• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.11a
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• pp.709-712
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• 2004

A study of the coupled flexural-torsional vibrations of thin-walled beams with monosymmetric cross-section is presented. The governing differential equations for free vibration of such beams are solved numerically to obtain natural frequencies and their corresponding mode shapes. The beam model is based on the Bernoulli-Euler beam theory and the effect of warping is taken into consideration. Numerical results are given for two specific examples of beams with free-free, clamped-free, hinged-hinged, clamped-hinged and clamped-clamped end constraints both including and excluding the effect of warping stiffness. The effect of warping stiffness on the natural frequencies and mode shapes is discussed and it is concluded that substantial error can be incurred if the effect is ignored.

• Steel and Composite Structures
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• v.32 no.6
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• pp.795-806
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• 2019

In this paper, free vibration of sandwich beam with flexible core resting on orthotropic Pasternak is investigated. The top and bottom layers are reinforced by carbon nanotubes (CNTs). This sandwich structural is modeled by Euler and Frostig theories. The effect of agglomeration using Mori-Tanaka model is considered. The Eringen's theory is applied for size effect. The structural damping is investigated by Kelvin-voigt model. The motion equations are calculated by Hamilton's principle and energy method. Using analytical method, the frequency of the structure is obtained. The effect of agglomeration and CNTs volume percent for different parameter such as damping of structure, thickens and spring constant of elastic medium are presented on the frequency of the composite structure. Results show that with increasing CNTs agglomeration, frequency is decreased.

• Steel and Composite Structures
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• v.46 no.3
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• pp.335-344
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• 2023

This investigation presents nonlinear free vibration of a carbon nanotube reinforced composite beam based on the Von Kármán nonlinearity and the Euler-Bernoulli beam theory The material properties of the structure is considered as made of a polymeric matrix by reinforced carbon nanotubes according to different material distributions. The governing equations of the nonlinear vibration problem is delivered by using Hamilton's principle and the Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The nonlinear natural frequency and the nonlinear free response of the system is obtained with the effect of different patterns of reinforcement.

• Steel and Composite Structures
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• v.19 no.2
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• pp.369-384
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• 2015

This work presents a free vibration analysis of functionally graded metal-ceramic (FG) beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a simple displacement field based on higher order shear deformation theory is implemented. The proposed theory is based on the assumption that the 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 most interesting feature of this theory is that it accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. In addition, it has strong similarities with Euler-Bernoulli beam theory in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. By employing the Hamilton's principle, governing equations of motion for coupled axial-shear-flexural response are determined. The validity of the present theory is investigated by comparing some of the present results with those of the first-order and the other higher-order theories reported in the literature. Illustrative examples are given also to show the effects of varying gradients, porosity volume fraction, aspect ratios, and thickness to length ratios on the free vibration of the FG beams.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.28 no.5
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• pp.459-465
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• 2015

There exist different kinds of aircrafts, such as conventional airplane, rotorcraft, fighter, and unmanned aerial vehicle. Their shape and feature are dependent upon their own assigned mission. One of the fundamental analyses performed during the aircraft design is the structural analysis. It becomes more complicated and requires severe computations because of the recent complex trends in aircraft structure. In order for efficiency in the structural analysis, a simplified approach, such as equivalent beam or plate model, is preferred. However, it is not clear which analysis will be appropriate to analyze the realistic configuration, such as an aircraft wing, i.e., between an equivalent beam and plate analysis. It is necessary to assess the limitation for both the one-dimensional beam analysis and the two-dimensional plate theory. Thus, in this paper, the static structural analysis results obtained by EDISON solvers were compared with the three-dimensional results obtained from MSC NASTRAN. Before that, EDISON program was verified by comparing the results with those from MSC NASTRAN program and other analytic solutions.

