• Structural Engineering and Mechanics
• /
• v.37 no.4
• /
• pp.367-383
• /
• 2011

In this paper, nonlinear partial differential equations of motion for a hybrid composite plate subjected to initial stresses on elastic foundations are established to investigate its nonlinear vibration behavior. Pasternak foundation and Winkler foundations are used to represent the plate-foundation interaction. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the example problems. The governing equations of motion are reduced to the time-dependent ordinary differential equations by the Galerkin's method. Then, the Runge-Kutta method is used to evaluate the nonlinear vibration frequency and frequency ratio of hybrid composite plates. The nonlinear vibration behavior is affected by foundation stiffness, initial stress, vibration amplitude and the thickness ratio of layer. The effects of various parameters on the nonlinear vibration of hybrid laminated plate are investigated and discussed.

• Magazine of the Korean Society of Agricultural Engineers
• /
• v.43 no.5
• /
• pp.106-115
• /
• 2001

This paper deals with the free vibrations of horizontally curved beams partially supported on elastic foundations. Taking account of the effects of rotatory inertia and shear deformation, differential equations governing the free vibrations of such beams are derived, in which the Pasternak foundation model is considered as the elastic foundation. Differential equations are numerically solved to calculate natural frequencies and mode shapes. The experiments were performed in which the free vibration frequencies of such curved beams in laboratorial scale were measured and these results agreed quite well with the present studies. In numerical examples, the circular, parabolic, sinusoidal and elliptic curved members are considered. The parametric studies are performed and the lowest four frequency parameters are reported in tables and figures as the non-dimensional forms. Also the typical mode shapes are presented.

• Steel and Composite Structures
• /
• v.32 no.2
• /
• pp.281-292
• /
• 2019

Nonlocal elasticity and Reddy plant theory are used to study the vibration response of functionally graded (FG) nanoplates resting on two parameters elastic medium called Pasternak foundation. Nonlocal higher order theory accounts for the effects of both scale and the effect of transverse shear deformation, which becomes significant where stocky and short nanoplates are concerned. It is assumed that the properties of FG nanoplate follow a power law through the thickness. In addition, Poisson's ratio is assumed to be constant in this model. Both Winkler-type and Pasternak-type foundation models are employed to simulate the interaction of nanoplate with surrounding elastic medium. Using Hamilton's principle, size-dependent governing differential equations of motion and corresponding boundary conditions are derived. A differential quadrature approach is being utilized to discretize the model and obtain numerical solutions for various boundary conditions. The model is validated by comparing the results with other published results.

• Steel and Composite Structures
• /
• v.27 no.2
• /
• pp.177-184
• /
• 2018

This study contributes to evaluate multiphase topology optimization design of plate-like structures on elastic foundations by using classic plate theory. Multi-material optimal topology and shape are produced as an alternative to provide reasonable material assignments based on stress distributions. Multi-material topology optimization problem is solved through an alternative active-phase algorithm with Gauss-Seidel version as an optimization model of optimality criteria. Stiffness and adjoint sensitivity formulations linked to thin plate potential strain energy are derived in terms of multiphase design variables and Winkler-Pasternak parameters considering elastic foundation to apply to the current topology optimization. Numerical examples verify efficiency and diversity of the present topology optimization method of elastic thin plates depending on multiple materials and Winkler-Pasternak parameters with the same amount of volume fraction and total structural volume.

• Structural Engineering and Mechanics
• /
• v.34 no.3
• /
• pp.319-334
• /
• 2010

The dynamic response of a finite Bernoulli-Euler beam resting on a tensionless Pasternak foundation and subjected to a concentrated harmonic load is investigated in this study. This load may be applied at the center of the beam, or it may be offset from the center. Since the elastic foundation is assumed to be tensionless, the beam may lift off the foundation, resulting in contact and non-contact regions in the system. An analytical/numerical solution is obtained from the governing equations of the contact and non-contact regions to determine the coordinates of the lift-off points. Although there is no nonlinear term in the equations, the problem appears to be nonlinear since the contact regions are not known in advance. Due to that nonlinearity, the essentials of the problem (the coordinates of the lift-off points) are calculated numerically using the Newton-Raphson technique. The results, which represent the symmetric and asymmetric responses of the beam, are presented graphically in this work. They illustrate the effects of the forcing frequency and the beam length on the extent of the contact regions and displacements.

