2009
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Bending Analysis of Laminated Composite Plates with Arbitrary Boundary Conditions
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2
It is well known that for laminated composite plates a Levytype solution exists only for crossply and antisymmetric angleply laminates. Numerous investigators have used the Levy method to solve the governing equations of various equivalent singlelayer plate theories. It is the intension of the present study to introduce a method for analytical solutions of laminated composite plates with arbitrary lamination and boundary conditions subjected to transverse loads. The method is based on separation of spatial variables of displacement field components. Within the displacement field of a firstorder shear deformation theory (FSDT), a laminated plate theory is developed. Two systems of coupled ordinary differential equations with constant coefficients are obtained by using the principle of minimum total potential energy. Since the procedure used is simple and straightforward it can, therefore, be adopted in developing higherorder shear deformation and layerwise laminated plate theories. The obtained equations are solved analytically using the statespace approach. The results obtained from the present method are compared with the Levytype solutions of crossply and antisymmetric angleply laminates with various admissible boundary conditions to verify the validity and accuracy of the present theory. Also for other laminations and boundary conditions that there exist no Levytype solutions the present results may be compared with those obtained from finite element method. It is seen that the present results have excellent agreements with those obtained by Levytype method.
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1
13


A.M
Naserian Nik
Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad
Department of Mechanical Engineering, Faculty
Iran


M
Tahani
Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad
Department of Mechanical Engineering, Faculty
Iran
mtahani@ferdowsi.um.ac.ir
Laminated plates
Analytical solution
Arbitrary boundary conditions
Firstorder shear deformation theory
[[1] Smith C.S., 1990, Design of Marine Structures in Composite Materials, Elsevier Applied Science, London.##[2] Haresceugh R.I. 1989, Aircraft and aerospace applications of composites, in: Concise Encyclopedia of Composite Materials, edited by A. Kelly,Pergamon Press, Oxford.##[3] Beardman P., 1989, Automotive components: fabrication, in: Concise Encyclopedia of Composite Materials, edited by A. Kelly, Pergamon Press, Oxford.##[4] Bowen D.H., 1989, Applications of composites: an overview, in: Concise Encyclopedia of Composite Materials, edited by A. Kelly,Pergamon Press, Oxford.##[5] Fujihara K., Teo K., Gopal R., Loh P.L., Ganesh V.K., Ramakrishna S., Foong K.W.C., Chew C.L., 2004, Fibrous composite materials in dentistry and orthopedics: review and applications, Composites Science and Technology 64: 775788.##[6] Bhaskar K., Kaushik B., 2004, Simple and exact series solutions for flexure of orthotropic rectangular plates with any combination of clamped and simply supported edges, Composite Structures 63: 63–68.##[7] Bose P., Reddy J.N., 1998, Analysis of composite plates using various plate theories, Part 1: Formulation and analytical solutions, Structural Engineering and Mechanics, An International Journal 6(6): 583612.##[8] Bose P., Reddy J.N., 1998, Analysis of composite plates using various plate theories, Part 2: Finite element model and numerical results, Structural Engineering and Mechanics, An International Journal 6(6): 727746.##[9] Kant T., Swaminathan K., 2002, Analytical solutions for static analysis of laminated composite and sandwich plates based on a higher order refined theory, Composite Structures 56(4): 329344.##[10] Nelson R.B., Lorch D.R., 1974, A refined theory for laminated orthotropic plates, ASME Journal of Applied Mechanics 41: 177183.##[11] Pagano N.J., 1970, Exact solutions for rectangular bidirectional composites and sandwich plates, Journal of Composite Materials 4: 2034.