Dynamic responses of supported beams with intermediate supports …downloads.hindawi.com/journals/sv/2012/849298.pdf · B. Zhang and S. Shepard / Dynamic responses of supported beams

Shock and Vibration 19 (2012) 1403–1413 1403DOI 10.3233/SAV-2012-0682IOS Press

Dynamic responses of supported beams withintermediate supports under moving loads

Biaobiao Zhang∗ and Steve ShepardDepartment of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL, USA

Received 24 August 2011

Revised 31 October 2011

Abstract. In this paper, a new beam shape function configuration method for determining transient responses of a finite Euler-Bernoulli beam with two intermediate supports excited by moving pressure wave loads is developed. To clarify this method,this beam structure is excited by the moving sinusoidal loads as an example. Transient responses of this beam structure areinvestigated and verified by the traditional finite element method. This method can be used to solve transient response problemsof moving pressure loads exciting the beam structure with intermediate support. Actually it can be extended to solve othercomplicated beam structure problems.

Keywords: Transient response, moving load, Euler-Bernoulli beam, intermediate supports, shape functions configuration

1. Introduction

Wide application of beam structures with various boundary conditions can be found in many branches of engi-neering such as in bridge and railway designs. Research work on these problems has been ongoing for many years.In order to achieve a deep understanding of work done to find the dynamic response of beam structures, it is usefulfor us to first review previous publications in the literature. Specifically, Fryba [1] contributed his initial work bystudying transient responses of simply-supported beams under a moving load or continuous random load with aconstant velocity. Lee [2] established an intermediate supported beam model through setting point constraints tobe linear springs of large stiffness. By using the Euler beam theory and the assumed mode method, he analyzedtransverse vibration of a beam subjected to a moving point load. Zhu and Law [3] applied Hamilton’s principleto study the vibration behavior of a Timoshenko beam with non-uniform cross-section subjected to moving loads.The beam structure had multi-span supports with the intermediate point constraints being simplified with very stifflinear springs. Hilal and Zibdeh [4] studied the transverse vibration of an Euler-Bernoulli beam under differentclassical boundary conditions excited by a constant force experiencing acceleration, deceleration and constant ve-locity movement. Tang [5] used hyperbolic sine and cosine functions as base functions to model the free-vibrationnormal modes of a uniform beam. Good accuracy of calculated normal modes was achieved up to at least the 100thmode. To investigate the dynamic behavior of a multi-span non-uniform bridge excited by a moving vehicle, Lawand Zhu’s efforts [6] involved modeling the bridge structure as a continuous Euler-Bernoulli beam with multi-spanlinear large stiffness spring supports. Chan and Ashebo [7] tried to identify moving forces on a continuous bridgethrough using singular value decomposition method, assuming the bridge as an Euler-Bernoulli beam. In his model,exact mode shape functions of the vibrating beam were obtained from the boundary conditions. Stcioiu et al. [8]theoretically and experimentally investigated the dynamic characteristics of a four-span continuous beam under a

∗Corresponding author: Biaobiao Zhang, Department of Mechanical Engineering, The University of Alabama, Tuscaloosa, AL 35487, USA.Tel.: +1 205 348 0048; Fax: +1 205 348 6419; E-mail: [email protected].

ISSN 1070-9622/12/$27.50 2012 – IOS Press and the authors. All rights reserved

1404 B. Zhang and S. Shepard / Dynamic responses of supported beams with intermediate supports under moving loads

Fig. 1. Beam with intermediate supports and free ends under moving load.

moving mass. Good agreement between the theoretical and experimental results was obtained. Ricciardelli andBriatico [9] studied the case of a simply supported beam excited by a sinusoidal harmonic load moving at a constantspeed. The approximated responses at and away from resonance were presented.

