• Steel and Composite Structures
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• 제41권6호
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• pp.787-803
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• 2021

The size-dependent nonlinear thermomechanical vibration analysis of pre- and post-buckled tapered two-directional functionally graded (2D-FG) microbeams is presented in this study. In the context of the modified couple stress theory, the formulations are derived based on the parabolic shear deformation beam theory and von Karman nonlinear strains. Different thermomechanical material properties are assumed to be temperature-dependent and smoothly vary in both length and thickness directions using the power law and the physical neutral axis concept is employed. The nonlinear governing equations are derived using the Hamilton principle and the resulting variable coefficient equations of motion are solved using the differential quadrature method (DQM) and iterative Newton's method for clamped-clamped and simply supported boundary conditions. Comparison studies are presented to validate the derived model and solution procedure. The impacts of induced thermal moments, temperature power index, two gradient indices, nonuniform cross-section, and microstructure length scale parameter on the frequency-temperature configurations are explored for both clamped and simply supported microbeams.

• Advances in nano research
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• 제14권5호
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• pp.475-493
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• 2023

In the context of nonclassical nonlocal strain gradient elasticity, this article studies the free and forced responses of functionally graded material (FGM) porous nanoplates exposed to thermal and magnetic fields under a moving load. The developed mathematical model includes shear deformation, size-scale, miscorstructure influences in the framework of higher order shear deformation theory (HSDT) and nonlocal strain gradient theory (NSGT), respectively. To explore the porosity effect, the study considers four different porosity models across the thickness: uniform, symmetrical, asymmetric bottom, and asymmetric top distributions. The system of quations of motion of the FGM porous nanoplate, including the effects of thermal load, Lorentz force, due to the magnetic field and moving load, are derived using the Hamilton's principle, and then solved analytically by employing the Navier method. For the free and forced responses of the nanoplate, the effects of nonlocal elasticity, strain gradient elasticity, temperature rise, magnetic field intensity, porosity volume fraction, and porosity distribution are analyzed. It is found that the forced vibrations of FGM porous nanoplates under thermal and live loads can be damped by applying a directed magnetic field.

• Steel and Composite Structures
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• 제16권4호
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• pp.389-415
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• 2014

This study deals with dynamic model of composite sandwich panels with functionally graded flexible cores under low velocity impacts of multiple large or small masses using a new improved higher order sandwich panel theory (IHSAPT). In-plane stresses were considered for the functionally graded core and face sheets. The formulation was based on the first order shear deformation theory for the composite face sheets and polynomial description of the displacement fields in the core that was based on the second Frostig's model. Fully dynamic effects of the functionally graded core and face-sheets were considered in this study. Impacts were assumed to occur simultaneously and normally over the top and/or bottom of the face-sheets with arbitrary different masses and initial velocities. The contact forces between the panel and impactors were treated as internal forces of the system. Nonlinear contact stiffness was linearized with a newly presented improved analytical method in this paper. The results were validated by comparing the analytical, numerical and experimental results published in the latest literature.

• Steel and Composite Structures
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• 제41권6호
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• pp.899-910
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• 2021

A 4-unknown shear deformation theory is applied to investigate the vibration of functionally graded plates under thermal environment. The plate is fabricated from a functionally graded material mixed of ceramic and metal with continuously varying material properties through the plate thickness. Three types of thermal loadings, uniform, linear and nonlinear temperature rises along the plate thickness are taken into account. The present theory contains four unknown functions as against five or more in other higher order shear deformation theories. The through-the-thickness distributions of transverse shear stresses of the plate are considered to vary parabolically and vanish at upper and lower surfaces. The present model does not require any problem dependent shear correction factor. Analytical solutions for the free vibration analysis are derived based on Fourier series that satisfy the boundary conditions (Navier's method). Benchmark solutions are firstly considered to evaluate the accuracy of the proposed model. Comparisons with the solutions available in literature revealed the good capabilities of the present model for the simulations of vibration responses of FG plates. Some parametric studies are carried out for the frequency analysis by varying the volume fraction profile and the temperature distribution across the plate thickness.

• Structural Engineering and Mechanics
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• 제90권1호
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• pp.83-96
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• 2024

This paper suggests an analytical approach to investigate the free vibration and stability of functionally graded (FG) beams with both perfect and imperfect characteristics using a quasi-3D higher-order shear deformation theory (HSDT) with stretching effect. The study specifically focuses on FG beams resting on variable elastic foundations. In contrast to other shear deformation theories, this particular theory employs only four unknown functions instead of five. Moreover, this theory satisfies the boundary conditions of zero tension on the beam surfaces and facilitates hyperbolic distributions of transverse shear stresses without the necessity of shear correction factors. The elastic medium in consideration assumes the presence of two parameters, specifically Winkler-Pasternak foundations. The Winkler parameter exhibits variable variations in the longitudinal direction, including linear, parabolic, sinusoidal, cosine, exponential, and uniform, while the Pasternak parameter remains constant. The effective material characteristics of the functionally graded (FG) beam are assumed to follow a straightforward power-law distribution along the thickness direction. Additionally, the investigation of porosity includes the consideration of four different types of porosity distribution patterns, allowing for a comprehensive examination of its influence on the behavior of the beam. Using the virtual work principle, equations of motion are derived and solved analytically using Navier's method for simply supported FG beams. The accuracy is verified through comparisons with literature results. Parametric studies explore the impact of different parameters on free vibration and buckling behavior, demonstrating the theory's correctness and simplicity.

