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MIDAS UK Expert Engineer Webinar Series
Dynamic Analysis of
High Speed Two (HS2)
Pere Alfaras,
Principal Bridge Engineer
ARCADIS UK
1. About
2. Introduction to the problem
3. Eurocode requirements
4. Stiffness & Mass
5. Eigenvalue analysis
6. Time history analysis
7. Results interpretation
8. Conclusion
CONTENTS
1. About
About myself
About Arcadis
About HS2
MIDAS UK Expert Engineer Webinar Series
About myself
BLABLABLA
MIDAS UK Expert Engineer Webinar Series
Pere Alfaras
→ Principal Bridge Engineer
→ Arcadis UK
→ 10+ years of experience
→ Bridge and Structural design
→ Lecturer / Finite Element Method
@ UPC/BarcelonaTech for 5 years
Lower Thames Crossing - UK High Speed Two (HS2) - UK
Jeddah Metro - KSA Abi Bakr Al-Siddiq Highway - KSA
High Speed Line Warsaw-Poznan-
Wroclaw - Poland “Eix Diagonal” Motorway - Spain
About Arcadis
→ Global Design & Consultancy
→ Natural and Built Assets
→ NCE100 Company of the Year
→ We are 27,000 people
→ Over 300 offices
→ Active in over 70 countries
→ €3.2 billion in revenues
→ Extensive bridge expertise
→ UK based team circa 100 staff
→ www.arcadis.com
MIDAS UK Expert Engineer Webinar Series
Millau Viaduct - France Tower Bridge, London - UK
Albert Bridge, London - UK Rotherhithe Brige, London - UK
Te Wero Bridge – New Zealand Vasco de Gama Bridge - Portugal
About HS2
MIDAS UK Expert Engineer Webinar Series
Overall budget: £55.7bn
→ Design JV: Arcadis, Setec & COWI
→ Consortium with Eiffage Kier JV
→ Scope: packages C2 & C3
→ 80km of railway line
→ 86 Bridges
→ 14 Viaducts (3.6km total)
→ 3 green tunnels (5.0km total)
→ Civils works complete by 2022
2. Introduction to the problem
Background
Resonance and dynamic magnification
MIDAS UK Expert Engineer Webinar Series
Background
Classic code requirements
First French HSL: Paris-Lyon
ERRI D214 Committee studies
→ Static Analysis with Dynamic amplification factor
→ Following issues were observed:
• Resonance phenomena
• Ballast degradation
• Rapid track deterioration
• Short-span structures specially affected
→ Concluded that for speeds over 200 km/h:
• Likelihood of resonance effects
• Dynamic amplification factor unable to predict resonance
• Deck acceleration must be assessed
→ Established rules for dynamic assessment - now implemented in
Eurocodes
MIDAS UK Expert Engineer Webinar Series
Resonance and dynamic magnification
Ballasted Vs Ballastless tracks
Simply supported Vs Continuous
Resonant speed
→ Ballast grains loose its grain interlock when a > 0.7g
→ Ballastless tracks wheel-rail contact is reduced beyond
acceptable limits when a > g
→ Single-span structures specially susceptible to resonance
→ Resonance effects are significantly reduced on continuous
structures
→ Resonance speed usually 200km/h < v < design speed
MIDAS UK Expert Engineer Webinar Series
3. Eurocode requirements
Dynamic amplification factor
Requirements for a static or dynamic analysis
Acceleration check
Load models, combinations, design speed
Structural damping
MIDAS UK Expert Engineer Webinar Series
Dynamic amplification factor
If dynamic analysis not required
If dynamic analysis required
Ф x (LM71”+”SW/0)
• Ф depends on track irregularities and determinant length LФ
Most unfavourable value of:
Ф x (LM71”+”SW/0)
or
1 + 𝜑′𝑑𝑦𝑛 + Τ𝜑′′ 2 𝑥
𝐻𝑆𝐿𝑀𝑜𝑟𝑅𝑇
+ Acceleration check
MIDAS UK Expert Engineer Webinar Series
Maximum dynamic response
Increase resulting fromtrack defects andvehicle imperfections
Is a dynamic analysis required? (simple structures)
MIDAS UK Expert Engineer Webinar Series
Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
Accepted by relevant authority
nT > 1.2n0
Is Skew < 15 Redesign
Use eigenforms for torsion and for bending
May use the eigenforms for bending only
Dynamic analysis required Dynamic analysis not required
n0 within limitsof Fig. NA.14
Y
N
Y
N
N N
Y
N Y
Y
N
AcceptedY
Not accepted
NA to BS EN 1991-2:2003 Figure NA.12
Figure NA.14
Acceleration check
Maximum peak values
[EN 1990-2002 A2.4.4.2.1]
→ To ensure traffic safety, the Eurocodes recommend:
- bt = 3.5 m/s2 for ballasted track (ballast stability)
- df = 5.0 m/s2 for ballast-less track (wheel-rail contact)
→ EN 1990-2002 UK Annex: The maximum peak values of bridge
deck acceleration and the associated frequency limits should be
determined for the individual project.
