Dynamic Modelling of Railway Bridge Train Interaction for ...

Dynamic Modelling of Railway Bridge Train Interactionfor Fatigue Damage Assessment

Dr Khosru Rahman CEng, FIMechE, CMath, CSci, FIMA, BEM

Royal Academy of Engineering Visiting Professor (University of Greenwich)

On-Line PresentationDate: 24th February 2021Time: 6.30pm

Picture Source - https://i.ytimg.com/vi/xMhIgvMBuds/hqdefault.jpg

Railway Bridges – The challenges driving the need for research

• Increasing volume of traffic, increases in weight and train speeds.

• Age corrosion, pitting, and impact damage degrade the structures.

• Operational bridges must remain structurally safe to full fill their economic and social function.

• The prioritisation of maintenance and ultimate replacement is now becoming an area of significantinterest.

- The importance of being able to establish accurate fatigue life predictions thus becomes apparent -

Railway Bridge Fatigue Damage – Parameters Influencing Damage

Vibration Modelling – What is Modal Analysis ?

There is always much confusion about what modalanalysis actually means and how best it can beexplained.

Vibration Modelling – Physical, Analytical & Modal Space Models

Any mechanical structure, whether it’s a bridge,building, aircraft, a flight control computer or a PCBboard can be characterised by its dynamic properties;

- Natural Frequencies- Damping- Mode Shapes

In simple terms the structure’s weight and stiffness arerepresented by the above dynamic properties. In amodal analysis, the modes of vibration of the structureare used and analysed to find out its response when anexternal force acts upon it.

The response of a structure is the summation ofresponses/deformation patterns at different naturalfrequencies. This is the principal of modalsuperposition. We can analyse the data in the time orfrequency domains. They are one and the same, we justswitch between the two for ease of interpretation.

Vibration Modelling – The Washing Machine

The washing machine represents a simple structure where the vibrations are excited by an un-balanced system –the rotating drum. For the purpose of this discussion I have represented the system in mathematical form as asimple one degree of freedom system. In reality the dynamics is much more complex.

𝑀𝑦 + 𝐶𝑦 + 𝐾𝑦 = 𝐹0 𝑆𝑖𝑛 𝜔𝐹𝑡

𝐹0 = 𝑚𝑅𝜔𝐹2

(3.63)

𝑦𝑝 𝑡 = 𝑌 𝑆𝑖𝑛 ( 𝜔𝐹𝑡 − ) (3.63)

𝑌 = 𝐹0

𝑘 − 𝑚𝜔𝐹2 2 + 𝑐2𝜔𝐹

2 (3.71)

= 𝑇𝑎𝑛−1 𝑐𝜔𝐹

𝑘 − 𝑚𝜔𝐹2 (3.72)

𝑦𝑝 𝑡 = 𝐹0

𝐾 − 𝑀𝜔𝐹2 2 + 𝐶2𝜔𝐹

2 𝑆𝑖𝑛 𝜔𝐹𝑡 − 𝑇𝑎𝑛−1

𝐶𝜔𝐹

𝐾 − 𝑀𝜔𝐹2 (3.73)

𝑌

𝛿𝑠𝑡 =

1

1 − 𝜔𝐹

𝜔𝑛

2

2

+ 2𝜔𝐹

𝜔𝑛

2

1 2

(3.74)

𝑀𝑦 + 3𝑐 𝑠𝑖𝑛𝛽 2𝑦 + 2𝑘 𝑠𝑖𝑛𝜃 2𝑦 = 𝐹𝑜𝑆𝑖𝑛𝜔𝐹𝑡

𝑦𝑝 𝑡 = 𝑌1𝑆𝑖𝑛 𝜔𝐹𝑡 + 𝑌2𝐶𝑜𝑠 𝜔𝐹𝑡

Vibration Modelling – The Washing Machine

𝐸 𝐼 𝜕4𝑦(𝑥 ,𝑡)

𝜕𝑥4+ 𝜇

𝜕2𝑦(𝑥 ,𝑡)

𝜕𝑡2+ 2 𝜇 𝜔𝑑

𝜕𝑦(𝑥 ,𝑡)

𝑑𝑡 − 𝜌𝐼 1 +

𝐸

𝑘𝐺 𝜕4𝑦 𝑥 ,𝑡

𝜕𝑥2𝜕𝑡2 +

𝜌2𝐼

𝑘𝐺 𝜕4𝑦 𝑥 ,𝑡

𝜕𝑡4 = 𝑃(𝑥 ,𝑡)

Timoshenko Beam (TB) Theory

Euler-Bernoulli Beam (EBB) Theory

𝐸 𝐼 𝜕4𝑦(𝑥 ,𝑡)

𝜕𝑥4 + 𝜇

𝜕2𝑦(𝑥 ,𝑡)

𝜕𝑡2 + 2 𝜇 𝜔𝑑

𝜕𝑦(𝑥 ,𝑡)

𝑑𝑡 = 𝑃 𝑥 ,𝑡

EBB Equation for a Series of Moving Loads

Bridge Interaction Problem – The Moving Load Problem

The dynamic response of a bridge, undermoving loads, can be modelled usingeither the Timoshenko or Euler-Bernoullibeam theories.

