Sinusoidal Beam Theory Research Articles

Finite element model for free vibration analysis of curved zigzag nanobeams

By using sinusoidal beam theory and doublet mechanics formulation, this paper investigates free vibrations of curved zigzag nanobeams. A finite element doublet mechanics (FEDM) model is developed and corresponding material-length-scale-parameter (MLSP, ηFEM) is defined to solve the problems. The proposed model is validated by comparing the results with those from molecular dynamic simulations (MDS), DM and Eringen’s nonlocal theory. For armchair single-walled CNTs, it is observed from previous study that while the MLSP of Eringen’s nonlocal theory should be calibrated with the variation of length/tube diameter (L/d) for better accuracy with MDS, this value of proposed FEDM does not change. Besides, it does not depend on the boundary conditions unlike the Eringen’s nonlocal theory. The effects of MLSP, slenderness ratio, open angle and boundary conditions on the natural frequencies of curved nanobeams are studied. Some new results are represented for benchmark purpose.

Hygro-thermal buckling of porous FG nanobeams considering surface effects

Hygro-thermal buckling of the porous FG nanobeam incorporating the surface effect is investigated. The even distribution of porosities is assumed in this paper. Various porous FG nanobeam models including classical beam theory (CBT), Timoshenko beam theory (TBT), Reddy beam theory (RBT), sinusoidal beam theory (SBT), hyperbolic beam theory (HBT) and exponential beam theory (EBT) are developed in this paper. The nonlocal strain gradient theory with material length scale and nonlocal parameters is adopted to examine the buckling behavior. The governing equations of the porous FG nanobeam are derived from principle of minimum potential energy. In the numerical examples, the effect of the nonlocal parameter, material length scale parameter, the temperature rise, the moisture concentration, surface effect, material gradient index, and porosity volume fraction on the buckling temperature and moisture are analyzed and discussed in detail. The results show that the stiffness of the beam depends on the relation of size between nonlocal parameter and length scale parameter. The paper will be helpful for the design and manufacture of the FG nanobeam under complex environments.

Comment on the Navier’s solution in “A sinusoidal beam theory for functionally graded sandwich curved beams” (Composite Structures 226 (2019) 111246)

In “A sinusoidal beam theory for functionally graded sandwich curved beams” (Composite Structures 226 (2019) 111246), the Navier’s solution was obtained for a so-called simply-supported curved beam at two ends. However, this case is mathematically indeterminate and hence not worked out. Based on the Timoshenko theory, the governing differential equations are formulated for curved beam problems and then analytically solved. Excellent agreement with the elasticity theory validates the solution for a free-clamped curved beam. Careful study on the simply-supported curved beam shows that the Navier’s solution in the literature is the one when the inappropriate constraint is assumed, which causes a significant error (up to 42%) for a relative large-curvature beam. In this paper, the Navier’s solution is therefore assessed.

A sinusoidal beam theory for functionally graded sandwich curved beams

To the best of the authors’ knowledge, there is no research work available on FG sandwich curved beams in the whole variety of literature. Therefore, the analysis presented on static behaviour of FG sandwich curved beams in this study is the main contribution and the novelty of the present study. A sinusoidal beam theory considering the effects of transverse normal stress/strain is applied for the bending analysis of FG sandwich beams curved in elevation. The beam has functionally graded skins at top and bottom and isotropic core at the middle. Material properties of FG skins are varied through the thickness according to the power law distribution. The present theory accounts for sinusoidal variation of axial strain and cosine distribution of transverse shear strain through the thickness of the beam. Governing equations of equilibrium are derived using Hamilton’s principle. Analytical solutions for simply supported curved beam are obtained using Navier’s solution procedure. The non-dimensional numerical results are obtained for different radii of curvature and various power law coefficients. The present study contributes many new results on the bending analysis of functionally graded sandwich beams curved in elevation for the reference of future research in this area.

