Respuesta Dinamica de Puentes Atirantados

7/30/2019 Respuesta Dinamica de Puentes Atirantados

1/6

Dynamic Response of Cable-Stayed Bridges Subjected to Moving Vehicles

Raid KAROUMI

M.Sc., Ph.D. student

Dept. of Structural Eng.

Royal Inst. of Technology

Stockholm, Sweden

E-mail:

[email protected]

Raid Karoumi, born 1964,

received his M.Sc. degree in Civil

Engineering in 1990 from the

Royal Institute of Technology in

Stockholm. Between 1990 and

1993 he worked as a consultant

and was involved in a variety of

special projects. Since 1993, he

has been working with his Ph.D.

on the dynamic response of cable-

stayed bridges.

Summary

The response of a cable-stayed bridge under a moving vehicle is complex due to the

interaction between the bridge and the vehicle. This paper describes a method of evaluating

this response by idealizing the bridge as a Bernoulli-Euler beam on elastic supports with

varying support stiffness. The analysis uses the mode superposition technique and calculates

the response in time domain, utilizing an iterative scheme. To illustrate the efficiency of the

solution methodology and to highlight the dynamic effects, a numerical example is presented.

1. Introduction

With increasing span length and increasing slenderness of the stiffening girder, the dynamic

behaviour of cable-stayed bridges requires a lot of consideration. The dynamic response of

bridge structure subjected to moving vehicles is complicated. This is because the dynamic

effects induced by traversing vehicles on the bridge are greatly influenced by the interaction

between vehicles and the bridge structure.

In recent years the dynamic behaviour of cable-stayed bridges has been a source of interesting

research. This includes free vibration and forced vibration due to wind and earthquakes

[1,3,4]. However, literature dealing with the dynamic of these bridges due to the vehicular

movements are relatively few. Fleming and Egeseli [3] investigated the dynamic response of a

cable-stayed bridge to seismic and wind loads as well as the load of a single moving constant

force. Chatterjee et al. [2] used a continuum approach for finding the vibration of cable-stayed

bridges due to vehicular movement. Wang and Huang [7] conducted a similar study but used

the finite element method for the analysis. Miyazaki et al. [6] used the finite element method

and studied the running quality of prestressed concrete multicable-stayed railway bridges.

In all the aforementioned studies, authors either used very simple vehicle models [3] (constantforce) or very complicated and time-consuming vehicle and bridge models [2,6,7].


2/6

The objective of this study is to analyze the cable-stayed bridge dynamic response using a very

simple bridge model, namely a continuous beam on elastic supports and a simple vehicle

model consisting of a single sprung mass. The numerical results are compared with the results

obtained using the commercial finite element codeABAQUSto validate the model. The results

indicate that the model presented in this paper is very effective for calculating the bridgedynamic response to moving vehicles.

2. Vehicle and Structure Modelling

2.1 Vehicle Model

The vehicle model used in this study is a suspension model with two degrees of freedom. This

model is sufficient if the real vehicle length is small compared with the bridge length. The

author believes that the use of simplified models may be more effective in identifying

correlation between the governing vehicle-bridge interaction parameters and the bridge

response.

If the contact force between the bridge and the vehicle is denoted by ( )F t , the following

equations of motion can be established

( ) ( ) ( ) + + +

+ =m m gt

k w w ct t

F tS S1 2 11

2 12 1m

w w wd

d

d

d

d

d

2

20 (1)

( )m t k w w c t tS S22

2 12 1d

d

d

d

d

d

2

2 0

w w w

+ +

= (2)

where ( )w t1 and ( )w t2 are the displacements of the vehicle unsprung mass m1and the vehiclesprung mass m2 , respectively, kS the stiffness of the spring connecting the two masses, cSthe damping coefficient of the viscous damper, and g the acceleration of gravity.

Assuming that the vehicle never loses contact with the bridge (that is ( )F t > 0) , the following

coupling equations for the point of contact ( x t x tv= ) must be fulfilled:

( ) ( )( )w t y x t ,t 1 = , ( )w ty

x vy

t1 = +

, ( )w t

y

xv

y

x t vy

x ay

t1 = + + +

2

2 2

2 2

22 , (3a-c)

where ( )w t1 and ( )w t1 denote the unsprung mass vertical velocity and acceleration,respectively, v and a the vehicle velocity and acceleration, respectively, and y x t ,t the

bridge displacement.

