Bending and vibration analyses of coupled axially functionally graded tapered beams

The nonlinear bending and vibrations of tapered beams made of axially functionally graded (AFG) material are analysed numerically. For a clamped–clamped boundary conditions, Hamilton’s principle is employed so as to balance the potential and kinetic energies, the virtual work done by the damping, and that done by external distributed load. The nonlinear strain–displacement relations are employed to address the geometric nonlinearities originating from large deflections and induced nonlinear tension. Exponential distributions along the length are assumed for the mass density, moduli of elasticity, Poisson’s ratio, and cross-sectional area of the AFG tapered beam; the non-uniform mechanical properties and geometry of the beam along the length make the system asymmetric with respect to the axial coordinate. This non-uniform continuous system is discretised via the Galerkin modal decomposition approach, taking into account a large number of symmetric and asymmetric modes. The linear results are compared and validated with the published results in the literature. The nonlinear results are computed for both static and dynamic cases. The effect of different tapered ratios as well as the gradient index is investigated; the numerical results highlight the importance of employing a high-dimensional discretised model in the analysis of AFG tapered beams.     

Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams

The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the corresponding homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The deflection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams (FGMLBs) are given analytically in terms of the deflection of the reference homogenous Euler-Bernoulli beams (HEBBs) with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily determined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be easily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.     

Nonlinear dynamic instability of edge-cracked functionally graded graphene-reinforced composite beams

Nonlinear Dynamics - This work investigates nonlinear dynamic instability of edge-cracked functionally graded (FG) graphene nanoplatelet (GNP)-reinforced composite (GNPRC) beams consist of...     

Efficient higher-order shear deformation theories for bending and free vibration analyses of functionally graded plates

In this paper, various efficient higher-order shear deformation theories are presented for bending and free vibration analyses of functionally graded plates. The displacement fields of the present theories are chosen based on cubic, sinusoidal, hyperbolic, and exponential variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theories is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton’s principle. Analytical solutions for deflections, stresses, and frequencies are obtained for simply supported rectangular plates. The accuracy of the present theories is verified by comparing the obtained results with the exact three-dimensional (3D) and quasi-3D solutions and those predicted by higher-order shear deformation theories. Numerical results show that all present theories can archive accuracy comparable to the existing higher-order shear deformation theories that contain more number of unknowns.     

High-order shear theory for static analysis of functionally graded plates with porosities

The bending responses of porous functionally graded (FG) thick rectangular plates are investigated according to a high-order shear deformation theory. Both the effect of shear strain and normal deformation are included in the present theory and so it does not need any shear correction factor. The equilibrium equations according to the porous FG plates are derived. The solution to the problem is derived by using Navier's technique. Numerical results have been reported and compared with those available in the open literature for non-porous plates. The effects of the exponent graded and porosity factors are investigated.     

Exact solution for in-plane static problems of circular beams made of functionally graded materials

Exact analytical solutions of in-plane static problems of circular beams with uniform cross-section made of functionally graded material (FGM) are obtained. Material properties are assumed to be varying arbitrarily through the thickness. The effects of axial extension and shear deformations are considered. The differential equation system is solved exactly using the initial values method. The circumferential stress distribution on the cross-section is also obtained. The results are compared with those of rather complex approaches in the literature, such as elasticity approach, and the comparison shows an excellent agreement. Effects of power law exponent and radius-to-height ratio of the beam on circumferential stress distribution and displacements are investigated.     

Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation

This paper presents a geometric nonlinear analysis formulation for beams of functionally graded cross-sections by means of a Total Lagrangian formulation. The influence of material gradation on the numerical response is investigated in detail. Two examples are given that illustrate the main features of the formulation, in which the behavior of beams of graded cross-sections is compared with homogeneous material beams. A motivation for this work is the potential development of functionally graded risers for the offshore oil exploration industry.     

Geometric nonlinear analyses of functionally graded beams using a tailored Lagrangian formulation

This paper presents a geometric nonlinear analysis formulation for beams of functionally graded cross-sections by means of a Total Lagrangian formulation. The influence of material gradation on the numerical response is investigated in detail. Two examples are given that illustrate the main features of the formulation, in which the behavior of beams of graded cross-sections is compared with homogeneous material beams. A motivation for this work is the potential development of functionally graded risers for the offshore oil exploration industry.     

Accurate dynamic analysis of functionally graded beams under a moving point load

Based on the physical neutral surface, an N-node novel weak form quadrature beam element is proposed and the explicit formulas for computing the stiffness and mass matrices are given. The proposed element is then used to analyze the dynamic behavior of the functionally graded material (FGM) beams under a moving point load. Both elasticity modulus and mass density vary exponentially across the thickness. Investigations show that the maximum dynamic magnification factors are independent of the power-law exponent k at a fixed nondimensional parameter α. This finding may be useful in design and engineering applications.     

