• Steel and Composite Structures
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• v.25 no.6
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• pp.671-684
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• 2017

Elastic-plastic shear buckling behaviors of the web-post in a Castellated Steel Beam (CSB) with hexagonal web openings under vertical shear force were investigated further using Finite Element Model (FEM) based on a sub-model, which took the upper part of the web-post under horizontal shear force to represent the whole web-post under vertical shear force. A simplified design method for the web-post elastic-plastic shear buckling strength was proposed based on simulation results of the sub-model. Proper boundary conditions were applied to the sub-model to assure that its behaviors were identical to those of the whole web-post. The equation to calculate the thin plate elastic shear buckling strength was adopted as the basic form to build the design equation for elastic-plastic buckling strength of the sub-model. Parameters that might affect the elastic-plastic shear buckling strength of the whole web-post were studied. After obtaining the vertical shear buckling strength of a sub-model through FEM, the shear buckling coefficient k can be obtained through the back analysis. A practical calculation method for k was proposed through curving fitting the parameter study results. The elastic-plastic shear buckling strength of the web-post calculated using the proposed shear buckling coefficient k agreed well with that obtained from the FEM and test results. And it was more precise than those obtained from EC3 based on the strut model.

• Structural Engineering and Mechanics
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• v.31 no.2
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• pp.181-210
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• 2009

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of isotropic flat plates. The so-called exact finite strip is assumed to be simply supported out-of-plane at the loaded ends. The strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. It is noted that in the present method, only one of the calculated out-of-plane buckling deflection modes, corresponding to the lowest buckling load, i.e., the first mode is used for the initial post-buckling study. Thus, the postbuckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman's compatibility equation governing the behavior of isotropic flat plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial postbuckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

• Structural Engineering and Mechanics
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• v.43 no.2
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• pp.211-223
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• 2012

This paper deals with large deformation post-buckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling by an alternative method. Analytical approximate solutions for the geometrically nonlinear problem are presented. The solution for the limiting case of a string is also obtained. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation, and supplementary condition with cosinoidal nonlinearity can also be simplified to be a polynomial integral equation. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. Two approximate formulae for load along axis, potential strain for free hygrothermal expansion and periodic solution are established for small as well as large angle of rotation at the end of the beam. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

• Steel and Composite Structures
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• v.46 no.5
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• pp.637-647
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• 2023

Nonlinear free vibration and stability responses of a carbon nanotube reinforced composite beam under temperature rising are investigated in this paper. The material of the beam is considered as a polymeric matrix by reinforced the single-walled carbon nanotubes according to different distributions with temperature-dependent physical properties. With using the Hamilton's principle, the governing nonlinear partial differential equation is derived based on the Euler-Bernoulli beam theory. In the nonlinear kinematic assumption, the Von Kármán nonlinearity is used. The Galerkin's decomposition technique is utilized to discretize the governing nonlinear partial differential equation to nonlinear ordinary differential equation and then is solved by using of multiple time scale method. The critical buckling temperatures, the nonlinear natural frequencies and the nonlinear free response of the system is obtained. The effect of different patterns of reinforcement on the critical buckling temperature, nonlinear natural frequency, nonlinear free response and phase plane trajectory of the carbon nanotube reinforced composite beam investigated with temperature-dependent physical property.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.31 no.1A
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• pp.1-7
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• 2011

This paper deals with buckling loads of the column with the constant surface area. The shape function of variable column depth is chosen as the linear taper. The ordinary differential equation governing buckled shapes of the column is derived based on the dynamic equilibrium equation of such column subjected to an axial load. Three kinds of end constraint of hinged-hinged, hinged-clamped and clamped-clamped are considered in numerical examples. Effects of the column parameters on buckling loads are extensively discussed. Especially, section ratios of the strongest column are calculated, under which the maximum, i.e. strongest, buckling loads are achieved. Also the buckled shapes are obtained for searching the nodal points where the inner transverse supports are simply installed to increase the buckling loads.

• Steel and Composite Structures
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• v.19 no.3
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• pp.679-695
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• 2015

A numerical solution using finite difference method to evaluate the thermal buckling of simply supported FGM plate with variable thickness is presented in this research. First, the governing differential equation of thermal stability under uniform temperature through the plate thickness is derived. Then, the governing equation has been solved using finite difference method. After validating the presented numerical method with the analytical solution, the finite difference formulation has been extended in order to include variable thickness. The accuracy of the finite difference method for variable thickness plate has been also compared with the literature where a good agreement has been found. Furthermore, a parametric study has been conducted to analyze the effect of material and geometric parameters on the thermal buckling resistance of the FGM plates. It was found that the thickness variation affects isotropic plates a bit more than FGM plates.