Time march analysis using Matlab provides an insight into extending the life of the structure by optimization of:Rolling masses - Speed - Mass distribution

Bridge Design & Assessment Data & BS-5400 Traffic Selection

BS-5400 Fatigue Class & SN-Curve

acc = annual cycle count, ni = No. of Stress Cycles , Nf = No. of Cycles to failure

Existing bridge design codes used to characterise

Fatigue Class (Class C used for assessment) Dynamic Amplification for Quasi-Static Analysis Quasi-Static & Dynamic fatigue damage

Miner’s Cumulative Damage Index

Load Series Arrangement – Bridge Train Axle Load Set-up

MATLAB Model – Flow Diagram

• EBB theory implemented in a Matlab Model.

• Assessment of any bridge for different train types and mixes.

• Model contains all train types from BS-5400.

• Additional bridges and train types can be added to the database in standard format.

• All Fatigue Classes from BS-5400 are implemented within the Matlab model.

• Matlab code approximately -3000 lines with additional post processing scripts for data processing and assessment.

Matlab Model – Verification

The MATLAB model was tested by performing an initial comparison of the dynamic response with that of the static responsewith a unit load traversing the bridge.

For the dynamic part of the MATLAB model the speed of the moving load is set to a low value, 10km/h, this slow enough forthe beam to behave statically, with no oscillations.

Bridge Train Interaction Response – Trains 1, 5, 7 & 8 (BS-5400 Medium Traffic)

DAF According to UK Network

Rail Bridge Assessment

NR/GN/CIV/025 The Structural Assessment of Underbridges

Damping According to Fryba (1999)

Rain-flow Counting & Stress Histogram

S Train - 1 DHP Train - 5

HF Train - 7 HF Train - 8

Fatigue Cumulative Damage Index – Trains 1, 5, 7 & 8 (BS-5400 Medium Traffic)

Bridge Vertical Frequency 5.3Hz

Stress/Acceleration Response & FFT’s for TRAIN 7 (Heavy Freight Train)

Bridge Vertical Frequency 5.3Hz

120km/h

180km/h

170km/h

Bridge Vertical Frequency 5.3HzBridge Vertical Frequency 5.3Hz

189km/h

126km/h

54km/h

Bridge Frequency 5.3Hz

J = 1

J = 2

J = 7

J = 3

BS-5400 Train 2 – Electric Multiple Unit (EMU-T2) – Wagon Pass Frequencies

J = 7

175km/h

126

116km/h

J = 2

J = 1

J = 3

Heavy Freight Train 7 (HF-T7)

Bridge Frequency 5.3Hz

BS-5400 Train 7 – Heavy Freight Train (HF-T7) – Wagon Pass Frequencies

Wagon Pass FrequencyBy setting Wpf = fn, (bridge vertical frequency), and solving for v we can calculate the (critical) train speed (vc) at which the bridge resonant frequency will be excited.

175km/h

116km/h

116km/h 175km/h

Key Points from Wagon Pass Frequency Assessment

CDI & DAF spike when speed excites bridge resonant frequency CDI Spikes occur at multiples of wagon pass frequencies Quasi-Static Analysis is unable to predict these spikes

Fatigue Life Summary & Conclusion

Fatigue Life Summary & Conclusion

Fatigue Life Summary & Conclusion

Fatigue Life Summary & Conclusion

Beam theory provides a 1st order method for estimating bridgesfatigue damage accumulation when subjected to a full range ofrolling stock traffic.

Estimates show:

Fatigue damage accumulation increases with speed.

Damage is significantly higher when speeds are close to thebridges resonant frequency.

Example: for train speeds of 120km/h the fatigue lifeestimated is 40% lower than that estimated bytraditional Quasi-static (DAF applied) methods.

Avoiding resonances will result in longer fatigue life.

Estimates suggest the dynamic analysis results in 9%and 18% higher fatigue lives at 50km/h and 70km/hfor the medium train mix considered.

The Quasi-static method for calculating the stresses areconservative suggesting a lower fatigue life.