Elasticity solutions for plane anisotropic functionally graded beams

This paper considers the plane stress problem of generally anisotropic beams with elastic compliance parameters being arbitrary functions of the thickness coordinate. Firstly, the partial differential equation, which is satisfied by the Airy stress function for the plane problem of anisotropic functionally graded materials and involves the effect of body force, is derived. Secondly, a unified method is developed to obtain the stress function. The analytical expressions of axial force, bending moment, shear force and displacements are then deduced through integration. Thirdly, the stress function is employed to solve problems of anisotropic functionally graded plane beams, with the integral constants completely determined from boundary conditions. A series of elasticity solutions are thus obtained, including the solution for beams under tension and pure bending, the solution for cantilever beams subjected to shear force applied at the free end, the solution for cantilever beams or simply supported beams subjected to uniform load, the solution for fixed–fixed beams subjected to uniform load, and the one for beams subjected to body force, etc. These solutions can be easily degenerated into the elasticity solutions for homogeneous beams. Some of them are absolutely new to literature, and some coincide with the available solutions. It is also found that there are certain errors in several available solutions. A numerical example is finally presented to show the effect of material inhomogeneity on the elastic field in a functionally graded anisotropic cantilever beam.     

Semi-analytical elasticity solutions for bi-directional functionally graded beams

Elasticity solutions are presented for bending and thermal deformations of functionally graded beams with various end conditions, using the state space-based differential quadrature method. The beams are assumed to be macroscopically isotropic, with Young’s modulus varying exponentially along the thickness and longitudinal directions, while Poisson’s ratio remaining constant. The state space method is adopted to obtain analytically the thickness variation of the elastic field and, when coupled with differential quadrature, the longitudinal discretization can be analyzed in an approximate manner. This approach is then validated by comparing the numerical results with the exact solutions for a special functionally graded beam and with finite element solutions. The influences of material gradient indices on the response of bi-directional functionally graded beams are finally investigated.     

Static and vibration analysis of functionally graded beams using refined shear deformation theory

Static and vibration analysis of functionally graded beams using refined shear deformation theory is presented. The developed theory, which does not require shear correction factor, accounts for shear deformation effect and coupling coming from the material anisotropy. Governing equations of motion are derived from the Hamilton’s principle. The resulting coupling is referred to as triply coupled axial-flexural response. A two-noded Hermite-cubic element with five degree-of-freedom per node is developed to solve the problem. Numerical results are obtained for functionally graded beams with simply-supported, cantilever-free and clamped-clamped boundary conditions to investigate effects of the power-law exponent and modulus ratio on the displacements, natural frequencies and corresponding mode shapes.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

A high-order theory for functionally graded axially symmetric cylindrical shells

A high-order theory for functionally graded axially symmetric cylindrical shell based on expansion of the axially symmetric equations of elasticity for functionally graded materials into Legendre polynomials series has been developed. The axially symmetric equations of elasticity have been expanded into Legendre polynomials series in terms of a thickness coordinate. In the same way, functions that describe functionally graded relations has been also expanded. Thereby, all equations of elasticity including Hook’s law have been transformed to corresponding equations for coefficients of Legendre polynomials expansion. Then system of differential equations in terms of displacements and boundary conditions for the coefficients of Legendre polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in more details. For obtained boundary-value problems’ solution, a finite element has been used and numerical calculations have been done with COMSOL MULTIPHYSICS and MATLAB.     

Microstructure-dependent couple stress theories of functionally graded beams

A microstructure-dependent nonlinear Euler-Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler-Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.     

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of the major threats to their service safety in engineering applications. This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams. First, based on the Euler-Bernoulli beam theory and von Kármán geometric nonlinearity, a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang's two-variable method is formulated. Second, an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finally, based on the free energy theorem, it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermore, the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.     

New assessment on the Saint-Venant solutions for functionally graded beams

A symplectic approach is proposed to investigate the Saint-Venant problem of functionally graded beams with Young's modulus varying exponentially in the axial direction and constant Poisson radio. A matrix state equation is derived with a shift-Hamiltonian operator matrix whose particular eigenvalues are proved to compose the basic solutions of the Saint-Venant problem. The present analyses demonstrate that the Saint-Venant solutions under simple extension and pure bending can be derived using either the direct expansion method or the rigid motion removing method.     

Cross-sectional warping and precision of the first-order shear deformation theory for vibrations of transversely functionally graded curved beams

The warping may become an important factor for the precise transverse vibrations of curved beams. Thus, the first aim of this study is to specify the structural design parameters where the influence of cross-sectional warping becomes great and the first-order shear deformation theory lacks the precision necessary. The outof-plane vibrations of the first-order shear deformation theory are compared with the warping-included vibrations as the curvature and/or thickness increase for symmetric and as...     

A spectrally formulated finite element for wave propagation analysis in functionally graded beams

In this paper, spectral finite element method is employed to analyse the wave propagation behavior in a functionally graded (FG) beam subjected to high frequency impulse loading, which can be either thermal or mechanical. A new spectrally formulated element that has three degrees of freedom per node (based upon the first order shear deformation theory) is developed, which has an exact dynamic stiffness matrix, obtained by exactly solving the homogeneous part of the governing equations in the frequency domain. The element takes into account the variation of thermal and mechanical properties along its depth, which can be modeled either by explicit distribution law like the power law and the exponential law or by rule of mixture as used in composite. Ability of the element in capturing the essential wave propagation behavior other than predicting the propagating shear mode (which appears only at high frequency and is present only in higher order beam theories), is demonstrated. Propagation of stress wave and smoothing of depthwise stress distribution with time is presented. Dependence of cut-off frequency and maximum stress gradient on material properties and FG material (FGM) content is studied. The results are compared with the 2D plane stress FE and 1D Beam FE formulation. The versatility of the method is further demonstrated through the response of FG beam due to short duration highly transient temperature loading.