Dynamic analysis of multi-span functionally graded beams subjected to a variable speed moving load

Transcription

1 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 Porto Portuga 3 June - Juy 4 A. Cunha E. Caetano P. Ribeiro G. Müer (eds.) ISS: 3-9; ISB: Dynamic anaysis of muti-span functionay graded beams subjected to a variabe speed moving oad Buntara Stheny Gan guyen Dinh Kien Department of Architecture Coege of Engineering ihon University -akagawara okusada Koriyama-shi Fukushima-ken Japan e-mai: buntara@arch.ce.nihon-u.ac.jp Department of Soid Mechanics Institute of Mechanics Vietnam Academy of Science and echnoogy 8 Hoang Quoc Viet Hanoi Vietnam e-mai: ndkien@imech.ac.vn ABSRAC: A finite eement formuation based on the first-order shear deformabe beam theory for dynamic anaysis of mutispan beams made of functionay graded materia (FGM) subjected to a variabe speed moving oad is presented. he beam materia is assumed to be graded in the thickness direction by a power-aw distribution. Exact soution of static part of the governing differentia equations of a FGM beam segment is empoyed to interpoate the dispacements and rotation. he shift in the neutra axis position arose from the materia inhomogeneity is taken into consideration in the formuation. he dynamic response of the beam is computed with the aid of the direct integration ewmark method. he numerica resuts show that the proposed formuation is capabe to give accurate dynamic characteristics of the beam by using a sma number of eements. A parametric study is carried out to highight the effect of the materia inhomogeneity and oading parameters on the dynamic behaviour of the beams. he infuence of the aspect ratio on the dynamic response is aso examined and highighted. KEY WORDS: Functionay graded materia; Muti-span beam; Moving oad; Dynamic anaysis; Finite eement method. IRODUCIO Functionay graded materias (FGMs) have received much attention from engineers and researchers since they were first initiated by Japanese scientists in 984. FGMs are formed by graduay varying the voume fraction of constituent materias usuay ceramics and metas in a desired spatia direction. As a resut the effective properties of FGMs exhibit continuous change thus eiminating interface probems and mitigating therma stress concentrations. FGMs are now widey used as a structura materia and anaysis of structures made of FGMs has become an important topic in the fied of structura mechanics. Many investigations on anaysis of FGM structures subjected to different oadings can be found in the iterature ony contributions that are most reevant to the present work are briefy discussed beow. Praveen and Reddy [] derived a pate eement for evauating the static and dynamic response of FGM pates. Chakraborty et a. [] formuated a shear deformabe beam eement for anayzing the thermo-eastic behaviour of FGM beams. Based on the higher-order shear deformabe beam theory Kadoi et a. [3] deveoped a beam eement for studying the static behaviour of FGM beams under ambient temperature. In [4] Li presented a unified approach for studying static and dynamic behaviour of FGM beams. Based on a new beam theory Sina et a. [5] proposed an anaytica method for evauating the free vibration of FGM beams. Zhang and Zhou [6] studied the bucking and noninear bending of thin FGM pates by adopting the neutra surface as the reference pane. aking the shift in the neutra axis position into account Kang and Li [7 8] derived soutions for FGM cantiever beams subjected to a concentrated oad or moment at the free end. In [9] Ashorbagy et a. studied the free vibration of FGM Bernoui beams by using the finite eement method. Shahba et a. [] empoyed the exact soution of a homogeneous beam eement to derive a finite eement formuation for computing the critica oads and natura frequencies of tapered imoshenko beams made of axiay FGM. Şimşek and his co-worker [ ] used the poynomia series as approximation functions for dispacements and rotation to study the vibration of FGM beams subjected to a moving oad. Etaher et a. [3] determined the neutra axis position and then derived a beam eement for computing the natura frequencies of FGM macro/nanobeams. Based on the physica neutra surface and the third-order shear deformation beam theory Zhang [4] studied the noninear bending of FGM beams under therma and mechanica oadings. guyen et a. [5] studied the vibration of non-uniform FGM beams under a moving point oad by using the finite eement method. Recenty guyen and his co-worker [6 7] used the exact soution of a homogeneous imoshenko beam eement to formuate finite eement