• Wind and Structures
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• v.28 no.1
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• pp.19-30
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• 2019

This article present the free vibration analysis of simply supported perfect and imperfect (porous) FG beams using a high order trigonometric deformation theory. It is assumed that the material properties of the porous beam vary across the thickness. Unlike other theories, the number of unknown is only three. This theory has a parabolic shear deformation distribution across the thickness. So it is useless to use the shear correction factors. The Hamilton's principle will be used herein to determine the equations of motion. Since, the beams are simply supported the Navier's procedure will be retained. To show the precision of this model, several comparisons have been made between the present results and those of existing theories in the literature.

• Structural Engineering and Mechanics
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• v.69 no.4
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• pp.439-455
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• 2019

This research deals with thermo-electro-mechanical buckling analysis of the sandwich nano-beams with face-sheets made of functionally graded carbon nano-tubes reinforcement composite (FG-CNTRC) based on the nonlocal strain gradient elasticity theory (NSGET) considering various higher-order shear deformation beam theories (HSDBT). The sandwich nano-beam with FG-CNTRC face-sheets is subjected to thermal and electrical loads while is resting on Pasternak's foundation. It is assumed that the material properties of the face-sheets change continuously along the thickness direction according to different patterns for CNTs distribution. In order to include coupling of strain and electrical field in equation of motion, the nonlocal non-classical nano-beam model contains piezoelectric effect. The governing equations of motion are derived using Hamilton principle based on HSDBTs and NSGET. The differential quadrature method (DQM) is used to calculate the mechanical buckling loads of sandwich nano-beam as well as critical voltage and temperature rising. After verification with validated reference, comprehensive numerical results are presented to investigate the influence of important parameters such as various HSDBTs, length scale parameter (strain gradient parameter), the nonlocal parameter, the CNTs volume fraction, Pasternak's foundation coefficients, various boundary conditions, the CNTs efficiency parameter and geometric dimensions on the buckling behaviors of FG sandwich nano-beam. The numerical results indicate that, the amounts of the mechanical critical load calculated by PSDBT and TSDBT approximately have same values as well as ESDBT and ASDBT. Also, it is worthy noted that buckling load calculated by aforementioned theories is nearly smaller than buckling load estimated by FSDBT. Also, similar aforementioned structure is used to building the nano/micro oscillators.

• Steel and Composite Structures
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• v.39 no.5
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• pp.615-630
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• 2021

The main purpose of the present work was to study the dynamic instability of a three-layered, thick composite sandwich beam with the functionally graded (FG) flexible core subjected to an axial compressive follower force. Flutter instability of a sandwich cantilever beam was analyzed using the high-order theory of sandwich beams, for the first time. The governing equations in general for sandwich beams with an FG core were extracted and could be used for all types of sandwich beams with any types of face sheets and cores. A polynomial function is considered for the vertical distribution of the displacement field in the core layer along the thickness, based on the results of the first Frosting's higher order model. The governing partial differential equations and the equations of boundary conditions of the dynamic system are derived using Hamilton's principle. By applying the boundary conditions and numerical solution methods of squares quadrature, the beam flutter phenomenon is studied. In addition, the effects of different geometrical and material parameters on the flutter threshold were investigated. The results showed that the responses of the dynamic instability of the system were influenced by the follower force, the coefficients of FGs and the geometrical parameters like the core thickness. Comparison of the present results with the published results in the literature for the special case confirmed the accuracy of the proposed theory. The results showed that the follower force of the flutter phenomenon threshold for long beams tends to the corresponding results in the Timoshenko beam.

• Structural Engineering and Mechanics
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• v.58 no.3
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• pp.515-532
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• 2016

The elastoplastic response of functionally graded material (FGM) beams resting on a nonlinear elastic foundation to an eccentric axial load is investigated by using the finite element method. The FGM is assumed to be formed from ceramic and metal phases with their volume fraction vary in the thickness direction by a power-law function. A bilinear elastoplastic behavior is assumed for the metallic phase, and the effective elastoplastic properties of the FGM are evaluated by Tamura-Tomota-Ozawa (TTO) model. Based on the classical beam theory, a nonlinear finite beam element taking the shift in the neutral axis position into account is formulated and employed in the investigation. An incremental-iterative procedure in combination with the arc-length control method is employed in computing the equilibrium paths of the beams. The validation of the formulated element is confirmed by comparing the equilibrium paths obtained by using the present element and the one available in the literature. The numerical results show that the elastoplastic post-buckling of the FGM beams is unstable, and the post-buckling strength is higher for the beams associated with a higher ceramic content. Different from homogeneous beams, yielding in the FGM beam occurs in the layer near the ceramic layer before in the layer near metal surface. A parametric study is carried out to highlight the effect of the material distribution, foundation support and eccentric ratio on the elastoplastic response of the beams.

