scholarly journals Boundary Conditions in 2D Numerical and 3D Exact Models for Cylindrical Bending Analysis of Functionally Graded Structures

2016 ◽  
Vol 2016 ◽  
pp. 1-17 ◽  
Author(s):  
F. Tornabene ◽  
S. Brischetto ◽  
N. Fantuzzi ◽  
M. Bacciocchi
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Numerical Models ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Cylindrical Bending ◽  
Exact Model ◽  
Graded Material ◽  
Generalized Differential ◽  
Graded Structures

The cylindrical bending condition for structural models is very common in the literature because it allows an incisive and simple verification of the proposed plate and shell models. In the present paper, 2D numerical approaches (the Generalized Differential Quadrature (GDQ) and the finite element (FE) methods) are compared with an exact 3D shell solution in the case of free vibrations of functionally graded material (FGM) plates and shells. The first 18 vibration modes carried out through the 3D exact model are compared with the frequencies obtained via the 2D numerical models. All the 18 frequencies obtained via the 3D exact model are computed when the structures have simply supported boundary conditions for all the edges. If the same boundary conditions are used in the 2D numerical models, some modes are missed. Some of these missed modes can be obtained modifying the boundary conditions imposing free edges through the direction perpendicular to the direction of cylindrical bending. However, some modes cannot be calculated via the 2D numerical models even when the boundary conditions are modified because the cylindrical bending requirements cannot be imposed for numerical solutions in the curvilinear edges by definition. These features are investigated in the present paper for different geometries (plates, cylinders, and cylindrical shells), types of FGM law, lamination sequences, and thickness ratios.

scholarly journals Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates

Materials ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 3820
Author(s):  
Chiung-Shiann Huang ◽  
S. H. Huang
Keyword(s):  
Boundary Conditions ◽  
Material Properties ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Series Solutions ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Graded Material

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

Supersonic Aeroelasticity and Dynamic Instability of Functionally Graded Porous Cylindrical Shells Using a Unified Solution Formulation

2020 ◽  
Vol 20 (12) ◽  
pp. 2050132
Author(s):  
Mohammad Rahmanian ◽  
Masoud Javadi
Keyword(s):  
Boundary Conditions ◽  
Power Law ◽  
Dynamic Instability ◽  
Cylindrical Shells ◽  
Free Vibrations ◽  
Computational Effort ◽  
Functionally Graded ◽  
Clear Understanding ◽  
Stability Margins ◽  
Graded Material

This study provides an overview on the effect of porosity on free vibrations and more importantly aeroelastic stability margins of cylindrical shells. A general formulation for cylindrical shells is first developed including the effects of shear deformation and rotary inertia along with Sander’s rigid body rotation modification. Two porosity distributions of even and uneven are considered for functionally graded shells. The most general form of power-law model which is known as four-parameter power-law is utilized to provide a clear understanding for the qualification of functionally graded material. A Ritz-based solution algorithm being capable of representing all combinations of symmetric and asymmetric boundary conditions by a penalty method is also introduced. In addition to the capability of satisfying all boundary conditions, the presented solution method is very fast in terms of convergence and computational effort. Various parametric studies are provided and practical results are reported.

scholarly journals Third-order nonlocal elasticity in buckling and vibration of functionally graded nanoplates on Winkler-Pasternak media

2020 ◽  
Author(s):  
A. Cutolo ◽  
V. Mallardo ◽  
M. Fraldi ◽  
E. Ruocco
Keyword(s):  
Boundary Conditions ◽  
Nonlocal Elasticity ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Sensitivity Analyses ◽  
Third Order ◽  
Graded Material ◽  
Benchmark Solutions

Abstract The focus of the present work is to present an analytical approach for buckling and free vibrations analysis of thick functionally graded nanoplates embedded in a Winkler-Pasternak medium. The equations of motion are derived according to both the third-order shear deformation theory, proposed by Reddy, and the nonlocal elasticity Eringen's model. For the first time, the equations are solved analytically for plates with two simply supported opposite edges, the solutions also turning helpful as shape functions in the analysis of structures with more complex geometries and boundary conditions. Sensitivity analyses are finally performed to highlight the role of nonlocal parameters, aspect and side-to-thickness ratios, boundary conditions, and functionally graded material properties in the overall response of plates and cylindrical shells. It is felt that the proposed strategy could be usefully adopted as benchmark solutions in numerical routines as well as for predicting some unexpected behaviors, for instance, in terms of buckling load, in thick nanoplates on elastic foundations.