formuations for arge dispacement anaysis of tapered FGM beams subjected to end forces. Moving oad probem in which the position of the externa oads continuousy varies with time is a topic of investigation for a ong time. Many investigations on the topic have been reported in the iterature among which the monograph by Frýba [8] is an exceent reference. In addition to the traditiona anaytica methods the finite eement method with its versatiity in spatia discretization has aso been widey empoyed in the moving oad probem. In this ine of work Fiho [9] Hino et a. [] are pioneers in deveopment of finite eement formuations for anayzing structures subjected to moving oads. Lin and rethewey [] derived the finite eement formuation for a Bernoui beam excited by different types of moving oad. Osson [] presented an anaytica soution for the fundamenta moving oad probem and then discussed the accuracy of the finite eement soution. Rieker et a. [3] investigated the effect of discretization on the accuracy of beams subjected to a moving oad. hambiranam 3879

2 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 and Zhuge [4] took the effect of the eastic foundation into their finite eement formuation in studying the dynamic response of a Bernoui beam subjected to moving oads. Based on the first-order shear deformabe beam theory guyen [5] formuated a beam eement for computing the dynamic response of prestressed beams resting on a twoparameter foundation traversed by a moving harmonic oad. In this paper the dynamic anaysis of muti-span FGM beams subjected to a variabe speed moving oad is carried out by using the finite eement method. o this end a finite eement formuation based on the first-order shear deformabe beam theory is derived by using the exact soution of a static part of governing differentia equations of a FGM beam segment as interpoation functions. he formuation as in Refs. [ ] takes the shift in the neutra axis position into account and thus there is no axia-bending couping term in the stiffness matrix. Using the derived formuation the dynamic characteristics of the beams are computed with the aid of the direct integration ewmark method. he infuence of the materia inhomogeneity oading parameters as we as the aspect ratio on the dynamic response of the beam is investigated and highighted. FGM BEAM Consider a muti-span FGM beam with rectanguar crosssection (tota ength L height h and width b) subjected to a oad P(t) moving from eft to right as depicted in Figure. In the figure the Cartesian coordinate systems (x z) is chosen with the x and z-axes parae to the ongitudina direction aong the mid-pane of the beam and norma to this pane respectivey. he FGM is assumed to be composed of ceramic and meta with the effective materia properties (ike Young s mods shear mods and mass density) P(z) vary aong the beam thickness according to a power-aw distribution as z Pz ( ) = ( Pc Pm) + + Pm h () where P c and P m denote the materia properties of ceramic and meta respectivey; n is the non-negative power-aw index dictating the materia variation profie. As seen from Eq. () the bottom surface contains ony meta and the top surface is pure ceramic. Ceary due to the variation of the Young s mods the neutra axis where the norma stress and strain are zeros for a beam segment in pure bending is no onger at the mid-pane. he distance from the neutra axis to the midpane of the beam h can be expicity expressed in term of the Young s modui of the constituent materias and the index n as foows [3 4] h/ Ezzdz ( ) h/ hn( Ec Em) h = = h/ ( n+ )( Ec + nem) Ezdz ( ) h/ where E c and E m are respectivey the Young s modui of the ceramic and meta and E(z) denotes the effective Young s mods of the beam as defined by Eq. (). Having the neutra axis position to be determined a new coordinate system (x z ) with x=x and z=z +h is introduced. n () Based on the first-order shear deformabe beam theory the axia dispacement u(xzt) and the transverse dispacement w(xzt) at any point distanced z from the neutra axis are given by u( x z) t = u() x t zθ() x t = u() x t ( z h) θ() x t (3) wxzt ( ) = w( xt ) where u and w are respectivey the axia and transverse dispacements of the corresponding point on the neutra axis and θ is the cross-section rotation. he axia strain ε x and the shear strains γ xz resuted from Eq. (3) are as foows ε = u ( z h ) θ γ = w θ (4) x x x xz x where a comma denotes the differentiation with respect to the spatia variabe x. Figure. A muti-span beam with generic beam eement. Assuming eastic behavior the stresses associated with the strains in Eq. (4) are as foows σ = Ez ( ) ε τ = Gz ( ) γ (5) x x xz xz where G(z) is the effective shear mods defined by Eq. () in the same manner as E(z). From Eqs. (4) and (5) the strain energy for the beam is given by U = ( ) σε x x + ψτxzγ xz dv V (6) L [ = Au x Au xθ x + A θ x + ψ A33 ( w x θ ) ] dx where V denotes the beam voume; ψ is the shear correction factor; A A A A 33 are respectivey the axia axiabending couping bending and shear rigidities defined as h/ ( ) = ( )[( )( ) ] h/ h/ A A A b E z z h z h dz A = b G( z) dz 33 h/ It shoud be noted that due to Eq. () the axia-bending rigidity A defined by Eq. (7) is aways equas to zero and the second term under the integra in Eq. (6) is vanished. he kinetic energy of the beam is given by = ρ( z)( u + w ) dv & & L = [ I ( u + w ) Iu & θ + I & θ ] dx & & & V (7) (8) 388

3 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 In Eq. (8) a over dot denotes the derivative with respect to time; ρ(z) is the effective mass density; I I I are the mass moments defined as I I I z z h z h da (9) ( ) = ρ( )[( )( ) ] A It shoud be noted that different from the couping rigidity the axia-bending couping moment I is not equa to zero. he potentia of the moving oad P(t) is simpy given by V = P w δ ( x s( t)) () where P denotes the ampitude of the oad; the abscissa x is measured from the eft end of the beam; δ(.) is the Dirac deta function; s(t) is the function describing motion of the force P(t) at time t which given by s() t v t at = + () with v denotes the initia speed of the moving oad and a is its acceeration which assumed to be constant in the present work. When a= a uniform motion with constant speed is resumed. Appying Hamiton s principe one can obtain the foowing equations of motion for the beam Iu&& I && θ A u xx = Iw&& ψ A33( w xx θ x ) = P δ( x s( t)) I && θ Iu&& A θ xx ψ A33( w x θ) = () In Eq. () as above mentioned the term reated with the couping axia-bending rigidity A is vanished. he associated non-essentia boundary conditions for the beam are as foows A u = ψ A ( w θ) = Q A θ = M (3) x 33 x x where Q M are the prescribed axia shear forces and moment at the beam ends respectivey. he essentia boundary conditions require the axia dispacement at eft end and transverse dispacements at a the support equa to zeros. 3 FIIE ELEME FORMULAIO Assuming the beam is divided into a number of two-node beam eements with ength of. A generic eement is depicted in the upper right corner of Figure. he vector of noda dispacements for the eement d contains six components as { u w θ u w θ } d = (4) where u w θ are respectivey the axia transverse dispacements and rotation at node ; u w θ are the corresponding quantities at node. In Eq. (4) and hereafter a superscript is used to indicate the transpose of a vector or a matrix. An interpoation scheme is now necessary to introduce to the dispacements and rotation. he present work used the exact soution of static part of the homogeneous equations of motion Eq. () to interpoate the dispacements and rotation. he exact soution of the equiibrium equations for a beam eement can easiy be obtained with the aid of a symboic software for exampe by using the dsove command in Mape [6] one gets the soution in the forms u = C 5x+C6 w = C x + C x 6 + C x+ C θ = A C x Cx C3 ψ A33 (5) In the above equation C C C 6 are the constants which can be determined from the foowing eement end conditions u = u w = w θ = θ when x = (6) u = u w = w θ = θ when x = Eqs. (5) and (6) ead to the interpoation functions for u w and θ in the forms u = u w = w θ = θ d d d (7) where u = { u u } w = { w w w3 w4} θ = { θ θ θ3 θ4} are respectivey the matrices of interpoation functions for u w θ with the foowing components u = x x u = (8) and w w w3 w4 3 x x x = 3 λ + + λ + λ 3 x λ x λ x = λ 3 x x x = 3 λ + λ 3 x λ x λ x = + + λ θ θ θ 3 θ 4 6 x x = ( + λ) x x = 3 ( 4+ λ) + λ+ + λ 6 x x = ( + λ) x x = 3 ( λ) + λ (9) () In Eqs. (8)-() the abscissa x is measured from the node and the shear deformation parameter λ is defined as λ A = () ψ A33 388

