Abstract
Effectively evaluating the fatigue behavior of aging bridge structures is an urgent task for engineers and researchers, as more and more existing bridges approach the end of their service life. Determining the fatigue load effects due to moving trains is a key aspect of the available assessment frameworks (e.g., the SN approach). In this paper, a vehiclebridge system model is integrated into the fatigue assessment procedure employing SN lines. The system model is used to obtain fatigue stresses generated by moving train loads. The integration is proven able to aid in effective prediction of fatigue damage and remaining fatigue life of railway bridges.
1. Introduction
More and more existing bridges are approaching the end of their service life worldwide. In this context, it is an urgent task for engineers and researchers to effectively evaluate the fatigue behavior of these aging structures. Determining the load effects (i.e., fatigue stresses) due to moving trains is a key aspect of the fatigue assessment of railway bridges. Even small differences in stress predictions can affect fatigue life estimates considerably, which can lead to inappropriate decisions regarding assessment and maintenance of existing bridges [1]. Consequently, it is crucial for the reliable assessment to obtain fatigue stress ranges which reflect the real load effects as much as possible.
In addition, significant dynamic effects have been caused in trainbridge systems, since increasing axle load and train speed were implemented for the railway transportation in many countries. These effects have to be accounted for in the fatigue evaluation, as coupling vibrations of the train and bridge can have a considerable impact on the stress response of bridge members. Recently, a numerical approach for the stress analysis of railway bridges based on trainbridge coupled dynamics has been proposed in Li, et al. [2].
In this paper, a coupled vehiclebridge system model is integrated into the fatigue assessment procedure employing SN lines. The system model developed is verified by field data from a steel railway bridge and utilized to compute the fatigue stresses of critical members. Fatigue damage and remaining fatigue life of the investigated bridge detail are estimated with various SN lines provided by current codes.
2. SN approach
Over the last several decades, the SN approach, with the use of the Miner’s rule, has been well developed and adopted to design and assess fatigueprone bridge components. An SN line defines the relationship between the constantamplitude stress range due to external loading and the applied number of stress cycles which lead to the fatigue failure of structural details. Several fatiguerelated codes proposed SN lines for different categories and/or classifications of commonly used bridge details in engineering practice [36]. A singleslope SN line can be expressed as [3]:
where $\mathrm{\Delta}\sigma $ is the constantamplitude stress range; $N$ is the applied number of stress cycles corresponding to a specific $\mathrm{\Delta}\sigma $; C and $m$ are, respectively, the fatigue detail coefficient for each category and material constant. The material constant $m$ represents the slope of the SN line on logarithmic scale, while $N$ indicates the fatigue resistance and/or life of the specific structural detail when the constantamplitude stress range $\mathrm{\Delta}\sigma $ is applied.
For a bilinear SN line, Eq. (1) can be revised as [4]:
where CAFL is the constantamplitude fatigue limit corresponding to a specific number of cycles $N$; ${m}_{1}$ and ${m}_{2}$ are, respectively, the slopes of the SN line above and below the CAFL.
Combining the SN line (i.e., Eq. (1)) and the Miner’s rule, the cumulative fatigue damage caused by variableamplitude stress ranges can be given by:
where $D$ is the fatigue damage accumulation index; ${n}_{i}$ is the number of applied loading cycles in the predefined stress range bin $\mathrm{\Delta}{\sigma}_{i}$; ${N}_{i}$ is the critical number of cycles under stress range $\mathrm{\Delta}{\sigma}_{i}$.
Based on the equivalence principle, the equivalent stress range ${S}_{re}$ can be determined, which results in the same fatigue damage as that caused by all the variableamplitude stress ranges $\mathrm{\Delta}{\sigma}_{i}$ in a stress range bin histogram. Considering Eqs. (1) and (3), ${S}_{re}$ for singleslope SN lines can be expressed as:
where $\sum {n}_{i}$ is the total number of stress cycles in a stress range bin histogram. Similarly, ${S}_{re}$ for the bilinear SN lines can be given by:
where ${n}_{j}$ is the number of cycles in the stress range bin $\mathrm{\Delta}{\sigma}_{j}$, which is greater than the CAFL; ${n}_{k}$ is the number of cycles in the stress range bin $\mathrm{\Delta}{\sigma}_{k}$ smaller than the CAFL; and $\sum {n}_{j}^{}+\sum {n}_{k}^{}$ is the total number of stress cycles in the stress range bin histogram. Consequently, the fatigue damage index $D$ for both singleslop and bilinear SN lines can be calculated by a uniform formula as:
where $\sum n$=$\sum {n}_{i}$ for Eq. (4) or $\sum {n}_{j}^{}+\sum {n}_{k}^{}$ for Eq. (5); and $m={m}_{1}$ in bilinear SN lines.