Although many previous papers discussed complicate beam-loads interaction problems about moving loads orvehicles across multi-span bridge by using various methods, Most of previous works just describe general accuracytrends in their results by simply referring to results from other papers published. Rare discussed high resultsaccuracy through strict verification from conventional method for example, finite element method. Besides, thereis an important configuration of interest that has not been specifically addressed. None of the works presented todate have addressed the problem of an intermediately supported beam, but with free ends, excited by a transientsinusoidal pressure load.

This paper describes a new method to solve the problem of moving loads on a beam with intermediate supports.The advantage of this method is to solve beam transient response problems without their own explicit shape functionsby using newly shape function configured. In order to ensure the new shape functions satisfy the structure boundaryconditions, some original general constraints should be relieved and new constraints will be fixed. Note that in thiswork, New shape functions are based on clamped-free beam shape functions due to the fact that the cantileveredbeam functions ready satisfy the geometric boundary conditions on the end of the beam at x = L. If shape functionswere generated by using those of other beam structures such as an end-supported beam as basis functions, theywould work as well through using following analytical method.

2. Theoretical model

Consider a finite-length Euler-Bernoulli beam with two intermediate supports and free ends under a movingtransient sinusoidal pressure wave load as illustrated in Fig. 1. The function f(x, t) represents the transient sinusoidalwave loads and T is the single half-cycle period. w(x, t) is the displacement from this beam structure and parametersa and b are the distances from both ends of the beam structure of length L, respectively. The pressure load is movingacross the beam at constant speed c. For the beam with two free ends and two intermediate constraints, the boundaryconditions are

∂2W (0, t)∂t2

=∂2W (L, t)

∂t2= 0, (1a)

∂3W (0, t)∂t3

=∂3W (L, t)

∂t3= 0, (1b)

and

B. Zhang and S. Shepard / Dynamic responses of supported beams with intermediate supports under moving loads 1405

w(a, t) = w(L − b, t) = 0. (1c)

Note that unlike a simply supported beam or a free to free beam, the beam considered here has six conditions. Twoof those conditions are at the supports while the other four are at the free ends. To solve the dynamic responseof a beam with intermediate constraints under moving wave loads, the shape functions of this beam structure aregenerated by using the functions of the clamp-free beam structure. However, the clamped-free boundary conditionsare quite different from the beam with intermediate supported conditions described above. As a result, the boundaryconstraint at the Euler-Bernoulli beam’s clamped end needs to be released. Furthermore, two constraint points needto be added to the initial clamped-free beam to simulate a beam with intermediate support conditions. To do this, alinear polynomial function is added to the basis shape functions obtained from the beam with clamped-free boundaryconditions so that new intermediate-supported beam shape functions can be realized. For a clamped-free beamstructure, its shape functions [10] are

∅n(x) = coshβnx− cosβnx− sinhβnL− sinβnL

coshβnL+ cosβnL(sinh βnx− sinβnx) (n = 1, 2 . . .). (2)

By starting with these basis functions and adding a linear polynomial function, a new shape functions for the beamwith two intermediate constraints are given by

u(x) ≈N+2∑n=1

An∅n(x) +B1x+B0, (3)

whereAn, B1 andB0 are initial unknown constants, andN + 2 is the total number of truncated terms. Note that thereason for using N + 2 as the upper limit on the series will become apparent below. While other shape functionswould be valid, automatically satisfying the boundary conditions at x = L is one of the advantages of using functionschosen here. Suppose that the dynamic deflection of the intermediate-supported beam structure in free vibrationstate can be expressed as

W (x, t) = u(x)q(t), (4)

where q(t) is the generalized coordinate. Based on Eqs (1c) and (4), the constraint equations are

u(a) =N+2∑n=1

An∅n(a) +B1a+B0 = 0, (5a)

and

u(L− b) =N+2∑n=1

An∅n(L− b) +B1(L− b) +B0 = 0. (5b)

By rearranging Eqs (5a) and (5b), constants B1 and B0 can be represented as

B1 =−1

L− b− a

(N+2∑n=1

An∅n(L− b) −N+2∑n=1

An∅n(a)