4 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 It shoud be noted that the interpoation functions for the transverse dispacement and the rotation given in Eq. (9) and () respectivey have the same forms as that derived by Kosmatka [7] for a homogeneous imoshenko beam eement. his due to the fact that the couping rigidity is vanished and thus the static part of the equations of motion for the FGM beam eement has the same form as that of the homogeneous beam. herefore the interpoation functions for the FGM beam in the present work differ from that of the homogeneous beam in Ref. [7] ony in the way of definition of the shear deformation parameter λ Eq. (). Using Eqs. (8)-() one can rewrite the expressions for the strain energy Eq. (6) in the form nele nele U = dkd= d( k + k + kγγ ) d () i= i= In () the symbo denotes the assemby over the tota number of eements nele; k is the eement stiffness matrix which composed of the foowing axia stiffness matrix k bending stiffness k and shear stiffness k γγ = ux ux = θ x θ x k A dx k A dx γγ = wx θ ψ A33 wx θ k ( ) ( ) dx Simiary the kinetic energy for the beam can be written as nele nele = = + ww + uθ + i= i= (3) dmd & & d& ( m m m m ) d& (4) with m is the mass matrix for the eement and it composes of the axia mass matrix m the transverse mass matrix m ww the axia-bending mass matrix m uθ and the bending mass matrix with the foowing forms m = u u ww = w w m I dx m I dx uθ = u θ = θ θ m I dx m I dx he potentia of the moving oad is now given by (5) V = P d w δ ( x s( t)) (6) From Eqs. ()-(6) the system of equations of motion for the beam can be written in the context of the finite eement anaysis as MD && + KD = F (7) where D and D && denote the structura vectors of noda dispacements and noda acceerations respectivey; M and K are the structura mass and stiffness matrices formed by assemby of the above formuated eement mass and stiffness matrices respectivey; F is the structura vector of externa noda forces with the foowing form F = P{ } (8) w w w3 w4 w5 w6 he oad vector F in Eq. (8) consists of a zero coefficients except for those reating the eement under oading. he system of equations of motion (7) can be soved by the direct integration ewmark method [8 9]. he average constant acceeration method which ensures numericay unconditiona stabiity is used in the present work. For free vibration anaysis the right hand side of Eq. (7) is set to zero and in this case Eq. (7) eads to an eigenvaue probem which can be sove by a standard method described in [9]. 4 UMERICAL RESULS AD DISCUSSIO A muti-span FGM beam composed of stee and aumina with the bottom surface is pure stee and the top surface contains ony aumina is considered. Otherwise stated the beam is assumed to be formed from four equa-ength spans with L= m and b=.5 m h= m. he materia properties adopted from Refs. [ ] are as foows: E s = GPa E a =38 GPa ρ s =78 kg/m 3 ρ a =396 kg/m 3 ν s =.377 ν a =.3 where the subscripts s and a stand for stee and aumina respectivey. he ampitude of the moving oad is P = k. A shear correction factor ψ=5/6 as used in Ref. [] is adopted herewith. hree types of motions namey uniform (constant speed) acceeration and deceeration are considered. In the acceeration the speed of the oad at the eft end of the beam is assumed to be zero and it graduay increases to a speed v at the time the oad exits the beam. On the contrary in the deceeration motion the speed of the oad at the eft end of the beam is assumed to be zero and it then graduay increases to v at the time of exiting the beam. With this assumption Eq. () gives a tota time Δt necessary for the oad P(t) to traverse across the beam is L /v for the uniform motion and L /v for the case of acceeration and deceeration motions. A tota 8 time steps are used in the ewmark method in a the computations reported beow. 4. Formuation verification he fundamenta frequency of a singe span FGM beam composed of auminum (A) and aumina (A O 3 ) previousy studied by an anaytica method in Ref. [5] and by a numerica method in Ref. [] is computed to verify the accuracy of the formuated eement. he materia properties of the constituent materias are given in Refs. [5 ]. abe. Frequency parameter μ of one span beam. Index n Source L/h= L/h=3 L/h=. Present Ref. [5] Ref. [] Present Ref. [5] Ref. [] abe ists the fundamenta frequency parameter μ for the beam where the parameter obtained by an anaytica method in Ref. [5] and by a numerica method in Ref. [] is aso given. he abe shows a good agreement between the resut of the present work with that of Refs. [5 ]. he resut isted in abe has been obtained by using tweve eements the minimum number of eements for convergence. In this regard tweve eements are used to discrete a span in the 388

