FGM Beams Research Articles

Free Vibration of Functionally Graded Material Beam under Thermal Loads

The material properties of FGM beams are assumed to vary continuously through the thickness according to power law distribution. Based on the assumption that plane section remains plane and normal to the beam axis, the differential equations of motion were established for free vibration of FGM elastic beams under thermal loads and mechanical loads on the premise of the middle plane strain caused by heterogeneity of the materials. Natural frequency and dominant mode shape of FGM beams with four kinds of boundary conditions were obtained.

Dynamic buckling of suddenly heated or compressed FGM beams resting on nonlinear elastic foundation

In this paper, static and dynamic buckling of an FGM beam subjected to uniform temperature rise loading and uniform compression are studied. Material properties of the beam are assumed to be graded across the thickness. Each thermo-mechanical property of the beam is assumed to be temperature dependent. Initial imperfection of the beam is also taken into account. The beam is resting over a three-parameter elastic foundation with hardening/softening cubic nonlinearity which acts in tension as well as in compression. Nonlinear governing equations are obtained based on the static version of virtual displacements and are solved via the multi-term Galerkin method. Dynamic buckling load levels are estimated based on the well-known Hoff-Simitses criterion. Results reveal that for sufficiently stiff softening elastic foundation, post-buckling equilibrium path becomes unstable. Furthermore, when the thermal post-buckling equilibrium path is stable, no dynamic buckling occurs according to this criterion.

Low velocity impact response of thick FGM beams with general boundary conditions in thermal field

Low velocity impact response of an FGM beam with through-the-thickness distributed properties is studied in this paper. Third order shear deformation beam theory is implemented to derive the governing equations. A modification of the Hertz contact law which is suitable for graded materials is adopted. A Ritz-based solution in space domain appropriate for various in-plane and out-of-plane boundary conditions is developed. Solution of the resulted system of equations in time domain is accomplished via the fourth-order Runge–Kutta method. The developed method is validated with the results of commercial finite elements software. The effects of various involved parameters, such as graded property profile, impactor velocity, impactor mass, boundary conditions, thermal environment and impact position are studied in detail.

Implementation of Penalty Method for Optimization of FGM Beams

In this study, optimization of four-parameter power-law distribution of functionally graded (FG) beams resting on elastic foundations for the density with constraint on the frequency parameter is presented. To perform optimization, Genetic Algorithm (GA) is used to find the optimal solution. Genetic Algorithms is most directly suited to unconstrained optimization. Thus, penalty method is implemented for handling the existing constraints. A proper artificial neural network (ANN) is trained by training data sets obtained from generalized differential quadrature (GDQ) method and then is applied to reproduce the behavior of the structure both in free vibration and density for improving the speed of the optimization process.

Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams

By using mathematical similarity and load equivalence between the governing equations, bending solutions of FGM Timoshenko beams are derived analytically in terms of the homogenous Euler–Bernoulli beams. The deflection, rotational angle, bending moment and shear force of FGM Timoshenko beams are expressed in terms of the deflection of the corresponding homogenous Euler–Bernoulli beams with the same geometry, the same loadings and end constraints. Consequently, solutions of bending of the FGM Timoshenko beams are simplified as the calculation of the transition coefficients which can be easily determined by the variation law of the gradient of the material properties and the geometry of the beams if the solutions of corresponding Euler–Bernoulli beam are known. As examples, solutions are given for the FGM Timoshenko beams under S–S, C–C, C–F and C–S end constraints and arbitrary transverse loadings to illustrate the use of this approach. These analytical solutions can be as benchmarks in the further investigations of behaviors of FGM beams.

Nonlinear bending analysis of FGM beams based on physical neutral surface and high order shear deformation theory

Model of the FGM beams is first put forward by using on physical neutral surface and high-order shear deformation theory. Nonlinear von Kármán strain–displacement relationships are adopted, and material properties are assumed to be temperature dependent and vary along the thickness. It is worth noting that physical neutral surface will be changed with temperature. Therefore, the displacements have special forms, there are no stretching–bending couplings in constitutive equations, and governing equations have the simple forms, so the solution procedure is similar as homogeneous isotropic beams. The validity of physical neutral surface higher-order shear deformation beam theory can be confirmed by comparison with related researchers’ results. Nonlinear bending approximate solutions are given out using Ritz method. Physical neutral surface theory has many merits in the engineering application due to its easiness and simplicity.

