• 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.35 no.5
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• pp.671-685
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• 2020

This manuscript presents impacts of gradation of material functions and axial load functions on critical buckling loads and mode shapes of functionally graded (FG) thin and thick beams by using higher order shear deformation theory, for the first time. Volume fractions of metal and ceramic materials are assumed to be distributed through a beam thickness by both sigmoid law and symmetric power functions. Ceramic-metal-ceramic (CMC) and metal-ceramic-metal (MCM) symmetric distributions are proposed relative to mid-plane of the beam structure. The axial compressive load is depicted by constant, linear, and parabolic continuous functions through the axial direction. The equilibrium governing equations are derived by using Hamilton's principles. Numerical differential quadrature method (DQM) is developed to discretize the spatial domain and covert the governing variable coefficients differential equations and boundary conditions to system of algebraic equations. Algebraic equations are formed as a generalized matrix eigenvalue problem, that will be solved to get eigenvalues (buckling loads) and eigenvectors (mode shapes). The proposed model is verified with respectable published work. Numerical results depict influences of gradation function, gradation parameter, axial load function, slenderness ratio and boundary conditions on critical buckling loads and mode-shapes of FG beam structure. It is found that gradation types have different effects on the critical buckling. The proposed model can be effective in analysis and design of structure beam element subject to distributed axial compressive load, such as, spacecraft, nuclear structure, and naval structure.

• Computers and Concrete
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• v.34 no.2
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• pp.179-193
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• 2024

The bending and buckling effect for carbon nanotube-reinforced composite (CNTRC) beams can be evaluated by developing the theory of third shear deformation (TSDT). This study examines beams supported by viscoelastic foundations, where single-walled carbon nanotubes (SWCNTs) are dispersed and oriented within a polymer matrix. Four patterns of reinforcement are used for the CNTRC beams. The rule of mixtures is assessed for the material properties of CNTRC beams. The effective functionally graded materials (FGM) properties are studied by considering three different uneven distribution types of porosity. The damping coefficient is considered to investigate the viscosity effect on the foundation in addition to Winkler's and Pasternak's parameters. The accuracy of the current theory is inspected with multiple comparison works. Moreover, the effects of different beam parameters on the CNTRC beam bending and buckling over a viscoelastic foundation are discussed. The results demonstrated that the O-beam is the weakest type of CNTRC beam to resist buckling and flexure loads, whereas the X-beam is the strongest. Moreover, it is indicated that the presence of porosity in the beams decreases the stiffness and increases deflection. In comparison, the deflection was reduced in the presence of a viscoelastic foundation.

• Computers and Concrete
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• v.34 no.3
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• pp.297-305
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• 2024

This study investigates the buckling behavior of CNTRC beams on a Winkler-Pasternak elastic foundation, considering their stiffness. To achieve the highest accuracy, the shear stiffness is taken into account based on the Higher-order Shear Deformation Theory (HSDT). A novel exponential power-law distribution of the CNT volume fraction across the beam thickness is employed to model CNTRC beams. Various reinforcement patterns are incorporated into the polymer matrix, featuring single-walled carbon nanotubes (SWCNT) that are both aligned and distributed. The effective mechanical properties of the CNTRC beam are predicted using the rule of mixtures. Hamilton's principle is applied to derive the differential equations of motion. This theoretical framework enables the validation of the approach by comparing numerical simulation results with previous studies. The impact of the exponent order (n), CNT volume fraction, geometrical ratio, and Winkler-Pasternak parameters on buckling analysis is thoroughly presented and discussed. The results indicate that, among the different types of analyzed CNTRC beams, the X-Beam pattern demonstrates the highest buckling load capacity.

• Computers and Concrete
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• v.33 no.3
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• pp.241-250
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• 2024

This article investigates the thermal and post-buckling problems of graphene platelets reinforced metal foams (GPLRMF) beams with initial geometric imperfection. Three distribution forms of graphene platelet (GPLs) and foam are employed. This article utilizes the mixing law Halpin Tsai model to estimate the physical parameters of materials. Considering three different boundary conditions, we used the Euler beam theory to establish the governing equations. Afterwards, the Galerkin method is applied to discretize these equations. The correctness of this article is verified through data analysis and comparison with the existing articles. The influences of geometric imperfection, GPL distribution modes, boundary conditions, GPLs weight fraction, foam distribution pattern and foam coefficient on thermal post-buckling are analyzed. The results indicate that, perfect GPLRMF beams do not undergo bifurcation buckling before reaching a certain temperature, and the critical buckling temperature is the highest when both ends are fixed. At the same time, the structural stiffness of the beam under the GPL-A model is the highest, and the buckling response of the beam under the Foam-II mode is the lowest, and the presence of GPLs can effectively improve the buckling strength.

