Natural Frequencies of Beams with Axial Material Gradation Resting on Two Parameter Elastic Foundation

Sigmoid functionally graded plates embedded on Winkler-Pasternak foundation: Free vibration analysis by dynamic stiffness method

ABSTRACTMany experimental observations have shown that most nanostructures, such as carbon nanotubes, are often characterized by a certain degree of waviness along their axial direction. This geometrical imperfection has profound effects on the mechanical behavior of carbon nanotubes. In the present work, stability of a slightly curved carbon nanotube under lateral loading is investigated based on Eringen's nonlocal elasticity theory. Euler Bernoulli and Timoshenko beam theories are employed to obtain equilibrium equations. Winkler-Pasternak elastic foundation is used to approximate the effect of matrix. Effects of initial curvature, nonlocal parameter, beam length, and elastic foundation parameters on initiation of critical conditions are investigated.

Static bending and free vibration of organic solar cell resting on Winkler-Pasternak elastic foundation through the modified strain gradient theory

The present paper deals with the development of a dynamic stiffness matrix to evaluate the free vibration response of functionally graded nanoplate (FG-nP) resting on the Winkler-Pasternak elastic foundation. The complete mathematical modeling of the dynamic stiffness matrix for nanostructures is given for the first time. The equation of motion for rectangular FG-nP plates supported on an elastic foundation is derived using Hamilton’s principle in conjunction with nonlocal elasticity theory. The non-local theory is incorporated to account for the size effect in the small-scale plate. The effective material property of the porous FG-nP has been calculated using three recently developed models of porosity. The developed dynamic stiffness matrix is solved using the Wittrick-Williams algorithm to extract the natural frequencies of the FG-nP. The variation of natural frequencies with the change of numerical values, such as nonlocal parameter, aspect ratio, elastic foundation parameters, and porosity volume fraction is analyzed. The validity and accuracy of the results are confirmed through comparison with the available literature. The use of non-local theory in dynamic stiffness analysis is shown to be effective in predicting the natural frequency of the FG-nP on a Winkler-Pasternak elastic foundation, providing new insights into the dynamic behavior of small-scale structures.

Size-dependent functionally graded beam model based on an improved third-order shear deformation theory

The present work deals with the thermomechanical bending response of functionally graded plates resting on Winkler-Pasternak elastic foundations. Theoretical formulations are based on a recently developed refined trigonometric shear deformation theory (RTSDT). The theory accounts for trigonometric distribution of transverse shear stress, and satisfies the free transverse shear stress conditions on the top and bottom surfaces of the plate without using shear correction factor. Unlike the conventional trigonometric shear deformation theory, the present refined trigonometric shear deformation theory contains only four unknowns as against five in case of other shear deformation theories. The material properties of the functionally graded plates are assumed to vary continuously through the thickness, according to a simple power law distribution of the volume fraction of the constituents. The elastic foundation is modelled as twoparameter Pasternak foundation. The results of the shear deformation theories are compared together. Numerical examples cover the effects of the gradient index, plate aspect ratio, side-to-thickness ratio and elastic foundation parameters on the thermomechanical behavior of functionally graded plates. It can be concluded that the proposed theory is accurate and efficient in predicting the thermomechanical bending response of functionally graded plates.

In this paper, free vibration of two-dimensional functionally graded (2D-FG) sectorial plate with variable thickness resting on Winkler–Pasternak elastic foundation has been studied. It is assumed that the plate properties vary continuously through its both circumference and thickness according to power law distribution of the volume fraction. Primarily, the motion equations have been derived based on first order shear deformation theory (FSDT). The numerical two-dimensional differential quadrature method (2D-DQM) has been employed to solve the motion equations. Four different kinds of boundary condition are considered. The effects of geometrical and elastic foundation parameters along with 2D-FG power indices effects on the natural frequencies have been studied. Also the frequency parameters of the sectorial plate with uniform, linear and nonlinear variation of thickness for various boundary conditions have been computed and the effects of variable thickness parameters on natural frequency have been investigated.

Nonlinear free vibration of functionally graded graphene platelets reinforced porous nanocomposite plates resting on elastic foundation

ABSTRACTThe thermal buckling analysis of nanoplates is based on nonlocal elasticity theory with four-unknown shear deformation theory resting on Winkler–Pasternak elastic foundation. The nanoplate is assumed to be under three types of thermal loadings, namely uniform temperature rise, linear temperature rise, and nonlinear temperature rise through the thickness. The theory involves four unknown variables with small-scale effects, as against five in the case of other higher-order theories and first-order shear deformation theory. Closed-form solution for theory was also presented. Results are presented to discuss the influences of the nonlocal parameter, aspect ratio, side-to-thickness ratio, and elastic foundation parameters on the thermal buckling characteristics of analytical rectangular nanoplates.

This paper is devoted to investigate the nonlinear dynamic characteristics of lattice sandwich composite panels resting on Winkler–Pasternak elastic foundations under simultaneous aerodynamic and thermal loads in supersonic airflow. The first-order shear deformation and von Karman large deflection theories are applied in the structural modeling. The supersonic piston theory is used to model the aerodynamic pressure acting on the lattice sandwich composite panel. The equation of motion of the structure is established by Hamilton’s principle with the assumed mode method. The nonlinear vibration responses of the lattice sandwich composite panel under different aerodynamic pressures are computed. In addition, the influences of several significant parameters including the ply angle of laminated face sheets, elastic foundation parameters, aerodynamic pressure and temperature change on the nonlinear aerothermoelastic characteristics and the route to chaos for the lattice sandwich composite panel are investigated. Time histories, phase maps, Poincare plots and fast Fourier transform frequency spectra are presented to identify the periodic, quasi-periodic and chaotic motions.