Nonlinear vibration of edged cracked FGM beams using differential quadrature method

This paper investigated the nonlinear vibration of functionally graded beams containing an open edge crack based on Timoshenko beam theory. The cracked section is modeled by a massless elastic rotational spring. It is assumed that material properties follow exponential distributions through the beam thickness. The differential quadrature (DQ) method is employed to discretize the nonlinear governing equations which are then solved by a direct iterative method to obtain the nonlinear vibration frequencies of beams with different boundary conditions. The effects of the material gradient, crack depth and boundary conditions on nonlinear free vibration characteristics of the cracked FGM beams are studied in detail.

Relationship between Bending Solutions of FGM Timoshenko Beams and those of Homogenous Euler-Bernoulli Beams

Relationship between bending solutions of functionally graded Timoshenko beams (FGMTB) and those of homogenous Euler-Bernoulli beams (HEBB) were studied in this presentation. The deflection and the rotational angle of FGMTB are expressed in terms of the deflection of the corresponding HEBB with the same geometry, loadings and end constraints. Consequently, solutions of bending of the FGMTB are simplified as the calculation of the transition coefficients which can be easily determined by the variation law of the gradient of the material properties and the geometry of the beams because the solutions of corresponding HEBB are well known. As examples, solutions for the FGM Timoshenko beams under S-S, C-C, C-F and C-S end constraints and subjected to arbitrary transverse loadings were presented. These analytical solutions can be as benchmarks in the further investigations of behaviors of FGM beams.

Bending of Thermoviscoelastic Functionally Graded Materials Beams

According to the constitutive relation of linear thermovisoelasticity, with the help of Laplace transformation method and the introduction of structure functions and thermal functions, the mathematical model and its corresponding variational principle for thermoviscoelastic FGM beams are set up on the basis of the assumption that plane section remains plane and normal to the beam axis. Using Laplace transformation method, the deflection and the stress distribution are discussed.

Post-Buckling Analysis of FGM Beam Subjected to Non-Conservative Forces and In-Plane Thermal Loading

Based on the Kirchhoff large deformation theory, the post-buckling behavior of right movable simply supported FGM beam subjected to non-conservative forces and in-plane thermal loading was analyzed in this paper. The temperature-dependent and spatially dependent material properties of the FGM beam were assumed to vary through the thickness. The nonlinear governing equations of FGM beam subjected to a uniform distributed tangential load along the central axis and in-plane thermal loading were derived. Then, a shooting method and Runge-kutta method are employed to numerically solve the resulting equations. The post-buckling equilibrium paths of the FGM beam with different parameters were plotted, and the effects of non-conservative force, temperature, gradient index of FGM on the post-buckling behavior of right movable simply supported FGM beams were analyzed.

Deformation analysis of functionally graded beams by the direct approach

In this paper we employ the direct approach to the theory of rods and beams, which is based on the deformable curve model with a triad of rotating directors attached to each point. We show that this model (also called directed curve) is an efficient approach for analyzing the deformation of elastic beams with a complex material structure. Thus, we consider non-homogeneous, composite and functionally graded beams made of isotropic or orthotropic materials and we determine the effective stiffness properties in terms of the three-dimensional elasticity constants. We present general analytical expressions of the effective stiffness coefficients, valid for beams of arbitrary cross-section shape. Finally, we apply this method for FGM beams made of metal foams and compare our analytical results with the numerical results obtained by a finite element analysis.

Comparison of Post-Buckling Behaviors of a S-S FGM Beam under Conservative and Non-Conservative Distributed Forces

Based on the large deformation theory and considering the axial extension of the beam, the governing equations of post-buckling of a simply supported elastic FGM beam subjected to conservative and non-conservative distributed forces were established. In the analysis, it was assumed that the material properties of the beam vary continuously as a power function of the thickness coordinate. By using shooting method, the nonlinear boundary-value problem was solved numerically and the equilibrium paths as well as the post- buckling configurations of the deformed beam were presented. A comparison between the results of conservative system and that of non-conservative systems were given. The results shows that the features of the equilibrium paths of the the functionally graded beam under non-conservative are evidently different from those to a conservative one.

Analysis of FGM Beams by Means of Classical and Advanced Theories

This paper proposes several axiomatic refined theories for the linear static analysis of beams made of materials whose properties are graded along one or two directions. Via a unified formulation, a generic N-order approximation is assumed for the displacement unknown variables over the beam cross-section. The governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier type, closed form solution is adopted. Classical beam theories, such as Euler-Bernoulli's and Timoshenko's, are obtained as particular cases. Beams that undergo bending and torsional loadings are investigated. Several values of the span-to-height ratio are considered. Slender as well as deep beams are, therefore, investigated. Comparisons with elasticity solutions and three-dimensional finite element models are given. The numerical investigation shows that the proposed unified formulation yields the complete three-dimensional displacement and stress fields as long as the appropriate approximation order is considered. The accuracy of the solution depends upon the geometrical parameters of the beam and the loading conditions.

