	Stresstestresults (MPa)	Relative error (%)	Finite element results (MPa)	Stresstestresults (MPa)	Relative error (%)	Finite element results (MPa)	Stresstestresults (MPa)	Relative error (%)
Vertical loads	20.84	18.60	12.04	23.84	21.00	13.52	74.58	66.20	12.65	66.91	63.10	6.04
Vertical loads and lateralloads	16.25	13.50	20.37	33.95	32.20	5.43	73.74	67.50	9.25	72.49	69.25	4.68
Vertical loads and lateral loads in the opposite direction	22.99	22.10	4.03	52.15	46.90	11.19	67.98	65.85	3.24	60.34	57.90	4.21
Vertical loads and torsional load	26.68	25.60	4.22	40.22	38.10	5.56	73.64	71.00	3.72	68.38	65.10	5.05
Vertical load and torsional load in the opposite direction	30.44	28.40	7.18	14.77	11.90	24.12	57.67	54.90	5.04	57.68	51.40	12.21

Based on the above-mentioned method, the time histories of the stress components of each node were generated. Figure 11 shows the time history of each stress component of node 196178 at the welded joint between the transom and inner web of the side frame. The time histories of the three stress components of this node exhibited random distributions. The stress component ${\sigma }_{x\prime }$ was mainly affected by the vertical load, so its distribution characteristics were similar to those of the vertical load of the air springs. The stress component ${\sigma }_{y\prime }$ was jointly affected by the vertical and torsional loads of the air springs. The torsional load borne by the bogie frame was the main influencing factor of the stress component ${\tau }_{x\prime y\prime }$ . Therefore, the time history of the stress component ${\tau }_{x\prime y\prime }$ was similar to that of the torsional load. Varying independently with time, the three stress components exhibited non-proportional variations.

Figure 11.

Time history of each stress component of the sample node. (a) Time history of the stress component ${\sigma }_{x\prime }$ ,(b) time history of the stress component ${\sigma }_{y\prime }$ , and (c) time history of the stress component ${\tau }_{x\prime y\prime }$ .

Fatigue strengths of welds using different methods

The time history of the stress components of each node were processed, and the node’s degree of utilization of the typical welds of the bogie frame were calculated based on the method proposed by the IIW Recommendation (hereinafter referred to as the “IIW method”) and the method recommended by the FKM Guideline (hereinafter referred to as the “FKM method”). Figures 12–17 show the degree of utilization of the welds between the outer web of the side frame and the upper and lower cover plates (hereinafter referred to as “weld A” and “weld B”), the welds between the inner web and the upper and lower cover plates (hereinafter referred to as “weld C” and “weld D”), and the welds between the transom and the outer and inner webs of the side frame (hereinafter referred to as “weld E” and “weld F”). These welds are marked in Figure 3.

Figure 12.

Distribution of node’s degree of utilization of weld A.

Figure 13.

Distribution of node’s degree of utilization of weld B.

Figure 14.

Distribution of node’s degree of utilization of weld C.

Figure 15.

Distribution of node’s degree of utilization of weld D.

Figure 16.

Distribution of node’s degree of utilization of weld E.

Figure 17.

Distribution of node’s degree of utilization of weld F.

The results showed a significant difference between the results obtained by the FKM Guideline and the IIW method. For all welded joints except welded joint B, the node degree of utilization obtained by the FKM method was greater than that gained by the IIW method. In the middle part of welded joint A, there was the largest difference between node degree of utilizations obtained using the two methods. For node 159533, the node degree of utilization obtained using the FKM method was 221.62% larger than that obtained using the IIW method. For most areas of welded joint B, the node degree of utilization obtained based on the IIW method was larger than that obtained by the FKM method. Considering node 169910 in the middle as an example, the node degree of utilization obtained based on the IIW method was 39.15% larger than that obtained by the FKM method. The FKM Guideline applied a simple but overly conservative method for evaluating the structural fatigue strength in the non-proportional stress state, which was the main cause of the above difference.

Under a single load, the time history of each stress component of the weld joints changed synchronously. According to the FKM Guideline, the node degree of utilization should be calculated based on the maximum principal stress criterion. For structures in the non-proportional stress state, the FKM Guideline suggests that the node degree of utilization under a single load should be calculated first, after which the structural fatigue strength should be evaluated through linear superposition based on the algebraic sum of the node degree of utilization under each load. If the time history of each stress component of the nodes continues to change synchronously under different loads, the time histories of the three stress components calculated according to equation (10) under the integrated action of various loads also change synchronously. At this time, the obtained node degree of utilizations will have the same value, whether it is calculated based on the stress-time history under the integrated action of various loads or based on the stress-time history under each load alone. The difference between the node degree of utilization obtained using the IIW method and that obtained using the FKM method is determined only by the criterion value, CV. For general cases, with the simultaneous action of different loads, the loads are not correlated with time. Under the combined action of various loads, the time histories of the three stress components of the nodes exhibit non-proportional variations. If the calculation is based on the superposition method determined by the FKM Guideline, the time history of the stress component under different loads should be characterized by proportional variations. The node degree of utilization obtained based on this simplified method tends to be conservative. The more significant the synchronism is between the time histories of the three stress components obtained according to equation (10), the closer the results obtained by the FKM method are to the actual fatigue damage level of the welded joints. Hence, without considering the criterion value, CV, the node degree of utilization obtained based on the FKM method will be larger than that obtained based on the IIW method.

For some nodes, the criterion value $\mathrm{CV}$ will change the distribution characteristics of the results obtained based on different methods. For the nodes in most areas of welded joint B, the vertical load of the air springs is the primary factor affecting the structural fatigue strength, whereas the other loads contribute less to the node degree of utilization. Therefore, under the joint action of various loads, the time histories of the three stress components are well synchronized. Without considering the criterion value $\mathrm{CV}$ , the node degree of utilizations obtained by the FKM method and the IIW method are quite similar. When CV = 1.0, the node degree of utilization of weld B is shown in Figure 13. Considering node 169910 as an example, there was only a relative error of 1.60% between the results obtained based on the different methods when the influence of CV was not considered. After introducing $\mathrm{CV}=0.5$ , the node degree of utilization obtained based on the IIW method was larger than that obtained using the FKM method.

Conclusion

1. Accounting for the influence of small amplitude cycles on structural fatigue damage, the S–N curve recommended by the FKM Guideline and the equivalent constant amplitude stress calculation method were corrected.

2. Based on node stress components in the weld coordinate system, a method for evaluating the fatigue strengths of bogie frames under random loads was determined based on the IIW and FKM methods to evaluate the fatigue strengths of typical welds. For different welded joints, there were differences in the distribution of node degree of utilizations obtained based on the different methods.

3. For welded joints with significant non-proportional characteristics between stress components, the node degree of utilization obtained using the FKM method was larger than that obtained using the IIW method. When the stress components of the weld joints were synchronized, the difference between the node degree of utilizations obtained using the FKM and IIW methods was mainly determined by the criterion value CV.

4. The FKM Guideline provides a conservative approach to structural fatigue strength analysis. The fatigue test results of typical welded joints confirm the effectiveness of the IIW method. Therefore, the IIW method is used in the fatigue strength analysis of the welded structure under the spectrum load, which can effectively balance the lightweight design and reliability of the structure.