Journal of Computational Science and Technology

Journal of Computational Science and Technology

Vol. 7, No. 1, 2013

Dynamic Response of Non-Uniform Functionally Graded Beams Subjected to a Variable Speed Moving Load* Dinh Kien NGUYEN**, Buntara Sthenly GAN*** and Thi Ha LE**** ** Institute of Mechanics, Vietnam Academy of Science and Technology 18 Hoang Quoc Viet, Hanoi, Vietnam *** Department of Architecture, College of Engineering, Nihon University 1-Nakagawara, Tokusada, Koriyama-shi, Fukushima 963-8642, Japan E-mail: [email protected] **** Theoretical Mechanics Group, Hanoi University of Transport and Communications Cau Giay, Dong Da, Hanoi, Vietnam

Abstract The dynamic response of non-uniform functionally graded beams subjected to a variable speed moving point load is studied by using the finite element method. The material properties of the beams are assumed to be graded in the thickness direction by a power law. A beam element, taking the effects of shear deformation, cross-sectional variation and the shift in the neutral axis position, is formulated by using exact polynomials obtained from solutions of the governing differential equations of a homogeneous Timoshenko beam element. The dynamic responses of the beams are computed by using the implicit Newmark method. The numerical results show that the dynamic characteristics of the beams, including the maximum mid-span deflection, mid-span axial stress distribution are greatly influenced by the acceleration and deceleration of the moving load. The effects of the moving speed, material non-homogeneity, cross-section variation as well as aspect ratio on the dynamic response of the beams are investigated in detail and highlighted. Key words: Functionally Graded Material, Non-Uniform Beam, FEM, Moving Load, Dynamic Response

1. Introduction

*Received 25 Sep., 2012 (No. 12-0403) [DOI: 10.1299/jcst.7.12]

Copyright © 2013 by JSME

Analysis of structures subjected to moving loads is an important topic in civil engineering and it has been drawn much attention from engineers and researchers for a long time. A large number of publications on the topic can be found in the literature; only the main contributions are briefly discussed herein. The early and excellent reference is the monograph of Fryba(1), in which a number of closed-form solutions for the moving load problems has been derived by using Fourier and Laplace transforms. Also by using the Fourier transform, Chonan(2) derived the solutions for the deflection and moment of an elastically supported Timoshenko beam subjected to an axial force and a moving point load. Abu-Hilal et al.(3), (4) investigated the dynamic response of Bernoulli beam with various boundary conditions subjected to an accelerating and decelerating point load, and a harmonic point load. Though a number of closed-form solutions have been reported for simple moving load problems on beams, solving complex problems must be done numerically, especially the finite element method should be used. In this line of work, Hino et al.(5), (6) developed the

12


Vol. 7, No. 1, 2013

Galerkin finite element method for analyzing the moving load problems of bridge engineering. Lin and Trethewey(7) derived the finite element equations for a Bernoulli beam subjected to different types of moving load, and then solved the obtained governing equation by the Runge-Kutta integration method. Thambiratnam and Zhuge(8) computed the dynamic deflection of beams on a Winkler elastic foundation subjected to a constant speed moving load by using the traditional planar Bernoulli beam element. Also by using the finite element method, Nguyen and his coworker(9), (10) computed dynamic response of Bernoulli and Timoshenko beams subjected to a moving load or a moving harmonic load. Based on the analytical and finite element solutions to a fundamental moving load problem, Olsson(11) provided an interesting discussion and the referent data for further studies of the moving load problem. Rieker et al. (12) discussed the effect of discretization and boundary conditions on the accuracy of the finite element solution of beams subjected to moving loads. Adopting polynomials as trail functions for the kinematic variables in solving the Lagrange equations, Kocatürk and Şimşek(13), (14) studied the dynamic response of viscoelastic Bernoulli and Timoshenko beams subjected to an eccentric compressive force and a moving harmonic load. Functionally graded materials (FGMs) invented by Japanese scientists in 1984(15) have received great interest from researchers. The FGMs are formed by varying percentage of constituents in any desired spatial direction, and as a result the specific physical and mechanical properties of the formed material can be obtained. With this feature, FGMs offer great potential for use as a structural material, and analysis of FGM structures has become a main topic in structural mechanics. A comprehensive list of publications on the analysis of FGM structures subjected to different loadings is given in a review paper by Birman and Byrd(16), only the papers that are most relevant to the problem addressed in the present work are discussed below. Sankar(17) proposed an elasticity solution for FGM beams subjected to a static transverse load by assuming that the Young’s modulus varies exponentially through the beam thickness. Employing polynomials obtained by solving the governing different equations as interpolation functions, Chakraborty et al.(18), (19) formulated a beam element based on the first-order shear deformation theory for analyzing the thermoelastic behavior of FGM beams. The wave propagation behavior of FGM beams under high frequency impulse loading was investigated by using the spectral finite element method. Based on classical and higher order shear deformation beam theories, Aydogdu and Taskin(20) studied the free vibration of simply supported FGM beams with Young’s modulus varies in thickness direction according to power and exponential laws. Using the third order shear deformation beam theory, Kadoli et al.(21) formulated a beam element to study the static behavior of metal ceramic beams under ambient temperature. Benatta et al.(22) derived an analytical solution to the bending problem of an FGM beam taking the warping effect into consideration. Li(23) described a unified approach for studying static and dynamic behavior of FGM Timoshenko and Bernoulli beams with the material properties are arbitrary functions through the thickness. Xiang and Yang(24) investigated the free and forced vibration of laminated FGM Timoshenko beam of variable thickness under thermally induced initial stress. Pradhan and Murmu(25) studied the thermo-mechanical vibration of FGM beam and FGM sandwich beam resting on variable Winkler and two-parameter foundations by employing the modified differential quadrature method in solving the governing differential equations. Sina et al.(26) presented an analytical method based on a new beam theory for free vibration analysis of FGM beams with different boundary conditions. Huang and Li(27) studied the free vibration of non-uniform cross-section beams made of axially FGM. Using the finite element method, Alshorbagy et al.(28) investigated the free vibration characteristics of Euler-Bernoulli beams with material gradation in both axial and transversal directions. Shahba et al.(29) employed the exact shape functions from a

