Functionally Graded Beams Research Articles

Rational approach for higher-order shear deformation beam theories

The studies on the higher-order beam theory and its application to functionally graded (FG), laminated composite and sandwich beams have received much research attention over recent years. While significant progress has been made, there is an issue with the selection of the appropriate distribution function for describing section warping due to transverse loading. In the present study, a rational approach is proposed for the determination of correct warping functions for symmetric cross sections. Two new conditions on warping function are suggested: first, the warping function should vanish on the neutral plane; and second, the first derivative of warping function should take unity value on the neutral plane. A rigorous procedure is then developed by using the equilibrium condition and three conditions. An exact higher-order theory is then developed for creating accurate analysis models to consider both material and geometry variations over the beam cross section. A finite element analysis model is presented and a two-node beam element with bubble displacement modes is implemented. Results for sandwich and FG beam problems are shown and numerical results are compared with those of plane stress continuum models to demonstrate the correctness and application of the new theory and finite element model.

Size-dependent dynamics of a FG Nanobeam near nonlinear resonances induced by heat

This study explores heat-induced nonlinear vibration of a functionally graded (FG) capacitive nanobeam within the framework of nonlocal strain gradient theory (NLSGT). The elastic FG beam, which is firstly deflected by a DC voltage, is driven to vibrate about its deflected position by a periodic heat load. The nano-structure, which consists of a clamped-clamped nanobeam, is modeled assuming Euler–Bernoulli beam assumption which accounts for the nonlinear von-Karman strain and the electrostatic and intermolecular forcing. To simulate the static and dynamic responses, a model reduction procedure is carried out by employing the Galerkin method. The method of Averaging as a regular semi-analytic perturbation method is applied to obtain governing equations of the steady-state responses. With the purpose of establishing the validity of the solution, a Shooting technique in conjunction with the Floquet theory is used to capture the periodic motions and then examine their stability. The nonlinear resonance frequency of the FG nanobeam near its fundamental natural frequency (primary resonance) and near principal parametric resonance is investigated while the emphasis is placed on studying the effect of various parameters including DC voltage, amplitude of the periodic heat source, material index, damping ratio, and small scale parameters. The main objective of this study is to model a miniature structure which can be used as either a sensitive remote temperature sensor or a high-efficiency thermal energy harvester.

Evaluation of stress distributions in the geometrical nonlinear regime of functionally graded structures

In this study, stresses distributions in the large deflection field of Functionally Graded (FG) structures are investigated. Numerical simulations are carried out using one-dimensional (1D) models combining the Finite Element Method (FEM) and Carrera Unified Formulation (CUF) to evaluate the three-dimensional (3D) displacement field over the cross-section, employing Lagrange and Taylor polynomials. Then, the geometrical nonlinear relations are solved with the Newton-Raphson linearization and constrained with an arc-length method. The nonlinear behavior of cantilever beams undergoing bending and compression loading conditions is calculated. The results show the influence of the adopted theory to approximate the displacement field over the cross-section on FG beams. According to the results of this study, it is emphasized that geometrical nonlinearities should be taken into consideration in the evaluation of stress distributions in FG structures.

Application of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation

The present study is dealt with the applicability of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique on free vibration of functionally graded (FG) beam with uniformly distributed porosity along the thickness of the beam. The material properties such as Young’s modulus, mass density, and Poisson’s ratio are also considered to vary along the thickness of the FG beam as per the power-law exponent model. The porous FG beam is embedded in an elastic substrate; namely, the Kerr elastic foundation and the displacement field of the beam are governed by a refined higher-order shear deformation theory. The effectiveness of the Rayleigh–Ritz method is due to the use of the shifted Chebyshev polynomials as a shape function. The orthogonality of shifted Chebyshev polynomial makes the technique more computationally efficient and avoid ill-conditioning for the higher number of terms of the polynomial. Hinged–hinged, clamped–hinged, clamped–clamped, and clamped-free boundary conditions have been taken into account for the parametric study. Validation of the present model is examined by comparing it with the existing literature in special cases showing remarkable agreement. A pointwise convergence study is also carried out for shifted Chebyshev polynomial-based Rayleigh–Ritz method, and the effect of power-law exponent, porosity volume fraction index, and elastic foundation on natural frequencies is studied comprehensively.

