• Structural Engineering and Mechanics
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• v.40 no.4
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• pp.583-594
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• 2011

In the present study, free vibration of an axially functionally graded (AFG) pile embedded in Winkler-Pasternak elastic foundation is analyzed within the framework of the Euler-Bernoulli beam theory. The material properties of the pile vary continuously in the axial direction according to the power-law form. The frequency equation is obtained by using Lagrange's equations. The unknown functions denoting the transverse deflections of the AFG pile is expressed in modal form. In this study, the effects of material variations, the parameters of the elastic foundation on the fundamental frequencies are examined. It is believed that the tabulated results will be a reference with which other researchers can compare their results.

• Computers and Concrete
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• v.19 no.6
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• pp.677-687
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• 2017

The nonlinear buckling response of nano composite anti-symmetric functionally graded polymeric microplate reinforced by single-walled carbon nanotubes (SWCNTs) rested on orthotropic elastomeric foundation with temperature dependent properties is investigated. For the carbon-nanotube reinforced composite (CNTRC) microplate, a uniform distribution (UD) and four types of functionally graded (FG) distribution are considered. Based on orthotropic Mindlin plate theory, von Karman geometric nonlinearity and Hamilton's principle, the governing equations are derived. Generalized differential quadrature method (GDQM) is employed to calculate the non-linear buckling response of the plate. Effects of FG distribution type, elastomeric foundation, aspect ratio (thickness to width ratio), boundary condition, orientation of foundation orthotropy and temperature are considered. The results are validated. It is found that the critical buckling load without elastic medium is significantly lower than considering Winkler and Pasternak medium.

• Steel and Composite Structures
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• v.32 no.6
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• pp.795-806
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• 2019

In this paper, free vibration of sandwich beam with flexible core resting on orthotropic Pasternak is investigated. The top and bottom layers are reinforced by carbon nanotubes (CNTs). This sandwich structural is modeled by Euler and Frostig theories. The effect of agglomeration using Mori-Tanaka model is considered. The Eringen's theory is applied for size effect. The structural damping is investigated by Kelvin-voigt model. The motion equations are calculated by Hamilton's principle and energy method. Using analytical method, the frequency of the structure is obtained. The effect of agglomeration and CNTs volume percent for different parameter such as damping of structure, thickens and spring constant of elastic medium are presented on the frequency of the composite structure. Results show that with increasing CNTs agglomeration, frequency is decreased.

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

• Smart Structures and Systems
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• v.18 no.6
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• pp.1125-1143
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• 2016

Forced vibration analysis of a simple supported viscoelastic nanobeam is studied based on modified couple stress theory (MCST). The nanobeam is excited by a transverse triangular force impulse modulated by a harmonic motion. The elastic medium is considered as Winkler-Pasternak elastic foundation.The damping effect is considered by using the Kelvin-Voigt viscoelastic model. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new non-classical beam model reduces to the classical beam model when the length scale parameter is set to zero. The considered problem is investigated within the Timoshenko beam theory by using finite element method. The effects of the transverse shear deformation and rotary inertia are included according to the Timoshenko beam theory. The obtained system of differential equations is reduced to a linear algebraic equation system and solved in the time domain by using Newmark average acceleration method. Numerical results are presented to investigate the influences the material length scale parameter, the parameter of the elastic medium and aspect ratio on the dynamic response of the nanobeam. Also, the difference between the classical beam theory (CBT) and modified couple stress theory is investigated for forced vibration responses of nanobeams.