13


Vol. 7, No. 1, 2013

uniformed homogeneous Timoshenko beam segment to formulate a finite element formulation for computing natural frequencies and buckling loads of tapered Timoshenko beams made of axially FGM. Using the same method as in Refs (13) and (14), Şimşek and Kocatürk(30) studied the free and forced vibration of an FGM Bernoulli beam under a concentrated moving harmonic load. The work is then extended to the Timoshenko beam subjected to a moving mass(31), and the nonlinear Timoshenko beam(32) subjected to a moving harmonic load. In this paper, the dynamic response of non-uniform FGM beams subjected to a variable speed moving point load is investigated. The material properties of the beams are assumed to vary in the thickness direction by a power law. With the non-uniform beams considered herein the analytical methods are difficult to deal with, and the finite element method is employed instead of. To this end, a first order shear deformable beam element, taking the variation of cross section and the shift in the neutral axis position is formulated by using exact polynomials obtained from solutions of the governing differential equation of a homogeneous Timoshenko beam segment as interpolation functions for the transverse displacement and rotation. Regarding the cited work on dynamic analysis of FGM beams subjected to moving loads, two new features are considered in the present paper. Firstly, the cross section area of the beam is assumed to vary along the beam axis, and secondly the effect of acceleration and deceleration of the moving load is taken into consideration. Using the formulated element, the equations of motion are constructed and solved by the implicit Newmark integration method. The influence of the non-uniform cross section, moving load parameters as well as the length to height ratio on the dynamic response of the beams is examined and highlighted in detail.

2. Non-uniform FGM Beams Fig. 1(a) shows a simply supported FGM beam subjected to a point load Q0 which moves from left to right in a uniform (constant speed), accelerated or decelerated speed motion. In the figure, L, h, Eb, Et, Gb, Gt, ρb, ρt denote the total length, cross section height, Young’s modulus, shear modulus and mass density of the material constituents at the bottom and top surfaces of the beam, respectively. The cross section area, A(x) and moment of inertia I(x) are assumed to vary longitudinally in two following manners:     x x • Type A: A( x) = Am  1 − α − 0.5  , I ( x) = I m  1 − α − 0.5  L L     2 2       x x A( x) = Am 1 − α  − 0.5   , I ( x) = I m 1 − α  − 0.5   • Type B:     L L   where Am, Im denote the area and moment of inertia of a mid-span section, respectively; α is a parameter defined how the cross section varies, and it will be called a non-uniform parameter in the below. Mathematically, α varies in a range of [0; 2), but a value of α very near 2 is impractical. The homogeneous beam of the Type A profile subjected to a moving load has been studied by Hino et al.(6). Figure 1(b) shows the longitudinal variation of the beam width corresponding to the two section types. The beam material is assumed to be formed from two materials with the effective property P continuously varies in the thickness direction according to a power law as

n

 z′  P ( z ′) = ( Pt − Pb )  + Pb h

(1)

where n is the nonnegative power law index, defined the distribution of the constituents through the thickness; the coordinate system (x’, z’) is chosen as well as the x’ axis is on the lower surface, and z’ in the above equation is measured from the bottom surface; Pb and Pt denote the material property such as Young’s modulus, shear modulus or mass density of the material at the bottom and top surfaces, respectively. As evidence from Eq. (1), the

14


Vol. 7, No. 1, 2013

bottom and top surfaces respectively corresponding to z’ = 0 and z’ = h, contain pure one material constituent. Due to the variation of Young’s modulus through the thickness of the beam, the neutral axis is no longer at the mid-plane, but it shifts from the mid-plane unless for the case of symmetric Young’s modulus. The position of the neutral axis can be determined by solving the following equation(33), (34), Et − Eb Eb

(1 + β ) n ∫0 (1 + β ) ξ dξ + 1

2

n

 1  E − Eb Γ(n + 1)  1 − 2  = t Eb β 2 Γ(n + 3 ) β  

(2)

where Г(.) is the Gamma function with its value can readily be defined for a given power law index; β = (h − h0)/h0, where h0 denotes the distance from the bottom surface to the neutral axis; ξ is a non-dimensional integration parameter in the thickness direction of the beam cross section. The solution of Eq. (2), namely β, and thus the position of the neutral axis h0, depends on the ratio between the Young’s modulus of the material constituents, Et/Eb, and the power law index n. Fig. 2 shows the effect of the power law index n on the position of neutral axis of the FGM beam for various values of the Et/Eb ratio. The figure shows an increase in the axial axis position with respect to the lower surface when increasing the Et/Eb ratio, regardless of the index n.