Static bending analysis of beams made of functionally graded porous materials

This paper constitutes a first attempt to explore the influence of porosity on bending static analysis of functionally graded (FG) beams using a refined mixed finite element beam model. The material properties of functionally graded porous beams are estimated using a modified power law distribution with two different types of porosity namely even and uneven distributions. The potential of the proposed model is highlighted via a comparison study. Then, a parametric study is carried out to show the effects of power law index, porosity coefficient, boundary conditions and types of porosity distributions on deflections and stresses of the studied FG beams.

A Comprehensive Review on Vibration Analysis of Functionally Graded Beams

Beams and beam structures are structural components commonly used in mechanical, aerospace, nuclear, and civil engineering. To meet the different engineering design limitations such as operational conditions, weight, and vibrational characteristics, these components may be made of various materials such as functionally-graded materials (FGMs), composites, and homogeneous materials. Functionally-graded (FG) beams play a key role not only in classical structural applications, but also have vast applications in thermal, electric-structural and electric-thermal-structural systems, e.g. in the form of FG beam energy harvesters, sensors and actuators. In all these applications, using new materials like FGMs can greatly improve the efficiency of the structural components and systems. Since FG beams are mostly used as moving components in engineering structures, vibration analysis of these components has been studied by numerous researchers. In order to solve the governing equation and related boundary conditions of the FG beams, powerful numerical methods with a high level of accuracy and fast rate of convergence are often required. The differential quadrature method (DQM) is a powerful and reliable numerical method which has been extensively used by researchers to perform the vibration analyses of FG structures in the last decade. In this paper, firstly various mathematical models which have been used to express the material properties of FGMs are reviewed. Secondly different elasticity theories which have been applied in vibration analysis of FG beams are summarized. In addition, a review on the DQM and its applications is presented. At the next step, a comprehensive review on free vibration analyses of FG beams based on different elasticity theories and in particular those using the DQM is performed. In continue, a brief review on the application of other numerical methods in vibration analysis of FG beams is presented. Moreover, because of the importance of nonlinear vibration analysis of FG beams, a review on the application of various numerical methods and different elasticity theories on nonlinear vibration analysis of FG beams is performed. Finally, a brief review on linear and nonlinear vibration analysis of FG microbeams, as a special type of FG beams, is presented.

A material-based higher-order shear beam model for accurate analyses of FG beams with arbitrary material distribution

A material-based higher-order shear beam model is proposed for accurate analyses of functionally graded (FG) beams with arbitrary material distribution through thickness. Distinct from the existing higher-order shear beam models, the shear function of the present model is dependent on the material distribution. The specific form of this material-based shear function is described by a piecewise linear interpolation field and determined by ensuring the consistency of transverse shear stress distributions between Euler-Bernoulli beam theory and the higher-order shear beam theory. Based on the proposed beam model, a 3-node beam finite element is developed for static and buckling analyses. Numerical examples are studied to visualize the accuracy and effectiveness of the proposed model and finite element. Solutions of static displacements and critical loads obtained by various beam elements including the proposed element, the existing displacement-based beam elements and the mixed higher-order shear beam element are compared. The accuracy loss of the existing higher-order shear beam models is discussed. As demonstrated by the results, the proposed beam model and finite element have excellent solution accuracy and desirable applicability.

Forced vibration analysis of blade after selective laser shock processing based on Timoshenko’s beam theory

The main purpose of this paper is to theoretically explore the effect of selective laser shock processing (LSP) on the forced vibration response of blade based on the functionally graded (FG) Timoshenko’s beam theory. Considering the complexity of direct solution caused by gradient distribution of elastic modulus, pre-stress, and variable cross-section, the actual blade after selective LSP is simplified into a series of uniform segments. The pre-stress is transformed into an additional excitation load by an integral technique, and thus an analytical solution for this type of segment can be obtained. Then, in order to avoid solving the additional integration constants, a simple semi-analytical solution for the forced vibration of rotating pre-stressed FG beam with arbitrary cross-section is presented by using the continuity condition of deformation, and the effects of strengthened position, thickness of strengthened layer and single-sided/double-sided shock mode on the amplitude of vibration response are discussed, respectively. The results show that the selective LSP can effectively suppress the vibration deflection by enlarging the strengthened zone, deepening the strengthened layer and employing the double-sided shock.

