[american institute of aeronautics and astronautics 49th aiaa/asme/asce/ahs/asc structures,...

American Institute of Aeronautics and Astronautics

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Finite Element Approach for Vehicle-Structure Interaction of a Uniform Timoshenko Bridge

Mahsa Moghaddas1 and Ramin Sedaghati2

Department of Mechanical and Industrial Engineering, Concordia University, Montreal, Quebec, H3G 1M8, Canada

Ebrahim Esmailzadeh3

Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, 2000 Simcoe Street North, Oshawa, Ontario, L1H 7K4, Canada

and

Peyman Khosravi4 Department of Civil Engineering, Dalhousie University, Halifax, Nova Scotia, B3J 1Z1, Canada

Abstract In this study the finite element formulation for the dynamics of a bridge traversed by moving vehicles is presented. The vehicle including the driver and the passenger is modeled as a half-car-planner model with six degrees of freedom, traveling on the bridge with constant velocity. The bridge is a uniform with simply supported end conditions, and it obeys the Timoshenko beam theory. The equations of motion are derived using the extended Hamilton’s principle, and transformed to the finite element form using the weak-form formulation. Newmark’s method is applied to solve the governing equations and the results are compared with those available in the literature. Also, the maximum deflections in Timoshenko and Euler-Bernoulli beams have been compared. The results illustrated that as the velocity of the vehicle increases, the difference between the maximum beam deflection in two beam models becomes more significant.

I. Introduction

The dynamic response of bridge structures subjected to moving vehicles or trains is an important problem encountered in their design. Numerous studies have been done concerning this topic. In early studies a constant moving load model was used where the inertia of the vehicle was small compared to the beam, and the interaction between the vehicle and the bridge was ignored1,2,3,45. With the great raise in the percentage of heavy trucks and high speed vehicles in highway and railway traffic, the dynamic interaction problem between vehicles and bridge structures has attracted a great amount of attention during the last three decades. Other models have been also considered regarding bridges under a moving mass which allow for the inertia of the vehicle 5,6,7. Although these simulations are more complete than the earlier model as they allows for the inertia of the vehicle, they suffer from inability to consider the bouncing effect of the moving mass. In the more inclusive models the vehicle has been simulated as a series of mass-spring-damper elements, providing detailed consideration of vehicle-bridge dynamic analysis8,9. Esmailzadeh and Jalili10 have investigated the dynamics of the vehicle-structure interaction of a bridge traversed by moving vehicles taking into account the passenger dynamics for a more realistic simulation. The vehicle, containing the driver and the passenger, was modeled as a mobile half planer model traveling on a wide span uniform bridge modeled in the form of a simply supported Euler-Bernoulli beam. The response of the beam and the vehicles were studied by modal expansion (Galerkin approximation).

In this paper the finite element formulation of a simply supported Timoshenko beam under a half car planner model is presented. Linear beam elements with two degrees of freedom at each node were utilized to model the beam. The equations of motion in matrix form for the Timoshenko beam element carrying the moving half car model with six degrees of freedom are derived using Hamilton’s principle. The entire equations of motion for the 1 M.Sc. Student, E-mail: [email protected] 2 Associate Professor, AIAA member, E-mail: [email protected] 3 Professor, AIAA member, E-mail: [email protected] 4 Postdoctoral Fellow , AIAA member, E-mail : [email protected]

49th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference <br> 16t7 - 10 April 2008, Schaumburg, IL

AIAA 2008-2310

Copyright © 2008 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


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system can be obtained by assembling the matrices of all conventional Timoshenko beam elements and the Timoshenko beam element with the moving vehicle on it. The problem is solved by direct integration using Newmark-β method11 to obtain the dynamic response of the Timoshenko beam and the vehicle components. Results are validated by a verification example.

II. Theory and formulation

Figure 1 shows the suspension system of a 6-degree of freedom half car model moving on a bridge. It is assumed that the vehicle moves on the bridge with the constant velocity ( )u t& , where ( )u t is the location of the center of gravity (c.g.) of the vehicle body measured from the left end support of the bridge, and both front and rear tires remain in contact with the bridge surface constantly.