##[12] Qianl L.F., Batra R.C., Chen L.M., 2003, Elastostatic deformations of a thick plate by using a higherorder shear and normal deformable plate theory and two meshless local PetrovGalerkin (MLPG) methods, CMES 4(l): 161175.##[13] Ren J.G., 1986, Bending theory of laminated plate, Composites Science and Technology 27: 225248.##[14] Savithri S., Varadan T.K., 1992, Laminated plates under uniformly distributed and concentrated loads, ASME Journal of Applied Mechanics 59: 211214.##[15] Soldatos K.P., Watson P., 1997, Accurate stress analysis of laminated plates combining a twodimensional theory with the exact threedimensional solution for simply supported edges, Mathematics and Mechanics of Solids 2: 459489.##[16] Soni S.R., Pagano N.J., 1982, Elastic response of composite laminates, in: Mechanics of Composite MaterialsRecent Advances, edited by Z. Hashin, C.T. Herakovich, Pergamon Press, New York.##[17] Shu X.P. Soldatos K.P., 2000, Cylindrical bending of angleply laminates subjected to different sets of edge boundary conditions, International Journal of Solids and Structures 37: 42894307.##[18] Swaminathan K., Ragounadin D., 2004, Analytical solutions using a higherorder refined theory for the static analysis of antisymmetric angleply composite and sandwich plates, Composite Structures 64: 405417.##[19] Wenbin Yu., 2005, Mathematical construction of a Reissner–Mindlin plate theory for composite laminates, International Journal of Solids and Structures 42: 66806699.##[20] Timoshenko S.P., Krieger S.W., 1959, Theory of Plates and Shells, McGrawHill, New York, Second Edition.##[21] Bhaskar K., Kaushik B., 2004, Simple and exact series solutions for flexure of orthotropic rectangular plates with any combination of clamped and simply supported edges, Composite Structure 63: 6368.##[22] Khalili M.R., Malekzadeh K., Mittal R.K., 2005, A new approach to static and dynamic analysis of composite plates with different boundary conditions, Composite Structure 69: 149155.##[23] Green A.E., 1944, Double fourier series and boundary value problems, in: Proceedings of Cambridge Philosophical Society 40: 222228.##[24] Kabir H.R.H., Chaudhuri R.A., 1992, Boundarycontinuous fourier solution for clamped mindlin plates, ASCE Journal of Engineering Mechanics 118: 14571467.##[25] Green A.E., Hearmon R.F.S., 1945, The buckling of flat rectangular plywood plates, Philosophical Magazine 36: 659687.##[26] Chaudhuri R.A., Kabir H.R.H., 1992, Fourier analysis of clamped moderately thick arbitrarily laminated plates, AIAA Journal 30(11): 27962798.##[27] Vel S.S., Batra R.C., 1999, Analytical solutions for rectangular thick laminated plates subjected to arbitrary boundary conditions, AIAA Journal 37: 14641473.##[28] Eshelby J.D., Read W.T., Shockley W., 1953, Anisotropic elasticity with applications to dislocation theory, Acta Metallurgica 1: 251259.##[29] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, FL, Second Edition.##[30] Fung Y.C., 1965, Foundation of Solid Mechanics, PrenticeHall, Englewood Cliffs, New Jersey, First Edition.##[31] Goldberg J.L., Schwartz A.J., 1972, Systems of Ordinary Differential Equations, An Introduction, Harper and Row, New York.##]
The Buckling of NonHomogeneous Truncated Conical Shells under a Lateral Pressure and Resting on a Winkler Foundation
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In this paper, the buckling of nonhomogeneous isotropic truncated conical shells under uniform lateral pressure and resting on a Winkler foundation is investigated. The basic relations and governing equations have been obtained for nonhomogeneous truncated conical shells. The critical uniform lateral pressures of nonhomogeneous isotropic truncated conical shells with or without a Winkler foundation are obtained. Finally, carrying out some computations, effects of the variations of truncated conical shell characteristics, the nonhomogeneity and the Winkler foundation on the critical uniform lateral pressures have been studied. The results are compared with other works in open literature.
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A.H
Sofiyev
Department of Civil Engineering, Suleyman Demirel University, Isparta 32260, Turkey
Department of Civil Engineering, Suleyman
Iran
asofiyev@mmf.sdu.edu.tr


