비균질 Pasternak지반에 의해 지지된 박판의 안정 해석

이용수,김광서 한국전산구조공학회 2001 한국전산구조공학회논문집 Vol.14 No.3

본 논문은 등분포 면내응력을 받고 비균질 Pasternak지반에 의해 지지된 장방형판의 진동해석을 다룬 것이다. Winkler 지반계수와 전단지반계수가 고려된 2 변수지반을 Pasternak 지반이라 불리운다. 판의 중앙부분과 가장자리부분의 Winkler 지반계수값을 각각 k₁과 k₂로 선택하였고 전단지반계수값은 판의 전지역에 대해 일정한 값을 취하였다. 전체 휨강성행렬, 기하강성행렬, 질량 행렬 및 Pasternak 지반의 강성행렬을 조합하고 이 행렬들로 이루어지는 고유값 문제를 푼다. 그 결과 지반강성을 고려할 때 전단지반계수가 무시되면 안된다는 것으로 나타났다. This paper deals with the vibration analysis of the rectangular plates which are subjected to uniform in-plane stresses and supported on In-homogeneous Pasternak foundation. Two parametric foundation which Winkler foundation parameter and shear foundation parameter considered, is called by the Pasternak foundation. The values of Winkler foundation parameter of central and border zone of plate are chosen as k₁and k₂respectively, and the value of shear foundation is chosen as constant about all zone of plate. After composing global flexural stiffeness matrix, geometrical stiffeness matrix, mass matrix, and the stiffeness matrix of the Pasternak foundation, eigenvalue problems which are composed of these matrices are solved. The result shows that the shear foundation parameter must not be ignore when considering the stiffeness of foundation.

Coupled effect of variable Winkler–Pasternak foundations on bending behavior of FG plates exposed to several types of loading

Nabil Himeur,Belgacem Mamen,Soumia Benguediab,Abdelhakim Bouhadra,Abderrahmane Menasria,Benattou Bouchouicha,Fouad Bourada,Mohamed Benguediab 국제구조공학회 2022 Steel and Composite Structures, An International J Vol.44 No.3

This study attempts to shed light on the coupled impact of types of loading, thickness stretching, and types of variation of Winkler–Pasternak foundations on the flexural behavior of simply- supported FG plates according to the new quasi– 3D high order shear deformation theory, including integral terms. A new function sheep is used in the present work. In particular, both Winkler and Pasternak layers are non-uniform and vary along the plate length direction. In addition, the interaction between the loading type and the variation of Winkler–Pasternak foundation parameters is considered and involved in the governing equilibrium equations. Using the virtual displacement principle and Navier’s solution technique, the numerical results of nondimensional stresses and displacements are computed. Finally, the non-dimensional formulas’ results are validated with the existing literature, and excellent agreement is detected between the results. More importantly, several complementary parametric studies with the effect of various geometric and material factors are examined. The present analytical model is suitable for investigating the bending of simply-supported FGM plates for special technical engineering applications.

Correlation between beam on Winkler-Pasternak foundation and beam on elastic substrate medium with inclusion of microstructure and surface effects

Suchart Limkatanyu,Paitoon Ponbunyanon,Woraphot Prachasaree,Kittisak Kuntiyawichai,권민호 대한기계학회 2014 JOURNAL OF MECHANICAL SCIENCE AND TECHNOLOGY Vol.28 No.9

A novel beam-elastic substrate element with inclusion of microstructure and surface energy effects is proposed in this paper. Themodified couple stress theory is employed to account for the microstructure-dependent effect of the beam bulk material while Gurtin-Murdoch surface theory is used to capture the surface energy-dependent size effect. Interaction mechanism between the beam and thesurrounding substrate medium is represented by the Winkler foundation model. The governing differential equilibrium and compatibilityequations of the beam-elastic substrate system are consistently derived based on virtual displacement and virtual force principles, respectively. Both essential and natural boundary conditions of the system are also obtained. Two modified Tonti’s diagrams are presented toprovide the big picture of both displacement-based and force-based formulations of the system. Due to similarity between the currentproblem and the one related to the beam on Winkler-Pasternak foundation, the so-called “natural” beam-Winkler-Pasternak foundationelement coined by the authors is employed to perform two numerical simulations to study the characteristics and behaviors of a beamsubstratesystem with inclusion of microstructure and surface effects.

Thermo-mechanical behavior of porous FG plate resting on the Winkler-Pasternak foundation

Rabia, Benferhat,Tahar, Hassaine Daouadji,Abderezak, Rabahi Techno-Press 2020 Coupled systems mechanics Vol.9 No.6

The effect of porosity on the thermo-mechanical behavior of simply supported functionally graded plate reposed on the Winkler-Pasternak foundation is investigated analytically in the present paper using new refined hyperbolic shear deformation plate theory. Both even and uneven distribution of porosity are taken into account and the effective properties of FG plates with porosity are defined by theoretical formula with an additional term of porosity. The present formulation is based on a refined higher order shear deformation theory, which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the FG plate and takes into account the various distribution shape of porosity. The elastic foundation is described by the Winkler-Pasternak model. Anew modified power-law formulation is used to describe the material properties of FGM plates in the thickness direction. The closed form solutions are obtained by using Navier technique. The present results are verified in comparison with the published ones in the literature. The results show that the dimensionless and stresses are affected by the porosity volume fraction, constituent volume fraction, and thermal load.

