In this study, the buckling and free vibration of Timoshenko beams resting on variable elastic foundation analyzed by means of a new finite element formulation. The Winkler model has been applied for elastic foundation. A two-node element with four degrees of freedom is suggested for finite element formulation. Displacement and rotational fields are approximated by cubic and quadratic polynomial interpolation functions, respectively. The length of the element is assumed to be so small, so that linear variation could be considered for elastic foundation through the length of the element. By these assumptions and using energy method, stiffness matrix, mass matrix and geometric stiffness matrix of the proposed beam element are obtained and applied to buckling and free vibration analysis. Accuracy of obtained formulation is approved by comparison with the special cases of present problem in other studies. Present formulation shows faster convergence in comparison with conventional finite element formulation. The effects of different parameters on the stability and free vibration of Timoshenko beams investigated and results are completely new.

This paper develops a framework for buckling and free vibration analysis of in-plane heterogeneous orthotropic nanoplates, considering nonlocal elasticity, surface effects and elastic foundation, by formulating a simple boundary method whose basis functions are set to approximately satisfy the governing equilibrium equation, as in Trefftz methods. The novelty of the work is in two points: first, the surface effects based on Gurtin-Murdoch model are formulated considering variable thickness of the nanoplate; second, for the first time, simultaneous effect of surface layer, elastic foundation and in-plane heterogeneity are investigated on the behavior of orthotropic nanoplates along with nonlocal effects, considering simple, clamped, free and guided edges. The boundary conditions are imposed by collocation, which enhances the versatility of the method, while the solution has complete continuity over the entire domain. Verification with the literature reflects very good accuracy of the implemented method. In the numerical study, it was observed that the ratio of the buckling load and the free vibration frequency, with and without nonlocal and surface effects, is larger for the cases with constant thickness than those with variable thickness. Moreover, nanoplates with free or guided edges showed less variation of the ratio with respect to the nonlocal effect, than those with simple and clamped edges.

In this paper, free vibration of an Euler-Bernoulli beam with variable cross-section resting on elastic foundation and under axial tensile force is considered. Beam’s constant height and exponentially varying width yields variable cross-section. The problem is handled for three different boundary conditions: clamped-clamped, simply supported-simply supported and clamp-free beams. First, the equation of motion that governs the free vibration is derived and then dimensionless frequencies are determined using differential transform method (DTM). DTM is a semi-analytical approach based on Taylor expansion series that is a powerful tool in solution ordinary and partial differential equations. The effects of axial force, elastic foundation coefficient and non-uniformity parameter on dimensionless frequencies are investigated. Wherever possible, comparisons are made with the studies in open literature. Results show, the DTM yields rapid convergence without any frequency missing although convergence rate depends on boundary conditions. Also, dimensionless frequencies are sensitive to axial force rather than other parameters.

The large amplitude free vibration behaviour of an Euler-Bernoulli beam with immovable ends subjected to the thermal loads is investigated. Applying the Hamilton’s principle, the beam governing equation of motion is derived. By implementing the Galerkin’s method, the ordinary nonlinear differential equation is derived and because of the large coefficient of the nonlinear term, the Homotopy Perturbation Method (HPM) is used to solve the governing nonlinear equation and the accuracy of the mentioned method is investigated. The results are validated by comparing them with those available in the literature. Moreover, the effects of the system parameters including foundation coefficients, thermal load and vibrational modes on the system nonlinear vibration behaviour are investigated.

In the present study, the finite element method is developed for the static analysis of nano-beams underthe Winkler foundation and the uniform load. The small scale effect along with Eringen's nonlocal elasticity theoryis taken into account. The governing equations are derived based on the minimum potential energy principle. Galerkin weighted residual method is used to obtain the finite element equations. The validity and novelty of theresults for bending are tested and comparative results are presented. Deflections according to different Winklerfoundation parameters and small scale parameters are tabulated and plotted. As it can be seen clearly from figuresand tables, for simply-supported boundary conditions, the effect of small scale parameter is very high when theWinkler foundation parameter is smaller. On the other hand, for clamped-clamped boundary conditions, the effectof small scale parameter is higher when the Winkler foundation parameter is high. Although the effect of the smallscale parameter is adverse on deflection for simply-supported and clamped-clamped boundary conditions.

In this work, the bending of a sheet on elastic pad has been analyzed. The governing equations for bending of plate were derived and analytically solved. A formulation was derived for the indentation depth at which the plastic deformation of the sheet initiates. The equations were also numerically solved using finite difference method to verify the analytic solution. Additionally, some experiments were performed to verify these results.

During the past decade researchers have investigated vibrations of piezoelectric and piezomagnetic nanobeams. In addition to displacement fields, usually the electromagnetic fields are also considered one dimensional because the small ratio of height to length of the beam geometry. This makes the implication of electromagnetic boundary conditions on the upper and lower faces of the beam rigorous. In this study free vibration of Timoshenko beam having Magneto-Electro-elastic properties laying on elastic foundation has been studied considering one dimensional displacement fields and two dimensional electromagnetic fields. State equations which are a Integro-Differential couple has been discretized using a modified radial point interpolation function. Frequency response and the mode shapes have been queried for Dirichlet and Neumman boundary conditions with two conditions open circuit and closed circuit. . Obtained frequencies have been compared with results of the corresponding fully one dimensional problem.

In this paper, frequency analysis of Bi-directional Porous functionally graded beams with variable cross section which are resting on elastic foundation based on Reddy third order shear deformation theory is studied. Mechanical property gradients defined in accordance with two models of exponential and volume fraction power law. Governing equation which is obtained with the aid of third order shear deformation theory and by considering elastic foundation effect in conjunction with Hamilton’s principle. Due to intrinsic closed form solution of equations, differential equations solved with using Generalized Differential Quadrature Method by considering various end conditions. In order to validate the results comparisons are made with solutions which are available for other papers. This study reveals that the difference between the results of this paper and the results of others is negligible. Eventually the effects of geometrical parameters, power and exponential law indexes and elastic foundation coefficients on natural frequencies of Bi-directional FGM beams is studied. The results reveal that non dimensional frequencies increase with the rise of elastic foundation coefficients and the soar of porosities and material gradients in two directions causes a sharp decrease in non dimensional frequencies.The results of this study can be used in optimal design, vibration control and detection of failure structure of functional graded structures

The following article investigates nonlinear symmetric buckling of moderately thick circular Nano plates with an orthotropic property under uniform radial compressive in-plane mechanical load. Taking into account Eringen nonlocal elasticity theory, principle of virtual work, first order shear deformation plate theory (FSDT) and nonlinear Von-Karman strains, the governing equations are obtained based on displacements. The differential quadrature method (DQM) as a numerical procedure is applied for solving the equations. In this analysis, for solving the stability equations, adjacent equilibrium methodis employed. In nonlinear buckling analyses and for obtaining the buckling load, generally the available nonlinear terms of the stability equation are neglected. However, in this study, for getting the most accurate data, nonlinear terms are considered and the non-dimensional buckling load is compared with the condition of considering or neglecting that of terms and the effect of that of terms are also studied. The accuracy of the present results is validated by comparing the solutions with available studies. The effects of nonlocal parameter, thickness, radiusand elastic foundation are investigated on non-dimensional buckling loads. The results of analyses based on local and non-local theories are compared. From the results, it can be seen that the effect of nonlocal parameter on simply support condition is less than clamped condition. It can be observed that with increasing the radius of the plate, the difference between local and non-local analyses, increases.