3. Dynamic analysis of trainbridge system
In order to obtain the dynamic stresses induced by moving trains for the fatigue assessment, a coupled vehiclebridge system model is herein developed and verified with the use of field data from a steel truss bridge, i.e., the Baihe Bridge in China.
3.1. Vehiclebridge system model
The coupled vehiclebridge system model consists of a 3D vehicle model, a 3D bridge model, and a wheelrail interaction model. The carbody, bogie, and wheelset are considered as rigid bodies. These rigidbody components are connected by the primary and/or secondary suspension systems, which are modeled as springs and dashpots. To take into account the spatial actions on the bridge structure by vehicles, it is assumed that each carbody or bogie has five DOFs (degreesoffreedom), i.e., swaying, rolling, yawing, floating, and pitching; while each wheelset has four DOFs, i.e., swaying, rolling, yawing, and floating. Consequently, the equation of motion of the vehicle model can be established by employing the Lagrange’s equation [7].
The bridge model is established by using the finite element (FE) method, and the equation of motion of the bridge is formulated through the direct stiffness method (DSM). Using original mass, stiffness, and damping matrices of the structure in geometric coordinates, the DSM can avoid the selection of bridge mode shapes and take into account contributions of all the modes to the dynamic stress response.
The wheelrail interaction model defines dynamic forces applied on each other by the vehicle and bridge subsystems. In this study, it is assumed that in the vertical direction, the motion of a wheelset can be considered as the superposition of motion of the bridge deck and track irregularities. As a result, the floating and rolling movements of a wheelset are not independent and are related to the bridge movements. In the horizontal direction, the Kalker’s creep theory, indicating that the lateral wheelrail interactions rely on their relative motion, is adopted. Based on the abovementioned assumptions, considering the equilibrium of forces and compatibility of displacements at the wheelrail contact, the dynamic equation of motion of the coupled vehiclebridge system can be expressed as:
where $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ are, respectively, the mass, damping, and stiffness matrices; ${\mathbf{X}}_{v}$, ${\dot{\mathbf{X}}}_{v}$, and ${\ddot{\mathbf{X}}}_{v}$ represent, respectively, the displacement, velocity, and acceleration vectors; $\mathbf{F}$ is the vector of the dynamic force; the subscripts $v$ and $b$ stand for, respectively, the vehicle subsystem and the bridge subsystem. Eq. (7) can be solved by using the Newmark$\beta $ method.
3.2. Fatigue stress
After obtaining the nodal displacements of the bridge (i.e., ${\mathbf{X}}_{b}\left(t\right)$ in Eq. (7)), the dynamic element stresses can be calculated as:
where ${\mathbf{S}}_{b}\left(t\right)$ is the vector of dynamic stress time histories; $\mathbf{\epsilon}\left(t\right)$ is the vector of dynamic strains; $\mathbf{E}$ is the elastic matrix, which depends on material properties of the bridge structure in question; and $\mathbf{B}$ is the displacementstrain matrix, which can be evaluated through derivative operations of the element shape functions with respect to geometric coordinates.
In order to estimate the fatigue damage accumulation and fatigue life, stress time histories obtained by Eq. (8) have to be processed. In the present analysis, the rainflow counting method is applied to the raw stress data to generate stress range bin histograms which are next used for the fatigue assessment.
3.3. Verification of the system model
The Baihe Bridge is a continuous steel truss structure with three spans, 128 m each. The two main trusses are 13.4 m high and 8.0 m apart (see Fig. 1). As part of the BeijingTongliao Railway Line opened for traffic in 1980, it spans the Miyun Reservoir in the northeast of Beijing. In 2011, a routine inspection was performed on the bridge by the Beijing Railway Bureau, in association with Beijing Jiaotong University. The inspection includes modal analysis, and static and dynamic load tests.