Numerical Solution of Bending Problem of a FGM Cantilever Beam with Large Deformation

Nonlinear bending problem of FGM cantilever beam under distributed load are discussed in this paper. Based on the large deformation theory and considering the axial extension of the beam, the equilibrium equations with geometric nonlinearity of a FGM beam subjected to distributed load are established. They consist of a boundary value problem of ordinary differential equations with strong non-linearity, in which seven unknown functions are included and the arc length of the deformed axis is considered as one of the basic unknown functions. By using shooting method and analytical continuation, the nonlinear boundary-value problem is numerically solved as well as the non-linear bending characteristic curves of the deformed beam versus the load are presented.

Analysis of FGM beams by means of a unified formulation

This paper proposes several axiomatic refined theories for the linear static analysis of beams made of functionally graded materials. A bi-directional variation upon the cross-section is accounted for. Via a unified formulation, a generic N-order approximation is assumed for the displacement unknown variables over the beam cross-section. The governing differential equations and the boundary conditions are derived in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier type, closed form solution is adopted. Beams undergo bending and torsional loadings. Deep beams are investigated. Comparisons with three-dimensional finite element models are given. The numerical investigation shows that the proposed unified formulation yields the complete three-dimensional displacement and stress fields as long as the appropriate approximation order is considered.

Efficient modeling of smart piezoelectric composite laminates: a review

Current research issues in the development of efficient analysis models and their efficient numerical implementation for smart piezoelectric laminated structures are discussed in this paper. The improved zigzag theories with a layerwise quadratic variation of electric potential have emerged as the best compromise between accuracy and cost for hybrid composite, sandwich and FGM beams and plates. The concept of associating surface potentials to electric nodes and internal potentials to physical nodes is very effective in modeling the equipotential electroded surfaces. Unified formulations for shear and extension mode actuation, and modeling of piezoelectric composite actuators and sensors are discussed. Future challenge lies in developing efficient theories capable of predicting the interlaminar transverse shear stresses in hybrid laminates directly from the constitutive equations.

Electro-elastic stresses in composite active beams with functionally graded layer

Analytical expressions have been derived for the through thickness stresses of a composite active FGM beam subjected to electrical excitation. The structure is comprised of a substrate, an electro-elastically graded layer and an active layer. Continuous gradation of the volume fraction in the FGM layer is modelled in the form of an m th power polynomial of the coordinate axis in thickness direction of the beam. A numerical scheme of discretizing the continuous FGM layer (in sub-layers) and treating the beam as a discretely graded structure has also been developed. Appropriate expressions for the solution have been derived for the case of continuous power law gradation ( m th power) of the FGM layer. The discretized FGM layer scheme has been shown to yield results that practically match those predicted analytically by the closed-form model.

Analyses of tapered fgm beams with nonlocal theory

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.

Theoretical Modeling and Experimental Validation of Thermal Response of Metal-Ceramic Functionally Graded Beams

A third-order zigzag theory based finite element model in conjunction with the modified rule of mixtures and Wakashima–Tsukamoto model for estimating effective modulus of elasticity and coefficient of thermal expansion, respectively, is presented for layered functionally graded beams under thermal loading. The model is validated through experiments with two systems, Al/SiC and Ni/Al2O3, fabricated using powder metallurgy and thermal spraying techniques, respectively. The predicted thermal deflections for simply supported and cantilever FGM beams are found to be in good agreement with the experimental values for both systems. For nonlinear variation of FGM composition across the thickness, two models for thickness discretization with equal thickness and equal change in volume fraction, respectively, are evaluated in terms of magnitude of axial stress and its jump at the interfaces. The effect of inhomogeneity parameter and number of layers in the FGM on the reduction of thermal stress and its jump at the interfaces is investigated.

Two-dimensional elasticity solutions for functionally graded beams resting on elastic foundations

Abstract Exact solutions for bending and free vibration of functionally graded beams resting on a Winkler–Pasternak elastic foundation are presented based on the two-dimensional theory of elasticity. The beam is assumed orthotropic at any point, while material properties varying exponentially along the thickness direction. The system of governing partial differential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into an infinite trigonometric series. The problem is finally solved using the state space method, which is validated by comparing the present results to those available in the literature. Effects of several parameters, such as gradient index, aspect ratios, and foundation parameters on mechanical behavior of FGM beams are investigated. Numerical results are presented to serve as benchmarks for future analyses of such beams.