• Steel and Composite Structures
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• v.28 no.5
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• pp.589-606
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• 2018

The previous studies reflected the significant effect of neutral-axis position and coupling of in-plane and out-of-plane displacements on behavior of functionally graded (FG) nanobeams. In thin FG beam, this coupling can be eliminated by a proper choice of the reference axis. In shear deformable FG nanobeam, not only this coupling can't be eliminated but also the position of neutral-axis is dependent on through-thickness distribution of shear strain. For the first time, in this paper it is avoided to guess a shear strain shape function and the exact shape function and consequently the exact position of neutral axis for arbitrary gradation of higher order nanobeam are obtained. This paper presents new methodology based on differential transform and collocation methods to solve coupled partial differential equations of motion without any simplifications. Using exact position of neutral axis and higher order beam kinematics as well as satisfying equilibrium equations and traction-free conditions without shear correction factor requirement yields to better results in comparison to the previously published results in literature. The classical rule of mixture and Mori-Tanaka homogenization scheme are considered. The Eringen's nonlocal continuum theory is applied to capture the small scale effects. For the first time, the dependency of exact position of neutral axis on length to thickness ratio is investigated. The effects of small scale, length to thickness ratio, Poisson's ratio, inhomogeneity of materials and various end conditions on vibration and buckling of local and nonlocal FG beams are investigated. Moreover, the effect of axial load on natural frequencies of the first modes is examined. After degeneration of the governing equations, the exact new formulas for homogeneous nanobeams are computed.

• Smart Structures and Systems
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• v.20 no.6
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• pp.709-728
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• 2017

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric 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 piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important 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.

• 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.

• Smart Structures and Systems
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• v.22 no.1
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• pp.105-120
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• 2018

In this paper, the nonlinear free and forced vibration responses of sandwich nano-beams with three various functionally graded (FG) patterns of reinforced carbon nanotubes (CNTs) face-sheets are investigated. The sandwich nano-beam is resting on nonlinear Visco-elastic foundation and is subjected to thermal and electrical loads. The nonlinear governing equations of motion are derived for an Euler-Bernoulli beam based on Hamilton principle and von Karman nonlinear relation. To analyze nonlinear vibration, Galerkin's decomposition technique is employed to convert the governing partial differential equation (PDE) to a nonlinear ordinary differential equation (ODE). Furthermore, the Multiple Times Scale (MTS) method is employed to find approximate solution for the nonlinear time, frequency and forced responses of the sandwich nano-beam. Comparison between results of this paper and previous published paper shows that our numerical results are in good agreement with literature. In addition, the nonlinear frequency, force response and nonlinear damping time response is carefully studied. The influences of important parameters such as nonlocal parameter, volume fraction of the CNTs, different patterns of CNTs, length scale parameter, Visco-Pasternak foundation parameter, applied voltage, longitudinal magnetic field and temperature change are investigated on the various responses. One can conclude that frequency of FG-AV pattern is greater than other used patterns.

• Computers and Concrete
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• v.33 no.1
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• pp.91-102
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• 2024

Beam-shaped components commonly rotate along a fixed axis when massive mechanical structures like rotors, jet engine blades, motor turbines, and rotating railway crossings perform their functions. For these structures to be useful in real life, their mechanical behavior is essential. Therefore, this is the first article to use the modified shear deformation theory type hyperbolic sine functions theory and the FEM to study the static bending response of rotating functionally graded GPL-reinforced composite (FG-GPLRC) beams with initial geometrical deficiencies in thermal media. Graphene platelets (GPLs) in three different configurations are woven into the beam's composition to increase its strength. By comparing the numerical results with those of previously published studies, we can assess the robustness of the theory and mechanical model employed in this study. Parameter studies are performed to determine the effect of various geometric and physical variables, such as rotation speed and temperature, on the bending reactions of structures.

• Steel and Composite Structures
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• v.36 no.2
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• pp.179-186
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• 2020

In this paper, the forced resonance vibration of porous functionally graded (FG) curved nanobeam is examined. In order to capture the hardening and softening mechanisms of nanostructure, the nonlocal strain gradient theory is employed to build the size-dependent model. Using the Timoshenko beam theory together with the Hamilton principle, the equations of motion for the curved nanobeam are derived. Then, Navier series are used in order to obtain the dynamical deflections of the porous FG curved nanobeam with simply-supported ends. It is found that the resonance position of the nanobeam is very sensitive to the nonlocal and strain gradient parameters, material variation, porosity coefficient, as well as geometrical conditions. The results indicate that the resonance position is postponed by increasing the strain gradient parameter, while the nonlocal parameter has the opposite effect on the results. Furthermore, increasing the opening angle or length-to-thickness ratio will result in resonance position moves to lower-load frequency.

