Copyrigt 15 American-Eurasian Network for Scientific Information publiser JOURNA OF APPIED SCIENCES RESEARCH ISSN: 1819-5X EISSN: 181-157X JOURNA ome page: ttp://www.aensiweb.com/jasr 15 Special; 11(17): pages 1-7. Publised Online 3 August 15. Researc Article Free Vibration Analysis of Functionally Gradedeuler-Bernoulli and Timosenko Beams Using evy-type Solution 1 uan C. Trin 1 Adelaja I. Osofero 1 Tuc P. Vo Kien T. Nguyen 1 Faculty of Engineering and Environment Nortumbria University Newcastle upon Tyne NE1 8ST UK. Faculty of Civil Engineering and Applied Mecanics University of Tecnical Education Ho Ci Min City 1 Vo Van Ngan Street Tu Duc District Ho Ci Min City Vietnam. Received: 3 June 15; Accepted: 5 July 15 15 AENSI PUBISHER All rigts reserved ABSTRACT In tis paper fundamental frequency of Functionally Graded (FG) beams wit various boundary conditions is presented based on Classical Beam Teory (CBT) and First-order Beam Teory (FOBT). Te material properties tat vary across te tickness are determined by te power law. Governing equations ofmotion and boundary conditions are derived from te Hamilton s principle. evy-type solution is applied to analyse te effect of span-to-tickness ratio power-law index and boundary conditions on te vibration beaviour of FG beams. Present results sow tat natural frequency decreases wit anincrease in power-law index and a decreasein span-to-tickness ratio. Tis work also corroborates te suggestiontatte sear effect sould be considered in studying natural vibration of FG moderate tick beams especially for Clamped-Clamped or Clamped-SimplySupport boundary conditions. Keywords:Functionally Graded (FG) beam;free vibration;evy-type solution; Arbitrary boundary conditions. INTRODUCTION Functionally Graded Materials (FGMs)are a class of composite materials tat are microscopically eterogeneous and in wic te mecanical properties vary gradually and continuously from one layer to te oter. Suc materials are created from te exploitation of basic material elements into various organic and inorganic compounds to produce advanced polymers and elastomers alloys glasses and ceramics. Due to te gradual transformation in te properties of FGMs delamination wic occurs in conventional composite structures is eliminated. In addition te material properties can be customized to meet specific demands in engineering design. Aircraft aerospace structures and electronic facilities as well as biomedical installations are goodexamples of applications of FG material.atest researc into FGM involve te applications of tis material simplified approaces omogenizing to an equivalent isotropic material and te development of accurate teories and tecniques to analyse FG structures beaviours. As far as te teoretical models of FGMs and analysis tecniques are concerned literature can be foundon te development of sear deformation teories and metodologies.tese teories can be classified into tree main categories: te Classical Beam Teory (CBT) te First-order Beam Teory (FOBT) and te Higer-order BeamTeory (HOBT). Te Euler-Bernoulli teory known as CBT wic neglects te transverse sear deformation effects provides acceptable results for tin beams. Tis teory was applied to study te effects of different material distribution slenderness ratios and boundary conditions on te dynamic caracteristics of FG beam by Alsorbagy et al.[1]. In teirwork te principle of virtual work was applied to establis governing equations of motion and finite element model was employed. Sarkar and Ganguli[] Corresponding Autor: Adelaja I. Osofero Nortumbria University Faculty of Engineering and Environment NE1 8ST. Newcastle upon Tyne. UK. Tel: +-191 7 3981 E-mail: adelaja.osofero@nortumbria.ac.uk
