dinâmica estrutural ii_pontes_01

8/13/2019 Dinâmica Estrutural II_Pontes_01

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Universidade do Estado do Rio de Janeiro, UERJ

Centro de Tecnologia e Ciências, CTC

Faculdade de Engenharia, FEN

Análise Dinâmica de Tabuleiros Rodoviários Análise no Domínio do Tempo

Efeito do Peso dos VeículosEfeito das Irregularidades da Pista

Prof. José Guilherme Santos da Silva, DSc.Coordenador da Área de Concentração de EstruturasPrograma de Pós-graduação em Engenharia Civil, PGECIV



ÍNDICE

1. Introdução2. Modelagem da carga móvel

3. Modelagem do sistema veículo-ponte

4. Exemplos: efeito do peso dos veículos5. Conclusões



1. INTRODUCTION

Number of vehicles on the bridge is an important point in the evaluationof their dynamical actions;

Effect of the interaction of their masses with the bridge deck mass;

Crossing frequency of the moving load which can produce a steady-state response situation;

The moving load is taken as an infinite series of vehicles, regularlyspaced and moving at constant velocity, to allow the observation of thesteady state response of the structure.



2. INFINITE TRAIN OF VEHICLES

3. MODEL OF THE VEHICLE-BRIDGE SYSTEM



4. EXAMPLES

4.1. Vehicles Properties

Two classes: 450kN and 120kN;Regularly spaced: l=5.0m;

Frequencies: 3Hz and 20Hz;Damping factor: =0.1.



0

0

u

u

k k k

k k

u

u

ccc

cc

u

u

m0

0m

1

v

vpvsvs

vsvs

1

v

vpvsvs

vsvs

1

v

ns

s

00

0

0

θu

u

u

dk k dk dk dk k d-k k k 0k

dk 0k k k

dk k k k k k

θ

u

u

u

dccdcdcdcc

dccc0c

dc0ccc

dcccccc

θ

u

u

u

I000

0m00

00m0

000m

v

2

1

v

2

vs2vs1vs2vs1vs2vs1

vs2vp2vs2vs2

vs1vp1vs1vs1

vs2vs1vs2vs1vs2vs1

v

2

1

v

2

vs2vs1vs2vs1vs2vs1

vs2cp2vs2vs2

vs1vp1vs1vs1

vs2vs1vs2vs1vs2vs1

v

2

1

v

v

ns2

ns1

s



0

0

0

0

0

θ

u

u

u

u

dk k dk 0dk dk k

d-k k k 00k

00k k 0k

dk 00k k k

dk k k k k k k k

θ

u

u

u

u

dccdc0dcdcc

dccc00c

00cc0c

dc00ccc

dcccccccc

θu

u

u

u

I00000m000

00m00

000m0

0000m

v

3

2

1

v

2

vs3vs1vs3vs1vs3vs1

vs3vp3vs3vs3

vp2vs2vs2

vs1vp1vs1vs1

vs3vs1vs3vs2vs1vs3vs2vs1

v

3

2

1

v

2

vs3vs1vs3vs1vs3vs1

vs3cp3vs3vs3

cp2vs2vs2

vs1vp1vs1vs1

vs3vs1vs3vs2vs1vs3vs2vs1

v

3

2

1

v

v

ns3

ns2

ns1

s

The moving load is modelled by an infinite series of equal vehicles, regularly spaced, and running

at constant velocity, . Each vehicle is considered by a two, three or four masses model,respectively, with in plane degrees of freedom (translational and rotational) and one, two or even

three axles. If l is the distance between two successive vehicles and as these cars enter one byone into the bridge deck, it is created a time repeated movement variation governed by the

frequency, f t = /l (traversing frequency), associated with the movement of the vehicles on thebridge. After a certain time period, t1 (crossing period), the first vehicle in the train reaches the farend of the bridge and from this instant, on the total mass of the vehicles on the bridge remains

practically constant. Under these conditions the bridge will soon reach a steady-state responsesituation, which includes repetition of maximum values directly related to the fatigue strength.



4.2. Simple Supported Beam Deck

Inertia moment: I=3.98m

4

;Young’s modulus: E=3x107kN/m

2;

Distributed mass: m=9200kg/m;Frequency (unloaded deck): 6.3Hz;Frequency (loaded deck: vehicles 120kN and 450kN): 6.6Hz and 7.2Hz;

Damping factor: =0.03;Finite Element Model.

