Dynamic analysis of metro rail bridge subjected to moving loads considering soil–structure interaction
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Abstract
This paper investigates the dynamic response of a metro rail over-bridge, subjected to moving loads. Track irregularity and train inertia effects are not considered. Bridge superstructure, piers and substructure are modelled using shell element, rails are modelled with frame elements and the interaction between bridge deck and piers is simulated using link supports (bearings) in SAP2000 (2014). Moving load analysis is performed for two models namely fixed base model and complete pile model. For complete pile model, the piles are modelled using frame elements. IS 2911: 2010 is considered to evaluate the soil stiffness properties. Modal damping ratio of 5% is adopted. Finite element method is used to perform the dynamic analysis and Newmark-β method is considered to solve the equations of motion. From the comparative study between two models, and for two loading cases, it was noted that the speed of the train is a very important parameter influencing the dynamic response of the bridge. Moreover, the resonance phenomenon for the complete pile model was observed at lower speed compared to the fixed base model for both the loading cases. From this study, it can be stated that a full three-dimensional (3D) multi-span simply supported bridges’ dynamic analysis is important to obtain the transient response of the bridge structure.
Keywords
Soil–structure interaction Resonance Dynamic analysis Complete pile model Newmark-βIntroduction
The development of high-speed railways in various countries has increased the interest in dynamic behaviour of railway bridges. Under the loads of high speed, the bridges are subjected to high impacts. The dynamic aspects are of special interest and have often shown to be the governing factor in the structural design. Vehicle speed is an important parameter which influences the dynamic behaviour of bridge. In addition to vehicle speed, the characteristics of the bridge structure and the vehicle, the rail surface unevenness, varying vehicular travel lanes are the different parameters that govern the dynamic behaviour of bridge.
Wang et al. (2007) investigated the resonance response of a multi-span continuous beam and simply supported beam under the action of load moving at constant velocities. Ju and Lin (2003) have provided solution to reduce the resonance phenomenon of a simply supported bridge by studying three-dimensional (3D) vehicle–bridge interaction (VBI) analysis. Fryba (2001) investigated an elementary theoretical model of a bridge using the integral transformation method. The analysis gives the critical speed at which the resonance vibration may occur. Zhang et al. (2010) studied a numerical solution for the dynamic response of a train bridge interaction subjected to multi-support seismic loads. Salcher and Adam (2012) have presented the results of a numerical study which aimed at a quick and accurate assessment of the dynamic railway bridge response subjected to high-speed trains. Li et al. (2013) have studied two different bridge models which were considered as simplified small-scale models. These models showed fair agreement with the experimental study.
Romero et al. (2013) have put in a lot to understand and explain dynamic soil–bridge interaction under high-speed train. The parametric study conducted for various types of soils resulted in concluding the fact that the resonance response is attained at lower speeds for loose soils. Tavares (2007) has studied the effect of support stiffness in vertical direction under high-speed moving trains. Zeng et al. (2015) investigates the random vibration and the dynamic reliability of train moving over slab track on bridge under track irregularities and earthquakes by the pseudo-excitation method (PEM). A numerical solution for the dynamic response of train–track–bridge coupled system considering the influence of soil–structure interaction (SSI) is studied and verified with the results of field experiments, by Li et al. (2013). Zehsaz et al. (2009) have presented a new method for dynamic analysis for railway, as a beam with limited length, lying on a viscoelastic bed and subjected to moving load is presented. Ülker-Kaustell et al. (2010) has presented a qualitative analysis of the dynamic soil–structure interaction (SSI) of a portal frame railway bridge based on the linear theory of elasticity. Xia et al. (2014) has presented the study of the train–bridge system under collision loads. A continuous bridge with box girders is considered as a case study. A model for dynamic analysis of the vehicle–track nonlinear coupling system is established by the finite element method in Lei et al. (2016). Feenstra and Isenberg (2012) have studied a detailed, three-dimensional finite element model to evaluate the dynamic amplification factor for light rail vehicles.
SAP2000 (2014) is used to analyse the 3D rail bridge. Gharad and Sonparote (2017) have studied and validated the dynamic response of two-dimensional (2D) and three-dimensional (3D) bridge models under the action of moving loads, using the same software. Following sections discuss the modelling, analysis and comparison of different loading conditions using the finite element approach.
