• Structural Engineering and Mechanics
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• 제20권4호
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• pp.465-476
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• 2005

Free vibration of orthotropic functionally graded beams, whose material properties can vary arbitrarily along the thickness direction, is investigated based on the two-dimensional theory of elasticity. A hybrid state-space/differential quadrature method is employed along with an approximate laminate model, which allows us to obtain the semi-analytical solution easily. With the introduction of continuity conditions at each fictitious interface and boundary conditions at the top and bottom surfaces, the frequency equation for an inhomogeneous beam is derived. A completely exact solution of an FGM beam with material constants varying in exponential way through the thickness is also presented, which serves a benchmark to verify the present method. Numerical results are performed and discussed.

• Earthquakes and Structures
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• 제17권1호
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• pp.31-37
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• 2019

This work presents a free vibration analysis of functionally graded beams by employing an original high order shear deformation theory (HSDBT). This theory use only three unknowns, but it satisfies the stress free boundary conditions on the top and bottom surfaces of the beam without requiring any shear correction factors. The mechanical properties of the beam are assumed to vary continuously in the thickness direction by a simple power-law distribution in terms of the volume fractions of the constituents. In order to investigate the free vibration response, the equations of motion for the dynamic analysis are determined via the Hamilton's principle. The Navier solution technique is adopted to derive analytical solutions for simply supported beams. The accuracy and effectiveness of proposed model are verified by comparison with previous research.

• Structural Engineering and Mechanics
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• 제78권5호
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• pp.529-543
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• 2021

Inevitable source-uncertainties in geometry configuration, boundary condition, and material properties may deviate the structural dynamics from its expected responses. This paper aims to examine the influence of these uncertainties on the vibration of functionally graded beams. Finite element procedures are presented for Timoshenko beams and utilized to generate reliable datasets. A prerequisite to the uncertainty quantification of the beam vibration using Monte Carlo simulation is generating large datasets, that require executing the numerical procedure many times leading to high computational cost. Utilizing artificial neural networks to model beam vibration can be a good approach. Initially, the optimal network for each beam configuration can be determined based on numerical performance and probabilistic criteria. Instead of executing thousands of times of the finite element procedure in stochastic analysis, these optimal networks serve as good alternatives to which the convergence of the Monte Carlo simulation, and the sensitivity and probabilistic vibration characteristics of each beam exposed to randomness are investigated. The simple procedure presented here is efficient to quantify the uncertainty of different stochastic behaviors of composite structures.

• Structural Engineering and Mechanics
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• 제72권3호
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• pp.293-304
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• 2019

In this paper, a general model is developed to predict the distribution of interfacial shear and normal stresses of FG beam reinforced by porous FGM plates under mechanical loading. The beam is assumed to be isotropic with a constant Poisson's ratio and power law elastic modulus through the beam thickness. Stress distributions, depending on an inhomogeneity constant, were calculated and presented in graphicals forms. It is shown that both the normal and shear stresses at the interface are influenced by the material and geometry parameters of the composite beam, and it is shown that the inhomogeneities play an important role in the distribution of interfacial stresses. The results presented in the paper can serve as a benchmark for future analyses of functionally graded beams strengthened by imperfect varying properties plates. Numerical comparisons between the existing solutions and the present new solution enable a clear appreciation of the effects of various parameters. The results of this study indicated that the imperfect functionally graded panel strengthening systems are effective in enhancing flexural behavior of the strengthened FGM beams. This research is helpful in understanding the mechanical behaviour of the interface and design of hybrid structures.

• Geomechanics and Engineering
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• 제24권6호
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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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• 제43권6호
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• pp.797-808
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• 2022

This paper presents an investigation on the free vibration characteristics of functionally graded nanocomposite double-beams reinforced by single-walled carbon nanotubes (SWCNTs). The double-beams coupled by an interlayer spring, resting on the elastic foundation with a linear layer and shear layer, and is simply supported in thermal environments. The SWCNTs gradient distributed in the thickness direction of the beam forms different reinforcement patterns. The materials properties of the functionally graded carbon nanotube-reinforced composites (FG-CNTRC) are estimated by rule of mixture. The first order shear deformation theory and Euler-Lagrange variational principle are employed to derive the motion equations incorporating the thermal effects. The vibration characteristics under several patterns of reinforcement are presented and discussed. We conducted a series of studies aimed at revealing the effects of the spring stiffness, environment temperature, thickness ratios and carbon nanotube volume fraction on the nature frequency.

• Steel and Composite Structures
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• 제15권5호
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• pp.467-479
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• 2013

In this paper, a new first-order shear deformation beam theory based on neutral surface position is developed for bending and free vibration analysis of functionally graded beams. The proposed theory is based on assumption that the in-plane and 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 neutral surface position for a functionally graded beam which its material properties vary in the thickness direction is determined. Based on the present new first-order shear deformation beam theory and the neutral surface concept together with Hamilton's principle, the motion equations are derived. To examine accuracy of the present formulation, several comparison studies are investigated. Furthermore, the effects of different parameters of the beam on the bending and free vibration responses of functionally graded beam are discussed.

• Structural Engineering and Mechanics
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• 제75권6호
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• pp.737-746
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• 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.

• Coupled systems mechanics
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• 제6권1호
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• pp.97-111
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• 2017

The present paper reports an analytical study of delamination fracture in the Mixed Mode Flexure (MMF) functionally graded beam with considering the material non-linearity. The mechanical behavior of MMF beam is modeled by using a non-linear stress-strain relation. It is assumed that the material is functionally graded along the beam height. Fracture behavior is analyzed by the J-integral approach. Non-linear analytical solution is derived of the J-integral for a delamination located arbitrary along the beam height. The J-integral solution derived is verified by analyzing the strain energy release rate with considering the non-linear material behavior. The effects of material gradient, crack location along the beam height and material non-linearity on the fracture are evaluated. It is found that the J-integral value decreases with increasing the upper crack arm thickness. Concerning the influence of material gradient on the non-linear fracture, the analysis reveals that the J-integral value decreases with increasing the ratio of modulus of elasticity in the lower and upper edge of the beam. It is found also that non-linear material behavior leads to increase of the J-integral value. The present study contributes for the understanding of fracture in functionally graded beams that exhibit material non-linearity.

• Steel and Composite Structures
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• 제24권1호
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• pp.65-77
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• 2017

The purpose of this research is to study the nonlinear free vibration and post-buckling analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams resting on a nonlinear elastic foundation. Uniformly and functionally graded distributions of single walled carbon nanotubes as reinforcing phase are considered in the polymeric matrix. The modified form of rule of mixture is used to estimate the material properties of CNTRC beams. The governing equations are derived employing Euler-Bernoulli beam theory along with energy method and Hamilton's principle. Applying von $K\acute{a}rm\acute{a}n's$ strain-displacement assumptions, the geometric nonlinearity is taken into consideration. The developed governing equations with quadratic and cubic nonlinearities are solved using variational iteration method (VIM) and the analytical expressions and numerical results are obtained for vibration and stability analysis of nanocomposite beams. The presented comparative results are indicative for the reliability, accuracy and fast convergence rate of the solution. Eventually, the effects of different parameters, such as foundation stiffness, volume fraction and distributions of carbon nanotubes, slenderness ratio, vibration amplitude, coefficients of elastic foundation and boundary conditions on the nonlinear frequencies, vibration response and post-buckling loads of FG-CNTRC beams are examined. The developed analytical solution provides direct insight into parametric studies of particular parameters of the problem.