• Proceedings of the Computational Structural Engineering Institute Conference
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• 2000.10a
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• pp.369-376
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• 2000

For the general loading condition and boundary condition, it is very difficult to obtain closed-form solutions for buckling loads and natural frequencies of thin-walled structures because its behaviour is very complex due to the coupling effect of bending and torsional behaviour. Consequently most of previous finite element formulations introduced approximate displacement fields using shape functions as Hermitian polynomials, isoparametric interpoation function, and so on. The purpose of this study is to calculate the exact displacement field of a thin-walled straight beam element with the non-symmetric cross section and present a consistent derivation of the exact dynamic stiffness matrix. An exact dynamic element stiffness matrix is established from Vlasov's coupled differential equations for a uniform beam element of non-symmetric thin-walled cross section. This numerical technique is accomplished via a generalized linear eigenvalue problem by introducing 14 displacement parameters and a system of linear algebraic equations with complex matrices. The natural frequencies are evaluated for the non-symmetric thin-walled straight beam structure, and the results are compared with available solutions in order to verify validity and accuracy of the proposed procedures.

• Structural Engineering and Mechanics
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• v.33 no.6
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• pp.697-708
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• 2009

Stresses are solved for two parallel cracks in an infinite orthotropic plate during passage of incoming shock stress waves normal to their surfaces. Fourier transformations were used to reduce the boundary conditions with respect to the cracks to two pairs of dual integral equations in the Laplace domain. To solve these equations, the differences in the crack surface displacements were expanded to a series of functions that are zero outside the cracks. The unknown coefficients in the series were solved using the Schmidt method so as to satisfy the conditions inside the cracks. The stress intensity factors were defined in the Laplace domain and were inverted numerically to physical space. Dynamic stress intensity factors were calculated numerically for selected crack configurations.

• Structural Engineering and Mechanics
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• v.62 no.4
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• pp.431-441
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• 2017

In this article, an analytical-numerical approach is presented in order to determine the dynamic response of thin plates resting on multiple elastic point supports with time-varying stiffness. The proposed method is essentially based on transforming a familiar governing partial differential equation into a new solvable system of linear ordinary differential equations. When dealing with time-invariant stiffness, the solution of this system of equations leads to a symmetric matrix, whose eigenvalues determine the natural frequencies of the point-supported plate. Moreover, this method proves to be applicable for any plate configuration with any type of boundary condition. The results, where possible, are verified upon comparison with available values in the literature, and excellent agreement is achieved.

• Proceedings of the KSME Conference
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• 2001.06b
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• pp.355-360
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• 2001

By using Frenet formulation and Timoshenko beam theory, the partial differential equations of motion are derived for a helical spring having a doubly symmetrical cross section subjected to the pre-load axially. These equations of motion are solved to give the dispersion relationship and dynamic stiffness matrix is assembled. Natural frequencies are obtained from the receptance of the system. The results of the dynamic stiffness method are compared with those of the transfer matrix method from published examples and finite element method.

• Structural Engineering and Mechanics
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• v.19 no.5
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• pp.503-517
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• 2005

The parametric dynamic stability of an asymmetric sandwich beam with viscoelastic core on viscoelastic supports at the ends and subjected to an axial pulsating load is investigated. A set of Hill's equations are obtained from the non-dimensional equations of motion by the application of the general Galerkin method. The zones of parametric instability are obtained using Saito-Otomi conditions. The effects of shear parameter, support characteristics, various geometric parameters and excitation force on the zones of instability are investigated.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2000.11a
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• pp.563-567
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• 2000

This research is concerned with the dynamic modeling of solar arrays equipped with strain energy hinges(SEH). It is found from experiments that the SEH has nonlinear dynamic characteristics and complex buckling behavior, which is difficult to explain theoretically. In this paper, we use an equivalent one-dimensional nonlinear torsional spring for the SEH. Assuming that solar panels are rigid, we developed the systematic approach for the derivation of the theoretical model for the solar arrays equipped with the multitudes of the SEH. To this end, the kinematic relation of the displacement vector of each body is derived and then applied to the equations of motion. Lagrangian equations of motion are used for the derivations.

• Advances in materials Research
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• v.9 no.1
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• pp.33-48
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• 2020

With the use of differential quadrature method (DQM), forced vibrations and resonance frequency analysis of functionally graded (FG) nano-size beams rested on elastic substrate have been studied utilizing a shear deformation refined beam theory which contains shear deformations influence needless of any correction coefficient. The nano-size beam is exposed to uniformly-type dynamical loads having partial length. The two parameters elastic substrate is consist of linear springs as well as shear coefficient. Gradation of each material property for nano-size beam has been defined in the context of Mori-Tanaka scheme. Governing equations for embedded refined FG nano-size beams exposed to dynamical load have been achieved by utilizing Eringen's nonlocal differential law and Hamilton's rule. Derived equations have solved via DQM based on simply supported-simply supported edge condition. It will be shown that forced vibrations properties and resonance frequency of embedded FG nano-size beam are prominently affected by material gradation, nonlocal field, substrate coefficients and load factors.

• Steel and Composite Structures
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• v.10 no.2
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• pp.151-168
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• 2010

The subject of the paper is an isotropic metal foam rectangular plate. Mechanical properties of metal foam vary continuously through plate of the thickness. A nonlinear hypothesis of deformation of plane cross section is formulated. The system of partial differential equations of the plate motion is derived on the basis of the Hamilton's principle. The system of equations is analytically solved by the Bubnov-Galerkin method. Numerical investigations of dynamic stability for family rectangular plates with respect analytical solution are performed. Moreover, FEM analysis and theirs comparison with results of numerical-analytical calculations are presented in figures.

• Steel and Composite Structures
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• v.32 no.3
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• pp.293-304
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• 2019

In the present paper, the element free Galerkin (EFG) method is developed for geometrically nonlinear analysis of deep beams considering small scale effect. To interpret the behavior of structure at the nano scale, the higher-order gradient elasticity nonlocal theory is taken into account. The radial point interpolation method with high order of continuity is used to construct the shape functions. The nonlinear equation of motion is derived using the principle of the minimization of total potential energy based on total Lagrangian approach. The Newmark method with the small time steps is used to solve the time dependent equations. At each time step, the iterative Newton-Raphson technique is applied to minimize the residential forces caused by the nonlinearity of the equations. The effects of nonlocal parameter and aspect ratio on stiffness and dynamic parameters are discussed by numerical examples. This paper furnishes a ground to develop the EFG method for large deformation analysis of structures considering small scale effects.

• Advances in aircraft and spacecraft science
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• v.6 no.4
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• pp.297-314
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• 2019

Dynamic stability of a porous metal foam nano-dimension plate on elastic substrate exposed to bi-axial time-dependent forces has been studied via a novel 3-variable plate theory. Various pore contents based on uniform and non-uniform models have been introduced. The presented plate model contains smaller number of field variables with shear deformation verification. Hamilton's principle will be utilized to deduce the governing equations. Next, the equations have been defined in the context of Mathieu-Hill equation. Correctness of presented methodology has been verified by comparison of derived results with previous data. Impacts of static and dynamical force coefficients, non-local coefficient, foundation coefficients, pore distributions and boundary edges on stability regions of metal foam nanoscale plates will be studied.