A
Valiyev
Chair of Mathematics and General Technical Subjects of Odlar Yurdu University, Baku, Azerbaijan
Chair of Mathematics and General Technical
Iran


P
Ozyigit
Department of Civil Engineering, Suleyman Demirel University, Isparta 32260, Turkey
Department of Civil Engineering, Suleyman
Iran
Buckling
Nonhomogeneous material
Truncated conical shell
Winkler foundation
Critical uniform lateral pressure
[[1] Pasternak P.L., 1954, On a New Method of Analysis of an Elastic Foundation by Means of Two Foundation Constants, Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu i Arkhitekture, Moscow, USSR (in Russian).##[2] Kerr A.D., 1964, Elastic and viscoelastic foundation models, ASME Journal of Applied Mechanics 31: 491498.##[3] Bajenov V.A., 1975, The Bending of the Cylindrical Shells in an Elastic Medium, Kiev, Visha Shkola (in Russian).##[4] Sun B., Huang Y., 1988, The exact solution for the general bending problems of conical shells on the elastic foundation, Applied Mathematics and Mechanics 9(5): 455469.##[5] Eslami M.R., Ayatollahi M.R., 1993, Modalanalysis of shell of revolution on elasticfoundation, International Journal of Pressure Vessels and Piping 56(3):351368.##[6] Paliwal D.N., Pandey R.K., Nath T., 1996, Free vibration of circular cylindrical shell on Winkler and Pasternak foundation, International Journal of Pressure Vessels and Piping 69: 7989.##[7] Ng T.Y., Lam K.Y., 2000, Free vibrations analysis of rotating circular cylindrical shells on an elastic foundation, Journal of Vibration and Acoustics 122: 8589.##[8] Tj H.G., Mikami T., Kanie S., Sato M., 2006, Free vibration characteristics of cylindrical shells partially buried in elastic foundations, Journal of Sound and Vibration 290: 785793.##[9] Lomakin V.A., 1976,The Elasticity Theory of Nonhomogeneous Materials, Nauka, Moscow (in Russian).##[10] Awrejcewicz J., Krysko V.A., Kutsemako A.N., 1999, Free vibrations of doubly curved inplane nonhomogeneous shells, Journal of Sound and Vibration 225(4): 701722.##[11] Shen H.S., 2003, Postbuckling analysis of pressure loaded functionally graded cylindrical shells in thermal environments, Engineering Structures 25: 487497.##[12] Goldfeld Y., 2007, Elastic buckling and imperfection sensitivity of generally stiffened conical shells, AIAA Journal 45(3): 721–729.##[13] Sofiyev A.H., Omurtag M., Schnack E., 2009, The vibration and stability of orthotropic conical shells with nonhomogeneous material properties under a hydrostatic pressure, Journal of Sound and Vibration 319(35): 963983.##[14] Najafizadeh M.M., Hasani A., Khazaeinejad P., 2009, Mechanical stability of functionally graded stiffened cylindrical shells, Applied Mathematical Modelling 33(2):11511157.##[15] Tomar J., Gupta D., Kumar V., 1986, Natural frequencies of a linearly tapered nonhomogeneous isotropic elastic circular plate resting on an elastic foundation, Journal of Sound and Vibration 111: 18.##[16] Sofiyev A.H., 1987, The stability of nonhomogeneous cylindrical shells under the effect of surroundings, Soviet Scientific and Technical Research Institute (VINITI), Moscow 3(189): 19 (in Russian).##[17] Sofiyev a.H., Keskin S.N., Sofiyev A.L.H., 2004, Effects of elastic foundation on the vibration of laminated nonhomogeneous orthotropic circular cylindrical shells, Journal of Shock and Vibration 11: 89101.##[18] Sheng G.G., Wang X., 2008, Thermal vibration, buckling and dynamic stability of functionally graded cylindrical shells embedded in an elastic medium, Journal of Reinforced Plastics and Composites 27(2): 117134.##[19] Singer j., 1962, The effect of axial constraint on the instability of thin conical shells under external pressure, ASME Journal of Applied Mechanics, 212214.##[20] Agamirov V.L., 1990, Dynamic Problems of Nonlinear Shells Theory, Moscow, Nauka (in Russian).##[21] Tong L., 1996, effect of axial load on free vibration of orthotropic conical shells, Journal of Vibration and Acoustics 118: 164168.##[22] Liew K.M., Ng T.Y., Zhao X., 2005, Free vibration analysis of conical shells via the elementfree KPRitz method, Journal of Sound and Vibration 281: 627645.##]
Free Vibration of Functionally Graded Beams with Piezoelectric Layers Subjected to Axial Load
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This paper studies free vibration of simply supported functionally graded beams with piezoelectric layers subjected to axial compressive loads. The Young's modulus of beam is assumed to be graded continuously across the beam thickness. Applying the Hamilton’s principle, the governing equation is established. Resulting equation is solved using the Euler’s Equation. The effects of the constituent volume fractions, the influences of applied voltage and axial compressive loads on the vibration frequency are presented. To investigate the accuracy of the present analysis, a compression study is carried out with a known data.
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28