z/ Adelaja I. Osofero et al 15 /Journal Of Applied Sciences Researc 11(17) Special Pages: 1-7 developed a close-form solutions of free vibration for non-uniform Euler-Bernoulli beams wit a free-free end condition. iu et al. [3] investigated natural frequencies of exponential FG beams wit single delamination analytically. Jin and Wang [] analysed te free vibration of FG beams under nine different boundary conditionsbased on CBT and differential quadrature rule. It is wort noting tat solutions based on te CBT underestimate deflection and overestimate natural frequencies of moderately tick beams. To overcome te limitation of te CBT many sear deformation teories ave been proposed. One of tese teories is Timosenko beam teory wic is also called FOBT. Tis teoryaccounts for te transverse sear deformation effects by te way of linear variation of in-plane displacements troug te tickness. By using tis teory Sari [5]studied nonrotating androtating Timosenko beams wit damaged boundaries based on Cebysev collocation metod. Aydogdu[] presented Navier solution of natural frequencies and some mode sapes for FG beams. Pradan and Cakraverty[7] applied Rayleig-Ritz metod using te CBT and FOBT to analyse natural vibration of FG beams. Simsek[8] applied agrange approac to analyse fundamental frequency analysis of FG beams wit several z y boundary conditions using classical first and various iger-order sear deformation beam teories. Altoug several studies focused on vibration analysis of FG Euler-Bernoulli and Timosenko beams can be found few studies are based on analytical approaces suc as evy-type solution. In te present work evy-type solution is applied to examinefree vibration beaviour of FGEuler-Bernoulli and Timosenko beams wit different boundary conditions.formulations of governing equations are establised using Hamilton s principle.te effects of boundary conditions powerlaw index and te span-to-tickness ratio on te natural frequency of FG beams are investigated.. Teoretical formulation: Consider a FG beam wit te cross-section and span beingb and respectively as sown in Fig.1. It is assumed tat Young s modulus E(z)and mass density (z) varies gradually across te tickness following te power-law form stated by [7] and given in Eq. (1) P z = P c P z + 1 p m + Pm (1) were subscripts c and m stand for ceramic and metal P z / x / Fig. 1: Geometry and coordinate of a FG beam. signifies eiter E(z) or (z) and p is te power-law index determining te volume of ceramic and metal troug te tickness. It can be identified from Eq. (1) tat te material continuously canges from pure metal at z = / to pure ceramic atz = / b Fig. illustrates tevariation of te Young s modulus E(z) troug te beam tickness for different values of te power-law index p. It is seen from te figure tat te beam becomes fully ceramic atp = wile it comes closer to fully metalasp increases..5..3..1 -.1 p=1 p=5 p=5 p= p=1 p=.5 p=. p=.1 p= -. -.3 -. -.5 5 115 15 15 5 315 35 Young's modulus E(z) Fig. : Variation of Young's modulus E(z) troug te tickness.