scholarly journals Analytical Study of Free Vibrations of Fluid Coupling and Structure in Collision of Turbulent Fluid with FGM Plate

2021 ◽  
Vol 39 (1) ◽  
pp. 145-154
Author(s):  
Masoud Rahmani ◽  
Amin Moslemi Petrudi ◽  
Mohammad Reza Pourdavood
Keyword(s):  
Boundary Conditions ◽  
Natural Frequency ◽  
Plate Thickness ◽  
Volume Ratio ◽  
Order Theory ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Turbulent Fluid ◽  
Graded Material ◽  
Mathematica Software

In this paper, the free and forced vibration of a functional rectangular plate in contact with a turbulent fluid is investigated. Functional plates have been considered due to their high thermal resistance to residual stresses. The geometry of the problem is that one side of the reservoir in which the fluid is placed is covered with a plate of Functionally Graded Material (FGM). In order to approximate the displacement of the plate, assuming the third-order theory of shear deformation, trigonometric harmonic test functions are used, which determine the boundary conditions of the simple and fixed plate support. In the equations governing fluid oscillating behavior, the potential velocity of the fluid is obtained by determining the boundary conditions of the fluid in the form of February series functions. To achieve the natural frequency of the plate in contact with turbulent fluid and the shape of the vibrating mode, the Rayleigh-Ritz energy method is used based on the minimum potential energy. In order to check the accuracy of the method used, the results of analytical solution after solving the equations by coding in Wolfram Mathematica software have been compared with numerical solution of Abaqus software and then with accurate results in references, which shows the appropriate accuracy of the solution. Finally, the effect of volumetric coefficient parameters, volume ratio, length ratio, plate thickness ratio, fluid height, reservoir width and boundary conditions on the natural frequency of the plate in contact with turbulent fluid has been investigated and analyzed.

Vibration analysis of a thin functionally graded plate having an out of plane material inhomogeneity resting on Winkler–Pasternak foundation under different combinations of boundary conditions

2018 ◽  
Vol 233 (8) ◽  
pp. 2636-2662 ◽  
Author(s):  
Piyush Pratap Singh ◽  
Mohammad Sikandar Azam ◽  
Vinayak Ranjan
Keyword(s):  
Boundary Conditions ◽  
Power Law ◽  
Functionally Graded Material ◽  
Functionally Graded ◽  
Pasternak Foundation ◽  
Functionally Graded Plate ◽  
Material Inhomogeneity ◽  
Power Law Exponent ◽  
Out Of Plane ◽  
Graded Material

In the present research article, classical plate theory has been adopted to analyze functionally graded material plate, having out of plane material inhomogeneity, resting on Winkler–Pasternak foundation under different combinations of boundary conditions. The material properties of the functionally graded material plate vary according to power law in the thickness direction. Rayleigh–Ritz method in conjugation with polynomial displacement functions has been used to develop a computationally efficient mathematical model to study free vibration characteristics of the plate. Convergence of frequency parameters (nondimensional natural frequencies) has been attained by increasing the number of polynomials of displacement function. The frequency parameters of the functionally graded material plate obtained by proposed method are compared with the open literature to validate the present model. Firstly, the present model is used to calculate first six natural frequencies of the functionally graded plate under all possible combinations of boundary conditions for the constant value of stiffness of Winkler and Pasternak foundation moduli. Further, the effects of density, aspect ratio, power law exponent, Young’s modulus on frequency parameters of the functionally graded plate resting on Winkler–Pasternak foundation under specific boundary conditions viz. CCCC (all edges clamped), SSSS (all edges simply supported), CFFF (cantilever), SCSF (simply supported-clamped-free) are studied extensively. Furthermore, effect of stiffness of elastic foundation moduli (kp and kw) on frequency parameters are analyzed. It has been observed that effects of aspect ratios, boundary conditions, Young’s modulus and density on frequency parameters are significant at lower value of the power law exponent. It has also been noted from present investigation that Pasternak foundation modulus has greater effect on frequency parameters as compared to the Winkler foundation modulus. Most of the results presented in this paper are novel and may be used for the validation purpose by researchers. Three dimensional mode shapes for the functionally graded plate resting on elastic foundation have also been presented in this article.

