Stability of cantilever on elastic foundation under a subtangential follower force via shear deformation beam theories

In this paper, a refined quasi-three-dimensional shear deformation theory for thermo-mechanical analysis of functionally graded sandwich plates resting on a two-parameter (Pasternak model) elastic foundation is developed. Unlike the other higher-order theories the number of unknowns and governing equations of the present theory is only four against six or more unknown displacement functions used in the corresponding ones. Furthermore, this theory takes into account the stretching effect due to its quasi-three-dimensional nature. The boundary conditions in the top and bottoms surfaces of the sandwich functionally graded plate are satisfied and no correction factor is required. Various types of functionally graded material sandwich plates are considered. The governing equations and boundary conditions are derived using the principle of virtual displacements. Numerical examples, selected from the literature, are illustrated. A good agreement is obtained between numerical results of the refined theory and the reference solutions. A parametric study is presented to examine the effect of the material gradation and elastic foundation on the deflections and stresses of functionally graded sandwich plate resting on elastic foundation subjected to thermo-mechanical loading.

Download Full-text

Thermal Buckling and Free Vibration Analysis of Functionally Graded Plate Resting on an Elastic Foundation According to High Order Shear Deformation Theory Based on New Shape Function

Applied Sciences ◽

10.3390/app10124190 ◽

2020 ◽

Vol 10 (12) ◽

pp. 4190

Author(s):

Aleksandar Radaković ◽

Dragan Čukanović ◽

Gordana Bogdanović ◽

Milan Blagojević ◽

Blaža Stojanović ◽

...

Keyword(s):

Shear Deformation ◽

Elastic Foundation ◽

Power Law ◽

Deformation Theory ◽

Natural Frequencies ◽

Shape Function ◽

Shear Deformation Theory ◽

Functionally Graded ◽

High Order ◽

Functionally Graded Plate

Functionally graded square and rectangular plates of different thicknesses placed on the elastic foundation modeled according to the Winkler-Pasternak theory have been studied. The thermal and mechanical characteristics, apart from Poisson’s ratio, are considered to continuously differ through the thickness of the studied material as stated in a power-law distribution. A mathematical model of functionally graded plate which include interaction with elastic foundation is defined. The equilibrium and stability equations are derived using high order shear deformation theory that comprises various kinds of shape function and the von Karman nonlinearity. A new analytically integrable shape function has been introduced. Hamilton’s principle has been applied with the purpose of acquiring the equations of motion. An analytical method for identifying both natural frequencies and critical buckling temperature for cases of linear and nonlinear temperature change through the plate thickness has been established. In order to verify the derived theoretical results on numerical examples, an original program code has been implemented within software MATLAB. Critical buckling temperature and natural frequencies findings are shown below. Previous scientific research and papers confirms that presented both the theoretical formulation and the numerical results are accurate. The comparison has been made between newly established findings based on introduced shape function and the old findings that include 13 different shape functions available in previously published articles. The final part of the research provides analysis and conclusions related to the impact of the power-law index, foundation stiffness, and temperature gradient on critical buckling temperature and natural frequencies of the functionally graded plates.

Download Full-text

Nonlinear thermo-mechanical buckling of higher-order shear deformable porous functionally graded material plates reinforced by orthogonal and/or oblique stiffeners

Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical Engineering Science ◽

10.1177/0954406219861658 ◽

2019 ◽

Vol 233 (17) ◽

pp. 6177-6196 ◽

Cited By ~ 4

Author(s):

Vu Hoai Nam ◽

Nguyen Thi Phuong ◽

Dang Thuy Dong ◽

Nguyen Thoi Trung ◽

Nguyen Van Tue

Keyword(s):

