• Structural Engineering and Mechanics
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• v.81 no.4
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• pp.461-479
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• 2022

A finite element method analysis framework is introduced for the free vibration analyses of functionally graded porous beam structures by employing two variables trigonometric shear deformation theory. Both Young's modulus and material density of the FGP beam element are simultaneously considered as grading through the thickness of the beam. The finite element approach is developed using a nonlocal strain gradient theory. The governing equations derived here are solved introducing a 3-nodes beam element. A comprehensive parametric study is carried out, with a particular focus on the effects of various structural parameters such as the dispersion patterns of GPL reinforcements and porosity, thickness ratio, boundary conditions, nonlocal scale parameter and strain gradient parameters. The results indicate that porosity distribution and GPL pattern have significant effects on the response of the nanocomposite beams.

• Wind and Structures
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• v.28 no.1
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• pp.19-30
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• 2019

This article present the free vibration analysis of simply supported perfect and imperfect (porous) FG beams using a high order trigonometric deformation theory. It is assumed that the material properties of the porous beam vary across the thickness. Unlike other theories, the number of unknown is only three. This theory has a parabolic shear deformation distribution across the thickness. So it is useless to use the shear correction factors. The Hamilton's principle will be used herein to determine the equations of motion. Since, the beams are simply supported the Navier's procedure will be retained. To show the precision of this model, several comparisons have been made between the present results and those of existing theories in the literature.

• Composite Materials and Engineering
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• v.3 no.3
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• pp.179-199
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• 2021

A simple solution for free vibration of cross-ply and angle-ply laminated composite plates in a thermal environment is investigated using a basic trigonometric shear deformation theory. By application of trigonometric four variable plate theory, the transverse displacement is subdivided into bending and shear components, the present theory's number of unknowns and governing equations is reduced, making it easier to use. Hamilton's Principle is extended to derive the equations of motion of the plates using Navier's double trigonometric series, a closed-form solution is obtained; the primary conclusion is that simple solution is obtained with good results accuracy when compared with previously published results, and the natural frequency will differ depending on, environment temperature, thickness ratio, and lamination angle, as well as the aspect ratio of the plate.

• Steel and Composite Structures
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• v.20 no.5
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• pp.963-981
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• 2016

A nonlocal trigonometric shear deformation beam theory based on neutral surface position is developed for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The present model is capable of capturing both small scale effect and transverse shear deformation effects of FG nanobeams, and does not require shear correction factors. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived by employing Hamilton's principle, and the physical neutral surface concept. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

• Structural Engineering and Mechanics
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• v.64 no.4
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• pp.391-402
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• 2017

In this paper, a new nonlocal trigonometric shear deformation theory is proposed for thermal buckling response of nanosize functionally graded (FG) nano-plates resting on two-parameter elastic foundation under various types of thermal environments. This theory uses for the first time, undetermined integral variables and it contains only four unknowns, that is even less than the first shear deformation theory (FSDT). It is considered that the FG nano-plate is exposed to uniform, linear and sinusoidal temperature rises. Mori-Tanaka model is utilized to define the gradually variation of material properties along the plate thickness. Nonlocal elasticity theory of Eringen is employed to capture the size influences. Through the stationary potential energy the governing equations are derived for a refined nonlocal four-variable shear deformation plate theory and then solved analytically. A variety of examples is proposed to demonstrate the importance of elastic foundation parameters, various temperature fields, nonlocality, material composition, aspect and side-to-thickness ratios on critical stability temperatures of FG nano-plate.

• Advances in nano research
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• v.10 no.3
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• pp.235-251
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• 2021

In the current study, the free vibrational behavior of a Porous Micro (PM) beam which is integrated with Functionally Graded Piezoelectric (FGP) layers with initial curvature is considered based on the two trigonometric shear deformation theories namely SSDBT and Tan-SDBT. The structure's mechanical properties are varied through its thicknesses following the given functions. The curved microbeam is exposed to electro-mechanical preload and also is rested on a Pasternak type of elastic foundation. Hamilton's principle is used to extract the motion equations and the MCST is used to capture the size effect. Navier's solution method is selected as an analytical method to solve the motion equations for a simply supported ends case and by validating the results for a simpler state with previously published works, effects of different important parameters on the behavior of the structure are considered. It is found that although increasing the porosity reduces the natural frequency, but enhancing the volume fraction of CNTs increasing it. Also, by increasing the central angle of the curved beam the vibrations of the structure increases. Designing and manufacturing more efficient smart structures such as sensors and actuators are of the aims of this study.

• Steel and Composite Structures
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• v.27 no.3
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• pp.311-325
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• 2018

This article presents the bending analysis of FGM rectangular plates resting on non-uniform elastic foundations in thermal environment. Theoretical formulations are based on a recently developed refined shear deformation theory. The displacement field of the present theory is chosen based on nonlinear variations in the in-plane displacements through the thickness of the plate. The present theory satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional trigonometric shear deformation theory, the present refined shear deformation theory contains only four unknowns as against five in case of other shear deformation theories. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a simple power law distribution of the volume fraction of the constituents. The elastic foundation is modeled as non-uniform foundation. The results of the shear deformation theories are compared together. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio and elastic foundation parameters on the thermo-mechanical behavior of functionally graded plates. Numerical results show that the present theory can archive accuracy comparable to the existing higher order shear deformation theories that contain more number of unknowns.

• 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.

• Steel and Composite Structures
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• v.53 no.1
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• pp.77-90
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• 2024

This paper focuses on trigonometric porosity distribution to analyze its effect on the free vibration frequencies of porous orthotropic multi-layered composite plates. Three types of porosity distributions are considered. The governing equations of the free vibration response of porous orthotropic multi-layered composite plates are derived from the Hamilton's principle using higher-order shear deformation theory. The free vibration frequency relation of the problem is obtained by performing Galerkin's method. After the validation process of the relation under the available literature, a few parametric analyses are performed to observe the influence of shear deformation, porosity distribution, orthotropy, layer sequence, and different geometric properties on the frequencies.

• Smart Structures and Systems
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• v.22 no.1
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• pp.121-132
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• 2018

This research investigates the free vibration analysis of advanced composite plates such as functionally graded plates (FGPs) resting on a two-parameter elastic foundations using a hybrid quasi-3D (trigonometric as well as polynomial) higher-order shear deformation theory (HSDT). This present theory, which does not require shear correction factor, accounts for shear deformation and thickness stretching effects by a sinusoidal and parabolic variation of all displacements across the thickness. Governing equations of motion for FGM plates are derived from Hamilton's principle. The closed form solutions are obtained by using Navier technique, and natural frequencies are found, for simply supported plates, by solving the results of eigenvalue problems. The accuracy of the present method is verified by comparing the obtained results with First-order shear deformation theory, and other predicted by quasi-3D higher-order shear deformation theories. It can be concluded that the proposed theory is efficient and simple in predicting the natural frequencies of functionally graded plates on elastic foundations.