Structural behavior analysis of a continuous steel truss arch railway bridge integrating monitoring data

Abstract

Using theoretical modeling combined with monitoring data, the typical dynamic and static behaviors of a continuous steel truss arch railway bridge are evaluated. The dynamic behavior involves an impact factor induced by the action of running trains, and the static behavior refers to axial bending performance and the stress distribution among different planes of the truss. The transverse position, length, and speed of running trains are introduced to conduct an analysis of their influences on the dynamic and static behaviors of the bridge superstructure. A structural safety evaluation is also conducted by comparison with the provisions recommended by design codes and by analysis of absolute stress. It is concluded that three types of members present different dynamic behaviors and that the value of the impact factor for chords B exceeds the provision recommended by the design codes. Chords C present the greatest ratio of bending stress versus axial stress. An imbalance of stress distribution exists among the three planes of the truss, and the difference is the smallest when trains run along the middle railways. Because the train-induced stress is considerably lower, the dynamic and static performances of the bridge are within the scope of safety.

Keywords

Railway bridges steel truss arch impact factor axial bending behavior stress distribution

Introduction

The construction of high-speed railways is limited by a variety of environmental factors. Various bridges have to be constructed to overcome factors such as rivers and valleys. Unlike highway bridges, the requirements for high-speed railway bridges are considerably stricter with respect to the stiffness and stability of the superstructure. The parameter of span range is one of the typical factors that determine the design of bridge superstructures. Special spans can be divided into two categories based on the length of the main span: medium length (100–200 m) and large length (200–500 m).¹ Tied steel arch, rigid frame, and arch-rigid frame hybrid are three commonly used structural forms for medium-length spans.^2,3 Compared with medium-length bridges, the main girder of a long-span bridge is known to be considerably more flexible. The structural forms of steel truss arch and cable-stayed are two means to overcome the problem.⁴

In this article, the structural behaviors of a steel truss arch system are studied by integrating the typical dynamic and static properties of strain responses. Dynamic behavior has attracted more attention for bridges with large span lengths.⁵ Due to the higher serviceability limits compared to conventional railway bridge design, the technical issues associated with the dynamic response of high-speed railway bridges have been investigated in many previous studies, such as seismic performance,^6–10 track structure, interaction,^11,12 creep effect,^13,14 and thermal effect.¹⁵ However, few investigations have focused on the effect of the impact factor by comparing with and further verifying the design codes. The impact factor is calculated to evaluate the dynamic magnitude effect of stress under moving loads in the design period. Therefore, the responses of stress calculated based on static load have to be multiplied by the impact factor. Various investigations have been conducted on the dynamic behavior of many forms of bridges. Related forms of bridges include simple railway bridges with a steel plate girder, multi-girder concrete bridges, continuous bridges with a skew box girder, composite concrete-steel cellular straight bridges, and curved continuous bridges with a composite multiple-box girder.^16–20 However, the newly emerging bridge form of steel truss arch has not been investigated because of its complex structural composition. Therefore, different types of members may present distinct dynamic behavior performance. In addition to its property of complex structural composition, the act of trains running at high speed and different transverse positions are another two significant influencing factors.

Because of the complex structure composed of members distributed spatially, the static behavior of the steel truss arch system is more difficult to understand and master. Some investigations have been conducted to obtain the stress distribution under dead loads and the rules for how stiffness variation influences the stress redistribution.^21,22 However, few studies have been conducted on the stress distribution under live loads. For railway bridges, running trains are one of the most significant live loads. Thus, the mechanism of stress distribution under running trains must be taken into consideration.