• Earthquakes and Structures
• /
• v.19 no.2
• /
• pp.91-101
• /
• 2020

This work presents an efficient and original hyperbolic shear deformation theory for the bending and dynamic behavior of functionally graded (FG) beams resting on Winkler - Pasternak foundations. The theory accounts for hyperbolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the beam without using shear correction factors. Based on the present theory, the equations of motion are derived from Hamilton's principle. Navier type analytical solutions are obtained for the bending and vibration problems. The accuracy of the present solutions is verified by comparing the obtained results with the existing solutions. It can be concluded that the present theory is not only accurate but also simple in predicting the bending and vibration behavior of functionally graded beams.

• Structural Engineering and Mechanics
• /
• v.88 no.5
• /
• pp.439-449
• /
• 2023

Non-linear vibration characteristics of functionally graded CNT-reinforced composite (FG-CNTRC) cylindrical shell panel on elastic foundation have not been sufficiently examined. In this situation, this study aims at the profound numerical investigation of the non-linear vibration response of FG-CNTRC cylindrical panels on Winkler-Pasternak foundation by introducing an accurate and effective 2-D meshfree-based non-linear numerical method. The large-amplitude free vibration problem is formulated according to the first-order shear deformation theory (FSDT) with the von Karman non-linearity, and it is approximated by Laplace interpolation functions in 2-D natural element method (NEM) and a non-linear partial derivative operator H_NL. The complex and painstaking numerical derivation on the curved surface and the crucial shear locking are overcome by adopting the geometry transformation and the MITC3+ shell elements. The derived nonlinear modal equations are iteratively solved by introducing a three-step iterative solving technique which is combined with Lanczos transformation and Jacobi iteration. The developed non-linear numerical method is estimated through the benchmark test, and the effects of foundation stiffness, CNT volume fraction and functionally graded pattern, panel dimensions and boundary condition on the non-linear vibration of FG-CNTRC cylindrical panels on elastic foundation are parametrically investigated.

• Journal of the Computational Structural Engineering Institute of Korea
• /
• v.14 no.3
• /
• pp.371-379
• /
• 2001

This paper deals with the free vibrations of horizontally curved members resting on elastic foundations with continuity effect. Taking into account the effects of rotatory inertia and shear deformation, differential equations governing the free vibrations of such beams are derived, in which the Pasternak foundation model is considered as the elastic foundation with continuity effect. The differential equations are solved numerically to calculate natural frequencies and mode shapes. The experiments were performed in which the natural frequencies of such curved beams in laboratorial scale were measured and these results agree quite well with the present numerical studies. In numerical examples, the circular, parabolic, sinusoidal and elliptic curved members with the hinged-hinged, hinged-clamped and clamped end constraints are considered. The parametric studies are conducted and the lowest four frequency parameters are reported in tables and figures as the non-dimensional forms. Also the typical mode shapes are presented.

• Coupled systems mechanics
• /
• v.11 no.4
• /
• pp.357-371
• /
• 2022

This paper presents a parametric study on the free vibration analysis of a functionally graded material (FGM) circular plate with non-uniform thickness resting on a variable Pasternak elastic foundation. The mechanical properties of the material vary in the transverse direction through the thickness of the plate according to the power-law distribution to represent the constituent components. The equation of motion of the circular plate has been carried out based on the classical plate theory (CPT), and the differential quadrature method (DQM) is employed to solve the governing equations as a semi-analytical method. The grid points are chosen based on Chebyshev-Gauss-Lobatto distribution to achieve acceptable convergence and better accuracy. The influence of geometric parameters, variable elastic foundation, and functionally graded variation for clamped and simply supported boundary conditions on the first three natural frequencies are investigated. Comparisons of results with similar studies in the literature have been presented and two-dimensional mode shapes for particular plates have been plotted to illustrate the effect of variable thickness profile.

• Advances in nano research
• /
• v.6 no.3
• /
• pp.279-298
• /
• 2018

Present investigation deals with the free vibration characteristics of nanoscale-beams resting on elastic Pasternak's foundation based on nonlocal strain-gradient theory and a higher order hyperbolic beam model which captures shear deformation effect without using any shear correction factor. The nanobeam is lying on two-parameters elastic foundation consist of lower spring layers as well as a shear layer. Nonlocal strain gradient theory takes into account two scale parameters for modeling the small size effects of nanostructures more accurately. Hamilton's principal is utilized to derive the governing equations of embedded strain gradient nanobeam and, after that, analytical solutions are provided for simply supported conditions to solve the governing equations. The obtained results are compared with those predicted by the previous articles available in literature. Finally, the impacts of nonlocal parameter, length scale parameter, slenderness ratio, elastic medium, on vibration frequencies of nanosize beams are all evaluated.