PASTERNAK ELASTİK ZEMİNE OTURAN HETEROJEN ANİZOTROPİK PLAKLARIN DİNAMİK ANALİZİ

Bu makalede, elastik zeminin heterojen ortotropik plakların (HTOP) titreşim frekansları üzerindeki etkileri kayma deformasyon teorisi (KDT) kullanılarak incelenmektedir. İki parametreli elastik ortamın plak üzerindeki reaksiyonunu tanımlamak için Pasternak elastik zemin (PEZ) modeli kullanılmaktadır. Problemin formülasyonu Donnell tipi teoriye dayanır. Heterojen ortotropik malzemenin Young modüllerinin üstel fonksiyon olarak değiştiği, Poisson oranı ve yoğunluğu sabit kabul edilmektedir. Temel denklemler, Galerkin yöntemi kullanılarak zamana bağlı geometrik kısmi türevli diferansiyel denklemler adi diferansiyel denklemlere indirgenmektedir. Türetilen denklemden heterojen ortotropik plakların frekansı için kapalı çözüm elde edilmektedir. Elde edilen değerler literatürdeki benzer çalışmalar ile karşılaştırılarak sonuçlar doğrulanmıştır. Son olarak, heterojenliğin, kayma gerilmelerinin ve PEZ’in frekans parametrelerine etkilerini göstermek için parametrik çalışma gerçekleştirilmiştir.

Anahtar Kelimeler:

Heterojen anizotrop malzemeler, Pasternak elastik zemini, Plaklar, Titreşim, Frekans parametresi

DYNAMIC ANALYSIS OF HETEROGENEOUS ANISOTROPIC PLATES RESTING ON THE PASTERNAK ELASTIC FOUNDATION

In this study, the effects of elastic foundation on the frequencies of the heterogeneous orthotropic plates using shear deformation plate theory are investigated. Pasternak elastic foundation model is used to define the reaction of two-parameter elastic media on the plate. The formulation of the problem is based on the Donnell type plate theory. The Young's moduli of heterogeneous orthotropic material change as exponential function, Poisson's ratio and density are considered constant. The basic partial differential equations are reduced to ordinary differential equations using Galerkin method and closed-form solution is obtained for the frequency of heterogeneous orthotropic plates. The obtained values are compared with those in the current literature and the results were confirmed. Finally, a parametric study is performed to show the effects of heterogeneity, shear stresses and elastic foundations on the frequency parameters.

Keywords:

Heterogeneous anisotropic materials, Pasternak elastic foundation, Plates, Vibration, Frequency parameter,

