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Multi-Span Laminated Composite Beam Traversed by Moving Non-rigid Masses

M.T. Ahmadian 1, R.A. Jafari-Talookolaei 2

and

E. Esmailzadeh 3

A laminated composite beam, with intermediate point constraints, resting on viscoelastic foundation subjected to moving non-rigid masses is analyzed. First-order shear deformation theory and Galerkin's method are employed to analyze the model. The bending moment and deflection of the beam at its centre and just below the location of the moving masses are examined and the corresponding velocity for the maximum values of those parameters is determined. The influence of the load velocity and the span number of the laminated composite beam on the maximum shear force and the bending moment are also investigated. The possibility of separation of moving masses from the beam during the motion is studied by monitoring the contact forces between them. It is found that separation of the moving masses from the slender laminated composite beam may occur with a high stiffness of the non-rigid masses and by having either a low or a high axial momentum of the oscillator. This separation can be suppressed by an elastic foundation with a relatively large stiffness and increasing span number.

Nomenclature dD = Maximum deflection at the beam

center ,f fk c = Normal stiffness and damping

coefficients of foundation mD = Maximum bending moment at the

beam center μ = Shear viscosity coefficient of

foundation fdD = Maximum deflection under the

oscillator ,

x xk cψ ψ = Rocking stiffness and damping

coefficients of foundation fmD = Maximum bending moment under

the oscillator fT = Fundamental period of the beam

( ), ( )aq t q t

= Vertical displacements of the masses measured from their equilibrium position

τ = Travel time of the oscillator moving from left end of the beam to the right

(u, w) = Mid-plane displacement in the x and z directions

L = Beam length

( , )x yψ ψ = Mid-plane bending slopes S = Oscillator position b = Beam width

cF = Normalized contact force

v = Axial velocity of the oscillator cF = Contact force

1 2( , )I I = Inertia coefficients ( , )am m = Masses of oscillator

uk , wk = Linear springs in the x and z directions, respectively. sn = Number of spans

δ = The Dirac-delta function

1 Professor, Centre of Excellence for Design, Robotics and Automation, School of Mechanical Engineering, Sharif University of Technology, Tehran, 11155-9567, Iran (ahmadian@sharif.edu) 2 Graduate Student, School of Mechanical Engineering, Sharif University of Technology, Tehran, 11155-9567, Iran (ramazanali@gmail.com) 3 Professor, Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Ontario, L1H 7K4 Canada, (ezadeh@uoit.ca)

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

AIAA 2008-1783

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

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I. Introduction n today's modern industries (aerospace, robotics, civil, marine, railway, etc.) the use of composite materials is becoming quite important due to its lightweight and high mechanical strength. Structures constructed in the field of transportation, say, bridges, guide-ways, railway tracks and roadways are

subjected to moving loads. In contrast to other dynamical loads, the moving forces due to the oscillatory non-rigid masses vary in both magnitude and position. Hence, the topic of moving oscillatory non-rigid masses makes a unique and interesting field of research in structural dynamics.

The influences of the shear deformation, rotary inertia and the location of the load distribution on the vibration of a Timoshenko beam subjected to a traveling mass has been studied by Esmailzadeh [1]. Lee investigated the possibility of the separation of a concentrated mass from the isotropic Timoshenko beam during the course of travel. He showed that the separation of mass from the beam may occur at high axial speeds of the mass [2]. Kadivar studied the finite element dynamic response of an un-symmetric laminated composite beam (LCB) subjected to moving loads based on the classical laminated theory (CLT), first-order shear deformation theory and the higher-order shear deformation theory (HSDT) [3].

The purpose of this paper is to study the effects of the load velocity on both the bending moment and the deflection of the beam at its centre and just below the oscillator due to the load velocity and position. The corresponding velocities for the maximum values of those parameters are presented. The influences of the load velocity and span number of LCB on the maximum shear force are investigated. The possibility of separation of the moving oscillatory masses from the LCB during the course of the motion is studied by monitoring the contact forces between the oscillator and LCB. Such a separation would violate the tacit assumption in all the reported studies that the beam and the oscillator remain in contact at all times. The intermediate point constraints are located equally along the LCB and modeled as linear springs with very large stiffness. These linear springs of sufficiently large stiffness will ensure that the points where the springs are attached will remain stationary during the transverse deformation of the LCB.

II. Mathematical Formulation A multi-span LCB resting on an elastic foundation and subjected to the moving oscillatory masses

is illustrated in Fig. 1. The parameters and a

q q are the displacements of the oscillator measured from

their equilibrium position such that when ,0== aqq the oscillator is at rest subjected to its own weight.