The Dashengguan Railway Bridge is selected as the case study. This bridge spans across the Yangtze River in China, and its superstructure is a steel truss arch system. To investigate the performance under actual service conditions, structural monitoring systems have been installed at typical locations for online evaluation of various structural behaviors. By analyzing the characteristics of strain curves of different members under various running trains, this article specifically focuses on the dynamic and static behaviors of the superstructure through monitoring combined with finite element (FE) modeling. Investigation is first conducted to obtain the impact factor property of different members under various running trains. Another investigation involves the axial bending behavior of typical members under running trains either. In addition, the stress distribution among three planes of the truss is finally investigated to verify integrality of the spatial structure. Influencing analysis is also involved during above investigations. Related parameters include the length, the transverse position, and the speed of running trains.

Bridge description and its structural health monitoring system

Description of Dashengguan Railway Bridge

Figure 1 presents the actual view of the Dashengguan Railway Bridge as well as the side and cross-sectional views with the geometrical dimensions in detail. The Dashengguan Railway Bridge provides a passageway with four railway lines for trains along the Beijing–Shanghai and Shanghai–Wuhan–Chengdu railways. In addition, the bridge will support an urban light rail with two lines for Nanjing City in the future. Figure 1(c) shows the transverse distribution of the aforementioned six railway lines. In addition, there are two types of high-speed trains that run along these four rails with 8 and 16 carriages. The design speeds for trains along the Beijing–Shanghai and Shanghai–Wuhan–Chengdu railway lines are 300 and 200 km/h, respectively. Additionally, the design speed for the urban light rail is 80 km/h.

Figure 1.

Dashengguan Railway Bridge: (a) actual view, (b) side view with geometrical dimensions (unit: m), and (c) cross-section configuration (unit: m).

A continuous steel truss arch was adopted as the superstructure of the bridge. There are six spans that compose the main bridge in the longitudinal direction. The distribution of spans is symmetrical, and the lengths of the half-side spans are sequentially 180, 192, and 336 m from the side to the middle. In addition, the bridge superstructure consists of three planes of steel truss arches by connection of transverse trusses. A two-tube combined truss is applied as the arch ring of the steel truss arch. The maximum height of the combined truss is 16.0 m at the mid-span section. The ratio of maximum height versus span length is 0.25. Vertical steel hangers are applied to connect the steel truss arch with the chord members supporting the bridge deck. An orthotropic steel deck is adopted as the bridge deck, and it is directly welded to the chord members. Three grades of steel material are applied to produce the steel members. The members of the main trusses are applied with the steel material of 370 MPa.

Overview of structural monitoring system

To monitor the structural response and the environmental actions, a structural monitoring system was equipped during the construction period. The monitoring data include wind speed, temperature, strain, displacement, and accelerated velocity. In this article, strain data in the mid-span cross section are acquired and applied for the evaluation of the dynamic and static behaviors. The location of the mid-span cross section is presented in Figure 1(b).

Figure 2 presents the configuration of the strain sensors at the section. For each truss member, two strain sensors are installed at the same cross section, such as DYB-11-11 and DYB-11-12. These two strain sensors are in the same horizontal plane. Fiber Bragg Grating (FBG) strain sensors are applied for strain measurements. Acquisition instruments with a recording rate of 50 samples per second are used for collecting the dynamic strain data.

Figure 2.

Configuration of strain sensors at the mid-span section: (a) cross-section diagram, (b) 1-1 section, and (c) cross section of truss members.

Measured strain curves of typical members

The strain curves of typical members are analyzed first in this section. Typical members include the chord members of three layers, which are labeled as chords A, B, and C from top to bottom. Figure 3 presents the distribution of railway lines and the locations of the three-layer members. Railways are named as Railways 1, 2, and 3 and from the downstream direction to the upstream direction. Members are also labeled with suffixes of 1, 2, and 3 in the same order. In the figure, chords A belong to the highest layer, such as members gauged with DYB-11-11 and DYB-11-12. Chords B are the second highest layer, such as members gauged with DYB-11-17 and DYB-11-18. Finally, chords C are located in the same level of the bridge deck, such as members gauged with DYB-11-21 and DYB-11-22.

Figure 3.

Distribution of railway lines and label of truss members.