Figure 1: (a) A simply supported FGM beam under a moving load; (b) Longitudinal variation of beam cross section. 0.75

0.7

h0/h

0.65

0.6

0.55

Et/Eb=2 Et/Eb=5 E /E =10

0.5

t

0.45 0

2

4

6

b

8

10

n

Figure 2: Effect of power law index n on the position of neutral axis.

15


Vol. 7, No. 1, 2013

3. Finite Element Formulation Considering an FGM beam in a coordinate system (x, z) as shown in Fig. 1(a), where the x axis coincides with the beam neutral axis. Adopting the first order Timoshenko beam theory, the axial and transverse displacements at any point of the beam are given by u ( x, z , t ) = − zθ ( x, t ) , w( x, z , t ) = w( x, t )

(3)

where w(x,t) and θ(x,t) denote the transverse displacement and cross section rotation correspond to a point with an abscissa x, respectively; z is the distance from the considering point to the x axis. Assuming the linearly elastic behavior, the strains and stresses associated with the displacement field in Eq. (3) are given by ∂θ ∂w , γ xz = −θ ∂x ∂x σ x = E ( z ) ε x , τ xz = ψ G ( z ) γ xz

ε x = −z

(4)

where ψ is the correction factor, and its value depends on the geometry of the beam cross section. From Eq. (4), the strain energy of the beam is given by U=

1 2

L

L



0 A( x )

0



1  ∂θ  ∫ ∫ (σ xε x + τ xzγ xz )dAdx = 2 ∫  Dxx  ∂x 

2

 ∂w  + ψAxz  −θ   ∂x 

2

  dx 

(5)

where Dxx and Axz are the bending and shear rigidities, respectively and computed as(33)

Dxx =

h  h− h  z 2 E ( z ) dA = b( x)  z 2 E (h0 + z ) dz + z 2 E (h0 − z ) dz  0 A( x )  0 

∫

∫

0

0

∫

h   h− h Axz = G ( z ) dA = b( x)  G (h0 + z ) dz + G (h0 − z ) dz  A( x ) 0   0

∫

∫

0

(6)

0

∫

with b(x) denotes the beam width, which varies longitudinally in the present work. In a similar way, the kinetic energy of the beam is given by 2 2 L L  ∂u  2  ∂w  2  1   ∂θ  1  ∂w   ρ ( z )   +    dAdx = T= I D   + IA    dx 2 0   ∂t  2 0 A( x )  ∂t    ∂t   ∂t   where h   h− h I D = z 2 ρ ( z ) dA = b( x)  z 2 ρ (h0 + z ) dz + z 2 ρ (h0 − z ) dz    0 0 A( x )

∫∫

∫

∫

∫

0

0

∫

h   h−h I A = ρ ( z ) dA = b( x)  ρ (h0 + z ) dz + ρ (h0 − z ) dz    0 A( x ) 0

∫

∫

0

(7)

(8)

0

∫

The potential energy of the moving load is simply given by

V = −Q0 w( x, t ) δ ( x − s (t ))

(9)

where δ(.) is the Dirac delta function, and s(t) is the function describing the motion of the load Q0 at time t, and given by

16


Vol. 7, No. 1, 2013

s (t ) = v0 t + 12 a t 2

(10)

with v0 is the speed of the moving load when it is at the left end of the beam, and a denotes the acceleration of the load Q0, which assumed to be constant in the present work. Applying Hamilton’s principle, Eqs. (5), (7) and (9) lead to the following governing differential equations IA

∂2w ∂   ∂w  + ψ Axz  − θ  = Q0 2 ∂t ∂x   ∂x 

(11)

∂ 2θ ∂  ∂θ   ∂w  I D 2 +  Dxx −θ  = 0  + ψ Axz  ∂t ∂x  ∂x   ∂x  and the essential and natural boundary conditions for the simply supported beam w(0, t ) = w( L, t ) = 0 Dxx

∂θ  ∂w  = 0 , ψ Axz  − θ  = 0 at ∂x ∂ x  

x = 0 and

(12)

x=L

Assuming the beam is being divided into nel elements with length of l. The vector of nodal displacements for a standard two-node beam element adopted herein is given by

d = { w1 θ 1

w2 θ 2

}T

(13)

where and hereafter (.)T denotes the transpose of a vector or a matrix inside the brackets. The transverse displacement and cross section rotation inside the element are interpolated from the nodal displacements according to

w = N wT d

, θ = N θT d

(14)

where N w = { N w1 N w 2 N w3 N w 4 } and N θ = { Nθ 1 Nθ 2 Nθ 3 Nθ 4 } are the matrices of interpolating functions for w and θ, respectively. The following polynomials obtained by solving the governing differential equations of a homogeneous Timoshenko beam segment(35), (36) are used in the present work, T

