• 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.

• Structural Engineering and Mechanics
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• v.57 no.6
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• pp.969-988
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• 2016

Vibration analysis of the beams on elastic foundation has gained the great interest of many researchers. In the literature, there are many studies that focus on the free vibration analysis of the beams on one or two parameter elastic foundations. On the other hand, there are no sufficient studies especially focus on the comparison of dynamic response including the bending moment and shear force of the beams resting on Winkler and two parameter foundations. In this study, dynamic response of the axially loaded Timoshenko beams resting on modified Vlasov type elastic soil was investigated by using the separation of variables method. Governing equations were obtained by assuming that the material had linear elastic behaviour and mass of the beam was distributed along its length. Numerical analysis were provided and presented in figures to find out the differences between the modified Vlasov model and conventional Winkler type foundation. Furthermore, the effect of shear deformation of elastic soil on the dynamic response of the beam was investigated.

• Steel and Composite Structures
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• v.27 no.2
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• pp.177-184
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• 2018

This study contributes to evaluate multiphase topology optimization design of plate-like structures on elastic foundations by using classic plate theory. Multi-material optimal topology and shape are produced as an alternative to provide reasonable material assignments based on stress distributions. Multi-material topology optimization problem is solved through an alternative active-phase algorithm with Gauss-Seidel version as an optimization model of optimality criteria. Stiffness and adjoint sensitivity formulations linked to thin plate potential strain energy are derived in terms of multiphase design variables and Winkler-Pasternak parameters considering elastic foundation to apply to the current topology optimization. Numerical examples verify efficiency and diversity of the present topology optimization method of elastic thin plates depending on multiple materials and Winkler-Pasternak parameters with the same amount of volume fraction and total structural volume.

• 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.

• Structural Engineering and Mechanics
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• v.66 no.5
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• pp.665-676
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• 2018

In this study, a four-node quadrilateral partial mixed plate element with low degrees of freedom (dofs) is developed for static and free vibration analysis of functionally graded material (FGM) plates rested on Winkler-Pasternak elastic foundations. The formulation of the presented finite element model is based on a parametrized mixed variational principle which is developed recently by the first author. The presented finite element model considers the effects of shear deformations and normal flexibility of the FGM plates without using any shear correction factor. It also fulfills the boundary conditions of the transverse shear and normal stresses on the top and bottom surfaces of the plate. Beside these capabilities, the number of unknown field variables of the plate is only six. The presented partial mixed finite element model has been validated through comparison with the results of the three-dimensional (3D) theory of elasticity and the results obtained from the classical and high-order plate theories available in the open literature.

• Steel and Composite Structures
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• v.39 no.1
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• pp.95-107
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• 2021

In this work, the dynamic response of functionally graded beams on variable elastic foundations is studied using a novel higher-order shear deformation theory (HSDT). Unlike the conventional HSDT, the present one has a new displacement field which introduces undetermined integral variables. The FG beams were assumed to be supported on Winkler-Pasternak type foundations in which the Winkler modulus is supposed to be variable in the length of the beam. The variable rigidity of the elastic foundation is assumed to be linear, parabolic and sinusoidal along the length of the beam. The material properties of the FG porous beam vary according to a power law distribution in terms of the volume fraction of the constituents. The equations of motion are determined using the virtual working principle. For the analytical solution, Navier method is used to solve the governing equations for simply supported porous FG beams. Numerical results of the present theory for the free vibration of FG beams resting on elastic foundations are presented and compared to existing solutions in the literature. A parametric study will be detailed to investigate the effects of several parameters such as gradient index, thickness ratio, porosity factor and foundation parameters on the frequency response of porous FG beams.

