The intention of this study is the analysis of thermal behavior of Functionally Graded Beam (FGB). The distribution of material properties is imitated exponential function. For thermal loading the steady state of heat conduction with exponentially and hyperbolic variations through the thickness of FGB, is considered. With comparing of thermal behavior of both isotropic Beam and FGB, it is appeared that the quality of temperature distribution plays very important part in thermal resultant distribution of stresses and strains for FGB. So that, for detecting the particular thermal behavior of FGB, the function of heat distribution must be same as function of material properties distribution. In addition, In the case of exponential distribution of heat with no mechanical loads, in spite of the fact that the bending is accrued, the neutral surface does not come into existence.

Nonlinear behavior of a Functionally Graded cantilever Beam is analyzed under non-uniform hygrothermal effect. To solve this problem, finite element method is applied within plane solid continua. Total Lagrangian approach is utilized in the nonlinear kinematic relations. Newton-Raphson method with incremental displacement is used in nonlinear solution. Comparison study is performed. Effects of material distribution, temperature and moisture changes on nonlinear deflections of the Functionally Graded Beam are presented and discussed.

In this paper, vibration control of a cracked, functoinally Graded, uncertain Beam allocated in a thermal environment has been investigated. For this purpose, piezoelectric patches are used as sensors to measure the displacement of the Beam and also as actuators to apply control forces. In this way, firstly, partial differential equation governing the dynamics of the system is derived by considering the Euler-Bernoulli assumption using Lagrange method. Approximate solution of eigenvalue equation is achieved using Rayleigh–Ritz method. After that, time dependent ordinary differential equations is obtained using Galerkin projection scheme and then represented in the state-space form. Based on this model, a robust observer based output feedback controller is designed for this continuous-time model. In this regard, controller and observer gains are designed by a Lyapunov-based method. This procedure is done by solving a set on linear matrix inequalities. Simulation studies show the effectiveness of the proposed method.

In this paper, the buckling and vibration behaviour of Functionally Graded flexoelectric nanoBeam is examined. The vibration and buckling formulations of Functionally Graded nanoBeam are developed by using a new theory that’ s presented exclusively for flexoelecteric nano-materials. So by considering Von-Karman strain and forming enthalpy equation based on displacement, polarization and electric potential, electromechanical coupling equations are developed base on Hamilton’ principle. By considering boundary condition of simply support and clamped-clamped and also Euler-Bernoulli Beam model, pre-buckling, buckling and the vibration behavior of Functionally Graded nanoBeam affected by flexoelectric will be investigated.

Traditional engineering materials lack the necessary properties that are needed in aerospace as well as other modern industries. In order to address the aforementioned problem, a number of different materials are used in concert. With the help of Functionally Graded material, all the necessary characteristics can be achieved. A fast transition between two different materials can lead to debonding, thermal stresses, residual stresses, and stress concentrations; a gradual change in material properties might mitigate these problems. This work provides a comparison of the analytical solutions for the deflections in Two Dimensional Functionally Graded Taper Beam (2D-FGTB) under a uniformly distributed load, adapting Reddy’s higher-order shear deformation theory. All the material properties of the Beam are Graded along the thickness and length dimensions using the power-law formula. The thickness of the Beam is assumed to change linearly along its length. Equations of motion are derived based on Hamilton's principle and Navier’s solutions. A parametric investigation is conducted to explore the effects of various material and geometrical parameters on the mechanics of 2D-FGTB. These parameters are found to be very significant in studying the static responses of 2D-FGTB.

In this article, in reference to the modified couple stress theory and Euler-Bernoulli Beam theory, the free lateral vibration response of a micro-Beam carrying a moveable attached mass is investigated. This is a decent model for biological and biomedical applications beneficial to the early-stage diagnosis of diseases and malfunctions of human body organs and enzymes. The micro-cantilever Beam is composed of Functionally Graded materials (FGMs). The material properties are supposed to show variations through-thickness of the Beam in consonance to the power of law. Rayleigh-Ritz method is applied in order to explore the natural frequencies of the first three vibration modes. In order to manifest the accuracy of the proposed method, the results are established and juxtaposed with technical literature. Influences of the material length-scale parameter that captures the size-dependency, ratio of the mass of the Beam to the mass of the attached mass and power index of the Graded material consequent to the vibrational behavior of the system are contemplated. This technical research denotes the value of the material gradation besides to the inertia of an attached mass in the dynamic behavior of the bio-micro-systems. As a result, the adoption of suitable power index, mass ratio and position of the attached mass lead to the superior design of bio-micro-systems persuading early-stage diagnostics.