Fig. 1: Multi-span LCB resting on viscoelastic foundation

subjected to moving non-rigid masses

Using the Hamilton principle and employing the first-order shear deformation theory, one can obtain the governing differential equations of the motion as:

, , ,a a tt a a t a a a t am q c q k q c q k q+ + = + , , , ,( ) ( ) ( , ) ( , )tt a t a t a a t a amq c c q k k q kw t cw t c q k qξ ξ+ + + + = + + +

I

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1

1 , 11 , 11 , 16 ,1

( , ) ( )sn

utt xx x xx y xx i i

i

kI u A u B B u s t x sb

ψ ψ δ−

=

= + + − −∑ 1

1 , 55 , , , , , ,1

( )( ) ( ( , ) ( )) ( ( , ) ( )) ( ) ( , ) ( )sn

a wf ftt x x x t t t txx i i

i

m m g kk ck cI w A w w w t q t w t q t x w w w s t x sb b b b b b b

μψ ξ ξ δ ξ δ−

=

+⎡ ⎤= + − − + − + − − − + − −⎢ ⎥⎣ ⎦∑

2 , 11 , 11 , 16 , 55 , ,( ) x xx tt xx x xx y xx x x x x t

k cI B u D D A w

b bψ ψψ ψ ψ ψ ψ ψ= + + − + − −

2 , 16 , 16 , 66 ,y tt xx x xx y xxI B u D Dψ ψ ψ= + +

(1)

The possibility of separation of the oscillator from the LCB during the course of motion can be

detected by changes in the sign of the contact force between the oscillator and LCB. This contact force is defined to be positive if the force acting on the LCB is pointing downward in the z direction. A change of sign from the positive to negative indicates that the oscillator has separated from LCB and the equation of motion is no longer valid to describe the ensuing motion.

The expression for the contact force is given by:

, ,( ) ( ( , ) ( )) ( ( , ) ( ))c a t tF m m g k w t q t c w t q tξ ξ= + + − + − (2)

III. Analytical Solution The Galerkin’s method is used at this stage to separate the spatial coordinates from the temporal

variables and to yield a set of ordinary differential equations in time domain. Based on this method, the displacements of the beam for three different boundary conditions are considered to be as:

Clamped-Clamped Beam (CC)

Simply-Supported Movable

Beam (SSM) Simply-Supported Immovable

Beam (SSIM)

1 1

1 1

( , ) ( )sin( ), ( , ) ( )sin( )

( , ) ( )sin( ), ( , ) ( )sin( )

n n

n nn nn n

x x n y y nn n

n nu x t u t x w x t w t xL Ln nx t t x x t t xL L

π π

π πψ ψ ψ ψ

= =

= =

⎧ = =⎪⎪⎨⎪ = =⎪⎩

∑ ∑

∑ ∑

1 1

1 1

( , ) ( ) cos( ), ( , ) ( ) sin( )

( , ) ( ) cos( ), ( , ) ( )cos( )

n n

n nn nn n

x x n y y nn n

n nu x t u t x w x t w t xL Ln nx t t x x t t xL L

π π

π πψ ψ ψ ψ

= =

= =

⎧ = =⎪⎪⎨⎪ = =⎪⎩

∑ ∑

∑ ∑

1 1

1 1

( , ) ( )sin( ), ( , ) ( )sin( )

( , ) ( ) cos( ), ( , ) ( )sin( )

n n

n nn nn n

x x n y y nn n

n nu x t u t x w x t w t xL Ln nx t t x x t t xL L

π π

π πψ ψ ψ ψ

= =

= =

⎧= =⎪⎪

⎨⎪ = =⎪⎩

∑ ∑

∑ ∑

IV. Results and Discussion Dynamic magnification factor (DMF) for the mid-span deflection

dD and the momentmD are the

ratio of the maximum magnitude of the dynamic deflection and the moment at the beam centre to the corresponding static values of the beam centre in terms of the dimensionless parameter /

fT τ . The

symbol f

T denotes the fundamental period of the beam and τ represents the travel time of the oscillator

moving from the left end of the beam to the right end /L vτ = . From the static analysis, the static deflections are 3 / (48 )MgL EI and 3 / (192 )MgL EI for the

SS and CC boundary conditions, respectively. Also, the static moment considered in this study is / 4mgL .

The materials steel and AS4/3501 Graphite-Epoxy, having the following mechanical properties, are used for the computer simulation study.