• Structural Engineering and Mechanics
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• v.72 no.2
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• pp.217-227
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• 2019

The methodology known as "shape sensing" allows the reconstruction of the displacement field of a structure starting from strain measurements, with considerable implications for structural monitoring, as well as for the control and implementation of smart structures. An approach to shape sensing is based on the inverse Finite Element Method (iFEM) that uses a variational principle enforcing a least-squares compatibility between measured and analytical strain measures. The structural response is reconstructed without the knowledge of the mechanical properties and load conditions but based only on the relationship between displacements and strains. In order to efficiently apply iFEM to the most common structural typologies of civil engineering, its formulation according to the kinematical assumptions of the Bernoulli-Euler theory is presented. Two beam inverse finite elements are formulated for different loading conditions. Depending on the type of element, the relationship between the minimum number of required measurement stations and the interpolation order is defined. Several examples representing common applications of civil engineering and involving beams and frames are presented. To simulate the experimental strain data at the station points and to verify the accuracy of the displacements obtained with the iFEM shape sensing procedure, a direct FEM analysis of the considered structures is performed using the LUSAS software.

• Structural Engineering and Mechanics
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• v.23 no.1
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• pp.15-27
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• 2006

A locally varying temperature field or a mixture of two or more different materials can cause local variation of elasticity properties of a beam. In this paper, a new Euler-Bernoulli beam element with varying Young's modulus along its longitudinal axis is presented. The influence of axial forces according to the linearized 2nd order beam theory is considered, as well. The stiffness matrix of this element contains the transfer constants which depend on Young's modulus variation and on axial forces. Occurrence of the polynomial variation of Young's modulus has been assumed. Such approach can be also used for smooth local variation of Young's modulus. The critical loads of the straight slender columns were studied using the new beam element. The influence of position of the local Young's modulus variation and its type (such as linear, quadratic, etc.) on the critical load value and rate of convergence was investigated. The obtained results based on the new beam element were compared with ANSYS solutions, where the number of elements gradually increased. Our results show significant influence of the locally varying Young's modulus on the critical load value and the convergence rate.

• Steel and Composite Structures
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• v.44 no.5
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• pp.721-740
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• 2022

In this paper, an accurate kinematic model has been developed to study the mechanical response of functionally graded (FG) sandwich beams, mainly covering the bending, buckling and free vibration problems. The studied structure with homogeneous hardcore and softcore is considered to be simply supported in the edges. The present model uses a new refined shear deformation beam theory (RSDBT) in which the displacement field is improved over the other existing high-order shear deformation beam theories (HSDBTs). The present model provides good accuracy and considers a nonlinear transverse shear deformation shape function, since it is constructed with only two unknown variables as the Euler-Bernoulli beam theory but complies with the shear stress-free boundary conditions on the upper and lower surfaces of the beam without employing shear correction factors. The sandwich beams are composed of two FG skins and a homogeneous core wherein the material properties of the skins are assumed to vary gradually and continuously in the thickness direction according to the power-law distribution of volume fraction of the constituents. The governing equations are drawn by implementing Hamilton's principle and solved by means of the Navier's technique. Numerical computations in the non-dimensional terms of transverse displacement, stresses, critical buckling load and natural frequencies obtained by using the proposed model are compared with those predicted by other beam theories to confirm the performance of the proposed theory and to verify the accuracy of the kinematic model.

• 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.

• Advances in nano research
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• v.7 no.6
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• pp.379-390
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• 2019

Putting emphasis on the effect of existence of porosity in the functionally graded materials (FGMs) on the dynamic responses of waves scattered in FG nanobeams resulted in implementation of a novel porosity-based homogenization method for FGMs and show its applicability in a wave propagation problem in the presence of axial pre-load for the first time. In the employed porosity-dependent method, the coupling between density and Young's moduli is included to consider for the effective moduli of the FG nanobeam by the means of a more reliable homogenization technique. The beam-type element will be modeled via the classical theory of beams, namely Euler-Bernoulli beam theory. Also, the dynamic form of the principle of virtual work will be extended for such nanobeams to derive the motion equations. Applying the nonlocal constitutive equations of Eringen on the obtained motion equations will be resulted in derivation of the nanobeam's governing equations. Depicted results reveal that the dispersion responses of FG nanobeams will be decreased as the porosity volume fraction is increased which must be noticed by the designers of advanced nanosize devices who are interested in employment of wave dispersion approach in continuous systems for specific goals.