2.2 Bridge Structure

For the present study, the self anchored fan-shaped cable-stayed bridge scheme with only the

main span suspended as shown in Fig. 1 is adopted. The following assumptions are made:


3/6

Fig. 1. Idealized vehicle in contact with a cable-stayed bridge

uniform mass and flexural rigidity of the stiffening girder

cable forces under dead load are so adjusted that all displacements remain zero

axial girder forces have negligible effect on frequencies and mode shapes

negligible cable mass

bridge damping is small and therefore neglected

bridge/vehicle system is at rest when the vehicle enters the main span of the bridge

road surface is in excellent shape. This means that the excitation of the dynamic system iscaused only by the elastic displacement of the bridge.

The governing equation of motion for vertical vibration of the bridge at any section of the

stiffening girder (idealized as a Bernoulli-Euler beam on elastic supports) is given by

( )( ) ( )

( )( ) ( )E I

y x,t

xk x y x,t m

y x,t

tx x F tg g

4

4 c g

2

2 v

+ + = (4)

where is the Dirac delta function, Eg the modulus of elasticity, Ig the moment of inertia,mg the mass per unit length, and kc the spring stiffness. The effects of rotatory inertia and

shear deformation are neglected as the cross-sectional dimensions of the stiffening girder are

small in comparison with its length and the higher vibration modes are not significantly

excited.

This paper will present only the simple case with a fixed pylon top, which roughly simulates a

cable-stayed bridge with short side spans and very stiff anchorage cables. (A more realistic

case is presented in [5]). If the cables are approximated by springs and fixed pylon top is

assumed, the following spring stiffness can be derived:

( )( )

k xE x q

H x

H

0 xL

cc

a

=

+

1

122

(5)


4/6

where a is the allowable cable stress and q the dead load plus the live load per unit length.The cable geometric nonlinearity, due to the change of the sag and shape under varying

stresses (forces), is approximately taken into account by using an equivalent tangent modulus

of elasticity

( )E x

cin equation (5). The derivation of equation (5) and an expression that

includes the effect of the elongation of the cables in the main span, the elongation of the

anchorage cables, and stiffness of the pylons, is given in [5].

Equations (1-5) together with the boundary and initial conditions, now provide all the

information necessary for a solution of the problem.

3.0 Dynamic Analysis

Using mode superposition technique, the solution of equation (4) can be written as a product

of eigenfunctions (mode shape functions) ( )z xi and generalized coordinates ( )i t giving

( ) ( ) ( )y x t z x tii

i, ==

1

(6)

where i is the mode number. The first natural step will be to study the free vibration of the

bridge structure by using equation (4) and setting the right-hand side equal to zero. By

replacing y(x,t) in equation (4) with the product of the two functions mentioned above,

dividing the beam into n equal segments each of length h, giving n-1 unknowns, and replacing

the derivative in (4) with its finite difference approximation, the following eigenvalueproblem is obtained:

A Z Z= (7)

where A is a symmetric banded matrix with a bandwidth of 5. The first diagonal elements

contain functions of the spring stiffnesses, the second diagonal elements contain -4, and the

third diagonal elements 1. Z is a matrix with the mode shapes stored in the different columns,

and a diagonal matrix containing the eigenvalues. Mode shapes and corresponding circularfrequencies are now obtained numerically by solving equation (7).

The next step is to study the forced vibration. By substituting (6) into (4), using theorthogonality properties of the eigenfunctions, and setting the normalization constant equal to

unity, the following equation of motion in generalized coordinates is obtained:

( ) + 2i i ig

i v

F

mz x= (8)

When dividing the beam into n equal segments, the term z xi v in equation (8) will be the

value in the ith column in Z and the row number corresponding to the point of contact.

Solving the simple second order differential equation (8), the bridge displacement is now

obtained from the sum


5/6

( ) ( )( )y z= +

++

=

i i vi g

i v i i g

i g

ii

i

i

i

n F z x

m

F z x m

mt t

2

02

20

1

1

cos

sin (9)

The bridge vertical acceleration is obtained by differentiating equation (9) twice. The dynamic

interaction between the bridge and the vehicle is included by utilizing an iterative scheme

where the recalculation of the bridge displacement is repeated in each step until convergence

is obtained.

4. Numerical Example

Results obtained from the present model and from the FEM code ABAQUS are shown in

Fig. 2 below.