In this paper, free vibration of a metal foam core sandwich (MFCS) beam embedded in Winkler–Pasternak elastic foundation is studied using the Chebyshev collocation method (CCM). This method can achieve high precision within the range allowed by the effective number of bits of computers. Three foam distribution types along the thickness direction are considered for the core. The Timoshenko beam theory is adopted and Hamilton’s principle is utilized to derive the boundary conditions and governing equations of the model. The numerical results show that natural frequencies of the sandwich beam initially increase and then decrease with the rise in thickness of metal foam core. By arranging the foam distribution in the core, natural frequencies of the sandwich beam can be significantly changed. Moreover, natural frequencies of the uniform foam distribution beam are insensitive to the foam coefficient. For the beam with non-uniform foam distribution, however, the natural frequencies increase or decrease with the foam coefficient, depending closely on the foam type. In addition, the present method is validated by comparing with the published ones for special cases.

In the present study, the free vibration of functionally graded beams resting on two parameter elastic foundation was examined. The properties of the functionally graded materials were presumed to vary continuously along the thickness direction. The foundation medium was assumed to be linear, homogeneous, and isotropic, and it was modeled by the Winkler-Pasternak model with two parameters for describing the reaction of the elastic foundation on the beam. The functionally graded beam was modeled with classical beam theory. The governing equation including the effects of functionally graded material properties, Winkler-Pasternak elastic foundation was solved using separation of variables. The eigenvalues of yielding fundamental equation versus clamped-clamped, clamped-free, clamped-simply supported, and simply supported-simply supported boundary conditions were found. To corroborate the results, comparisons were carried out with available results for homogeneous and functionally graded beams. The effects of Winkler-Pasternak type elastic foundation and functionally graded material properties on the values of dimensionless frequency parameter of beams were discussed. Briefly, it was found that the dimensionless frequency parameters of beam change according to material properties, presence of elastic foundation, and boundary conditions; moreover, the separate effects of these quantities on each other are interesting.

The buckling behavior of laminated composite plates reinforced by carbon nanotubes (CNTs) resting on Winkler–Pasternak elastic foundations under in-plane loads is investigated using reproducing kernel particle method (RKPM) based on first-order shear deformation theory. The minimum potential energy approach is utilized to obtain the governing equations and the stiffness matrices. The single-walled CNTs and poly-co-vinylene are used for the fibers and the matrix, respectively. The carbon nanotube fibers are uniformly distributed in the polymer matrix. The material properties of a carbon nanotube-reinforced composite (CNTRC) plate are estimated through a micromechanical model based on the extended rule of mixture. Full transformation approach is employed to enforce essential boundary conditions. The accuracy and convergency of the RKPM method is established by comparing the obtained results with the available literature. Then, the effects of volume fraction and orientation of CNTs, plate aspect ratio, plate width-to-thickness ratio and the elastic foundation parameters on the critical buckling load are investigated. The obtained results demonstrate that the geometric and mechanical properties and boundary conditions have noticeable effects on the buckling behavior of laminated CNTRC plates.

This study introduces a model capable of investigating the vibrational response of nanobeams that simultaneously incorporate structural perforations and axially functionally graded materials while being supported by elastic foundations. By using Euler-Bernoulli beam theory with the nonlocal elasticity framework, the model captures nanoscale phenomena often neglected in classical analysis and governing equation of motion is derived. The objective of this study is to examine the coupled effects of axial material gradation, periodic perforations, and elastic foundation parameters on the vibrational behavior of nanobeams, highlighting how variations in perforation geometry and material gradation jointly influence the mode shape and frequency response. The modified equivalent modeling strategy is employed to account for the periodic perforations. The Galerkin method is utilized to handle and extract natural frequencies and mode shapes. The proposed numerical scheme exhibits high accuracy, as verified by comparison with existing results from the literature, with the maximum deviation limited to just 0.081%. Perforation and axial material gradation significantly impact the vibrational characteristics, governed by the nonlocal effects and geometric configuration. Numerical findings highlight the sensitivity of dynamic response to filling ratio, perforation geometry, and foundation interaction across different boundary conditions.

This study explores the free vibration behavior of multi-cracked functionally graded (FG) nanobeams situated on Winkler–Pasternak elastic foundations. Cracks are modeled as rotational springs, effectively dividing the beam into sub-beams. The present model is formulated for metal–ceramic FG nanobeams composed of two isotropic phases with material properties varying continuously across the thickness based on a power-law distribution. The analysis incorporates Euler–Bernoulli beam theory and Eringen’s nonlocal elasticity theory to capture size-dependent effects. A fourth-order differential equation with constant coefficients is formulated, and an analytical approach is utilized to calculate the frequency parameter values. Validation with existing literature in specific cases confirms the accuracy of the obtained results. The effects of various factors, including the transverse gradient index, nonlocal parameter, Winkler and Pasternak foundation parameters, crack severity, and crack locations, on the first four frequency parameters are investigated. Unlike existing studies, the proposed model allows for any number of cracks (multiple cracks), offering a generalized framework for realistic damage scenarios. This is the first study to analytically address the combined effects of multiple cracks with elastic foundations in FG nanobeams. This framework provides a robust tool for analyzing FG nanobeams with cracks under diverse boundary conditions and material gradations. These findings have significant implications for the design and optimization of nanostructures in aerospace, mechanical, and biomedical applications.