Nonlinear responses of an arbitrary FGP circular plate resting on the Winkler-Pasternak foundation

Mohammad Arefi,M.N.M. Allam 국제구조공학회 2015 Smart Structures and Systems, An International Jou Vol.16 No.1

This paper presents nonlinear analysis of an arbitrary functionally graded circular plate integrated with two functionally graded piezoelectric layers resting on the Winkler-Pasternak foundation. Geometric nonlinearity is considered in the strain-displacement relation based on the Von-Karman assumption. All the mechanical and electrical properties except Poisson\'s ratio can vary continuously along the thickness of the plate based on a power function. Electric potential is assumed as a quadratic function along the thickness direction. After derivation of general nonlinear equations, as an instance, numerical results of a functionally graded material integrated with functionally graded piezoelectric material obeying two different functionalities is investigated. The effect of different parameters such as parameters of foundation, non homogenous index and boundary conditions can be investigated on the mechanical and electrical results of the system. A comprehensive comparison between linear and nonlinear responses of the system presents necessity of this study. Furthermore, the obtained results can be validated by using previous linear and nonlinear analyses after removing the effect of foundation.

Effects of elastic medium on the electric potential of neural tissue by using spherical bidomain Pasternak model

Taj, Muhammad,Khadimallah, Mohamed A.,Hussain, Muzamal,Elbahar, Mohamed,Safeer, Muhammad,Asghar, Sehar,Mujalli, Mohammed,Qazaq, Amjad,Ahmad, Monzoor,Elaloui, Elimame,Tounsi, Abdelouahed Techno-Press 2021 Advances in nano research Vol.10 No.6

An isotropic Pasternak-Like model is considered to elaborate the behavior of electric potential of neural tissue when immersed within elastic medium. Special attention is paid to the non dimensionlization of the parameters of the surrounding elastic medium. We found that shear layer potential per unit area and is the Winkler foundation-like constant, are two constants of Pasternak-like model and they actually effect on the potential of neural tissue. It is found that elastic medium affects the potential behavior of neural tissue. Elastic medium effect on the potential behavior of neural tissue is due to the Pasternak foundation parameters. Therefore the surrounding medium affects the potential of neural tissue during its transportation of functioning.

Nonlinear electromechanical analysis of a functionally graded square plate integrated with smart layers resting on Winkler-Pasternak foundation

Mohammad Arefi 국제구조공학회 2015 Smart Structures and Systems, An International Jou Vol.16 No.1

This paper presents nonlinear analysis of a functionally graded square plate integrated with two functionally graded piezoelectric layers resting on the Winkler-Pasternak foundation. Geometric nonlinearity was considered in the strain-displacement relation based on the Von-Karman assumption. All the mechanical and electrical properties except Poisson\'s ratio can vary continuously along the thickness of the plate based on a power function. Electric potential was assumed as a quadratic function along the thickness direction and trigonometric function along the planar coordinate. The effect of non homogeneous index was investigated on the responses of the system. Furthermore, a comprehensive investigation has been performed for studying the effect of two parameters of assumed foundation on the mechanical and electrical components. A comparison between linear and nonlinear responses of the system presents necessity of this study.

Free vibration of an axially functionally graded pile with pinned ends embedded in Winkler-Pasternak elastic medium

Cetin, Dogan,Simsek, Mesut Techno-Press 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.40 No.4

In the present study, free vibration of an axially functionally graded (AFG) pile embedded in Winkler-Pasternak elastic foundation is analyzed within the framework of the Euler-Bernoulli beam theory. The material properties of the pile vary continuously in the axial direction according to the power-law form. The frequency equation is obtained by using Lagrange's equations. The unknown functions denoting the transverse deflections of the AFG pile is expressed in modal form. In this study, the effects of material variations, the parameters of the elastic foundation on the fundamental frequencies are examined. It is believed that the tabulated results will be a reference with which other researchers can compare their results.

Size dependent vibration of embedded functionally graded nanoplate in hygrothermal environment by Rayleigh-Ritz method

Singh, Piyush P.,Azam, Mohammad S. Techno-Press 2021 Advances in nano research Vol.10 No.1

In this article, the vibration behavior of embedded Functionally Graded Nanoplate (FGNP) employing nonlocal Kirchhoff's plate theory has been investigated under hygrothermal environment. The FGNP is considered to be supported by Winkler-Pasternak foundation. The Eringen's differential theory is used for size effect on the vibration of the FGNP. Rayleigh-Ritz method with orthogonal polynomials are employed for the governing equations and edge constraints. The advantage of this method is that it overcomes all the drawbacks of edge constraints and can easily handle any combinations of mixed edge constraints. The coefficients viz. moisture expansion, thermal expansion and elastic coefficients are considered to be transversely graded across the FGNP. The similarity of the calculated natural frequencies is examined with the previous research, and a good concurrency is seen. The objective of this article is to analyze the parameters' effect on the nondimensionalized frequency of embedded FGNP under hygrothermal environment subjected to all possible edge constraints. For this, uniform and linear rise of temperature and moisture concentration are considered. The study highlights that the nonlocal effect is pronounced for higher modes. Moreover, the effect of the Pasternak modulus is seen to be prominent compared to the Winkler modulus on non dimensionalized frequencies of FGNP.

Free vibration of an axially functionally graded pile with pinned ends embedded in Winkler-Pasternak elastic medium

Dogan Çetin,Mesut Simsek 국제구조공학회 2011 Structural Engineering and Mechanics, An Int'l Jou Vol.40 No.4

In the present study, free vibration of an axially functionally graded (AFG) pile embedded in Winkler-Pasternak elastic foundation is analyzed within the framework of the Euler-Bernoulli beam theory. The material properties of the pile vary continuously in the axial direction according to the power-law form. The frequency equation is obtained by using Lagrange’s equations. The unknown functions denoting the transverse deflections of the AFG pile is expressed in modal form. In this study, the effects of material variations, the parameters of the elastic foundation on the fundamental frequencies are examined. It is believed that the tabulated results will be a reference with which other researchers can compare their results.