Using members in the side plane as examples, Figures 4 and 5 present typical strain curves of A3, B3, and C3 under 8-carriage and 16-carriage trains. The speed of these two types of trains is uniformly 160 km/h. The total length and standard gross weight of each carriage are, respectively, 25.0 m and 535 kN. In the figure, the strain curves of A3 are the simplest, with only one U-shaped curve. Noting that the minimum strain values are approximately −31 µε and −32 µε under 8- and 16-carriage trains, respectively, the extreme value of the strain curve is independent of the train length.

Figure 4.

Strain curves of (a) A3, (b) B3, and (c) C3 under 8-carriage train.

Figure 5.

Strain curves of (a) A3, (b) B3, and (c) C3 under 16-carriage train.

Compared with the strain curves of A3, the strain curves of B3 are more complex. The number of peaks in the strain curves is considerably greater. In addition, the overall tendency is similar between the curves induced by 8- and 16-carriage trains. However, the extreme values of the strain peaks are approximately 27.0 and 35.0 µε under 8- and 16-carriage trains, respectively. Thus, longer trains may induce a greater value of the strain peaks for B3.

As shown in Figures 4(c) and 5(c), the action of running trains induces many strain peaks for C3. Under the action of a single 8-carriage train, the number of strain peaks is approximately 9.0. This value is 17.0 when under the action of a 16-carriage train. The maximum strain values are approximately 21.0 and 25.0 µε under 8- and 16-carriage trains, respectively. Similar to B3, a greater train length may induce a considerably higher value of strain. In addition, the number of strain peaks corresponds to the number of carriages well. Thus, the strain data of chords C can be applied to identify the types of trains. The difference for C3 is the most significant that spurs of dynamic strain corresponds to the number of wheels much better. The reason is that the length of the influence lines of chords C is considerably shorter than chords A and B. Therefore, the action of each axle induces a single strain peak.

Dynamic behavior may be observed from the profile of the strain curves. Under the action of running trains, the strains of different members fluctuate with distinct regulations. Because the impact factor indicates the extreme value versus the mean value, B3 may present the greatest dynamic magnitude of strain response. According to the design codes, members that belong to the same span are defined with a unique impact factor value. Thus, the dynamic property of different members needs to be further evaluated.

Monitoring of the strain curves indicates that a train-induced strain of one cross section is observable at three types of members. For slender steel members, additional bending stress may induce worse performance of static-load capacity or stability. This indicates that the bending stress cannot be eliminated and should be evaluated in detail.

FE model for dynamic and static evaluations

FE modeling is introduced for dynamic and static evaluation for strain response. First, the primary FE model of the bridge superstructure is established according to the design drawing, as presented by Figure 6. In the figure, the model of the bridge superstructure is created by commercial FE software ANSYS 15.0. The truss members of the model are simulated by Beam 188, and the bridge deck is simulated by Shell 118. The elastic modulus and Poisson ratio of the steel are selected as 206 GPa and 0.30, respectively. In addition, the damping ratio of the structure is 0.03. Method of modal superposition is used for calculation of train-induced responses.

Figure 6.

Finite element model of the Dashengguan Railway Bridge.

The load model used for the FE model analysis consists of a series of moving loads. The time of an integral step is 0.00758 s. Node forces in two rows are introduced to simulate the train actions from wheels upon the bridge deck. Node forces consist of two parts: dead loads of the carriage weight and live loads of containing passengers. The total weight of one carriage is applied as 535 kN, which includes the self-weight of the train and live load of the passengers. The average weight of each passenger is adopted as 650 N, and the number of total passengers is applied as 200 persons per carriage.

The restrain of the superstructure from the substructure is simulated as fixed for simplification. Similar to the previous sections, only the calculated strain curves of A3, B3, and C3 under 8-carriage train are selected and presented for illustration. Figures 7 –9 present the static strain and the dynamic strain from the measured and calculated results for positions equipped with strain sensors. In the figures, the static strain is obtained by processing the monitored data using a low-pass filter. The selected filter order and cut-off frequency are 3.0 and 2.5, respectively. Without considering the stiffness of the bearing supports, the calculated strain results fit well with the measured data. Therefore, the FE model is utilized for the dynamic and static evaluation of the Dashengguan Railway Bridge.