Fig. 1Positions of bridge members for measurement in the field test
By analyzing the ambient vibration data, two fundamental vibration modes were identified, namely, the first lateral and vertical bending modes. The corresponding natural frequencies are, respectively, 1.04 Hz and 1.55 Hz [8]. In addition, the bridge FE model was established using BEAM 188 elements in the software ANSYS. In the bridge, high strength bolts are utilized for member connections, which are assumed as fixed in the FE model. In addition, considering that concrete piers and abutments of the bridge are much more rigid compared to the truss, only the superstructure was modeled. The calculated natural frequencies of the two fundamental modes are found to be 1.03 Hz and 1.55 Hz. Thus, very good agreement is achieved between the measured and computational modal parameters.
The dynamic stresses of major bridge members were monitored during the passage of a passenger train (see Fig. 1). This specific train, consisting of one locomotive and 12 passenger cars, passed through the bridge at the speed of 67 km/h. The length of the train is 340 m, and static axle loads of the locomotive and passenger car are, respectively, 230 kN and 154 kN.
The Class 5 track irregularities proposed by the U.S. Federal Railway Administration were included in the vehiclebridge system model to predict the stress time histories. Fig. 2 shows the predicted and measured stresses in bridge members highlighted in Fig. 1. The field data have been processed with a low pass filter of 30 Hz. It can be observed that for all the investigated members, good agreement is achieved between the numerical result from the vehiclebridge system model and the measured counterpart from the field inspection. This coincidence indicates that the system model established is capable of accurately predicting the dynamic stress response of railway bridges under moving train loads.
4. Results: fatigue damage and remaining life
Fatigue trains running in the history of the Baihe Bridge are identified based on the traffic survey in [9]. According to this survey, passenger trains, consisting of 1 locomotive and 20 cars, passed the bridge 22 times per day, while freight trains, composed of 2 locomotives and 43 cars, passed the bridge 28 times per day. Considering the movement to raise the speed of railway travel in China which started in 1997, the loading history of the Baihe Bridge is divided into two different periods, namely, period I (19801997) and period II (19982014). Moreover, it is assumed that speeds of the passenger and freight trains during time period I are, respectively, 50 km/h and 40 km/h, whereas the corresponding values for time period II are 70 km/h and 60 km/h, respectively. These speeds, along with the Class 5 track irregularities, are used to perform dynamic analyses of the coupled trainbridge system.
Fig. 2Predicted and measured dynamic stress time histories in bridge members
a) Upper chord
b) Diagonal
c) Vertical web member
d) Lower chord
e) Stringer
f) Floor beam
It can be observed from Fig. 2 that the stringer is the fatiguecritical member, experiencing most significant stress variations. The associated structural detail is the high strength bolted floorbeamtostringer connection. The fatigue resistance of this detail can be represented by SN lines proposed in several current codes (see Fig. 3).
Fig. 3SN lines in different codes for the high strength bolted connection
The accumulated fatigue damage can be calculated by using Eq. (6). Based on the obtained damage index, the remaining fatigue life of the investigated detail is estimated, assuming that no traffic evolutions occur in the future (i.e., after 2014). The associated results are reported in Table 1. Significant differences in the predicted fatigue life can be observed. Most of the stress ranges in the Baihe Bridge generated by passenger and freight trains lie in the range of 030 MPa, where SN lines in Fig. 3 significantly differ from one another. In particular, the British line leads to over 500 years of remaining life, which is far more than those from the other three lines. This indicates that SN lines for fatigue evaluation should be properly selected, since they can have a significant impact on the results.
Table 1Fatigue damage and remaining life of the stringer
SN line  Fatigue damage per year  Remaining life  
Period I (19801997)  Period II (19982014)  
China [3]  0.01979  0.01245  35 
UK [4]  0.00304  0.00075  >500 
EU [5]  0.01633  0.00504  123 
USA [6]  0.01324  0.00833  74 
5. Conclusions
This paper presented a procedure for fatigue assessment of steel railway bridges, integrating a vehiclebridge system model into the identification of load effects. The model was verified with the field data obtained from a real bridge, and good agreement is achieved. By employing the verified model for coupled dynamic analyses of the trainbridge system, fatigue stresses were computed. The fatigue damage and remaining fatigue life of the bridge were subsequently estimated based on the SN approach. The inclusion of the system model enables accurately predicting the fatigue load effects, and therefore, effectively assessing fatigue behavior of existing bridges. In addition, SN lines can have a significant impact on the projected fatigue life. Consequently, they should be properly selected.
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About this article
This study was sponsored by the Fundamental Research Funds for the Central Universities (Grant No. C15JB00300) of China.