3 Adelaja I. Osofero et al 15 /Journal Of Applied Sciences Researc 11(17) Special Pages: 1-7.1. Kinematics: Assume tat te deformation of FG beam is only in x z plane and let u x z t and w x z t be te axial and transverse displacement components at an arbitrary point x z.tese components can be expressed as stated in[9] and given in Eq. () w xzt u x z t = U x t + z ψ + ψ x 1 φ x t w x z t = W x t () wereu x t and W x t represent te displacement components of a point on te beam s neutral axiswile φ x t is te rotation angle of te crosssection about y-axis. ψ z ψ 1 z sape functionsaccounting for te transverse sear stress and strain over te ticknesscan be used to differentiate various sear deformation teories applied to studying te beaviours of beams. For te CBT and FOBTψ = 1 ψ 1 = and ψ = ψ 1 = 1 are employed respectively. Te strain field is tus given by: ε x = U x + z ψ 1 φ x + ψ φ x γ xz = ψ 1 φ + 1 + ψ w x. (3).. Variational formulation: To derive te governing equations and boundary conditions for te displacement field in Eq. () Hamilton s principle will be used: t δπ δk = () t 1 wereδπ and δkdenote te virtual variation of te strain energy and kinetic energy. Te virtual variation of te strain energy is: δπ = T b σ x δε x + σ xz δγ xz dzdydx = N x δu + M x ψ 1 δφ + δw + Qxzψ1δφ+1+ψδW dxdt. (5) weren x M x and Q xz are te stress resultants and can be described by: N x = M x = σ x bdz σ x bzdz Q xz = σ xz bdz. () Te virtual variation of te kinetic energy can be determined by: δk = T T u δu + w δw ρ z dxdt = U δu + WδW I + ψ W + ψ1φxδu+ ψ1uδφx+ψuδw I1+ψψ1W +ψ1φxδφx+ ψw +ψψ1φxδw Idxdt (7) were / / I I 1 I = 1 z z ρ z dz. (8).3. Constitutive equations: Te constitutive relation is given as: σ x = Eε x σ xz = E γ 1+υ xz. (9) By substitutingeq. (3) into Eq. (9) and te subsequent results into Eq. () te stress resultants are obtained as: N x M x = Q xz were A B B D A s A B D = E 1 z z A S = A A E 1+υ U ψ 1 φ + ψ W (1) ψ 1 φ + 1 + ψ W da da. (11).. Governing equations of motion: By substituting Eqs. (5) and (7) into Eq. () and integrating te equation by part collecting te coefficients of δu δφ and δw te equations of motion of te beam wit te displacement field in Eq. () are obtained as: N x = I U + I 1 ψ 1 φ + ψ W ψ 1 M x ψ 1 Q xz = ψ 1 I 1 U + ψ 1 I φ + ψ ψ 1 W ψ M x 1 + ψ Q xz = I W + I 1 ψ U +I ψ ψ 1 φ + ψ W. (1) Te natural and essential boundary conditions are specified as in Table 1. Table 1: Boundary conditions (BC). Specify Natural BC N x ψ 1 M x ψ M x ψ M x + 1 + ψ Q xz Essential BC U φ W W Te evy-type approac is used to study te natural frequency of FG beam wit various boundary conditions. Te displacement components can be expressed for bot CBT and FOBT as: U x t φ x t W x t = U x φ x W x wereω is te eigen-frequency. e iωt (13) By substituting Eq. (13) into Eq. (1) a system of ordinary differential equations along te x-axis is obtained for CBT and FOBT. For te CBT: U = e 1 U + e W + e 3 W W iv = e U + e 5 W + e W (1) Were coefficients e i are: e 1 = I ω e A = I 1ω e A 3 = B A
Adelaja I. Osofero et al 15 /Journal Of Applied Sciences Researc 11(17) Special Pages: 1-7 e = Be 1+I 1 ω Be 3 D e 5 = I ω Be 3 D e = I ω Be Be 3 D. For te FOBT: U = c 1 U + c φ + c 3 W φ = c U + c 5 φ + c W W = c 7 φ + c 8 W (15) were te coefficients c i are: c 1 = e a e 3 e e 3a C c = e a e e e a C c 3 = e e 5a C c = e 1e 3a e 1a e 3 c C 5 = e 1e a e 1a e c C = e 1e 5a C c 7 = A s = 1 c A 8 = I ω s A s e 1 = A e = B e 3 = I ω e = I 1 ω e 1a = e e a = D e 3a = e e a = I ω + A s e 5a = A s C = e 1 e a e 1a e. By applying te concept of te state-space approac te systems of Eqs. (1) and (15) can be converted into a matrix form as: Z x = TZ x (1) were Z x = U U W W W W for CBT Z x = U U φ φ W W forfobt and [T] is defined for CBT and FOBT as: CBT: 1 e 1 e e 3 1 1 1 e e 5 e FOBT: 1 c 1 c c 3 1. c c 5 c 1 c 7 c 8 A formal solution of Eq. (1) is given by: Z x = e Tx K (17) were K is a constant column vector determined from te boundary conditions at x = ±/; and e Tx is te general matrix solution of Eq. (1) wic is given as: e Tx eλ 1x = E E 1 (18) e λ x wereλ i i = 1 and E are eigenvalues and corresponding matrix of eigenvectors respectively associated wit te matrix T. Te boundary conditions can be expressed in terms of unknown functions Z x using Eq. (13) as: CBT: Clamped (C): U = W = W = Support (S): U = W = M x = Free (F): N x = M x = Q xz = (19) FOBT: Clamped (C): U = φ = W = Support (S): U = W = M x = Free (F): N x = M x = Q xz = () Substituting Eq. (17) into Eqs. (19) and () a omogene-oussystem of equations is obtained as: G ij K j = i j = 1 were eλ 1x G x = E E 1. e λ x By setting te determinant of G ij to zero te natural frequencyω can be determined. It is noted tat a trial and error procedure need to be used to obtain te natural frequency values due to te attendant of unknown ω in matrix T. Fig. 3 summarises te iteration procedure wic as been used in te present study to calculate te natural frequency ω.. Numerical results and discussion: In tis section numerical examples are carried out to demonstrate te accuracy and applicability of evy-type solution in investigating natural frequency of FG beams wit various boundary conditions. A FG beam composed of ceramic (Al O 3 ) and metal (Al) wose properties are E c =38 GPa υ c =.3 ρ c =39 kg/m 3 and E m =7 GPa υ m =.3 ρ m =7 kg/m 3 respectively is examined. Te power-law index p varies from to 1 wile te span-to-tickness ratio varies from.5 to. Four boundary conditions including Clamped Clamped (C-C); Clamped Simply Support (C-S); Simply Support Simply Support (S-S) and Clamped Free (C-F) are considered. Bot CBT and FOBT are applied to validate te applicability of evy-type approac in te analysis of te vibration beaviours of FG beams. It is wort noting tat for FOBT te sear correction factor is taken to be 5/. For simplicity te non-dimensional natural frequency is defined asω = ω ρ m. E m For verification purpose Tables -5 present te natural frequencies for various power-law index values and boundary conditions wit /=5 and /=using CBT and FOBT. It is found tat te present results from evy-type solution using te CBT are in good agreement wit tose attained from agrange multiplier metod in [8] and te differential quadrature rule in []. Te results using te FOBT are also in line wit tose obtained from finite element model [1] and in [8]. Fig. presents te variation of te fundamental frequency wit respect to te power-law index in bot tin and moderately tick FG beams using CBT and FOBT. It can be seen from te figure tat te igest and lowest values of natural frequencies are identified by C-C and C-F cases. It is also sown on te figure tat te fundamental frequency inversely relates to te values of power-law index. Tis beaviour is somewat expected as te iger values of p cause an increase in te volumes of metal wic leads to lower Young s modulus. Te results from Fig. also bring to te conclusion tat te CBT