• Steel and Composite Structures
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• v.20 no.5
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• pp.1119-1131
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• 2016

As a first attempt, an inverse hybrid numerical method for small scale parameter estimation of functionally graded (FG) nanobeams using measured frequencies is presented. The governing equations are obtained with the Eringen's nonlocal elasticity assumptions and the first-order shear deformation theory (FSDT). The equations are discretized by using the differential quadrature method (DQM). The discretized equations are transferred from temporal domain to frequency domain and frequencies of the nanobeam are obtained. By applying random error to these frequencies, measured frequencies are generated. The measured frequencies are considered as input data and inversely, the small scale parameter of the beam is obtained by minimizing a defined functional. The functional is defined as root mean square error between the measured frequencies and calculated frequencies by the DQM. Then, the conjugate gradient (CG) optimization method is employed to minimize the functional and the small scale parameter is obtained. Efficiency, convergence and accuracy of the presented hybrid method for small scale parameter estimation of the beams for different applied random error, boundary conditions, length-to-thickness ratio and volume fraction coefficients are demonstrated.

• Structural Monitoring and Maintenance
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• v.10 no.4
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• pp.315-337
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• 2023

During the design phase, it is crucial to determine the interface stresses between the reinforcing plate and the concrete base in order to predict plate end separation failures. In this work, a simple theoretical study of interface shear stresses in beams reinforced with P-FGM and E-FGM plates subjected to an arbitrarily positioned point load, or two symmetrical point loads, was presented using the linear elastic theory. The presence of pores in the reinforcing plate distributed in several forms was also taken into account. For this purpose, we analyze the effects of porosity and its distribution shape on the interracial normal and shear stresses of an FGM beam reinforced with an FRP plate under different types of load. Comparisons of the proposed model with existing analytical solutions in the literature confirm the feasibility and accuracy of this new approach. The influence of different parameters on the interfacial behavior of reinforced concrete beams reinforced with functionally graded porous plates is further examined in this parametric study using the proposed model. From the results obtained in this study, we can say that interface stress is significantly affected by several factors, including the pores present in the reinforcing plate and their distribution shape. Additionally, we can conclude from this study that reinforcement systems with composite plates are very effective in improving the flexural response of reinforced RC beams.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.31 no.3
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• pp.127-132
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• 2018

In this paper, a scheme for the evaluation of variability in the eigen-modes of functionally graded material(FGM) beams is proposed within the framework of perturbation-based stochastic analysis. As a random parameter, the spatially varying elastic modulus of FGM along the axial direction at the mid-surface of the beam is chosen, and the thru-thickness variation of the elastic modulus is assumed to follow the original form of exponential variation. In deriving the formulation, the first order Taylor expansion on the eigen-modes is employed. As an example, a simply supported FGM beam having symmetric elastic modulus with respect to the mid-surface is chosen. Monte Carlo analysis is also performed to check if the proposed scheme gives reasonable outcomes. From the analyses it is found that the two schemes give almost identical results of the mean and standard deviation of eigen-modes. With the propose scheme, the standard deviation shape of respective eigen-modes can be evaluated easily. The deviated mode shape is found to have one more zero-slope points than the mother modes shapes, irrespective of order of modes. The amount of deviation from the mean is found to have larger values for the higher modes than the lower modes.

• Computers and Concrete
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• v.31 no.3
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• pp.267-276
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• 2023

The main purpose of the current research is to develop an efficient two variables trigonometric shear deformation beam theory to investigate the buckling behavior of symmetric and non-symmetric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) beam resting on an elastic foundation with various boundary conditions. The proposed theory obviates the use to shear correction factors as it satisfies the parabolic variation of through-thickness shear stress distribution. The composite beam is made of a polymeric matrix reinforced by aligned and distributed single-walled carbon nanotubes (SWCNTs) with different patterns of reinforcement. The material properties of the FG-CNTRC beam are estimated by using the rule of mixture. The governing equilibrium equations are solved by using new analytical solutions based on the Galerkin method. The robustness and accuracy of the proposed analytical model are demonstrated by comparing its results with those available by other researchers in the existing literature. Moreover, a comprehensive parametric study is presented and discussed in detail to show the effects of CNTs volume fraction, distribution patterns of CNTs, boundary conditions, length-to-thickness ratio, and spring constant factors on the buckling response of FG-CNTRC beam. Some new referential results are reported for the first time, which will serve as a benchmark for future research.