Static Analysis of FG Beams via Complementary Functions Method

Static analysis of Functionally Graded (FG) beams is studied by the Complementary Functions Method (CFM). The material properties, Young’s modulus, of the straight beams, are graded in the thickness direction based on a power law distribution while the Poisson’s ratio is supposed to be constant. Governing equations of the considered problem are obtained with the aid of minimum total potential energy principle based on Timoshenko’s beam theory (FSDT). The main purpose of this paper is the infusion of the CFM to the static analysis of FG straight beams. The effectiveness and accuracy of the proposed method are confirmed by comparing its numerical results with those available in the literature. Application of this efficient method provides accurate results of static response for FGM beams with different variations of material properties in the thickness of the beam.

A DSC Regularized Dirac-Delta Method for Flexural Vibration of Elastically Supported FG Beams Subjected to a Moving Load

This research presents a numerical approach to address the moving load problem of functionally graded (FG) beams with rotational elastic edge constraints, in which the regularized Dirac-delta function is used to describe a time-dependent moving load source. The governing partial differential equations of the system, derived in accordance with the classical Euler–Bernoulli beam theory, are approximated by the discrete singular convolution (DSC) method. The resulting set of algebraic equations can then be solved by the Newmark-β integration scheme. Such a singular Dirac-delta formulation is also employed as the kernel function of the DSC method. In this work, the material properties of FG beams are assumed to be changed in the thickness direction. A convergence study is performed to validate the accuracy and reliability of the numerical results. In addition, the effects of moving load velocity and material distribution on the dynamic behavior of elastically restrained FG beams are also studied to serve as new benchmark solutions. By comparing with the available results in the existing literature, the present results show good agreement. More importantly, the major finding of this work indicates that the DSC regularized Dirac-delta approach is a good candidate for moving load problems, since the equally spaced grid system adopted in the DSC scheme can achieve a preferable representation of moving load sources.

Finite element model based on an efficient layerwise theory for dynamics and active vibration control of smart functionally graded beams

In this work, we present a two-noded efficient finite element (FE) model incorporating the layer-wise mechanics for the dynamics and active vibration control of smart functionally graded (FG) beams. The material properties in the FG beam are assumed to vary smoothly in the thickness direction according to power law variation. The effective properties are computed using Mori-Tanaka homogenization scheme. Electric potential profile in the electroded piezoelectric layers/patches is assumed quadratic across its thickness. The equations of motion are derived using extended Hamilton’s principle. Due to the complex algebraic expressions involved in the effective properties of FG system, the inertia and stiffness parameters are computed numerically using six point Gauss integration method. To fulfill the convergence requirements for weak integration of various energy terms in the variational formulation, the transverse displacement is interpolated with Hermite interpolation function possessing C1-continuity while the inplane displacement, shear rotation and quadratic component of the electric potential are interpolated with linear Lagrangian functions of C0-continuity. The equipotential condition of the electroded piezoelectric sensors and actuators is conveniently modelled using electric node concept. The control system is designed for constant gain velocity feedback (CGVF), and optimal LQR and LQG control strategies for a reduced order model using state space approach. The control performance is studied for single-input-single-output (SISO) and multi-input-multi-output (MIMO) configurations under various excitations. The effect of material inhomogeneity on stability/instability of the closed-loop response in the CGVF control has been discussed.

Evaluation of the bending response of compact and thin-walled FG beams with CUF

The study deals with the stress evaluations of the functionally graded beams (FGBs) using refined beam models. The refined beam models are obtained within the framework of the Carrera unified formulation (CUF), where expansions are used across the cross-section to enrich the kinematics of the finite elements and obtain 3 D-like accuracy of the solution. Analyses of FGBs are performed by using different structural theories based on Taylor-like and Lagrange expansions. The results are compared with the literature and 3 D finite element solutions. It is observed that calculation costs are significantly decreased in finite element analyses performed with CUF.

