• Structural Engineering and Mechanics
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• v.86 no.1
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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.

• Structural Engineering and Mechanics
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• v.53 no.1
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• pp.27-40
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• 2015

The rational analytical solutions are presented for functionally graded beams subjected to arbitrary tractions on the upper and lower surfaces. The Young's modulus is assumed to vary exponentially along the thickness direction while the Poisson's ratio keeps unaltered. Within the framework of symplectic elasticity, zero eigensolutions along with general eigensolutions are investigated to derive the homogeneous solutions of functionally graded beams with no body force and traction-free lateral surfaces. Zero eigensolutions are proved to compose the basic solutions of the Saint-Venant problem, while general eigensolutions which vary exponentially with the axial coordinate have a significant influence on the local behavior. The complete elasticity solutions presented here include homogeneous solutions and particular solutions which satisfy the loading conditions on the lateral surfaces. Numerical examples are considered and compared with established results, illustrating the effects of material inhomogeneity on the localized stress distributions.

• Structural Engineering and Mechanics
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• v.26 no.2
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• pp.151-162
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• 2007

Rigorous analytical solutions are obtained for the plane stress problem of a rectangular plate subjected to non-linearly distributed bending loads on two opposite edges. They are then used in a Galerkin type solution to obtain the corresponding convergent buckling loads. It is shown that the critical bending moment depends significantly on the actual edge load distribution and further the number of nodal lines of the buckled configuration can also be different from that corresponding to a linear antisymmetric distribution of the bending stresses. Results are tabulated for future use while judging approximate numerical solutions.

• Steel and Composite Structures
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• v.25 no.2
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• pp.127-139
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• 2017

In this paper, an exact analytical solution for simply supported sandwich plate which considers the permeation effect of adhesives is presented. The permeation layer is described as functionally graded material (FGM), the elastic modulus of which is assumed to be graded along the thickness following the exponential law. Based on the exact three-dimensional (3-D) elasticity theory, the solution of stresses and displacements for each layer is derived. By means of the recursive matrix method, the solution can be efficiently obtained for plates with many layers. The present solution obtained can be used as a benchmark to access other simplified solutions. The comparison study indicates that the finite element (FE) solution is close to the present one when the FGM layer in the FE model is divided into a series of homogeneous layers. However, the present method is more efficient than the FE method, with which the mesh division and computation are time-consuming. Moreover, the solution based on Kirchhoff-Love plate theory is greatly different from the present solution for thick plates. The influence of the thickness of the permeation layer on the stress and displacement fields of the sandwich plate is discussed in detail. It is indicated that the permeation layer can effectively relieve the discontinuity stress at the interface.

• Smart Structures and Systems
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• v.8 no.4
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• pp.413-431
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• 2011

Under the external shearing stress, the external radial stress and the electric potential simultaneously, the piezoelectric hollow cylinder transducer is studied. With the Airy stress function method, the analytical solutions of this transducer are obtained based on the theory of piezo-elasticity. The solutions are compared with the finite element results of Ansys and a good agreement is found. Inherent properties of this piezoelectric cylinder transducer are presented and discussed. It is very helpful for the design of the bearing controllers.

• Structural Engineering and Mechanics
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• v.36 no.6
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• pp.669-678
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• 2010

In general means, the stress concentration problem of elastic plate with a rectangular hole can be investigated by numerical methods, and only approximative results are derived. This paper deduces an analytical study of the stress concentration due to a rectangular hole in an elastic plate under bending loads. Base on classical elasticity theory and FEM applying the U-transformation technique, the uncoupled governing equations with 3-DOF are established, and the analytical displacement solutions of the finite element equations are derived in series form or double integral form. Therefore, the stress concentration factor can then be discussed easily and conveniently. For the plate subjected to unidirectional bending loads, the non-conforming plate bending element with four nodes and 12-DOF is taken as examples to demonstrate the application of the proposed method. The inner force distribution is obtained. The solutions are adequate for the condition when the hole is far away from the edges and the thin plate subjected to any transverse loadings.

• Structural Engineering and Mechanics
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• v.10 no.4
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• pp.337-357
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• 2000

Analytical formulations and solutions for the first time, to the stability analysis of a simply supported composite and sandwich plates based on a higher order refined theory, developed by the first author and already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of inplane displacements with respect to the thickness coordinate - thus modelling the warping of transverse cross sections more accurately and eliminating the need for shear correction coefficients. The equations of equilibrium are obtained using the Principle of Minimum Potential Energy (PMPE). The comparison of the results using this higher order refined theory with the available elasticity solutions and the results computed independently using the first order and the other higher order theories developed by other investigators and available in the literature shows that this refined theory predicts the critical buckling load more accurately than all other theories considered in this paper. New results for sandwich laminates are also presented which may serve as a benchmark for future investigations.

• Steel and Composite Structures
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• v.39 no.3
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• pp.291-306
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• 2021

Based on Reissner's mixed variational theorem (RMVT), the authors develop a semi-analytical finite element (FE) method for a three-dimensional (3D) bending analysis of nonhomogeneous orthotropic, complete and incomplete toroidal shells subjected to uniformly-distributed loads. In this formulation, the toroidal shell is divided into several finite annular prisms (FAPs) with quadrilateral cross-sections, where trigonometric functions and serendipity polynomials are used to interpolate the circumferential direction and meridian-radial surface variations in the primary field variables of each individual prism, respectively. The material properties of the toroidal shell are considered to be nonhomogeneous orthotropic over the meridianradial surface, such that homogeneous isotropic toroidal shells, laminated cross-ply toroidal shells, and single- and bi-directional functionally graded toroidal shells can be included as special cases in this work. Implementation of the current FAP methods shows that their solutions converge rapidly, and the convergent FAP solutions closely agree with the 3D elasticity solutions available in the literature.

• Journal of the Korean Society of Safety
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• v.11 no.1
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• pp.46-52
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• 1996

In this paper, an analytical method is proposed to find the dimensions of impact stresses with using the dimensions of impact loading parameter regardless of mass of impactor, velocity of impactor, and plate thickness. In analytical method of Impulsive stresses, the three-dimensional dynamic theory of elasticity using rectangular coordinates and the potential theory of displacement are utilized, and when the measurement of Impact loading is difficult especially for a steel ball colliding on an infinite plate, the impact loading can be obtained by using the classical plate theory and Hertz’s contact theory. And in the numerical analysis, the fast Fourier transform (F. F. T.) algorithm and the numerical inverse Laplace transformation are used because the analysis of impact loading Is difficult to obtain solutions by using the thress-dimensional dynamic theory of elasticity.

• Structural Engineering and Mechanics
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• v.59 no.1
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• pp.101-122
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• 2016

Elasticity solutions for bi-directional functionally graded beams subjected to arbitrary lateral loads are conducted, with emphasis on the end effects. The material is considered macroscopically isotropic, with Young's modulus varying exponentially in both axial and thickness directions, while Poisson's ratio remaining constant. In order to obtain an exact analysis of stress and displacement fields, the symplectic analysis based on Hamiltonian state space approach is employed. The capability of the symplectic framework for exact analysis of bi-directional functionally graded beams has been validated by comparing numerical results with corresponding ones in open literature. Numerical results are provided to demonstrate the influences of the material gradations on localized stress distributions. Thus, the material properties of the bi-directional functionally graded beam can be tailored for the potential practical purpose by choosing suitable graded indices.