• Smart Structures and Systems
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• 제22권1호
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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.

• Structural Engineering and Mechanics
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• 제86권1호
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• pp.69-83
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• 2023

Analytical solutions to problems are crucial because they provide high-quality comparison data for assessing the accuracy of numerical solutions. Benchmark analytical solutions for the vibrations of cracked functionally graded material (FGM) plates are not available in the literature because of the high level of complexity of such solutions. On the basis of first-order shear deformation plate theory (FSDT), this study proposes analytical series solutions for the vibrations of FGM rectangular plates with side or internal cracks parallel to an edge of the plates by using Fourier cosine series and the domain decomposition technique. The distributions of FGM properties along the thickness direction are assumed to follow a simple power law. The proposed analytical series solutions are validated by performing comprehensive convergence studies on the vibration frequencies of cracked square plates with various crack lengths and under various boundary condition combinations and by performing comparisons with published results based on various plate theories and the theory of three-dimensional elasticity. The results reveal that the proposed solutions are in excellent agreement with literature results obtained using the Ritz method on the basis of FSDT. The paper also presents tabulations of the first six nondimensional frequencies of cracked rectangular Al/Al2O3 FGM plates with various aspect ratios, thickness-to-width ratios, crack lengths, and FGM power law indices under six boundary condition combinations, the tabulated frequencies can serve as benchmark data for assessing the accuracy of numerical approaches based on FSDT.

• 한국해양공학회지
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• 제24권6호
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• pp.61-70
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• 2010

This study presents free vibration analyses of perforates steel plates with various cutouts. Four different parameters (shape, size, curvature radius ratio, and rotation of cutouts) were considered to investigate the effects of those parameters on the free vibration characteristics, such as natural frequencies of the perforated steel plates. Three different shapes of cutouts are circle, square, and triangle, and the considered sizes are 5, 10, 15, 20, and 25 mm. For the triangular and square cutouts, the characteristic radii of the inscribed circles of those cutouts were defined. In addition, the curvature radius ratio was defined as the ratio of curvature radius of bluntness and the characteristic radius. Then, total seven different curvature radius ratios (0, 0.1, 0.3, 0.5, 0.7, 0.9, and 1) were considered. To investigate the rotation effect of the cutouts, it was considered four rotations ($0^{\circ}$, $15^{\circ}$, $30^{\circ}$, and $45^{\circ}$) for the square cutouts and three rotations (0, 15, and 30) for the triangular cutouts. All the free vibration analyses were conducted using a general purpose finite element program. From the analyses we found that the most influential parameter for the free vibration response of the perforated plates is the size of cutout. The other factors such as the shape, curvature radius ratio, and rotation are minors; they mainly change the natural frequency as long as the size effect is accompanied.

• 대한기계학회논문집
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• 제16권12호
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• pp.2306-2314
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• 1992

본 연구에서는 Lagrangian 방정식을 이용해 알루미늄, CFRP, GFRP, BFRP 등으 로 혼합적층된 cross-ply 사각판에 대해 굽힘-신장연성을 고려하여 Runge-Kutta Gill 법을 적용하여 수치적으로 비선형진동해석을 수행하였다.그리고 여러가지 적층방법 에 따라 비선형 진동에 어떠한 영향이 미치는가를 검토하였으며, 형상비(a/b), 모우드 의 변화 그리고 탄성계수비에 따른 비선형진동 거동을 규명하였다. 한편, 기본진동 수에 대해서는 상용 유한요소프로그램인 ABAQUS의 결과와 비교하였으며, 단일 적층된 판의 비선형진동거동에 대해서는 Singh의 결과와 비교 검토하였다.

• Structural Engineering and Mechanics
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• 제70권5호
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• pp.601-621
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• 2019

This study presents a comprehensive nonlinear dynamic approach to investigate the linear and nonlinear vibration of sandwich plates fabricated from functionally graded materials (FGMs) resting on an elastic foundation. Higher-order shear deformation theory and Hamilton's principle are employed to obtain governing equations. The Runge-Kutta method is employed together with the commercially available mathematical software MAPLE 14 to solve the set of nonlinear dynamic governing equations. Method validity is evaluated by comparing the results of this study and those of previous research. Good agreement is achieved. The effects of temperature change on frequencies are investigated considering various temperatures and various volume fraction index values, N. As the temperature increased, the plate frequency decreased, whereas with increasing N, the plate frequency increased. The effects of the side-to-thickness ratio, c/h, on natural frequencies were investigated. With increasing c/h, the frequencies increased nonlinearly. The effects of foundation stiffness on nonlinear vibration of the sandwich plate were also studied. Backbone curves presenting the variation of maximum displacement with respect to plate frequency are presented to provide insight into the nonlinear vibration and dynamic behavior of FGM sandwich plates.