), (6a)

and

B0 = −N+2∑n=1

An∅n(a) +a

L− b− a

(N+2∑n=1

An∅n(L − b) −N+2∑n=1

An∅n(a)

). (6b)

By substituting Eqs (6a) and (6b) into Eq. (3) and rearranging them, we get

u(x) =N+2∑n=1

An

{∅n(x) +

−xL− b− a

(∅n(L− b) − ∅n(a)) +a

L− b− a(∅n(L− b) − ∅n(a)

}. (7)

The boundary conditions at both ends are free for the beam structure with intermediate supports as indicated inEqs (1a–1b). By using Eq. (3), we obtain


{∑N+2n=1 An∅′′

n(0) = 0∑N+2n=1 An∅′′

n(L) = 0, (8a)

and {∑N+2n=1 An∅′′′

n (0) = 0∑N+2n=1 An∅′′′

n (L) = 0. (8b)

Due to the fact that ∅n(x) already satisfies the clamped-free conditions, it is not necessary to reconsider the freeend at x = L again. Besides, it is suitable to choose N + 2, provided that convergence in the dynamic responsecalculation is ensured. From Eqs (8a–b) above, constants A1 and A2 can be represented by other constants, basedon the consideration that ∅1 and ∅2’s second and third derivatives are not zeros, such that⎧⎪⎨

⎪⎩A1 = A3(∅′′′

3 ∅′′2 −∅′′

3 ∅′′′2 )+...+AN+2(∅′′

2 ∅′′′N+2−∅′′′

2 ∅′′N+2)

∅′′′2 ∅′′

1 −∅′′2 ∅′′′

1

A2 = A3(∅′′1 ∅′′′

3 −∅′′3 ∅′′′

1 )+...+AN+2(∅′′1 ∅′′′

N+2−∅′′′1 ∅′′

N+2)

∅′′′2 ∅′′

1 −∅′′2 ∅′′′

1

. (8c)

By substituting Eq. (8c) into Eq. (7), we get

u(x) ≈N+2∑n=3

AnΦn(x), (9)

where

Φn(x) = ∅n(x) +−x

L− b− a(∅n(L− b) − ∅n(a)) +

a

L− b− a(∅n(L− b) − ∅n(a))

+∅′′2∅

′′′n − ∅′′′

2 ∅′′n

∅′′′2 ∅′′

1 − ∅′′2∅′′′

1

(∅n(x) +−x

L− b− a(∅n(L − b) − ∅n(a)))

+(

a

L− b− a(∅n(L− b) − ∅n(a))

)

+∅′′1∅

′′′n − ∅′′′

1 ∅′′n

∅′′′2 ∅′′

1 − ∅′′2∅′′′

1

(∅n(x) +−x

L− b− a(∅n(L − b) − ∅n(a)))

+(

a

L− b− a(∅n(L− b) − ∅n(a))

).

For the beam with two intermediate supports, strain energyU and kinetic energy T of an Euler beam are obtained as

U =12

∫ l

0

EIW 2xxdx, (10a)

and

T =12

∫ l

0

μW 2,tdx, (10b)

where subscripts x and t denote differentiation with respect to location and time, respectively. In order to find thenatural frequencies ω and mode shapes, a periodic transverse motion function is assumed

W (x, t) = u(x) sinωt. (11)

By substituting Eq. (11) into Eq. (10b), one can obtain an expression for kinetic energy as

T =12

∫ L

0

ω2μu2(x) cos2(ωt)dx =12

∫ L

0

ρAω2u2(x)1 + cos(2ωt)

2dx. (12a)

Therefore, the corresponding maximum kinetic energy is


Tmax =ρAω2

2

∫ L

0

(N+2∑n=3

AnΦn(x)

)2

dx. (12b)

Similarly, the potential energy is

U =EI

2

∫ L

0

(N+2∑n=3

AnΦ”n(x) sinωt

)2

dx, (12c)

and the maximum potential energy is

Umax =EI

2

∫ L

0

(N+2∑n=3

AnΦ”n(x)

