• Geomechanics and Engineering
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• v.9 no.3
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• pp.361-372
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• 2015

In this paper, a refined exponential shear deformation theory for free vibration analysis of functionally graded beam with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. For this purpose, a new displacement field based on refined shear deformation theory is implemented. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. Based on the present refined shear deformation beam theory, the equations of motion are derived from Hamilton's principle. The rule of mixture is modified to describe and approximate material properties of the FG beams with porosity phases. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions. 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.

• Structural Engineering and Mechanics
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• v.32 no.6
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• pp.811-833
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• 2009

In the present article bending, buckling and vibration analyses of tapered beams using Eringen non-local elasticity theory are being carried out. The associated governing differential equations are solved employing Rayleigh-Ritz method. Both Euler-Bernoulli and Timoshenko beam theories are considered in the analyses. Present results are in good agreement with those reported in literature. Beam material is considered to be made up of functionally graded materials (fgms). Non-local analyses for tapered beam with simply supported - simply supported, clamped - simply supported and clamped - free boundary conditions are carried out and discussed. Further, effect of length to height ratio on maximum deflections, vibration frequencies and critical buckling loads are studied.

• Coupled systems mechanics
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• v.2 no.3
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• pp.215-230
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• 2013

This paper investigates the pull-in instability of functionally graded poly-SiGe micro-beams under the combined electrostatic and intermolecular forces and temperature change. The exponential distribution model and Voigt model are used to analyze the functionally graded materials (FGMs). Principle of virtual work is used to derive the nonlinear governing differential equation which is then solved using differential quadrature method (DQM). A parametric study is conducted to show the significant effects of material composition, geometric nonlinearity, temperature change and intermolecular Casimir force.

• Advances in nano research
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• v.7 no.5
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• pp.351-364
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• 2019

In this work, dynamic behavior of functionally graded (FG) porous nano-beams is studied based on nonlocal nth-order shear deformation theory which takes into the effect of shear deformation without considering shear correction factors. It has been observed that during the manufacture of "functionally graded materials" (FGMs), micro-voids and porosities can occur inside the material. Thus, in this work, the investigation of the dynamic analysis of FG beams taking into account the influence of these imperfections is established. Material characteristics of the FG beam are supposed to be vary continuously within thickness direction according to a "power-law scheme" which is modified to approximate material characteristics for considering the influence of porosities. A comparative study with the known results in the literature confirms the accuracy and efficiency of the current nonlocal nth-order shear deformation theory.

• Advances in nano research
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• v.10 no.3
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• pp.281-293
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• 2021

This paper presents a new nonlocal Hyperbolic Shear Deformation Beam Theory (HSDBT) for the free vibration of porous Functionally Graded (FG) nanobeams. A new displacement field containing integrals is proposed which involves only three variables. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect and its account for shear deformation by a hyperbolic variation of all displacements through the thickness without using the shear correction factor. It has been observed that during the manufacture of Functionally Graded Materials (FGMs), micro-voids and porosities can occur inside the material. Thus, in this work, the investigation of the free vibration analysis of FG beams taking into account the influence of these imperfections is established. Four different porosity types are considered for FG nanobeam. Material characteristics of the FG beam are supposed to vary continuously within thickness direction according to a power-law scheme which is modified to approximate material characteristics for considering the influence of porosities. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the nanobeam are derived using Hamilton's principle. The effects of nonlocal parameter, aspect ratio, and the porosity types on the dynamic responses of the nanobeam are discussed.

• Smart Structures and Systems
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• v.26 no.2
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• pp.253-262
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• 2020

In this paper, a new higher order shear deformation model is developed for static analysis of functionally graded beams with considering porosities that may possibly occur inside the functionally graded materials (FGMs) during their fabrication. The model account for higher-order variation of transverse shear strain through the depth of the beam and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. The present work aims to study the effect of the distribution forms of porosity on the bending of simply supported FG beam. Based on the present higher-order shear deformation model, the equations of motion are derived by the principle of virtual works. Navier type solution method was used to obtain displacement and stresses, and the numerical results are compared with those available in the literature. A comprehensive parametric study is carried out to assess the effects of volume fraction index, porosity fraction index, and geometry on the bending of imperfect FG beams. It can be concluded that the proposed model is simple and precise for the resolution of the behavior of flexural FGM beams while taking into account the shape of distribution of the porosity.

• Structural Engineering and Mechanics
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• v.76 no.1
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• pp.115-127
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• 2020

This study presents an analytical approach to investigate the thermodynamic behavior of functionally graded beam resting on elastic foundations. The formulation is based on a refined deformation theory taking into consideration the stretching effect and the type of elastic foundation. The displacement field used in the present refined theory contains undetermined integral forms and involves only three unknowns to derive. The mechanical characteristics of the beam are assumed to be varied across the thickness according to a simple exponential law distribution. The beam is supposed simply supported and therefore the Navier solution is used to derive analytical solution. Verification examples demonstrate that the developed theory is very accurate in describing the response of FG beams subjected to thermodynamic loading. Numerical results are carried out to show the effects of the thermodynamic loading on the response of FG beams resting on elastic foundation.

• Geomechanics and Engineering
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• v.24 no.6
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• pp.545-557
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• 2021

Since the functionally graded materials (FGMs) are used extensively as thermal barriers in many of applications. Therefore, the current article focuses on studying and presenting dynamic responses of multilayer functionally graded (FG) deep beams placed in a thermal environment that is not addressed elsewhere. The material properties of each layer are proposed to be temperature-dependent and vary continuously through the height direction based on the Power-Law function. The deep layered beam is exposed to harmonic sinusoidal load and temperature rising. In the modelling of the multilayered FG deep beam, the two-dimensional (2D) plane stress continuum model is used. Equations of motion of deep composite beam with the associated boundary conditions are presented. In the frame of finite element method (FEM), the 2D twelve-node plane element is exploited to discretize the space domain through the length-thickness plane of the beam. In the solution of the dynamic problem, Newmark average acceleration method is used to solve the time domain incrementally. The developed procedure is verified and compared, and an excellent agreement is observed. In numerical examples, effects of graduation parameter, geometrical dimension and stacking sequence of layers on the time response of deep multilayer FG beams are investigated with temperature effects.

• 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.

• Steel and Composite Structures
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• v.31 no.5
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• pp.469-488
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• 2019

We in this paper study nonlinear bending of a functionally graded porous nanobeam subjected to multiple physical load based on the nonlocal strain gradient theory. For more reasonable analysis of nanobeams made of porous functionally graded magneto-thermo-electro-elastic materials (PFGMTEEMs), both constituent materials and the porosity appear gradient distribution in the present expression of effective material properties, which is much more suitable to the actual compared with the conventional expression of effective material properties. Besides the displacement function regarding physical neutral surface is introduced to analyze mechanical behaviors of beams made of FGMs. Then we derive nonlinear governing equations of PFGMTEEMs beams using the principle of Hamilton. To obtain analytical solutions, a two-step perturbation method is developed in nonuniform electric field and magnetic field, and then we use it to solve nonlinear equations. Finally, the analytical solutions are utilized to perform a parametric analysis, where the effect of various physical parameters on static bending deformation of nanobeams are studied in detail, such as the nonlocal parameter, strain gradient parameter, the ratio of nonlocal parameter to strain gradient parameter, porosity volume fraction, material volume fraction index, temperature, initial magnetic potentials and external electric potentials.