• Steel and Composite Structures
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• 제20권3호
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• pp.623-649
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• 2016

Most of the early studies on plates vibration are focused on two-dimensional theories, these theories reduce the dimensions of problems from three to two by introducing some assumptions in mathematical modeling leading to simpler expressions and derivation of solutions. However, these simplifications inherently bring errors and therefore may lead to unreliable results for relatively thick plates. The main objective of this research paper is to present 3-D elasticity solution for free vibration analysis of continuously graded carbon nanotube-reinforced (CGCNTR) rectangular plates resting on two-parameter elastic foundations. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this study, an equivalent continuum model based on the Eshelby-Mori-Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The proposed rectangular plates have two opposite edges simply supported, while all possible combinations of free, simply supported and clamped boundary conditions are applied to the other two edges. The formulations are based on the three-dimensional elasticity theory. A semi-analytical approach composed of differential quadrature method (DQM) and series solution is adopted to solve the equations of motion. The fast rate of convergence of the method is demonstrated and comparison studies are carried out to establish its very high accuracy and versatility. The 2-D differential quadrature method as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The convergence of the method is demonstrated and to validate the results, comparisons are made between the present results and results reported by well-known references for special cases treated before, have confirmed accuracy and efficiency of the present approach. The novelty of the present work is to exploit Eshelby-Mori-Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various coefficients of foundation and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR rectangular plates. The new results can be used as benchmark solutions for future researches.

• Computers and Concrete
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• 제31권2호
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• pp.139-149
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• 2023

This article investigates the statically analysis regarding the thermal buckling behavior of a nonuniform small-scale nanobeam made of functionally graded material based on classic beam theories along with the nonlocal Eringen elasticity. The material distribution of functionally graded structures is composed of temperature-dependent ceramic and metal phases in axial and thickness directions, called two-dimensional functionally graded (2D-FG). The partial differential (PD) formulations and end conditions are extracted by using to the conservation energy method. The porosity voids are assumed in the nonuniform functionally graded (FG) structure. The thermal loads are in the axial direction of the beam. The extracted nonlocal PD equations are also solved by employing generalized differential quadrature method (GDQM). Last but not least, the information acquired is used to produce miniature sensors, providing a unique perspective on the growth of nanoelectromechanical systems (NEMS).

• Steel and Composite Structures
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• 제38권5호
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• pp.477-496
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• 2021

The main objective of this paper is to study vibration of sandwich open cylindrical panel with damaged core and FG face sheets based on three-dimensional theory of elasticity. The structures are made of a damaged isotropic core and two external face sheets. These skins are strengthened at the nanoscale level by randomly oriented Carbon nanotubes (CNTs) and are reinforced at the microscale stage by oriented straight fibers. These reinforcing phases are included in a polymer matrix and a three-phase approach based on the Eshelby-Mori-Tanaka scheme and on the Halpin-Tsai approach, which is developed to compute the overall mechanical properties of the composite material. Three complicated equations of motion for the panel under consideration are semi-analytically solved by using 2-D differential quadrature method. Several parametric analyses are carried out to investigate the mechanical behavior of these multi-layered structures depending on the damage features, through-the-thickness distribution and boundary conditions. It is seen that for the large amount of power-law index "P", increasing this parameter does not have significant effect on the non-dimensional natural frequency parameters of the FG sandwich curved panel. Results indicate that by increasing the value of isotropic damage parameter "D" up to the unity (fully damaged core) the frequency would tend to become zero. One can dictate the fiber variation profile through the radial direction of the sandwich panel via the amount of "P", "b" and "c" parameters. It should be noticed that with increase of volume fraction of fibers, the frequency parameter of the panels does not increase necessarily, so by considering suitable amounts of power-law index "P" and the parameters "b" and "c", one can get dynamic characteristics similar or better than the isotropic limit case for laminated FG curved panels.

• Steel and Composite Structures
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• 제25권4호
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• pp.415-426
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• 2017

The thermo-mechanical vibration behavior of two dimensional functionally graded (2D-FG) porous nanobeam is reported in this paper. The material properties of the nanobeam are variable along thickness and length of the nanobeam according to the power law function. The nanobeam is modeled within the framework of Timoshenko beam theory. Eringen's nonlocal elasticity theory is used to develop the governing equations. Using the generalized differential quadrature method (GDQM) the governing equations are solved. The effect of porosity, temperature distribution, nonlocal value, L/h, FG power indexes along thickness and length and are investigated using parametric studies.

• Steel and Composite Structures
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• 제45권5호
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• pp.729-747
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• 2022

The critical buckling loads and buckling modes of bi-directional functionally graded porous unified higher order shear plate with elastic foundation are investigated. A mathematical model based on neutral axis rather than midplane is developed in comprehensive way for the first time in this article. The material constituents form ceramic and metal are graded through thickness and axial direction by the power function distribution. The voids and cavities inside the material are proposed by three different porosity models through the thickness of plate. The constitutive parameters and force resultants are evaluated relative to the neutral axis. Unified higher order shear plate theories are used to satisfy the zero-shear strain/stress at the top and bottom surfaces. The governing equilibrium equations of bi-directional functionally graded porous unified plate (BDFGPUP) are derived by Hamilton's principle. The equilibrium equations in the form of coupled variable coefficients partial differential equations is solved by using numerical differential integral quadrature method (DIQM). The validation of the present model is presented and compared with previous works for bucking. Deviation in buckling loads for both mid-plane and neutral plane are developed and discussed. The numerical results prove that the shear functions, distribution indices, boundary conditions, elastic foundation and porosity type have significant influence on buckling stability of BDFGPUP. The current mathematical model may be used in design and analysis of BDFGPU used in nuclear, mechanical, aerospace, and naval application.

