• Steel and Composite Structures
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• v.34 no.5
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• pp.643-655
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• 2020

In the present work, the buckling behavior of a single-layered graphene sheet (SLGS) embedded in visco-Pasternak's medium is studied using nonlocal four-unknown integral model. This model has a displacement field with integral terms which includes the effect of transverse shear deformation without using shear correction factors. The visco-Pasternak's medium is introduced by considering the damping effect to the classical foundation model which modeled by the linear Winkler's coefficient and Pasternak's (shear) foundation coefficient. The SLGS under consideration is subjected to compressive in- plane edge loads per unit length. The influences of many parameters such as nonlocal parameter, geometric ratio, the visco-Pasternak's coefficients, damping parameter, and mode numbers on the buckling response of the SLGSs are studied and discussed.

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

• Structural Engineering and Mechanics
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• v.30 no.1
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• pp.21-36
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• 2008

The static response of a finite beam resting on a tensionless Pasternak foundation and subjected to a concentrated vertical load is assessed in this study. The concentrated vertical load may be applied at the center of the beam, or it may be offset from the center. The tensionless character of the foundation results in the creation of lift-off regions between the beam and the foundation. An analytical/ numerical solution is obtained from the governing equations of the contact and lift-off regions to determine the extent of the contact region. Although there is no nonlinear term in the equations, the problem shows a nonlinear character since the contact region is 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 numerical results are presented in figures to illustrate the behaviours of the free-free and pinned-pinned beams under symmetric or asymmetric loading. The figures illustrate the effects of the shear foundation parameter and the symmetric and asymmetric loading options on the variation of the contact lengths and the displacement of the beam.

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

• Journal of the Computational Structural Engineering Institute of Korea
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• v.28 no.1
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• pp.85-92
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• 2015

This article presents the dynamic instability response of sigmoid functionally graded material plates on elastic foundation using the higher-order shear deformation theory. The higher-order shear deformation theory has ability to capture the quadratic variation of shear strain and consequently shear stress through the plate thickness. The governing equations are then written in the form of Mathieu-Hill equations and then Bolotin's method is employed to determine the instability regions. The boundaries of the instability regions are represented in the dynamic load and excitation frequency plane. The results of dynamic instability analysis of sigmoid functionally graded material plate are presented using the Navier's procedure to illustrate the effect of elastic foundation parameter on dynamic response. The relations between Winkler and Pasternak elastic foundation parameter are discussed by numerical results. Also, the effects of static load factor, power-law index and side-to-thickness ratio on dynamic instability analysis are investigated and discussed. In order to validate the present solutions, the reference solutions are used and discussed. The theoretical development as well as numerical solutions presented herein should serve as reference for the dynamic instability study of S-FGM plates.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.29 no.6
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• pp.529-538
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• 2016

The simplified plate theory is presented for static and free vibration analysis of power-law(P) and sigmoid(S) Functionally Graded Materials(FGM) plates. This theory considers the parabolic distribution of the transverse shear stress, and satisfies the condition that requires the transverse shear stress to be zero on the upper and lower surfaces of the plate, without the shear correction factor. The simplified plate theory uses only four unknown variables and shares strong similarities with classical plate theory(CPT) in many aspects such as stress-resultant expressions, equation of motion and boundary conditions. The material properties of the plate are assumed to vary according to the power-law and sigmoid distributions of the volume fractions of the constituents. The Hamilton's principle is used to derive the equations of motion and Winkler-Pasternak elastic foundation model is employed. The results of static and dynamic responses for a simply supported FGM plate are calculated and a comparative analysis is carried out. The results of the comparative analysis with the solutions of references show relevant and accurate results for static and free vibration problems of FGM plates. Analytical solutions for the static and free vibration problems are presented so as to reveal the effects of the power law index, elastic foundation parameter, and side-to-thickness ratio.

• Steel and Composite Structures
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• v.46 no.1
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• pp.1-13
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• 2023

Elastic bending of imperfect functionally graded sandwich plates (FGSPs) laying on the Winkler-Pasternak foundation and subjected to sinusoidal loads is analyzed. The analyses have been established using the quasi-3D sinusoidal shear deformation model. In this theory, the number of unknowns is condensed to only five unknowns using integral-undefined terms without requiring any correction shear factor. Moreover, the current constituent material properties of the middle layer is considered homogeneous and isotropic. But those of the top and bottom face sheets of the graded porous sandwich plate (FGSP) are supposed to vary regularly and continuously in the direction of thickness according to the trigonometric volume fraction's model. The corresponding equilibrium equations of FGSPs with simply supported edges are derived via the static version of the Hamilton's principle. The differential equations of the system are resolved via Navier's method for various schemes of FGSPs. The current study examine the impact of the material index, porosity, side-to-thickness ratio, aspect ratio, and the Winkler-Pasternak foundation on the displacements, axial and shear stresses of the sandwich structure.

• Steel and Composite Structures
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• v.47 no.6
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• pp.693-707
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• 2023

This article investigates the static thermo-mechanical response of anisotropic thick laminated composite plates on Visco-Pasternak foundations under various thermal load conditions (linear, non-linear, and uniform) along the transverse direction (thickness) of the plate, while keeping the mechanical load constant. The governing equations, which represent the thermo-mechanical behavior of the composite plate, are derived from the principle of virtual displacements. Using Navier's type solution, these equations are solved for the composite plate with simply supported condition. The Visco-Pasternak foundation type is included by considering the impact of the damping on the classical foundation model, which is modeled by Winkler's linear modulus and Pasternak's shear modulus. The excellent accuracy of the present solution is confirmed by comparing the results with those available in the literature. The study investigates the impact of geometric ratios, thermal expansion coefficient ratio, damping coefficient and foundation parameters on the thermo-mechanical flexural response of the composite plate. Overall, this article provides insights into the behavior of composite plates on visco-Pasternak foundations and may be useful for designing and analyzing composite structures in practical applications.

• 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.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 2004.05a
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• pp.794-798
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• 2004

This paper is analysis of stiffened opening thick plate on foundation. This paper has the object of investigating natural frequencies of opening thick plates on Pasternak foundation by means of finite element method and providing Kinematic design data for mat of building structures. In this paper, vibration analysis of rectangular opening thick plate is done by use of Serendipity finite element with 8 nodes by considering shearing strain of plate. And vibration analysis of stiffener is done by used of Timoshenko beam-column element wit 3 nodes. It is shown that natural frequencies depend on not only Winkler foundation parameter but also shear foundation parameter, opening position, opening size, stiffener size.