• Proceedings of the Computational Structural Engineering Institute Conference
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• 1996.10a
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• pp.3-10
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• 1996

Numerical methods are developed for solving the buckling loads and the elastica of clamped- free columns of circular cross-section with constant volume. The column model is based rut the Timoshenko beam theory. The Runge-Kutta and Regula-Falsi methods, respectively, are used to solve the governing differential equations and to compute the eigenvalues. Extensive numerical results, including buckling loads, elastica of buckled shapes and effects of shear de-formation, are presented in non-dimensional form for elastic columns whose radius of circular cross-section varies both linearly and parabolically with column length.

• Structural Engineering and Mechanics
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• v.57 no.6
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• pp.1107-1124
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• 2016

Consistent finite strain Plate constitutive relations are derived based on a hyperelastic formulation for an isotropic material. Plate equilibrium equations under finite strain are derived following a static kinematic approach. Three Euler angles and four shear angles, based on Timoshenko beam theory, represent the kinematics of the deformations in the plate cross section. The Green deformation tensor has been expressed in term of a deformation tensor associated with the deformation and stretches of an embedded plate element. Buckling formulation includes the in-plane axial deformation prior to buckling and transverse as well as in-plane shear deformations. Numerical results for a simply supported thick plate under uni-axial compression force are presented.

• Advances in materials Research
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• v.7 no.1
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• pp.1-16
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• 2018

Thermal bifurcation buckling behavior of fully clamped Euler-Bernoulli nanobeam built of a through thickness functionally graded material is explored for the first time in the present paper. The variation of material properties of the FG nanobeam are graded along the thickness by a power-law form. Temperature dependency of the material constituents is also taken into consideration. Eringen's nonlocal elasticity model is employed to define the small-scale effects and long-range connections between the particles. The stability equations of the thermally induced FG nanobeam are derived via the principal of the minimum total potential energy and solved analytically for clamped boundary conditions, which lead for more accurate results. Moreover, the obtained buckling loads of FG nanobeam are validated with those existing works. Parametric studies are performed to examine the influences of various parameters such as power-law exponent, small scale effects and beam thickness on the critical thermal buckling load of the temperature-dependent FG nanobeams.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2001.10a
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• pp.369-374
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• 2001

This paper deals with the buckling loads of column with rotation end restricted by rotational spring. The ordinary differential equations governing the buckling loads of such column is derived as nondimensional forms, and also its boundary conditions are derived. The buckled column model is based on the classical Bemoulli-Euler beam theory. The Runge-Kutta method and Regula-Falsi method are used to perform the integration of the differential equations and to determine the eigenvalue. The numerical methods developed herein for the buckling loads of the such column are found to be efficient and reliable. It is expected that the results obtained herein can be practically utilized in the structural engineering field.

• Computational Structural Engineering
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• v.9 no.3
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• pp.187-197
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• 1996

An explicit direct time integration method based solution algorithm is presented to predict dynamic buckling response of Euler column. On the basis of large deflection beam theory, a plane frame finite element is formulated and implemented into the solution algorithm. The element formulation takes into account geometrical nonlinearity and overall buckling of steel structural frames. The solution algorithm employs the central difference method. Using the computer program developed by the author, dynamic instability behavior of Euler column under impact loading is investigated by considering the time variation of load, load magnitude, and load duration. The free vibration of Euler column caused by a short duration impact load is also studied. The validity and efficiency of the present formulation and solution algorithm are verified through illustrative numerical examples.

• Smart Structures and Systems
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• v.25 no.6
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• pp.707-717
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• 2020

In this paper, analysis of thermal post-buckling behaviors of sandwich nanobeams with two layers of multi-phase magneto-electro-thermo-elastic (METE) composites have been presented considering geometric imperfection effects. Multi-phase METE material is composed form piezoelectric and piezo-magnetic constituents for which the material properties can be controlled based on the percentages of the constituents. Nonlinear governing equations of sandwich nanobeam are derived based on nonlocal elasticity theory together with classic thin beam model and an analytical solution is provided. It will be shown that post-buckling behaviors of sandwich nanobeam in thermo-electro-magnetic field depend on the constituent's percentages. Buckling temperature of sandwich nanobeam is also affected by nonlocal scale factor, magnetic field intensity and electrical voltage.