Figure 7.

Comparison between calculated and monitoring results of A3: (a) static strain and (b) dynamic strain.

Figure 8.

Comparison between calculated and monitoring results of B3: (a) static strain and (b) dynamic strain.

Figure 9.

Comparison between calculated and monitoring results of C3: (a) static strain and (b) dynamic strain.

Dynamic behavior of continuous steel truss arch

Impact factor calculation method based on strain curves

The structural response may be considerably greater under moving loads than under static loads with a similar value. This phenomenon is the so-called dynamic behavior of a structure. The impact factor is generally applied to profile the dynamic behavior of a structure by comparing the structural stress under a dynamic load with the responses under an equivalent static load.²³ The impact factor is generally defined as follows

D L F = 1 + D L A

(1)

where DLF denotes the dynamic load factor and DLA denotes the dynamic load allowance given by

D L A = \frac{R_{dyn} - R_{stat}}{R_{stat}}

(2)

where R_dyn denotes the maximum amplitude of dynamic strain data and R_stat denotes the maximum amplitude of static strain data.

The measured strain data consist of static load–induced strain and the dynamic amplitude. Based on the distinction of frequency between dynamic and static strain data, a low-pass filter is applied to eliminate the dynamic components of strain data and obtain the static component. The filter function is applied as Wavelet Daubechies. The sampling frequency and the frequency of Nyquist is 50 and 25 Hz. The number of layers is selected as 2. The selected filter order and cut-off frequency are 3.0 and 2.5 because of the lowest natural frequency of this bridge. As an example, Figure 10 presents an original dynamic strain curve and its filtered curve.

Figure 10.

An example of dynamic strain curve and filtered curve.

The impact factor can be calculated by applying equations (1) and (2). For FE evaluation, the factor can be directly obtained by comparison of the dynamic and static calculations. Taking the 8-carriage train with a speed of 240 km/h as an example, the impact factors are calculated under four different railways for three types of members. In addition, monitoring strain data of trains with a speed of approximately 240 km/h are selected to calculate the impact factor. Table 1 presents the average impact factors calculated using the monitoring strain data and the FE evaluations. As shown in this table, the calculated impact factors of A3, B3, and C3 correspond well with the monitoring data. Based on the FE model, further investigation is conducted to acquire the dynamic behavior of the bridge superstructure.

Table 1.

Comparison of impact factors between the theoretical and monitoring results.

Member	Railway 1		Railway 2		Railway 3		Railway 4
Member	Monitoring results	Calculated results	Monitoring results	Calculated results	Monitoring results	Calculated results	Monitoring results	Calculated results
A3	1.0087	1.0099	1.0062	1.0091	1.0058	1.0098	1.0055	1.0093
B3	1.1058	1.1119	1.0925	1.0976	1.0928	1.0989	1.0285	1.0758
C3	1.0287	1.0426	1.0105	1.0441	1.0094	1.0097	1.0105	1.0272

Dynamic behavior of the main truss members

Table 1 also shows that the impact factors of different members are distinct. Among the three different members, the impact factor of B3 is the greatest, reaching almost 1.06. A3 possesses the lowest impact factor value. Thus, the member of chords B should be given more attention. Additionally, the table also shows that the impact factor value is significantly affected by the positions of the railways. With the transverse positions of running trains approaching the truss members, the impact factor is considerably greater. However, the difference of impact factors between trains in different railways is not very great.

An investigation is also conducted to obtain the influence of train length on the impact factor. Assuming that the speed of the train is 240 km/m, Figure 11 presents the calculated results of the impact factors for 8- and 16-carriage running trains. As shown in this figure, a greater number of carriages induces a smaller impact factor for all three types of members. However, the influence of train length is limited. For chord B3, whose impact factor is the most sensitive to the train length, the greatest difference between the two types of trains is only approximately 0.03.

