• 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.

• 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.

• Journal of Korean Society of Steel Construction
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• v.12 no.3 s.46
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• pp.303-315
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• 2000

The main object of this study is to analyze mechanical behaviors as anisotropic three-dimensional body under various static loads. This paper presents the applicability of the finite difference method to three dimensional problem of anisotropic body. The finite difference method as applied here is generalized to anisotropic three-dimensional problem of elastic body where the governing differential equations of equilibrium of such bodies are expressed in terms of the displacement u, v, and w in the coordinates axes x, y and z, care being taken to modify the finite difference expressions to satisfy the appropriate boundary conditions. By adopting a new three dimensional finite difference modelling including elimination of pivotal difference points in the case of free boundary condition, the three dimensional problem of anisotropic body was successfully completed. Several numerical results show quick convergence and numerical validity of finite difference technique in three dimensional problem.

• Steel and Composite Structures
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• v.26 no.4
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• pp.513-531
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• 2018

In this article, static, buckling and free vibration analyses of a sinusoidal micro composite beam reinforced by single-walled carbon nanotubes (SWCNTs) with considering temperature-dependent material properties embedded in an elastic medium in the presence of magnetic field under transverse uniform load are presented. This system is used at micro or sub micro scales to enhance the stiffness of micro composite structures such as bar, beam, plate and shell. In the present work, the size dependent effects based on surface stress effect and modified strain gradient theory (MSGT) are considered. The generalized rule of mixture is employed to predict temperature-dependent mechanical and thermal properties of micro composite beam. Then, the governing equations of motions are derived using Hamilton's principle and energy method. Numerical results are presented to investigate the influences of material length scale parameters, elastic foundation, composite fiber angle, magnetic intensity, temperature changes and carbon nanotubes volume fraction on the bending, buckling and free vibration behaviors of micro composite beam. There is a good agreement between the obtained results by this research and the literature results. The obtained results of this study demonstrate that the magnetic intensity, temperature changes, and two parameters elastic foundations have important effects on micro composite stiffness, while the magnetic field has greater effects on the bending, buckling and free vibration responses of micro composite beams. Moreover, it is shown that the effects of surface layers are important, and observed that the changes of carbon nanotubes volume fraction, beam length-to-thickness ratio and material length scale parameter have noticeable effects on the maximum deflection, critical buckling load and natural frequencies of micro composite beams.

• Structural Engineering and Mechanics
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• v.23 no.3
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• pp.293-308
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• 2006

This paper presents a complete and consistent formulation to study the seismic response of a free-standing ship supported by an arrangement of n keel blocks which are all located in a dry dock. It is considered that the foundation of the system is subjected to both horizontal and vertical in plane excitation. The motion of the system is classified in eight different modes which are Rest (relative), Sliding of keel blocks, Rocking of keel blocks, Sliding of the ship, Sliding of both keel blocks and the ship, Sliding and rocking of keel blocks, Rocking of keel blocks with sliding of the ship, and finally Sliding and rocking of keel blocks accompanied with sliding of the ship. For each mode of motion the governing equations are derived, and transition conditions between different modes are also defined. This formulation is based on a number of fundamental assumptions which are 2D idealization for motion of the system, considering keel blocks as the rigid ones and the ship as a massive rigid block too, allowing the similar motion for all keel blocks, and supposing frictional nature for transmitted forces between contacted parts. Also, the rocking of the ship is not likely to take place, and the complete ship separation from keel blocks or separation of keel blocks from the base is considered as one of the failure mode in the system. The formulation presented in this paper can be used in its entirety or in part, and they are suitable for investigation of generalized response using suitable analytical, or conducting a time-history sensitivity analysis.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.15 no.3
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• pp.463-469
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• 2002

The governing equation and force-displacement rotations of a beam-column element on elastic foundation we derived based on variational approach of total potential energy. An exact static and dynamic 4×4 element stiffness matrix of the beam-column element is established via a generalized lineal-eigenvalue problem by introducing 4 displacement parameters and a system of linear algebraic equations with complex matrices. The structure stiffness matrix is established by the conventional direct stiffness method. In addition the F. E. procedure is presented by using Hermitian polynomials as shape function and evaluating the corresponding elastic and geometric stiffness and the mass matrix. In order to verify the efficiency and accuracy of the beam-column element using exact dynamic stiffness matrix, buckling loads and natural frequencies are calculated for the continuous beam structures and the results are compared with F E. solutions.

• Advances in nano research
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• v.12 no.1
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• pp.81-99
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• 2022

The aim of current work is to evaluate thermo-electrical characteristics of graphene nanoplatelets Reinforced Composite (GNPRC) coupled with magneto-electro-elastic (MEE) face sheet. In this regard, a cylindrical smart nanocomposite made of GNPRC with an external MEE layer is considered. The bonding between the layers are assumed to be perfect. Because of the layer nature of the structure, the material characteristics of the whole structure is regarded as graded. Both mechanical and thermal boundary conditions are applied to this structure. The main objective of this work is to determine critical temperature and critical voltage as a function of thermal condition, support type, GNP weight fraction, and MEE thickness. The governing equation of the multilayer nanocomposites cylindrical shell is derived. The generalized differential quadrature method (GDQM) is employed to numerically solve the differential equations. This method is integrated with Deep Learning Network (DNN) with ADADELTA optimizer to determine the critical conditions of the current sandwich structure. This the first time that effects of several conditions including surrounding temperature, MEE layer thickness, and pattern of the layers of the GNPRC is investigated on two main parameters critical temperature and critical voltage of the nanostructure. Furthermore, Maxwell equation is derived for modeling of the MEE. The outcome reveals that MEE layer, temperature change, GNP weight function, and GNP distribution patterns GNP weight function have significant influence on the critical temperature and voltage of cylindrical shell made from GNP nanocomposites core with MEE face sheet on outer of the shell.

• 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.

• Computers and Concrete
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• v.32 no.1
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• pp.61-74
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• 2023

This article investigates the wave propagation analysis of the imperfect functionally graded (FG) sandwich plates based on a novel simple four-variable integral quasi-3D higher-order shear deformation theory (HSDT). The thickness stretching effect is considered in the transverse displacement component. The presented formulation ensures a parabolic variation of the transverse shear stresses with zero-stresses at the top and the bottom surfaces without requiring any shear correction factors. The studied sandwich plates can be used in several sectors as areas of aircraft, construction, naval/marine, aerospace and wind energy systems, the sandwich structure is composed from three layers (two FG face sheets and isotropic core). The material properties in the FG faces sheet are computed according to a modified power law function with considering the porosity which may appear during the manufacturing process in the form of micro-voids in the layer body. The Hamilton principle is utilized to determine the four governing differential equations for wave propagation in FG plates which is reduced in terms of computation time and cost compared to the other conventional quasi-3D models. An eigenvalue equation is formulated for the analytical solution using a generalized displacements' solution form for wave propagation. The effects of porosity, temperature, moisture concentration, core thickness, and the material exponent on the plates' dispersion relations are examined by considering the thickness stretching influence.