International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

137

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Transient Responses of Beam with Elastic Foundation Supports under

Moving Wave Load Excitation

1Yi Wang,

2Yong Wang,

3 Biaobiao Zhang,

4Steve Shepard

1,2The Enjoyor Company,No.2, Xiyuan 8th Rd, Hangzhou City,

Zhejiang Province, 310030, P.R.China 3,4

Department of Mechanical Engineering, University of Alabama, Tuscaloosa,

AL35487,USA

ABSTRACT

In this paper, the responses of a free to free boundary constrained beam on elastic foundation to various moving loads are

studied by the application of finite element method. We apply the Newmark integration method for numerical simulation.

Results are verified by using commercial ANSYS package through modeling the beam with elastic foundation supports excited

by moving point load at the constant speed. The effects of the following parameters on the dynamic behavior of the beam

under both moving point load and pressure wave load are evaluated: the traveling speed of load, stiffness of the elastic

foundation base and viscous damping. This study offers a good basis for research work on the beam under moving acoustic

wave loads in the future.

Key words: Free to free beam, Elastic Foundation, Moving loads.

1. INTRODUCTION

Beam structures with elastic foundation supports

are widely used in many areas, such as mechanical and

aerospace engineering, long historical research work in

them can be seen in the literature of engineering

mechanics. Most of these beam problems focused on beam

issues subjected to time and space varying loads. Moving

loads have considerable effects on the dynamic behavior

of the structures. In order to make a better understanding

of dynamic responses from moving loads on beam

structure, it is necessary for us to review previous papers

in the literature. Olsson [1] studied the dynamics of a

simple supported beam subjected to the point load moving

at a constant speed by using both analytical and the finite

element method. Jaiswal and Iyengar [2] studied the

dynamics of the infinite beam on a finite elastic foundation

base subjected to a moving load. Effects of various

parameters, such as foundation mass, velocity of the

moving load, damping and axial force on the beam are

investigated. Thambiratnam and Zhuge[3]analyzed the

dynamic analysis of beams on an elastic foundation

subjected to moving point loads through simplifying the

foundation with springs of variable stiffness. Effects of

some parameters such as the speed of the moving load, the

foundation stiffness and the length of the beam on the

response of the beam were investigated. Chen et al. [4]

studied the response of an infinite Timoshenko beam on a

viscoelastic foundation under a harmonic moving load.

Kim and Roesset [5] analyzed dynamic response of an

infinitely long beam on the frequency-independent linear

hysteretic damping foundation base by using the constant

amplitude or a harmonic moving load as an excitation

force. In the model developed by Kim [6], the vibration

and stability of an infinite Euler - Bernoulli beam on a

Winkler foundation were investigated through applying a

static axial force and a moving load with either constant or

harmonic amplitude variations to excite the system. Some

important results were obtained from the process of

changing relative parameters. Bogacz, etc.,[7] studied

dynamical problems caused by a distributed load which

was acting on a beam on an elastic foundation at a moving

velocity, The load was represented by the Heaviside

function(or its linear superposition) and by a moving load

harmonically distributed in space. Shi,etc.[8] established

the mathematic model which was used to describe the

issue about the elastic foundation beam with cantilevered

support excited by the moving load velocity, the transient

response under the moving load velocity was obtained.

Nayyeriamiri and Onyango[9] studied the problem of a

simply-supported beam on elastic foundation to repeated

moving concentrated loads by means of the Fourier sine

transformation method in order to obtain the analytical

forced responses.

The objective of this paper is to develop a

numerical procedure for evaluating the dynamic response

of the elastic foundation beam when subjected to

sinusoidal moving loads. The finite element method is

used for modeling beam elements. We use the Newmark

integration method to obtain the dynamic response. Effects

of the following parameters on the dynamic behavior of

the beam are evaluated: the traveling speed of the load; the

stiffness of the elastic foundation base; viscous damping.

This will make a good basis for developing beam acoustic

sensor with the help of force reconstruction method which

is used to identify moving acoustic loads through their

excitation process on the beam structure.