M
Karami Khorramabadi
Faculty of Engineering, Islamic Azad University, Khorramabad Branch
Faculty of Engineering, Islamic Azad University,
Iran
mehdi_karami2001@yahoo.com
Free vibration
Functionally graded beam
Piezoelectric layer
[[1] Bisegna P., Maceri F., 1996, An exact threedimensional solution for simply supported rectangular piezoelectric plates, ASME Journal of Applied Mechanics 63: 628638.##[2] Kapuria S., Dumir P.C., Sengupta S., 1996, Exact piezothermoelastic axisymmetric solution of a finite transversely isotropic cylindrical shell, Computers and Structures 61:10851099.##[3] Ding H.J., Chen W.Q., Guo Y.M., Yang Q.D., 1997, Free vibrations of piezoelectric cylindrical shells filled with compressible Fluid, International Journal of Solids and Structures 34: 20252034.##[4] Shul'ga N.A., 1993, Harmonic electroelastic oscillation of spherical bodies, Soviet Applied Mechanics 29: 812817.##[5] Chen W.Q., Ding H.J., 1998, Exact static analysis of a rotating piezoelectric spherical shell, Acta Mechanica Sinica 14: 257265.##[6] Chen W.Q., Ding H.J., Xu R.Q., 2001, Three dimensional free vibration analysis of a fluidfilled piezoceramic hollow sphere, Computers and Structures 79: 653663.##[7] Tanigawa Y., 1995, Some basic thermo elastic problems for nonhomogeneous structural materials, Applied Mechanics Reviews 48: 287300.##[8] Loy C.T., Lam K.Y., Reddy J.N., 1999, Vibration of functionally graded cylindrical shells, International Journal of Mechanical Sciences 41: 309324.##[9] Cheng Z.Q., Batra R.C., 2000, Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plates, Journal of Sound and Vibration 229: 879895.##[10] Chen W.Q., Wang X., Ding H.J., 1999, Free vibration of a fluidfilled hollow sphere of a functionally graded material with spherical isotropy, Journal of the Acoustical Society of America 106: 25882594.##[11] Chen W.Q., 2000, Vibration theory of nonhomogeneous, spherically isotropic piezoelastic bodies, Journal of Sound and Vibration 229: 833860.##[12] Ootao Y., Tanigawa Y., 2000, Threedimensional transient piezothermoelasticity in functionally graded rectangular plate bonded to a piezoelectric plate, International Journal of Solids and Structures 37: 43774401.##[13] Wang B.L., Han J.C., Du S.Y., 1999, Functionally graded pennyshaped cracks under dynamic loading, Theoretical and Applied Fracture Mechanics 32: 165175.##[14] Chen W.Q., Liang J., Ding H.J., 1997, Three dimensional analysis of bending problems of thick piezoelectric composite rectangular plates, Acta Materiale Compositae Sinica 14: 108115 (in Chinese).##[15] Chen W.Q., Xu R.Q., Ding H.J., 1998, On free vibration of a piezoelectric composite rectangular plate, Journal of Sound and Vibration 218: 741748.##[16] Ding H.J., Xu R.Q., Guo F.L., 1999, Exact axisymmetric solution of laminated transversely isotropic piezoelectric circular plates (I) exact solutions for piezoelectric circular plate, Science in China (E) 42: 388395.##[17] Wang J.G., 1999, State vector solutions for nonaxisymmetric problem of multilayered half space piezoelectric medium, Science in China (A) 42: 13231331.##[18] Reddy J.N., Praveen G.N., 1998, Nonlinear transient thermoelastic analysis of functionally graded ceramicmetal plates, International Journal of Solids and Structures 35: 44674476.##[19] Bolotin V.V., 1964, The dynamic Stability of Elastic Systems, Holden Day, San Francisco.## ##]
Free Vibration Analysis of a Nonlinear Beam Using Homotopy and Modified LindstedtPoincare Methods
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2
In this paper, homotopy perturbation and modified LindstedtPoincare methods are employed for nonlinear free vibrational analysis of simply supported and doubleclamped beams subjected to axial loads. Midplane stretching effect has also been accounted in the model. Galerkin's decomposition technique is implemented to convert the dimensionless equation of the motion to nonlinear ordinary differential equation. Homotopy and modified LindstedtPoincare (HPM) are applied to find analytic expressions for nonlinear natural frequencies of the beams. Effects of design parameters such as axial load and slenderness ratio are investigated. The analytic expressions are valid for a wide range of vibration amplitudes. Comparing the semianalytic solutions with numerical results, presented in the literature, indicates good agreement. The results signify the fact that HPM is a powerful tool for analyzing dynamic and vibrational behavior of structures analytically.