Bending and free vibration analyses of functionally graded material nanoplates via a novel nonlocal single variable shear deformation plate theory

2020 ◽  
pp. 095440622096452
Author(s):  
Le Kha Hoa ◽  
Pham Van Vinh ◽  
Nguyen Dinh Duc ◽  
Nguyen Thoi Trung ◽  
Le Truong Son ◽  
...  
Keyword(s):  
Shear Deformation ◽  
Functionally Graded Material ◽  
Numerical Solutions ◽  
Scale Effects ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Small Scale ◽  
Parameter Study ◽  
Graded Material

A novel nonlocal shear deformation theory is established to investigate functionally graded nanoplates. The significant benefit of this theory is that it consists of only one unknown variable in its displacement formula and governing differential equation, but it can take into account both the quadratic distribution of the shear strains and stresses through the plate thickness as well as the small-scale effects on nanostructures. The numerical solutions of simply supported rectangular functionally graded material nanoplates are carried out by applying the Navier procedure. To indicate the accuracy and convergence of this theory, the present solutions have been compared with other published results. Furthermore, a deep parameter study is also carried out to exhibit the influence of some parameters on the response of the functionally graded material nanoplates.

Bi-Directional Functionally Graded Nanotubes: Fluid Conveying Dynamics

2018 ◽  
Vol 10 (04) ◽  
pp. 1850041 ◽  
Author(s):  
Ye Tang ◽  
Tianzhi Yang
Keyword(s):  
Boundary Conditions ◽  
Numerical Solutions ◽  
Differential Quadrature Method ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Graded Materials ◽  
Critical Flow Velocity ◽  
Dynamic Behaviors ◽  
Fundamental Frequencies ◽  
Material Gradation

In the paper, a novel model of fluid-conveying nanotubes made of bi-directional functionally graded materials is presented for investigating the dynamic behaviors and stability. For the first time, the material properties of the nanotubes along both radical and axial directions are under consideration. Based on Euler–Bernoulli beam and Eringen’s nonlocal elasticity theories, the governing equation of the nanotubes and associated boundary conditions are developed using Hamilton’s principle. Differential quadrature method (DQM) is applied for discretizing the equation to determine the numerical solutions of the nanotubes with different boundary conditions. Numerical examples are presented to examine the effects of the material gradation, nonlocal parameter, and mode order on the dynamics and stability. It is shown that the two-directional materials distribution can significantly change the critical flow velocity, fundamental frequencies and stability. Comparing with traditional one-directional distribution, such 2D is more flexible to tune overall dynamic behaviors, this may provide new avenues for smart pipes.

Non-linear thermo-mechanical cylindrical bending of functionally graded plates

2008 ◽  
Vol 222 (3) ◽  
pp. 305-318 ◽  
Author(s):  
F Fallah ◽  
A Nosier
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Functionally Graded ◽  
Power Law Distribution ◽  
Cylindrical Bending ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Fg Plates ◽  
Non Linear ◽  
The One

Based on the first-order non-linear von Karman theory, cylindrical bending of functionally graded (FG) plates subjected to mechanical, thermal, and combined thermo-mechanical loadings are investigated. Analytical solutions are obtained for an FG plate with various clamped and simply-supported boundary conditions. The closed form solutions obtained are very simple to be used in design purposes. The material properties are assumed to vary continuously through the thickness of the plate according to a power-law distribution of the volume fraction of the constituents. The effects of non-linearity, material property, and boundary conditions on various response quantities are studied and discussed. It is found that linear analysis is inadequate for analysis of simply-supported FG plates even in the small deflection range especially when thermal load is present. Also it is shown that bending—extension coupling can not be seen in response quantities of clamped FG plates. Also an exact solution is developed for the one-dimensional heat conduction equation with variable heat conductivity coefficient.

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