Transcript Railway Bridge Dynamics

Overview of Lecture Series
Dermot O’Dwyer
Material to be Covered
• Identify factors that Influence bridge
response
• Identifying the types of problem that
structural engineers have to address
• Identify suitable analysis methods for the
different types of analysis
• Review structural dynamics
Resources
• Lectures on disk
– Railway bridge dynamics
– Dynamics primer
• Prototype analysis programs
– Moving force program
– Moving vehicle program
What Will You Be Able to Do?
• Perhaps nothing new but maybe these
sessions will clarify the origin of some of
the code requirements
• Will be able to give feedback on what you
would like an analysis package to deliver
Lecture 1
Overview of Railway Bridge
Dynamics
Factors that affect the dynamic
response of a railway bridge
1.
2.
3.
4.
5.
Bridge stiffness
Bridge mass
Train mass
Train speed
Rail and wheel irregularities and the
presence of track irregularities
6. Train suspension characteristics
Multiple problems
• Longitudinal Response
• Response of Transverse members
• Liquefaction of ballast due to high vertical
accelerations
• Critical speeds
• Fatigue
• Resonance
Lecture 2
Qualitative Response of Longitudinal
Members
Qualitative Response of
Longitudinal members
The rolling load moves at a slow
rate across the bridge
The rolling load moves at a slow
rate across the bridge
• If the load is moving slowly then the response of
the bridge at any time is equal to the static
deflection.
• If you need to ensure that there are no dynamic
effects then by reducing the speed of the train the
dynamic effects can be removed
• How slow is slow? Depends
• Static response can be used to check a dynamic
program
Rolling load moves
instantaneously to mid-span
Rolling load moves
instantaneously to mid-span
• This case shows that dynamic effects can be
significant
• Classic dynamic case of suddenly applied to
a spring
• Deflection is twice the static deflection for
this case
Rolling load moves rapidly from
mid-span to end
Rolling load moves rapidly from
mid-span to end
• Rolling load must climb out of the dip – this
requires a vertical movement
• The faster the load moves the greater the vertical
acceleration required
• The lower the bridge stiffness the greater the dip
and hence the greater the vertical acceleration
required
• The contact force will be dependent on the vertical
acceleration and the mass of the load
• The faster the load moves the less time the contact
force acts on the bridge
Rolling load traverses the bridge
rapidly
Rolling load traverses the bridge
rapidly
• Complex response
• Non-linear
• Response depends on
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–
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Mass of moving load
Mass of bridge
Velocity of moving load
Stiffness of bridge
Assumptions
• Contact between the wheel and the rail is
perfectly smooth
• Vertical velocity and acceleration are zero
when the wheel begins to cross the bridge
Lecture 3
Quantitative Response of Longitudinal
Members
Mathematical Models
• Mathematical models involve some
simplification. Generally, analysts attempt
to identify the simplest models capable of
predicting the system response with
sufficient accuracy
• It is vitally important that the analyst
understands the implications of the various
simplifications
Hirearcy of Models
• Moving forces model
• Moving mass model
• Moving train model
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–
–
–
Include non-linear contact spring
Include track
Include track and wheel irregularities
Include initial vertical velocities and
accelerations
Moving Force Model
• The moving force model replaces the
moving masses (i.e. the train) as a series of
constant vertical forces that move across the
bridge
• Advantages
– Significant simplification – linearises problem
– Reduced requirement for vehicle data
Ladislav Fryba’s Solution
• Response of a simply supported beam
subjected to a moving force
What is so great about Fryba’s
solution?
• Why linear is important
– Superposition
• Analyse respons to a single moving load
• Separate analysis required for each speed
but not each train
• Ideal for a fatigue analysis
Fryba - Free Response
• The closed form solution is only valid for the
period while the load is on the bridge
• The solution is to model the free response of the
bridge after the load has left
• Use the velocity and displacement of the bridge at
the instant the load leaves the bridge
• Requires a modal approach
Limitations of Fryba Formula
• Theoretical limitations
– Ignores vehicle characteristics
• More complex bridge forms would require
different closed form solutions
– Multiple spans
– Variable sections
• Can’t incorporate track and wheel
irregularities
Lecture 4
Numerical Modelling – Time-Stepping
Advantages of Numerical
Approach
•
•
•
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Bridges of all types can be modelled
Track irregularites can be incorporated
Wheel irregularities can be incorporated
Vehicle response can be incorporated
• But!
– Each run is unique and if track and wheel
irregularities are incorporated a large amount of
data is required
Moving Mass Models
• Potentially less accurate than moving force
because the sprung mass will be lumbed
with the unsprung mass
Moving Mass Term
• Moving mass terms show the errors
involved in using moving forces
Moving Train Models
Moving Train Models