5 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 computations reported beow. It shoud be noted that the frequency parameter in abe is normaized according to Ref. [5] as foows h/ ρ( zdz ) μ = ωl = ωl h E( z) dz h/ h/ ha h/ I (9) where L is the ength of span and ω is the fundamenta frequency of the beam. abe. Maximum defection and corresponding speed of singe span FGM beam under a constant speed moving oad. Index n..5 Pure aumina Pure stee Present work Ref. [] ext the maximum defection at the mid-span caed the defection factor beow and the corresponding speed of a singe span FGM beam composed of stee and aumina subjected to a constant speed moving oad are computed. he probem has been studied by Şimşek and Kocatürk in Ref. [] for a beam with L= m b=.4 m h=.9 m and a moving oad with ampitude P = k. abe ists the computed vaues of the defection factor and the corresponding speed of the moving oad where the corresponding data of Ref. [] are aso given. In the abe the defection factor is defined as =max(w(l/t)/w s ) with w s =PL 3 /48E s I is the static defection of the homogeneous stee beam under a static oad P acting at the mid-span of the beam. he exceent agreement between the numerica resut of the present with that of Ref. [] is noted from the abe. Since no data on muti-span FGM beams are avaiabe in the iterature a muti-span homogeneous beam with the materia and geometric data of Ref. [3] has been used to vaidate the derived formuation. Both the natura frequency and dynamic response of the beam under a constant moving oad computed by using tweve formuated eements for each span are in good agreement with that of Ref. [3]. In order to keep the paper in the required ength the comparison on the muti-span homogeneous beam mentioned above is not shown herewith. 4. Dynamic defection he normaized defections at the midpoints of the first and s for the beam under uniform motion are depicted in Fig. for various vaues of the index n and for v=3 m/s. he figure ceary shows the effect of the index n on the defections where the maximum defection is arge for a beam associated with a higher index n. his due to the fact that as seen from Eq. () the beam with higher index n contains more stee and thus it is softer. he defection for the third and s of the beam is the same as that of the second and s respectivey due to the symmetry when the beam under uniform motion. In Figs. 3 and 4 the defections at the Defection w(l/)/w s Defection w(l/)/w s Defection w(l/)/w s Defection w(l/)/w s Position of oad s/l Position of oad s/l n=.3 n= n=5 Figure. ormaized defection at midpoint of the first two spans of beam under uniform motion (v=3m/s). Defection w(l/)/w s Defection w(l/)/w s Position of oad s/l Position of oad s/l uniform acceerated deceerated Figure 3. ormaized defection at midpoint of the first two spans of beam under different motions (n=3 v=3m/s) uniform acceerated deceerated Position of oad s/l Position of oad s/l Figure 4. ormaized defection at midpoint of the ast two spans of beam under different motions (n=5 v=3m/s). 3883

6 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY n=.3 n= n= Figure 5. Moving oad speed versus defection parameter of beam under uniform motion uniform acceerated deceerated Figure 6. Moving oad speed versus defection parameter of beam under different motions (n=.5). midpoints of the spans are shown for the beam under different motions and for n=5 v=3 m/s. As seen from the figures whie the time at which the maximum defection occurred is consideraby atered by the motion type the maximum defection is hardy changed by the type of motions. In addition the beam vibrates much more under the deceerated motion than when it is subjected to the uniform and acceerated motions. 4.3 Dynamic defection factor In Fig. 5 the dynamic defection factor of the beam under uniform motion is potted again the moving speed for various vaues of the index n. he corresponding curves for the beam subjected to different motion types are shown in Fig. 6. As seen from the figures dependence of the defection factor on the moving speed is very different between the spans and the motion type. Whie the dependence of the defection factor on the moving speed of the is simiar to that of the singe span beam for the case of uniform and deceerated motions that is quite different for the remaining spans and for the beam under acceerated motion. he effect of the index n on the defection factor is ceary seen from Fig. 5 where the higher index n is the arger defection factor is regardess of the moving speed n=.3.5 n= n= Figure 7. ormaized axia stress distribution at the mid spans for the beam under acceerated motion (v=3m/s) n=.3.5 n= n= Figure 8. ormaized axia stress distribution at the mid spans for the beam under deceerated motion (v=3m/s). 4.4 Axia stresses In Fig. 7 and 8 the distribution of the normaized stress at the midpoints of the spans through the beam thickness is shown for the beam under acceerated and deceerated motions. In the figures σ s is the static axia stress of a singe simpy supported beam with ength of L acted by a oad P at the mid-span that is σ s =P hl/8i. he axia stresses shown in the figures have been computed at the time when the oad just arrives to the midpoints of the spans. he stress distribution as seen from the figures is strongy affected by the motion type regardess of the index n. When the beam subjected to the acceerated motion the difference in the stress distribution between the spans as seen from Fig. 7 is hardy recognized and the ampitude of the maximum and 3884