Figure 1 Suspension system of a 6-dof car model moving on a bridge

The vehicle is modeled as a 6-dof system which consists of a body (sprung mass), two axles (unsprung masses),

driver and a passenger. Each of the masses has only vertical oscillation and the body is considered to have the angular motion (pitch) in addition. The compliance of the suspension system, the tires and the passenger seats are modeled by combination of linear springs and viscous dampers connected in parallel configurations. The bridge is

2pm 1pm

1pK 1pC 2pK 2pC

,sm J

Vehicle Passengers

Vehicle Suspension System

Vehicle Tire

2tm 1tm

2C 2K

2tC 2tK

1C 1K

1tC 1tK

2d 1d

2F 1F 2b

1b

θ sy

1py

1ty2ty

2py

X

2F 1F

( )u t

2 ( )tξ

1( )tξ

2b 1b

Z

Y


3

contemplated initially free of any load or deflection and therefore at the equilibrium under its own weight. The steady state displacements of the vehicle are also measured from their static equilibrium position. In order to generate the governing equations of motion of the presented model, the Hamilton’s principle is applied:

2 2

1 1

( ) 0t t

nct tT U dt W dtδ δ− + =∫ ∫ (1)

where Tδ is the virtual total kinetic energy, Uδ is the virtual potential energy and ncWδ is the virtual non-conservative forces of the system. The equations of motion for the Timoshenko beam can be written as12:

2

2

2

2

( , ) 0

0

s

s

y yA k AG f x tt x x

yI EI k AGt x x x

ρ ψ

ψ δψ δρ ψδ δ

⎡ ⎤∂ ∂ ∂⎛ ⎞− + − =⎜ ⎟⎢ ⎥∂ ∂ ∂⎝ ⎠⎣ ⎦∂ ∂ ⎛ ⎞ ⎛ ⎞− + + =⎜ ⎟ ⎜ ⎟∂ ∂ ⎝ ⎠ ⎝ ⎠

(2)

where ρ is the beam volumetric density, I is the cross-sectional moment of inertia, A is the cross-sectional area, E

is Young’s modulus of elasticity, sk is the shear correction factor in Timoshenko beam theory and G is the shear

modulus. ( , )y x t and ( , )x tψ are the traverse elastic deflection and the orientation of the beam cross-section, respectively.

The total Kinetic energy of the system is consisted of the kinetic energy of the beam and the kinetic energy of the vehicle:

2 2

0 0

12

L L

vehicleyT A dx I dx Tt t

δ δψρ ρδ δ

⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞= + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∫ ∫ (3)

And the system virtual kinetic energy is defined as

2 2 2

1 1 1

2 2

2 20 0

t t L L t

vehiclet t t

yTdt A dx y I dx dt T dtt t

ψδ ρ δ ρ δψ δ⎧ ⎫⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪= − + +⎨ ⎬⎜ ⎟ ⎜ ⎟∂ ∂⎪ ⎪⎝ ⎠ ⎝ ⎠⎩ ⎭

∫ ∫ ∫ ∫ ∫ (4)

Besides, the total potential energy of the system includes the potential energy of the beam and the kinetic energy

of the vehicle: 2 2

0 0

12

L L

s vehicleyU EI dx k AG dx U

x xδψ δψδ δ

⎧ ⎫⎪ ⎪⎛ ⎞ ⎛ ⎞= + + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∫ ∫ (5)

The system virtual potential energy is defined as:

2 2

1 1

2

1

2 2

2 20 0

t t L L

s st t

t

vehiclet

y yUdt EI k AG dx k AG dx y dtx x x x

U dt

ψ δ δψδ ψ δψ δδ δ

δ

⎧ ⎫⎛ ⎞⎛ ⎞ ⎛ ⎞∂ ∂⎪ ⎪⎛ ⎞= − − + − + +⎜ ⎟⎨ ⎬⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂⎝ ⎠⎝ ⎠ ⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭∫ ∫ ∫ ∫

∫ (6)

and the virtual work of the non-conservative forces in the system can be written as:

2 2 2

1 1 10 0( , )

vehicle

t t L L t

nc nct t t

yW dt c dx y f x t dx y dt W dtt

δδ δ δ δδ

⎧ ⎫⎛ ⎞= − − +⎨ ⎬⎜ ⎟⎝ ⎠⎩ ⎭

∫ ∫ ∫ ∫ ∫ (7)

where c is the damping coefficient of the beam and ( , )f x t is the vehicle weight acting of the bridge which is

consisted as 1F and 2F shown in Figure 1. For the finite element analysis of the beam the Lagrange linear interpolation of y and ψ are considered in the

form:


4

2

1( ) ( )j j

jy x w tϕ

=

=∑ (8)

2

1( ) ( )j j

jx tψ ϕ

=

= Ψ∑ (9)

where 1( )xϕ and 2 ( )xϕ are the linear interpolation functions.