N w1

3 2  1  x x x = 2   − 3   − λ   + (1 + λ ) 1 + λ   l  l l 

N w2

3 2 λ   x   λ   x  l  x   =   −  2 +    +  1 +    1 + λ  l   2  l   2   l 

N w3

3 2 1  x  x  x =− 2   − 3   − λ   1 + λ   l   l  l

N w4

3 2 λ   x  λ  x  l  x   =   −  1 −    −   1 + λ  l   2  l  2  l 

T

(15)

and

17


Vol. 7, No. 1, 2013

 x  2  x    −    l   l 

Nθ 1 =

6 (1 + λ ) l

Nθ 2 =

2  1  x x 3  + (4 + λ )   + (1 + λ ) 1 + λ   l  l 

6 =− (1 + λ ) l

Nθ 3

(16)

 x  2  x     −     l   l 

2 1  x  x  3   − (2 − λ )   1 + λ   l   l 

Nθ 4 =

In Eqs. (15) and (16), λ is the shear deformation parameter, defined as

λ=

12 E0 I 0 l 2 ψ G0 A0

(17)

with A0, I0, E0, G0 are the cross section area, moment of inertia, Young’s and shear moduli of the homogeneous uniform beam, respectively. In the numerical computation performed in the next section, A(x), I(x) corresponding to the section at the left node of the element, and Eb, Gb are assigned as A0, I0, E0, G0 respectively. It is necessary to note that, as given in Eqs. (15) and (16), when λ = 0 the interpolation functions Nwi (i =1..4) deduce to the well-known Hermite polynomials, and Nθi (i =1..4) deduce to the derivative of Nwi with respect to the space variable x. The Hermite polynomials are used in formulating the traditional Bernoulli beam element, and thus the Timoshenko element of the present work reduces to the Bernoulli beam element when λ = 0. Substituting Eqs. (14)-(16) into Eqs. (5) and (7), one can write the strain and kinetic energies for the element in the forms

U=

1 n 1 n T = d k d ∑ ∑ d T (kb + k s ) d 2 i =1 2 i =1 el

el

(18)

1 n 1 n T = ∑ d T k d = ∑ d T (m w + mθ ) d 2 i =1 2 i =1 el

el

where l

kb =

∫ 0

l

T

∂N θ ∂N θT  ∂N w   ∂N w  − N θ  Axz  − N θ  dx Dxx dx, k s =  ∂x ∂x ∂ ∂ x x     0

∫

(19)

are the stiffness matrices stemming from the bending and shear deformation, respectively, and l

∫

l

∫

m w = N w I A N wT dx , mθ = N θ I D N θT dx 0

(20)

0

respectively are the transverse and rotation mass matrices. In order to improve the convergences of the numerical results, the exact variation of the cross section profiles is used in computing the element stiffness and mass matrices. From Eqs. (9), (14) and (18), the discrete equation of motion for the beam can be written in the form

18


Vol. 7, No. 1, 2013

+ K D = F MD ex

(21)

where M, K are the structural mass and stiffness matrices, obtained by assembling the formulated element mass and stiffness matrices; Fex is the external nodal load vector, which having a simply form T

    Fex = Q0  0 0 0 ... N w1 N w 2 N w3 N w4 ... 0 0 0 

  element under loading

(22)

where the interpolating functions Nwi (i=1..4) are evaluated at the current position of the moving load. The system of Eq. (21) can be solved by using the direct integration method. The average acceleration implicit Newmark method, which ensures the unconditional convergence, is adopted in the present work(37). In the free vibration analysis, the right hand side of Eq. (21) is set to zeros, and a harmonic response, D = D sin ωt is assumed, so that Eq. (21) deduces to

(K − ω

2

)

M D =0

(23)

where ω is the circular frequency, and D is the vibration amplitude. Eq. (23) can be solved by a standard method of the eigenvalue problem as described in the Ref. (37).

4. Numerical Results A simply supported beam made of FGM subjected to a variable speed moving load is considered. The geometric data for the beam are as follows: bm = 0.5 m, h = 1.0 m, L = 5 and 20 m, where bm , h , L are the width of mid-span cross section, height and total length of the beam, respectively. Two values of length are chosen in order to study the effect of the aspect ratio, L/h. Otherwise stated, the beam is assumed to be formed from steel and alumina. The Young’s modulus and the mass density of steel respectively are 210 GPa and 7,800 kg/m3, and that of alumina are 390 GPa and 3,960 kg/m3 (30). The amplitude of the moving load is Q0 = 100 kN. A Poisson’s ratio υ = 0.3 is adopted for both the material constituents. Since the cross section of both the two types of the beam is rectangular, a shear correction factor ψ = 5/6 is used in the computation. Three types of motion, namely constant speed v, acceleration and deceleration are considered. In the acceleration motion, it is assumed that the speed of the load at the left end of the beam is zero, and it exits the beam with a speed of v. For the decelerated motion, the speed of the load at the left end is v and that at the right end is zero. With this assumption, Eq. (10) gives a total time ∆T necessary for the load to traverse across the beam of L / v for the constant speed motion and L / 2v , where v is the speed of the moving load corresponding to the time when the load exists and enters the beam for the accelerated and decelerated motions, respectively. In the computation reported below, 500 time steps are used for the Newmark method, and thus a uniform time increment width of dt = ∆T / 500 is employed for the method. Following the work in Ref. (11), and in order to facilitate the numerical results, the following dimensionless parameters representing the maximum mid-span deflection and the moving load speed are introduced as

19


Vol. 7, No. 1, 2013

 w(L / 2, t )   , f D = max  w0  

fv =

πv ω10

(24)

where w0 = Q0 L3 / 48 Eb I m is the static deflection of the uniform steel beam under a static load Q0 acting at the mid-span of the beam, and ω10 = π 2 / L2 Eb I m / ρ b Am is the fundamental frequency of the simply supported uniform steel beam(1). The parameter fD is similar to the dynamic magnification factor defined in the moving load problem of homogeneous beams(1),(11), but for the FGM beams considered in the present work, fD is affected by both the dynamic effect and the material non-homogeneity, and it will be called deflection factor in the following.