5 Adelaja I. Osofero et al 15 /Journal Of Applied Sciences Researc 11(17) Special Pages: 1-7 results in a iger value of natural frequencies compared to te FOBT and tat tis difference is more significant for arder boundaries and ticker beams i.e. C-C and C-S for / = 5 and insignificant for S-S and C-F in tin beams (/ = ). Fig. 5 sows te relationsip between nondimensional frequency and span-to-tickness ratio wit respect to material parameters for C-C and S-S boundary conditions using CBT and FOBT respectively. It can be seen from tis figure tat for various boundary conditions te results using CBT seem to be constant wit various slenderness ratios(/) greater tan 7.5.Wile te results using FOBT exibit a significant difference on te span-totickness ratio of up to. Tis corroborates te suggestion tat FOBT gives more reliable results in te study of vibration beaviour of FG beams and tat te sear effect sould be accounted to te moderate tick beams wit slenderness ratios less tan. Initial ω ini Matrix T Increase ω Eigenvalues λ i and eigenvectors E of T Matrix G Increase ω No Te determinants of [G] canges sign? Yes Error = ω n+1 ω n Error Tolerance No Yes Fig. 3: Process of state-space approac. End Table : Non-dimensional fundamental frequency of FG beams wit various boundary conditions and p index (/ = CBT). BC Reference..5 1 5 1 C-C Simsek[8] 1.1 11.5537 1.5713 9.555 8.718 8.3 8.55 Jin and 1.13 11.558 1.573 9.555 8.718 8.3 8.55 Present 1.13 11.551 1.57 9.555 8.7189 8.39 8.559 C-S Jin and 8.5558 7.995 7.319.77.131 5.818 5.9 Present 8.5559 7.999 7.33.8.1319 5.8 5.7 S-S Simsek[8] 5.777 5.98..13 3.87 3.8 3.557 Jin and 5.777 5.97.1.13 3.87 3.8 3.557 Present 5.778 5.971.1.18 3.875 3.31 3.558 C-F Simsek[8] 1.955 1.8171 1.7 1.59 1.371 1.357 1.71 Jin and 1.955 1.817 1.5 1.59 1.371 1.357 1.71 Present 1.9538 1.817 1.3 1.535 1.3718 1.3 1.79
Non-dimensional fundamental frequency Non-dimensional fundamental frequency Adelaja I. Osofero et al 15 /Journal Of Applied Sciences Researc 11(17) Special Pages: 1-7 Table 3: Non-dimensional fundamental frequency of FG beams wit various boundary conditions and p index(/ = 5 CBT). BC Reference..5 1 5 1 C-C Simsek[8] 1.18 11.3398 1.3718 9.3 8.577 8.19 7.8797 Present 1.18 11.339 1.378 9.3 8.578 8.19 7.8797 C-S Present 8. 7.833 7.189.539 5.9983 5.898 5.883 S-S Simsek[8] 5.3953 5.19.593.18 3.7793 3.599 3.91 Present 5.3953 5.7.593.18 3.7793 3.599 3.91 C-F Simsek[8] 1.9385 1.8 1.5 1.91 1.3599 1.9 1.55 Present 1.9385 1.837 1.5 1.91 1.3599 1.9 1.55 Table : Non-dimensional fundamental frequency of FG beams wit various boundary conditions and p index(l/ = FOBT). BC Reference..5 1 5 1 C-C Simsek[8] 1.35 11.385 1.3 9.31 8. 8.199 7.918 Vo et al. [1] 1. 11.3795 1.38 9.311 8.7 8.198 7.9115 Present 1. 11.379 1.8 9.9 8. 8.175 7.913 C-S Present 8.81 7.895 7.85.5373 5.9 5.98 5.91 S-S Simsek[8] 5.3 5.87.51.51 3.838 3.59 3.51 Vo et al. [1] 5.3 5.81.5.39 3.839 3.9 3.55 Present 5.3 5.81.53.37 3.837 3.88 3.53 C-F Simsek[8] 1.99 1.81 1. 1.51 1.397 1.338 1.5 Vo et al. [1] 1.89 1.78 1.19 1.8 1.333 1. 1.37 Present 1.89 1.78 1.19 1.8 1.3335 1. 1.37 Table 5: Non-dimensional fundamental frequency of FG beams wit various boundary conditions and p index(/ = 5 FOBT). BC Reference..5 1 5 1 C-C Simsek[8] 1.3 9.17 8.75 7.953 7.113.7.3 Vo et al.[1] 9.998 9.383 8.717 7.915 7.191.7.311 Present 9.9975 9.385 8.75 7.8998 7.188.8.318 C-S Present 7.5.987.39 5.8 5.315.935.757 S-S Simsek[8] 5.155.8.83 3.99 3.3 3.31 3.313 Vo et al. [1] 5.15.833.399 3.9711 3.5 3.5 3.93 Present 5.155.83.3989 3.979 3.7 3.3 3.91 C-F Simsek[8] 1.898 1.755 1.17 1.3 1.3338 1.5 1. Vo et al. [1] 1.89 1.78 1.19 1.8 1.333 1. 1.37 Present 1.89 1.78 1.19 1.8 1.3335 1. 