• Structural Engineering and Mechanics
• /
• v.59 no.3
• /
• pp.579-599
• /
• 2016

In this study, static bending of edge cracked micro beams is studied analytically under uniformly distributed transverse loading based on modified couple stress theory. The cracked beam is modelled using a proper modification of the classical cracked-beam theory consisting of two sub-beams connected through a massless elastic rotational spring. The deflection curve expressions of the edge cracked microbeam segments separated by the rotational spring are determined by the Integration method. The elastic curve functions of the edge cracked micro beams are obtained in explicit form for cantilever and simply supported beams. In order to establish the accuracy of the present formulation and results, the deflections are obtained, and compared with the published results available in the literature. Good agreement is observed. In the numerical study, the elastic deflections of the edge cracked micro beams are calculated and discussed for different crack positions, different lengths of the beam, different length scale parameter, different crack depths, and some typical boundary conditions. Also, the difference between the classical beam theory and modified couple stress theory is investigated for static bending of edge cracked microbeams. It is believed that the tabulated results will be a reference with which other researchers can compare their results.

• Advances in nano research
• /
• v.4 no.4
• /
• pp.251-264
• /
• 2016

In this paper, a simple and refined nonlocal hyperbolic higher-order beam theory is proposed for bending and vibration response of nanoscale beams. The present formulation incorporates the nonlocal scale parameter which can capture the small scale effect, and it considers both shear deformation and thickness stretching effects by a hyperbolic variation of all displacements across the thickness without employing shear correction factor. The highlight of this formulation is that, in addition to modeling the displacement field with only two unknowns, the thickness stretching effect (${\varepsilon}_z{\neq}0$) is also included in the present model. By utilizing the Hamilton's principle and the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanoscale beam are reformulated. Verification studies demonstrate that the developed theory is not only more accurate than the refined nonlocal beam theory, but also comparable with the higher-order shear deformation theories which contain more number of unknowns. The theoretical formulation proposed herein may serve as a reference for nonlocal theories as applied to the static and dynamic responses of complex-nanobeam-system such as complex carbon nanotube system.

• Structural Engineering and Mechanics
• /
• v.75 no.6
• /
• pp.737-746
• /
• 2020

This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

• Earthquakes and Structures
• /
• v.13 no.3
• /
• pp.255-265
• /
• 2017

In this paper, an efficient shear deformation theory is developed for wave propagation analysis in a functionally graded beam. More particularly, porosities that may occur in Functionally Graded Materials (FGMs) during their manufacture are considered. The proposed shear deformation theory is efficient method because it permits us to show the effect of both bending and shear components and this is carried out by dividing the transverse displacement into the bending and shear parts. Material properties are assumed graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents; but the rule of mixture is modified to describe and approximate material properties of the functionally graded beams with porosity phases. The governing equations of the wave propagation in the functionally graded beam are derived by employing the Hamilton's principle. The analytical dispersion relation of the functionally graded beam is obtained by solving an eigenvalue problem. The effects of the volume fraction distributions, the depth of beam, the number of wave and the porosity on wave propagation in functionally graded beam are discussed in details. It can be concluded that the present theory is not only accurate but also simple in predicting the wave propagation characteristics in the functionally graded beam.

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.18 no.11
• /
• pp.1157-1169
• /
• 2008

This paper presents spectral element formulation which approximates Lamb wave propagation by PZT transducers bonded on a thin plate. A two layer beam model under 2-D plane strain condition is introduced to simulate high-frequency dynamic responses induced by a piezoelectric (PZT) layer rigidly bonded on a base plate. Mindlin-Herrmann and Timoshenko beam theories are employed to represent the first symmetric and anti-symmetric Lamb wave modes on a base plate, respectively. The Euler-Bernoulli beam theory and 1-D linear piezoelectricity are used to model the electro-mechanical behavior of a PZT layer. The equations of motions of a two layer beam model are derived through Hamilton's principle. The necessary boundary conditions associated with the electro-mechanical properties of a PZT layer are formulated in the context of dual functions of a PZT layer as an actuator and a sensor. General spectral shape functions of response field and the associated boundary conditions are obtained through equations of motions converted into frequency domain. Detailed spectrum element formulation for composing the dynamic stiffness matrix of a two layer beam model is presented as well. The validity of the proposed spectral element is demonstrated through numerical examples.