(

)

4.1 Verification of formulation In order to verify the accuracy of the proposed formulation and numerical procedure, the fundamental frequency of a uniform FGM beam formed from aluminum and alumina is computed and compared with the analytical result of Sina et al.(26), and the numerical result of Şimşek(31). The properties of the material constituents are given in Refs. (26), (31). Table 1 lists values of the non-dimensional fundamental frequencies of the beam with various values of the aspect ratio, L/h, where the results of Refs. (26) and (31) are also given. Very good agreements between the natural frequencies obtained in the present work with that of Refs. (26) and (31) is seen from table, and an error just less than 0.2% between the frequency parameter computed in the present work compared to that obtained by the analytical solution of Ref. (26) is noted. The non-dimensional fundamental frequency in the table is defined in accordance with Ref. (26) as follows

µ = ω1 L2

IA

h

2

∫

h − h0

− h0

(25)

E ( z ) dz

where ω1 is the fundamental frequency. It is noted that the frequencies listed in the table have been computed by using 10 elements. More elements have been employed, but improvement in the accuracy of the computed frequency parameter compared to the analytical solution of Ref. (26) was not observed. In order to verify the formulation in more further, the maximum dynamic magnification factor and the corresponding speed of a uniform FGM beam with L = 20 m, h = 0.9 m and b = 0.4 m, previously studied by Şimşek and Kocatürk (30) are considered. The beam is formed from steel and alumina with the Young’s moduli and mass densities given above. Since the work in Ref.(30) is based on the Bernoulli beam theory, and thus in order to enable to compare the numerical results, the shear deformation parameter λ, defined by Eq. (17), is set to zero in the derived formulation in this computation. Table 2 lists the maximum dynamic deflection factors and the corresponding speeds for various values of the index n, where the numerical results of Ref. (30) are also given. Table 1: Comparison of non-dimensional fundamental frequencies of uniform FGM beam

n 0.0

0.3

L/h

Present

Ref. (26)

Ref. (31)

10

2.8026

2.797

2.804

30

2.8438

2.843

2.843

100

2.8486

2.848

2.848

10

2.6992

2.695

2.701

30

2.7368

2.737

2.738

100

2.7412

2.742

2.742

20


Vol. 7, No. 1, 2013

Table 2: Maximum deflection factor and corresponding speed of uniform FGM beam subjected to a constant speed moving load

n 0.2

Present v (m/s) max(fD)

Ref. (30) max(fD) v (m/s)

1.0346

1.0344

222

222

0.5

1.1444

197

1.1444

198

1

1.2504

178

1.2503

179

2

1.3776

164

1.3376

164

Pure alumina

0.9328

252

0.9328

252

Pure steel

1.7324

132

1.7324

132

Very good agreement between the numerical results of the present work with that of Ref. (30) is seen from the table. The numerical results listed in Table 2 have been computed by using only 10 elements also, and because of this excellent convergence, a mesh of 10 elements is employed in the computation reported below. 4.2 Effect of material non-homogeneity The material non-homogeneity distribution through the beam thickness is defined by Eq. (1) through the power law index n, and the effect of this index on the dynamic response of the beam is examined in this subsection. Fig. 3 depicts the relation between the deflection factor and the speed parameter of the Type A beam having a non-uniform parameter α = 0.5, subjected to a constant speed moving load. For the low values of the speed parameter fv, the deflection factors in Fig. 3 both increase and decrease with increasing fv, and this phenomenon is associated with the oscillations of the beam subjected to a low speed moving load as in case of the homogeneous beams(11). For a given value of the index n and when the parameter fv exceeds a certain value, the deflection factor steadily increases with increasing speed parameter, and it then decreases after reaching a peak value. The index n clearly affects the deflection factor in Fig. 3, where an increase in the index n results in a higher factor fD, regardless of the speed parameter and the aspect ratio. This increase in the factor fD can be seen from Eq. (1), where the effective Young’s modulus and shear modulus decrease with the increasing of the index n, and as a result both the bending and shear rigidities, defined by Eq. (6), are decreased. The increase in the deflection factor by raising the power law index can also be seen clearly from the relation between this factor and the index n as depicted in Fig. 4 for the Type A beam having α = 0.5. The figure shows a steady increase in the deflection factor with the increasing of the index n, regardless of the moving speed and the aspect ratio. The effect of the index n on the relation between the deflection factor with the speed factor and the power law index is similar for both the aspect ratios, except that the factor fD is considerably higher for the beam having the lower aspect ratio. Fig. 5 shows the normalized mid-span axial stress distribution through the thickness of the Type A beam under a constant speed moving load. In the figure, σ 0 = Q0 h L / 8 I m is the maximum mid-span axial stress of the uniform beam under a static load Q0 acting at the mid-span, and the axial stress is computed when the moving load is at the mid-span of the beam. It is seen from the figure that the axial stress of the homogeneous steel beam is linear and the magnitudes of both the compressive stress and tensile stress are equal to each other as expected. The stress of the FGM beam is, however not linear, and the magnitude of the compressive stress is higher than that of the tensile stress, regardless of the power law index and the moving speed. The magnitude of the compressive stress tends to increase when raising the index n. The higher moving speed results in larger magnitudes of both the tensile and compressive stresses, but it hardly changes the distribution of the stress through the beam thickness.