• Structural Engineering and Mechanics
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• 제44권2호
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• pp.139-161
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• 2012

In this paper, the static behavior of bi-directional functionally graded (FG) non-uniform thickness circular plate resting on quadratically gradient elastic foundations (Winkler-Pasternak type) subjected to axisymmetric transverse and in-plane shear loads is carried out by using state-space and differential quadrature methods. The governing state equations are derived based on 3D theory of elasticity, and assuming the material properties of the plate except the Poisson's ratio varies continuously throughout the thickness and radius directions in accordance with the exponential and power law distributions. The stresses and displacements distribution are obtained by solving state equations. The effects of foundation stiffnesses, material heterogeneity indices, geometric parameters and loads ratio on the deformation and stress distributions of the FG circular plate are investigated in numerical examples. The results are reported for the first time and the new results can be used as a benchmark solution for future researches.

• Geomechanics and Engineering
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• 제36권4호
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• pp.407-416
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• 2024

Dynamic response of a rock tunnels by laminated porous concrete beam reinforced by nanoparticles subjected to harmonic transverse dynamic load is investigated considering structural damping. The effective nanocomposite properties are evaluated on the basis of Mori-Tanaka model. The concrete beam is modeled by the exponential shear deformation theory (ESDT). Utilizing nonlinear strains-deflection, energy relations and Hamilton's principal, the governing final equations of the concrete laminated beam are calculated. Utilizing differential quadrature method (DQM) as well as Newmark method, the dynamic displacement of the concrete laminated beam is discussed. The influences of porosity parameter, nanoparticles volume percent, agglomeration of nanoparticles, boundary condition, geometrical parameters of the concrete beam and harmonic transverse dynamic load are studied on the dynamic displacement of the laminated structure. Results indicated that enhancing the nanoparticles volume percent leads to decrease in the dynamic displacement about 63%. In addition, with considering porosity of the concrete, the dynamic displacement enhances about 2.8 time.

• Computers and Concrete
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• 제30권2호
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• pp.85-97
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• 2022

Using a combination of nonlocal Eringen as well as classical beam theories, this research explores the thermal buckling of a bidirectional functionally graded nanobeam. The formulations of the presented problem are acquired by means on conserved energy as well as nonlocal theory. The results are obtained via generalized differential quadrature method (GDQM). The mechanical properties of the generated material vary in both axial and lateral directions, two-dimensional functionally graded material (2D-FGM). In nanostructures, porosity gaps are seen as a flaw. Finally, the information gained is used to the creation of small-scale sensors, providing an outstanding overview of nanostructure production history.

• Steel and Composite Structures
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• 제41권6호
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• pp.787-803
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• 2021

The size-dependent nonlinear thermomechanical vibration analysis of pre- and post-buckled tapered two-directional functionally graded (2D-FG) microbeams is presented in this study. In the context of the modified couple stress theory, the formulations are derived based on the parabolic shear deformation beam theory and von Karman nonlinear strains. Different thermomechanical material properties are assumed to be temperature-dependent and smoothly vary in both length and thickness directions using the power law and the physical neutral axis concept is employed. The nonlinear governing equations are derived using the Hamilton principle and the resulting variable coefficient equations of motion are solved using the differential quadrature method (DQM) and iterative Newton's method for clamped-clamped and simply supported boundary conditions. Comparison studies are presented to validate the derived model and solution procedure. The impacts of induced thermal moments, temperature power index, two gradient indices, nonuniform cross-section, and microstructure length scale parameter on the frequency-temperature configurations are explored for both clamped and simply supported microbeams.

• Advances in nano research
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• 제10권1호
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• pp.1-14
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• 2021

In the present computational approach, thermal buckling and frequency characteristics of a doubly curved laminated nanopanel with the aid of Two-Dimensional Generalized Differential Quadrature Method (2D-GDQM) and Nonlocal Strain Gradient Theory (NSGT) are investigated. Additionally, the temperature changes along the thickness direction nonlinearly. The novelty of the current study is in considering the effects of laminated composite and thermal in addition of size effect on frequency, thermal buckling, and dynamic deflections of the laminated nanopanel. The acquired numerical and analytical results are compared by each other to validate the results. The results demonstrate that some geometrical and physical parameters, have noticeable effects on the frequency and pre-thermal buckling behavior of the doubly curved open cylindrical laminated nanopanel. The favorable suggestion of this survey is that for designing the laminated nano-sized structure should pay special attention to size-dependent parameters because nonlocal and length scale parameters have an important role in the static and dynamic behaviors of the laminated nanopanel.