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M.T
Ahmadian
Center of Excellence in Design, Robotics and Automation, School of Mechanical Engineering, Sharif University of Technology
Center of Excellence in Design, Robotics
Iran
ahmadian@mech.sharif.ir


M
Mojahedi
School of Mechanical Engineering, Sharif University of Technology
School of Mechanical Engineering, Sharif
Iran


H
Moeenfard
School of Mechanical Engineering, Sharif University of Technology
School of Mechanical Engineering, Sharif
Iran
Free vibration
Nonlinear beam
Homotopy Perturbation method
LindstedtPoincare method
Axial load
[[1] Pirbodaghi T., Ahmadian M.T., Fesanghary M., 2008, On the homotopy analysis method for nonlinear vibration of beams, Mechanics Research Communications, in press, doi:10.1016/j.mechrescom.2008.08.001.##[2] Nayfeh A.H., Mook D.T., 1979, Nonlinear Oscillations, Wiley, New York.##[3] Shames I.H., Dym C.L., 1985, Energy and Finite Element Methods in Structural Mechanics, McGrawHill, New York.##[4] Malatkar P., 2003, Nonlinear Vibrations of Cantilever Beams and Plates, Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Engineering Mechanics.##[5] Pillai S.R.R., Rao B.N., 1992, On nonlinear free vibrations of simply supported uniform beams, Journal of Sound and Vibration 159(3): 527531.##[6] Ramezani A., Alasty A., Akbari J., 2006, Effects of rotary inertia and shear deformation on nonlinear free vibration of microbeams, ASME Journal of Vibration and Acoustics 128(5): 611615.##[7] Foda M.A., 1999, Influence of shear deformation and rotary inertia on nonlinear free vibration of a beam with pinned ends, Computers and Structures 71: 663670.##[8] Liao S.J., 1995, An approximate solution technique which does not depend upon small parameters: a special example, International Journal of Nonlinear Mechanics 30: 371380.##[9] He J.H., 2000, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, International Journal of NonLinear Mechanics 35: 3743.##[10] Abbasbandy S., 2006, The application of homotopy analysis method to nonlinear equations arising in heat transfer, Physics Letters A 360: 109113.##[11] Abbasbandy S., 2007, The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation, Physics Letters A 361: 478483.##[12] Hayat T., Sajid M., 2007, On analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Physics Letters A 361: 316–322.##[13] HE J.H., 2000, A new perturbation technique which is also valid for large parameters, Journal of Sound and Vibration 229(5): 12571263.##[14] He J.H., 2003, Homotopy perturbation method: a new nonlinear analytical technique, Applied Mathematics and Computation 135: 7379.##[15] Belendez A., Hernandez A., Belendez T., Neipp C., Marquez A., 2007, Application of the homotopy perturbation method to the nonlinear Pendulum, European Journal of Physics 28: 93104.##[16] Belendez A., Belendez T., Marquez A., Neipp C., 2006, Application of He’s homotopy perturbation method to conservative truly nonlinear oscillators, Chaos, Solitons & Fractals, doi:10.1016/j.chaos.2006.09.070.## ##]
Transverse Vibration of Clamped and Simply Supported Circular Plates with an Eccentric Circular Perforation and Attached Concentrated Mass
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2
In this investigation RayleighRitz variational method has been applied to determine the least natural frequency coefficient for the title problem. Classical plate theory assumptions have been used to calculate strain energy and kinetic energy. Coordinate functions are combination of polynomials which satisfy boundary conditions at the outer boundary and trigonometric terms. In the second part of this study ABAQUS software is used to compute vibration natural frequency for some special combinations of geometrical and mechanical parameters. Then results of RayleighRitz method have been obtained for the mentioned special cases. It can be seen that the agreement between them is acceptable.
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44