21


Vol. 7, No. 1, 2013

1.6

1.4

1.4

1.2

1.2

f

fD

D

1.6

1

1 n=0.2 n=0.5 n=1 n=2

0.8

0.6 0

0.5

1

f

n=0.2 n=0.5 n=1 n=2

0.8

0.6 0

1.5

0.5

1

fv

v

1.5

1.8

1.6

1.6

1.4

1.4

1.2

1.2

fD

fD

(a) L/h = 5 (b) L/h = 20 Figure 3: Relation between deflection factor and moving speed parameter of Type A beam under constant speed moving load (α = 0.5).

f =0.125 v f =0.25 v f =0.375 v f =0.5

1 0.8

0.8

2

4

n

6

8

fv=0.125 f =0.25 v f =0.375 v f =0.5 v

0.6

v

0.6 0

1

0.4 0

10

2

4

6

8

10

n

(a) L/h = 5 (b) L/h = 20 Figure 4: Relation between deflection factor and power law index n of Type A beam under constant speed moving load (α = 0.5). 0.5

0

−0.25

−0.5 −1.5

n=0.2 n=2 n=10 pure steel

0.25

z/h

0.25

z/h

0.5

n=0.2 n=2 n=10 pure steel

0

−0.25

−1

−0.5

0 σ/σ0

0.5

1

1.5

−0.5 −2

−1.5

−1

−0.5 0 σ/σ0

0.5

1

1.5

(a) fv = 0.125 (b) fv = 0.5 Figure 5: Normalized mid-span axial stress distribution through thickness of Type A beam subjected to a constant speed moving load (L/h = 20, α = 0.5).

4.3 Effect of acceleration and deceleration Fig. 6 shows the relation between the deflection factor and the speed parameter of the Type A beam under different motions of the moving load for n = 0.5 and α = 0.5. As shown in the figure, the relation between the factor fD and the parameter fv is strongly influenced by the acceleration and deceleration of the moving load. At the given values of the index n and the non-uniform parameter α, the factor fD obtained in the decelerated motion of the moving load is always higher than that obtained in the accelerated motion. Furthermore, the deflection factor obtained in the decelerated motion of the moving load will surpass the factor obtained in the constant speed motion when the moving speed exceeds a certain value.

22

Journal of Computational Science and Technology 1.5

1.5

1.3

1.3

1.1

1.1

fD

fD

Vol. 7, No. 1, 2013

0.9

0.7 0

0.9 uniform acceleration deceleration

uniform acceleration deceleration 0.5

1 fv

1.5

0.7 0

2

0.5

1

fv

1.5

(a) L/h = 5 (b) L/h = 20 Figure 6: Relation between deflection factor and speed parameter of Type A beam under different motions of moving load (n = 0.5, α = 0.5).

The effect of the acceleration and deceleration can also be observed in the time histories for the normalized mid-span deflection of the Type A beam as displayed in Fig. 7 for the case α = 1, n = 2 and fv = 0.25. As seen from Fig. 7, while the beam executes more vibration cycles under the decelerated motion, it is more “silent” under the accelerated motion of the moving load. The aspect ratio, as seen from Fig. 6 and 7, alters the magnitude of the deflection factor fD, but it hardly affects the relation between the factor fD and the speed parameter fv as well as the time histories of the beam. 1.5

1.2

0.9

0.9 w(L/2,t)/w0

w(L/2,t)/w0

1.2

0.6

0.6

0.3

0.3

−0.3 0

0

uniform acceleration deceleration

0

0.2

0.4

0.6 t/∆ T

0.8

1

−0.3 0

uniform acceleration deceleration 0.2

0.4

0.6

0.8

1

t/∆ T

(a) L/h = 5 (b) L/h = 20 Figure 7: Time histories for normalized mid-span deflection of Type A beam under different motions of moving load (α = 1, n = 2, fv = 0.25).

