Introduction
The modern approach to railway dynamics is shifting from quasi-static calculations with dynamic coefficients to more accurate dynamic analyses, especially when it comes to high-speed railways with speeds exceeding 200 km/h. The danger of resonance at critical speeds is real, and a simple dynamic factor may not always capture it. Excessive vertical acceleration could lead to:
Instability of the railway track bed and rapid degradation of its geometry.
Inadequate contact between the rail and wheel.
Increased fatigue stress on the main structure of the bridge.
Discomfort for passengers.
Studies by the ERRI D214 Committee, part of the European rail research institute, have influenced the guidelines and recommendations embedded in current Eurocodes.
Criteria and Requirements
Dynamic analysis and acceleration checks are governed by standards such as ČSN EN 1992-2 and ČSN EN 1991-2 pertaining to bridge loads. The maximum peak values of vertical acceleration are crucial for:
Railways with gravel beds (to ensure the stability of the gravel bed).
Directly driven bridge decks (to maintain wheel-rail contact).
In these calculations, certain frequency shapes, as defined in ČSN EN 1990 A2.4.4.2.1, need to be included. These take into account bending and torsional shapes, especially in cases of eccentric tracks.
Structural Stiffness and Mass
The structure's stiffness and mass play a significant role in its dynamic behavior. Resonance peaks occur when the frequency of the load aligns with the structure's natural frequency. Overestimating bridge stiffness can lead to inaccurate predictions of these resonance speeds. The stiffness for composite sections should consider the cracked section properties.
In terms of the bridge's mass, maximum dynamic effects from loading happen at resonance peaks. The railway bed should be considered with a minimal volumetric mass, dry, clean, and of minimal thickness for lower estimates, and with maximum volumetric mass, saturated, dirty, and maximum thickness for upper estimates.
Train Loading Models
Loading models as per ČSN EN 1991-2 6.4.6.1 include:
HLSM-A for spans > 7 m, continuous and complex structures.
HLSM-B for simple spans < 7 m.
Real trains, with speeds guided by EN 1991-2 6.4.5.2 parameters.
Modal Analysis
The Lanczos method is recommended as it's compatible with the consistent matrix. The recommended time step for linear time-dependent analysis should be smaller than the maximum natural frequency of the highest shape used in the modal analysis.
Damping
Modal damping is consistent across all frequencies.
Train Load Generator in Midas Civil
This is dependent on:
The system choice from A1-A10.
The train's speed.
Distance between nodes/elements.
The wheel function which exhibits a linear distribution of force over time.
Assessment
The vertical acceleration assessment, using the example provided, indicates that it's below the threshold value, making it satisfactory. Internal design forces and dynamic factors are governed by ČSN EN 1991-2 6.4.6.1.
Conclusion
High train speeds can easily destabilize structures. Small, single-span bridges are predominantly susceptible. Inputs of stiffness and mass are crucial to the final structural response.
During modal superposition, selecting an adequate number of natural shapes is vital since the final response depends on their linear combination. Use modal damping as per standards unless precise data is available. The train load generator significantly speeds up the analysis process.
By optimizing the design and analysis based on these principles, engineers can ensure that bridges and tracks are not only safe but also comfortable for passengers even at high speeds. This integration of high-tech tools like Midas Civil with the principles from established standards ensures a reliable and efficient design process for high-speed railways.
Lukáš Juříček, a Product Engineer at IDEA StatiCa since September 2016, boasts over 7 years of experience in engineering software development. His academic background includes Civil Engineering from Brno University of Technology and a Master's in Applied Mechanics from VSB - Technical University of Ostrava. Lukáš has significantly contributed to the field with articles and webinars on advanced engineering topics, demonstrating expertise in software tools like Abaqus, AutoCAD, and ANSYS