Thermo-hygro-mechanical bending and vibration of functionally graded material microbeams with microporosity defect

In this study, the thermo-hygro-mechanical bending and vibration of functionally graded material (FGM) microbeams with microporosity defect are investigated. The mathematical model of the system is developed by using the sinusoidal beam theory and the modified couple stress theory. Based on Hamilton’s principle, the governing equations and boundary conditions of imperfect FGM microbeams are derived. Then, Navier’s method is utilized to calculate the deflections and natural frequencies of the system. Analytical results are presented to illustrate the effects of the microporosity volume fraction, the microporosity distribution type, the power-law index, the temperature and the moisture on the static bending and free vibration of imperfect FGM microbeams.

A Porous Microbeam Model for Bending and Vibration Analysis Based on the Sinusoidal Beam Theory and Modified Strain Gradient Theory

In this research, we analyze size-dependent bending and vibration of microbeams made of porous metal foams. The porous microbeam model is developed based on the sinusoidal beam theory and the modified strain gradient theory. Hamilton’s principle is employed to obtain the governing equations and boundary conditions of the porous microbeam. Analytical solutions are presented for deflections and natural frequencies of the porous microbeam by using Navier’s method. The influences of the porosity distribution, the porosity coefficient, the slenderness ratio, and the microbeam thickness are clarified on the static bending and free vibration of porous microbeams. These findings can be applied to the design of metal foam microstructures in engineering.

A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation

Microstructure-dependent buckling behavior of single-walled carbon nanotubes (SWCNTs) surrounded by a two-parameter elastic foundation is investigated. The governing equations and corresponding boundary conditions in buckling are achieved by implementing minimum total potential energy principle via modified strain gradient theory and several beam theories. The resulting equations are analytically solved by employing Navier’s solution procedure for simply supported boundary conditions. A detailed parametric study is performed to indicate effects of diameter-to-length scale parameter ratio, diameter-to-length ratio, shear deformation, shear correction factor and foundation parameters on buckling loads of SWCNTs. Numerical results reveal that the classical buckling loads evaluated by all shear deformation beam theories agree well with each other while a discrepancy occurs between the size-dependent buckling loads of parabolic beam theory (PBT), sinusoidal beam theory (SBT) and Timoshenko beam theory with proposed shear correction factor (TBT∗), and those of Timoshenko beam theory (TBT).

Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory

This article examines the application of nonlocal strain gradient elasticity theory to wave dispersion behavior of a size-dependent functionally graded (FG) nanobeam in thermal environment. The theory contains two scale parameters corresponding to both nonlocal and strain gradient effects. A quasi-3D sinusoidal beam theory considering shear and normal deformations is employed to present the formulation. Mori–Tanaka micromechanical model is used to describe functionally graded material properties. Hamilton’s principle is employed to obtain the governing equations of nanobeam accounting for thickness stretching effect. These equations are solved analytically to find the wave frequencies and phase velocities of the FG nanobeam. It is indicated that wave dispersion behavior of FG nanobeams is significantly affected by temperature rise, nonlocality, length scale parameter and material composition.

Size-dependent behaviour of functionally graded microbeams using various shear deformation theories based on the modified couple stress theory

This study investigates the mechanical behaviours of functionally graded (FG) microbeams based on the modified couple stress theory. The material properties of these beams are varied through beam’s depth and calculated by using classical rule of mixture and Mori–Tanaka scheme. The displacement fields are presented by using a unified framework which covers various theories including classical beam theory, first-order beam theory, third-order beam theory, sinusoidal beam theory, and quasi-3D beam theories. The governing equations of bending, vibration and buckling problems are derived using the Hamilton’s principle and then solved by using Navier solutions with simply-supported boundary conditions. A number of numerical examples are conducted to show the validity and accuracy of the proposed approaches. Effects of Poisson’s ratio, material length scale parameter, power-law index, estimation methods of material properties and slenderness ratio on deflections, stresses, natural frequencies and critical buckling loads of FG microbeams are examined.