For steel: 3206.8 , 79 , 0.3, 7800 /E Gpa G Gpa kg mυ ρ= = = =

For AS4/3501 Graphite-Epoxy [0/90/90/0]: 11 22 12 13

323 12

144.8 , 9.65 , 4.14 , 4.144.14 , 0.3, 13.89.23 /

E Gpa E Gpa G Gpa G GpaG Gpa kg mυ ρ

= = = == = =

Unless mentioned otherwise, the beam geometric dimensions, the oscillator and the foundation parameters are taken to be as:

/ 15, 0.065 , 0.25 , 1000 / , 100 /a a aL h b m m m kg k k N m c c Ns m= = = = = = = =

0, 0, 0, 0, 0x xf fk c k cψ ψμ= = = = =

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The fundamental period of these beams for the CC and SS boundary conditions are: For the laminated composite beam [4]:

3 3: ( / 15 1.999 10 ), :( / 15 3.689 10 ),f fCC L h T s SS L h T s− −= → = × = → = × For the steel beam [5]:

3 3: ( / 15 2.8 10 ), :( / 15 6.4 10 ),f fCC L h T s SS L h T s− −= → = × = → = × The comparisons of dynamic magnification factor DMF versus /fT τ for the steel and graphite

beams are shown in Figures 2 and 3. It can be seen that the DMF of SSIM beams for the moving oscillator can basically be divided into two regions: i) under the critical and ii) over the critical regions. When under the critical region, the DMF both increases and decreases with an increase in /

fT τ . In this

region, the DMF increases until /f

T τ = 0.3 and it reaches to a minimum value of DMF at /f

T τ = 0.4.

The main increase in DMF occurs only between 0.4 and 1.2. In the over critical region /f

T τ > 1.2 the

DMF decreases as /f

T τ increases.

Fig. 2: Variations of dD with respect to /fT τ for

the SSIM boundary condition Fig. 3: Variations of dD with respect to /fT τ for

the CC boundary condition The dynamic deflections at the beam centre versus the position of the moving oscillator at the critical velocity (i.e., the velocity at which, the maximum value of

dD occurs) are investigated. The

maximum deflection at the beam centre occurs when the moving oscillator reaches the values of 0.75 and 0.7 of the beam length for the SSIM and the CC boundary conditions, respectively. The critical velocity of the LCB is greater than that of the steel beam as can be seen from these figures. This is due to the difference in the fundamental period and the inertia of the beam. The effect of the oscillator velocity /

fT τ on

mD is presented in Figures 4 and 5 with the SSIM and CC

boundary conditions, respectively. It can be seen that the maximummD occurs at / 0.8fT τ = and

/ 0.7fT τ = for the SSIM and the CC boundary conditions, respectively.

Fig. 4: Variations of mD with respect to /fT τ for

the SSIM boundary condition Fig. 5: Variations of mD with respect to /fT τ for

the CC boundary condition In a similar manner and for the beam deflection, the maximum bending moment occurs when the

oscillator reaches the values of 0.56 and 0.52 of the beam length for the SSIM and the CC boundary conditions, respectively.

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The normalized deflection at the only intermediate point constraint /c c sw w w= for a two-span

Graphite-Epoxy beam and the normalized deflection /f f sw w w= under the moving oscillator traveling at the three constant axial velocities: i) under critical, ii) critical and iii) over critical velocities are shown in Figures 6 to 8. These three cases correspond to the axial velocities of 250 m/s, 325.27 m/s and 400 m/s, respectively. The stiffness wk used for the modeling of the point constraint is varied

between 910 and 1210 N/m. The deflection under the moving oscillator is found to be insensitive to these prescribe values of wk for all the considered cases. However, the deflection at the point constraint

is found to vary drastically with the assumed stiffness of these points. The value of 1010 /N m is found for wk to give a point constraint deflection, which is at least 1000 times smaller than the

corresponding maximum deflection under the moving oscillator. Also, the stiffness uk considered in

this paper is 1010 /N m .

Fig. 6: Deflections under the moving oscillator and at the intermediate

point constraint for a two-span LCB (v = 250m/s)

Fig. 7: Deflections under the moving oscillator and at the intermediate

point constraint for a two-span LCB (v = 325.27m/s)

Fig. 8: Deflections under the moving oscillator and at the intermediate

point constraint for a two-span LCB (v = 400m/s)

The normalized contact forces / (( ) )c c aF F m m g= + between the oscillator and the single span LCB for two cases of low and high axial momentum of the oscillator are shown in Fig. 9. The negative value of the contact force in these figures represents the separation of the oscillator from LCB while the positive value guaranties that the oscillator remains on LCB. For Cases 1 and 2, the separation of the oscillator from the beam is found to occur at the value of S/L approximately equal to 0.81(0.96) and 0.81(0.82) for the Graphite (steel) beams, respectively.