S.M
Mirkhalaf Valashani
Department of Mechanical Engineering, Islamic Azad University, Arak Branch
Department of Mechanical Engineering, Islamic
Iran
mohsen_61m@yahoo.com
Vibration
Circular plate
Eccentric circular perforation
Concentrated mass
[[1] Jacquot R.G., Lindsay J.E., 1977, On the influence of Poisson’s ratio on circular plates natural frequencies, Journal of Sound and Vibration 52: 603605.##[2] Laura P.A.A., Grossi R.O., 1978, Influence of Poisson’s ratio on the lower natural frequencies of transverse vibration of a circular plate of linearly varying thickness and with an edge elastically restrained against rotation, Journal of Sound and Vibration 60(4): 587590.##[3] Gutierrez R.H., Laura P.A.A., 1977, A note on the determination of the fundamental frequency of vibration of rectangular and circular plates supporting masses distributed over a finite area, Applied Acoustics 10: 303313.##[4] Narita Y., Leissa A.W., 1980, Transverse vibration of simply supported circular plates having partial elastic constraints, Journal of Sound and Vibration 70(1): 103116.##[5] Laura P.A.A., Paloto J.C., Santos R.D., 1975, A note on the vibration and stability of a circular plate elastically restrained against rotation, Journal of Sound and Vibration 41(2): 177180.##[6] Laura P.A.A., Luisoni L.E., Lopez J.J., 1976, A note on free and forced vibrations of circular plates: the effect of support flexibility, Journal of Sound and Vibration 47(2): 287291.##[7] Irie T., Yamada G., Tanaka K., 1983, Free vibration of circular plate elastically restrained along some radial segments, Journal of Sound and Vibration 89(3): 295308.##[8] Grossi R.O., Laura P.A.A., 1987, Additional results of transverse vibrations of polar orthotropic circular plates carrying concentrated masses, Applied Acoustics 21: 225233.##[9] Laura P.A.A., Gutierrez R.H., 1991, Free vibration of solid circular plate of linearly varying thickness and attached to a winkler foundation, Journal of Sound and Vibration 144(1): 149161.##[10] Bercin A.N., 1996, Free vibration solution for clamped orthotropic plates using the Kantorovich method, Journal of Sound and Vibration 196(2): 243247.##[11] Ranjan V., Gosh M.K., 2006, Transverse vibration of thin solid and annular circular plate with attached discrete masses, Journal of Sound and Vibration 292: 9991003.##[12] Kang W., Lee N.H., Pang Sh., Chung W.Y., 2005, Approximate closed form solutions for free vibration of polar orthotropic circular plates, Applied Acoustics 66: 11621179.##[13] Park Ch., 2008, Frequency equation for the inplane vibration of a clamped circular plate, Journal of Sound and Vibration 313: 325333.##[14] Avalos D.R., Larrondo H.A., Laura P.A.A., Sonzogni V., 1998, Transverse vibrations of circular plate with a concentric square hole with free edges, Journal of Sound and Vibration 209(5): 889891.##[15] Bambill D.V., La Malfa S., Rossit C.A., Laura P.A.A., 2004, Analytical and experimental investigation on transverse vibrations of solid, circular and annular plates carrying a concentrated mass at an arbitrary position with marine applications, Ocean Engineering 31: 127138.##[16] Laura P.A.A., Masia U., Avalos D.R., 2006, Small amplitude transverse vibrations of circular plates elastically restrained against rotation with an eccentric circular perforation with a free edge, Journal of Sound and Vibration 292: 10041010.## ##]
Mechanical Behavior of an ElectrostaticallyActuated Microbeam under Mechanical Shock
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2
In this paper static and dynamic responses of a fixedfixed microbeam to electrostatic force and mechanical shock for different cases have been studied. The governing equations whose solution holds the answer to all our questions about the mechanical behavior is the nonlinear elastoelectrostatic equations. Due to the nonlinearity and complexity of the derived equations analytical solution are not generally available; therefore, the obtained differential equations have been solved by using of a step by step linearization method (SSLM) and a Galerkin based reduced order model. The pullin voltage of the structure and the effect of shock forces on the mechanical behavior of undeflected and electrostatically deflected microbeam have been investigated. The proposed models capture the other design parameters such as intrinsic residual stress from fabrication processes and the nonlinear stiffening or stretching stress due to beam deflection.
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45
57