1.37 5. Conclusions: Analytical solution based on evy-type approac is applied to study te natural vibration of FG beams. Formulations for bot CBT and FOBT are establised and numerical results are obtained to study te effect of boundary condition slenderness ratio and material property on te free vibration of FG beam. Te efficiency of te present resultsigligts te potential of compu-tational feasibility for analytical approac to understand-ding FGM s buckling static and dynamic beaviours. 1 1 1 1 8 C-C(CBT) C-S(CBT) S-S(CBT) C-F(CBT) C-C(FOBT) C-S(FOBT) S-S(FOBT) C-F(FOBT) 1 1 8 C-C(CBT) C-S(CBT) S-S(CBT) C-F(CBT) C-C(FOBT) C-S(FOBT) S-S(FOBT) C-F(FOBT) 8 1 Power law index - p (a) / = 5 (b) / = Fig. : Variation of non-dimensional fundamental frequency of FG beams wit respect to te span-to-tickness ratio / by CBTandFOBT. 8 1 Power law index - p
Non-dimensional fundamental frequency Non-dimensional fundamental frequency Trin et al. 15/Journal Of Applied Sciences Researc(X)August Pages:x-x 1 1 5 1 8 3 p= (CBT) p= (FOBT) p=1 (CBT) p=1 (FOBT) p=1 (CBT) p=1 (FOBT) 1 p= (CBT) p= (FOBT) p=1 (CBT) p=1 (FOBT) p=1 (CBT) p=1 (FOBT) 5 1 15 5 3 35 Span-to-tickness ratio (/) (a) C-C 5 1 15 5 3 35 Span-to-tickness ratio (/) (b) S-S Fig. 5: Variation of non-dimensional fundamental frequency of FG beams wit respect to te span-to-tickness ratio / for C-C and S-S boundary conditions. Acknowledgments Tere is no acknowledgment. Autors Contribution: Dr. Tuc P. Vo Dr. Adelaja I. Osofero ave developed te ideas and play an important role in outlining and reviewing te numerical results and writing.mr. uan C. Trin as developed te Matlab code and finised writing up te paper. Dr. Kien T. Nguyen as contributed to coding. Financial Disclosure: Tere is no conflict of interest. Funding/Support: Mr uan C. Trin is supported by Nortumbria University Researc Development Fund. References 1. Alsorbagy A.E. M.A. Eltaer and F.F. Mamoud 11. Free vibration caracteristics of a functionally graded beam by finite element metod. Applied Matematical Modelling 35(1): 1-5.. Sarkar K. and R. Ganguli 13. Closed-form solutions for non-uniform Euler Bernoulli free free beams. Journal of Sound and Vibration 33(3): 78-9. 3. iu Y. J. Xiao and D. Su 1. Free Vibration of Exponential Functionally Graded Beams wit Single Delamination. Procedia Engineering 75: 1-18.. Jin C. and X. Wang 15. Accurate free vibration analysis of Euler functionally graded beams by te weak form quadrature element metod. Composite Structures 15: 1-5. 5. Sari M.e.S. and E.A. Butcer 1. Free vibration analysis of non-rotating and rotating Timosenko beams wit damaged boundaries using te Cebysev collocation metod. International Journal of Mecanical Sciences (1): 1-11.. Aydogdu M. and V. Taskin 7 Free vibration analysis of functionally graded beams wit simply supported edges. Materials & Design 8(5): 151-15. 7. Pradan K.K. and S. Cakraverty 13. Free vibration of Euler and Timosenko functionally graded beams by Rayleig Ritz metod. Composites Part B: Engineering 51: 175-18. 8. Şimşek M. 1. Fundamental frequency analysis of functionally graded beams by using different iger-order beam teories. Nuclear Engineering and Design (): 97-75. 9. Kdeir A.A.R. J.N. 199. Free vibration of cross-fly laminated beams wit arbitrary boundary conditions. Int. J. Eng. Sci. 3: 1. 1. Vo T.P. et al. 1. Finite element model for vibration and buckling of functionally graded sandwic beams based on a refined sear deformation teory. Engineering Structures : 1-.