• Advances in nano research
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• v.6 no.4
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• pp.339-356
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• 2018

In this article, the critical buckling of a single-walled carbon nanotube (SWCNT) embedded in Kerr's medium is studied. Based on the nonlocal continuum theory and the Euler-Bernoulli beam model. The governing equilibrium equations are acquired and solved for CNTs subjected to mechanical loads and embedded in Kerr's medium. Kerr-type model is employed to simulate the interaction of the (SWNT) with a surrounding elastic medium. A first time, a comparison with the available results is made, and another comparison between various models Winkler-type, Pasternak-type and Kerr-type is studied. Effects of nonlocal parameter and aspect ratio of length to diameter of nanobeam, as well as the foundation parameters on buckling of CNT are investigated. These results are important in the mechanical design considerations of nanocomposites based on carbon nanotubes.

• Coupled systems mechanics
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• v.11 no.2
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• pp.167-198
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• 2022

In this paper we deal with classical instability problems of heterogeneous Euler beam under conservative loading. It is chosen as the model problem to systematically present several possible solution methods from simplest deterministic to more complex stochastic approach, both of which that can handle more complex engineering problems. We first present classical analytic solution along with rigorous definition of the classical Euler buckling problem starting from homogeneous beam with either simplified linearized theory or the most general geometrically exact beam theory. We then present the numerical solution to this problem by using reduced model constructed by discrete approximation based upon the weak form of the instability problem featuring von Karman (virtual) strain combined with the finite element method. We explain how such numerical approach can easily be adapted to solving instability problems much more complex than classical Euler's beam and in particular for heterogeneous beam, where analytic solution is not readily available. We finally present the stochastic approach making use of the Duffing oscillator, as the corresponding reduced model for heterogeneous Euler's beam within the dynamics framework. We show that such an approach allows computing probability density function quantifying all possible solutions to this instability problem. We conclude that increased computational cost of the stochastic framework is more than compensated by its ability to take into account beam material heterogeneities described in terms of fast oscillating stochastic process, which is typical of time evolution of internal variables describing plasticity and damage.

• Journal of the Korea Academia-Industrial cooperation Society
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• v.20 no.12
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• pp.614-621
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• 2019

A generalized laminated composite beam element is presented for the flexural and buckling analysis of laminated composite beams with double and single symmetric cross-sections. Based on shear-deformable beam theory, the present beam model accounts for transverse shear and warping deformations, as well as all coupling terms caused by material anisotropy. The plane stress and plane strain assumptions were used along with the cross-sectional stiffness coefficients obtained from the analytical technique for different cross-sections. Two types of one-dimensional beam elements with seven degrees-of-freedom per node, including warping deformation, i.e., three-node and four-node elements, are proposed to predict the flexural behavior of symmetric or anti-symmetric laminated beams. To alleviate the shear-locking problem, a reduced integration scheme was employed in this study. The buckling load of laminated composite beams under axial compression was then calculated using the derived geometric block stiffness. To demonstrate the accuracy and efficiency of the proposed beam elements, the results based on three-node beam element were compared with those of other researchers and ABAQUS finite elements. The effects of coupling and shear deformation, support conditions, load forms, span-to-height ratio, lamination architecture on the flexural response, and buckling load of composite beams were investigated. The convergence of two different beam elements was also performed.

• Steel and Composite Structures
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• v.8 no.2
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• pp.159-178
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• 2008

An analytical procedure is presented for the determination of the buckling load and the buckling temperature of a straight, slender, geometrically perfect, axially loaded, translationally and rotationally restrained steel column exposed to fire. The exact kinematical equations of the column are considered, but the shear strain is neglected. The linearized stability theory is employed in the buckling analysis. Behaviour of steel at the elevated temperature is assumed in accordance with the European standard EC 3. Theoretical findings are applied in the parametric analysis of restrained columns. It is found that the buckling length factor decreases with temperature and depends both on the material model and stiffnesses of rotational and translational restraints. This is in disagreement with the buckling length for intermediate storeys of braced frames proposed by EC 3, where it is assumed to be temperature independent. The present analysis indicates that this is a reasonable approximation only for rather stiff rotational springs.