Transient response of 2D functionally graded beam structure

The objective of this article is investigation of dynamic response of thick multilayer functionally graded (FG) beam under generalized dynamic forces. The plane stress problem is exploited to describe the constitutive equation of thick FG beam to get realistic and accurate response. Applied dynamic forces are assumed to be sinusoidal harmonic, sinusoidal pulse or triangle in time domain and point load. Equations of motion of deep FG beam are derived based on the Hamilton principle from kinematic relations and constitutive equations of plane stress problem. The numerical finite element procedure is adopted to discretize the space domain of structure and transform partial differential equations of motion to ordinary differential equations in time domain. Numerical time integration method is used to solve the system of equations in time domain and find the time responses. Numerical parametric studies are performed to illustrate effects of force type, graduation parameter, geometrical and stacking sequence of layers on the time response of deep multilayer FG beams.

Static stability and of symmetric and sigmoid functionally graded beam under variable axial load

This manuscript presents impacts of gradation of material functions and axial load functions on critical buckling loads and mode shapes of functionally graded (FG) thin and thick beams by using higher order shear deformation theory, for the first time. Volume fractions of metal and ceramic materials are assumed to be distributed through a beam thickness by both sigmoid law and symmetric power functions. Ceramic–metal–ceramic (CMC) and metal–ceramic–metal (MCM) symmetric distributions are proposed relative to mid-plane of the beam structure. The axial compressive load is depicted by constant, linear, and parabolic continuous functions through the axial direction. The equilibrium governing equations are derived by using Hamilton\'s principles. Numerical differential quadrature method (DQM) is developed to discretize the spatial domain and covert the governing variable coefficients differential equations and boundary conditions to system of algebraic equations. Algebraic equations are formed as a generalized matrix eigenvalue problem, that will be solved to get eigenvalues (buckling loads) and eigenvectors (mode shapes). The proposed model is verified with respectable published work. Numerical results depict influences of gradation function, gradation parameter, axial load function, slenderness ratio and boundary conditions on critical buckling loads and mode-shapes of FG beam structure. It is found that gradation types have different effects on the critical buckling. The proposed model can be effective in analysis and design of structure beam element subject to distributed axial compressive load, such as, spacecraft, nuclear structure, and naval structure.

Bending, Buckling and Free Vibration Analysis of Size-Dependent Nanoscale FG Beams Using Refined Models and Eringen’s Nonlocal Theory

In this study, a theoretical unification of twenty-one nonlocal beam theories are presented by using a unified nonlocal beam theory. The small-scale effect is considered based on the nonlocal differential constitutive relations of Eringen. The present unified theory satisfies traction free boundary conditions at the top and bottom surface of the nanobeam and hence avoids the need of shearing correction factor. Hamilton’s principle is employed to derive the equations of motion. The present unified nonlocal formulation is applied for the bending, buckling and free vibration analysis of functionally graded (FG) nanobeams. The elastic properties of FG material vary continuously by gradually changing the volume fraction of the constituent materials in the thickness direction. Closed-form analytical solutions are obtained by using Navier’s solution technique. Non-dimensional displacements, stresses, natural frequencies and critical buckling loads for FG nanobeams are presented. The numerical results presented in this study can be served as a benchmark for future research.

Thermo-Mechanical Vibration Analysis of Imperfect Inhomogeneous Beams Based on a Four-Variable Refined Shear Deformation Beam Theory Considering Neutral Surface Position

