• Structural Engineering and Mechanics
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• v.48 no.3
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• pp.351-365
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• 2013

In this paper, unified nonlocal shear deformation theory is proposed to study bending, buckling and free vibration of nanobeams. This theory is based on the assumption that the in-plane and transverse displacements consist of bending and shear components in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. In addition, this present model is capable of capturing both small scale effect and transverse shear deformation effects of nanobeams, and does not require shear correction factors. The equations of motion are derived from Hamilton's principle. Analytical solutions for the deflection, buckling load, and natural frequency are presented for a simply supported nanobeam, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory and Reddy beam theories.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.3 no.4
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• pp.109-122
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• 1983

An accurate thick beam element (TB4) which includes the effects of the shear deformation and rotatory inertia has been degenerated from the three dimensional continuum by employing the Timoshenko beam assumptions. The proposed TB4 element has four nodes and two degrees of freedom at each node, totally eight degrees of freedom. The transverse deflection W and plane rotation ${\theta}$ with the cubic interpolation functions are selected as nodal variables. The element characteristics are formulated by discretizing the beam equations of motion, using the Galerkin weighted residual method, and are numerically integrated by the reduced shear integration technique, using the three-point Gauss quadrature with the various shear coefficients. Several numerical examples are analyzed to demonstrate the accuracy and the monotonic convergence behavior of the proposed TB4 beam element. The result indicates that the TB4 element shows the more excellent performance and the monotonic convergence behavior than the other existing Timoshenko beam type elements for the whole range of the beam aspect ratios, in both static and free vibration analyses.

• Geomechanics and Engineering
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• v.14 no.6
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• pp.519-532
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• 2018

In this work, two dimensional (2D) and quasi three-dimensional (quasi-3D) HSDTs are proposed for bending and free vibration investigation of functionally graded (FG) plates using hyperbolic shape function. Unlike the existing HSDT, the proposed theories have a novel displacement field which include undetermined integral terms and contains fewer unknowns. The material properties of the plate is inhomogeneous and are considered to vary continuously in the thickness direction by three different distributions; power-law, exponential and Mori-Tanaka model, in terms of the volume fractions of the constituents. The governing equations which consider the effects of both transverse shear and thickness stretching are determined through the Hamilton's principle. The closed form solutions are deduced by employing Navier method and then fundamental frequencies are obtained by solving the results of eigenvalue problems. In-plane stress components have been determined by the constitutive equations of composite plates. The transverse stress components have been determined by integrating the 3D stress equilibrium equations in the thickness direction of the FG plate. The accuracy of the present formulation is demonstrated by comparisons with the different 2D, 3D and quasi-3D solutions available in the literature.

• Steel and Composite Structures
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• v.18 no.4
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• pp.1063-1081
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• 2015

In this paper, a new nonlocal hyperbolic refined plate model is presented for free vibration properties of functionally graded (FG) plates. This nonlocal nano-plate model incorporates the length scale parameter which can capture the small scale effect. The displacement field of the present theory is chosen based on a hyperbolic variation in the in-plane displacements through the thickness of the nano-plate. By dividing the transverse displacement into the bending and shear parts, the number of unknowns and equations of motion of the present theory is reduced, significantly facilitating structural analysis. The material properties are assumed to vary only in the thickness direction and the effective properties for the FG nano-plate are computed using Mori-Tanaka homogenization scheme. The governing equations of motion are derived based on the nonlocal differential constitutive relations of Eringen in conjunction with the refined four variable plate theory via Hamilton's principle. Analytical solution for the simply supported FG nano-plates is obtained to verify the theory by comparing its results with other available solutions in the open literature. The effects of nonlocal parameter, the plate thickness, the plate aspect ratio, and various material compositions on the dynamic response of the FG nano-plate are discussed.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.11 no.5
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• pp.89-100
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• 2001

The Ritz method Is applied In a three-dimensional (3-D) analysis to obtain accurate frequencies for thick. linearly tapered. annular plates. The method is formulated for annular plates haying any combination of free or fixed boundaries at both Inner and outer edges. Admissible functions for the three displacement components are chosen as trigonometric functions in the circumferential co-ordinate. and a1gebraic polynomials in the radial and thickness co-ordinates. Upper bound convergence of the non-dimensional frequencies to the exact values within at least four significant figures is demonstrated. Comparisons of results for annular plates with linearly varying thickness are made with ones obtained by others using 2-D classical thin place theory. Extensive and accurate ( four significant figures ) frequencies are presented 7or completely free. thick, linearly tapered annular plates haying ratios of average place thickness to difference between outer radius (a) and inner radius (b) radios (h$_{m}$/L) of 0.1 and 0.2 for b/L=0.2 and 0.5. All 3-D modes are included in the analyses : e.g., flexural, thickness-shear. In-plane stretching, and torsional. Because frequency data liven is exact 7o a\ulcorner least four digits. It is benchmark data against which the results from other methods (e.g.. 2-D 7hick plate theory, finite element methods. finite difference methods) and may be compared. Throughout this work, Poisson\`s ratio $\upsilon$ is fixed at 0.3 for numerical calculations.s.