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Case 1:

Case 2:

6 6

0, 1000 / , 2.5 , 0.5 /

4.5(10 ) / , 2.5(10 ) / ,a a a

Steel Graphite Epoxy

c c k N m m m gr v m s

k N m k N m−

= = = = = =

= =

7 7

0, 1000 / , 1

: 2(10 ) / , 750 / , : 10 / , 850 /a a ac c k N m m m kg

Steel k N m v m s Graphite Epoxy k N m v m s

= = = = =

= = − = =

Fig. 9: The normalized contact force between the oscillator and the beam (L/h = 16.67)

The effects of varying the stiffness of the elastic foundation and the span number ( sn ) on the contact force versus the values of /fT τ for LCB are shown in Figures 10 and 11, respectively. It should be noted that increasing the span number will not raise the length and weight of the beam. The separation of the oscillator from the LCB can be, therefore, suppressed by the presence of an elastic foundation of large stiffness and increasing the span number as shown in these figures.

Fig. 10: Effect of elastic foundation on

the contact force for LCB Fig. 11: Effect of span number on the

contact force for LCB

The variations of the normal damping, the shear viscosity, the rocking stiffness and damping of the

foundation together with the damping of the oscillator and the slender ratio of the LCB on the contact force against /fT τ for the LCB were studied. It could be observed that the separation of the oscillator from the LCB can be suppressed by the increase of any of the foundation parameters, namely,

, , , ,x xf fk c k cψ ψμ or by decreasing the slenderness ratio of the LCB.

The effect of higher span number on the maximum shear force and the bending moment along the beam versus axial velocity of the oscillator are illustrated in Figures 12 and 13, respectively. The bending moment and the shear force are normalized by their corresponding values of / ( / 2)xM Mx mgL= and / ( / 2)xz xzQ Q mg= . It can be seen, from Fig. 12, that there exists a main peak that the maximum shear force increases and after reaching to a maximum value then decreases. Also, as the span number increases, the maximum shear force occurs at a lower velocity of the oscillator and its corresponding value of the shear force decreases.

Furthermore, there are two main peaks, shown in Fig. 13, and as the span number increases these two peaks approach to one another and the maximum bending moment decreases.

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Fig. 12: Effect of higher span number on

the maximum shear force vs. /fT τ Fig. 13: Effect of higher span number on the

maximum bending moment vs. /fT τ

V. Conclusion In this paper, the dynamic response of a multi-span laminated composite beam LCB resting on the

viscoelastic foundation under the action of moving non-rigid masses with the constant axial velocity is presented by using the first-order shear deformation theory and Galerkin’s method. The possibility of the separation of the moving oscillatory mass from the LCB during the course of the motion is studied by monitoring the contact force between the oscillator and the LCB. It has been shown that the separation of the oscillator from the slender LCB may happened with a high stiffness of the oscillator and either a low or a high axial momentum of the oscillator. This separation can be suppressed by having an elastic foundation with a relatively large stiffness and with an increase of the span number. As indicated above, the maximum shear force occurs at the lower velocity of the oscillator, and by increasing the span number the corresponding value of shear force decreases. Furthermore, two main peaks in the maximum bending moment diagram versus /fT τ approach towards each other when the span number is increased.

References 1 Esmailzadeh, E. and Ghorashi, M., “Vibration analysis of a Timoshenko beam subjected to a traveling

mass”, Journal of Sound and Vibration, Vol. 199, 1997, pp. 615-628 2 Lee, H.P. “Dynamic response of a Timoshenko beam on a Winkler foundation subjected to a moving mass”,

Applied Acoustics, Vol. 55, No. 3, 1998, pp. 203-215 3 Kadivar, M. and Mohebpour, S.R. “Finite element dynamic analysis of unsymmetric composite laminated

beams with shear effect and rotary inertia under the action of moving loads”, Finite Elements in Analysis and Design, 29, 1998, pp. 259-273

4 Ahmadian, M.T. and Jafari-Talookolaei, R.A. “A new approach in finite element modeling of generally laminated composite beams”, ASME IMECE Conference, Chicago, Illinois, USA, 2006, November 6-10

5 Timoshenko, S. “Vibration Problems in Engineering”, Wiley, New York, 1974