• Structural Engineering and Mechanics
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• 제75권2호
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• pp.157-174
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• 2020

In the present paper, a simple analytical model is developed based on a new refined parabolic shear deformation theory (RPSDT) for free vibration and buckling analysis of orthotropic rectangular plates with simply supported boundary conditions. The displacement field is simpler than those of other higher-order theories since it is modeled with only two unknowns and accounts for a parabolic distribution of the transverse shear stress through the plate thickness. The governing differential equations related to the present theory are obtained from the principle of virtual work, while the solution of the eigenvalue problem is achieved by assuming a Navier technique in the form of a double trigonometric series that satisfy the edge boundary conditions of the plate. Numerical results are presented and compared with previously published results for orthotropic rectangular plates in order to verify the precision of the proposed analytical model and to assess the impacts of several parameters such as the modulus ratio, the side-to-thickness ratio and the geometric ratio on natural frequencies and critical buckling loads. From these results, it can be concluded that the present computations are in excellent agreement with the other higher-order theories.

• Steel and Composite Structures
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• 제35권4호
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• pp.567-578
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• 2020

Based on third-order shear deformation shell theory, the present paper investigates post-buckling properties of eccentrically stiffened metal foam curved shells/panels having initial geometric imperfectness. Metal foam is considered as porous material with uniform and non-uniform models. The single-curve porous shell is subjected to in-plane compressive loads leading to post-critical stability in nonlinear regime. Via an analytical trend and employing Airy stress function, the nonlinear governing equations have been solved for calculating the post-buckling loads of stiffened geometrically imperfect metal foam curved shell. New findings display the emphasis of porosity distributions, geometrical imperfectness, foundation factors, stiffeners and geometrical parameters on post-buckling properties of porous curved shells/panels.

• Structural Engineering and Mechanics
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• 제86권3호
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• pp.291-309
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• 2023

This paper addresses the finite element modeling of functionally graded porous (FGP) beams for free vibration and buckling behaviour cases. The formulated finite element is based on simple and efficient higher order shear deformation theory. The key feature of this formulation is that it deals with Euler-Bernoulli beam theory with only three unknowns without requiring any shear correction factor. In fact, the presented two-noded beam element has three degrees of freedom per node, and the discrete model guarantees the interelement continuity by using both C⁰ and C¹ continuities for the displacement field and its first derivative shape functions, respectively. The weak form of the governing equations is obtained from the Hamilton principle of FGP beams to generate the elementary stiffness, geometric, and mass matrices. By deploying the isoparametric coordinate system, the derived elementary matrices are computed using the Gauss quadrature rule. To overcome the shear-locking phenomenon, the reduced integration technique is used for the shear strain energy. Furthermore, the effect of porosity distribution patterns on the free vibration and buckling behaviours of porous functionally graded beams in various parameters is investigated. The obtained results extend and improve those predicted previously by alternative existing theories, in which significant parameters such as material distribution, geometrical configuration, boundary conditions, and porosity distributions are considered and discussed in detailed numerical comparisons. Determining the impacts of these parameters on natural frequencies and critical buckling loads play an essential role in the manufacturing process of such materials and their related mechanical modeling in aerospace, nuclear, civil, and other structures.

• Advances in nano research
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• 제6권2호
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• pp.147-162
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• 2018

A new nonlocal higher order shear deformation theory (HSDT) is developed for buckling properties of single graphene sheet. The proposed nonlocal HSDT contains a new displacement field which incorporates undetermined integral terms and contains only two variables. The length scale parameter is considered in the present formulation by employing the nonlocal differential constitutive relations of Eringen. Closed-form solutions for critical buckling forces of the graphene sheets are obtained. Nonlocal elasticity theories are used to bring out the small scale influence on the critical buckling force of graphene sheets. Influences of length scale parameter, length, thickness of the graphene sheets and shear deformation on the critical buckling force have been examined.

• Earthquakes and Structures
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• 제15권3호
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• pp.285-294
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• 2018

In this study, nonlinear dynamic response of a concrete plate retrofit with Aluminium oxide ($Al_2O_3$) under seismic load and magnetic field is investigated. The plate is a composite reinforced by Aluminium oxide with characteristics of the equivalent composite being determined using Mori-Tanka model considering agglomeration effect. The plate is simulated with higher order shear deformation plate model. Employing nonlinear strains-displacements, stress-strain, the energy equations of column was obtained and using Hamilton's principal, the governing equations were derived. Differential quadrature method (DQM) in conjunction with Newark method is applied for obtaining the dynamic response of structure. The influences of magnetic field, volume percent of nanoparticles, geometrical parameters of column, agglomeration and boundary conditions on the dynamic response were investigated. Results showed that with increasing volume percent of nanoparticles, the dynamic deflection decreases.