• Geomechanics and Engineering
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• v.37 no.5
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• pp.453-465
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• 2024

This paper presents a logarithmic shear deformation theory to study the thermal buckling response of power-law FG one-dimensional structures in thermal conditions with different boundary conditions. It is assumed that the functionally graded material and thermal properties are supposed to vary smoothly according to a contentious function across the vertical direction of the beams. A P-FG type function is employed to describe the volume fraction of material and thermal properties of the graded (1D) beam. The Ritz model is employed to solve the thermal buckling problems in immovable boundary conditions. The outcomes of the stability analysis of FG beams with temperature-dependent and independent properties are presented. The effects of the thermal loading are considered with three forms of rising: nonlinear, linear and uniform. Numerical results are obtained employing the present logarithmic theory and are verified by comparisons with the other models to check the accuracy of the developed theory. A parametric study was conducted to investigate the effects of various parameters on the critical thermal stability of P-FG beams. These parameters included support type, temperature fields, material distributions, side-to-thickness ratios, and temperature dependency.

• Steel and Composite Structures
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• v.45 no.1
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• pp.67-82
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• 2022

In this paper, buckling and free vibration of imperfect, functionally graded beams, including porosities, are investigated, using a higher order shear strain theory. Due to defects during the manufacturing process, micro porosities may appear in the material, hence the appearance of this imperfection in the structure. The material properties of the beams are assumed to vary regularly, with power and sigmoid law, in the direction of thickness. A novel porosity distribution affecting the functionally graded volume fraction is presented. For the compact formulation used for cementite-based materials and already used in P-FGM, we have adapted it for the distribution of S-FGM. The equations of motion in the FG beam are derived using Hamilton's principle. The boundary conditions for beam FG are assumed to be simply supported. Navier's solution is used to obtain the closed form solutions of the FG beam. The numerical results of this work are compared with those of other published research to verify accuracy and reliability. The comparisons of different shear shape functions, the influence of porosity, thickness and inhomogeneity parameters on buckling and free vibration of the FG beam are all discussed. It is established that the present work is more precise than certain theories developed previously.

• Advances in nano research
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• v.4 no.2
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• pp.85-111
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• 2016

In this paper, the classical and non-classical boundary conditions effect on free vibration characteristics of functionally graded (FG) size-dependent nanobeams are investigated by presenting a semi analytical differential transform method (DTM) for the first time. Three kinds of mathematical models, namely; power law (P-FGM), sigmoid (S-FGM) and Mori-Tanaka (MT-FGM) distribution are considered to describe the material properties in the thickness direction. The nonlocal Eringen theory takes into account the effect of small size, which enables the present model to become effective in the analysis and design of nanosensors and nanoactuators. Governing equations are derived through Hamilton's principle and they are solved applying semi analytical differential transform method. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as small scale effects, spring constant factors, various material compositions and mode number on the normalized natural frequencies of the FG nanobeams in detail. It is explicitly shown that the vibration of FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

• Advances in nano research
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• v.8 no.4
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• pp.293-305
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• 2020

In the current research, the free vibrational behavior of the FG nano-beams integrated in the hygro-thermal environment and reposed on the elastic foundation is investigated using a novel integral Timoshenko beam theory (ITBT). The current model has only three variables unknown and requires the introduction of the shear correction factor because her uniformed variation of the shear stress through the thickness. The effective properties of the nano-beam vary according to power-law and symmetric sigmoid distributions. Three models of the hygro-thermal loading are employed. The effect of the small scale effect is considered by using the nonlocal theory of Eringen. The equations of motion of the present model are determined and resolved via Hamilton principle and Navier method, respectively. Several numerical results are presented thereafter to illustrate the accuracy and efficiency of the actual integral Timoshenko beam theory. The effects of the various parameters influencing the vibrational responses of the P-FG and SS-FG nano-beam are also examined and discussed in detail.

• Structural Engineering and Mechanics
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• v.71 no.2
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• pp.185-196
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• 2019

In this investigation, study of the static and dynamic behaviors of functionally graded beams (FGB) is presented using a hyperbolic shear deformation theory (HySDT). The simply supported FG-beam is resting on the elastic foundation (Winkler-Pasternak types). The properties of the FG-beam vary according to exponential (E-FGB) and power-law (P-FGB) distributions. The governing equations are determined via Hamilton's principle and solved by using Navier's method. To show the accuracy of this model (HySDT), the current results are compared with those available in the literature. Also, various numerical results are discussed to show the influence of the variation of the volume fraction of the materials, the power index, the slenderness ratio and the effect of Winkler spring constant on the fundamental frequency, center deflection, normal and shear stress of FG-beam.

• Steel and Composite Structures
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• v.30 no.6
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• pp.603-615
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• 2019

In the present work, free vibration of beams made of imperfect functionally graded materials (FGMs) including porosities is investigated. Because of faults during process of manufacture, micro voids or porosities may arise in the FGMs, and this situation causes imperfection in the structure. Therefore, material properties of the beams are assumed to vary continuously through the thickness direction according to the volume fraction of constituents described with the modified rule of mixture including porosity volume fraction which covers two types of porosity distribution over the cross section, i.e., even and uneven distributions. The governing equations of power law FGM (P-FGM) and sigmoid law FGM (S-FGM) beams are derived within the frame works of classical beam theory (CBT) and first order shear deformation beam theory (FSDBT). The resulting equations are solved using separation of variables technique and assuming FG beams are simply supported at both ends. To validate the results numerous comparisons are carried out with available results of open literature. The effects of types of volume fraction function, beam theory and porosity volume fraction, as well as the variations of volume fraction index, span to depth ratio and porosity volume fraction, on the first three non-dimensional frequencies are examined in detail.