)2

dx. (12d)

Finally, we make conventional definition

Θ = (Tmax − Umax). (13)

According to Eqs (12b), (12d) and (13), Θ can be represented by An : n = 3, 4 . . .N+2. Applying the stationaryconditions of Θ with regard to An (such that partial derivatives are zero), then an explicit relation for An can befound using

∂Θ∂An

=EI

2

∫ L

0

2

(N+2∑n=3

AnΦ”n(x)

)Φ”

n(x)dx − ρAω2

2

∫ L

0

2

(N+2∑n=3

AnΦn(x)

)Φn(x)dx = 0. (14a)

By appropriately grouping terms and using matrix notation, the result is

EI

⎡⎢⎣

∫ L

0Φ

′′3Φ

′′3dx− ρAω2

∫ L

0Φ3Φ3dx · · · ∫ L

0Φ

′′N+2Φ

′′3dx − ρAω2

∫ L

0ΦN+2Φ3dx

.... . .

...∫ L

0 Φ′′3Φ

′′N+2dx− ρAω2

∫ L

0 Φ3ΦN+2dx · · · ∫ L

0 Φ′′N+2Φ

′′N+2dx− ρAω2

∫ L

0 ΦN+2ΦN+2dx

⎤⎥⎦

(14b)

×⎡⎣ A3

. . .AN+2

⎤⎦ = 0.

This equation can be summarized as

(Kα − ω2Mα)ψ = 0. (15)

The beam structure with immediate support’s natural frequencies of ω can be therefore obtained using

Kα = EI

⎛⎜⎝

∫ L

0 Φ3(x)Φ3(x)dx · · · ∫ L

0 Φ3(x)ΦN+2(x)dx...

. . ....∫ L

0ΦN+2(x)Φ3(x)dx · · · ∫ L

0ΦN+2(x)ΦN+2(x)dx

⎞⎟⎠ ,

Mα = ρA

⎛⎜⎝

∫ L

0Φ3(x)Φ3(x)dx · · · ∫ L

0Φ3(x)ΦN+2(x)dx

.... . .

...∫ L

0 ΦN+2(x)Φ3(x)dx · · · ∫ L

0 ΦN+2(x)ΦN+2(x)dx

⎞⎟⎠ , (16)

and

Ψ =

⎡⎢⎢⎢⎢⎣

A3

A4

. . .AN+1

AN+2

⎤⎥⎥⎥⎥⎦ .


It should be noted that the above development is irrespective of the forcing function, the developed shape functionsas well as the resulting natural frequencies and mode shapes depend on the boundary conditions described above, inthe following section, the effects of the forcing function on the following beam response will be considered.

In the following section, the dynamic equation of a beam with intermediate supports will be established withthe help of the Lagrangian formulation, based on the beam shape functions developed above. From Eq. (15), weget circular frequencies ωi(i = 1, 2, . . .). Mode shape functions of the beam are obtained by substituting ψ fromEq. (15) into Eq. (9).

3. Forced response of the beam structure with intermediate supports

The assumed basis function approach employed in this section is used to determine the forced response of thebeam with intermediate supports described above. So the dynamic deflection of the beam structure can be expressedas

W (x, t) ≈N+2∑n=3

un(x)qn(t), (17)

where N is the total number of truncated terms, qn(t) is the generalized coordinate, and un(x) represents shapefunctions of the beam. Note that un(x) = AnΦn(x) is obtained in Eq. (9). The kinetic energy T , the bendingpotential energy V are obtained respectively, as

T =12

∫ L

0

μ

(∂W (x, t)

∂t

)2

dx =12

N+2∑i=3

N+2∑j=3

∫ L

0

AiAjμqi(t)Φi(x)Φj(x)qj(t)dx

(18)

=12

N+2∑i=3

N+2∑j=3

mi,j qi(t)qj(t),

wheremi,j=∫ L

0 AiAjμΦi(x)Φj(x)dx, and

V =12

∫ L

0

EI

(∂2W (x, t)

∂2x

)2

dx =12

N+2∑i=3

N+2∑j=3

∫ L

0

AiAjEIΦ”j(x)Φ

”j (x)qi(t)qj(t)dx

(19)

=12

N+2∑i=3

N+2∑j=3

ki,jqi(t)qj(t),

where ki,j =∫ L

0EIAiAjΦ”

j (x)Φ”j (x)dx.