Figure 11.

Impact factors under different numbers of train carriages: (a) chord A3, (b) chord B3, and (c) chord C3.

It is commonly known that faster trains induce more serious dynamic behaviors. Figure 12 presents the measured running speeds of the trains. As shown in this figure, the majority of the speed of running trains distributes within the range from 160 to 240 km/h. Thus, these two speeds, 160 and 240 km/h, are set as the moving speed of the node force for the dynamic calculation. Taking the 8-carriage train as an example, Figure 13 presents the calculated impact factor for the train with speeds of 160 and 240 km/h. As shown in this figure, the value of the impact factor for the three members under the train speed of 240 km/h is significantly greater than that under 160 km/h. Similarly, the impact factor of B3 is the most sensitive to the speed of the running train. The maximum value is approximately 0.042 between the two aforementioned different speeds. This result indicates that the speed of running trains plays a significant role in the dynamic behavior of the truss arch system. Therefore, the influencing factor of train speed has to be considered for the dynamic evaluation of the bridge.

Figure 12.

Speed distribution of running trains.

Figure 13.

Impact factors under different speeds of running trains: (a) chord A3, (b) chord B3, and (c) chord C3.

Comparison between designed codes and analytical results

According to most bridge design codes, the dynamic magnitude effect is taken into account by means of multiplying the impact factor with static response. However, the above design codes lack consideration of the individuation of different members. For simplification, the impact factor for a typical bridge is unique and only related to the span of the bridge.

However, the above investigation indicates that different members of the truss arch system possess distinct dynamic behaviors. Therefore, investigations have to be conducted to compare the monitoring-based calculation results with the design codes. The design of the Dashengguan Railway Bridge is based on the specification for the design of the Beijing–Shanghai high-speed railway. According to this code, the impact factor for a typical bridge is described as follows

D L F = 1 + \frac{28}{40 + L}

(3)

where L (in meters) denotes the length of the located span. Applying 336 m in the equation, the designed impact factor of the Dashengguan Railway Bridge is calculated and obtained. The calculated result of the designed impact factor is 1.0745.

It is observed that a shorter length and higher speed of the running train result in higher impact factor values. Table 2 presents the calculated impact factors of members under a single running train with eight carriages moving at 240 km/h. As shown in this table, the highest impact factor is 1.1212, and the lowest impact factor is 1.0064. By comparing the designed and calculated results, members of chords B have exceeded the limit established by the design codes. In addition, the dynamic behavior may deteriorate as the structural stiffness decreases and the train speed increases. Therefore, persistent monitoring and further dynamic evaluation are extremely essential for the safety of the main members.

Table 2.

Summary of impact factors under an 8-carriage train with a speed of 240 km/m.

Member	A1	A2	B1	B2	C1	C2
Railway 1	1.0074	1.0079	1.1162	1.1155	1.0121	1.0185
Railway 2	1.0069	1.0085	1.1083	1.1212	1.0109	1.0198
Railway 3	1.0067	1.0085	1.1057	1.1153	1.0096	1.0198
Railway 4	1.0064	1.0079	1.0981	1.1088	1.0074	1.0185

Axial bending and stress distribution behavior

Axial bending structural behavior of the trusses

Static behavior should be taken into account for structures with complex compositions. There are two aspects that should be given particular attention for the bridge. The first aspect is the axial bending behavior, and the second is the distribution of stress between the three different planes of the truss. To eliminate the influence of dynamic magnitude, a static load analysis was only conducted on the FE model.

