• Structural Engineering and Mechanics
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• v.64 no.3
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• pp.381-390
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• 2017

Present paper deals with the large amplitude flexural vibration of carbon nanotube reinforced composite (CNTRC) plates. Distribution of CNTs as reinforcements may be uniform or functionally graded (FG). The equivalent material properties of the composite media are obtained according to a refined rule of mixtures which contains efficiency parameters. To account for the large deformations, von $K{\acute{a}}rm{\acute{a}}n$ type of geometrical nonlinearity is included into the formulation. The matrix representation of the governing equations is obtained according to the Ritz method where the basic shape functions are written in terms of the Chebyshev polynomials. Time dependency of the problem is eliminated by means of the Galerkin method and the resulting nonlinear eigenvalue problem is solved employing a direct displacement control approach. Results are obtained for completely clamped and completely simply supported plates. Results are first validated for the especial cases of FG-CNTRC and cross-ply laminated plates. Afterwards, parametric studies are given for FG-CNTRC plates with different boundary conditions. It is shown that, nonlinear frequencies are highly dependent to the volume fraction and dispersion profiles of CNTs. Furthermore, mode redistribution is observed in both simply supported and clamped FG-CNTRC plates.

• Structural Engineering and Mechanics
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• v.44 no.4
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• pp.431-448
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• 2012

In this paper, the first-order shear deformation theory (FSDT) (Mindlin) for continuum incorporating surface energy is exploited to study the static behavior of ultra-thin functionally graded (FG) plates. The size-dependent mechanical response is very important while the plate thickness reduces to micro/nano scales. Bulk stresses on the surfaces are required to satisfy the surface balance conditions involving surface stresses. Unlike the classical continuum plate models, the bulk transverse normal stress is preserved here. By incorporating the surface energies into the principle of minimum potential energy, a series of continuum governing differential equations which include intrinsic length scales are derived. The modifications over the classical continuum stiffness are also obtained. To illustrate the application of the theory, simply supported micro/nano scaled rectangular films subjected to a transverse mechanical load are investigated. Numerical examples are presented to present the effects of surface energies on the behavior of functionally graded (FG) film, whose effective elastic moduli of its bulk material are represented by the simple power law. The proposed model is then used for a comparison between the continuum analysis of FG ultra-thin plates with and without incorporating surface effects. Also, the transverse shear strain effect is studied by a comparison between the FG plate behavior based on Kirchhoff and Mindlin assumptions. In our analysis the residual surface tension under unstrained conditions and the surface Lame constants are expected to be the same for the upper and lower surfaces of the FG plate. The proposed model is verified by previous work.

• Structural Engineering and Mechanics
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• v.71 no.6
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• pp.627-641
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• 2019

The paper focuses on bending analysis of the functionally graded (FG) plates with arbitrary shapes and boundary conditions. The material property of FG plates is modelled by using the power law distribution. Based on the first order shear deformation plate theory (FSDT), the governing equations as well as boundary conditions are formulated and obtained by using the principle of virtual work. The coupled Boundary Element-Radial Basis Function (BE-RBF) method is established to solve the complex FG plates. The proposed methodology is developed by applying the concept of the analog equation method (AEM). According to the AEM, the original governing differential equations are replaced by three Poisson equations with fictitious sources under the same boundary conditions. Then, the fictitious sources are established by the application of a technique based on the boundary element method and approximated by using the radial basis functions. The solution of the actual problem is attained from the known integral representations of the potential problem. Therefore, the kernels of the boundary integral equations are conveniently evaluated and readily determined, so that the complex FG plates can be easily computed. The reliability of the proposed method is evaluated by comparing the present results with those from analytical solutions. The effects of the power index, the length to thickness ratio and the modulus ratio on the bending responses are investigated. Finally, many interesting features and results obtained from the analysis of the FG plates with arbitrary shapes and boundary conditions are demonstrated.

• Steel and Composite Structures
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• v.12 no.6
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• pp.491-504
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• 2012

In this paper, the nonlinear cylindrical bending of simply supported, functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes (SWCNTs), is studied. The plates are subjected to uniform pressure loading in thermal environments and their geometric nonlinearity is introduced in the strain-displacement equations based on Von-Karman assumptions. The material properties of SWCNTs are assumed to be temperature-dependent and are obtained from molecular dynamics simulations. The material properties of functionally graded carbon nanotube-reinforced composites (FG-CNTCRs) are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are reduced to linear differential equation with nonlinear boundary conditions yielding a simple solution procedure. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.

• Steel and Composite Structures
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• v.18 no.1
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• pp.187-212
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• 2015

In this paper, various four variable refined plate theories are presented to analyze vibration of temperature-dependent functionally graded (FG) plates. By dividing the transverse displacement into bending and shear parts, the number of unknowns and governing equations for the present model is reduced, significantly facilitating engineering analysis. These theories account for parabolic, sinusoidal, hyperbolic, and exponential distributions of the transverse shear strains and satisfy the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Power law material properties and linear steady-state thermal loads are assumed to be graded along the thickness. Uniform, linear, nonlinear and sinusoidal thermal conditions are imposed at the upper and lower surface for simply supported FG plates. Equations of motion are derived from Hamilton's principle. Analytical solutions for the free vibration analysis are obtained based on Fourier series that satisfy the boundary conditions (Navier's method). Non-dimensional results are compared for temperature-dependent and temperature-independent FG plates and validated with known results in the literature. Numerical investigation is conducted to show the effect of material composition, plate geometry, and temperature fields on the vibration characteristics. It can be concluded that the present theories are not only accurate but also simple in predicting the free vibration responses of temperature-dependent FG plates.