M
Fathalilou
Sama Organization (Affiliated with Islamic Azad University), Khoy Branch
Sama Organization (Affiliated with Islamic
Iran


A
Motallebi
Department of Mechanical Engineering, Islamic Azad University, Khoy Branch
Department of Mechanical Engineering, Islamic
Iran


H
Yagubizade
Department of Mechanical Engineering, Urmia University
Department of Mechanical Engineering, Urmia
Iran


Gh
Rezazadeh
Department of Mechanical Engineering, Urmia University
Department of Mechanical Engineering, Urmia
Iran
g.rezazadeh@urmia.ac.ir


K
Shirazi
Sama Organization (Affiliated with Islamic Azad University), Khoy Branch
Sama Organization (Affiliated with Islamic
Iran


Y
Alizadeh
Sama Organization (Affiliated with Islamic Azad University), Khoy Branch
Sama Organization (Affiliated with Islamic
Iran
MEMS
Microbeam
Electrostatic actuation
Pullin voltage
Mechanical shock
[[1] Basso M., Giarre L., Dahleh M., Mezic I., 1998, Numerical analysis of complex dynamics in atomic force microscopes, in: Proceedings of the IEEE International Conference on Control Applications, Trieste, Italy, 14 September: 10261030.##[2] Fritz J., Baller M.K., Lang H.P., Rothuizen H., Vettiger P., Meyer E., Gntherodt H.J., Gerber C., Gimzewski J.K., 2001, Translating biomolecular recognition into nanomechanics, Science 288: 316318.##[3] Sidles J.A., 1991, Noninductive detection of single protonmagnetic resonance, Applied Physics Letters 58(24): 28542856.##[4] Nabian A., Rezazadeh Gh., HaddadDerafshi M., Tahmasebi A., 2008, Mechanical behavior of a circular micro plate subjected to uniform hydrostatic and nonuniform electrostatic pressure, Microsystem Technologies 14: 235240.##[5] Senturia S., 2001, Microsystem Design, Kluwer, Norwell, MA, USA.##[6] Rezazadeh Gh., Sadeghian H., Abbaspour E., 2008, A comprehensive model to study nonlinear behaviour of multilayered micro beam switches, Microsystem Technologies 14(1): 143.##[7] Sadeghian H., Rezazadeh Gh., Osterberg P.M., 2007, Application of the generalized differential quadrature method to the study of pullin phenomena of mems switches, Journal of Microelectromechanical Systems 16(6).##[8] Osterberg P.M., Senturia S.D., 1997, MTest: a test chip for MEMS material property measurement using electrostatically actuated test structures, Journal of Microelectromechanical Systems 6: 107118.##[9] AbdelRahman E.M., Younis M.I., Nayfeh A.H., 2002, Characterization of the mechanical behavior of an electrically actuated microbeam, Journal of Micromechanical Microengineering 12: 759766.##[10] Younis M.I., Jordy D., Pitarresi J.M., 2007, Computationally efficient approaches to characterize the dynamic response of microstructures under mechanical shock, Journal of Microelectromechanical Systems 16(3).##[11] Tas N., Sonnenberg T., Jansen H., Legtenberg R., Elwenspoek M., 1996, Stiction in surface micromachining, Journal of Micromechanical Microengineering 6(4): 385397.##[12] Tanner D.M., Walraven J.A., Helgesen K., Irwin L.W., Smith N.F., Masters N., 2000, MEMS reliability in shock environments, in: Proceedings of the IEEE International Reliability Physics Symposium, 129138.##[13] Meirovitch L., 2001, Fundamentals of Vibrations, McGrawHill, Boston, USA.##[14] Béliveau A., Spencer G.T., Thomas K.A., Roberson S.L., 1999, Evaluation of MEMS capacitive accelerometers, Design & Test of Computers, IEEE 16: 4856.##[15] Brown T.G., Davis B., Hepner D., Faust J., Myers C., Muller P., Harkins T., Hollis M., Miller C., Placzankis B., 2001, Strapdown microelectromechanical (MEMS) sensors for highG munition applications, IEEE Transactions on Magnetics 37: 336342.##[16] Lim B.B., Yang J.P., Chen S.X., Mou J.Q., Lu Y., 2002, Shock analysis of MEMS actuator integrated with HGA for operational and nonoperational HDD, In: Digest of the AsiaPacific Magnetic Recording Conference, WEP1801WEP1802.##[17] Wagner U., Franz J., Schweiker M., Bernhard W., MullerFiedler R., Michel B., Paul O., 2001, Mechanical reliability of MEMSstructures under shock load, Microelectronics Reliability. 41: 16571662.##[18] Fan M.S., Shaw H.C., Dynamic response assessment for the MEMS accelerometer under severe shock loads, In: National Aeronautics and Space Administration NASA, Washington, DC, TP200120997.##[19] Li G.X., Shemansky J.R., 2000, Drop test and analysis on micromachined structures, Sensors Actuators A 85: 280286.##[20] Srikar V.T., Senturia S.D., 2002, The reliability of microelectromechanical systems (MEMS) in shock environments, Journal of Microelectromechanical Systems 11: 206214.##[21] Yee J.K., Yang H.H., Judy J.W., 2003, Shock resistance of ferromagnetic micromechanical magnetometers, Sensors Actuators A 103: 242252.##[22] Millet O., Collard D., Buchaillot L., 2002, Reliability of packaged MEMS in shock environments: crack and stiction modeling, In: Design, Test, Integration and Packaging of MEMS/MOEMS, Cannes, 696703.##[23] Coster J.D., Tilmans H.C., Van Beek J.T.M., Rijks T.G.S.M., Puers R., 2004, The influence of mechanical shock on the operation of electrostatically driven RFMEMS switches, Journal of Micromechanical Microengineering 14: 549554.##[24] Mukherjee T., Fedder G.K., Ramaswamy D., White J., 2000, Emerging simulation approaches for micromachined devices, IEEE ComputerAided Design of Integrated Circuits and Systems 19: 15721589.##[25] Senturia S.D., Aluru N., White J., 1997, Simulating the behavior of MEMS devices, IEEE Computational Science & Engineering 4(1): 3043.##[26] Gupta R.K., 1997, Electrostatic pullin test structure design for insitu mechanical property measurement of microelectromechanical systems (MEMS), PhD dissertation, MIT, Cambridge, MA, 1027.##[27] Rezazadeh Gh., Tahmasebi A., Zubtsov M., 2006, Application of piezoelectric layers in electrostatic mem actuators: Controlling of pullin voltage, Microsystem Technologies 12(12): 11631170.##[28] Nayfeh A.H., Mook D.T., 1979, Nonlinear Oscillations, Wiley, New York.##[29] de Silva Clarence W., 2005, Vibration and Shock Handbook, CRC Press, Taylor & Francis Group.## ##]
Buckling Analyses of Rectangular Plates Composed of Functionally Graded Materials by the New Version of DQ Method Subjected to NonUniform Distributed InPlane Loading
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2
In this paper, the new version of differential quadrature method (DQM), for calculation of the buckling coefficient of rectangular plates is considered. At first the differential equations governing plates have been calculated. Later based on the new version of differential quadrature method, the existing derivatives in equation are converted to the amounts of function in the grid points inside the region. Having done that, the equation will be converted to an eigen value problem and the buckling coefficient is obtained. Solving this problem requires two kinds of loading: (1) unaxial halfcosine distributed compressive load and (2) uniaxial linearly varied compressive load. Having considered the answering in this case and the analysis of the effect of number of grid points on the solution of the problem, the accuracy of answering is considered, and also the effect of power law index over the buckling coefficient is investigated. Finally, if the case is an isotropic type, the results will be compared with the existing literature.
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58
72