• Smart Structures and Systems
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• v.18 no.4
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• pp.755-786
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• 2016

The hygro-thermo-mechanical bending behavior of sigmoid functionally graded material (S-FGM) plate resting on variable two-parameter elastic foundations is discussed using a four-variable refined plate theory. The material characteristics are distributed within the thickness direction according to the two power law variation in terms of volume fractions of the constituents of the material. By employing a four variable refined plate model, both a trigonometric distribution of the transverse shear strains within the thickness and the zero traction boundary conditions on the top and bottom surfaces of the plate are respected without utilizing shear correction factors. The number of independent variables of the current formulation is four, as against five in other shear deformation models. The governing equations are deduced based on the four-variable refined plate theory incorporating the external load and hygro-thermal influences. The results of this work are compared with those of other shear deformation models. Various numerical examples introducing the influence of power-law index, plate aspect ratio, temperature difference, elastic foundation parameters, and side-to-thickness ratio on the static behavior of S-FGM plates are investigated.

• Structural Engineering and Mechanics
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• v.69 no.5
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• pp.511-525
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• 2019

This paper presents an analytical study of wave propagation in simply supported graduated functional plates resting on a two-parameter elastic foundation (Pasternak model) using a new theory of high order shear strain. Unlike other higher order theories, the number of unknowns and governing equations of the present theory is only four unknown displacement functions, which is even lower than the theory of first order shear deformation (FSDT). Unlike other elements, the present work includes a new field of motion, which introduces indeterminate integral variables. The properties of the materials are assumed to be ordered in the thickness direction according to the two power law distributions in terms of volume fractions of the constituents. The wave propagation equations in FG plates are derived using the principle of virtual displacements. The analytical dispersion relation of the FG plate is obtained by solving an eigenvalue problem. Numerical examples selected from the literature are illustrated. A good agreement is obtained between the numerical results of the current theory and those of reference. A parametric study is presented to examine the effect of material gradation, thickness ratio and elastic foundation on the free vibration and phase velocity of the FG plate.

• Earthquakes and Structures
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• v.22 no.5
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• pp.447-460
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• 2022

In this article, wave propagation in functional gradation plates (FG) resting on an elastic foundation with two parameters is studied using a new quasi-three-dimensional (3D) higher shear deformation theory (HSDT). The new qausi-3D HSOT has only five variables in fields displacement, which means has few numbers of unknowns compared with others quasi-3D. This higher shear deformation theory (HSDT) includes shear deformation and effect stretching with satisfying the boundary conditions of zero traction on the surfaces of the FG plate without the need for shear correction factors. The FG plates are considered to rest on the Winkler layer, which is interconnected with a Pasternak shear layer. The properties of the material graded for the plates are supposed to vary smoothly, with the power and the exponential law, in the z-direction. By based on Hamilton's principle, we derive the governing equations of FG plates resting on an elastic foundation, which are then solved analytically to obtain the dispersion relations. Numerical results are presented in the form of graphs and tables to demonstrate the effectiveness of the current quasi-3D theory and to analyze the effect of the elastic foundation on wave propagation in FG plates.

• Structural Engineering and Mechanics
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• v.27 no.2
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• pp.199-222
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• 2007

Composite laminated structures supported on elastic foundations are being increasingly used in a great variety of engineering applications. Composites exhibit larger dispersion in their material properties compared to the conventional materials due to large number of parameters associated with their manufacturing and fabrication processes. And also the dispersion in elastic foundation stiffness parameter is inherent due to inaccurate modeling and determination of elastic foundation properties in practice. For a better modeling of the material properties and foundation, these are treated as random variables. This paper deals with effects of randomness in material properties and foundation stiffness parameters on the free vibration response of laminated composite plate resting on an elastic foundation. A $C^0$ finite element method has been used for arriving at an eigen value problem. Higher order shear deformation theory has been used to model the displacement field. A mean centered first order perturbation technique has been employed to handle randomness in system properties for obtaining the stochastic characteristic of frequency response. It is observed that small amount of variations in random material properties and foundation stiffness parameters significantly affect the free vibration response of the laminated composite plate. The results have been compared with those available in the literature and an independent Monte Carlo simulation.