• Journal of Korean Society of Steel Construction
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• v.19 no.5
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• pp.539-548
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• 2007

This paper deals with flexural-torsional free vibrations of the circular strip foundation with the variable breadth on Pasternak soil. The cross-section of the strip foundation is chosen as the rectangular one with the constant thickness and variable breadth, which is symmetrical about the mid-arc. Also, the foundation that supports the circular strip is modeled as the Pasternak soil with the shear layer. Ordinary differential equations accompanying the boundary conditions are derived. In the governing equations, the transverse, rotatory and torsional inertias are included. These equations are solved numerically and four lowest frequencies are obtained. In the numerical results, the effects of foundation parameters on frequencies are extensively investigated. It is expected that the theories and numerical results of this study can be used in the dynamic design of strip foundations.

• 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.

• Advances in nano research
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• v.4 no.3
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• pp.229-249
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• 2016

In this paper, buckling characteristics of nonhomogeneous functionally graded (FG) nanobeams embedded on elastic foundations are investigated based on third order shear deformation (Reddy) without using shear correction factors. Third-order shear deformation beam theory accounts for shear deformation effects by a parabolic variation of all displacements through the thickness, and verifies the stress-free boundary conditions on the top and bottom surfaces of the FG nanobeam. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam in deformation, which acts in tension as well as in compression. The material properties of FG nanobeam are supposed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. Nonlocal equations of motion are derived through Hamilton's principle and they are solved applying analytical solution. Comparison between results of the present work and those available in literature shows the accuracy of this method. The obtained results are presented for the buckling analysis of the FG nanobeams such as the effects of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

• Advances in nano research
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• v.16 no.6
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• pp.549-564
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• 2024

This paper aims to analyze the dynamic response of thermoelectric carbon nanotube-reinforced composite (CNTRC) beams under moving harmonic load resting on Pasternak elastic foundation. The governing equations of thermoelectric CNTRC beam are obtained based on the Karama shear deformation beam theory. The beams are resting on the Pasternak foundation. Previous articles have not performed the moving load mode with the analytical method. The exact solution for the transverse and axial dynamic response is presented using the Laplace transform. A comparison of previous studies has been published, where a good agreement is observed. Finally, some examples were used to analyze, such as excitation frequency, voltage, temperature, spring constant factors, the volume fraction of Carbon nanotubes (CNTs), the velocity of a moving harmonic load, and their influence on axial and transverse dynamic and maximum deflections. The advantages of the proposed method compared to other numerical methods are zero reduction of the error percentage that exists in numerical methods.

• Steel and Composite Structures
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• v.47 no.5
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• pp.551-568
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• 2023

In this research, a new two-dimensional (2D) and quasi three-dimensional (quasi-3D) higher order shear deformation theory is devised to address the bending problem of functionally graded plates resting on an elastic foundation. The displacement field of the suggested theories takes into account a parabolic transverse shear deformation shape function and satisfies shear stress free boundary conditions on the plate surfaces. It is expressed as a combination of trigonometric and exponential shear shape functions. The Pasternak mathematical model is considered for the elastic foundation. The material properties vary constantly across the FG plate thickness using different distributions as power-law, exponential and Mori-Tanaka model. By using the virtual works principle and Navier's technique, the governing equations of FG plates exposed to sinusoidal and evenly distributed loads are developed. The effects of material composition, geometrical parameters, stretching effect and foundation parameters on deflection, axial displacements and stresses are discussed in detail in this work. The obtained results are compared with those reported in earlier works to show the precision and simplicity of the current formulations. A very good agreement is found between the predicted results and the available solutions of other higher order theories. Future mechanical analyses of three-dimensionally FG plate structures can use the study's findings as benchmarks.

• 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.

• Steel and Composite Structures
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• v.46 no.6
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• pp.839-856
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• 2023

This work presents a simple four-unknown refined integral plate theory for mechanical and thermal buckling behaviors of functionally graded (FG) plates resting on Visco-Pasternak foundations. The proposed refined high order shear deformation theory has a new displacement field which includes indeterminate integral variables and contains only four unknowns in which any shear correction factor not used, with even less than the conventional theory of first shear strain (FSDT). Governing equations are deduced from the principle of minimum total potential energy and a Navier type analytical solution is adopted for simply supported FG plates. The Visco-Pasternak foundations is considered by adding the impact of damping to the usual foundation model which characterized by the linear Winkler's modulus and Pasternak's foundation modulus. The accuracy of the present model is demonstrated by comparing the computed results with those available in the literature. Some numerical results are presented to show the impact of material index, elastic foundation type, and damping coefficient of the foundation, on the mechanical and thermal buckling behaviors of FG plates.

• Wind and Structures
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• v.22 no.3
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• pp.329-348
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• 2016

This work presents a simple hyperbolic shear deformation theory for analysis of functionally graded plates resting on elastic foundation. The proposed model contains fewer number of unknowns and equations of motion than the first-order shear deformation model, but the transverse shear stresses account for a hyperbolic variation and respect the tangential stress-free boundary conditions on the plate boundary surface without introducing shear correction factors. Equations of motion are obtained from Hamilton's principle. The Navier-type analytical solutions for simply-supported plates are compared with the existing solutions to demonstrate the accuracy of the proposed theory.

• Structural Engineering and Mechanics
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• v.61 no.6
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• pp.721-736
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• 2017

In this paper, free vibration characteristics of functionally graded (FG) nanobeams embedded on elastic medium are investigated based on third order shear deformation (Reddy) beam theory by presenting a Navier type solution for the first time. The material properties of FG nanobeam are assumed to vary gradually along the thickness and are estimated through the power-law and Mori-Tanaka models. A two parameters elastic foundation including the linear Winkler springs along with the Pasternak shear layer is in contact with beam. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal equations of motion are derived based on third order shear deformation beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared to some cases in the literature. The obtained results are presented for the vibration analysis of the FG nanobeams such as the influences of foundation parameters, gradient index, nonlocal parameter and slenderness ratio in detail.

• Structural Engineering and Mechanics
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• v.54 no.6
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• pp.1061-1073
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• 2015

This paper presents a nonlocal sinusoidal shear deformation beam theory (SDBT) for the nonlinear vibration of single walled carbon nanotubes (CNTs). The present model is capable of capturing both small scale effect and transverse shear deformation effects of CNTs, and does not require shear correction factors. The surrounding elastic medium is simulated based on Pasternak foundation. Based on the nonlocal differential constitutive relations of Eringen, the equations of motion of the CNTs are derived using Hamilton's principle. Differential quadrature method (DQM) for the natural frequency is presented for different boundary conditions, and the obtained results are compared with those predicted by the nonlocal Timoshenko beam theory (TBT). The effects of nonlocal parameter, boundary condition, aspect ratio on the frequency of CNTs are considered. The comparison firmly establishes that the present beam theory can accurately predict the vibration responses of CNTs.