MIDAS Civil TUTORIAL Dynamic Analysis of High Speed Rail ... Academy...[EN 1990-2002 A2.4.4.2.1] → To ensure traffic safety, the Eurocodes recommend: - bt = 3.5 m/s2 for ballasted

MIDAS Civil TUTORIAL

Dynamic Analysis of

High Speed Rail Bridges

with Eurocodes

1. Introduction to the problem

2. Eurocode provisions

3. Eigenvalue analysis

4. Time history analysis

5. Dynamic loads

6. Case study

7. Conclusions

CONTENTS

1. Introduction to the problem

Background

Resonance and dynamic magnification

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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

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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

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2. Eurocode provisions

Requirements for a static or dynamic analysis

Dynamic amplification factor

Acceleration check

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Is a dynamic analysis required? (simple structures)

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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

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 𝑥𝐻𝑆𝐿𝑀𝑜𝑟𝑅𝑇

+ Acceleration check

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Maximum dynamic response

Increase resulting fromtrack defects andvehicle imperfections

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)

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3. Eigenvalue analysis

Fundamental frequencies

Stiffness

Mass

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Fundamental frequencies

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

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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

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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

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4. Time history analysis

Setting up the Time History Analysis

Time step

Structural Damping

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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

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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.

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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= +

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5. Dynamic loads

Train Load Models

Model input

Load distribution

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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

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Dynamic nodal loads

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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

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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

Load distribution

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Load models for the acceleration

check and dynamic factor

[BS EN 1991-2:2003 6.4.6.4 (3)]

Transverse distribution

[BS EN 1991-2:2003 6.3.6.3]

→ The representation of each axle by a single point force tends to

overestimate dynamic effects for loaded lengths of less than 10m.

In such cases, the load distribution effects of rails, sleepers and

ballast may be taken into account.

→ A point force in (…) HSLM (except for HSLM-B) (…) may be

distributed over three rail support points (…)

a is the distance between rail support points

Load distribution

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Longitudinal distribution

[BS EN 1991-2:2003 6.3.6.1 (1)]

Load distribution

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Longitudinal distribution

tj+2

tj+2

tj+2

tj+2

0.25F

0.5F

0.25F 0.25F

0.5F

0.25F

0.50F

Time function

for node i

tj+2

tj

tj+1

tj+2

tj

tj+1

tj+2

tj+2

tj+2

0.25F 0.25F

i i i

tj

tj+1

t = tj

t = tj+1

t = tj+2

0.5F

Load distribution

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0

1

2

3

4

5

6

7

8

140 190 240 290 340 390

Deck

peak a

ccele

ration [m

/s²]

Train speed [kph]

A1 - w/o distribution

A1 - with longitudinal and

transversal distribution

6. Case study

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Case Study

General Arrangement

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Case Study

Materials and loads

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Materials

Loads

→ Concrete C45,

• Ec = 35.22 GPa

• ν = 0.20

→ Concrete self-weight: 25 kN/m³

→ Permanent loads (ballast, rails, sleepers…): 45 kN/m

→ Traffic loads: HSLM-A1 to HSLM-A10

Example – Is a dynamic analysis required?

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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

Case Study – Eigenvalue analysis results

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Mode No Frequency Displacement Rotation

(Hz)

X Y Z X Y Z

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

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

Acceleration envelopes

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Can be used to identify the critical locations on the deck

Acceleration Time History

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Time Domain Response – to ensure that the critical time has been captured

Free vibrationForced excitation

Acceleration Response Spectrum

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Frequency Domain Response - to identify critical modes/frequencies

Deflection vs. time animation

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Animation may help to spot irregularities

Acceleration check

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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 a

ccele

ration [m

/s2]

Train speed [km/h]

A1

A2

A3

A4

A5

A6

A7

A8

A9

A10

Dynamic amplification factor

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-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

[kN

m]

Train speed

[km/h]

Ф x (LM71”+”SW/0)

1 + 𝜑′𝑑𝑦𝑛

+ Τ𝜑′′ 2 𝑥𝐻𝑆𝐿𝑀𝑜𝑟𝑅𝑇

→ Dynamic responses of all deck

members must be checked and

compared to the equivalent static

responses

7. Conclusions

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Conclusions

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

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Thank youContact:http://[email protected]