High Speed 2 is a new high speed railway linking up London, the Midlands, the North and Scotland serving over 25 stations, including eight of Britain’s 10 largest cities and connecting around 30 million people.
Since high-speed railway bridges are subjected to cyclic loading by the continuous wheel loads traveling at high speed and regular spacing, their dynamic behavior is of extreme importance and has significant influence on the riding safety of the trains. High speed trains impose significant dynamic actions to bridges and viaducts. For speeds above 200 km/h the effect of resonance must be considered, since in many cases can be the governing factor when deciding the structural form. The classic approach of design of bridge is to use static analysis to find forces, flexure under certain moving load and consider dynamic amplification factor to account the dynamic effects of train. In HS1 this approach was used to take into account the dynamic effects. However certain issues were observed in the First French HSL project: Paris-Lyon:
Rapid track deterioration
Short Span Structures specially affected
In a joint effort in Europe a committee ‘ERRI D214’ was constituted to study these problems. Following conclusions were drawn by the committee for train speeds over 200 km/h:
Likelihood of resonance effects
Dynamic amplification factor unable to predict resonance
Deck acceleration must be assessed
The committee established a series of rules and guidelines for dynamic assessment of bridges. These guidelines have been implemented in Eurocode. What we see in the Eurocode is basically the conclusions of the committee on this particular issue.
Resonance and dynamic magnification:
The problem of resonance manifests in different ways depending on the type of structure. If ballasted tracks are used and deck acceleration is greater than 0.7g (g: acceleration due to gravity), in that case the ballast grains loose its grain interlock. Due to this there is loss in horizontal and vertical strength resulting in issues with track alignment, quick deterioration of track and risk of derailment of train. In case of ballast less track when deck acceleration is greater than g the contact between the wheel and rail is lost which also leads to quick deterioration of track as well as risk of deterioration. Usually it is assumed that single span simply supported structures have less or no dynamic effect of high speed train. But contrary to this single span structures are especially susceptible to resonance. Resonance effects are significantly reduced on continuous structures.
When to go for Dynamic analysis?
Eurocode provides flow chart for simple and complex structures to determine whether dynamic analysis is needed or not. Below shown is a flow chart given in the code.
Simple structures are those which behave like beam between supports. For structures with behavior such as grillage, orthotropic, cable stayed or more complex behavior are classified as complex structure.
Maximum peak values are given in EN 1990-2002 A220.127.116.11.1. To ensure traffic safety the recommended values are:
γbt = 3.5 m/s2 for ballasted track (ballast stability)
γdf = 5.0 m/s2 for ballast-less track (wheel-rail contact)
The above mentioned values are w.r.t stability of ballast, track maintenance as well as to avoid derailment. Passenger comfort criteria is not cover in this clause. Passenger comfort criteria is covered in EN 1990-2002 A18.104.22.168.1.
Frequencies to be considered should be up to the greater of [BS EN 1990-2002 A22.214.171.124.1]:
5 times the frequency of the fundamental mode of vibration of the member being considered
Frequency of the third mode of vibration of the member
Bending and torsional need to be identified to assess n0 and nT. Mass participation factors can be used to identify the relevant modes.
Mass and stiffness considerations:
Any overestimation of bridge stiffness will overestimate the natural frequency of the structure and speed at which resonance occurs. A lower bound estimate of the stiffness throughout the structure shall be used. Regarding the cracked stiffness, assessment of cracked stiffness is essential, since a reduced cracked stiffness lead to a lower fundamental frequencies hence lower resonant speed. For the estimation of mass, a lower bound estimation predicts maximum deck accelerations. An upper bound estimate of mass is used to predict the lowest speed at which resonant effects are likely to occur.
Time History analysis:
Time history analysis need to be performed to mimic the dynamic effect of train load. Linear time history can be considered as generally the structural behavior is within linear range. Modal integration (modal superposition method) is generally used with the first mode of the structure in accordance with BS EN 1990-2002 A126.96.36.199.1.
Eurocode does not give any recommendation on the the time step. However ERRI D214 provides guidelines on time step which are shown below. Time step value should not be greater than:
h1 = 1/8fmax h2 = Lmin/200ν h3 = Lmin/4nν h4 = 0.001s
fmax: maximum frequency used on the modal analysis
Lmin: minimum span
N: number of modes used on modal analysis
ν : speed of train
If very high value of time step is considered it will affect the amplitude of the dynamic analysis:
Structural damping can be considered as per the Eurocode recommendations [BS EN 1991-2:2003 188.8.131.52].
A 30 m span simply supported psc box girder bridge is considered with nodes at every 0.5 m. The structural arrange ment of the bridge is shown below:
From the above flow chart it was concluded that the dynamic analysis is required for this bridge. The results from the dynamic analysis are discussed below.
Peak values must be plotted against speeds to identify resonance/critical speed
It can be seen that the bridge is safe from the risk of derailment as the maximum peak acceleration is less than 3.5m/s2 which is specified by the euro code for the ballasted track.
The dynamic response of the deck members must be checked and compared to the equivalent static responses.
The conclusions for the dynamic analysis are tabulated below:
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