• Steel and Composite Structures
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• v.21 no.1
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• pp.1-36
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• 2016

In the present study, the nonlinear magneto-electro-mechanical free vibration behavior of rectangular double-bonded sandwich microbeams based on the modified strain gradient theory (MSGT) is investigated. It is noted that the top and bottom sandwich microbeams are considered with boron nitride nanotube reinforced composite face sheets (BNNTRC-SB) with electrical properties and carbon nanotube reinforced composite face sheets (CNTRC-SB) with magnetic fields, respectively, and also the homogenous core is used for both sandwich beams. The connections of every sandwich beam with its surrounding medium and also between them have been carried out by considering Pasternak foundations. To take size effect into account, the MSGT is introduced into the classical Timoshenko beam theory (CT) to develop a size-dependent beam model containing three additional material length scale parameters. For the CNTRC and BNNTRC face sheets of sandwich microbeams, uniform distribution (UD) and functionally graded (FG) distribution patterns of CNTs or BNNTs in four cases FG-X, FG-O, FG-A, and FG-V are employed. It is assumed that the material properties of face sheets for both sandwich beams are varied in the thickness direction and estimated through the extended rule of mixture. On the basis of the Hamilton's principle, the size-dependent nonlinear governing differential equations of motion and associated boundary conditions are derived and then discretized by using generalized differential quadrature method (GDQM). A detailed parametric study is presented to indicate the influences of electric and magnetic fields, slenderness ratio, thickness ratio of both sandwich microbeams, thickness ratio of every sandwich microbeam, dimensionless three material length scale parameters, Winkler spring modulus and various distribution types of face sheets on the first two natural frequencies of double-bonded sandwich microbeams. Furthermore, a comparison between the various beam models on the basis of the CT, modified couple stress theory (MCST), and MSGT is performed. It is illustrated that the thickness ratio of sandwich microbeams plays an important role in the vibrational behavior of the double-bonded sandwich microstructures. Meanwhile, it is concluded that by increasing H/lm, the values of first two natural frequencies tend to decrease for all amounts of the Winkler spring modulus.

• Advances in nano research
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• v.7 no.4
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• pp.249-263
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• 2019

In the current paper, an exact solution method is carried out for analyzing the thermo-mechanical vibration of curved FG nano-beams subjected to uniform thermal environmental conditions, by considering porosity distribution via nonlocal strain gradient beam theory for the first time. Nonlocal strain gradient elasticity theory is adopted to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field is considered. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Material properties of curved porous FG nanobeam are assumed to be temperature-dependent and are supposed to vary through the thickness direction of beam which modeled via modified power-law rule. Since variation of pores along the thickness direction influences the mechanical and physical properties, porosity play a key role in the mechanical response of curved FG nano-structures. The governing equations and related boundary condition of curved porous FG nanobeam under temperature field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved nanobeam supposed to thermal loading. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, porosity volume fractions, thermal effect, gradient index, opening angle and aspect ratio on the natural frequency of curved FG porous nanobeam are successfully discussed. It is concluded that these parameters play key roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

• Structural Engineering and Mechanics
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• v.78 no.2
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• pp.117-124
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• 2021

This work focused on the novel numerical tool for the bending responses of carbon nanotube reinforced composites (CNTRC) beams. The higher order shear deformation beam theory (HSDT) is used to determine strain-displacement relationships. A new exponential function was introduced into the carbon nanotube (CNT) volume fraction equation to show the effect of the CNT distribution on the CNTRC beams through displacements and stresses. To determine the mechanical properties of CNTRCs, the rule of the mixture was employed by assuming that the single-walled carbon nanotubes (SWCNTs)are aligned and distributed in the matrix. The governing equations were derived by Hamilton's principle, and the mathematical models presented in this work are numerically provided to verify the accuracy of the present theory. The effects of aspect ratio (l/d), CNT volume fraction (Vcnt), and the order of exponent (n) on the displacement and stresses are presented and discussed in detail. Based on the analytical results. It turns out that the increase of the exponent degree (n) makes the X-beam stiffer and the exponential CNTs distribution plays an indispensable role to improve the mechanical properties of the CNTRC beams.

