• Steel and Composite Structures
• /
• v.24 no.5
• /
• pp.579-589
• /
• 2017

The objective of this work is to analyze post-buckling of functionally graded (FG) beams with porosity effect under compression load. Material properties of the beam change in the thickness direction according to power-law distributions with different porosity models. It is known that post-buckling problems are geometrically nonlinear problems. In the nonlinear kinematic model of the beam, total Lagrangian finite element model of two dimensional (2-D) continuum is used in conjunction with the Newton-Raphson method. In the study, the effects of material distribution, porosity parameters, compression loads on the post-buckling behavior of FG beams are investigated and discussed with porosity effects. Also, the effects of the different porosity models on the FG beams are investigated in post-buckling case.

• Smart Structures and Systems
• /
• v.13 no.4
• /
• pp.637-657
• /
• 2014

This paper presents a dynamic analysis of three-dimensional beams. Structures made of functionally graded materials are considered. Several higher-order as well as classical theories are derived by means of a compact notation for the a-priori expansion order of the displacement field over the beam cross-section. The governing differential equations and boundary conditions are obtained in a condensed nuclear form that does not depend on the kinematic hypotheses. The problem is, then, exactly solved in space by means of a Navier-type solution, whereas time integration is performed by means of Newmark's solution scheme. Slender and short simply supported beams are investigated. Results are validated towards three-dimensional FEM results obtained via the commercial software ANSYS. Numerical investigations show that good accuracy can be obtained through the proposed formulation provided that the appropriate expansion order is considered.

• Structural Engineering and Mechanics
• /
• v.20 no.4
• /
• pp.465-476
• /
• 2005

Free vibration of orthotropic functionally graded beams, whose material properties can vary arbitrarily along the thickness direction, is investigated based on the two-dimensional theory of elasticity. A hybrid state-space/differential quadrature method is employed along with an approximate laminate model, which allows us to obtain the semi-analytical solution easily. With the introduction of continuity conditions at each fictitious interface and boundary conditions at the top and bottom surfaces, the frequency equation for an inhomogeneous beam is derived. A completely exact solution of an FGM beam with material constants varying in exponential way through the thickness is also presented, which serves a benchmark to verify the present method. Numerical results are performed and discussed.

• Structural Engineering and Mechanics
• /
• v.69 no.6
• /
• pp.637-649
• /
• 2019

In the current research paper, a quasi-3D beam theory is developed for free vibration analysis of functionally graded microbeams. The volume fractions of metal and ceramic are assumed to be distributed through a beam thickness by three functions, power function, symmetric power function and sigmoid law distribution. The modified coupled stress theory is used to incorporate size dependency of micobeam. The equation of motion is derived by using Hamilton's principle, however, Navier type solution method is used to obtain frequencies. Numerical results show the effects of the function distribution, power index and material scale parameter on fundamental frequencies of microbeams. This model provides designers with guidance to select the proper distributions and functions.

• Earthquakes and Structures
• /
• v.17 no.1
• /
• pp.31-37
• /
• 2019

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's principle. The Navier solution technique is adopted to derive analytical solutions for simply supported beams. The accuracy and effectiveness of proposed model are verified by comparison with previous research.

• Computers and Concrete
• /
• v.30 no.4
• /
• pp.243-256
• /
• 2022

The static analysis of spinning functionally graded (FG) nanotube on the basis of the nonlocal strain gradient theory (NSGT) is presented. The high-order beam theory is employed for mathematical modeling of the tube structures according to the Sinusoidal shear deformation beam theory. The energy conservation principle is operated to generate the equations. The centrifugal force is assumed along the tube length due to the rotating of the tube, moreover, the nanotube is made of functionally graded material (FGM) composed of ceramic and metal phases along the tube radius direction. The generalized differential quadratic method (GDQM) is utilized to solve the formulations. Finally, the numerical results are discussed in detail to examine the impact of different relevant parameters on the bending the buckling behavior of the rotating nanotube.

• Steel and Composite Structures
• /
• v.44 no.4
• /
• pp.557-567
• /
• 2022

Nonlinear free vibration analysis of a functionally graded beam resting on the nonlinear viscoelastic foundation is studied with uniform temperature rising. The non-linear strain-displacement relationship is considered in the finite strain theory. The governing nonlinear dynamic equation is derived based on the finite strain theory with using of Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The influences of temperature rising, material distribution parameter, nonlinear viscoelastic foundation parameters on the nonlinear free response and phase trajectory are investigated. In this paper, it is aimed that a contribution to the literature for nonlinear thermal vibration solutions of a functionally graded beam resting on the nonlinear viscoelastic foundation by using of multiple time scale method.

• Steel and Composite Structures
• /
• v.44 no.4
• /
• pp.451-471
• /
• 2022

In the present paper, the static bending and buckling responses of functionally graded carbon nanotubes-reinforced composite (FG-CNTRC) beam under various boundary conditions are investigated within the framework of higher shear deformation theory. The significant feature of the proposed theory is that it provides an accurate parabolic distribution of transverse shear stress through the thickness satisfying the traction-free boundary conditions needless of any shear correction factor. Uniform (UD) and four graded distributions of CNTs which are FG-O, FG-X, FG- and FG-V are selected here for the analysis. The effective material properties of FG-CNTRC beams are estimated according to the rule of mixture. To model the FG-CNTRC beam realistically, an efficient Hermite-Lagrangian finite element formulation is successfully developed. The accuracy and efficiency of the present model are demonstrated by comparison with published benchmark results. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of CNTs volume fraction, distribution patterns of CNTs, boundary conditions, and length-to-thickness ratio on the bending and buckling responses of FG-CNTRC beam. Several new referential results are also reported for the first time which will serve as a benchmark for future studies in a similar direction. It is concluded that the FG-X-CNTRC beam is the strongest beam that carries the lowest central deflection and is followed by the UD, V, Λ, and FG-O-CNTRC beam. Besides, the critical buckling load belonging to the FG-X-CNTRC beam is the highest, followed by UD and FG-O.

• Structural Engineering and Mechanics
• /
• v.63 no.5
• /
• pp.683-689
• /
• 2017

In this paper, a hyperbolic shear deformation theory is presented for bending analysis of functionally graded beams. This theory used in displacement field in terms of thickness co-ordinate to represent the shear deformation effects and does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The governing equations are derived by employing the virtual work principle and the physical neutral surface concept. A simply supported functionally graded beam subjected to uniformly distributed loads and sinusoidal loads are consider for detail numerical study. The accuracy of the present solutions is verified by comparing the obtained results with available published ones.

• Structural Engineering and Mechanics
• /
• v.81 no.6
• /
• pp.705-714
• /
• 2022

The aim of this paper is to investigate nonlinear dynamic responses of functionally graded composite beam resting on the nonlinear viscoelastic foundation subjected to moving mass with temperature rising. The non-linear strain-displacement relationship is considered in the finite strain theory and the governing nonlinear dynamic equation is obtained by using the Hamilton's principle. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then the governing equation is solved by using of multiple time scale method. The influences of temperature rising, material distribution parameter, nonlinear viscoelastic foundation parameters, magnitude and velocity of the moving mass on the nonlinear dynamic responses are investigated. Also, the buckling temperatures of the functionally graded beams based on the finite strain theory are obtained.