• Steel and Composite Structures
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• v.20 no.2
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• pp.227-249
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• 2016

The objective of this work is to present a zeroth-order shear deformation theory for free vibration analysis of functionally graded (FG) nanoscale plates resting on elastic foundation. The model takes into consideration the influences of small scale and the parabolic variation of the transverse shear strains across the thickness of the nanoscale plate and thus, it avoids the employ use of shear correction factors. Also, in this present theory, the effect of transverse shear deformation is included in the axial displacements by using the shear forces instead of rotational displacements as in available high order plate theories. The material properties are supposed to be graded only in the thickness direction and the effective properties for the FG nanoscale plate are calculated by considering Mori-Tanaka homogenization scheme. The equations of motion are obtained using the nonlocal differential constitutive expressions of Eringen in conjunction with the zeroth-order shear deformation theory via Hamilton's principle. Numerical results for vibration of FG nanoscale plates resting on elastic foundations are presented and compared with the existing solutions. The influences of small scale, shear deformation, gradient index, Winkler modulus parameter and Pasternak shear modulus parameter on the vibration responses of the FG nanoscale plates are investigated.

• Structural Engineering and Mechanics
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• v.49 no.5
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• pp.569-595
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• 2014

In this paper, a novel semi-energy finite strip method (FSM) is developed based on the concept of first order shear deformation theory (FSDT) in order to attempt the post-buckling solution for thin and relatively thick functionally graded (FG) plates under uniform end-shortening. In order to study the effects of through-the-thickness shear stresses on the post-buckling behavior of FG plates, two previously developed finite strip methods, i.e., semi-energy FSM based on the concept of classical laminated plate theory (CLPT) and a CLPT full-energy FSM, are also implemented. Moreover, the effects of aspect ratio on initial post-buckling stiffness of FG rectangular plates are studied. It has been shown that the variation of the ratio of initial post-buckling stiffness to pre-buckling stiffness ($S^*/S$) with respect to aspects ratios is quite independent of volume fractions of constituents in thin FG plates. It has also been seen that the universal curve representing the variation of ($S^*/S$) with aspect ratio of a FG plate demonstrate a saw shape curve. Moreover, it is revealed that for the thin FG plates in contrast to relatively thick plates, the variations of non-dimensional load versus end-shortening is independent of ceramic-metal volume fraction index. This means that the post-buckling behavior of thin FG plates and the thin pure isotropic plates is similar. The results are discussed in detail and compared with those obtained from finite element method (FEM) of analysis. The study of the results may have a great influence in design of FG plates encountering post-buckling behavior.

• Computers and Concrete
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• v.26 no.1
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• pp.63-74
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• 2020

In this investigation, study of the bending response of functionally graded (FG) porous plates is presented using a cubic shear deformation theory. The properties of the FG-plate vary according to a power-law distribution which is modified to approximate material characteristics for considering the effect of porosities. The equilibrium equations are derived by using the principle of virtual work and solved by using Navier's procedure. Various numerical results are discussed to demonstrate the influence of the variation of the power index, the porosity parameter and the geometric ratios on the bending response of FG porous plates.

• Steel and Composite Structures
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• v.25 no.4
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• pp.389-401
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• 2017

In this work, a new higher shear deformation theory (HSDT) is developed for the free vibration and buckling of functionally graded (FG) sandwich plates. The proposed theory presents a new displacement field by using undetermined integral terms. Only four unknowns are employed in this theory, which is less than the classical first shear deformation theory (FSDT) and others HSDTs. Equations of motion are obtained via Hamilton's principle. The analytical solutions of FG sandwich plates are determined by employing the Navier method. A good agreement between the computed results and the available solutions of existing HSDTs is found to prove the accuracy of the developed theory.

• Steel and Composite Structures
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• v.44 no.2
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• pp.171-181
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• 2022

Functionally graded material (FGM) has been spotlighted as an advanced composite material due to its excellent thermo-mechanical performance. And the buckling of FGM resting on elastic foundations has been a challenging subject because its behavior is directly connected to the structural safety. In this context, this paper is concerned with a numerical buckling analysis of metal-ceramic FG plates resting on a two-parameter (Pasternak-type) elastic foundation. The buckling problem is formulated based on the neutral surface and the (1,1,0) hierarchical model, and it is numerically approximated by 2-D natural element method (NEM) which provides a high accuracy even for coarse grid. The derived eigenvalue equations are solved by employing Lanczos and Jacobi algorithms. The numerical results are compared with the reference solutions through the benchmark test, from which the reliability of present numerical method has been verified. Using the developed numerical method, the critical buckling loads of metal-ceramic FG plates are parametrically investigated with respect to the major design parameters.