International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

138

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2. THEORETICAL MODEL

Consider a finite-length Bernoulli – Euler beam

resting on an elastic foundation under moving sinusoidal

loads as illustrated in Fig. 1.

f(x,t)

The equation of motion is [10]

2 4

2 4

( , ) ( , ) ( , )2 ( , ) ( , )b f

W x t W x t W x tEI C W x t f x t

tt x

,

(1)

where the beam flexural rigidity is EI , the beam mass

density is , the foundation elasticity constant is fC , b

is the damping circular modal frequency, and ( , )f x t

represents moving loads. Note that the mass and bending

stiffness of the elastic foundation are being neglected. For

an elastically supported free to free end beam illustrated in

Fig.1 above, its boundary conditions are:

2 2

2 2

(0, ) ( , )0

W t W L t

t t

,

(2a) 3 3

3 3

(0, ) ( , )0

W t W L t

t t

.

(2b)

3. FE FORMULATION FOR THE BEAM

UNDER THE MOVING POINT LOAD

Generally the finite element analysis procedure is

to develop the model approximating the deformation

within an element through using nodal values of

displacement and rotation. Here it should be noted that the

energy contributions from each element depend on the

displacements and rotations at the nodes associated with

that element. By calculating the kinetic and strain energy

for each element, the contributions of each element will be

added together to obtain a global model of the structure.

The damping matrix is obtained through using the

dissipation function. The Euler-Bernoulli beam theory is

used for constituting the finite element matrices. The beam

illustrated in Fig.1 is modeled with 15 equally sized

elements. A straight beam element i with uniform cross

section under a moving point load is shown in Fig.2 as

following

Fig.2: Moving load on a straight beam element

A standard beam element is modeled using two

nodes (at the ends of the beam element), and two degrees

of freedom per node (translation and rotation), as shown in

Fig.2. The deformation within the ith element, ( , )W x t , is

approximated using cubic shape functions, the kinetic

energy and strain energy of a single element are:

2

0

( , ).

2

LA W x t

K E dxt

,

(3a) 2

2

2

0

( , ).

2

LEI W x t

S E dxx

.

(3b)

The shape functions of element i are represented as:

2 3

1

2 3

2

2 3

3

2 3

4

1 3 2

2

3 2

e

e e

e e e

e e

e

e e

e e e

e e

x xN

L L

x xN x L L

L L

x xN

L L

x xN L L

L L

.

(4)

Fig.1. Elastic foundation beam subjected to moving wave loads

Cf

x

c T

N=3 Beam

eL

x

E I ρA

Element i

2eW

2eW

1e

2e

F c

International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

139

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Mass matrix of this beam element is:

2 2

2 2

156 22 54 13

22 4 13 3

54 13 156 22420

13 3 22 4

e e

e e e ee

e e

e e e e

L L

L L L LALM

L L

L L L L

.

(5)

Stiffness matrix of the beam element is:

2 2

3

2 2

12 6 12 6

6 4 6 2

12 6 12 4

6 2 6 4

e e

e e e ee

e ee

e e e e

L L

L L L LELK

L LL

L L L L

.

(6)

Supposing that the beam structure is on the

Winkler's foundation base, the elastic foundation is

assumed to have constant linear spring modulus,

21 1 1

2 2 2

TT

foundation f f e f eU C W dA W C WdA d K d , (7)

in which the Winkler’s foundation stiffness matrix for the

element is:

T

f f e eK C N N dA .

(8)

Where matrix eN is shape function matrix

illustrated in the Eqn. (4), ed represents node

displacement vectors and dA is represented as Bdx ,

where B is the width of the beam face contact with the

foundation, so the foundation stiffness matrix is:

2 2

2 3 2 3

2 2

2 3 2 3

13 11 9 13

35 210 70 420

11 13

210 105 420 140

9 13 13 11

70 420 35 210

13 11

420 140 210 105

e f e f e f e f

e f e f e f e f

f

e f e f e f e f

e f e f e f e f

bL C bL C bL C bL C

bL C bL C bL C bL C

KbL C bL C bL C bL C

bL C bL C bL C bL C

. (9)

The expression of the beam element deflection is:

1 1 2 1 3 2 4 2( ) ( ) ( ) ( ) ( )e e e e e e e e eW x N x W N x N x W N x

, (10)

where ( )eiN x (i=1,2,3,4) are the interpolation functions

shown in Eqn. (4). After decouping equation (1) by using

equation (10), we get the equation of motion for a multiple

degree of freedom damped structural system is represented

as follows:

( )fM y C y K K y F t ,

(11)

where y , y and y are the respective acceleration,

velocity and displacement vectors for the whole structure

and ( )F t is the external force vector which is depended

on the moving point load. For the case that the moving

load is the pressure wave load rather than point load, the

problem will be more complicated, but the solution

procedure is similar.