R
Kazemi Mehrabadi
Department of Mechanical Engineering, Islamic Azad University, Arak Branch
Department of Mechanical Engineering, Islamic
Iran


V.R
Mirzaeian
Iran University of Science and Technology
Iran University of Science and Technology
Iran
mirzaeian224@yahoo.com
Buckling
Functionally Graded Materials
Rectangular plates
Differential quadrature method
Nonuniform distributed inplane loading
[[1] Brush D.O., Almroth B.O., 1975, Buckling of Bars, Plates and Shells, McGraw Hill, New York.##[2] Leissa A.W., 1992, Review of recent developments in laminated composite plate buckling analysis, Composite Material Technology 45: 17.##[3] Tauchert T.R., 1991, Thermally induced flexure, buckling and vibration of plates, Applied Mechanics Review 44(8): 347360.##[4] Biswas P., 1976, Thermal buckling of orthotropic plates, ASME Journal of Applied Mechanics 98:361363.##[5] Reddy J.N., Wang C.M., Kitipornachi S., 2001, Axisymmetric bending of functionally graded circular annular plates, European Journal of Mechanics A/Solid 20: 841855.##[6] Bellman R.E., Casti J., 1971, Differential quadrature and longterm integration, Journal of Mathematical Analysis and Applications 34: 235238.##[7] Wang X., Gu H., Liu B., 1996, On buckling analysis of beams and frame structures by differential quadrature elemnt method, ASCE Proceedings of Engineering Mechanics 1: 382385.##[8] Liu G.R., Wu T.Y., 2001, Vibration analysis of beam using the generalized differential quadrature rule and domain decomposition, Journal of Sound and Vibration 246: 461481.##[9] Bert C.W., Devarakonda K.K., 2003, Buckling of rectangular plates subjected to nonlinearly distributed inplane loading, International Journal of Solids and Structures 40:40974106.##[10] Sherbourne A.N., Pandey M.D., 1991, Differential quadrature method in the buckling analysis of beams and composite plates, Computers and Structures 40:903913.##[11] Bert C.W., Wang X., Striz A.G., 1993, Differential quadrature for static and free vibration analyses of anisotropic plates, International Journal of Solids and Structures 30(13): 17371744.##[12] Wang X., Bert C.W., 1993, A new approach in applying differential quadrature to static and free vibrational analyses of beams and plates, Journal of Sound and Vibration 162(3): 566–72.##[13] Wang X., Gu H., Liu B., On buckling analysis of beams and frame structures by the differential quadrature element method. Proceedings of Engineering Mechanics 1: 382385.##[14] Wang X., 1995, Differential quadrature for buckling analysis of laminated Plates, Computers and Structures 57(4): 715719.##[15] Wang X., Tan M., Zhou Y., 2003, Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method, ThinWalled Structures 41: 1529.##[16] Wang X., Shi X., Applications of differential quadrature method for solutions of rectangular plates subjected to nonuniformly distributed inplane loadings (unpublished manuscript).##[17] Wang X., Wang X., 2006, Differential quadrature buckling analyses of rectangular plates subjected to nonuniform distributed inplane loadings, Thinwalled Structures 44: 837843.##[18] Koizumi M., 1997, FGM activities in Japan, Composites 28(12): 14.##[19] Wang X., Liu F., Wang X., Gan L., 2005, New approaches in application of differential quadrature method to fourth order differential equations, Communications In Numerical Methods In Engineering 21: 6171.##[20] Praveen G.N., Reddy J.N., 1998, Nonlinear Transient thermoelastic analysis of functionally graded ceramicmetal plates, International Journal of Solids and Structures 35(33): 44574476.##[21] Van der Neut A., 1958, Buckling caused by thermal stresses, high temperature effects in aircraft structures. AGARDO graph 28: 215–47.##[22] Benoy M.B., 1969, An energy solution for the buckling of rectangular plates under nonuniform inplane loading, Aeronautical Journal 73: 974977.##[23] Young W.C., Budynas R.G., 2002, Roark’s Formulas for Stress & Strain, McGrawHill, New York, USA, Seventh Edition.## ##]
Comparison of Various Shell Theories for Vibrating Functionally Graded Cylindrical Shells
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2
The classical shell theory, firstorder shear deformation theory, and thirdorder shear deformation theory are employed to study the natural frequencies of functionally graded cylindrical shells. The governing equations of motion describing the vibration behavior of functionally graded cylindrical shells are derived by Hamilton’s principle. Resulting equations are solved using the Naviertype solution method for a functionally graded cylindrical shell with simply supported edges. The effects of transverse shear deformation, geometric size, and configurations of the constituent materials on the natural frequencies of the shell are investigated. Validity of present formulation was checked by comparing the numerical results with the Love’s shell theory.
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73
83


M
Javadinejad
Department of Mechanical Engineering, Islamic Azad University, Khomeinishahr Branch
Department of Mechanical Engineering, Islamic
Iran
mehdy.javady@yahoo.com
Functionally graded material
Cylindrical shell
Natural frequency
Various shell theories
[[1] Reddy J.N., Khdeir A.A., 1989, Buckling and vibration of laminated composite plates using various plate theories, AIAA Journal 27(12): 18081817.##[2] Sivadas K.R., Ganesan N., 1991, Vibration analysis of laminated conical shells with variable thickness, Journal of Sound and Vibration 148(3): 477491.##[3] Matsunaga H., 1999, Vibration and stability of thick simply supported shallow shells subjected to inplane stresses, Journal of Sound and Vibration 225(1): 4160.##[4] Loy C.T., Lam K.Y., Reddy J.N., 1999, Vibration of functionally graded cylindrical shells, International Journal of Mechanical Sciences 41: 309324.##[5] Pradhan S.C., Loy C.T., Lam K.Y., Reddy J.N., 2000, Vibration characteristics of functionally graded cylindrical shells under various boundary conditions, Applied Acoustics 61: 111129.##[6] Najafizadeh M.M., Isvandzibaei M.R., 2007, Vibration of functionally graded cylindrical shells based on higher order shear deformation plate theory with ring support, Acta Mechanica 191: 7591.##[7] Patel B.P., Gupta S.S., Loknath M.S., Kadu C.P., 2005, Free vibration analysis of functionally graded elliptical cylindrical shells using higherorder theory, Composite Structures 69(3): 259270.##[8] Haddadpour H., Mahmoudkhani S., Navazi H.M., 2007, Free vibration analysis of functionally graded cylindrical shells including thermal effects, ThinWalled Structures 45(6): 591599.##[9] Kadoli R., Ganesan N., 2006, Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperaturespecified boundary condition, Journal of Sound and Vibration 289(3): 450480.##[10] Zhiyuan C., Huaning W., 2007, Free vibration of FGM cylindrical shells with holes under various boundary conditions, Journal of Sound and Vibration 306(12): 227237.##[11] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Boca Raton, FL, Second Edition.## ##]