Dynamic analysis of the stress-strain state of a beam superstructure of a railway bridge under moving train loads

1. Introduction

The infrastructure of railway bridges is a key element of transport networks that ensure the movement of passengers and cargo. With increasing speeds and axle loads, accurate assessment of the dynamic behavior of structures is critically important for the safety and durability of bridges. Traditional static methods do not take into account temporary changes in the reaction of the structure during train movement. Dynamic effects (vibrations, resonance, etc.) lead to stresses and deflections exceeding the calculated static values. The study is devoted to the dynamic analysis of bridge spans by the finite element method using MIDAS Civil software. Tasks: 1) development of a finite element model; 2) assessment of stresses and deformations during train movement; 3) determination of dynamic coefficients and velocity effects; 4) identification of critical stress zones [1].

Dynamic analysis of railway bridges has been actively studied for several decades. Willis and Timoshenko laid the foundations for the dynamic response of beams under moving loads, and Fryba (1999) developed the theoretical foundations for analyzing structural vibrations. Experiments on existing bridges have made it possible to measure dynamic deformations and accelerations, revealing the dependence of the reaction on the characteristics of the train and the structure. Yang et al. (2004) showed that dynamic gain coefficients change with the speed of the train, and critical speeds cause resonance. Modern research (Xia et al., 2018; Zhai et al., 2019) uses 3D models using the finite element method to study the interaction of loads and the path-bridge system. At the same time, a detailed analysis of the stress-strain state of girder superstructures under real train configurations remains an urgent task [2].

2. Materials and methods

The novelty of the research lies in the complex application of the finite element method for the dynamic analysis of railway spans, taking into account real-world operating conditions. Unlike traditional static calculations, the work takes into account temporary changes in the reaction of the structure when trains are moving at high speeds and increased axial loads. The developed finite element model in the MIDAS Civil software makes it possible not only to evaluate stresses and deformations, but also to accurately determine dynamic coefficients and velocity effects. The study identifies areas of critical stress that arise due to vibrations and resonance, which is impossible with a static approach. The results obtained increase the accuracy of forecasting the durability and safety of bridge structures.

Bridge Model Description. The investigation employs a representative railway bridge configuration consisting of a simply-supported steel-concrete composite beam superstructure. The bridge span length measures 30.0 meters, reflecting typical dimensions encountered in operational railway infrastructure (Table 1). The superstructure comprises four longitudinal steel I-beams with a concrete deck slab, connected through shear studs to ensure composite action [3, 6, 7].

3. Results and discussion

The results of the modal analysis showed the dynamic characteristics of the bridge superstructure. The first natural frequency (symmetrical vertical bend) is 4.28 Hz, the second (antisymmetric bend) is 11.64 Hz, and the third (second symmetrical bend) is 17.32 Hz (Table 3). The main mode of transport provides about 78 % of the total structural response under vertical loads from the train [12].

3.3. Stress-strain state analysis

Studying the stress distribution provides insight into the behavior of the structure. At 120 km/h in the average span: longitudinal stresses range from +194.7 MPa (lower steel flange) to –11.8 MPa (upper concrete fiber). The distribution is almost linear (according to Euler-Bernoulli theory). The Von Mises stresses are up to 162.4 MPa (Table 6), and the elastic behavior is below the yield point (Fig. 2) [14].

Fig. 1The dynamics of movement depending on the speed of the train

a) Mid-span displacement time history at 120 km/h

b) Dynamic amplification factors versus train speed

Table 5Dynamic analysis results at various speeds [13]

Speed (km/h)	Max Disp. (mm)	DAF Disp.	Max Stress (MPa)	DAF Stress	Max Accel. (m/s2)
40	21.4	1.14	158.3	1.11	0.68
60	22.1	1.18	164.0	1.15	0.85
80	23.4	1.25	172.5	1.21	1.02
100	25.2	1.35	186.8	1.31	1.21
120	26.8	1.43	194.7	1.37	1.38
140	25.8	1.38	189.7	1.33	1.29
160	24.7	1.32	182.5	1.28	1.18

Fig. 2Longitudinal stresses and bending moment

a) Longitudinal stress distribution through depth

b) Bending moment envelopes

Table 6Stress components at mid-span (120 km/h) [15]

Location	${\sigma }_x$ (MPa)	${\tau }_{xy}$ (MPa)	${\sigma }_{vm}$ (MPa)
Concrete top	–11.8	0.5	11.9
Concrete bottom	–1.1	1.2	2.1
Steel top flange	–0.7	2.8	4.2
Steel web center	96.5	18.4	103.2
Steel bottom flange	194.7	3.2	162.4
Note: ${\sigma }_x$ – longitudinal normal stress, ${\tau }_{xy}$ – shear stress, ${\sigma }_{vm}$ – von Mises equivalent stress

3.4. Acceleration response analysis

Time diagrams of acceleration show an increase in frequency with increasing train speed. The maximum is 1.38 m/s² at 120 km/h, which is higher than the forecasts of quasi-static methods. Accelerations affect the path, the interaction of the wheel with the rail, and passenger comfort. The reaction peaks correspond to the passage of the axis groups and the natural frequencies of the structure (Fig. 3) [15].

Fig. 3Vertical acceleration time history at mid-span

4. Conclusions

The study carried out a dynamic analysis of the railway bridge superstructure (finite element method, MIDAS Civil). The effects of dynamic amplification and the relationship between the speed of the train and the behavior of the structure are revealed. Key findings: DAF: 1.14–1.43 (displacements), 1.11-1.37 (stresses) – effects exceed static forecasts; maximum gain – at ~120 km/h (resonance); 3D modeling reflects stress and strain distributions; The coefficients are consistent with the design code. Critical speeds require special attention. Stress data helps to estimate fatigue life and plan maintenance [16-20]. The prospects: analysis of multi-span structures, consideration of the interaction of track and bridge, validation by field measurements.