The workWEdue to the external force f(x, t) is

WE =N+2∑i=3

qi(t)∫ L

0

f(x, t)Φi(x)dx. (20)

The beam Lagrangian function L is T −V +WE , here V −WE is the total potential energy. It should be pointedout that in order to simplify our new analytical method, damping has been ignored in this work. The Euler-Lagrangeequation is expressed as

d

dt

(∂L

∂qi

)− ∂L

∂qi= 0. (21)


Substituting Eqs (18–20) into Eq. (21),one obtains

N+2∑j=3

mij qj(t)+N+2∑j=3

kijqj(t) = Qi(i = 3, . . .N + 2), (22)

whereQi =∫ L

0f(x, t)Φi(x)dx is the generalized force.

Using matrix form for Eq. (22),⎛⎜⎝

m3,3 · · · m3,N+2

.... . .

...mN+2,3 · · ·mN+2,N+2

⎞⎟⎠⎡⎣ q3

. . .qN+2

⎤⎦+

⎛⎜⎝

k3,3 · · · k3,N+2

.... . .

...kN+2,3 · · · kN+2,N+2

⎞⎟⎠⎡⎣ q3

. . .qN+2

⎤⎦ =

⎡⎣ Q3

. . .QN+2

⎤⎦ . (23)

Equation (23) can be summarized as

[M ][q] + [K][q] = [Q], (24)

where,

[M ] =

⎛⎜⎝

m3,3 · · · m3,N+2

.... . .

...mN+2,3 · · ·mN+2,N+2

⎞⎟⎠ , [q] =

⎡⎣ q3

. . .qN+2

⎤⎦ ,⎛⎜⎝K] = (

k3,3 · · · k3,N+2

.... . .

...kN+2,3 · · · kN+2,N+2

⎞⎟⎠ ,

[q] =

⎡⎣ q3

. . .qN+2

⎤⎦ , [Q] =

⎡⎣ Q3

. . .QN+2

⎤⎦ .

Transient responses can be calculated numerically in the time domain by using Newmark’s integration scheme [12],This is achieved by using the integration parameters α = 0.25 and β = 0.5, which lead to the constant-averageacceleration approximation. Here parameters α and β are used to control the stability and accuracy of the Newmarkmethod. By using this approach in Eq. (24), the beam displacement and velocity response at each position and forany time can be obtained.

As noted earlier, the goal here is to study an intermediate supported beam subjected to a traveling transient pressureload (distributed force) load. The development of specific formulation needed to find such response is given next.

4. Numerical example

Following is such a description, prior to describing the resulting response, the natural frequencies and mode shapesfor the unloaded beam will be described. Since the objective of this work is to examine the response of the previouslydescribed beam when subjected to a traveling pressure load, a detailed description of that load is needed first.

In this study, a finite set of traveling sinusoidal half-cycles as the force load to represent a transient movingpressure wave load is considered. To explain this continuous wave load f(x, t) traveling over the beam structure,three different time stages are described [11]. When this distributed force load begins to progressively step on thebeam until it is entirely on the beam, it is expressed by

f(x, t) = − sin{

2πcT

(x − ct)}{1 −H(x− ct)} , (0 � t � KET ), (25a)

whereKE is the number of half-cycles, T is single half-cycle time period as shown in Fig. 1, andH is the Heavisidestep function. In this and all subsequent discussions, the number of half-cycle always begins and ends with a valueof zero. Furthermore, an integer number of half-cycles of a sinusoidal wave with a full-cycle period of 2T will beused. The speed of the wave is denoted by c. Once the load is completely on the beam, the force is expressed as

f(x, t) = sin{

2πcT

(x− ct)}{H(x− c(t−KET )) −H(x− ct)} ,

(KET � t � L

c

), (25b)