According to the classic arch truss mechanism theory, the highest and second highest chords sustain axial forces that induce similar fiber strains at different edges. An additional bending moment is adverse for the stress state of the member of the truss arch. Imbalance of acting loads and spatial static behavior may induce unexpected bending stress. Taking the 8-carriage train as an example, Table 3 presents the maximum bending stress and the corresponding axial stress of the nine members. In addition, the ratio between the maximum bending and corresponding axial stress, which is labeled R, is also presented in this table.

Table 3.

Calculated axial stress and bending stress of members under 8-carriage train (units: MPa).

Member	Railway 1			Railway 2			Railway 3			Railway 4
Member	Bending	Axial	R (%)	Bending	Axial	R (%)	Bending	Axial	R (%)	Bending	Axial	R (%)
A1	−0.305	−2.74	11	−0.285	−2.56	11	−0.266	−2.39	11	−0.253	−2.29	11
A2	−0.293	−2.45	12	−0.296	−2.51	12	−0.295	−2.51	12	−0.293	−2.45	12
A3	−0.252	−2.29	11	−0.267	−2.39	11	−0.286	−2.56	11	−0.304	−2.73	11
B1	−0.152	−2.62	6	−0.127	−2.41	5	−0.11	−2.26	5	−0.109	−2.08	5
B2	−0.116	−2.26	5	−0.127	−2.31	5	−0.127	−2.33	5	−0.116	−2.29	5
B3	−0.099	−2.08	5	−0.111	−2.19	5	−0.127	−2.39	5	−0.151	−2.58	6
C1	1.36	3.8	36	1.7	1.31	130	1.11	1.25	89	1.02	1.21	84
C2	1.29	2.41	54	3.64	1.29	282	3.57	1.29	277	1.77	1.29	137
C3	1.21	0.93	130	1.1	1.25	88	1.78	1.31	136	3.61	1.36	265

As shown in this table, it is clear that different members possess distinctive axial bending behavior. For chords A and B, the R values are only approximately 11% and 5%, respectively, for all cases. Chords C are highly adverse in static performance because the value of R is considerably higher than that for the other members. The minimum and maximum values of R for chords C are 282% and 77%, respectively. This result indicates that chords C should be given more attention with respect to the static behavior combining axial and bending effects. Sustaining loads transferred directly by the bridge deck, chords C play roles in two levels of the structural system. The first role is acting as tension chords of the arch structure, and the second role is acting as the bending beams supporting the bridge deck. Therefore, the running trains induce a considerably bending moment, thereby making the value of R relatively large.

This table also shows that the value of R is relative to the transverse position of running trains. Chords C are the most sensitive to the positions of running trains. For members of C1, C2, and C3, the maximum values of R occur at running train positions of Railways 1, 2, and 4, respectively. Noting that these three railways are the closest railways to the corresponding members, the axial bending behavior of chords C is extraordinary susceptible to the transverse positions of running trains. In contrast, the sensitivity of R to the transverse positions of running trains is considerably smaller for chords A and B. The distinction among the four different railways is small enough to be neglected.

Although the value of R is relatively great for typical members such as chords C, the level of bending and axial stress induced by running train is not more than 4 MPa. Noting that the ultimate tensile strength of steel material is 370 MPa, the axial bending performance of main members is far lower. It consequently indicates that the axial bending behavior is not crucial factor for static behavior of the bridge.

Stress distribution between three planes of truss

In addition to the axial bending behavior, another concern for the arch truss system is the stress distribution between the three planes of the truss. Among the three types of members mentioned above, chords A and B are typical members of the arch truss system that sustain high-level axial compression stress. Therefore, chords A and B are selected for static behavior evaluation of stress distribution. In addition, axial compression stress is selected as the factor for investigating the stress distribution behavior because of its significance of static behavior.

The axial compression stress is calculated under actions of running trains. The speed of the trains is used as 240 km/h. In addition, the static stress is calculated and obtained by filtering either. Table 4 presents the normalized results of chords A and B in the three planes. As shown in this table, the difference in ratio between chords A in the three planes is approximately 0.16 when the train runs along the side railway. When the train runs along the middle railway, the difference in ratio is only approximately 0.07. In addition, similar regulation is obtained for chords B. However, the difference in ratio between the three planes reached 0.21 in the case of the train running along the side railways. Although the difference in ratio decreased to 0.09 in the case of the train running along the middle railways, the value is also greater than that of chords A.