Modelling of 3D bridge
Material properties and characteristic values of the railway bridge
Properties | Density (kN/m^{3}) | Modulus of elasticity (kN/m^{2}) | Poisson’s ratio | Grade |
---|---|---|---|---|
Deck (box-girder) | 25 | 34 × 10^{6} | 0.15 | M50 |
Pier and pier cap | 25 | 36 × 10^{6} | 0.15 | M60 |
Pile and pile cap | 25 | 31 × 10^{6} | 0.15 | M40 |
Rail | 77 | 210 × 10^{6} | 0.30 | 500 |
Expansion joint filler (Epoxy) | 18 | 6.52 × 10^{6} | 0.35 | – |
Cross-sectional properties of the various components of bridge
Component | Size (mm) | Depth (mm) |
---|---|---|
Box-girder: | 1650 | |
Top width | 8500 | |
Bottom width | 3186 | |
Pier cap | 3186 × 2200 | 500 |
Pier | 1700 × 2200 | 11,120 |
Pile cap | 5100 × 5100 | 1800 |
Pile diameter | 1200 | 11,800 |
Foundation stiffness of pile cap
Directions | R_{z} (MN/m) | R_{x} (MN/m) | M_{y} (MN m/rad) | R_{y} (MN/m) | M_{x} (MNm/rad) |
---|---|---|---|---|---|
Pile cap | 13,328.5 | 1681.59 | 189,480 | 1681.59 | 189,480 |
Two horizontal springs with spring constant value of 6000 kN/m each are assigned on middle point of the sections of pile and a vertical spring at the bottom of the pile with spring constant value calculated as 6000 kN/m is assigned. These values are evaluated based on IS 2911: 2010.
Rail fastener
Bridge parameters considered in the present study
Parameters | Unit | Value |
---|---|---|
Rail fastener | ||
Lateral stiffness | kN/m | 50,000 |
Vertical stiffness | kN/m | 30,000 |
Damping | 50 | |
Bearings | ||
Stiffness | 1,143,752 | |
Damping | kN/m | 80 |
Bearings
Moving train load model
Modal and moving load analyses
Natural frequencies of the bridge
Mode number | Complete pile model | Fixed base model | ||
---|---|---|---|---|
Frequency (Hz) | Characteristic | Frequency (Hz) | Characteristic | |
1 | 2.317 | Longitudinal symmetrical floating | 2.535 | Longitudinal symmetrical floating |
2 | 2.489 | Lateral symmetrical bending | 2.6222 | Lateral symmetrical bending |
3 | 6.256 | Longitudinal bending with antisymmetric vertical bending | 6.292 | Longitudinal bending with antisymmetric vertical bending |
4 | 6.54 | Lateral antisymmetric bending | 6.624 | Lateral antisymmetric bending (twist) |
Resonant conditions
Maximum dynamic acceleration in vertical direction with respect to speed for single train loading
Speed (km/h) (v) | Vertical acceleration (m/s^{2}) | |
---|---|---|
Fixed base model | Complete pile model | |
10 | 0.0056 | 0.02655 |
20 | 0.0106 | 0.02925 |
30 | 0.0091 | 0.03822 |
40 | 0.0110 | 0.03798 |
50 | 0.0119 | 0.0418 |
60 | 0.0117 | 0.04245 |
61.30 | 0.01587 | 0.04426 |
67.08 | 0.01692 | 0.04143 |
70 | 0.0126 | 0.03881 |
Double train loadings
Comparative studies
In the previous sections, the maximum acceleration and displacement responses at the centre of the mid-span deck due to single train and double train loadings were calculated. In this section, a comparison of the dynamic responses, obtained due to these load cases, for the two different bridge models (fixed base and complete pile) is done.
Fixed base model
Complete pile model
From the above comparison, it can be stated that for the double train loading case, a definite increase in the acceleration and displacement values compared to the single train loading case is evident. Thus, during dynamic analysis of such bridges, the contribution of various loading conditions, developing the critical vertical bridge deck responses, should be considered.
Limits for displacement
The excessive bridge deformations can endanger traffic by causing unacceptable changes in geometry of the track and in bridge structures, which leads to discomfort of passengers. The UIC-code (International Union of railways) 776-3 R (1989) recommends the limitations to be placed on bridge deformation to avoid risk to traffic and discomfort to passengers. The vertical deflection for the considered bridge, due to live load is obtained as 1.03 mm, which is within the limits specified in the code.
Conclusions
- 1.
The natural frequency of a multi-span simply supported bridge plays a vital role in identifying its vertical resonance response. The maximum vertical resonant speed of the bridge deck (under single train loading) corresponds to the fundamental frequency of the bridge structure.
- 2.
The dynamic behaviour of the bridge structure is governed by the soil–bridge interaction under moving loads. The vertical acceleration response of the mid span of bridge deck is obtained at lower speeds under moving loads when SSI is considered for both the loading conditions. The fixed base model does not represent the actual dynamic behaviour of the bridge structure. Hence, it becomes a prime responsibility to analyse any given structure under dynamic loading considering SSI.
- 3.
For the case when two trains were assumed to be moving in opposite directions, a definite increase in the acceleration and displacement values compared to the single train loading case was observed. Thus, during dynamic analysis of such bridges, the contribution of various loading conditions, developing the critical vertical bridge deck responses, should be considered.
- 4.
The vertical deformations caused due to live loads, for this particular bridge, are within safety allowances as per the guidelines of UIC code.
Notes
Acknowledgements
The authors are thankful to Nagpur Metro Rail Corporation Limited, Nagpur and special thanks to Dr. M. P. Ramnavas, General Manager/Design, Nagpur Metro Rail Corporation Limited, Nagpur, for providing the necessary information. The authors are also thankful to the anonymous reviewers for their essential suggestions.
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