0,12m

2,50m

10,00m

0,30m

0,12m

2,26m

9,20m 0,40m0,40m

...

17

13

L

m1 m2 m10 m11...



Time Response: displacement.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

t/t1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

v/vest

Displacement: load mobility effectSimple supported beam: L=30mInfinite train of vehicles: V=125km/h e P=450kNSteady-state phase: t/t1 > 0,80

Time Response: bending moment.

0.80

1.00

1.20

Bending moment: load mobility effectSimple supported beam: L=30mInifinite train of vehicles: V=125km/h e P=450kNSetady-state phase: t/t1 > 0,80



Time Response: reaction force.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

t/t1

0.00

0.20

0.40

0.60

0.80

1.00

1.20

R/Rest

Reaction force: load mobility effectSimple supported beam: L=30mInfinite train of vehicles: V=125km/h e P=450kNSteady-state phase: t/t1 > 0,80

Response Spectra: displacement.

1.08

1.10

1.12

Load mobility effect

Weight of the vehicles: 450kN


Response spectraLoad mobility effectDisplacement



Response Spectra: bending moment.

0.0 1.0 2.0 3.0

Beta=v/l.f01

1.00

1.02

1.04

1.06

1.08

1.10

1.12

FA=M/Mest

Response spectraLoad mobility effect

Bending moment




Response Spectra: reaction force.

1.08

1.10

1.12

Response spectraLoad mobility effectReaction force






4.3. Continuous Beam Deck

Finite Element Model

m1 m34 m35m2 ......

L Le = 0.83LLb = 0.25L Le = 0.83L Lb = 0.25L

1 2 34 2614 36

399 3120

Variations of Beam Fundamental Frequency with Bridge Geometry

MiddleSpan

End Span OverhangsSectionSpacing

Unloaded Deck Loaded Deck

L(m) Le(m) Lb(m) Ls(m)Lowest Mode

Frequency (Hz)Lowest Mode

Frequency (Hz)

24.00 20.00 6.00 2.00 9.12 9.22

30.00 25.00 7.50 2.50 7.43 7.78

36.00 30.00 9.00 3.00 5.20 5.86



Maximum Displacements: FA= v/vest.

BRIDGE: Continuous beams deck with overhangs. Spans 24m, 30m e 36m.CRITICAL VELOCITY (km/h): 130, 140 e 125, respectively.SECTIONS: S1 and S20.

Middle SpanL(m)

Maximum amplification factor: FA = v/vest

Section S1 Section S20

24.0m 2.68 1.54

30.0m 2.42 1.77

36.0m 1.98 1.54

Maximum Bending Moments: FA= M/Mest.

BRIDGE: Continuous beams deck with overhangs. Spans 24m, 30m e 36m.CRITICAL VELOCITY (km/h): 130, 140 e 125, respectively.SECTIONS: S4, S14 and S20.

Middle SpanL(m)

Maximum amplification factor: FA = M/MestSection S4 Section S14 Section S20

24.0m 1.15 1.29 1.42

30.0m 1.15 1.33 1.62

36.0m 1.12 1.30 1.44



Maximum Reaction Forces: FA= R/Rest.

BRIDGE: Continuous beams deck with overhangs. Spans 24m, 30m e 36m.

CRITICAL VELOCITY (km/h): 130, 140 e 125, respectively.SECTIONS: S4 and S14.

Middle SpanL(m)

Maximum amplification factor: FA = R/Rest

Section S4 Section S14

24.0m 1.23 1.2130.0m 1.25 1.20

36.0m 1.22 1.19



5. CONCLUSIONS

Infinite Train of Vehicles suitable solution to the computation ofmoving load response values for highwaybridge decks where the steady-stateresponse can be a major concern as in the

case of the bridge fatigue behaviour;

Response Values dynamic amplification factors (DAF), due to theload mobility, more than twice of those obtained

with only one moving vehicle;Irregular Pavement Surface the amplifications can be still much

larger than those due to the load

mobility;The Redundant Beam Problem deserves more attention to widely

be verified the DAF, when the staticvalues obtained according todesign codes are considered.

dinâmica estrutural ii_pontes_01

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