• Advances in materials Research
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• v.6 no.2
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• pp.93-128
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• 2017

In this paper, thermal vibration behavior of nanoscale beams made of functionally graded (FG) materials subjected to various types of thermal loading are investigated. A Reddy shear deformation beam theory which captures both the microstructural and shear deformation effects without the need for any shear correction factors is employed. Material properties of FG nanobeam are assumed to be temperature-dependent and vary gradually along the thickness according to the power-law form. The influence of small scale is captured based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. The comparison of the obtained results is conducted with those of nonlocal Euler-Bernoulli beam theory and it is demonstrated that the proposed modeling predict correctly the vibration responses of FG nanobeams. The effects of nonlocal parameter, material graduation, mode number, slenderness ratio and thermal loading on vibration behavior of the nanobeams are studied in detail.

• Structural Engineering and Mechanics
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• v.81 no.1
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• pp.29-37
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• 2022

The present paper deals with the application of one dimensional piezoelectric materials in particular piezo-thermoelastic nanobeam. The generalized piezo-thermo-elastic theory with two temperature and Euler Bernoulli theory with small scale effects using nonlocal Eringen's theory have been used to form the mathematical model. The ends of nanobeam are considered to be simply supported and at a constant temperature. The mathematical model so formed is solved to obtain the non-dimensional expressions for lateral deflection, electric potential, thermal moment, thermoelastic damping and frequency shift. Effect of frequency and nonlocal parameter on the lateral deflection, electric potential, thermal moment with generalized piezothermoelastic theory are represented graphically using the MATLAB software. Comparisons are made with the different theories of thermoelasticity.

• 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.

• Computers and Concrete
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• v.19 no.6
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• pp.701-710
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• 2017

In this study, the thermoelastic beam in modified couple stress theory due to laser source and heat flux is investigated. The beam are heated by a non-Guassian laser pulse and heat flux. The Euler Bernoulli beam theory and the Laplace transform technique are applied to solve the basic equations for coupled thermoelasticity. The simply-supported and isothermal boundary conditions are assumed for both ends of the beam. A general algorithm of the inverse Laplace transform is developed. The analytical results have been numerically analyzed with the help of MATLAB software. The numerically computed results for lateral deflection, thermal moment and axial stress due to laser source and heat flux have been presented graphically. Some comparisons have been shown in figures to estimate the effects of couple stress on the physical quantities. A particular case of interest is also derived. The study of laser-pulse find many applications in the field of biomedical, imaging processing, material processing and medicine with regard to diagnostics and therapy.

• Structural Engineering and Mechanics
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• v.48 no.2
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• pp.195-205
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• 2013

The buckling problem of linearly tapered micro-columns is investigated on the basis of modified strain gradient elasticity theory. Bernoulli-Euler beam theory is used to model the non-uniform micro column. Rayleigh-Ritz solution method is utilized to obtain the critical buckling loads of the tapered cantilever micro-columns for different taper ratios. Some comparative results for the cases of rectangular and circular cross-sections are presented in graphical and tabular form to show the differences between the results obtained by modified strain gradient elasticity theory and those achieved by modified couple stress and classical theories. From the results, it is observed that the differences between critical buckling loads achieved by classical and those predicted by non-classical theories are considerable for smaller values of the ratio of the micro-column thickness (or diameter) at its bottom end to the additional material length scale parameters and the differences also increase due to increasing of the taper ratio.

• Structural Engineering and Mechanics
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• v.65 no.1
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• pp.9-17
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• 2018

This study concerns about calculating exact natural frequencies of frames using a single variable shear deformation theory (SVSDT) which considers the parabolic shear stress distribution across the cross section. Free vibration analyses are performed for multi-bay, multi-storey and multi-bay multi-storey type frame structures. Dynamic stiffness formulations are derived and used to obtain first five natural frequencies of frames. Different beam and column cross sections are considered to reveal their effects on free vibration analysis. The calculated natural frequencies are tabulated with the results obtained using Euler-Bernoulli Beam Theory (EBT) and Timoshenko Beam Theory (TBT). Moreover, the effects of inner and outer columns on natural frequencies are compared for multi-bay frames. Several mode shapes are plotted.

• Wind and Structures
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• v.27 no.6
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• pp.431-438
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• 2018

This paper presents a free vibration analysis of size-dependent functionally graded (FG) nanobeams with all surface effects considerations on the basis of modified couple stress theory. The material properties of FG nanobeam are assumed to vary according to power law distribution. Based on the Euler-Bernoulli beam theory, the modeled nanobeam and its equations of motion are derived using Hamilton's principle. An analytical method is used to discretize the model and the equation of motion. The model is validated by comparing the benchmark results with the obtained results. Results show that the vibration behavior of a nanobeam is significantly influenced by surface density, surface tension and surface elasticity. Also, it is shown that by increasing the beam size, influence of surface effect reduces to zero, and the natural frequency tends to its classical value.

• Advances in materials Research
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• v.11 no.4
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• pp.279-298
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• 2022

In this paper, the important novelty and the defining a physical phenomenon of the resent research is the development of nonlocal stress and strain parameters on the porous sandwich beam with functionally graded materials in the top and bottom face sheets.Also, various beam models including Euler-Bernoulli, Reddy and the generalized formulation of two-variable beam theories are obtained in this research. According to a nonlocal strain elasticity theory, the strain at a reference point in the body is dependent not only on the stress state at that point, but also on the stress state at all of the points throughout the body. Thus, the nonlocal stress-strain elasticity theory is defined that can be actual at micro/nano scales. It can be seen that the critical buckling load and transverse deflection of sandwich beam by considering both nonlocal stress-strain parameters is higher than the nonlocal stress parameter. On the other hands, it is noted that by considering the nonlocal stress-strain parameters simultaneously becomes the actual case.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.30 no.4
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• pp.335-341
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• 2017

In this paper, beam finite elements with higher-order derivatives' continuity are formulated and evaluated for various boundary conditions. All the beam elements are based on Euler-Bernoulli beam theory. These higher-order beam elements are often required to analyze structures by using newly developed higher-order beam theories and/or non-classical beam theories based on nonlocal elasticity. It is however rare to assess the performance of such elements in terms of boundary and loading conditions. To this end, two higher-order beam elements are formulated, in which $C^2$ and $C^3$ continuities of the deflection are enforced, respectively. Three different boundary conditions are then applied to solve beam structures, such as cantilever, simply-support and clamped-hinge conditions. In addition to conventional Euler-Bernoulli beam boundary conditions, the effect of higher-order boundary conditions is investigated. Depending on the boundary conditions, the oscillatory behavior of deflections is observed. Especially the geometric boundary conditions are problematic, which trigger unstable solutions when higher-order deflections are prescribed. It is expected that the results obtained herein serve as a guideline for higher-order derivatives' continuous finite elements.

• Structural Engineering and Mechanics
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• v.64 no.6
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• pp.723-730
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• 2017

This paper studies on dynamic and stability behavior of a clamped-elastically restrained nanobeam under the action of a nonconservative force with an emphasis on the influence of surface properties on divergence and flutter instability. Using the Euler-Bernoulli beam theory incorporating surface effects, a governing equation for a clamped-elastically restrained nanobeam is derived according to Hamilton's principle. The characteristic equation is obtained explicitly and the force-frequency interaction curves are displayed to show the influence of the surface effects, spring stiffness of the elastic restraint end on critical loads including divergence and flutter loads. Divergence and flutter instability transition is analyzed. Euler buckling and stability of Beck's column are some special cases of the present at macroscale.