4.4 Effect of non-uniform parameter In Fig. 8, the effect of the non-uniform parameter on the maximum deflection factor of the Type A beam under the accelerated and decelerated motions of the moving load is depicted. As expected, the maximum deflection factor increases with the increasing of the non-uniform parameter α, regardless of the index n and the motion type. At a given value of the non-uniform parameter α, the maximum deflection factor of the beam under deceleration is always higher than that of the beam under acceleration, but the influence of the non-uniform parameter on the maximum deflection factor obtained in the decelerated motion is similar to that obtained in the accelerated motion. 4.5 Effect of section profile The effect of section profile on the relation between the maximum deflection factor and the non-uniform parameter of the beam under different motions of moving load is depicted in Fig. 9 for two different aspect ratios and for an index n = 2. As shown in the figure, the maximum deflection factor of the beam with Type A section is much more sensitive to the non-uniform parameter α compares to that of the beam with Type B section, regardless of

23


Vol. 7, No. 1, 2013

the motion type and the aspect ratio. For a given value of the parameter α, the maximum deflection factor of the Type A beam is higher than that of the Type B beam and the difference between the factors is larger for a higher value of α, regardless of the type of motion and the aspect ratio. The aspect ratio affects the amplitude of the maximum deflection factor, but it hardly changes the relation between this factor and the non-uniform parameter. 2.2

1.8 D

D

max(f )

1.8

2

max(f )

2

2.2 n=0.2 n=0.5 n=2 n=5

1.6 1.4

1.6 1.4

1.2

1.2

1

1

0.8 0

0.4

0.8 α

1.2

0.8 0

1.6

n=0.2 n=0.5 n=2 n=5

0.4

0.8 α

1.2

1.6

(a) Accelerated motion (b) Decelerated motion Figure 8: Effect of non-uniform parameter α on maximum deflection factor of Type A beam under different motions of moving load (L = 20 m).

2

2.1 type A, uni. type A, acc. type A, dec. type B, uni. type B, acc. type B, dec.

2

max(fD)

max(fD)

2.5 2.4

1.8

type A, uni. type A, acc. type A, dec. type B, uni. type B, acc. type B, dec.

1.6

1.6 1.4

1.2 0

0.4

0.8 α

1.2

1.6

1.2 0

0.4

0.8 α

1.2

1.6

(a) L = 5 m (b) L = 20 m Figure 9: Effect of section profile on relation between maximum deflection factor and non-uniform parameter of beam under different motions of moving load (n = 2).

5. Conclusions A finite element procedure for analyzing dynamic response of functionally graded beams under a variable speed moving point load has been described. A first order shear deformable beam element taking the effect of the material non-homogeneity, non-uniform cross section and the shift in the neutral axis into account has been formulated and employed in the analysis. The dynamic response of the beams, including the dynamic deflection factor, time histories and axial stress at the mid-span of the beam has been computed by using the implicit Newmark method. The numerical results have shown that the dynamic response of the beams is not only governed by the moving speed, but by the material distribution also. The acceleration and deceleration of the moving load greatly affect the dynamic response of the beams, and the dynamic deflection factor obtained in the deceleration can surpass the factor obtained in the uniform motion when the moving speed exceeds a certain value. The influence of the external load parameters, the aspect ratio as well as the section profile on the dynamic response of the beam has been investigated in detail.

24


Vol. 7, No. 1, 2013

Acknowledgments The financial support from Vietnam NAFOSTED to the first author is gratefully acknowledged.

References (1) Fryba, L., Vibration of solids and structures under moving loads, (1972), Academia, Prague. (2) Chonan, S., The elastically supported Timoshenko beam subjected to an axial force and a moving load. International Journal of Mechanical Science, Vol. 17, Issue 9, (1975), pp. 573–581. (3) Abu-Hilal, M. and Zibden, H. S., Vibration of beams with general boundary traversed by a moving force. Journal of Sound and Vibration, Vol. 229, Issue 2, (2000), pp. 377–388. (4) Abu-Hilal, M. and Mohsen, M., Vibration of beams with general boundary conditions due to a moving harmonic load, Journal of Sound and Vibration, Vol. 232, Issue 4, (2000), pp. 703–717. (5) Hino, J., Yoshimura, T., Konihi, K. and Ananthanarayana, N., A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load. Journal of Sound and Vibration, Vol. 96, Issue 1, (1984), pp. 45–53. (6) Hino, J., Yoshimura T. and Ananthanarayana, N., Vibration analysis of non-linear beams subjected to a moving load using the finite element method, Journal of Sound and Vibration, Vol. 100, Issue 4, (1985), pp. 477–491. (7) Lin, W.H. and Trethewey, M.W., Finite element analysis of elastic beams subjected to moving dynamic loads, Journal of Sound and Vibration, Vol. 136, Issue 2, (1990), pp. 323–342. (8) Thambiratnam, D. and Zhuge, Y., Dynamic analysis of beams on elastic foundation subjected to moving loads, Journal of Sound and Vibration, Vol. 198, Issue 2, (1996), pp. 149–169. (9) Nguyen Dinh Kien, Dynamic response of prestressed Timoshenko beams resting on two-parameter foundation to moving harmonic load, Technische Mechanik, Vol. 28, Issue 3-4, (2008), pp. 237–258. (10) Nguyen Dinh Kien and Le Thi Ha, Dynamic characteristics of elastically supported beam subjected to a compressive axial force and a moving load, Vietnam Journal of Mechanics, Vol. 33, Issue 2, (2011), pp. 113–131. (11) Olsson, M., On the fundamental moving load problem, Journal of Sound and Vibration, Vol. 145, Issue 2, (1991), pp. 299–307. (12) Rieker, J.R., Lin, Y.H. and Trethewey, M.W., Discretization considerations on moving load finite element beam models, Finite Elements in Analysis and Design, Vol. 21, Issue 3, (1986), pp.129–144. (13) Kocatürk, T. and Şimşek, M., Vibration of visoelastic beams subjected to an eccentric axial force and a concentrated moving harmonic force. Journal of Sound and Vibration, Vol. 291, Issues 1-2, (2006), pp. 302–322. (14) Kocatürk, T. and Şimşek, M., Dynamic analysis of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load, Computers & Structures, Vol. 84, Issues 31-32, (2006), pp. 2113–2117. (15) Koizumi, M., FGM activities in Japan, Composites: part B, Vol. 28, Issue 1-2, (1997), pp. 1–4. (16) Birman, V. and Byrd, L.W., Modeling and analysis of functionally graded materials and structures. Applied Mechanics Reviews, Vol. 60, Issue 5, (2007), pp. 195–216. (17) B.V. Sankar, An elasticity solution for functionally graded beams. Composites Science