7 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 minimum stress are amost the same for a the spans. he situation is different when the beam subjected to the deceerated motion where the ampitude of stress at the midpoint of the is sighty greater than that of the second and s. It shoud be noted that different from homogeneous beams the axia stress of the FGM beam is not zero at the mid-pane but at the neutra surface. hus the position where the axia stress vanished is sighty shifted upwards from the mid-pane. he effect of the materia distribution as defined through the index n on the axia stress can be seen from the figures where the maximum of both the compressive and tensie stresses is sighty increased for the beam associated with a higher index n. In Fig. 9 the distribution of the axia stress at the midpoints of the first two spans through the thickness of the beam under uniform speed motion is shown for various vaues of the moving speed and for n=.5. he axia stress as seen from the figure is ceary affected by the moving speed and the maximum stress is increased when raising the moving speed. Comparing to the the change in the axia stress caused by increasing the speed of the moving oad of the is much more pronounced. Due to the symmetry the stress distribution through the beam thickness of the third and fourth spans is not shown herewith. abiity of the formuated eement in modeing the shear deformation effect of the FGM beams. Defection w(l/)/w s Defection w(l/)/w s L/h=5 L/h= L/h= Position of oad s/l Figure. ormaized defection at midpoint of the first two spans of beam with various vaues of aspect ratio under acceerated motions (n=.5 v=3m/s) Defection w(l/)/w s.6.4. L/h=5 L/h= L/h= v= m/s v= m/s v=5 m/s Figure 9. ormaized axia stress distribution at the midpoints of two s for the beam under uniform motion (n=.5). 4.5 Effect of aspect ratio he normaized defections at the midpoints of the first two spans of the beam subjected to the acceerated and deceerated motions are respectivey depicted in Figs. and for various vaues of the aspect ratio L/h and for n=.5 v=3 m/s. he effect of the aspect ratio that is the shear deformation on the dynamic defections of the beam is noticeaby seen from the figures regardess of the motion type. At the given vaues of the moving speed and the materia parameter the maximum defection sighty increases for the beam having a ower aspect ratio. he figures show the importance of the shear deformation on the dynamic response of the beam and the effect of the shear deformation must be taken into account when study the dynamic behaviour of muti-span beams formed from spans having ow aspect ratio. he numerica resut obtained in this Sub-section shows the Defection w(l/)/w s Position of oad s/l Figure. ormaized defection at midpoint of the first two spans of beam with various vaues of aspect ratio under deceerated motions (n=.5 v=3m/s). 5 COCLUSIOS A finite eement formuation based on the first-order shear deformabe beam theory for dynamic anaysis of muti-span FGM beams subjected to a variabe speed moving oad has been presented. he eement formuation was derived by empoying the exact soution of static part of the governing differentia equations a FGM imoshenko beam segment to interpoate the dispacements and rotation. he effect of the shift in the neutra axis position has aso been taken into account in the derivation of the finite eement formuation. Using the derive formuation the dynamic response of the beams has been computed with the aid of the impicit ewmark method. A parametric study has been carried out for 3885

