• Journal of Korean Association for Spatial Structures
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• v.4 no.4 s.14
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• pp.113-128
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• 2004

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of solid paraboloidal and complete (that is, without a top opening) paraboloidal shells of revolution with variable wall thickness. Unlike conventional shell theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. The ends of the shell may be free or may be subjected to any degree of constraint. Displacement components $u_r,\;u_{\theta},\;and\;u_z$ in the radial, circumferential, and axial directions, respectively, are taken to be sinusoidal in time, periodic in ${\theta}$, and algebraic polynomials in the r and z directions. Potential (strain) and kinetic energies of the paraboloidal shells of revolution are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four digit exactitude is demonstrated for the first five frequencies of the complete, shallow and deep paraboloidal shells of revolution with variable thickness. Numerical results are presented for a variety of paraboloidal shells having uniform or variable thickness, and being either shallow or deep. Frequencies for five solid paraboloids of different depth are also given. Comparisons are made between the frequencies from the present 3-D Ritz method and a 2-D thin shell theory.

• Structural Engineering and Mechanics
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• v.66 no.5
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• pp.665-676
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• 2018

In this study, a four-node quadrilateral partial mixed plate element with low degrees of freedom (dofs) is developed for static and free vibration analysis of functionally graded material (FGM) plates rested on Winkler-Pasternak elastic foundations. The formulation of the presented finite element model is based on a parametrized mixed variational principle which is developed recently by the first author. The presented finite element model considers the effects of shear deformations and normal flexibility of the FGM plates without using any shear correction factor. It also fulfills the boundary conditions of the transverse shear and normal stresses on the top and bottom surfaces of the plate. Beside these capabilities, the number of unknown field variables of the plate is only six. The presented partial mixed finite element model has been validated through comparison with the results of the three-dimensional (3D) theory of elasticity and the results obtained from the classical and high-order plate theories available in the open literature.

• Transactions of the Korean Society of Mechanical Engineers A
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• v.21 no.6
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• pp.933-941
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• 1997

In the analysis of thin elastic structures such as plate and shell-like structures, classical lower-order theories like Kirchhoff and Reissner-Mindin theories are insufficient to describe the behavior of such structures in the region where the state of stresses is complex. On the other hand, the fully three dimensional theory of linear elasticity can provide desired analysis accuracy, but requires expensive computational implementation compared to the classical theories. This paper is concerned with the development of hierarchical models for elastic structures which can be used for hierarchical modeling for the analysis of such structures. Derivation and limit model analysis (when the thickness of structures tends to zero) of hierarchical models are presented together with a introduction of modeling error estimation. Also, numerical results supporting theoretical results are given.

• Structural Engineering and Mechanics
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• v.44 no.1
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• pp.15-34
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• 2012

In the present paper, an improved high-order theory is used for bending analysis of soft-core sandwich plates. Third-order plate assumptions are used for face sheets and quadratic and cubic functions are assumed for transverse and in-plane displacements of the orthotropic soft core. Continuity conditions for transverse shear stresses at the interfaces as well as the conditions of zero transverse shear stresses on the upper and lower surfaces of the plate are satisfied. Also, transverse flexibility and transverse normal strain and stress of the orthotropic core are considered. The equations of motion and boundary conditions are derived by principle of minimum potential energy. Analytical solution for bending analysis of simply supported sandwich plates under various transverse loads are presented using Navier's solution. Comparison of the present results with those of the three-dimensional theory of elasticity and some plate theories in the literature confirms the accuracy of the proposed theory.

• Steel and Composite Structures
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• v.28 no.5
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• pp.541-557
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• 2018

This paper presents the influence of carbon nanotubes (CNTs) waviness and aspect ratio on the vibrational behavior of functionally graded nanocomposite sandwich annular sector plates resting on two-parameter elastic foundations. The carbon nanotube-reinforced (CNTR) sandwich plate has smooth variation of CNT fraction along the thickness direction. The distributions of CNTs are considered functionally graded (FG) or uniform along the thickness and their mechanical properties are estimated by an extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes' effective aspect ratio as the filler content increases. Effects of CNT distribution, volume fraction, aspect ratio and waviness, and also effects of Pasternak's elastic foundation coefficients, sandwich plate thickness, face sheets thickness and plate aspect ratio are investigated on the free vibration of the sandwich plates with wavy CNT-reinforced face sheets. The study is carried out based on three-dimensional theory of elasticity and in contrary to two-dimensional theories, such as classical, the first- and the higher-order shear deformation plate theories, this approach does not neglect transverse normal deformations. The sandwich annular sector plate is assumed to be simply supported in the radial edges while any arbitrary boundary conditions are applied to the other two circular edges including simply supported, clamped and free.

