• Earthquakes and Structures
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• v.10 no.6
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• pp.1429-1449
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• 2016

The effect of porosity on bending and free vibration behavior of simply supported functionally graded plate reposed on the Winkler-Pasternak foundation is investigated analytically in the present paper. The modified rule of mixture covering porosity phases is used to describe and approximate material properties of the FGM plates with porosity phases. The effect due to transverse shear is included by using a new refined shear deformation theory. The number of unknown functions involved in the present theory is only four as against five or more in case of other shear deformation theories. The Poisson ratio is held constant. Based on the sinusoidal shear deformation theory, the position of neutral surface is determined and the equation of motion for FG rectangular plates resting on elastic foundation based on neutral surface is obtained through the minimum total potential energy and Hamilton's principle. The convergence of the method is demonstrated and to validate the results, comparisons are made with the available solutions for both isotropic and functionally graded material (FGM). The effect of porosity volume fraction on Al/Al2O3 and Ti-6Al-4V/Aluminum oxide plates are presented in graphical forms. The roles played by the constituent volume fraction index, the foundation stiffness parameters and the geometry of the plate is also studied.

• Steel and Composite Structures
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• v.51 no.6
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• pp.601-617
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• 2024

This work aims to explore the static behaviour of a tapered functionally graded porous plate (FGPP) with even and uneven porosity distributions resting on two parametric elastic foundations. The plate under investigation is subjected to bi-sinusoidal loading and the edges of the plate are exposed to different combinations of edge restrictions. In order to examin the static behaviour, bending factors (BF) related to bending and normal stresses have been evaluated using classical plate theory. To achieve this, the governing equations have been derived employing the energy concept. And to solve it, the Rayleigh-Ritz method with an algebraic function has been utilised; it is simple, precise, and computationally intensive. After convergence and validation analyses, new findings are made available. The BF of the plate have been exhaustively examined to explain the influence of aspect ratios, material property index, porosity factor, taper factor, and Winkler and Pasternak stiffness. It is observed that the BF of an elastically supported FGPP are influenced by the index of material propery and the aspect ratio. Findings also indicate that the impact of porosity is more when it is spread evenly, as opposed to when it is unevenly distributed. Further, the deformed plate's structure is significantly influenced by the different thickness variations. Examination of bending characteristics of FGPP having different new cases of thickness variations with different types of porosity distribution under fifteen different mixed edge constraints is the prime novality of this work. Results presented are reliable enough to be taken into account for future studies.

• Smart Structures and Systems
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• v.20 no.4
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• pp.509-524
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• 2017

In this article, a free vibration analysis of functionally graded (FG) plates resting on elastic foundations is presented using a quasi-3D hybrid-type higher order shear deformation theory. Undetermined integral terms are employed in the proposed displacement field and modeled based on a hybrid-type (sinusoidal and parabolic) quasi-3D HSDT with five unknowns in which the stretching effect is taken into account. Thus, it can be said that the significant feature of this theory is that it deals with only 5 unknowns as the first order shear deformation theory (FSDT). The elastic foundation parameters are introduced in the present formulation by following the Pasternak (two-parameter) mathematical model. Equations of motion are obtained via the Hamilton's principles and solved using Navier's method. Accuracy of the proposed theory is confirmed by comparing the results of numerical examples with the ones available in literature.

• Geomechanics and Engineering
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• v.22 no.5
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• pp.415-431
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• 2020

In this study, dynamics responses of advanced composite plates resting variable elastic foundations via a quasi-3D theory are developed using an analytical approach. This higher shear deformation theory (HSDT) is included the shear deformation theory and effect stretching that has five unknowns, which is even inferior to normal deformation theories found literature and other theories. The quasi-three-dimensional (quasi-3D) theory accounts for a parabolic distribution of the transverse shear deformation and satisfies the zero traction boundary conditions on the surfaces of the advanced composite plate without needing shear correction factors. The plates assumed to be rest on two-parameter elastic foundations, the Winkler parameter is supposed to be constant but the Pasternak parameter varies along the long side of the plate with three distributions (linear, parabolic and sinusoidal). The material properties of the advanced composite plates gradually vary through the thickness according to two distribution models (power law and Mori-Tanaka). Governing differential equations and associated boundary conditions for dynamics responses of the advanced composite plates are derived using the Hamilton principle and are solved by using an analytical solution of Navier's technique. The present results and validations of our modal with literature are presented that permitted to demonstrate the accuracy of the present quasi-3D theory to predict the effect of variables elastic foundation on dynamics responses of advanced composite plates.

• Earthquakes and Structures
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• v.17 no.5
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• pp.447-462
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• 2019

This present paper concerned with the analytic modelling for vibration of the functionally graded (FG) plates resting on non-variable and variable two parameter elastic foundation, based on two-dimensional elasticity using higher shear deformation theory. Our present theory has four unknown, which mean that have less than other higher order and lower theory, and we denote do not require the factor of correction like the first shear deformation theory. The indeterminate integral are introduced in the fields of displacement, it is allowed to reduce the number from five unknown to only four variables. The elastic foundations are assumed a classical model of Winkler-Pasternak with uniform distribution stiffness of the Winkler coefficient (kw), or it is with variables distribution coefficient (kw). The variable's stiffness of elastic foundation is supposed linear, parabolic and trigonometry along the length of functionally plate. The properties of the FG plates vary according to the thickness, following a simple distribution of the power law in terms of volume fractions of the constituents of the material. The equations of motions for natural frequency of the functionally graded plates resting on variables elastic foundation are derived using Hamilton principal. The government equations are resolved, with respect boundary condition for simply supported FG plate, employing Navier series solution. The extensive validation with other works found in the literature and our results are present in this work to demonstrate the efficient and accuracy of this analytic model to predict free vibration of FG plates, with and without the effect of variables elastic foundations.