→ Passenger comfort criteria is covered elsewhere in the code
(EN 1990-2002 A2.4.4.3.1)
MIDAS UK Expert Engineer Webinar Series
4. Stiffness & Mass
Stiffness
Mass
Mass control parameter
MIDAS UK Expert Engineer Webinar Series
Stiffness
Bridge stiffness
Young’s modulus
Shear deformation
Cracked stiffness
→ 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
→ Short term concrete elastic modulus for concrete elements
→ Should be considered
→ Assessment of cracked stiffness is essential, since a reduced
cracked stiffness lead to lower fundamental frequencies hence
lower resonant speeds
MIDAS UK Expert Engineer Webinar Series
Mass
Upper and lower bound estimates
of mass
Self-weight
Ballast
Other superimposed loads
→ a lower bound estimate to predict maximum deck accelerations
→ an upper bound estimate of mass to predict the lowest speeds at
which resonant effects are likely to occur
→ According to EN 1991-1-1 (enhanced density values may be
used if confirmed via testing and approved by relevant authority)
→ minimum likely dry clean density and minimum thickness of
ballast
→ maximum saturated density of dirty ballast with allowance for
future track lifts
→ rails, sleepers, parapets, OLE, others
MIDAS UK Expert Engineer Webinar Series
Case Study – General Arrangement
Span arrangement
Cross-section
MIDAS UK Expert Engineer Webinar Series
Example – Is a dynamic analysis required?
MIDAS UK Expert Engineer Webinar Series
Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
Accepted by relevant authority
nT > 1.2n0
Is Skew < 15 Redesign
Use eigenforms for torsion and for bending
May use the eigenforms for bending only
Dynamic analysis required Dynamic analysis not required
n0 within limitsof Fig. NA.14
Y
N
Y
N
N N
Y
N Y
Y
N
AcceptedY
Not accepted
Figure NA.14Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
5. Eigenvalue analysis
Frequencies to be considered
Mass participation factors
Bending and torsional modes
MIDAS UK Expert Engineer Webinar Series
Eigenvalue analysis
Frequencies to be considered
[BS EN 1990-2002 A2.4.4.2.1]
Bending and torsional modes
Mass participation factors
Up to the greater of:
→ 30 Hz
→ 1,5 times the frequency of the fundamental mode of vibration of
the member being considered
→ The frequency of the third mode of vibration of the member
→ Need to be identified to assess n0 and nT
→ Can be used to identify the relevant modes
MIDAS UK Expert Engineer Webinar Series
Case Study – Eigenvalue analysis results
MIDAS UK Expert Engineer Webinar Series
Displacement Rotation
Mode No Frequency X Y Z X Y Z
(Hz) mpm (%) mpm (%) mpm (%) mpm (%) mpm (%) mpm (%)
1 6.14 0 0 82.67 0 0 0
2 16.49 0 82.67 0 0 0 0
3 20.04 0 0 0 82.82 0 0
4 20.56 0 0 0 0 0 0
5 28.69 81.91 0 0 0 0 0
Case Study – Is a dynamic analysis required?
MIDAS UK Expert Engineer Webinar Series
Start V ≤ 200km/h
L ≥ 40m(see Note 1)
n0 within limitsof Fig. NA.14
Accepted by relevant authority
nT > 1.2n0
Is Skew < 15 Redesign
Use eigenforms for torsion and for bending
May use the eigenforms for bending only
Dynamic analysis required Dynamic analysis not required
n0 within limitsof Fig. NA.14
Y
N
Y
N
N N
Y
N Y
Y
N
AcceptedY
Not accepted
30
6.1nT > 1.2n0
Is Skew < 15
May use the eigenforms for bending only
Dynamic analysis required
6. Time history analysis
Time step
Structural Damping
Train Load Models
Model input
MIDAS UK Expert Engineer Webinar Series
Setting up the Time History Analysis
Linear or Non-linear?
Modal or Direct Integration?
Transient or Periodic?
→ Generally structural behaviour within linear range
→ Modal integration (modal superposition method) should
generally be used with the first modes of the structure (in
accordance to BS EN 1990-2002 A2.4.4.2.1)
→ This is a transient problem
MIDAS UK Expert Engineer Webinar Series
Time step
ERRI D214 (e), 1999 → recommends to choose a time step not greater than:
where:
𝑓𝑚𝑎𝑥 : maximum frequency used on the modal analysis;
𝐿𝑚𝑖𝑛: minimum span;
𝑛: number of modes used on the modal analysis;
𝑣: train speed.
