• Advances in nano research
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• v.13 no.1
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• pp.63-75
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• 2022

The nonlinear dynamic behavior of a nonuniform small-scale nonlocal beam is investigated in this work. The nanobeam is theoretically modeled using the nonlocal Eringen theory, as well as a few of Von-nonlinear Kármán's theories and the classical beam theory. The Hamilton principle extracts partial differential equations (PDE) of an axially functionally graded (AFG) nano-scale beam consisting of SUS304 and Si₃N₄ throughout its length, and an elastic Winkler-Pasternak substrate supports the tapered AFG nanobeam. The beam thickness is a function of beam length, and it constantly varies throughout the length of the beam. The numerical solution strategy employs an iteration methodology connected with the generalized differential quadratic method (GDQM) to calculate the nonlinear outcomes. The nonlinear numerical results are presented in detail to examine the impact of various parameters such as nonlinear amplitude, nonlocal parameter, the component of the elastic foundation, rate of cross-section change, and volume fraction parameter on the linear and nonlinear free vibration characteristics of AFG nanobeam.

• Journal of the Korean Society of Safety
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• v.12 no.3
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• pp.10-16
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• 1997

The problem of sound radiation from curved beam under the action of harmonic line forces is studied. The reaction due to fluid loading on the vibratory response of the curved beam is taken into account. The curved beam is assumed to occupy the plane y=0. The curved beam material and the elastic foundation are assumed to be lossless including a tension force(T), damping coefficient(C) and stiffness of foundation($k_s$) will be employed. The non-dimensional sound power is derived through integration of the surface intensity distribution over the entire curved beam. The expression for sound power is integrated numerically and the results are examined as a function of wavenumber ratio($\gamma$) and stiffness factor($\psi$).

• Journal of the Society of Naval Architects of Korea
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• v.28 no.1
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• pp.92-98
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• 1991

The structures such as railway bridges can be modelled as the multi-span beam on the elastic foundation. These structures are usually subject to the moving load, which has a great effect on dynamic stresses and can cause severe motions, especially at high velocities. In this paper, the dynamic responses of the multi-span beam on the elastic foundation were obtained by using the Galerkin's method and the numerical time integration technique. As trial functions, the same orthogonal polynomial functions obtained in part 1, were used. From the numerical results, it was found that the one term expansion of the assumed solution usually leads to the accurate solutions. However, in the case that the stiffness of the transnational spring is very high or the rotational spring is placed where the slope of the first mode is zero, the higher modes must be included to obtain the accurate solutions.

• Structural Engineering and Mechanics
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• v.61 no.2
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• pp.209-220
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• 2017

The purpose of this paper is to study the geometrically nonlinear free vibration of functionally graded nano/micro beams (FGNBs) based on the modified couple stress theory. For practical applications, some analytical expressions of nonlinear frequencies for FGNBs on a nonlinear Pasternak foundation are developed. Hamilton's principle is employed to obtain nonlinear governing differential equations in the context of both Euler-Bernoulli and Timoshenko beam theories for a comprehensive investigation. The modified continuum theory contains one material length scale parameter to capture the size effect. The variation of two-constituent material along the thickness is modeled using Reddy's power-law. Also, the Mori-Tanaka method as an accurate homogenization technique is implemented to estimate the effective material properties of the FGNBs. The results are presented for both hinged-hinged and clamped-clamped boundary conditions. The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method and then the powerful method of homotopy analysis is utilized to obtain the semi-analytical solutions. Eventually, the presented analytical expressions are used to examine the influences of the length scale parameter, material gradient index, and elastic foundation on the nonlinear free vibration of FGNBs.

• Structural Engineering and Mechanics
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• v.63 no.2
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• pp.171-180
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• 2017

Static and dynamic instability of a viscoelastic carbon nanotube (CNT) embedded on a thermo-elastic foundation are investigated, in this research. The CNT is modeled based on Euler-Bernoulli beam (EBB) and nonlocal small scale elasticity theory is utilized to analyze the structure. Governing equations of the system are derived using Hamilton's principle and differential quadrature (DQ) method is applied to solve the partial differential equations. The effects of variable axial load and diverse boundary conditions on static/vibration instability are studied. To verify the result of the DQ method, the Galerkin weighted residual approach is used for the instability analysis. It is observed appropriate agreement for results of two different solution methods and satisfactory accuracy with those obtained in prior studies. The results of this work could be useful for engineers and designers in order to produce and design nano/micro structures in thermo-elastic medium.