• Steel and Composite Structures
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• v.28 no.5
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• pp.541-557
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• 2018

This paper presents the influence of carbon nanotubes (CNTs) waviness and aspect ratio on the vibrational behavior of functionally graded nanocomposite sandwich annular sector plates resting on two-parameter elastic foundations. The carbon nanotube-reinforced (CNTR) sandwich plate has smooth variation of CNT fraction along the thickness direction. The distributions of CNTs are considered functionally graded (FG) or uniform along the thickness and their mechanical properties are estimated by an extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes' effective aspect ratio as the filler content increases. Effects of CNT distribution, volume fraction, aspect ratio and waviness, and also effects of Pasternak's elastic foundation coefficients, sandwich plate thickness, face sheets thickness and plate aspect ratio are investigated on the free vibration of the sandwich plates with wavy CNT-reinforced face sheets. The study is carried out based on three-dimensional theory of elasticity and in contrary to two-dimensional theories, such as classical, the first- and the higher-order shear deformation plate theories, this approach does not neglect transverse normal deformations. The sandwich annular sector plate is assumed to be simply supported in the radial edges while any arbitrary boundary conditions are applied to the other two circular edges including simply supported, clamped and free.

• Steel and Composite Structures
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• v.34 no.1
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• pp.75-89
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• 2020

The current paper illustrates the effect of in-plane varying compressive force on critical buckling loads and buckling modes of sandwich composite laminated beam rested on elastic foundation. To generalize a proposed model, unified higher order shear deformation beam theories are exploited through analysis; those satisfy the parabolic variation of shear across the thickness. Therefore, there is no need for shear correction factor. Winkler and Pasternak elastic foundations are presented to consider the effect of any elastic medium surrounding beam structure. The Hamilton's principle is proposed to derive the equilibrium equations of unified sandwich composite laminated beams. Differential quadrature numerical method (DQNM) is used to discretize the differential equilibrium equations in spatial direction. After that, eigenvalue problem is solved to obtain the buckling loads and associated mode shapes. The proposed model is validated with previous published works and good matching is observed. The numerical results are carried out to show effects of axial load functions, lamination thicknesses, orthotropy and elastic foundation constants on the buckling loads and mode shapes of sandwich composite beam. This model is important in designing of aircrafts and ships when non-uniform compressive load and shear loading is dominated.

• Steel and Composite Structures
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• v.48 no.3
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• pp.335-351
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• 2023

This research studies the primary resonance and nonlinear vibratory responses of multilayer functionally graded shallow (MFGS) shells under external excitations. The shells considered with functionally graded porous (FGP) core and resting on two types of nonlinear viscoelastic foundations (NVEF) governed by either a linear model with two parameters of Winkler and Pasternak foundations or a nonlinear model of hardening/softening cubic stiffness augmented by a Kelvin-Voigt viscoelastic model. The shells considered have three layers, sandwiched by functionally graded (FG), FGP, and FG materials. To investigate the influence of various porosity distributions, two types of FGP middle layer cores are considered. With the first-order shear deformation theory (FSDT), Hooke's law, and von-Kármán equation, the stress-strain relations for the MFGS shells with FGP core are developed. The governing equations of the shells are consequently derived. For the sake of higher accuracy and reliability, the P-T method is implemented in numerically analyzing the vibration, and the method of multiple scales (MMS) as one of the perturbation methods is used to investigate the primary resonance. The results of the present research are verified with the results available in the literature. The analytical results are compared with the P-T method. The influences of material, geometry, and nonlinear viscoelastic foundation parameters on the responses of the shells are illustrated.

• Steel and Composite Structures
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• v.43 no.1
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• pp.91-106
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• 2022

This paper has focused on presenting a three dimensional theory of elasticity for free vibration of 3D-graphene foam reinforced polymer matrix composites (GrF-PMC) cylindrical panels resting on two-parameter elastic foundations. The elastic foundation is considered as a Pasternak model with adding a Shear layer to the Winkler model. The porous graphene foams possessing 3D scaffold structures have been introduced into polymers for enhancing the overall stiffness of the composite structure. Also, 3D graphene foams can distribute uniformly or non-uniformly in the shell thickness direction. The effective Young's modulus, mass density and Poisson's ratio are predicted by the rule of mixture. Three complicated equations of motion for the panel under consideration are semi-analytically solved by using 2-D differential quadrature method. The fast rate of convergence and accuracy of the method are investigated through the different solved examples. Because of using two-dimensional generalized differential quadrature method, the present approach makes possible vibration analysis of cylindrical panels with two opposite axial edges simply supported and arbitrary boundary at the curved edges. It is explicated that 3D-GrF skeleton type and weight fraction can significantly affect the vibrational characteristics of GrF-PMC panel resting on two-parameter elastic foundations.

• Advances in nano research
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• v.17 no.1
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• pp.1-8
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• 2024

In this paper, stability analysis of sandwich toroidal shell segments (TSSs) with carbon nanotube (CNT)-reinforced face sheets featuring various types of auxetic cores, surrounded by elastic foundations under radial pressure is presented. Two distinct types of auxetic structures are considered for the core, including re-entrant auxetic structure and graphene origami (GOri)-enabled auxetic structure. The nonlinear stability equilibrium equations of the longitudinally shallow shells are formulated using the von Karman shell theory, in conjunction with Stein and McElman approximation while considering Winkler-Pasternak's elastic foundation to simulate the interaction between the shell and elastic foundation. The Galerkin method is employed to derive the nonlinear stability responses of the shells. The numerical investigations show the influences of various types of auxetic-core layers, CNT-reinforced face sheets, as well as elastic foundation on the stability of sandwich shells.