Table 1

Parameter definitions

Item Value Units Description

L 0.75 m Beam lengthB 0.05 m Beam widthD 0.05 m Beam thicknessC 343 m/s Wave speed

KE – – Number of half-cyclesT 0.002 sec Half-cycle durationρ 7820 kg/mˆ3 Beam densityE 2.068e+11 N/ mˆ2 Beam Young’s modulusν 0.29 – Beam Poisson’s ratioA 0.15 m Left constraint locationB 0.25 m Right constraint location

Table 2

Natural frequencies comparison (Hz)

No. FEM results Analytical results Error(%)

1 370.2 370.0 −0.0622 863.4 863.4 0.00553 1798.0 1797.2 −0.0444 3402.1 3401.3 −0.02355 4981.3 4981.4 0.0026 8626.1 8624.0 −0.02437 10219 10254 0.348 11686 11750 0.549 17187 17167 −0.11610 21473 21321 −0.70

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Beam length (m)

Am

plitu

de

FEMOur method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Beam length (m)

Am

plitu

deFEMOur method

Mode 1Mode 2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Beam length (m)

Am

plitu

de

FEMOur method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-5

-4

-3

-2

-1

0

1

2

3

4

Beam length (m)

Am

plitu

de

FEM Our method

Mode 3 Mode 4

Fig. 2. Mode pairs for corresponding frequencies in Table 2.

until it reaches the other end of the beam. Finally, the load begins to leave the beam as

f(x, t) = sin{

2πcT

(x− ct)}{H(x− c(t−KET )) −H(x− L)} ,

(L

c� t � L

c+KET

). (25c)

Note that Eqs (25a–c) are used to represent transient sinusoidal wave loads traveling across the beam structureparallel to the beam axis.

Initial parameters of the beam structure used for numeric study to be described below are listed in Table 1.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Beam length (m)

Am

plitu

de

FEMOur method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1.5

-1

-0.5

0

0.5

1

1.5

Beam length (m)

Am

plitu

de

FEMOur method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Beam length (m)

Am

plitu

de

FEMOur method

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8-1.5

-1

-0.5

0

0.5

1

1.5

Beam length (m)

Am

plitu

de

FEMOur method

Mode 5 Mode 6

Mode 7 Mode 8

Fig. 3. Mode pairs for corresponding frequencies in Table 2.

Fig. 4. Mid-beam location forced displacement vs time under sine wave loads.

In this work, to simplify the problem, damping has been ignored in this topic. The analytical analysis and finiteelement method are to be described below here, both conducted using Matlab .

4.1. Natural frequencies comparison

Table 2 shows the comparison of results for the first ten natural frequencies for the intermediately supported beamwith free ends. In the table, results when using methods described above are compared to those found using a finiteelement method. Relative errors of the natural frequencies are based on the FEM results. It can be noticed thatrelative error is very small and reasonable.


Fig. 5. Mid-beam location forced velocity vs time under sine wave loads.

Fig. 6. Mid-beam location forced acceleration vs time under sine wave loads.

Corresponding mode pairs are shown in Figs 2 and 3. It can be seen that the analytical mode shapes have verygood agreement with the FEM mode shapes. As a result, this new configuration method is suitable for solving beamtransient problems with explicit shape functions. But it should be noted that the analytical mode shapes do havesome deviation from the FEM mode shapes at the higher frequencies. For the cases considered here, however thatdeviation is not issue, according to our examination, the first ten natural frequencies contribute almost all vibrationenergies to the system. Now that the mode shapes have been verified, the forced response will be examined next.