• Steel and Composite Structures
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• v.49 no.5
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• pp.487-502
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• 2023

In the realm of nanotechnology, the nonlocal strain gradient theory takes center stage as it scrutinizes the behavior of spinning cantilever nanobeams and nanotubes, pivotal components supporting various mechanical movements in sport structures. The dynamics of these structures have sparked debates within the scientific community, with some contending that nonlocal cantilever models fail to predict dynamic softening, while others propose that they can indeed exhibit stiffness softening characteristics. To address these disparities, this paper investigates the dynamic response of a nonlocal cantilever cylindrical beam under the influence of external discontinuous dynamic loads. The study employs four distinct models: the Euler-Bernoulli beam model, Timoshenko beam model, higher-order beam model, and a novel higher-order tube model. These models account for the effects of functionally graded materials (FGMs) in the radial tube direction, giving rise to nanotubes with varying properties. The Hamilton principle is employed to formulate the governing differential equations and precise boundary conditions. These equations are subsequently solved using the generalized differential quadrature element technique (GDQEM). This research not only advances our understanding of the dynamic behavior of nanotubes but also reveals the intriguing phenomena of both hardening and softening in the nonlocal parameter within cantilever nanostructures. Moreover, the findings hold promise for practical applications, including drug delivery, where the controlled vibrations of nanotubes can enhance the precision and efficiency of medication transport within the human body. By exploring the multifaceted characteristics of nanotubes, this study not only contributes to the design and manufacturing of rotating nanostructures but also offers insights into their potential role in revolutionizing drug delivery systems.

• Structural Engineering and Mechanics
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• v.68 no.1
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• pp.53-66
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• 2018

This paper presents an analysis of the bending, buckling and free vibration of functionally graded sandwich beams resting on elastic foundation by using a refined quasi-3D theory in which both shear deformation and thickness stretching effects are included. The displacement field contains only three unknowns, which is less than the number of parameters of many other shear deformation theories. In order to homogenize the micromechanical properties of the FGM sandwich beam, the material properties are derived on the basis of several micromechanical models such as Tamura, Voigt, Reuss and many others. The principle of virtual works is used to obtain the equilibrium equations. The elastic foundation is modeled using the Pasternak mathematical model. The governing equations are obtained through the Hamilton's principle and then are solved via Navier solution for the simply supported beam. The accuracy of the proposed theory can be noticed by comparing it with other 3D solution available in the literature. A detailed parametric study is presented to show the influence of the micromechanical models on the general behavior of FG sandwich beams on elastic foundation.

• Steel and Composite Structures
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• v.25 no.2
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• pp.141-155
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• 2017

This paper presents an investigation into the magneto-thermo-mechanical vibration and damping of a viscoelastic functionally graded-carbon nanotubes (FG-CNTs)-reinforced curved microbeam based on Timoshenko beam and strain gradient theories. The structure is surrounded by a viscoelastic medium which is simulated with spring, damper and shear elements. The effective temperature-dependent material properties of the CNTs-reinforced composite beam are obtained using the extended rule of mixture. The structure is assumed to be subjected to a longitudinal magnetic field. The governing equations of motion are derived using Hamilton's principle and solved by employing differential quadrature method (DQM). The effect of various parameter like volume percent and distribution type of CNTs, temperature change, magnetic field, boundary conditions, material length scale parameter, central angle, viscoelastic medium and structural damping on the vibration and damping behaviors of the nanocomposite curved microbeam is examined. The results show that with increasing volume percent of CNTs and considering magnetic field, material length scale parameter and viscoelastic medium, the frequency of the system increases and critically damped situation occurs at higher values of damper constant. In addition, the structure with FGX distribution type of CNTs has the highest stiffness. It is also observed that increasing temperature, structural damping and central angle of curved microbeam decreases the frequency of the system.