• Structural Engineering and Mechanics
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• v.84 no.3
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• pp.423-435
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• 2022

In this article, we present torsion-bending analysis of a composite FGM beam with an open section, according to the advanced and refined theory of 1D / 3D beams based on the 3D Saint-Venant's solution and taking into account the edge effects. The (initially one-dimensional) model contains a set of three-dimensional (3D) displacement modes of the cross section, reflecting its 3D mechanical behaviour. The modes are taken into account depending on the mechanical characteristics and the geometrical form of the cross-section of the composite FGM beam. The model considered is implemented on the CSB (Cross-Section and Beam Analysis) software package. It is based on the RBT/SV theory (Refined Beam Theory on Saint-Venant principle) of FGM beams. The mechanical and physical characteristics of the FGM beam continuously vary, depending on a power-law distribution, across the thickness of the beam. We compare the numerical results obtained by the three-beam theories, namely: The Classical Beam Theory of Saint-Venant (Classical Beam Theory CBT), the theory of refined beams (Refined Beam Theory RBT), and the theory of refined beams, using the higher (high) modes of distortion of the cross-section (Refined Beam Theory using distorted modes RBTd). The results obtained confirm a clear difference between those obtained by the three models at the level of the supports. Further from the support, the results of RBT and RBTd are of the same order, whereas those of CBT remains far from those of higher-order theories. The 3D stresses, strains and displacements, obtained by the present study, reflect the 3D behaviour of FGM beams well, despite the initially 1D nature of the problem. A validation example also shows a very good agreement of the proposed models with other models (classical or higher-order beam theory) and Carrera Unified Formulation 1D-beam model with Lagrange Expansion functions (CUF-LE).

• 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.

• Steel and Composite Structures
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• v.18 no.2
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• pp.409-423
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• 2015

In the present work, a simple and refined trigonometric higher-order beam theory is developed for bending and vibration of functionally graded beams. The beauty of this theory is that, in addition to modeling the displacement field with only 3 unknowns as in Timoshenko beam theory, the thickness stretching effect (${\varepsilon}_Z{\neq}0$) is also included in the present theory. Thus, the present refined beam theory has fewer number of unknowns and equations of motion than the other shear and normal deformations theories, and it considers also the transverse shear deformation effects without requiring shear correction factors. The neutral surface position for such beams in which the material properties vary in the thickness direction is determined. Based on the present refined trigonometric higher-order beam theory and the neutral surface concept, the equations of motion are derived from Hamilton's principle. Numerical results of the present theory are compared with other theories to show the effect of the inclusion of transverse normal strain on the deflections and stresses.

• Steel and Composite Structures
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• v.35 no.4
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• pp.545-554
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• 2020

The present paper explores nonlinear dynamical properties of piezo-magnetic beams based on a nonlocal refined higher-order beam formulation and piezoelectric phase effect. The piezoelectric phase increment may lead to improved vibrational behaviors for the smart beams subjected to magnetic fields and external harmonic excitation. Nonlinear governing equations of a nonlocal intelligent beam have been achieved based upon the refined beam model and a numerical provided has been introduced to calculate nonlinear vibrational curves. The present study indicates that variation in the volume fraction of piezoelectric ingredient has a substantial impact on vibrational behaviors of intelligent nanobeam under electrical and magnetic fields. Also, it can be seen that nonlinear free/forced vibrational behaviors of intelligent nanobeam have dependency on the magnitudes of induced electrical voltages, magnetic potential, stiffening elastic substrate and shear deformation.

• Steel and Composite Structures
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• v.19 no.4
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• pp.829-841
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• 2015

In this paper, a refined exponential shear deformation beam theory is developed for bending analysis of functionally graded beams. The theory account for parabolic 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. Contrary to the others refined theories elaborated, where the stretching effect is neglected, in the current investigation this so-called "stretching effect" is taken into consideration. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Based on the present shear deformation beam theory, the equilibrium equations are derived from the principle of virtual displacements. Analytical solutions for static are obtained. Numerical examples are presented to verify the accuracy of the present theory.

• Coupled systems mechanics
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• v.4 no.1
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• pp.99-114
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• 2015

In this paper, a refined exponential shear deformation beam theory is developed for bending analysis of functionally graded beams. The theory account for parabolic 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. Contrary to the others refined theories elaborated, where the stretching effect is neglected, in the current investigation this so-called "stretching effect" is taken into consideration. The material properties of the functionally graded beam are assumed to vary according to power law distribution of the volume fraction of the constituents. Based on the present shear deformation beam theory, the equations of motion are derived from Hamilton's principle. Analytical solutions for static are obtained. Numerical examples are presented to verify the accuracy of the present theory.

• Advances in nano research
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• v.5 no.4
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• pp.281-301
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• 2017

In this disquisition, an exact solution method is developed for analyzing the vibration characteristics of magneto-electro-elastic functionally graded (MEE-FG) beams by considering porosity distribution and various boundary conditions via a four-variable shear deformation refined beam theory for the first time. Magneto-electroelastic properties of porous FG beam are supposed to vary through the thickness direction and are modeled via modified power-law rule which is formulated using the concept of even and uneven porosity distributions. Porosities possibly occurring inside functionally graded materials (FGMs) during fabrication because of technical problem that lead to creation micro-voids in FG materials. So, it is necessary to consider the effect of porosities on the vibration behavior of MEE-FG beam in the present study. The governing differential equations and related boundary conditions of porous MEE-FG beam subjected to physical field are derived by Hamilton's principle based on a four-variable tangential-exponential refined theory which avoids the use of shear correction factor. An analytical solution procedure is used to achieve the natural frequencies of porous-FG beam supposed to magneto-electrical field which satisfies various boundary conditions. A parametric study is led to carry out the effects of material graduation exponent, porosity parameter, external magnetic potential, external electric voltage, slenderness ratio and various boundary conditions on dimensionless frequencies of porous MEE-FG beam. It is concluded that these parameters play noticeable roles on the vibration behavior of MEE-FG beam with porosities. Presented numerical results can be applied as benchmarks for future design of MEE-FG structures with porosity phases.

• Wind and Structures
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• v.21 no.3
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• pp.273-287
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• 2015

In this paper, a free vibration analysis of functionally graded beam made of porous material is presented. The material properties are supposed to vary along the thickness direction of the beam according to the rule of mixture, which is modified to approximate the material properties with the porosity phases. 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.

• Advances in nano research
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• v.4 no.4
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• pp.251-264
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• 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 Monitoring and Maintenance
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• v.6 no.2
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• pp.147-159
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• 2019

In present article, a size-dependent refined thick beam element has been established based upon nonlocal elasticity theory. Next, it is used to explore vibration response of porous metal foam nanobeams on elastic medium. The established beam element introduces ten degrees of freedom. Different porosity distributions called uniform, symmetric and asymmetric will be employed. Herein, introduced thick beam element contains shear deformations without using correction factors. Convergence and verification studies of obtained results from finite element method are also provided. The impacts of nonlocality factor, foundation factors, shear deformation, slenderness ratio, porosity kinds and porosity factor on vibration frequencies of metal foam nano-sized beams have been explored.