Shear Deformation ◽

Elastic Foundation ◽

Functionally Graded Material ◽

Thermal Environment ◽

Plate Theory ◽

Higher Order ◽

Functionally Graded ◽

Shear Deformation Plate Theory ◽

Graded Material ◽

Post Buckling

In this paper, an analytical approach for nonlinear buckling and post-buckling behavior of stiffened porous functionally graded plate rested on Pasternak's elastic foundation under mechanical load in thermal environment is presented. The orthogonal and/or oblique stiffeners are attached to the surface of plate and are included in the calculation by improving the Lekhnitskii's smeared stiffener technique in the framework of higher-order shear deformation plate theory. The complex equilibrium and stability equations are established based on the Reddy's higher-order shear deformation plate theory and taken into account the geometrical nonlinearity of von Kármán. The solution forms of displacements satisfying the different boundary conditions are chosen, the stress function method and the Galerkin procedure are used to solve the problem. The good agreements of the present analytical solution are validated by making the comparisons of the present results with other results. In addition, the effects of porosity distribution, stiffener, volume fraction index, thermal environment, elastic foundation… on the critical buckling load and post-buckling response of porous functionally graded material plates are numerically investigated.

Download Full-text

A Novel Refined Plate Theory for Free Vibration Analyses of Single-Layered Graphene Sheets Lying on Winkler-Pasternak Elastic Foundations

Journal of Nano Research ◽

10.4028/www.scientific.net/jnanor.58.151 ◽

2019 ◽

Vol 58 ◽

pp. 151-164 ◽

Cited By ~ 2

Author(s):

Fatima Boukhatem ◽

Aicha Bessaim ◽

Abdelhakim Kaci ◽

Abderrahmane Mouffoki ◽

Mohammed Sid Ahmed Houari ◽

...

Keyword(s):

Free Vibration ◽

Shear Deformation ◽

Elastic Foundation ◽

Displacement Field ◽

Scale Effects ◽

Plate Theory ◽

Small Scale ◽

Graphene Sheets ◽

Refined Plate Theory ◽

Foundation Parameters

In this article, the analyses of free vibration of nanoplates, such as single-layered graphene sheets (SLGS), lying on an elastic medium is evaluated and analyzed via a novel refined plate theory mathematical model including small-scale effects. The noteworthy feature of theory is that the displacement field is modelled with only four unknowns, which is even less than the other shear deformation theories. The present one has a new displacement field which introduces undetermined integral variables, the shear stress free condition on the top and bottom surfaces of the plate is respected and consequently, it is unnecessary to use shear correction factors. The theory involves four unknown variables, as against five in case of other higher order theories and first-order shear deformation theory. By using Hamilton’s principle, the nonlocal governing equations are obtained and they are solved via Navier solution method. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, nonlocal parameter, and elastic foundation parameters are all examined. From this work, it can be observed that the small-scale effects and elastic foundation parameters are significant for the natural frequency.

Download Full-text

Buckling Temperature and Natural Frequencies of Thick Porous Functionally Graded Beams Resting on Elastic Foundation in a Thermal Environment

Advances in Acoustics and Vibration ◽

10.1155/2019/7986569 ◽

2019 ◽

Vol 2019 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Zakaria Ibnorachid ◽

Lhoucine Boutahar ◽

Khalid EL Bikri ◽

Rhali Benamar

Keyword(s):

Shear Deformation ◽

Elastic Foundation ◽

Power Law ◽

Natural Frequencies ◽

Volume Fraction ◽

Thermal Environment ◽

Slenderness Ratio ◽

Higher Order ◽

Functionally Graded ◽

Transverse Displacement

In this paper, free vibrations of Porous Functionally Graded Beams (P-FGBs), resting on two-parameter elastic foundations, and exposed to three forms of thermal field, uniform, linear, and sinusoidal, are studied using a Refined Higher-order shear Deformation Theory. The present theory accounts for shear deformation by considering a constant transverse displacement and a higher-order variation of the axial displacement through the thickness of the beam. The stress-free boundary conditions are satisfied on the upper and lower surfaces of the beam without using any shear correction factor. The material properties are temperature-dependent and vary continuously through the depth direction of the beam, based on a modified power-law rule, in which two kinds of porosity distributions, uniform, and nonuniform, through the cross-section area of the beam, are considered. Hamilton’s principle is applied to obtain governing equations of motion, which are solved using a Navier-type analytical solution for simply supported P-FGB. Numerical examples are proposed and discussed in detail, to prove the effect of the thermal environment, the porosity distribution, and the influence of several parameters such as the power-law index, porosity volume fraction, slenderness ratio, and elastic foundation parameters on the critical buckling temperatures and the natural frequencies of the P-FGB.

Download Full-text