By writing Hamilton’s principle and setting the coefficient of virtual displacement to zero, the governing differential equations can be derived. Then using Galerkin method and weak formulation, the set of equations of motion in the form of 6 discrete equations for the vehicle and 4 finite element equations for the Timoshenko element carrying the vehicle can be generated. The total equations of motion can be obtained by assembling the matrices of the element carrying the vehicle to rest of the element matrices.

The final equations of motion can be cast into the following form:

[ ]{ } [ ( )]{ } [ ( )]{ } [ ( )]M Q C t Q K t Q F t+ + =&& & (10)

where [M] is the total mass matrix of the system, [C(t)] is the total damping matrix of the system, [K(t)] is the system’s total stiffness matrix ,and [F(t)] is the total force matrix of the system. It should be noted that as the vehicle is moving on the beam, the total damping, stiffness and force matrices of the system are varying by time.

The equations of motion in Eq. (10) have been solved by Newmark-β method in order to obtain simultaneously the dynamic response of the Timoshenko beam and vehicle.

III. Numerical Example In order to validate the results, the numerical simulation is performed, with the same parameter values presented

in Ref. 10 as follows: Bridge:

4 3100 , 207 , 0.174 , 4050 / , 1750 / .L m E GPa I m kg m c Ns mρ= = = = = Vehicle

1 2 1 21794.4 , 87.15 , 140.4 , 75 ,s t t p pm kg m kg m kg m m kg= = = = = 2

1 2 1 23443.05 , 1.271 , 1.716 , 0.481 , 1.313 ,J kgm b m b m d m d m= = = = =

1 2 1 2 1 266824.4 / , 18615.0 / , 101115.0 / , 14000.0 /t t p pk N m k N m k k N m k k N m= = = = = =

1 2 1 2 1 21190 / , 1000 / , 14.6 / , 62.1 / .t t p pc Ns m c Ns m c c Ns m c c Ns m= = = = = = where L is the length of the beam and the vehicle parameters are presented in Figure 1.

It is noted that in Ref. 10, the beam has been modeled as Euler-Bernoulli and the governing equations have been solved analytically using mode superposition technique. Considering this in order to have a fair comparison, the formulation in the present works has been adjusted based on Euler-Bernoulli beam formulation. The equations of

motion for the vehicle-bridge interaction system is solved by Newmark-β method with 12

γ = and 14

β = and time

step 0.01t sΔ = .

Figure 2 shows the time history of the midspan deflection and Figure 3 illustrates the time history of the vehicle body bounce for the vehicle speed of 88 Km/h, with different number of elements. It can be observed that as the number of beam elements increases the results become closer to the results of analytical approach presented in Ref. 10 and excellent agreement exists between them for 50 number of beam elements.


5

0 1 2 3 4 5 6 7 8 9 10-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

mid

span

def

lect

ion(

cm)

time(s)

Ref. 1010 Elments40 elements50 elements

Figure 2. Time history of the beam midspan for V=88 Km/h

0 1 2 3 4 5 6 7 8 9 10-2.5

-2

-1.5

-1

-0.5

0

0.5

Veh

icle

Bod

y c.

g. B

ounc

e (c

m)

time(s)

Ref. 1010 Elments40 elements50 elements

Figure 3.Time history of the vehicle body bounce for V=88 Km/h

IV. Compassion between Timoshenko and Euler-Bernoulli The maximum beam deflection versus the vehicle velocity has been plotted for both Timoshenko and Euler-

Bernoulli beams and shown in Figure 4. The parameters of the beam models have been chosen the same as the previous example, and for the Timoshenko beam the following parameters have been used:


6

279.6 , 4.94 , 0.83sG GPa A m k= = =

It can be observed from the diagram that the maximum beam deflection has been underestimated in Euler-Bernoulli beam model. This difference is slight in lower velocities, but as the vehicle velocity increases, the disparity between the two beam models becomes more significant.