• Steel and Composite Structures
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• v.19 no.3
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• pp.521-546
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• 2015

A new refined hyperbolic shear and normal deformation beam theory is developed to study the free vibration and buckling of functionally graded (FG) sandwich beams under various boundary conditions. The effects of transverse shear strains as well as the transverse normal strain are taken into account. Material properties of the sandwich beam faces are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. Equations of motion are derived from Hamilton's principle. Analytical solutions for the bending, free vibration and buckling analyses are obtained for simply supported sandwich beams. Illustrative examples are given to show the effects of varying gradients, thickness stretching, boundary conditions, and thickness to length ratios on the bending, free vibration and buckling of functionally graded sandwich beams.

• Computers and Concrete
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• v.25 no.2
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• pp.155-166
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• 2020

In this work, the buckling and vibrational behavior of the composite beam armed with single-walled carbon nanotubes (SW-CNT) resting on Winkler-Pasternak elastic foundation are investigated. The CNT-RC beam is modeled by a novel integral first order shear deformation theory. The current theory contains three variables and uses the shear correction factors. The equivalent properties of the CNT-RC beam are computed using the mixture rule. The equations of motion are derived and resolved by Applying the Hamilton's principle and Navier solution on the current model. The accuracy of the current model is verified by comparison studies with others models found in the literature. Also, several parametric studies and their discussions are presented.

• Steel and Composite Structures
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• v.50 no.4
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• pp.459-474
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• 2024

Classical and first-order nonlocal beam theory are employed in this study to assess the thermal buckling performance of a small-scale conical, cylindrical beam. The beam is constructed from functionally graded (FG) porosity-dependent material and operates under the thermal conditions of the environment. Imperfections within the non-uniform beam vary along both the radius and length direction, with continuous changes in thickness throughout its length. The resulting structure is functionally graded in both radial and axial directions, forming a bi-directional configuration. Utilizing the energy method, governing equations are derived to analyze the thermal stability and buckling characteristics of a nanobeam across different beam theories. Subsequently, the extracted partial differential equations (PDE) are numerically solved using the generalized differential quadratic method (GDQM), providing a comprehensive exploration of the thermal behavior of the system. The detailed discussion of the produced results is based on various applied effective parameters, with a focus on the potential application of nanotubes in enhancing sports bikes performance.

• Steel and Composite Structures
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• v.44 no.1
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• pp.81-89
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• 2022

The purposes of the present work it to study the effect of shear deformation on the static post-buckling response of simply supported functionally graded (FGM) axisymmetric beams based on classical, first-order, and higher-order shear deformation theories. The behavior of postbuckling is introduced based on geometric nonlinearity. The material properties of functionally graded materials (FGM) are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The equations of motion and the boundary conditions derived using Hamilton's principle. This article compares and addresses the efficiency, the applicability, and the limits of classical models, higher order models (CLT, FSDT, and HSDT) for the static post-buckling response of an asymmetrically simply supported FGM beam. The amplitude of the static post-buckling obtained a solving the nonlinear governing equations. The results showing the variation of the maximum post-buckling amplitude with the applied axial load presented, for different theory and different parameters of material and geometry. In conclusion: The shear effect found to have a significant contribution to the post-buckling behaviors of axisymmetric beams. As well as the classical beam theory CBT, underestimate the shear effect compared to higher order shear deformation theories HSDT.

• Advances in aircraft and spacecraft science
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• v.11 no.1
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• pp.55-81
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• 2024

This paper presents the exact solutions and closed forms for of nonlinear stability and vibration behaviors of straight and curved beams with nonlinear viscoelastic boundary conditions, for the first time. The mathematical formulations of the beam are expressed based on Euler-Bernoulli beam theory with the von Karman nonlinearity to include the mid-plane stretching. The classical boundary conditions are replaced by nonlinear viscoelastic boundary conditions on both sides, that are presented by three elements (i.e., linear spring, nonlinear spring, and nonlinear damper). The nonlinear integro-differential equation of buckling problem subjected to nonlinear nonhomogeneous boundary conditions is derived and exactly solved to compute nonlinear static response and critical buckling load. The vibration problem is converted to nonlinear eigenvalue problem and solved analytically to calculate the natural frequencies and to predict the corresponding mode shapes. Parametric studies are carried out to depict the effects of nonlinear boundary conditions and amplitude of initial curvature on nonlinear static response and vibration behaviors of curved beam. Numerical results show that the nonlinear boundary conditions have significant effects on the critical buckling load, nonlinear buckling response and natural frequencies of the curved beam. The proposed model can be exploited in analysis of macrosystem (airfoil, flappers and wings) and microsystem (MEMS, nanosensor and nanoactuators).