4. RESULTS AND DISCUSSION

4.1 Moving point load at the constant speed

Table 1 provides a list of the initial modeling parameters,

including the geometry and material properties for both the

beam and elastic foundation. Here, E is the Young’s

modulus of beam material and FE is the Young’s

modulus of foundation material. The foundation elasticity

constant can be obtained from f F

BC E

H , where B is

width of the beam and h is the thickness of the foundation

base.

Table 1: Parameter Definition

The transverse point load F has a constant

velocityL

C

the beam and L is the total length of the beam. For the

Item Description Units Value

L Beam length M 0.5

B Beam width M 0.03

H Beam thickness M 0.001

c Moving load speed ms-1 343

ρ Beam density kgm-3 2700

E Beam young’s modulus Nm-2 70Gpa

μ Poisson’s ratio - 0.33

TH Foundation thickness m 1

EF Foundation Young’s

modulus

Nm-2 33.1MPa

International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

140

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forced vibration analysis, an implicit time integration

method named as the Newmark integration method [13] is

used with the integration parameters 1 4 and

1 2 , which lead to the constant-average acceleration

approximation. The time step is chosen as 20

20

Tt

during the beam transient response analysis to ensure that

all the 20 modes contribute to the dynamic response. Here

20T is the period of the 20th natural mode of the structure.

Responses of beam with elastic foundation under the

moving point load are obtained by using Matlab software

package and Ansys simulation results. As shown in Fig.3

and Fig.4, Matlab results show high agreement with Ansys

results.

Fig.3 Response of beam end point when the moving load is

leaving at 146m/s

Fig.4 Response of beam end point when the moving load is

leaving at 343m/s

For the beam with finite length under the static

load, its displacement at the middle location can be

calculated as following[14]

0

2 cosh( ) cos( )

2 sinh( ) sin( )f

F L LW

K L L

,

(12)

where

0.25

4

fK

EI

. In our case, the beam structure is

under the point load of 10N, so the static displacement at

mid-location is 05

0 2.5051 10W m under the point

load moving at the speed of 343m/s. Figure 5 shows the

effect of the Rayleigh damping on mid-point displacement

of beam structure vs time. Displacements are decreased

with increasing damping ratio. It should be noted that dD

represents transient response at mid-point divided by the

static displacement 0W .

Fig.5 Rayleigh damping effect on transient displacement at

the beam mid-location

Figure 6 shows foundation base stiffness effect on

transient displacement at the beam mid-location. In this

figure, we can find that with the decrease of foundation

stiffness value, the maximum transient displacement is

increased.

Fig.6 Foundation base stiffness effect on transient

displacement at the beam mid-location

0 0.5 1 1.5 2 2.5 3 3.5

x 10-3

-7

-6

-5

-4

-3

-2

-1

0

1

2x 10

-5

Time(s)

Dis

pla

cem

ent(

m)

point load moving speed 146m/s

ANSYSFEM

0 0.5 1 1.5

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-5

Time(s)

Dis

pla

cem

ent(

m)

Moving point load speed 343m/s

ANSYSFEM

0 0.5 1 1.5

x 10-3

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time(s)

Dd

damping=0.05damping=0.03damping=0.01damping=0.00

Dis

pla

cem

ent

(m)

0 0.5 1 1.5

x 10-3

-3

-2

-1

0

1

2

3

Time(s)

Dd

damping value=0.01

Kf=8.2759e+006Kf=1.6552e+007Kf=3.3103e+007

International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

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Figure 7 shows the deflection of the beam mid-

location under a moving load for different load velocities.

With the increase of moving load velocity, the maximum

deflection at the middle location has increased.

