• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.1031-1042
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• 2000

The equations prescribed in Ω⊂R(sup)n are called loaded, if they contain some operations of the traces of desired solution on manifolds (of dimension which is strongly less than n) from closure Ω. These equations result from approximations of nonlinear equations by linear ones, in the problems of optimal control when the control when the control actions depends on a part of independent variables, in investigations of the inverse problems and so on. In present work we study the nonlocal boundary value problems for first-order loaded differential operator equations. Criterion of unique solvability is established. We illustrate the obtained results by examples.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2000.06a
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• pp.712-717
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• 2000

The main purpose of this paper is to present both the fundamental and some higher natural frequencies of axially loaded Timoshenko beams resting on the elastic foundation. The non-dimensional differential equation governing the free vibrations of such beam is derived in which the effects of rotatory inertia and shear deformation are included. The Improved Euler method and Determinant Search method are used to perform the integration of the differential equation and to determine the natural frequencies, respectively. The hinged-hinged, hinged-clamped and clamped-clamped end constraints are applied in numerical examples. The relations between frequency parameters and both the foundation parameter and slenderness ratio are presented in figures. The effect of cross-sectional shapes is also investigated.

• Steel and Composite Structures
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• v.14 no.1
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• pp.73-83
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• 2013

In this paper Hamiltonian Approach (HA) have been used to analysis the nonlinear free vibration of Simply-Supported (S-S) and for the Clamped-Clamped (C-C) Euler-Bernoulli beams fixed at one end subjected to the axial loads. First we used Galerkin's method to obtain an ordinary differential equation from the governing nonlinear partial differential equation. The effect of different parameter such as variation of amplitude to the obtained on the non-linear frequency is considered. Comparison of HA with Runge-Kutta 4th leads to highly accurate solutions. It is predicted that Hamiltonian Approach can be applied easily for nonlinear problems in engineering.

• Structural Engineering and Mechanics
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• v.31 no.4
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• pp.453-475
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• 2009

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams on elastic soil is plenty, but the free vibration analysis of Reddy-Bickford beams on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded Reddy-Bickford beam on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton's principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of one end fixed and the other end simply supported Reddy-Bickford beam on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed and the mode shapes are presented in graphs.

• Structural Engineering and Mechanics
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• v.28 no.2
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• pp.129-152
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• 2008

This paper considers the solution of the stochastic differential equations (SDEs) with random operator and/or random excitation using the spectral SFEM. The random system parameters (involved in the operator) and the random excitations are modeled as second order stochastic processes defined only by their means and covariance functions. All random fields dealt with in this paper are continuous and do not have known explicit forms dependent on the spatial dimension. This fact makes the usage of the finite element (FE) analysis be difficult. Relying on the spectral properties of the covariance function, the Karhunen-Loeve expansion is used to represent these processes to overcome this difficulty. Then, a spectral approximation for the stochastic response (solution) of the SDE is obtained based on the implementation of the concept of generalized inverse defined by the Neumann expansion. This leads to an explicit expression for the solution process as a multivariate polynomial functional of a set of uncorrelated random variables that enables us to compute the statistical moments of the solution vector. To check the validity of this method, two applications are introduced which are, randomly loaded simply supported reinforced concrete beam and reinforced concrete cantilever beam with random bending rigidity. Finally, a more general application, randomly loaded simply supported reinforced concrete beam with random bending rigidity, is presented to illustrate the method.

• Structural Engineering and Mechanics
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• v.35 no.2
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• pp.191-203
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• 2010

To investigate the critical buckling load and post-buckling behavior of an axially loaded pile entirely embedded in soil, the non-linear large deflection differential equation for a pinned pile, based on the Winkler-model and the discretionary distribution function of the foundation coefficient along pile shaft, was established by energy method. Assuming that the deflection function was a power series of some perturbation parameter according to the boundary condition and load in the pile, the non-linear large deflection differential equation was transformed to a series of linear differential equations by using perturbation approach. By taking the perturbation parameter at middle deflection, the higher-order asymptotic solution of load-deflection was then found. Effect of ratios of soil depth to pile length, and ratios of pile stiffness to soil stiffness on the critical buckling load and performance of piles (entirely embedded and partially embedded) after flexural buckling were analyzed. Results show that the buckling load capacity increases as the ratios of pile stiffness to soil stiffness increasing. The pile performance will be more stable when ratios of soil depth to pile length, and soil stiffness to pile stiffness decrease.

• Structural Engineering and Mechanics
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• v.60 no.3
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• pp.455-470
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• 2016

In this paper, we investigate the free vibration of axially loaded non-uniform Rayleigh cantilever beams. The Rayleigh beams account for the rotary inertia effect which is ignored in Euler-Bernoulli beam theory. Using an inverse problem approach we show, that for certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation for Rayleigh beams. The derived property variation can serve as test functions for numerical methods. For the rotating beam case, the results have been compared with those derived using the Euler-Bernoulli beam theory.

• Journal of the Korean Society of Industry Convergence
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• v.5 no.1
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• pp.45-52
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• 2002

In this study, an exact solution of governing differential equation for the bending problem of partially loaded orthotropic rectangular plates is presented and also its computer program is developed. The method requires that two opposite edges be clamped or simply supported, or one edge clamped and the other simply supported. Any combination of boundary conditions could exist along the other edges. The plate could he subjected to uniform, partially uniform, and line loads. The solution for the deflection of rectangular plate is expressed as a Levy type single Fourier series and the loads arc expressed as a corresponding series. The advantage of the solution is that it overcomes the limitations of the previous Navier's and Levy's methods (limitation of boundary condition and loading conditions of plate), it is easy to program on a computer and it becomes fast to solve the bending problem with computer program. Calculations are presented for isotropic and orthotropic plates with different loading and boundary conditions. Comparisons are made for the isotropic plate with various boundary conditions between the result of this paper and the result of Navier, Levy and Szilard. The deflections were in excellent agreement.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.14 no.1
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• pp.56-63
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• 2004

Exact solutions are presented for the free vibration and buckling of rectangular plates haying two opposite edges ( x=0 and a) simply supported and the other two ( y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress $\sigma$$_{x}$=- $N_{0}$[1-a(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement ( w) to vary as sin(m$\pi$x/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients. for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and byields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters a= 0, 0.5, 1, 1.5. 2, for which a=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for a= 0, 1, 2 obtained by the method of integration of the differential equation (a=0) or the method of energy (a=1, 2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b =0.5, 1, 2 when a=2, with load intensities $N_{0}$ / $N_{cr}$ =0, 0.5, 0.8, 0.95, 1. where $N_{cr}$ is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes ate also shown.shown.

• Smart Structures and Systems
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• v.14 no.2
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• pp.225-246
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• 2014

The present paper develops piezo-thermo-elastic analysis of a thick spherical shell for generalized functionally graded piezoelectric material. The assumed structure is loaded under thermal, electrical and mechanical loads. The mechanical, thermal and electrical properties are graded along the radial direction based on a power function with three different non homogenous indexes. Primarily, the non homogenous heat transfer equation is solved by applying the general boundary conditions, individually. Substitution of stress, strain, electrical displacement and material properties in equilibrium and Maxwell equations present two non homogenous differential equation of order two. The main objective of the present study is to improve the relations between mechanical and electrical loads in hollow spherical shells especially for functionally graded piezoelectric materials. The obtained results can evaluate the effect of every non homogenous parameter on the mechanical and electrical components.