• Steel and Composite Structures
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• v.50 no.2
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• pp.135-148
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• 2024

This paper analytically investigates the free vibration analysis of functionally graded-carbon nanotube reinforced composite (FG-CNTRC) plates by dynamic stiffness method (DSM). The properties of CNTRC are determined with the extended rule of mixture. The governing differential equations of motion based on the first-order shear deformation theory of CNTRC plate are derived using Hamilton's principle. The FG-CNTRC plates are studied for a uniform and two different distributions of carbon nanotubes (CNTs). The accuracy and performance of the DSM are compared with the results obtained from closed closed-form and semi-analytical solution methods in previous studies. In this study, the effects of boundary condition, distribution type of CNTs, plate aspect ratio, plate length to thickness ratio, and different values of CNTs volume fraction on the natural frequencies of the FG-CNTRC plates are investigated. Finally, various natural frequencies of the plates in different conditions are provided as a benchmark for comparing the accuracy and precision of the other analytical and numerical methods.

• Advances in nano research
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• v.9 no.3
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• pp.211-220
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• 2020

This paper investigates the effect of linear and non-linear distribution of carbon nanotube volume fraction in the FG-CNTRC beams on the critical buckling by using higher-order shear deformation theories. Here, the material properties of the CNTRC beams are assumed to be graded in the thickness direction according to a new exponential power law distribution in terms of the carbon nanotube volume fractions. The single-walled carbon nanotube is aligned and distributed in the polymeric matrix with different patterns of reinforcement; the material properties of the CNTRC beams are described by using the rule of mixture. The governing equations are derived through using Hamilton's principle. The Navier solution method is used under the specified boundary conditions for simply supported CNTRC beams. The mathematical models provided in this work are numerically validated by comparison with some available results. New results of critical buckling with the non-linear distribution of CNT volume fraction in different patterns are presented and discussed in detail, and compared with the linear distribution. Several aspects of beam types, CNT volume fraction, exponent degree (n), aspect ratio, etc., are taken into this investigation. It is revealed that the influences of non-linearity distribution in the beam play an important role to improve the mechanical properties, especially in buckling behavior. The results show that the X-Beam configuration is the strongest among all different types of CNTRC beams in supporting the buckling loads.

• Steel and Composite Structures
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• v.36 no.6
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• pp.643-656
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• 2020

This article presents a comprehensive static analysis of simply supported cross-ply carbon nanotubes reinforced composite (CNTRC) laminated nanobeams under various loading profiles. The nonlocal strain gradient constitutive relation is exploited to present the size-dependence of nano-scale. New higher shear deformation beam theory with hyperbolic function is proposed to satisfy the zero-shear effect at boundaries and parabolic variation through the thickness. Carbon nanotubes (CNTs), as the reinforced elements, are distributed through the beam thickness with different distribution functions, which are, uniform distribution (UD-CNTRC), V- distribution (FG-V CNTRC), O- distribution (FG-O CNTRC) and X- distribution (FG-X CNTRC). The equilibrium equations are derived, and Fourier series function are used to solve the obtained differential equation and get the response of nanobeam under uniform, linear or sinusoidal mechanical loadings. Numerical results are obtained to present influences of CNTs reinforcement patterns, composite laminate structure, nonlocal parameter, length scale parameter, geometric parameters on center deflection ad stresses of CNTRC laminated nanobeams. The proposed model is effective in analysis and design of composite structure ranging from macro-scale to nano-scale.

• Advances in aircraft and spacecraft science
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• v.4 no.5
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• pp.585-611
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• 2017

This article covenants with the post buckling witticism of carbon nanotube reinforced composite (CNTRC) beam supported with an elastic foundation in thermal atmospheres with arbitrary assumed random system properties. The arbitrary assumed random system properties are be modeled as uncorrelated Gaussian random input variables. Unvaryingly distributed (UD) and functionally graded (FG) distributions of the carbon nanotube are deliberated. The material belongings of CNTRC beam are presumed to be graded in the beam depth way and appraised through a micromechanical exemplary. The basic equations of a CNTRC beam are imitative constructed on a higher order shear deformation beam (HSDT) theory with von-Karman type nonlinearity. The beam is supported by two parameters Pasternak elastic foundation with Winkler cubic nonlinearity. The thermal dominance is involved in the material properties of CNTRC beam is foreseen to be temperature dependent (TD). The first and second order perturbation method (SOPT) and Monte Carlo sampling (MCS) by way of CO nonlinear finite element method (FEM) through direct iterative way are offered to observe the mean, coefficient of variation (COV) and probability distribution function (PDF) of critical post buckling load. Archetypal outcomes are presented for the volume fraction of CNTRC, slenderness ratios, boundary conditions, underpinning parameters, amplitude ratios, temperature reliant and sovereign random material properties with arbitrary system properties. The present defined tactic is corroborated with the results available in the literature and by employing MCS.

• Steel and Composite Structures
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• v.27 no.2
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• pp.149-159
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• 2018

Thermo-mechanical buckling of sandwich beams with a stiff core and face sheets made of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) within the framework of Timoshenko beam theory is presented. The material properties of FG-CNTRC are supposed to vary continuously in the thickness direction and are estimated through the rule of mixture. Also the properties of these materials should be considered temperature dependent. The governing equations and boundary conditions are derived by using Hamilton's principle and solved using an efficient technique called the Differential Transform Method (DTM) to achieve the critical buckling of the sandwich beam in uniform thermal environment. A detailed parametric study is guided to investigate the effects of carbon nanotube volume fraction, slenderness ratio, core-to-face sheet thickness ratio, and clamped-clamped, simply-simply and clamped-simply end supports on the critical buckling behavior of sandwich beams with FG-CNTRC face sheets. Numerical results for comparison of sandwich beams with uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) face sheets with those with FG-CNTRC face sheets are also presented.