8 Proceedings of the 9th Internationa Conference on Structura Dynamics EURODY 4 a four-span beam formed from stee and aumina subjected to uniform speed acceerated and deceerated motions. he effect of the materia inhomogeneity the moving speed the motion type as we as the aspect ratio on the dynamic characteristics incuding the dynamic defection defection factor and the axia stress distribution has been examined and highighted in detai. he numerica resuts obtained in the present paper have shown that the proposed formuation is capabe to give accurate dynamic characteristics of the FGM beams by using a sma number of the formuated eements. It has been demonstrated numericay that as in case of homogeneous beams the shear deformation pays an important roe in the dynamic characteristics of the FGM beam. It has aso been shown the good abiity in modeing the shear deformation effect of the formuation derived in the present work. ACKOWLEDGMES he financia support from Vietnam afosted to the second author is gratefuy acknowedged. REFERECES [] G.. Praveen and J.. Reddy oninear transient thermoeastic anaysis of functionay graded ceramic-meta pates Int. J. Soids Struct. 35: [] A. Chakraborty S. Gopaakrishnan and J.R. Reddy A new beam finite eement for the anaysis of functionay graded materias Int. J. Mech. Sci. 45 () pp [3] R. Kadoi K. Akhtar and. Ganesan Static anaysis of funtionay graded beams using higher order shear deformation beam theory App. Math. Mode. 3 () pp [4] X.F. Li A unified approach for anayzing static and dynamic bahviours of functionay graded imoshenko and Euer-Bernoui beams J. Sound Vib. 38: [5] S.A. Sina H.M. avazi and H. Haddadpour An anaytica method for free vibration anaysis of functionay graded beams Mater. Des. 3 (3) pp [6] D.-G Zhang and Y.-H. Zhou A theoretica anaysis of FGM thin pates based on physica surface Comp. Mater. Sci. 44 (6) pp [7] Y.A. Kang and X.F. Li Bending of functionay graded cantiever beam with power-aw noninearity subjected to an end force Int. J. on-inear Mech. 44: [8] Y.A. Kang and X.F. Li Large defection of a non-inear cantiever functionay graded beam J. Reinf. Past. Comp. 9 () pp [9] A.E. Ashorbagy M.A. Etaher and F.F. Mahmoud Free vibration chatacteristics of a functionay graded beam by finite eement method App. Math. Mode. 35 () pp [] A. Shahba R. Attarnejad M.. Marvi and S. Hajiar Free vibration and stabiity anaysis of axiay functionay graded tapered imoshenko beams with cassica and non-cassica boundary conditions Compos. Pt B Eng. 4 (4) pp [] M. Şimşek and. Kocatürk Free and forced vibration of a functionay graded beam subjected to a concentrated moving harmonic oad Compos. Struct. 9 (4) pp [] M. Şimşek Vibration anaysis of a functionay graded beam under a moving mass by using different beam theories Compos. Struct. 9 (4) pp [3] M.A. Etaher A.E. Ashorbagy and F.F. Mahmoud Determination of neutra axis position and its effect on natura frequencies of functionay graded maco/nanobeams Compos. Struct. 99 pp [4] D.-J. Zhang oninear bending anaysis of FGM beams based on physica neutra surface and high order shear deformation theory Compos. Struct. pp [5] D.K. guyen Buntara S.Gan and.h. Le Dynamic response of nonuniform functionay graded beams subjected to a variabe speed moving oad. J. Comput. Mate. Sci. ech. JSME. 7 () pp [6] D.K. guyen Large dispacement response of tapered cantiever beams made of axiay functionay graded materia Compos. Pt B Eng. 55 pp [7] D.K. guyen and Buntara S. Gan Large defections of tapered functionay graded beams subjected to end forces App. Math. Mode. In press. [8] L. Fryba Vibration of soids and structures under moving oads homas eford Prague Czech Repubic third edition 97. [9] F.V. Fiho Finite eement anaysis of structures under moving oads Shock Vib. Digest pp [] J. Hino. Yoshimura K. Konihi and. Ananthanarayana A finite eement method prediction of the vibration of a bridge subjected to a moving vehice oad J. Sound Vib. 96 () pp [] W.H. Lin and M.W. rethewey Finite eement anaysis of eastic beams subjected to moving dynamic oads J Sound Vib. 36 () pp [] M. Osson On the fundamenta moving oad probem J. Sound Vib. 45 () pp [3] J.R. Rieker Y.H. Lin and M.W. rethewey Discretization considerations on moving oad finite eement beam modes Finite Eem. Ana. Des. (3) pp [4] D. hambiranam and Y. Zhuge Dynamic anaysis of beams on eastic foundation subjected to moving oads. J. Sound Vib. 98 () pp [5] D.K. guyen Dynamic response of prestressed imoshenko beams resting on two-parameter foundation to a moving oad echnische Mechanik 8 (3-4) pp [6] F. Garvan he Mape book Chapman & Ha/CRC Forida USA. [7] J.B. Kosmatka An improve two-node finite eement for stabiity and natura frequencies of axia-oaded imoshenko beams. Comput. Struct. 57 () pp [8] R.D. Cook D.S. Makus and M.E. Pesha Concepts and appications of finite eement anaysis John Wiey and Sons ew York USA third edition 989. [9] M. Géradin and R. Rixen Mechanica vibrations. heory and appication to structura dynamics John Wiey and Sons Chichester UK second edition 997. [3] M. Ichikawa Y. Miyakawa and A. Matsuda Vibration anaysis of the continuous beam subjected to a moving mass J. Sound Vib. 3 (3) pp