25


Vol. 7, No. 1, 2013

and Technology, Vol. 61, Issue 5, (2001), pp. 689–696. (18) Chakraborty, A., Gopalakrishnan, S. and Reddy, J.N., A new beam finite element for the analysis of functionally graded materials. International Journal of Mechanical Science, Vol. 45, Issue 3, (2003), pp. 519–539. (19) Chakraborty, A. and Gopalakrishnan, S., A spectrally formulated finite element for wave propagation analysis in functionally graded beams. International of Solids and Structures, Vol. 40, Issue 10, (2003), pp. 2421–2448. (20) Aydogdu, M. and Taskin, V., Free vibration analysis of functionally graded beams with simply supported edges, Materials & Design, Vol. 28, Issue 5, (2007), pp. 1651–1656. (21) Kadoli, R., Akhtar, K. and Ganesan, N., Static analysis of funtionally graded beams using higher order shear deformation beam theory, Applied Mathematical Modelling, Vol. 32, Issue 12, (2008), pp. 2509–2525. (22) Benatta, M.A., Mechab, I., Tounsi, A. and Adda Bedia, E.A., Static analysis of functionally graded short beams including warping and shear deformation effects. Computational Materials Science, Vol. 44, Issue 2, (2008), pp. 765–773. (23) Li, X.F., A unified approach for analyzing static and dynamic behaviors of functionally graded Timoshenko and Euler-Bernoulli beams. Journal of Sound and Vibration, Vol. 318, Issues 4-5, (2008), pp. 1210–1229. (24) Xiang, H.J. and Yang, J., Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Composites: part B, Vol. 39, Issue 2, (2009), pp. 292–303. (25) Pradhan, S.C. and Murmu, T., Thermo-mechanical vibration of FGM sandwich beam under variable elastic foundations using differential quadrature method, Journal of Sound and Vibration, Vol. 321, Issues 1-2, (2009), pp. 342–362. (26) Sina, S.A., Navazi, H.M. and Haddadpour, H., An analytical method for free vibration analysis of functionally graded beams, Materials & Design, Vol. 30, Issue 3, (2009), pp. 741–747. (27) Huang, Y. and Li, X.F., A new approach for free vibration of axially functionally graded beams with non-uniform cross-section, Journal of Sound and Vibration, Vol. 329, Issue 11, (2010), pp. 2291–2303. (28) Alshorbagy, A.E., Eltaher, M.A. and Mahmoud, F.F., Free vibration chatacteristics of a functionally graded beam by finite element method, Applied Mathematical Modelling, Vol. 35, Issue 1, (2011), pp. 412–425. (29) Shahba, A., Attarnejad, R., Marvi, M. T., and Hajilar, S., Free vibration and stability analysis of axially functionally graded tapered Timoshenko beams with classical and non-classical boundary conditions, Composites: part B, Vol. 42, Issue 4, (2011), pp. 801–808. (30) Şimşek, M. and Kocatürk, T. Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load, Composite Structures, Vol. 90, Issue 4, (2009), pp. 465–473. (31) Şimşek, M., Vibration analysis of a functionally graded beam under a moving mass by using different beam theories, Composite Structures, Vol. 92, Issue 4, (2010), pp. 904-917. (32) Şimşek, M., Non-linear vibration analysis of a functionally graded Timoshenko beam under action of a moving harmonic load, Composite Structures, Vol. 92, Issue 10, (2010), pp. 2532-2546. (33) Kang, Y.A. and Li, X.F., Bending of functionally graded cantilever beam with power-law nonlinearity subjected to an end force, International Journal of Non-Linear Mechanics, Vol. 44, Issue 6, (2009), pp. 693-703. (34) Kang, Y.A. and Li, X.F., Large deflection of a non-linear cantilever functionally graded beam, Journal of Reinforced Plastics and Composites, Vol. 29, Issue 4, (2010), pp.

26


Vol. 7, No. 1, 2013

1761-1774. (35) Kosmatka, J.B., An improve two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams. Computers & Structures, Vol. 57, Issue 1, (1995), pp. 141-149. (36) Yokoyama, T., Vibration analysis of Timoshenko beam-column on two-parameter elastic foundation, Computers & Structures, Vol. 61, Issue 6, (1996), pp. 995-1007. (37) Géradin, M. and Rixen, R., Mechanical vibrations. Theory and application to structural dynamics. 2nd edition, John Wiley and Sons, (1997), Chichester.

27

Journal of Computational Science and Technology - J-Stage

Journal of Computational Science and Technology - J-Stage

Suggest Documents