• Structural Engineering and Mechanics
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• v.76 no.3
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• pp.413-420
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• 2020

Considering inverse cotangential shear strain function, the present paper studies nonlinear stability of nonlocal higher-order refined beams made of metal foams based on Chebyshev-Ritz method. Based on inverse cotangential beam model, it is feasible to incorporate shear deformations needless of shear correction factor. Metal foam is supposed to contain different distributions of pores across the beam thickness. Also, presented Chebyshev-Ritz method can provide a unified solution for considering various boundary conditions based on simply-supported and clamped edges. Nonlinear effects have been included based upon von-karman's assumption and nonlinear elastic foundation. The buckling curves are shown to be affected by pore distribution, geometric imperfection of the beam, nonlocal scale factor, foundation and geometrical factors.

• Computers and Concrete
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• v.31 no.3
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• pp.267-276
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• 2023

The main purpose of the current research is to develop an efficient two variables trigonometric shear deformation beam theory to investigate the buckling behavior of symmetric and non-symmetric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beam resting on an elastic foundation with various boundary conditions. The proposed theory obviates the use to shear correction factors as it satisfies the parabolic variation of through-thickness shear stress distribution. The composite beam is made of a polymeric matrix reinforced by aligned and distributed single-walled carbon nanotubes (SWCNTs) with different patterns of reinforcement. The material properties of the FG-CNTRC beam are estimated by using the rule of mixture. The governing equilibrium equations are solved by using new analytical solutions based on the Galerkin method. The robustness and accuracy of the proposed analytical model are demonstrated by comparing its results with those available by other researchers in the existing literature. Moreover, a comprehensive parametric study is presented and discussed in detail to show the effects of CNTs volume fraction, distribution patterns of CNTs, boundary conditions, length-to-thickness ratio, and spring constant factors on the buckling response of FG-CNTRC beam. Some new referential results are reported for the first time, which will serve as a benchmark for future research.

• Magazine of the Korean Society of Agricultural Engineers
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• v.42 no.6
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• pp.122-129
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• 2000

The purpose of this study is to investigate both the fundamental and some higher natural frequencies and mode shapes of compressive members resting on the linear elastic foundation. The model of compressive member is based on the classical Bernoulli-Euler beam theory. The differential equation governing free vibrations of such members subjected to an axial load is derived and solved numerically for calculating the natural frequencies and mode shapes. The Improved Euler method is used to integrate the differential equation and the Determinant Search method combined with the Regula-Falsi method to determine the natural frequencies, respectively. In numerical examples, the hinged-hinged, hinged-clamped, clamped-hinged and clamped-clamped end constraints are considered. The convergence analysis is conducted for determining the available step size in the Improved Euler method. The validation of theories developed herein is also conducted by comparing the numerical results between this study and SAP 90. The non-dimensional frequency parameters are presented as the non-dimensional system parameters: section ratio, modulus parameter and load parameter. Also typical mode shapes are presented.

• Structural Engineering and Mechanics
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• v.55 no.4
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• pp.743-763
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• 2015

This work presents a new nonlocal hyperbolic shear deformation beam theory for the static, buckling and vibration of nanoscale-beams embedded in an elastic medium. The present model is able to capture both the nonlocal parameter and the shear deformation effect without employing shear correction factor. The nonlocal parameter accounts for the small size effects when dealing with nanosize structures such as nanobeams. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanoscale-beam are obtained using Hamilton's principle. The effect of the surrounding elastic medium on the deflections, critical buckling loads and frequencies of the nanobeam is investigated. Both Winkler-type and Pasternak-type foundation models are used to simulate the interaction of the nanobeam with the surrounding elastic medium. Analytical solutions are presented for a simply supported nanoscale-beam, and the obtained results compare well with those predicted by the other nonlocal theories available in literature.

• Journal of the Korean Society for Railway
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• v.16 no.3
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• pp.202-206
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• 2013

A submerged floating railway is an innovative tunnel infrastructure passing through the deep sea independent of wave and wind so that high speed trains can run on it. It doesn't depend on water depth and is cost effective due to modular construction on land. The construction period can be reduced drastically. This paper introduces the concept design of a submerged floating railway, and for securing safety, proposes a method to analyze the structural behavior of the body in case of collision with a submarine. The theory of a beam with an elastic foundation was used to calculate the equivalent mass of the body so that the perfect elastic collision could be applied to calculate the collision velocity. The maximum deformation and bending moment was analyzed based on energy conservation. To verify the results, a collision analysis using a finite element analysis code was made. Comparing the results confirmed that this simplified collision analysis method gives enough accurate deformation and bending moment to be used for actual estimation in the initial design stage.