Dynamics of beams made of axially grading material has been analyzed in present work. Shear deformation and rotational inertia of the rectangular cross-sectional beam have been considered using Timoshenko-Ehrenfest beam model. Material properties of the beam have been assumed as a power-law function. Solution of the vibration problem of the axially functionally graded Timoshenko-Ehrenfest beam has been carried out with Ritz formulation. Present model has been validated with the previous literature works. Effects of power-law index parameter and grading material properties on the dynamics of axially functionally graded Timoshenko-Ehrenfest beam have been investigated. Transverse deflection and slope of the beam have been depicted in various cases. Present study can give useful results for designing of axially graded structural elements.

This work presents forced vibration responses of a cantilever beam made of functionally graded material under a harmonic load. The material properties of beam vary along the axial direction. The kinematics of the beam are considered within Timoshenko beam theory. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method is used in the time history. In this study, free and forced vibration responses of the axially functionally graded beam are investigated in detail. In the numerical examples, the effects of material graduation, geometric and dynamic parameters on the free and forced vibration response of axially graded beam are investigated.

The aim of this paper is to investigate geometrically nonlinear static analysis of axially functionally graded cantilever beam subjected to transversal non follower load. The considered problem is solved by finite element method with total Lagrangian kinematic approach. The material properties of the beam vary along the longitudinal direction according to the power law function. The finite element model of the beam is considered in the three dimensional continuum approximation for an eight-node quadratic element. The geometrically nonlinear problem is solved by Newton-Raphson iteration method. In the numerical results, the effects of the material distribution on the geometrically nonlinear static displacements of the axially functionally graded beam are investigated. Also, the differences between of material distributions are investigated in geometrically analysis.

Owing to efficiency, rising stability of structures, reduction in structural weight and cost, and the improvements in fabrication process, non-uniform beams are extensively adopted in civil and mechanical structures. Furthermore, the use of functionally graded (FG) materials has been increasing in many mechanical components due to their conspicuous characteristics such as high strength, thermal resistance and optimal distribution of weight. In the present paper, a numerical model combining the power series expansions and the Rayleigh-Ritz method is adopted for stability and free vibration analyses of axially functionally graded (FG) columns with variable cross-section. The main purpose of this paper is calculating the critical buckling loads and natural frequencies concurrently for AFG members with exponentially-varying geometrical properties. For this, a mixed power series expansions and the principle of stationary total potential energy as a first endeavor is presented. In this study, the material properties of the non-prismatic beam including Young’s modulus of elasticity and density of material are assumed to be graded smoothly along the beam axis by a power-law distribution of volume fractions of metal and ceramic. Moreover, the cross-sectional area and moment of inertia vary exponentially over the member’s length. In this regard, the power series approximation is applied to solve the fourth order differential equation of motion, since in the presence of variable cross-section and axially non-homogeneous material, stiffness quantities are not constant. All geometrical and material properties and displacement component are developed based on power series of an identified degree. The natural frequencies of the AFG beam with variable cross-section are derived by imposing the boundary conditions and solving the eigenvalue problem. The explicit expression of vibrational shape function is then derived based on this rigorous numerical method. The vibrational mode shapes of an elastic member are similar to the buckling ones. Therefore, the obtained deflected shapes of the considered non-prismatic beams can be used as deformation shape of member for the linear buckling analysis. The critical buckling load of non-prismatic beam can be then estimated by adopting the principle of stationary total potential energy. According to the steps mentioned above, for measuring the accuracy and competency of the proposed numerical procedure, two numerical examples including axially non-homogeneous and homogeneous column with non-uniform section are represented. Numerical results of the critical buckling loads and natural frequencies for various boundary conditions, different gradient index and cross-section variation are represented. Due to lack of similar research for the stability and free vibration analyses of elastic AFG beams with exponential variation of the cross-sectional area and moment of inertia, outcomes of homogeneous members are compared with the results presented in other available numerical and analytical references and those related to non-prismatic beams with material variation are then reported. The accuracy of the method is then remarked. This method has many positive points consisting of efficiency, accuracy and simplicity contrasted with more complex numerical methods. It has to be noticed that the present numerical method can be applied to determine the critical buckling loads and natural frequencies of axially functionally graded (FG) prismatic beams as well as non-prismatic ones.

