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

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

• Steel and Composite Structures
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• v.14 no.1
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• pp.85-104
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• 2013

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 two-parameter 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.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.15 no.2
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• pp.367-377
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• 2002

This paper deals with the free vibrations of shallow arches resting on elastic foundations. Foundations we assumed to follow the hypothesis proposed by Pasternak. The governing differential equation is derived for the in-plane free vibration of linearly elastic arches of uniform stiffness and constant mass per unit length. Two arch shapes with hinged-hinged and clamped-clamped end constraints we considered in analysis. The frequency equations (lowest symmetrical and antisymmetrical frequency equations) we obtained by Galerkin's method. The effects of arch rise, Winkler foundation parameter and shear foundation parameter on the lowest two natural frequencies are investigated. The effect of initial arch shapes on frequencies is also studied.

• Geomechanics and Engineering
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• v.24 no.4
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• pp.371-388
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• 2021

A novel flexibility-based beam-foundation model for inelastic analyses of beams resting on foundation is presented in this paper. To model the deformability of supporting foundation media, the Winkler-Pasternak foundation model is adopted. Following the derivation of basic equations of the problem (strong form), the flexibility-based finite beam-foundation element (weak form) is formulated within the framework of the matrix virtual force principle. Through equilibrated force shape functions, the internal force fields are related to the element force degrees of freedom. Tonti's diagrams are adopted to present both strong and weak forms of the problem. Three numerical simulations are employed to assess validity and to show effectiveness of the proposed flexibility-based beam-foundation model. The first two simulations focus on elastic beam-foundation systems while the last simulation emphasizes on an inelastic beam-foundation system. The influences of the adopted foundation model to represent the underlying foundation medium are also discussed.

• Geomechanics and Engineering
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• v.31 no.1
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• pp.99-112
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• 2022

The present study examines the natural frequencies (NFs) of perfect/imperfect functionally graded sandwich beams (P/IP-FGSBs), which are composed of a porous core constructed of functionally graded materials (FGMs) and a homogenous isotropic metal and ceramic face sheets resting on elastic foundations. To accomplish this, the material properties of the FGSBs are assumed to vary continuously along the thickness direction as a function of the volume fraction of constituents expressed by the modified rule of the mixture, which includes porosity volume fraction represented using four distinct types of porosity distribution models. Additionally, to characterize the reaction of the two-parameter elastic foundation to the Perfect/Imperfect (P/IP) FGSBs, the medium is assumed to be linear, homogeneous, and isotropic, and it is described using the Winkler-Pasternak model. Furthermore, the kinematic relationship of the P/IP-FGSBs resting on the Winkler-Pasternak elastic foundations (WPEFs) is described using trigonometric shear deformation theory (TrSDT), and the equations of motion are constructed using Hamilton's principle. A closed-form solution is developed for the free vibration analysis of P/IP-FGSBs resting on the WPEFs under four distinct boundary conditions (BCs). To validate the new formulation, extensive comparisons with existing data are made. A detailed investigation is carried out for the effects of the foundation coefficients, mode numbers (MNs), porosity volume fraction, power-law index, span to depth ratio, porosity distribution patterns (PDPs), skin core skin thickness ratios (SCSTR), and BCs on the values of the NFs of the P/IP-FGSBs.

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

• Coupled systems mechanics
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• v.9 no.6
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• pp.499-519
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• 2020

The effect of porosity on the thermo-mechanical behavior of simply supported functionally graded plate reposed on the Winkler-Pasternak foundation is investigated analytically in the present paper using new refined hyperbolic shear deformation plate theory. Both even and uneven distribution of porosity are taken into account and the effective properties of FG plates with porosity are defined by theoretical formula with an additional term of porosity. The present formulation is based on a refined higher order shear deformation theory, which is based on four variables and it still accounts for parabolic distribution of the transverse shearing strains and stresses through the thickness of the FG plate and takes into account the various distribution shape of porosity. The elastic foundation is described by the Winkler-Pasternak model. Anew modified power-law formulation is used to describe the material properties of FGM plates in the thickness direction. The closed form solutions are obtained by using Navier technique. The present results are verified in comparison with the published ones in the literature. The results show that the dimensionless and stresses are affected by the porosity volume fraction, constituent volume fraction, and thermal load.

• Journal of Mechanical Science and Technology
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• v.20 no.4
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• pp.467-472
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• 2006

A study of the natural vibrations of beam resting on elastic foundation with finite number of transverse open cracks is presented. Frequency equations are derived for beams with different end restraints. Euler-Bernoulli beam on Pasternak foundation and Euler-Bernoulli beam on Pasternak foundation are investigated. The cracks are modeled by massless substitute spring. The effects of the crack location, size and its number and the foundation constants, on the natural frequencies of the beam, are investigated.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.19 no.8
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• pp.828-837
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• 2009

Recently, as high-rise buildings increase steeply, sub-structures of them are often supported on elastic foundation(in a case of pasternak foundation or winkler foundation). And there are many machines in sub-structures of buildings and slabs of sub-structures are affected by vibration which they make. This paper deals with vibration of plates on elastic foundation. Machines on plates are considered as concentrated mass. This paper has the object of investigating natural frequencies of tapered thick plate on pasternak foundation by means of finite element method and providing kinetic design data for mat of building structures. Free vibration analysis that tapered thick plate with Concentrated Masses in this paper. Finite element analysis of rectangular plate is done by use of rectangular finite element with 8-nodes. In order to analysis plate which is supported on pasternak foundation. The Winkler parameter is varied with 10, $10^2$, $10^3$ and the shear foundation parameter is 5, 10. This paper is analyzed varying thickness by taper ratio. The taper ratio is applied as 0.0, 0.25, 0.5, 0.75, 1.0. And the Concentrated Mass is applied as P1, Pc, P2 respectively.