Showing papers on "Timoshenko beam theory published in 2017"

Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs)

Abstract: This paper studies the nonlinear bending behavior of a novel class of multi-layer polymer nanocomposite beams reinforced with graphene platelets (GPLs) that are non-uniformly distributed along the thickness direction. Nonlinear governing equation is established based on Timoshenko beam theory and von Karman nonlinear strain-displacement relationship. The effective Young's modulus of the nanocomposites is determined by modified Halpin-Tsai micromechanics model. Ritz method is employed to reduce the governing differential equation into an algebraic system from which the static bending solutions can be obtained. A comprehensive parametric study is then conducted, with a particular focus on the influences of distribution pattern, weight fraction, geometry and size of GPLs together with the total number of layers on the linear and nonlinear bending performances of the beams. Numerical results demonstrate the significantly improved bending performance through the addition of a very small amount of GPLs into polymer matrix as reinforcements. It is found that dispersing more GPLs that are in square shape with fewer single graphene layers near the top and bottom surfaces of the beam is the most effective way to reduce bending deflections. Beams with a higher weight fraction of GPLs that are symmetrically distributed in such a way are also less sensitive to the nonlinear deformation.

Effect of thickness stretching and porosity on mechanical response of a functionally graded beams resting on elastic foundations

Abstract: The novelty of this paper is the use of an efficient beam theory for bending, free vibration and buckling analysis of functionally graded material (FGM) beams on two-parameter elastic foundation. The present theory accounts for both shear deformation and thickness stretching effects by a parabolic variation of all displacements across the thickness, and satisfies the stress-free boundary conditions on the upper and lower surfaces of the beam without requiring any shear correction factor. Due to porosities, possibly occurring inside FGMs during fabrication, it is therefore necessary to consider the vibration, bending and buckling behaviors of beams having porosities in this work. The equation of motion for FGM beams is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The validity of the present theory is investigated by comparing some of the present in literature. It can be concluded that the proposed theory is accurate and simple in solving the bending, free vibration and buckling behaviors of FGM sandwich beams.

Buckling Analysis of Smart Size-Dependent Higher Order Magneto-Electro-Thermo-Elastic Functionally Graded Nanosize Beams

Abstract: The present paper examines the thermal buckling of nonlocal magneto-electro-thermo-elastic functionally graded (METE-FG) beams under various types of thermal loading namely uniform, linear and sinusoidal temperature rise and also heat conduction. The material properties of nanobeam are graded in the thickness direction according to the power-law distribution. Based on a higher order beam theory as well as Hamilton's principle, nonlocal governing equations for METE-FG nanobeam are derived and are solved using Navier type method. The small size effect is captured using Eringen's nonlocal elasticity theory. The most beneficial feature of the present beam model is to provide a parabolic variation of the transverse shear strains across the thickness direction and satisfies the zero traction boundary conditions on the top and bottom surfaces of the beam without using shear correction factors. Various numerical examples are presented investigating the influences of thermo-mechanical loadings, magnetic potential, external electric voltage, power-law index, nonlocal parameter and slenderness ratio on thermal buckling behavior of nanobeams made of METE-FG materials.

Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments

Abstract: Modeling and nonlinear vibration analysis of graphene-reinforced composite (GRC) laminated beams resting on elastic foundations in thermal environments are presented. The graphene reinforcements are assumed to be aligned and are distributed either uniformly or functionally graded of piece-wise type along the thickness of the beam. The motion equations of the beams are based on a higher-order shear deformation beam theory and von Karman strain displacement relationships. The beam–foundation interaction and thermal effects are also included. The temperature-dependent material properties of GRCs are estimated through a micromechanical model. A two-step perturbation approach is employed to determine the nonlinear-to-linear frequency ratios of GRC laminated beams. Detailed parametric studies are carried out to investigate the effects of material property gradient, temperature variation, stacking sequence as well as the foundation stiffness on the linear and nonlinear vibration characteristics of the GRC laminated beams.

Static, free and forced vibration of functionally graded (FG) sandwich beams excited by two successive moving harmonic loads

Abstract: This paper examines static, free and forced vibration of functionally graded (FG) sandwich beams under the action of double moving harmonic loads travelling with constant velocities using Timoshenko beam theory (TBT). Three different sandwich beam models with various cross-sectional shape and various boundary conditions are considered. It is assumed that in FG part of sandwich beams, the material properties vary continuously through the thickness of the beam according to simple power-law form. The problem is formulated based on the energy approach. For this purpose, the unknown displacement functions are approximated by using the simple polynomials together with the auxiliary functions for satisfying the essential boundary conditions. The equations of the motion are obtained by using the Lagrange's equations, and solved with the help of the implicit time integration method of Newmark-. In this study, the effects of the different sandwich beam models, boundary conditions, gradient index, the velocity, excitation frequency and the phase angles of the two successive harmonic loads, and the distance between the loads on the mechanical behavior of sandwich beams are discussed in detail. At the same time, extensive static and free vibration results are presented to check the reliability of the present formulation. Good agreement is observed.