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[1] GRIGORENKO, Y.M., GRIGORENKO, A.Y., “Static and Dynamic Problems for Anisotropic Inhomogeneous Shells with Variable Parameters and Their Numerical Solution (review)”, International Applied Mechanics, 49, 123-193, 2013.
[2] SOFIYEV, A.H., OMURTAG, M.H., SCHNACK, E., “The Vibration and Stability of Orthotropic Conical Shells with Non-Homogeneous Material Properties Under A Hydrostatic Pressure”, Journal of Sound and Vibration, 319, 963-983, 2009.
[3] PAN, E., “Exact Solution for Functionally Graded Anisotropic Elastic Composite Laminates”, Journal of Composite Materials, 37, 1903-1920, 2003.
[4] AMBARTSUMIAN, S. A., Theory of Anisotropic Plates; Strength, Stability, Vibration., Technomic published by Stamford, 1964.
[5] REDDY, J.N., Mechanics of Laminated Composite Plates and Shells. Theory and Analysis., Boca Raton, CRC Press, 2004.
[6] AYDOGDU, M., “A New Shear Deformation Theory for Laminated Composite Plates”, Composite Structures, 89, 94–101, 2009.
[7] CHEN, W.Q., BIAN, Z.G., DING, H.J., “Three-dimensional Vibration Analysis of Fluid-Filled Orthotropic FGM Cylindrical Shells”, International Journal of Mechanical Sciences, 46, 159-171, 2004.
[8] BATRA, R.C., JIN, J., “Natural Frequencies of a Functionally Graded Anisotropic Rectangular Plate”, Journal of Sound and Vibration, 282, 509-516, 2005.
[9] OOTAO, Y, TANIGAWA, Y., “Three-dimensional Solution for Transient Thermal Stresses of An Orthotropic Functionally Graded Rectangular Plate”, Composites Structures, 80, 10-20, 2007.
[10] PENG, X.L., LI, X.F., “Elastic Analysis of Rotating Functionally Graded Polar Orthotropic Disks”, International Journal of Mechanical Sciences, 60, 84-91, 2012.
[11] ZERIN, Z., “On the Vibration of Laminated Nonhomogeneous Orthotropic Shells”, Meccanica, 48(7), 1557-1572, 2013.
[12] AVEY A., PINARLIK M., “Fonksiyonel Değişimli Ortotropik Plakların Dinamik Tepkisine Kayma Deformasyonu ve Dönel Eylemsizlik Etkilerinin İncelenmesi”, Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 20(2), 236-243, 2016.
[13] ZERIN, Z., TURAN, F., BASOGLU, M.F., “Examination of Non-homogeneity and Lamination Scheme Effects on Deflections and Stresses of Laminated Composite Plates”, Structural Engineering And Mechanics, 57(4), 603-616, 2016.
[14] PASTERNAK, P.L., “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, 1, 1–56 (in Russian), 1954.
[15] KERR, A.D., “A Study of a New Foundation Model”, Acta Mechanica, 1(2), 135-147, 1964.
[16] XIANG, Y., WANG, C.M., KITIPORNCHAI, S., “Exact Vibration Solution for Initially Stressed Mindlin Plates on Pasternak Foundation”, International Journal of Mechanical Sciences, 36, 311–316, 1994.
[17] OMURTAG, M.H., KADIOGLU, F., “Free Vibration Analysis of Orthotropic Plates Resting on Pasternak Foundation by Mixed Finite Element Formulation”, Computers and Structures 67, 253-265, 1998.
[18] ZHOU, D., CHEUNG, Y.K., LO, S.H., AU, F.T.K., “Three-dimensional Vibration Analysis of Rectangular Thick Plates on Pasternak Foundation”, International Journal of Numerical Methods for Engineering, 59, 1313–1334, 2004.
[19] FERREIRA, A.J.M., ROQUE, C.M.C., NEVES, A.M.A., JORGE, R.M.N., SOARES, C.M.M., “Analysis of Plates on Pasternak Foundations by Radial Basis Functions” Computational Mechanics, 46, 791–803, 2010.
[20] ARANI, A. G., JALAEI, M. H., “Transient Behavior of an Orthotropic Graphene Sheet Resting on Orthotropic Visco-Pasternak Foundation”, International Journal of Engineering Science, 103, 97-113, 2016.
[21] MORIMOTO, T., TANIGAWA, Y., “Elastic Stability of Inhomogeneous Thin Plates on An Elastic Foundation”, Archive of Applied Mechanics, 77, 653-674, 2007.
[22] BAHMYARI, E., KHEDMATI, M.R., “Vibration Analysis of Nonhomogeneous Moderately Thick Plates With Point Supports Resting on Pasternak Elastic Foundation Using Element Free Galerkin Method”, Engineering Analysis with Boundary Elements, 37, 1212-1238, 2013.
[23] LAL, R., “Effect of Nonhomogeneity on Vibration of Orthotropic Rectangular Plates of Varying Thickness Resting on Pasternak Foundation”, Journal of Vibration and Acoustics, 131(1), 2009.
[24] SHARIYAT, M., ASEMI, K., “Three-dimensional Non-linear Elasticity-based 3D Cubic B-spline Finite Element Shear Buckling Analysis of Rectangular Orthotropic FGM Plates Surrounded by Elastic Foundations”, Composites: Part B Engineering, 56, 934-947, 2014.
[25] MA’EN, S. S., AL-KOUZ, W. G., “Vibration Analysis of Non-uniform Orthotropic Kirchhoff Plates Resting on Elastic Foundation Based on Nonlocal Elasticity Theory”, International Journal of Mechanical Sciences, 114, 1-11, 2016.
[26] ASEMI, K., SHARIYET, M., “Three-dimensional Biaxal Post-Buckling Analysis of Heterogeneous Auxetic Rectangular Plates on Elastic Foundation by New criteria”, Computer Methods in Applied Mechanics and Engineering, 302, 1-26, 2016.
[27] MANSOURI, M.H., SHARIYAT, M., “Differential Quadrature Thermal Buckling Analysis of General Quadrilateral Orthotropic Auxetic FGM Plates on Elastic Foundations”, Thin-Walled Structures, 112, 194-207, 2017.
[28] SOFIYEV, A.H., KARACA, Z., ZERIN, Z., “Non-linear Vibration of Composite Orthotropic Cylindrical Shells on the Non-linear Elastic Foundations within the Shear Deformation Theory”, Composite Structures, 159, 53–62, 2017.
[29] HACIYEV, V.C., SOFIYEV, A.H., KURUOGLU, N., “Free Bending Vibration Analysis of Thin Bidirectionally Exponentially Graded Orthotropic Rectangular Plates Resting on Two-Parameter Elastic Foundations”, Composite Structures, 184, 372-377, 2018.