• 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.66 no.6
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• pp.771-781
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• 2018

This study deals with free vibrations analysis with stretching effect of nanocomposite beams reinforced by single-walled carbon nanotubes (SWCNTs) resting on an elastic foundation. Four different carbon nanotubes (CNTs) distributions including uniform and three types of functionally graded distributions of CNTs through the thickness are considered. The rule of mixture is used to describe the effective material properties of the nanocomposite beams. The significant feature of this model is that, in addition to including the shear deformation effect and stretching effect it deals with only 4 unknowns without including a shear correction factor. The governing equations are derived through using Hamilton's principle. Natural frequencies are obtained for nanocomposite beams. The mathematical models provided in this paper are numerically validated by comparison with some available results. New results of free vibration analyses of CNTRC beams based on the present theory with stretching effect is presented and discussed in details. The effects of different parameters of the beam on the vibration responses of CNTRC beam are discussed.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.11a
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• pp.898-901
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• 2004

Since soil-structure interactions are one of the most important subjects in the structural/foundation engineering, much study concerning the soil-structure interactions had been carried out. One of typical structures related to the soil-structure interactions is the strip foundation which is basically defined as the beam or strip rested on or supported by the soils. At the present time, lack of studies on dynamic problems related to the strip foundations is still found in the literature. From these viewpoint, this paper aims to theoretically investigate dynamics of the circular strip foundations and also to present the practical engineering data for the design purpose. Differential equations governing the free, out-of-plane vibrations of such strip foundations are derived, in which effects of the rotatory and torsional inertias and also shear deformation are included although the warping of the cross-section is excluded. Governing differential equations subjected to the boundary conditions of corresponding end constraints are numerically solved for obtaining the natural frequencies and mode shapes by using the numerical integration technique and the numerical method of non-linear equation.

• Steel and Composite Structures
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• v.45 no.5
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• pp.641-652
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• 2022

Fluid-conveying tubes are widely used to transport oil and natural gas in industries. As an advanced composite material, functionally graded carbon nanotube-reinforced composites (FG-CNTRC) have great potential to empower the industry. However, nonlinear free vibration of the FG-CNTRC fluid-conveying pipe has not been attempted in thermal environment. In this paper, the nonlinear free vibration characteristic of functionally graded nanocomposite fluid-conveying pipe reinforced by single-walled carbon nanotubes (SWNTs) in thermal environment is investigated. The SWCNTs gradient distributed in the thickness direction of the pipe forms different reinforcement patterns. The material properties of the FG-CNTRC are estimated by rule of mixture. A higher-order shear deformation theory and Hamilton's variational principle are employed to derive the motion equations incorporating the thermal and fluid effects. A two-step perturbation method is implemented to obtain the closed-form asymptotic solutions for these nonlinear partial differential equations. The nonlinear frequencies under several reinforcement patterns are presented and discussed. We conduct a series of studies aimed at revealing the effects of the flow velocity, the environment temperature, the inner-outer diameter ratio, and the carbon nanotube volume fraction on the nature frequency.

• Steel and Composite Structures
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• v.35 no.3
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• pp.449-462
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• 2020

This work aims to study effects of the crack and the surface energy on the free longitudinal vibration of axially functionally graded nanorods. The surface energy parameters considered are the surface stress, the surface density, and the surface Lamé constants. The cracked nanorod is modelled by dividing it into two parts connected by a linear spring in which its stiffness is related to the crack severity. The surface and bulk material properties are considered to vary in the length direction according to the power law distribution. Hamilton's principle is implemented to derive the governing equation of motion and boundary conditions. Considering the surface stress causes that the derived governing equation of motion becomes non-homogeneous while this was not the case in works that only the surface density and the surface Lamé constants were considered. To extract the frequencies of nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using the harmonic differential quadrature method. Since the present work considers effects of all the surface energy parameters, it can be claimed that this is a comprehensive work in this regard.

• Structural Engineering and Mechanics
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• v.83 no.6
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• pp.771-788
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• 2022

This research investigates the free vibration of porous advanced composite plates resting on Winkler/Pasternak/ Kerr foundations by using a new hyperbolic quasi three dimensional (quasi-3D) shear deformation theory. The present theory, which does not require shear correction factor, accounts for shear deformation and thickness stretching effects by parabolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the plate. In this work, we consider imperfect FG plates with porosities embedded within elastic Winkler, Pasternak or Kerr foundations. Implementing an analytical approach, the obtained governing equations from Hamilton's principle according to FG plates are derived. The closed form solutions are obtained by using Navier technique, and natural frequencies of FG plates are found, for simply supported plates, by solving the results of eigenvalue problems. A comprehensive parametric study is presented to evaluate effects of the geometry of material, mode numbers, porosity volume fraction, Power-law index and stiffness of foundations parameters on free vibration characteristics of FG plates.

• Steel and Composite Structures
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• v.24 no.1
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• pp.79-91
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• 2017

Free vibrations of steel-concrete composite beams are analyzed by using the dynamic stiffness approach. The coupled equations of motion of the composite beams are derived with help of the Hamilton's principle. The effects of the shear deformation and rotary inertia of the two beams as well as the transverse and axial deformations of the stud connectors are included in the formulation. The dynamic stiffness matrix is developed on the basis of the exact general solutions of the homogeneous governing differential equations of the composite beams. The use of the dynamic stiffness method to determine the natural frequencies and mode shapes of a particular steel-concrete composite beam with various boundary conditions is demonstrated. The accuracy and effectiveness of the present model and formulation are validated by comparison of the present results with the available solutions in literature.

• Architectural research
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• v.16 no.2
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• pp.67-74
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• 2014

Free vibration analysis of thin shells is carried out by using isogeometric approach. For this purpose, a thin shell element based on Kirchhoff-Love shell theory is developed. Non-uniform rational B-spline surface (NURBS) definition is introduced to represent the geometry of shell and also used to derive all terms required in the isogeometric element formulation. Gauss integration rule is used for stiffness and mass matrices. The present shell element is then applied to examine vibrational behaviours of thin plate and shell structures. From numerical results, it is found be that reliable natural frequencies and associated mode shapes of thin shell structures can be predicted by the present isogeometric shell element.

• Magazine of the Korean Society of Agricultural Engineers
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• v.42 no.1
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• pp.105-112
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• 2000

This paper deals with the free vibrations of columns immersed in fluid. The column model is based on the classical Bernoulli-euler theory which neblects the effects of rotatory inerital and shear deformation. The eccentricity and rotatory inertial of the concentrated mass at the top are taken into accuont. In the governing equation for the free vibration of column, thedensity of immersed part was midified to account for theadded fluid mass. The govering differential equations are solved numerically using the corresponding boundary conditions. The lowest four natural frequencies and corresponding mode shapes are calculated over a range of non-dimensional system parameters ; the mas density ration of fluid to column, the ratio of fluid depth to span length, the ratio of tip mass to total column mass, the dimensionless mass moment of inertia, and the eccentricity.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2001.05a
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• pp.875-880
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• 2001

This paper deals with the free vibrations curved beams with elastic springs. Taking into account the effects of rotatory inertia and shear deformation, differential equations governing the free vibrations of such beams are derived, in which each elastic spring is modeled as a discrete Winkler foundation with very short longitudinal length. Differential equations are solved numerically to calculate natural frequencies and mode shapes. In numerical examples, the circular, parabolic, sinusoidal and elliptic curved members are considered. The parametric studies are conducted and the lowest four frequency parameters are reported in tables and figures as the non-dimensional fonns. Also the typical mode shapes are presented.