Influence of surface effects on vibration behavior of a rotary functionally graded nanobeam based on Eringen's nonlocal elasticity

Abstract: This article presents a free vibration analysis of size-dependent functionally graded rotating nanobeams with all surface effects considerations on the basis of the nonlocal continuum model. By using constitutive differential model of Eringen, the nonlocal elastic behavior is described which enables the present model to become effective in design and analysis of nanoactuators and nanosensors. The material for this work is a functionally graded which according to power law distribution, it is assumed that its bottom surface is aluminum and the top one is silicon. Taking attention to Euler---Bernoulli beam theory, the modeled nanobeam and its equations of motion are derived using Hamilton's principle. Novillity of this work is considering the effects of rotation and surface effects in addition to considering various boundary conditions of the FG nanobeam. The generalized differential quadrature method is used to discretize the model and to get a numerical approximation of the equation of motion. The model is validated by comparing the benchmark results with the obtained ones. Then influence of surfaces effects, nonlocal parameter, angular velocity, volume fraction index and boundary conditions on natural frequency ratio of the rotating FG nanobeams are investigated.

Vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving load

Abstract: This paper studies the vibration of bi-dimensional functionally graded Timoshenko beams excited by a moving concentrated load. The volume fraction of constituent materials is assumed to vary in both the thickness and longitudinal directions by power-law functions. The governing equations of motion based on Timoshenko beam theory are constructed from Hamilton’s principle. A finite element formulation is derived and used in combination with the Newmark method in computing the vibration response. A parametric study is carried out to highlight the effect of the material distribution and moving load speed on the vibration characteristics of the beams. The numerical results show that the two grading indexes which govern the variation of the effective material properties have opposite effect on the natural frequencies, dynamic magnification factor and mid-span axial stress. The influence of the aspect ratio on the dynamic behavior of the beams is also examined and discussed.

Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions

Abstract: In this paper, a unified formulation which is based on a general refined shear deformation beam theory is presented to conduct free vibration analysis of composite laminated beams subjected to general boundary conditions. In the refined theory model, the displacement fields are chosen by including the high-order variation of transverse shear strain through the thickness of the beam and meeting the stress-free boundary conditions on both the top and bottom surfaces. With considering the material couplings and the Poisson's effect, the governing equations and appropriate boundary conditions are derived from the Hamilton's principle. Exact solutions are obtained by employing the method of reverberation ray matrix (MRRM). In order to implement general boundary conditions, the artificial spring boundary technique is introduced in the MRRM to make it suitable for different boundary cases. The present solutions are compared with those available in the literature to confirm their validity. A systematic parameter study for composite beams with various boundary conditions, fiber orientations, lamina numbers and orthotropic ratios is also performed. New results for free vibration involving composite laminated beams with various boundary constraints are also presented for the first time and they may be served as benchmark for researchers in this field.

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets based on a higher-order shear deformation beam theory

Abstract: This article proposes a higher-order shear deformation beam theory for free vibration analysis of functionally graded carbon nanotube-reinforced composite sandwich beams in a thermal environment. The temperature-dependent material properties of functionally graded carbon nanotube-reinforced composite beams are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. The governing equations and boundary conditions are derived by using Hamilton's principle, and the Navier solution procedure is used to achieve the natural frequencies of the sandwich beam in a thermal environment. A parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio, and core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with functionally graded carbon nanotube-reinforced composite face sheets. Numerical results are also presented in order to compare the behavior of sandwich beams including uniformly distrib...

Free Vibration of Edge Cracked Functionally Graded Microscale Beams Based on the Modified Couple Stress Theory

Abstract: In this study, the free vibration analysis of edge cracked cantilever microscale beams composed of functionally graded material (FGM) is investigated based on the modified couple stress theory (MCST). The material properties of the beam are assumed to change in the height direction according to the exponential distribution. The cracked beam is modeled as a modification of the classical cracked-beam theory consisting of two sub-beams connected by a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces to the classical one when the length scale parameter is zero. The problem considered is investigated using the Euler–Bernoulli beam theory by the finite element method. The system of equations of motion is derived by Lagrange’s equations. To verify the accuracy of the present formulation and results, the frequencies obtained are compared with the results available in the literature, for which good agreement is observed. Numerical results are presented to investigate the effect of crack position, beam length, length scale parameter, crack depth, and material distribution on the natural frequencies of the edge cracked FG microbeam. Also, the difference between the classical beam theory (CBT) and MCST is investigated for the vibration characteristics of the beam of concern. It is believed that the results obtained herein serve as a useful reference for research of similar nature.