Table 4.

Normalized results of absolute value of minimum stress.

Member		Railway 1	Railway 2	Railway 3	Railway 4
Chords A	A1	1.00	1.00	0.93	0.84
	A2	0.89	0.98	0.98	0.90
	A3	0.84	0.93	1.00	1.00
Chords B	B1	1.00	0.92	0.86	0.79
	B2	0.97	0.99	1.00	0.98
	B3	0.81	0.85	0.93	1.00

Although the difference in stress among the three planes of the truss is great in ratio, the absolute value of the stress difference is relatively small. The greatest absolute value of stress difference is approximately 2.8 MPa. Noting that the yield strength of steel is greater than 370 MPa, the ratio between the value and yield strength is only approximately 0.8%. Thus, the influence of stress distribution imbalance by the actions of running trains could be eliminated for the static safety of the bridge.

Conclusion

Based on the monitoring strain data and calibrated FE simulation, the impact factors of three typical types of members are calculated and obtained. In addition, the axial bending behaviors under the action of running trains are obtained for three typical types of members. Having selected two types of typical ring members in compression, the characteristics of stress distribution among the three planes of the truss are acquired to verify the integrity of the structural system. Influencing analyses are also alternated with dynamic and static behavior evaluations, including factors of the length, the speed, and the transverse position of running trains. The following main conclusions can be drawn.

The value of the impact factor of chords B is the greatest among the three typical types of members, reaching approximately 1.12. The impact factors for chords A and C are only approximately 1.01 and 1.04, respectively. According to the design codes, the recommended impact factor value is approximately 1.07 for all the members. For chords A and C, the impact factor results are within the permissible range according to the design codes. However, the design code underestimates the dynamic magnitude of chords B from running trains. Because the maximum value of absolute stress induced by running trains is relatively low, the underestimated increasing stress may insignificantly reduce the safety of the members.

The transverse position, the length, and the speed of running trains have particular impacts on the value of the impact factor. For all three types of members, the greatest value of impact factor occurs when the trains run along the nearest railways. The impact factor induced by trains with 8 carriages is greater than that induced by trains with 16 carriages. In addition, the relation between the impact factor and speed of running trains is also significant, with a higher speed of trains inducing more serious dynamic behavior.

Although the level of train-induced bending stress is considerably lower than the actual axial stress considering the dead loads, the ratio of train-induced bending versus train-induced axial stress, which is labeled R, is relatively large. Different members exhibit distinct axial bending behaviors. Chords C present the greatest value of R compared with the other two members with a maximum value of 3.37. Compared with chords C, chords A and B present smaller values of R, with the maximum values only reaching 0.12 and 0.06, respectively. In addition, only chords C are significantly influenced by the transverse positions of trains. Closer positions of the running trains induce higher values of R. Moreover, the difference ratio between the four railways could be approximately 200% for chords C.

Distinct stresses induced by trains exist among the three planes of the truss. The greatest value of difference ratio is approximately 16% and 22% for chords A and B. When trains run along the middle railways, the difference in stress among the three planes is the smallest, with the difference ratio being only approximately 7%. However, the influence of stress imbalance may insignificantly induce static behavior because the train-induced stress is far less than the yield strength of the steel material.

Footnotes

Academic Editor: Simon Laflamme

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the National Basic Research Program of China (973 Program) (no. 2015CB060000), the Key Program of National Natural Science Foundation (no. 51438002), the Program of National Natural Science Foundation (nos 51578138 and 51508251), the Natural Science Fund for Colleges and Universities in Jiangsu Province (no. 15KJB560005), and the Fundamental Research Funds for the Central Universities (no. 2242016K41066).

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