• Wind and Structures
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• v.22 no.4
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• pp.429-454
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• 2016

In this paper, a shear deformation plate theory based on neutral surface position is developed for free vibration analysis of functionally graded material (FGM) plates. The material properties of the FGM plates are assumed to vary through the thickness of the plate by a simple power-law distribution in terms of the volume fractions of the constituents. During manufacture, defects such as porosities can appear. It is therefore necessary to consider the vibration behavior of FG plates having porosities in this investigation. The proposed theory is based on assumption that the in-plane and transverse displacements consist of bending and shear components, in which the bending components do not contribute toward shear forces and, likewise, the shear components do not contribute toward bending moments. The neutral surface position for a functionally graded plate which its material properties vary in the thickness direction is determined. The equation of motion for FG rectangular plates is obtained through Hamilton's principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. Numerical results are presented and the influences of the volume fraction index and porosity volume fraction on frequencies of FGM plates are clearly discussed.

• Steel and Composite Structures
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• v.25 no.6
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• pp.693-704
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• 2017

In this research, a simple hyperbolic shear deformation theory is developed and applied for the bending, vibration and buckling of powerly graded material (PGM) sandwich plate with various boundary conditions. The displacement field of the present model is selected based on a hyperbolic variation in the in-plane displacements across the plate's thickness. By splitting the deflection into the bending and shear parts, the number of unknowns and equations of motion of the present formulation is reduced and hence makes them simple to use. Equations of motion are obtained from Hamilton's principle. Numerical results for the natural frequencies, deflections and critical buckling loads of several types of powerly graded sandwich plates under various boundary conditions are presented. The accuracy of the present formulation is demonstrated by comparing the computed results with those available in the literature. As conclusion, this theory is as accurate as other theories available in the literature and so it becomes more attractive due to smaller number of unknowns.

• International Journal of Aerospace System Engineering
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• v.3 no.1
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• pp.10-16
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• 2016

A trigonometric Layerwise higher order shear deformation theory (TLHSDT) is developed and implemented for free vibration and buckling analysis of laminated composite and sandwich plates by analytical and finite element formulation. The present model assumes parabolic variation of out-plane stresses through the depth of the plate and also accomplish the zero transverse shear stresses over the surface of the plate. Thus a need of shear correction factor is obviated. The present zigzag model able to meet the transverse shear stress continuity and zigzag form of in-plane displacement continuity at the plate interfaces. Hence, botheration of shear correction coefficient is neglected. In the case of analytical method, the governing differential equation and boundary conditions are obtained from the principle of virtual work. For the finite element formulation, an efficient eight noded $C^0$ continuous isoparametric serendipity element is established and employed to examine the dynamic analysis. Like FSDT, the considered mathematical model possesses similar number of variables and which decides the present models computationally more effective. Several numerical predictions are carried out and results are compared with those of other existing numerical approaches.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2005.05a
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• pp.158-163
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• 2005

In this paper the aeroacoustic characteristics of a tilt-rotor system is measured by using a sealed model tilt-rotor. With a microphone array system and the do-dopplerization algorithm, the location and the characteristics of rotor noise are successfully measured. The most of high frequency noise (4kHz) is found to be located at rotor blade tips, but the low frequency tonal noise is dominant on the middle of the rotor blades. The measured tonal noise characteristics are compared to the results of theoretical calculation. At 0.5m distance from the rotor plane, measured and calculated data are relatively well matched regardless of rotating speed and collective pitch angie for the azimuthal angles between $0^{\circ}\;and\;60^{\circ}$. However, the data on the azimuthal angles between $70^{\circ}\;and\;90^{\circ}$ are not quite comparable. In addition, the compared data for far-field case (1.5m) are quite different. This is probably due to the unsteady effect which if not included in the theoretical calculation.

• Smart Structures and Systems
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• v.22 no.3
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• pp.303-314
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• 2018

An original quasi-3D hyperbolic shear deformation theory for simply supported functionally graded plates is proposed in this work. The theory considers both shear deformation and thickness-stretching influences by a hyperbolic distribution of all displacements within the thickness, and respects the stress-free boundary conditions on the upper and lower surfaces of the plate without using any shear correction coefficient. By expressing the shear parts of the in-plane displacements with the integral term, the number of unknowns and equations of motion of the proposed theory is reduced to four as against five in the first shear deformation theory (FSDT) and common quasi-3D theories. Equations of motion are obtained from the Hamilton principle. Analytical solutions for dynamic problems are determined for simply supported plates. Numerical results are presented to check the accuracy of the proposed theory.