In this paper, the analysis of nonlinear free vibrations of Beams made of Functionally Graded materials with magnetorheological fluid as core is investigated. It is assumed that the Beam is made of three layers including constraining layer, magnetorheological fluid and base layer and is located on Simply-Simply, Clamped-Simply and Clamped–, Clamped supports. The governing equations of the Beam are derived using the Hamilton’, s principle. To obtain the vibrational frequencies, the theory of Timoshenko Beam is used by the Generalized Differential Quadrature method. The effects of magnetic field intensity, power law exponents, core thickness and constraining layer thickness and the length of the Beam on natural frequency and modal loss factor related to different frequencies modes for the three boundary conditions have been investigated. The results show the effects of physical and geometrical parameters regarding the natural frequency and modal loss factor of the sandwich Beam with different modes. Also, the frequency and loss factor values obtained from Generalized Differential Quadrature method are very close to the results obtained by the Finite Element method. This shows the accuracy and precision of this method.

In this paper buckling response of a sandwich (SW) Beam containing Functionally Graded skins and metal (Type-S) or ceramic core (Type-H) is investigated using a third-order zigzag theory. The variation of material properties in Functionally Graded (FG) layers is quantified through exponential and power laws. The displacements are assumed using higher-order terms along with the zigzag factors to evaluate the effect of shear deformation. In-plane loads are considered. The governing equations are derived using the principle of virtual work. The model achieves stress-free boundaries unlike higher-order shear deformation theories and is C0 continuous so, does not require any post-processing method. The present model shows an accurate variation of transverse stresses in thickness direction due to the inclusion zigzag factor in assumed displacements and is independent of the number of layers in computing the results. Numerical solutions are arrived at by using three noded finite elements with 7DOF/node for sandwich Beams. The novelty of the paper lies in presenting a zig-zag buckling analysis for the FGSW Beam with thickness stretching. This paper presents the effects of the power law factor, end conditions, aspect ratio, and lamination schemes on the buckling response of FGM sandwich Beams. The numerical results are found to be in accordance with the existing results. The buckling strength was improved by increasing the power law factor for Type S Beams while the opposite behavior was seen in type H Beams for all types of end conditions. The end conditions played a major role in deciding the buckling response of FGSW Beams. Exponential law governed FGSW Beam exhibited a little higher buckling resistance for Type S Beams, while a little lower buckling resistance was found for Type S Beams for almost all lamination schemes and end conditions. Some new results are also presented which will serve as a benchmark for future research in a parallel direction.

In this paper, robust vibration control of a thin Functionally Graded Beam with a variable cross-section has been investigated. For this purpose, piezoelectric patches are used as sensors to measure the displacement of the Beam and as actuators to apply control forces. In this way, firstly, Euler-Bernoulli theory is used to derive the governing dynamical partial differential equation, through the Hamilton’s principle. Approximate solution of these equations is achieved using finite difference method, and the proper orthogonal decomposition is then used to obtain vibration mode shapes. After that, time-dependent ordinary differential equations are attained using Galerkin projection scheme and then represented in the statespace form. Since the data measurement is done in sampling intervals, the system is considered as sampleddata. In this way, direct digital control design methodology is used. For this purpose, based on its zero-order hold equivalent model, a robust discrete-time, observer-based, output feedback controller is designed. In this regard, controller and observer gains are designed by a Lyapunov-based method. This procedure is done by solving a set of linear matrix inequalities. Simulation studies show the effectiveness of the proposed method.

Collocation methods are popular in providing numerical approximations to complicated governing equations owing to their simplicity in implementation. However, point collocation methods have limitations regarding accuracy and have been modified upon with the application of B-spline approximations. The present study reports the stress and deformation behavior of shear deformable Functionally Graded cantilever Beam using B-spline collocation technique. The material grading is along the Beam height and varies according to power law. Poisson’ s ratio is assumed to be a constant. The equations are derived using virtual work principle in the framework of Timoshenko Beams to obtain a unified formulation for such Beams. A sixth order basis function is used for approximation and collocation points are generated using Greville abscissa. Deformation and stresses; bending (axial) stresses and transverse (shear) stresses, and position of neutral axis are studied for a wide range of power law index values. The results are reported along the Beam cross-section and Beam length.