• Structural Engineering and Mechanics
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• v.58 no.5
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• pp.931-947
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• 2016

The effects of nanotechnology and smartness on the buckling reduction of pipes are the main contributions of present work. For this ends, the pipe is simulated with classical piezoelectric polymeric cylindrical shell reinforced by armchair double walled boron nitride nanotubes (DWBNNTs), The structure is subjected to combined electro-thermo-mechanical loads. The surrounding elastic foundation is modeled with a novel model namely as orthotropic nonhomogeneous Pasternak medium. Using representative volume element (RVE) based on micromechanical modeling, mechanical, electrical and thermal characteristics of the equivalent composite are determined. Employing nonlinear strains-displacements and stress-strain relations as well as the charge equation for coupling of electrical and mechanical fields, the governing equations are derived based on Hamilton's principal. Based on differential quadrature method (DQM), the buckling load of pipe is calculated. The influences of electrical and thermal loads, geometrical parameters of shell, elastic foundation, orientation angle and volume percent of DWBNNTs in polymer are investigated on the buckling of pipe. Results showed that the generated ${\Phi}$ improved sensor and actuator applications in several process industries, because it increases the stability of structure. Furthermore, using nanotechnology in reinforcing the pipe, the buckling load of structure increases.

• Geomechanics and Engineering
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• v.28 no.1
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• pp.49-64
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• 2022

In this work, the bending and dynamic behaviors of advanced composite plates resting on variable visco-Pasternak foundations are studied using a simple shear deformation integral plate model. The research is carried out with a view to a three-parameter foundation including the influences of the variable Winkler coefficient, the constant Pasternak coefficient and the damping coefficient of the elastic medium. The present theory uses a displacement field with integral terms instead of derivative terms by including also the shear deformation effect without introducing the shear correction factors. The equations of motion for advanced composite plates are obtained using the Hamilton principle. Analytical solutions for the bending and dynamic analysis are deduced for simply supported plates resting on variable visco-Pasternak foundations. Some numerical results are presented to demonstrate the impact of material index, elastic foundation type, and damping coefficient of the foundation, on the bending and dynamic responses of advanced composite plates.

• 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.

• Structural Engineering and Mechanics
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• v.65 no.6
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• pp.645-656
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• 2018

Importance of procuring adequate knowledge about the mechanical behavior of double-layered graphene sheets (DLGSs) incensed the authors to investigate wave propagation responses of mentioned element while rested on a visco-Pasternak medium under hygro-thermal loading. A nonlocal strain gradient theory (NSGT) is exploited to present a more reliable size-dependent mechanical analysis by capturing both softening and hardening effects of small scale. Furthermore, in the framework of a classical plate theory the kinematic relations are developed. Incorporating kinematic relations with the definition of Hamilton's principle, the Euler-Lagrange equations of each of the layers are derived separately. Afterwards, combining Euler-Lagrange equations with those of the NSGT the nonlocal governing equations are written in terms of displacement fields. Interaction of the each of the graphene sheets with another one is regarded by the means of vdW model. Then, a widespread analytical solution is employed to solve the derived equations and obtain wave frequency values. Subsequently, influence of each participant variable containing nonlocal parameter, length scale parameter, foundation parameters, temperature gradient and moisture concentration is studied by plotting various figures.

• Steel and Composite Structures
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• v.24 no.4
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• pp.499-512
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• 2017

This paper deals with nonlinear dynamic stability of embedded piezoelectric nano-composite separators conveying pulsating fluid. For presenting a realistic model, the material properties of structure are assumed viscoelastic based on Kelvin-Voigt model. The separator is reinforced with single-walled carbon nanotubes (SWCNTs) which the equivalent material properties are obtained by mixture rule. The separator is surrounded by elastic medium modeled by nonlinear orthotropic visco Pasternak foundation. The separator is subjected to 3D electric and 2D magnetic fields. For mathematical modeling of structure, three theories of classical shell theory (CST), first order shear deformation theory (FSDT) and sinusoidal shear deformation theory (SSDT) are applied. The differential quadrature method (DQM) in conjunction with Bolotin method is employed for calculating the dynamic instability region (DIR). The detailed parametric study is conducted, focusing on the combined effects of the external voltage, magnetic field, visco-Pasternak foundation, structural damping and volume percent of SWCNTs on the dynamic instability of structure. The numerical results are validated with other published works as well as comparing results obtained by three theories. Numerical results indicate that the magnetic and electric fields as well as SWCNTs as reinforcer are very important in dynamic instability analysis of structure.