• Steel and Composite Structures
• /
• v.21 no.5
• /
• pp.999-1016
• /
• 2016

This study presents free vibration of beam made of porous material. The mechanical properties of the beam is variable in the thickness direction and the beam is investigated in three situations: poro/nonlinear nonsymmetric distribution, poro/nonlinear symmetric distribution, and poro/monotonous distribution. First, the governing equations of porous beam are derived using principle of virtual work based on Euler-Bernoulli theory. Then, the effect of pores compressibility on natural frequencies of the beam is studied by considering clamped-clamped, clamped-free and hinged-hinged boundary conditions. Moreover, the results are compared with homogeneous beam with the same boundary conditions. Finally, the effects of poroelastic parameters such as pores compressibility, coefficients of porosity and mass on natural frequencies has been considered separately and simultaneously.

• Transactions of the Korean Society of Mechanical Engineers
• /
• v.18 no.8
• /
• pp.2123-2138
• /
• 1994

A study on the dynamic mechanical properties of the high strength carbon fiber epoxy composite beam was carried out. The macromechanical model was used for the theoretical analysis of the symmetric laminated composite beam. The anisotropic plate theory and Bernoulli-Euler beam theory were used to predict the effective flexural elastic modulus and the specific damping capacity of laminated composite beam. The free flexural vibration and torsional vibration tests were carried out to determine the specific damping capacities of the unidirectional laminated composite beam. The vibration tests were performed in a vacuum chamber with laser vibrometer system and electromagnetic hammer to obtain accurate experimental data. From the computational and experimental results, it was found that the theoretical values with the macromechanical analysis and the experimental data of symmetric laminated composite beam were in good agreement.

• Steel and Composite Structures
• /
• v.26 no.6
• /
• pp.733-743
• /
• 2018

This paper presents post-buckling responses of a simply supported laminated composite beam subjected to a non-follower axially compression loads. In the nonlinear kinematic model of the laminated beam, total Lagrangian approach is used in conjunction with the Timoshenko beam theory. In the solution of the nonlinear problem, incremental displacement-based finite element method is used with Newton-Raphson iteration method. There is no restriction on the magnitudes of deflections and rotations in contradistinction to von-Karman strain displacement relations of the beam. The distinctive feature of this study is post-buckling analysis of Timoshenko Laminated beams full geometric non-linearity and by using finite element method. The effects of the fibber orientation angles and the stacking sequence of laminates on the post-buckling deflections, configurations and stresses of the composite laminated beam are illustrated and discussed in the numerical results. Numerical results show that the above-mentioned effects play a very important role on the post-buckling responses of the laminated composite beams.

• Transactions of the Korean Society for Noise and Vibration Engineering
• /
• v.13 no.9
• /
• pp.720-729
• /
• 2003

An iterative modal analysis approach is developed to determine the effect of transverse open cracks on the dynamic behavior of simply supported Euler-Bernoulli beam with the moving mass. The influences of the depth and the position of the crack in the beam have been studied on the dynamic behavior of the simply supported beam system by numerical method. The crack section is represented by a local flexibility matrix connecting two undamaged beam segments i.e. the crack is modelled as a rotational spring. This flexibility matrix defines the relationship between the displacements and forces across the crack section and is derived by applying fundamental fracture mechanics theory. As the depth of the crack is increased the frequency of the simply supported beam with the moving mass is increased.

• Advances in materials Research
• /
• v.8 no.3
• /
• pp.219-235
• /
• 2019

This research is devoted to analyzing mechanical-thermal post-buckling behavior of a micro-size beam reinforced with graphene platelets (GPLs) based on geometric imperfection effects. Graphene platelets have three types of dispersion within the structure including uniform-type, linear-type and nonlinear-type. The micro-size beam is considered to be perfect (ideal) or imperfect. Buckling mode shape of the micro-size beam has been assumed as geometric imperfection. Modified couple stress theory has been used for describing scale-dependent character of the beam having micro dimension. Via an analytical procedure, post-buckling path of the micro-size beam has been derived. It will be demonstrated that nonlinear buckling characteristics of the micro-size beam are dependent on geometric imperfection amplitude, thermal loading, graphene distribution and couple stress effects.