In this paper, the critical buckling loads and natural frequencies of axially functionally graded non-prismatic Timoshenko beam with different boundary conditions are acquired using the Finite Difference Method (FDM). In the recent years, the use of functionally graded materials (FGMs) has been increasing in different mechanical components due to their conspicuous characteristics such as high strength, thermal resistance and optimal distribution of weight. The designer can thus produce structures with favorable stability and manage the distribution of material properties. In this study, the material properties of non-prismatic Timoshenko beams such as Young’s modulus of elasticity and density of material are described by a power-law formulation along the beam axis. Contemplating elastic behavior, the system of equilibrium equations of non-uniform Timoshenko beam and the related boundary conditions are coupled in terms of the vertical displacement and the bending rotation of the cross-section. Afterwards, the system of second-order differential equations with variable coefficients and end conditions are discretized by finite difference formulations with second-order accuracy. Finally, the system of finite difference equations culminates in a set of simultaneous and linear equations and the critical buckling loads and natural frequencies are calculated by solving an eigenvalue problem of the obtained algebraic system. In order to demonstrate the accuracy and reliability of this approach to calculate elastic buckling loads and natural frequencies of functionally graded (FG) Timoshenko beams, one comprehensive example including axially non-homogeneous and homogeneous members with non-uniform cross-section is expressed. The numerical results in clamped-clamped, simply supported and fixed-free boundary conditions are accomplished. Moreover, the effect of various parameters such as volume fraction index, end conditions and the section variation on the elastic buckling and free vibration behavior of AFG Timoshenko beam are investigated in detail.

The vibrational behavior of an axially graded micro-beam with both axial and rotational motion under axial load have been studied in the magnetic field. A detailed parametric study was performed to explain the effect of various factors such as range of axial graded of materials, type of material distribution, viscosity coefficient, and magnetic field strength, length scale parameter of material, rotation and coupling crossing motion on the dynamical characteristics of the system. It is assumed that the material properties of the system change linearly or exponentially in the longitudinal direction. The critical axial and rotational speeds of the system are obtained by using Galerkin discretization technique and eigenvalue analysis. An analytical method has also been used to identify system instability thresholds. System stability maps were tested, and it has been shown that by increasing the magnetic field strength and the length scale parameter, the system stability can be improved. The results also show that the simultaneous determination of density gradient and elastic modulus in the longitudinal direction can reduce the destructive effects of compressive axial load.

In this study, an efficient finite element model with two degrees of freedom per node is developed for buckling analysis of axially functionally graded (AFG) tapered Timoshenko beams resting on Winkler elastic foundation. For this, the shape functions are exactly acquired through solving the system of equilibrium equations of the Timoshenko beam employing the power series expansions of displacement components. The element stiffness matrix is then formulated by applying the developed shape functions to the total potential energy along the element axis. It is demonstrated that the resulting shape functions, in comparison with Hermitian cubic interpolation functions, are proportional to the mechanical features of the beam element, including the geometrical properties, material characteristics, as well as the critical axial load. An exhaustive numerical example is implemented to clarify the efficiency and simplicity of the proposed mathematical methodology. Furthermore, the effects of end conditions, material gradient, Winkler parameter, tapering ratio, and aspect ratio on the critical buckling load of AFG tapered Timoshenko beam are studied in detail. The numerical outcomes reveal that the elastic foundation enhances the stability characteristics of axially non-homogeneous and homogeneous beams with constant or variable cross-section. Moreover, the results show that the influence of non-uniformity in the cross-section and axially inhomogeneity in material characteristics play significant roles in the linear stability behavior of Timoshenko beams subjected to different boundary conditions.

This paper addresses the linear buckling behavior of tapered Timoshenko beams made of axially functionally graded material (AFGM). Material properties of the non-prismatic Timoshenko beam vary continuously along the length of the member according to power law as well as exponential law. Based on Timoshenko beam assumption and using small displacements theory, the linear equilibrium equations are adopted from the energy principle for functionally graded non-uniform Timoshenko beams related to constant compressive axial load. The resulting system of stability equations are strongly coupled due to the presence of transverse deflection and angle of rotation. The differential equations are then uncoupled and lead to a single homogeneous differential equation where only bending rotation is present. The finite difference method (FDM) is then selected to numerically solve the resulting second-order differential equation with variable coefficients and determine the critical buckling loads. Two numerical examples are finally conducted to demonstrate the performance and efficiency of this mathematical procedure. The obtained results are contrasted with accessible numerical and analytical benchmarks.

In this paper, in order to improve the efficiency of the moving systems, vibrations and stability of axially functionally graded Rayleigh moving micro-beams are studied. Also, to clarify the influences of various parameters such as axially functionally graded, the length of the material characteristics, and the whirling inertia on the stability boundaries of Rayleigh and Euler-Bernoulli beams, a detailed parametric study is done. It is assumed that the material characteristics of the system change linearly or exponentially in longitudinal direction continuously. To calculate the natural frequencies, dynamics configuration, and divergence and flutter instability thresholds of the system, the strain gradient theory, Galerkin discretization method, and an eigenvalue problem are utilized. In addition, the analytical relations are extracted for the critical velocity of the system. Critical velocity contours and stability maps are examined for different distributions. It is demonstrated that the exponential and linear changes lead to a more stable system in the variable state of density and elastic modulus, respectively. Also, the results indicated that increasing the elastic modulus gradient parameter or decreasing the density gradient parameter results in an increase in the natural frequency of the system and a development in the stability regions. Hence, the alteration in the density and elastic modulus gradient parameters has an opposite role in the dynamic behavior of the system.