Fig.7 Deflection of the beam mid-location under a

moving load for different load velocities

4.2 Moving sinusoidal load at the

constant speed

In this study, a finite set of traveling

sinusoidal half-cycles will be considered as the

loading on the structure. In modeling this

continuous wave load ( , )f x t traveling over the

beam sensor structure, three different time stages

must be considered [10]. When the distributed

force load begins to progressively step on the beam

until it is entirely on the beam, it is expressed by

2

( , ) sin ( ) 1 ( ) , 0f x t A x ct H x ct t NTcT

, (12a)

In the equation above, N is the number of

half-cycles, T is single half-cycle time period as

shown in Fig. 1 and H is the Heaviside step

function. The speed of the wave is denoted by c,

which for the cases considered here will be the

speed of sound in air. This equation represents a

discrete number of half-cycles in a traveling

sinusoidal wave. Once the load is completely on

the beam, the force is expressed as

2

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

f x t A x ct H x c t NT H x ct NT tcT c

(12b)

Until it reaches the other end of the beam.

Finally, the load begins to leave the beam as :

Note that Eq.(12a~c) are used to represent

sinusoid wave loads traveling across the beam

structure at parallel direction to the beam axial

direction. By discretizing the elastically supported

beam structure with the finite element method,

transient responses can be calculated numerically

in the time domain by using Newmark’s integration

scheme [13]. Specifically the elastically supported

beam is discretized into 15 beam elements with 16

nodal points. Of course, the response at any point

can be interpolated by using the shape functions

and the weight factors. Although not specifically

shown, 15 beam elements have been determined to

be sufficient for convergence of the FEM transient

response calculations. When increasing the number

of elements from 15 to 30, no appreciable increase

in the accuracy of the solution is obtained. Because

of the time T chosen for the excitation force, most

of the frequency content of that excitation is not

that much greater than the fundamental frequency

of the beam. In our work, the beam structure

damping ratio is assumed to be very small and

ignored initially. Damping will be addressed in

future work.

Similar to Table 1,Table 2 provides a list of

the initial modeling parameters for the beam

structure under wave loads excitation, including the

geometry and material properties for both the beam

and elastic foundation.

Table 2: Parameter Definition

Item Description Units Value

L Beam length M 0.5

B Beam width M 0.03

H Beam thickness M 0.001

C Moving load speed ms-1 343

T Half-cycle duration Sec 0.0004

A Force amplitude N 10

N The number of half-cycles 2

ρ Beam density kgm-3 2700

E Beam young’s modulus Nm-2 70Gpa

μ Poisson’s ratio - 0.33

T Foundation thickness Mm 1

EF Foundation Young’s modulus Nm-2 8.27586

MPa

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Time(s)

Dd

V=100m/sV=150m/sV=200m/s

International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

142

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72.3645 10fK

2N m

58.27586 10fK

2N m

Fig.8: Effect of foundation stiffness on beam transient responses under wave loads

Figure 8 shows the effect of foundation

stiffness on beam transient responses under

sinusoidal wave loads, it can be found that by

reducing beam elastic foundation base stiffness,

beam transient responses under wave loads

increase.

5. CONCLUSIONS

The response of a free to free beam on

elastic foundation to moving concentrated load and

sinusoidal wave load by means of the finite

element method has been presented in this paper.

Numerical results have been verified by

corresponding results obtained from ANSYS

commercial codes. This technique is attractive for

treating the problems of beams on an elastic

foundation under moving loads, which are

extended to develop the acoustic wave sensor by

using force identification method. The effect of

some important parameters, such as the foundation

stiffness, the travelling speed has been studied.

Numerical examples are given in order to

determine the effects of various parameters on the

response of the beam. Within the range of values

considered, an increase in velocity parameter of the

moving loads results in the increase in dynamic

deflections. This study contains useful

contributions to the literature on moving loads

problems relation to transportation systems,

furthermore, the technique and the findings can

offer a good basis in practical applications such as

acoustic wave beam sensor development by means

of force reconstruction method.

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[3] D. Thambiratnam and Y. Zhuge,(Nov.1996):

Dynamic analysis of beams on an elastic

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International Journal of Engineering and Technology Volume 1 No. 2, November, 2011

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Copyright IJET © 2011 - IJET Publications UK

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