• Structural Engineering and Mechanics
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• 제40권5호
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• pp.665-688
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• 2011

This paper presents a theoretical analysis for the free vibration of rectangular tanks partially filled with an ideal liquid. Wet dynamic displacements of the tanks are approximated by combining the orthogonal polynomials satisfying the boundary conditions, since the rectangular tanks are composed of four rectangular plates. The classical boundary conditions of the tanks at the top and bottom ends are considered, such as clamped, simply supported, and clamped-free boundary conditions. As the facing rectangular plates are assumed to be geometrically and structurally identical, the vibration modes of the facing plates of the tanks can be divided into two categories: symmetric and antisymmetric modes with respect to the planes passing through the center of the tanks and perpendicular to the free liquid surface. The liquid displacement potentials satisfying the Laplace equation and liquid boundary conditions are derived, and the wet dynamic modal functions of a quarter of the tanks can be expanded by the finite Fourier transform for compatibility requirements along the contacting surfaces between the tanks and liquid. An eigenvalue problem is derived using the Rayleigh-Ritz method. Consequently, the wet natural frequencies of the rectangular tanks can be extracted. The proposed analytical method is verified by observing an excellent agreement with three-dimensional finite element analysis results. The effects of the liquid level and boundary condition at the top and bottom edges are investigated.

• Structural Engineering and Mechanics
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• 제88권6호
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• pp.551-567
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• 2023

In this work, the free vibration analysis of functionally graded material (FGM) sandwich plates with porosity is conducted using the p-version of the finite element method (FEM), which is based on the first-order shear deformation theory (FSDT). The sandwich plate consists of two face-sheet layers of FGM and a homogeneous core layer. The obtained results are validated using convergence and comparison studies with previously published results. Five porosities distribution models of FGM sandwich plates are assumed and analyzed. The effect of the thickness ratio, boundary conditions, volume fraction exponents, and porosity coefficients of the top and bottom layers of FGM sandwich plates on the natural frequency are addressed.

• Structural Engineering and Mechanics
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• 제35권2호
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• pp.217-240
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• 2010

A semi-analytical method is presented for accurately prediction of the free vibration behavior of generally laminated composite plates with arbitrary boundary conditions. The method employs the technique of separation of spatial variables within Hamilton's principle to obtain the equations of motion, including two systems of coupled ordinary homogeneous differential equations. Subsequently, by applying the laminate constitutive relations into the resulting equations two sets of coupled ordinary differential equations with constant coefficients, in terms of displacements, are achieved. The obtained differential equations are solved for the natural frequencies and corresponding mode shapes, with the use of the exact state-space approach. The formulation is exploited in the framework of the first-order shear deformation theory to incorporate the effects of transverse shear deformation and rotary inertia. The efficiency and accuracy of the present method are demonstrated by obtaining solutions to a wide range of problems and comparing them with finite element analysis and previously published results.

• Structural Engineering and Mechanics
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• 제5권6호
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• pp.691-703
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• 1997

This paper investigates the optimal locations of internal point supports in a symmetric crossply laminated rectangular plate for maximum fundamental frequency of vibration. The method used for solving this optimization problem involves the Rayleigh-Ritz method for the vibration analysis and the simplex method of Nelder and Mead for the iterative search of the optimum support locations. Being a continuum method, the Rayleigh-Ritz method allows easy handling of the changing point support locations during the optimization search. Rectangular plates of various boundary conditions, aspect ratios, composed of different numbers of layers, and with one, two and three internal point supports are analysed. The interesting results on the optimal locations of the point supports showed that (a) there are multiple solutions; (b) the locations are dependent on both the plate aspect ratios and the number of layers (c) the fundamental frequency may be raised significantly with appropriate positioning of the point supports.

• Structural Engineering and Mechanics
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• 제67권4호
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• pp.369-383
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• 2018

In this work, a novel hyperbolic shear deformation theory is developed for free vibration analysis of the simply supported functionally graded plates in thermal environment and the FGM having temperature dependent material properties. This theory has only four unknowns, which is even less than the other shear deformation theories. The theory presented is variationally consistent, without the shear correction factor. The present one has a new displacement field which introduces undetermined integral variables. Equations of motion are obtained by utilizing the Hamilton's principles and solved via Navier's procedure. The convergence and the validation of the proposed theoretical model are performed to demonstrate the efficacy of the model.