• Steel and Composite Structures
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• v.26 no.6
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• pp.673-691
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• 2018

The main goal of this research is to examine the in-plane and out-of-plane forced vibration of a curved nanocomposite microbeam. The in-plane and out-of-plane displacements of the structure are considered based on the first order shear deformation theory (FSDT). The curved microbeam is reinforced by functionally graded carbon nanotubes (FG-CNTs) and thus the extended rule of mixture is employed to estimate the effective material properties of the structure. Also, the small scale effect is captured using the strain gradient theory. The structure is rested on a nonlinear orthotropic viscoelastic foundation and is subjected to concentrated transverse harmonic external force, thermal and magnetic loads. The derivation of the governing equations is performed using energy method and Hamilton's principle. Differential quadrature (DQ) method along with integral quadrature (IQ) and Newmark methods are employed to solve the problem. The effect of various parameters such as volume fraction and distribution type of CNTs, boundary conditions, elastic foundation, temperature changes, material length scale parameters, magnetic field, central angle and width to thickness ratio are studied on the frequency and force responses of the structure. The results indicate that the highest frequency and lowest vibration amplitude belongs to FGX distribution type while the inverse condition is observed for FGO distribution type. In addition, the hardening-type response of the structure with FGX distribution type is more intense with respect to the other distribution types.

• 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.

• Composites Research
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• v.16 no.3
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• pp.68-74
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• 2003

In this paper, intermetallic/metal laminated composites have been successfully produced that utilizes SHS reactions between Ni and Al elemental metal foils. The reaction between Ni and Al started from the nucleation and growth of NiA1$_3$ and was followed by the diffusional growth of Ni$_2$A1$_3$ between Ni and NiA1$_3$. The SHS reaction was thermodynamically analyzed through the final volume fraction of the non-reacted Al related with the initial thickness ratio of Ni:Al and prior heat treatment. Thermally aging these 1aminates resulted in formation of a functionally gradient series of intermetallic phases. Microstructure showed that the intermetallic volume percent was 82, 59.5, 40% in the 1:1, 2:1, 4:1 thickness ratio specimen. Main phases of the intermetallic were NiAl and Ni$_3$Al having higher strength at room and high temperatures.

• Steel and Composite Structures
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• v.39 no.1
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• pp.95-107
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• 2021

In this work, the dynamic response of functionally graded beams on variable elastic foundations is studied using a novel higher-order shear deformation theory (HSDT). Unlike the conventional HSDT, the present one has a new displacement field which introduces undetermined integral variables. The FG beams were assumed to be supported on Winkler-Pasternak type foundations in which the Winkler modulus is supposed to be variable in the length of the beam. The variable rigidity of the elastic foundation is assumed to be linear, parabolic and sinusoidal along the length of the beam. The material properties of the FG porous beam vary according to a power law distribution in terms of the volume fraction of the constituents. The equations of motion are determined using the virtual working principle. For the analytical solution, Navier method is used to solve the governing equations for simply supported porous FG beams. Numerical results of the present theory for the free vibration of FG beams resting on elastic foundations are presented and compared to existing solutions in the literature. A parametric study will be detailed to investigate the effects of several parameters such as gradient index, thickness ratio, porosity factor and foundation parameters on the frequency response of porous FG beams.

• Steel and Composite Structures
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• v.33 no.6
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• pp.805-822
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• 2019

In this work, a simple four-variable integral plate theory is employed for examining the thermal buckling properties of functionally graded material (FGM) sandwich plates. The proposed kinematics considers integral terms which include the effect of transverse shear deformations. Material characteristics and thermal expansion coefficient of the ceramic-metal FGM sandwich plate faces are supposed to be graded in the thickness direction according to a "simple power-law" variation in terms of the "volume fractions" of the constituents. The central layer is always homogeneous and consists of an isotropic material. The thermal loads are supposed as uniform, linear, and nonlinear temperature rises within the thickness direction. The influences of geometric ratios, gradient index, loading type, and type sandwich plate on the buckling properties are examined and discussed in detail.

• Steel and Composite Structures
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• v.41 no.6
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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.

• Steel and Composite Structures
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• v.49 no.6
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• pp.603-614
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• 2023

In this paper, frequency analysis of curved functionally graded (FG) nanobeam by consideration of deepness effect has been studied. Differential transform method (DTM) has been used to obtain frequency responses. The nonlocal theory of Eringen has been applied to consider nanoscales. Material properties are supposed to vary in radial direction according to power-law distribution. Differential equations and related boundary conditions have been derived using Hamilton's principle. Finally, by consideration of nonlocal theory, the governing equations have been derived. Natural frequencies have been obtained using semi analytical method (DTM) for different boundary conditions. In order to study the effect of deepness, the deepness term is considered in strain field. The effects of the gradient index, radius of curvature, the aspect ratio, the nonlocal parameter and interaction of aforementioned parameters on frequency value for different boundary conditions such as clamped-clamped (C-C), clamped-hinged (C-H), and clamped-free (C-F) have been investigated. In addition, the obtained results are compared with the results in previous literature in order to validate present study, a good agreement was observed in the present results.

• Steel and Composite Structures
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• v.42 no.6
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• pp.805-826
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• 2022

The free and live load-forced vibration behaviour of porous functionally graded (PFG) higher order nanobeams in the thermal and magnetic fields is investigated comprehensively through this work in the framework of nonlocal strain gradient theory (NLSGT). The porosity effects on the dynamic behaviour of FG nanobeams is investigated using four different porosity distribution models. These models are exploited; uniform, symmetrical, condensed upward, and condensed downward distributions. The material characteristics gradation in the thickness direction is estimated using the power-law. The magnetic field effect is incorporated using Maxwell's equations. The third order shear deformation beam theory is adopted to incorporate the shear deformation effect. The Hamilton principle is adopted to derive the coupled thermomagnetic dynamic equations of motion of the whole system and the associated boundary conditions. Navier method is used to derive the analytical solution of the governing equations. The developed methodology is verified and compared with the available results in the literature and good agreement is observed. Parametric studies are conducted to show effects of porosity parameter; porosity distribution, temperature rise, magnetic field intensity, material gradation index, non-classical parameters, and the applied moving load velocity on the vibration behavior of nanobeams. It has been showed that all the analyzed conditions have significant effects on the dynamic behavior of the nanobeams. Additionally, it has been observed that the negative effects of moving load, porosity and thermal load on the nanobeam dynamics can be reduced by the effect of the force induced from the directed magnetic field or can be kept within certain desired design limits by controlling the intensity of the magnetic field.

• Geomechanics and Engineering
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• v.32 no.2
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• pp.159-177
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• 2023

In the present work, a simple and refined shear deformation theory is used to analyze the effect of visco-elastic foundation on the buckling response of exponentially-gradient sandwich plates under various boundary conditions. The proposed theory includes indeterminate integral variables kinematic with only four generalized parameters, in which no shear correction factor is used. The visco-Pasternak's foundation is taken into account by adding the influence of damping to the usual foundation model which characterized by the linear Winkler's modulus and Pasternak's foundation modulus. The four governing equations for FGM sandwich plates are derived by employing principle of virtual work. To solve the buckling problem, Galerkin's approach is utilized for FGM sandwich plates for various boundary conditions. The analytical solutions for critical buckling loads of several types of powerly graded sandwich plates resting on visco-Pasternak foundations under various boundary conditions are presented. Some numerical results are presented to indicate the effects of inhomogeneity parameter, elastic foundation type, and damping coefficient of the foundation, on the critical buckling loads.

• 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.