• Earthquakes and Structures
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• v.13 no.5
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• pp.509-518
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• 2017

The objective of the present paper is to investigate the bending behavior with stretching effect of carbon nanotube-reinforced composite (CNTRC) beams. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are estimated by using the rule of mixture. The significant feature of this model is that, in addition to including the shear deformation effect and stretching effect it deals with only 4 unknowns without including a shear correction factor. The single-walled carbon nanotubes (SWCNTs) are aligned and distributed in polymeric matrix with different patterns of reinforcement. The material properties of the CNTRC beams are assessed by employing the rule of mixture. The equilibrium equations have been obtained using the principle of virtual displacements. The mathematical models provided in this paper are numerically validated by comparison with some available results. New results of bending analyses of CNTRC beams based on the present theory with stretching effect is presented and discussed in details. the effects of different parameters of the beam on the bending responses of CNTRC beam are discussed.

• Geomechanics and Engineering
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• v.33 no.4
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• pp.381-391
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• 2023

The aim of this work is to analyze and predict the wave propagation behavior of the carbon nanotube reinforced composites (CNTRC) beams within the framework of various higher order shear deformation beam theory. Using the Euler-Lagrange principle, the wave equations for CNTRC beams are derived, where the determining factor is to make the determinant equal to zero. Based on the eigenvalue method, the relationship between wave number and circular frequency is obtained. Furthermore, the phase and group velocities during wave propagation are obtained as a function of wave number, and the material properties of CNTRC beams are estimated by the mixture rule. In this paper, various higher order shear beam theory including Euler beam theory, Timoshenko beam theory and other beam theories are mainly adopted to analyze the wave propagation problem of the CNTRC beams, and by this way, we conduct a comparative analysis to verify the correctness of this paper. The mathematical model provided in this paper is verified numerically by comparing it with some existing results. We further investigate the effects of different enhancement modes of CNTs, volume fraction of CNTs, spring factor and other aspects on the wave propagation behaviors of the CNTRC beams.

• Computers and Concrete
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• v.32 no.2
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• pp.207-215
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• 2023

This paper investigates wave propagation in functionally graded carbon nano-reinforced composite (FG-CNTRC) plates under the influence of temperature based on Reddy' plate model. The material properties of Carbon Nanotubes (CNTs) are size-dependent, and the volume fraction of CNTs varies only along the thickness direction of the plate for different CNTs reinforcement modes. In addition, the material properties of CNTs can vary for different temperature parameters. By solving the eigenvalue problem, analytical dispersion relations can be derived for CNTRC plates. The partial differential equations for the system are derived from Lagrange's principle and higher order shear deformation theory is used to obtain the wave equations for the CNTRC plate. Numerical analyses show that the wave propagation properties in the CNTRC plate are related to the volume fraction parameters of the CNTRC plate and the distribution pattern of the CNTs in the polymer matrix. The effects of different volume fractions of CNTs and the distribution pattern of carbon nanotubes along the cross section (UD-O-X plate) are discussed in detail.

• Steel and Composite Structures
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• v.44 no.4
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• pp.451-471
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• 2022

In the present paper, the static bending and buckling responses of functionally graded carbon nanotubes-reinforced composite (FG-CNTRC) beam under various boundary conditions are investigated within the framework of higher shear deformation theory. The significant feature of the proposed theory is that it provides an accurate parabolic distribution of transverse shear stress through the thickness satisfying the traction-free boundary conditions needless of any shear correction factor. Uniform (UD) and four graded distributions of CNTs which are FG-O, FG-X, FG- and FG-V are selected here for the analysis. The effective material properties of FG-CNTRC beams are estimated according to the rule of mixture. To model the FG-CNTRC beam realistically, an efficient Hermite-Lagrangian finite element formulation is successfully developed. The accuracy and efficiency of the present model are demonstrated by comparison with published benchmark results. Moreover, comprehensive numerical results are presented and discussed in detail to investigate the effects of CNTs volume fraction, distribution patterns of CNTs, boundary conditions, and length-to-thickness ratio on the bending and buckling responses of FG-CNTRC beam. Several new referential results are also reported for the first time which will serve as a benchmark for future studies in a similar direction. It is concluded that the FG-X-CNTRC beam is the strongest beam that carries the lowest central deflection and is followed by the UD, V, Λ, and FG-O-CNTRC beam. Besides, the critical buckling load belonging to the FG-X-CNTRC beam is the highest, followed by UD and FG-O.

• Structural Engineering and Mechanics
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• v.64 no.3
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• pp.381-390
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• 2017

Present paper deals with the large amplitude flexural vibration of carbon nanotube reinforced composite (CNTRC) plates. Distribution of CNTs as reinforcements may be uniform or functionally graded (FG). The equivalent material properties of the composite media are obtained according to a refined rule of mixtures which contains efficiency parameters. To account for the large deformations, von $K{\acute{a}}rm{\acute{a}}n$ type of geometrical nonlinearity is included into the formulation. The matrix representation of the governing equations is obtained according to the Ritz method where the basic shape functions are written in terms of the Chebyshev polynomials. Time dependency of the problem is eliminated by means of the Galerkin method and the resulting nonlinear eigenvalue problem is solved employing a direct displacement control approach. Results are obtained for completely clamped and completely simply supported plates. Results are first validated for the especial cases of FG-CNTRC and cross-ply laminated plates. Afterwards, parametric studies are given for FG-CNTRC plates with different boundary conditions. It is shown that, nonlinear frequencies are highly dependent to the volume fraction and dispersion profiles of CNTs. Furthermore, mode redistribution is observed in both simply supported and clamped FG-CNTRC plates.