1

1,1

1,2

1,3

1,4

1,5

1,6

1,7

50 100 150 200 250 300

Speed (Km/h)

D

ynamicamplificationfactorfor

deflection,max(ydyn

)/max(ysta

) Ig=0,3

Ig=0,5

Ig=0,8

Frequencies (Hz) for Ig=0,5 m4

Mode

number

Present

analysis

ABAQUS

1 0,5662 0,5628

2 0,7849 0,7785

3 1,0164 1,0150

4 1,2747 1,2750

5 1,6164 1,6187

6 2,0899 2,0950

7 2,6951 2,7046

8 3,4219 3,4384

-0,035

-0,03

-0,025

-0,02

-0,015

-0,01

-0,005

0

0,005

0 0,2 0,4 0,6 0,8 1

Vehicle position, x v/L

Midpointd

eflection(m)

Present

ABAQUS

D namicStatic

Ig=0,5 m4

v=125 Km/h

-0,20

-0,15

-0,10

-0,05

0,00

0,05

0,10

0,15

0,20

0 0,2 0,4 0,6 0,8 1

Vehicle position, x v/L

Sprungmassvert

icalacceleration

(m/s

2)

Ig=0,5 m4

v=125 Km/h

Fig. 2. Results obtained using the present analysis method and comparison of natural

frequencies (Hz) and mid point deflection with results from the FEM code ABAQUS

Bridge and vehicle data:


6/6

L H I m qg g= = = 300 12 103

m; = 45 m; 0,3 -0,8 m ; kg / m; =17 10 N / m;4 4

( )a g cE E x n= 720 10 N / m ; N / m ;6 2 2 = = = =0 2 1 10 10011, ;

m m c k S S2 ,= = = = 32700 0 12 5 104kg; kg; Ns / m; 12 10 N / m1

6 .

8 modes are included in the present analysis and inABAQUSdirect time integration was used.

In [5], a more detailed investigation is done and figures for the bending moment are shown.

5. Conclusions

The dynamic response of cable-stayed bridges is much more complicated than that of

beam/girder bridges. It is clear that the modelling assumptions outlined above ignore several

factors that may significantly affect the response, particularly those related to stiffness of theanchorage cables, road surface roughness, and vehicle configuration. The author believes that

a study of the simplified model is very useful in identifying the important parameters.

Furthermore, once the basic parameters have been identified, it is possible to refine the model

to include other variables.

The simple model presented here gives accurate results and only consumes a c.p.u. time on the

order of a few minutes. The effects of road surface roughness, vehicle type, and structural

configuration on the dynamic response will be the topic of a future paper by the author.

6. References

[1] Abdel-Ghaffar A.M., Nazmy A.S., 3-D Nonlinear seismic behaviour of cable-stayed

bridges, J. Struct. Engng., ASCE, Vol. 117, No. 11, Nov. 1991, pp. 3456-3476.

[2] Chatterjee P.K., Datta T.K., Surana C.S., Vibration of cable-stayed bridges under

moving vehicles, Struct. Engng. Int., No. 2, 1994, pp. 116-121.

[3] Fleming J.F., Egeseli E.A., Dynamic behaviour of a cable-stayed bridge, Earthquake

Engng. Struct. Dynamics, Vol. 8, 1980, pp. 1-16.

[4] Karoumi R., Aerodynamic stability, wind forces on suspension and cable-stayed

bridges (in Swedish), TRITA-BKN Report No. 11, Dept. of Struct. Engng., RoyalInstitute of Technology, Stockholm, 1994.

[5] Karoumi R., Dynamic response of cable-stayed bridges subjected to moving vehicles,

Licentiate Thesis, Dept. of Struct. Engng., Royal Institute of Technology, (in printing).

[6] Miyazaki S., Kanamori M., Wakui H., Matsumoto N., Sogabe M., Analytical study on

dynamic response and riding comfort of PC multicable-stayed railway bridge, Int.

Conference on Speedup Technology for Railway and Maglev Vehicles, Yokohama, Nov.

1993, pp. 424-429.

[7] Wang T.L., Huang D., Cable-stayed bridge vibration due to road surface roughness,

J. Struct. Engng., ASCE, Vol. 118, No. 5, May. 1992, pp. 1354-1374.

Respuesta Dinamica de Puentes Atirantados

Documents

Transcript of Respuesta Dinamica de Puentes Atirantados