40 50 60 70 80 90 100 110 120 130 140 15035

-2.2

-2.1

-2

-1.9

-1.8

-1.7

-1.6

-1.5

-1.4

Vehicle Velocity(Km/h)

Max

imum

Mid

span

Def

lect

ion

(cm

)

Timoshenko beamEuler-Bernoulli beam

Figure 4. Maximum Beam Deflection versus Vehicle Velocity for Timoshenko and Euler_Bernoulli Beams

40 50 60 70 80 90 100 110 120 130 140 1504

6

8

10

12

14

16

Vehicle Velocity (km/h)

Per

cent

age

Diff

eren

ce B

etw

een

max

imum

Def

lect

ions

Figure 5. Percentage Difference between Maximum Beam Deflections in Timoshenko and Euler-Bernoulli

Beams versus Vehicle Velocity


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For more clarification, in Figure 5 the percentage difference between maximum deflections in Timoshenko and Euler-Bernoulli beams versus vehicle velocity has been plotted in the velocity range of 35-150 Km/h. It can be observed that the percentage difference increases by the rise of vehicle velocity and adopts an almost constant value with slight fluctuations after about V=110 Km/h.

V. Conclusion

The finite element formulation for the dynamics of a bridge traversed by a moving vehicle has been presented.

The vehicle including the two passengers is simulated as a half-car-planner model with six degrees of freedom, moving on the bridge with constant velocity. The bridge has been modeled as a uniform simply supported Timoshenko beam. The equations of motion have been derived and transformed to the finite element form using the Galerkin method and weak-form formulation. Using time integration Newmark’s method the governing equations have been solved and the obtained results are compared with those reported in the literature. Moreover, the comparison between the maximum deflections in Timoshenko and Euler-Bernoulli beams has been performed. The results show that by increasing the vehicle velocity, the difference between the maximum beam deflection in two beam models becomes more significant.

References

1. C. E. Inglis, A Mathematical Treatise on Vibration in Railway Bridges, Cambridge University Press, 1934.

2. S. Timoshenko, D.H. Young, W. Weaver, Vibration Problems in Engineering, 4th edition, Wiley, New York, 1974

3. F. V. Hilho , Finite Element analysis of Structures under Moving Loads, Shock and Vibration Digest ,Volume 10,1978, pp. 27-35

4. M. Olsson , Finite Element, Modal Co-ordinate Analysis of Structures Subjected to Moving Loads, Journal of sound and vibration Volume 99, Number 1,1985, pp. 1-12.

5 H. P. Lee , Dynamic Response of a Beam with Moving Mass, Journal of sound and vibration Volume 198, Number 2,1996, pp. 249-256.

6 E. Esmailzadeh, M. Ghorashi, Vibration Analysis of a Timoshenko Beam Subjected to a Traveling Mass, Journal of sound and vibration Volume 199, Number 4, 1997, pp. 615-628.

7 P. Lou, G.-L. Dai, Q.-Y Zeng, Finite Element Analysis for a Timoshenko Beam Subjected to a Moving Mass, Proceedings of the I MECH E Part C Journal of Mechanical Engineering Science, Volume 220, Number 5, 2006 , pp. 669-678 8 E. S. Hwang and A. S. Nowak, “Simulation of Dynamic Load for Bridges”, Journal of Structural Engineering (ASCE) Volume 117, Number 5, 1991, pp. 1413-1434.

9 P. K. Chatterjee, T.K. Datta, C.S. Surana, Vibration of continuous Bridges under Moving Vehicles, Journal of sound and vibration Volume 169, Number 5, 1994, pp. 619-632.

10 E. Esmailzadeh, N. jalili, Vehicle-Passenger-Structure Interaction of Uniform Bridges Traversed by Moving Vehicles, Journal of sound and vibration Volume 260, Number 4, 2003, pp. 611-635. 11 K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice Hall, Englewood Cliffs, N.J., 1982

12 J. N. Reddy, An Introduction to the Finite Element Method, 2nd Edition, McGraw-Hill, 1993.

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