• Steel and Composite Structures
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• v.25 no.6
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• pp.649-661
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• 2017

This paper is motivated by the lack of studies in the technical literature concerning to the influence of carbon nanotubes (CNTs) waviness and aspect ratio on the vibrational behavior of functionally graded nanocomposite annular sector plates resting on two-parameter elastic foundations. The carbon nanotube-reinforced (CNTR) plate has smooth variation of CNT fraction based on the power-law distribution in the thickness direction, and the material properties are also estimated by the extended rule of mixture. In this study, the classical theory concerning the mechanical efficiency of a matrix embedding finite length fibers has been modified by introducing the tube-to-tube random contact, which explicitly accounts for the progressive reduction of the tubes' effective aspect ratio as the filler content increases. Parametric studies are carried out to highlight the influence of CNTs volume fraction, waviness and aspect ratio, boundary conditions and elastic foundation on vibrational behavior of FG-CNT thick sectorial plates. The study is carried out based on three-dimensional theory of elasticity and in contrary to two-dimensional theories, such as classical, the first- and the higher-order shear deformation plate theories, this approach does not neglect transverse normal deformations. The annular sector plate is assumed to be simply supported in the radial edges while any arbitrary boundary conditions are applied to the other two circular edges including simply supported, clamped and free. For an overall comprehension on 3-D vibration of annular sector plates, some mode shape contour plots are reported in this research work.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2002.05a
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• pp.260-267
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• 2002

For an elastically suspended rigid body with the planes of symmetry in a three dimensional space, a novel analysis fur the vibration modes is presented. From the decompositions of the stiffness and inertia matrices, the conditions for the existence of pure translation and pure couple modes are analyzed for an elastically suspended rigid body with the planes of symmetry. From this analysis, it can be showed that how the structure of stiffness and inertia must be related in order to produce the pure translation and pure couple modes when an elastically suspended rigid body has one, two, or three planes of symmetry.

• Structural Engineering and Mechanics
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• v.6 no.5
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• pp.497-516
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• 1998

A relatively simple two-dimensional multilayered shell model is presented for predicting both global quantities and stress distributions across the thickness of multilayered thick shells, that is based on a third-order zig-zag approach. As for any zig-zag model, the layerwise kinematics is accounted for, with the stress continuity conditions at interfaces met a priori. Moreover, the shell model satisfies the zero transverse shear stress conditions at the upper and lower free surfaces of the shell, irrespective of the lay-up. By changing the parameters in the displacement model, some higher order shell models are obtained as particular cases. Although it potentially has a wide range of validity, application is limited to cylindrical shell panels in cylindrical bending, a lot of solutions of two-dimensional models based on rather different simplyfying assumptions and the exact three-dimensional elasticity solution being available for comparisons for this benchmark problem. The numerical investigation performed by the present shell model and by the shell models derived from it illustrates the effects of transverse shear modeling and the range of applicability of the simplyfying assumptions introduced. The implications of retaining only selected terms depending on the radius-to-thickness ratio are focused by comparing the present solutions to the exact one and to other two-dimensional solutions in literature based on rather different simplyfying assumptions.

• Structural Engineering and Mechanics
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• v.66 no.6
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• pp.761-769
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• 2018

In this paper, we study the Carbon/Glass hybrid laminated composite plates, where the buckling behavior is examined using an accurate and simple refined higher order shear deformation theory. This theory takes account the shear effect, where shear deformation and shear stresses will be considered in determination of critical buckling load under different boundary conditions. The most interesting feature of this new kind of hybrid laminated composite plates is that the possibility of varying components percentages, which allows us for a variety of plates with different materials combinations in order to overcome the most difficult obstacles faced in traditional laminated composite plates like (cost and strength). Numerical results of the present study are compared with three-dimensional elasticity solutions and results of the first-order and the other higher-order theories issue from the literature. It can be concluded that the proposed theory is accurate and simple in solving the buckling behavior of hybrid laminated composite plates and allows to industrials the possibility to adjust the component of this new kind of plates in the most efficient way (reducing time and cost) according to their specific needs.

• Structural Engineering and Mechanics
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• v.68 no.3
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• pp.325-334
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• 2018

The Nonlinear dynamic response of a sandwich plate subjected to the low velocity impact is theoretically and experimentally investigated. The Hertz law between the impactor and the plate is taken into account. Using the Extended High Order Sandwich Panel Theory (EHSAPT) and the Ritz energy method, the governing equations are derived. The skins follow the Third order shear deformation theory (TSDT) that has hitherto not reported in conventional EHSAPT. Besides, the three dimensional elasticity is used for the core. The nonlinear Von Karman relations for strains of skins and the core are adopted. Time domain solution of such equations is extracted by means of the well-known fourth-order Runge-Kutta method. The effects of core-to-skin thickness ratio, initial velocity of the impactor, the impactor mass and position of the impactor are studied in detail. It is found that these parameters play significant role in the impact force and dynamic response of the sandwich plate. Finally, some low velocity impact tests have been carried out by Drop Hammer Testing Machine. The results are compared with experimental data acquired by impact testing on sandwich plates as well as the results of finite element simulation.