4.2. Transient response analytical solutions and verification by FEM

In this section, the transient response is calculated numerically in the time domain by using the Newmark’sintegration scheme as previous mentioned [12]. The beam forced response with intermediate supports will beverified by the finite element solutions with the same parameters. In the case, the transient load moves along the


beam with speed c = 343m/s. Displacements of the beam at mid-position obtained by using analytical and finiteelement methods are compared in following results for a few different excitation cases. Figure 4 through 6 showthe responses of the beam in terms of displacement, velocity and acceleration respectively, the (a) figures show theresponse at the center of the beam as a single half-cycle loads travels over the beam, the (b) figures show the samebut when the force is a full-cycle. Recall that the half-cycle duration T , was defined in Table 1. Note that the timeaxis in the figures varies from zero to unity. This non-dimensional time represents the duration required for the forceto travel across the beam. As seen in the figures, the response solutions show high coincidence. Both solution resultsrepresent the beam response as almost the same. Although not shown, transient responses at other beam locationsshow the same correlation with FEM at the mid-location. The analytical approach is thus considered acceptable torepresent beam transient responses under a moving pressure wave load. This fact draws the conclusion that the newanalytical method for this intermediate support model can be used to simulate the beam transient response of twointermediate supports under moving load excitation.

5. Conclusions

This research work focuses on the development of analytical tools to be used in the shape functions generationstudy for the transient responses of a beam with intermediate supports under moving pressure wave loads. Theresponses due to transient sinusoidal pressure wave as a moving pressure wave load example are determined usingshape function configuration method based on clamped-free beam mode functions and the addition of a polynomialfunction. Analytical results obtained from the new method are well verified by comparing with the finite elementmethod when the beam is subjected to the same loads. Actually this analytical method presented in this work can beextended to solve the transient response of other beam problems under moving pressure wave loads.

References

[1] L. Fryba, Vibration of solids and structures under moving loads, Noordhoff International Publishing, Groningen, 1972.[2] H.P. Lee, Dynamic Responses of a beam with intermediate point constraints subject to a moving load, Journal of Sound and Vibration 171

(1994), 361–368.[3] X.Q. Zhu and S.S. Law, Moving forces identification on a multi-span continuous bridge, Journal of Sound and Vibration 228(2) (25

November 1999), 377–396.[4] M. Abu Hilal and H.S. Zibdeh, Vibration analysis of beams with General Boundary conditions traversed by a moving force, Journal of

Sound and vibration 229(2) (2000), 377–388.[5] Yu Tang, Numerical evaluation of uniform beam modes, Journal of Engineering Mechanics 129(12) (December 2003), 1475–1477.[6] S.S. Law and X.Q. Zhu, Bridge dynamic responses due to road surface roughness and braking of vehicle, Journal of Sound and Vibration

282 (2005), 805–830.[7] T.H.T. Chan and D.B. Ashebo, Theoretical study of moving force identification on continuous bridges, Journal of Sound and Vibration

295(3–5) (22 August 2006), 870–883.[8] D. Stancioiu, S. James, H. Ouyang and J.E. Mottershead, Vibration of a continuous beam excited by a moving mass and experimental

validation, 7th International Conference on Modern Practice in Stress and Vibration Analysis, Journal of Physics: Conference Series 181(2009), 012084.

[9] Francesco Ricciardelli and Carmelo Briatico, Transient response of supported beams to moving forces with sinusoidal time variation, JEng Mech 137 (2011), 422.

[10] Daniel J. Inman, Engineering vibration, Prentice Hall, Upper Saddle River, New Jersey, TS176.K34, 2000.[11] W.S. Shepard, Jr., B.B. Zhang and C.-C. Chen, Structural configuration study for an acoustic wave sensor, in: Proceedings of SPIE, 15th

Annual International Symposium on Smart Structures and Materials & Nondestructive Evaluation and Health Monitoring (San Diego,CA, March 10–14, 2008) Paper No. 6932–131.

[12] J.N. Reddy, An Introduction to the Finite Element Method (Third Edition), McGraw-Hill Higher Education, Boston, 2006.

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