• Computers and Concrete
• /
• v.24 no.6
• /
• pp.579-586
• /
• 2019

In this research paper, the free vibrational behavior of the simply supported FG nano-plate is studied using the nonlocal two variables integral refined plate theory. The present model takes into account the small scale effect. The effective's properties of the plate change according to the power law variation (P-FGM). The equations of motion of the system are determined and resolved via Hamilton's principle and Navier procedure, respectively. The validity and efficiency of the current model are confirmed by comparing the results with those given in the literature. At the last section, several numerical results are presented to show the various parameters influencing the vibrational behavior such as the small-scale effect, geometry ratio, material index and aspect ratio.

• Structural Engineering and Mechanics
• /
• v.82 no.6
• /
• pp.725-734
• /
• 2022

This work presents a simple exponential shear deformation theory for the flexural and free vibration responses of thick bridge deck. Contrary to the existing higher order shear deformation theories (HSDT) and the first shear deformation theory (FSDT), the proposed model uses a new displacement field which incorporates undetermined integral terms and involves only two variables. Governing equations and boundary conditions of the theory are derived by the principle of virtual work. The simply supported thick isotropic square and rectangular plates are considered for the detailed numerical studies. Results of displacements, stresses and frequencies are compared with those of other refined theories and exact theory to show the efficiency of the proposed theory. Good agreement is achieved of the present results with those of higher order shear deformation theory (HSDT) and elasticity theory. Moreover, results demonstrate that the developed two variable refined plate theory is simple for solving the flexural and free vibration responses of thick bridge deck and can achieve the same accuracy of the existing HSDTs which have more number of variables.

• Earthquakes and Structures
• /
• v.17 no.5
• /
• pp.447-462
• /
• 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
• /
• v.27 no.1
• /
• pp.109-122
• /
• 2018

In This work an analysis of the propagation of waves of functionally graduated plates is presented by using a high order hyperbolic (HSDT) shear deformation theory. This theory has only four variables, which is less than the theory of first order shear deformation (FSDT). Therefore, a shear correction coefficient is not required. Unlike other conventional shear deformation theories, the present work includes a new field of displacement which introduces indeterminate integral variables. The properties of materials are supposed classified in the direction of the thickness according to two simple distributions of a power law in terms of volume fractions of constituents. The governing equations of the wave propagation in the functionally graded plate are derived by employing the Hamilton's principle. The analytical dispersion relation of the functionally graded plate is obtained by solving an eigenvalue problem. The convergence and the validation of the proposed theoretical numerical model are performed to demonstrate the efficacy of the model.

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

• Advances in nano research
• /
• v.5 no.4
• /
• pp.393-414
• /
• 2017

Forced vibration behavior of porous metal foam nanoplates on elastic medium is studied via a 4-variable plate theory. Different porosity distributions called uniform, symmetric and asymmetric are considered. Nonlocal strain gradient theory (NSGT) containing two scale parameters is employed for size-dependent modeling of porous nanoplates. The present plate theory satisfies the shear deformation effect and it has lower field variables compared with first order plate theory. Hamilton's principle is employed to derive the governing equations. Obtained results from Galerkin's method are verified with those provided in the literature. The effects of nonlocal parameter, strain gradient, foundation parameters, dynamic loading, porosity distributions and porosity coefficient on dynamic deflection and resonance frequencies of metal foam nanoscale plates are examined.

• Smart Structures and Systems
• /
• v.25 no.4
• /
• pp.447-457
• /
• 2020

In this study, a simple two-dimensional shear deformation model is employed for buckling analysis of functionally graded (FG) plates. The proposed theory has a kinematic with integral terms which considers the influence of shear deformation without using "shear correction factors". The impact of varying material properties and volume fraction of the constituent on buckling response of the FG plate is examined and discussed. The benefit of this theory over other contributions is that a number of variables is reduced. The basic equations that consider the influence of transverse shear stresses are derived from the principle of virtual displacements. The analytical solutions are obtained utilizing the "Navier method". The accuracy of the proposed theory is proved by comparisons with the different solutions found in the literature.

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

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

• Smart Structures and Systems
• /
• v.19 no.6
• /
• pp.601-614
• /
• 2017

In this work, free vibration analysis of size-dependent functionally graded (FG) nanoplates resting on two-parameter elastic foundation is investigated based on a novel nonlocal refined trigonometric shear deformation theory for the first time. This theory includes undetermined integral variables and contains only four unknowns, with is even less than the conventional first shear deformation theory (FSDT). Mori-Tanaka model is employed to describe gradually distribution of material properties along the plate thickness. Size-dependency of nanosize FG plate is captured via the nonlocal elasticity theory of Eringen. By implementing Hamilton's principle the equations of motion are obtained for a refined four-variable shear deformation plate theory and then solved analytically. To show the accuracy of the present theory, our research results in specific cases are compared with available results in the literature and a good agreement will be demonstrated. Finally, the influence of various parameters such as nonlocal parameter, power law indexes, elastic foundation parameters, aspect ratio, and the thickness ratio on the non-dimensional frequency of rectangular FG nanoscale plates are presented and discussed in detail.