• Advances in biomechanics and applications
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• v.1 no.3
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• pp.211-225
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• 2014

The usual assumption that the increase of fractures in aging bone is due entirely to lower bone density is taken back with respect to the possibility that aging bone fractures result from a loss of stability, or buckling, in the structure of the bone lattice. Buckling is an instability mode that becomes likely in end-loaded structures when they become too slender and lose lateral support. The relative importance of bone density and architecture in etiology bone fractures are poorly understood and the need for improved mechanistic understanding of bone failure is at the core of important clinical problems such as osteoporosis, as well as basic biological issues such as bone formation and adaptation. These observations motivated the present work in which simplified adaptive-beam buckling model is formulated within the context of the adaptive elasticity (Cowin and Hegedus 1976, Hegedus and Cowin 1976). Our results indicate that bone loss activation process leads systematically to the apparition of new elastic instabilities that can conduct to bone-buckling mechanism of fracture.

• Wind and Structures
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• v.26 no.4
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• pp.205-214
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• 2018

In this paper, the thermo-mechanical buckling characteristics of functionally graded (FG) size-dependent Timoshenko nanobeams subjected to an in-plane thermal loading are investigated by presenting a Navier type solution for the first time. Material properties of FG nanobeam are supposed to vary continuously along the thickness according to the power-law form and the material properties are assumed to be temperature-dependent. The small scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. The nonlocal governing equations are derived based on Timoshenko beam theory through Hamilton's principle and they are solved applying analytical solution. According to the numerical results, it is revealed that the proposed modeling can provide accurate critical buckling temperature results of the FG nanobeams as compared to some cases in the literature. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of the several parameters such as material distribution profile, small scale effects and aspect ratio on the critical buckling temperature of the FG nanobeams in detail. It is explicitly shown that the thermal buckling of a FG nanobeams is significantly influenced by these effects. Numerical results are presented to serve as benchmarks for future analyses of FG nanobeams.

• Advances in nano research
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• v.6 no.4
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• pp.377-397
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• 2018

In the present investigation, thermal buckling and free vibration characteristics of functionally graded (FG) Timoshenko nanobeams subjected to nonlinear thermal loading are carried out by presenting a Navier type solution. The thermal load is assumed to be nonlinear distribution through the thickness of FG nanobeam. Thermo-mechanical properties of FG nanobeam are supposed to vary smoothly and continuously throughout the thickness based on power-law model and the material properties are assumed to be temperature-dependent. Eringen's nonlocal elasticity theory is exploited to describe the size dependency of nanobeam. Using Hamilton's principle, the nonlocal equations of motion together with corresponding boundary conditions based on Timoshenko beam theory are obtained for the thermal buckling and vibration analysis of graded nanobeams including size effect. Moreover, in following a parametric study is accompanied to examine the effects of the several parameters such as nonlocal parameter, thermal effect, power law index and aspect ratio on the critical buckling temperatures and natural frequencies of the size-dependent FG nanobeams in detail. According to the numerical results, it is revealed that the proposed modeling can provide accurate frequency results of the FG nanobeams as compared some cases in the literature. Also, it is found that the small scale effects and nonlinear thermal loading have a significant effect on thermal stability and vibration characteristics of FG nanobeams.

• Computers and Concrete
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• v.30 no.4
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• pp.243-256
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• 2022

The static analysis of spinning functionally graded (FG) nanotube on the basis of the nonlocal strain gradient theory (NSGT) is presented. The high-order beam theory is employed for mathematical modeling of the tube structures according to the Sinusoidal shear deformation beam theory. The energy conservation principle is operated to generate the equations. The centrifugal force is assumed along the tube length due to the rotating of the tube, moreover, the nanotube is made of functionally graded material (FGM) composed of ceramic and metal phases along the tube radius direction. The generalized differential quadratic method (GDQM) is utilized to solve the formulations. Finally, the numerical results are discussed in detail to examine the impact of different relevant parameters on the bending the buckling behavior of the rotating nanotube.

• Steel and Composite Structures
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• v.28 no.2
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• pp.195-207
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• 2018

First-order reliability method (FORM) is enhanced based on the search direction using relaxed conjugate reliability (RCR) approach for the embedded nanocomposite beam under buckling failure mode. The RCR method is formulated using discrete conjugate map with a limited scalar factor. A dynamical relaxed factor is proposed to control instability of proposed RCR, which is adjusted using sufficient descent condition. The characteristic of equivalent materials for nanocomposite beam are obtained by micro-electro-mechanical model. The probabilistic model of nanocomposite beam is simulated using the sinusoidal shear deformation theory (SSDT). The beam is subjected to external applied voltage in thickness direction and the surrounding elastic medium is modeled by Pasternak foundation. The governing equations are derived in terms of energy method and Hamilton's principal. Using exact solution, the implicit buckling limit state function of nanocomposite beam is proposed, which is involved various random variables including thickness of beam, length of beam, spring constant of foundation, shear constant of foundation, applied voltage, and volume fraction of ZnO nanoparticles in polymer. The robustness, accuracy and efficiency of proposed RCR method are evaluated for this engineering structural reliability problem. The results demonstrate that proposed RCR method is more accurate and robust than the excising reliability methods-based FORM. The volume fraction of ZnO nanoparticles and the applied voltage are the sensitive variables on the reliable levels of the nanocomposite beams.

• Structural Engineering and Mechanics
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• v.54 no.4
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• pp.693-710
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• 2015

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

• 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.

• Computers and Concrete
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• v.25 no.4
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• pp.283-291
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• 2020

The present paper researches post-buckling behaviors of geometrically imperfect concrete beam resting on elastic foundation reinforced with graphene oxide powders (GOPs) based on finite element method (FEM). Distribution of GOPs are considered as uniform and linearly graded through the thickness. Geometric imperfection is considered as first buckling mode shape of the beam, the GOP reinforced beam is rested in initial position. The material properties of GOP reinforced composite have been calculated via employment of Halpin-Tsai micromechanical scheme. The provided refined beam element verifies the shear deformation impacts needless of any shear correction coefficient. The post-buckling load-deflections relations have been calculated via solving the governing equations having cubic non-linearity implementing FEM. Obtained findings indicate the importance of GOP distributions, GOP weight fraction, matrix material, geometric imperfection, shear deformation and foundation parameters on nonlinear buckling behavior of GOP reinforced beam.

• Advances in nano research
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• v.10 no.5
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• pp.493-507
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• 2021

This article focused on studying the buckling behavior of two-dimensional functionally graded (2D-FG) nanosize tubes, including porosity based on first shear deformation and higher-order theory of tube. The nano-scale tube is simulated based on the nonlocal gradient strain theory, and the general equations and boundary conditions are derived using Hamilton's principle for the Zhang-Fu's tube model (as higher-order theory) and Timoshenko beam theory. Finally, the derived equations are solved using a numerical method for both simply-supported and clamped boundary conditions. The parametric study is performed to study the effects of different parameters such as axial and radial FG power indexes, porosity parameter, nonlocal gradient strain parameters on the buckling behavior of di-dimensional functionally graded porous tube.

• Journal of Korean Association for Spatial Structures
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• v.6 no.3 s.21
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• pp.77-86
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• 2006

The stress analysis of cold-formed channel section steel beams under transverse load is presented. The local buckling as well as the lateral buckling effects are included in the analysis. The analytical model is developed based on the thin-walled beam theory, and a one-dimensional finite element model is formulated to solve the analytical model. Numerical results are compared with AISI code. It shows that the proposed model is appropriate for predicting of stress as well as deflection of the cold-formed channel section beam.