MIDAS UK Expert Engineer Webinar Series
ℎ1 =1
8𝑓𝑚𝑎𝑥ℎ2 =
𝐿𝑚𝑖𝑛
200𝑣ℎ3 =
𝐿𝑚𝑖𝑛
4𝑛𝑣ℎ4 = 0.001𝑠
-1.25
-1
-0.75
-0.5
-0.25
0
0.25
0.5
0.75
1
1.25
0 ms 50 ms 100 ms 150 ms 200 ms 250 ms 300 ms 350 ms
Am
plit
ude
Δt=5ms
Δt=40ms
Structural damping
Eurocode recommendations
[BS EN 1991-2:2003 6.4.6.3]
→ Recommended damping values
→ Additional damping: TOTAL= +
MIDAS UK Expert Engineer Webinar Series
Train Load Models
Load models for the acceleration
check and dynamic factor
[BS EN 1991-2:2003 6.4.6.1]
Fatigue loads
[BS EN 1991-2:2003 Annex D]
Speeds to be considered
[BS EN 1991-2:2003 6.4.6.2]
→ HSLM-A: for spans over 7m or complex structures
10 variations (A1 to A10)
→ HSLM-B: for simple structures with spans less than 7m
→ Real train
→ 12 train types
→ traffic mixes
→ 40 m/s vi 1,2 x Maximum Line Speed
→ Reduced speed steps in the vicinity of resonant speeds
MIDAS UK Expert Engineer Webinar Series
Dynamic nodal loads
MIDAS UK Expert Engineer Webinar Series
How to transform a moving load to dynamic loads using time functions:
0
50
100
150
200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Load [kN
]
Time [s]
Dynamic nodal loads
MIDAS UK Expert Engineer Webinar Series
0
50
100
150
200
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
Load [kN
]
Time [s]
How to transform a series of moving loads to a time function:
delay
7. Results interpretation
MIDAS UK Expert Engineer Webinar Series
Graphical outputs
Time History Graph
Fourier transform
Case Study - Graphical outputs
MIDAS UK Expert Engineer Webinar Series
Envelope results - to identify the critical locations on the deck
Case Study -Acceleration Time History
MIDAS UK Expert Engineer Webinar Series
Time Domain Response – to ensure that the critical time has been captured
Free vibrationForced excitation
Case Study -Acceleration Response Spectrum
MIDAS UK Expert Engineer Webinar Series
Frequency Domain Response - to identify critical modes/frequencies
Additional Example – Continuous Structure
MIDAS UK Expert Engineer Webinar Series
Animation may help to spot irregularities
Case Study -Acceleration check
MIDAS UK Expert Engineer Webinar Series
Peak values must be plotted against speeds to identify resonant/critical speeds.
0
0.5
1
1.5
2
2.5
3
3.5
140 190 240 290 340 390
Peak
acc
ele
ration [m
/s2]
Train speed [km/h]
A1
A2
A3
A4
A5
A6
A7
A8
A9
A10
Case Study - Dynamic amplification factor
MIDAS UK Expert Engineer Webinar Series
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
Deflect
ion a
t m
idsp
an [m
m]
0
2000
4000
6000
8000
10000
12000
14000
16000
140 165 190 215 240 265 290 315 340 365 390 415
Bendin
g M
om
ent [k
Nm
]
Train speed
[km/h]
Ф x (LM71”+”SW/0)
1 + 𝜑′𝑑𝑦𝑛 + Τ𝜑′′ 2 𝑥
𝐻𝑆𝐿𝑀𝑜𝑟𝑅𝑇
→ Dynamic responses of all deck
members must be checked and
compared to the equivalent static
responses
8. Conclusion
MIDAS UK Expert Engineer Webinar Series
Conclusion
Resonance and dynamic
magnification
Resonant speed
Model properties
Analysis
Result interpretation
→ It is relevant for speeds over 200 km/h
→ Short span structures are particularly prone to resonance
→ Difficult to anticipate the resonant speeds for most structures
→ A dynamic analysis is required to assess acceleration and
dynamic amplification factor for a range of speeds
→ Bridge stiffness and mass have to be carefully assessed
→ Upper and lower bounds must be considered
→ Requires numerous time history cases, which is time consuming
→ Vital to ensure accurate results
MIDAS UK Expert Engineer Webinar Series
Thank youContact:http://globalsupport.midasuser.comuksupport@midasuser.com
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