• Proceedings of the Computational Structural Engineering Institute Conference
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• 2007.04a
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• pp.117-120
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• 2007

This study deals with the free vibrations and buckling loads of column with clamped-spring ends and constant volume. The column has the regular polygon cross-section whose depth is varied with the linear functional fashion. The differential equation governing the free vibration of such column is derived in which the effect of axial load is included. The differential equation is solved numerically for calculating frequencies. By using the relationship between loads and frequencies, the buckling loads are also obtained.

• Steel and Composite Structures
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• v.43 no.3
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• pp.341-351
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• 2022

This paper presents an inelastic buckling behavior analysis of rectangular hollow steel tubes with geometrical imperfections under elevated temperatures. The main variables are the temperature loads, slenderness ratios, and exposure conditions at high temperatures. The material and structural properties of steels at different temperatures are based on Eurocode (EN 1993-1-2, 2005). In the elastic buckling analysis, the buckling strength decreases linearly with the exposure conditions, whereas the inelastic buckling analysis shows that the buckling strength decreases in clusters based on the exposure conditions of strong and weak axes. The buckling shape of the rectangular steel column in the elastic buckling mode, which depicts geometrical imperfection, shows a shift in the position at which bending buckling occurs when the lower section of the member is exposed to high temperatures. Furthermore, lateral torsional buckling occurs owing to cross-section deformation when the strong axial plane of the model is exposed to high temperatures. The elastic buckling analysis indicates a conservative value when the model is exposed to a relatively low temperature, whereas the inelastic buckling analysis indicates a conservative value at a certain temperature or higher. The comparative results between the inelastic buckling analysis and Eurocode 3 show that a range exists in which the buckling strength in the design equation result is overestimated at elevated temperatures, and the shapes of the buckling curves are different.

• Steel and Composite Structures
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• v.26 no.1
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• pp.17-29
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• 2018

The thermal effects on the buckling, postbuckling and nonlinear vibration behaviors of composite laminated trapezoidal plates are studied. Aiming at the complex plate structure and to simulate the temperature distribution of the plate, a finite element method (FEM) is applied in this paper. In the temperature model, based on the thermal diffusion equation, the Galerkin's method is employed to establish the temperature equation of the composite laminated trapezoidal plate. The geometrical nonlinearity of the plate is considered by using the von Karman large deformation theory, and combining the thermal model and aeroelastic model, Hamilton's principle is employed to establish the thermoelastic equation of motion of the composite laminated trapezoidal plate. The thermal buckling and postbuckling of the composite laminated rectangular plate are analyzed to verify the validity and correctness of the present methodology by comparing with the results reported in the literature. Moreover, the effects of the temperature with the ply-angle on the thermal buckling and postbuckling of the composite laminated trapezoidal plates are studied, the thermal effects on the nonlinear vibration behaviors of the composite laminated trapezoidal plates are discussed, and the frequency-response curves are also presented for the different temperatures and ply angles.

• International journal of steel structures
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• v.18 no.4
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• pp.1440-1463
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• 2018

The Wagner coefficient is a key parameter used to describe the inelastic lateral-torsional buckling (LTB) behaviour of the I-beam, since even for a doubly-symmetric I-section with residual stress, it becomes a monosymmetric I-section due to the characteristics of the non-symmetrical distribution of plastic regions. However, so far no theoretical derivation on the energy equation and Wagner's coefficient have been presented due to the limitation of Vlasov's buckling theory. In order to simplify the nonlinear analysis and calculation, this paper presents a simplified mechanical model and an analytical solution for doubly-symmetric I-beams under pure bending, in which residual stresses and yielding are taken into account. According to the plate-beam theory proposed by the lead author, the energy equation for the inelastic LTB of an I-beam is derived in detail, using only the Euler-Bernoulli beam model and the Kirchhoff-plate model. In this derivation, the concept of the instantaneous shear centre is used and its position can be determined naturally by the condition that the coefficient of the cross-term in the strain energy should be zero; formulae for both the critical moment and the corresponding critical beam length are proposed based upon the analytical buckling equation. An analytical formula of the Wagner coefficient is obtained and the validity of Wagner hypothesis is reconfirmed. Finally, the accuracy of the analytical solution is verified by a FEM solution based upon a bi-modulus model of I-beams. It is found that the critical moments given by the analytical solution almost is identical to those given by Trahair's formulae, and hence the analytical solution can be used as a benchmark to verify the results obtained by other numerical algorithms for inelastic LTB behaviour.