In this disquisition, an exact solution method is developed for analyzing the vibration characteristics of porous functionally graded (FG) beams by considering neutral surface position and different thermal loadings via a four-variable shear deformation refined beam theory. Four types of environmental conditions through the z-axis direction are supposed as: uniform (UTR), linear (LTR), nonlinear (NLTR) and sinusoidal (STR) temperature rises. Mechanical properties of porous FG beams are supposed to vary through the thickness direction and are modeled via the modified power-law. The modified power-law is formulated using the concept of even and uneven porosity distributions. Since the variation of pores along the thickness direction influences the mechanical properties, porosity plays a key role in the mechanical response of FG structures. The governing differential equations and related boundary conditions of porous FG beams are subjected to temperature field that is derived by Hamilton's principle based on a four-variable refined theory which verifies shear deformation regardless of any shear correction factor. The Navier-type solution procedure is used to achieve the natural frequencies of porous-FG beams supposed to various thermal loadings which satisfies the simply-simply boundary condition. A parametric study is led to carry out the effects of material graduation exponent, porosity volume fraction, different porosity distribution, and thermal effect on dimensionless frequencies of porous FG beams. It is concluded that these parameters play noticeable roles in the vibration behavior of imperfect FG beams. Presented numerical results can be applied as benchmarks for future designs of imperfect FG structures with porosity phases.

Viscoelastic nonlinear dynamic behaviour of Timoshenko FG beams

Retaining rotary-inertia terms and translation/rotation, a nonlinear analysis is conducted on the dynamic behaviour of Timoshenko functionally graded (FG) beams. The shear/normal strain field components are formulated using the Timoshenko theory. The elastic/viscous stress components are related to the strain components, using the Kelvin-Voigt scheme, and the strain components to translation/rotation terms using the Timoshenko beam theory. The viscous effects are incorporated using a negative work expression. The through-thickness gradual changes in the properties are modelled using the Mori-Tanaka technique. Energy-loss/work/energy Hamilton’s principle for system’s energy balance is utilised and the coupled nonlinear sets of equations of motion are obtained. Reduced-order motion models are obtained using Galerkin’s discretisation method. Numerical simulations are conducted via a parameter-continuation method and the dynamical behaviour is examined.

Effect of the Multi Vibration Absorbers on the Nonlinear FG Beam Under Periodic Load with Various Boundary Conditions

A semi-analytical method is used to study the effects of the multi vibration absorbers on the nonlinear functionally graded (FG) Euler-Bernoulli beam subjected to periodic load. The material properties of the beam are assumed to be continuously graded in the thickness direction. The governing equations of functionally graded beam are obtained based on the Hamilton's principle and these equations are solved by using the Rayleigh-Ritz method. To validate the results, comparisons are made with the available solutions for the natural frequencies of isotropic beam. The effects of the multi vibration absorbers and material parameters on the vibration response of functionally graded beamare investigated. For case study the effect of two symmetrical vibration absorbers is considered, these absorbers are applicable in some of the mechanic systems. In those systems, two absorbers are used close to the beginning and end of the structures instead of using them in the middle of these structures. By considering the industrial applications, it is shown that using the two symmetrical vibration absorbers with lower mass is close to the end of functionally graded beam is better than the middle of one. Also, the effect of different numbers of the vibration absorbers on the nonlinear functionally graded beam with simply supported boundary condition is considered. The results shown that increasing the number of vibration absorbers leads to decreasing the maximum deflection.

Finite Element Approach of Bending and Buckling Analysis of FG Beams Based on Refined Zigzag Theory

This is the first attempt to utilize the refined zigzag theory (RZT) to study the bending and buckling behavior of a functional graded (metal/ceramic) thick beam. RZT, which has been exploited for the analysis of multilayered composite and sandwich beams does not employ shear correction factor. Furthermore, the number of kinematics variables of the RZT is not dependent to the number of layers in comparison with the layerwise theory. With regarding to the numerical solution, RZT, also requires C0 continuity interpolation, which leads to the development of this theory in finite element method. It is assumed that the mechanical properties of the beam varies through the thickness. According to the volume fraction of metal and ceramic, it is discretized across the thickness; consequently, the functionally graded beam (FGB) is modeled as a multi-layered one. The beam subjected to uniformly transverse and axial loadings. The equilibrium equations are established using the principle of virtual work. Using the shape functions of the first and second order forms, the non-isoparametric finite element consisting of three nodes and nine degrees of freedom are extracted. To confirm the excellent accuracy of the present approach, some numerical examples are provided and compared with those available in the literature that reveals that RZT is a trusty and validated theory to analyze the FG thick beams.

Numerical analysis for free vibration of functionallygraded beams using an original HSDBT

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