Critical buckling loads of the perfect Hollomon's power-law columns

In this work, we present analytic formulas for calculating the critical buckling states of some plastic axial columns of constant cross-sections. The associated critical buckling loads are calculated by Euler-type analytic formulas and the associated deformed shapes are presented in terms of generalized trigonometric functions. The plasticity of the material is defined by the Hollomon's power-law equation. This is an extension of the Euler critical buckling loads of perfect elastic columns to perfect plastic columns. In particular, critical loads for perfect straight plastic columns with circular and rectangular cross-sections are calculated for a list of commonly used metals. Connections and comparisons to the classical result of the Euler–Engesser reduced-modulus loads are also presented.     

Buckling of thin-walled columns accounting for initial geometrical imperfections

The paper is devoted to the effect of some geometrical imperfections on the critical buckling load of axially compressed thin-walled I-columns. The analytical formulas for the critical torsional and flexural buckling loads accounting for the initial curvature of the column axis or the twist angle respectively are derived. The classical assumptions of theory of thin-walled beams with non-deformable cross-sections are adopted. The non-linear differential equations are derived and the critical buckling loads are approximated by means of the Galerkin’s method. Comparison of analytical results to numerical analysis of simply supported I-columns by means of finite element method (FEM) is provided. Moreover the analytical formulas is adapted to I-columns with lipped flanges and satisfactory agreement of analytical and numerical results of stability analysis is observed.     

Extensibility Effects on Euler Elastica’s Stability

Based on the conjugate point theory in calculus of variations, the extensibility effects on the stability of Euler elasticas with one clamped end and the other clamped in rotation in the post-buckling are investigated. For a slender rod, it is shown that: (1) the buckling load is a little bigger than Euler critical load, (2) the addition of extensibility to the elastica does not affect its stability in the post-buckling, in the sense that those Euler elasticas with one inflexion point are stable while those Euler elasticas with more than one inflexion point are unstable.     

Buckling analysis of Euler columns embedded in an elastic medium with general elastic boundary conditions

In this article, an efficient analytical method for elastically restrained Euler columns embedded in an elastic medium has been proposed to calculate buckling loads. The lateral deflection function under compression is represented by a Fourier sine series. Stokes’ transformation is employed to develop the legitimized stability equations. Explicit analytical expressions are derived, which can be used for any type of boundary conditions. The efficiency of present formulation is demonstrated by comparing the results to those obtained by imposing three well-known boundary conditions available in the literature. A very good agreement has been obtained. The present formulation permits to have more efficient coefficient matrix for calculating the buckling loads of Euler columns with any desired boundary conditions.     

Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading

This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses.     

THE COMPUTATIONAL METHOD FOR STABILITY OF TRUSSES WITH SMALL DEFORMATION

基于桁架结构稳定性计算的经典理论,分析了利用特征值理论开展桁架结构屈曲分析的计算方法,以及利用欧拉临界载荷屈曲理论,采取杆件撤除的静力求解确定桁架结构稳定临界载荷的计算方法. 通过理论研究和相关算例分析,论证了利用特征值理论和临界载荷屈曲理论相结合的方法,判断小变形桁架结构的失稳模态,求解桁架结构稳定临界载荷的确定性.     

On out-of-plane buckling of anisotropic elastic plate subjected to a simple shear

Out-of-plane buckling of anisotropic elastic plate subjected to a simple shear is investigated. From exact 3-D equilibrium conditions of anisotropic elastic body with a plane of elastic symmetry at critical configuration, the eqution for buckling direction (buckling wave direction) parameter is derived and the shape functions of possible buckling modes are obtained. The traction free boundary conditions which must hold on the upper and lower surfaces of plate lead to a linear eigenvalue problem whose nontrivial solutions are just the possible buckling modes for the plate. The buckling conditions for both flexural and barreling modes are presented. As a particular example of buckling of anisotropic elastic plate, the buckling of an orthotropic elastic plate, which is subjected to simple shear along a direction making an arbitrary angle of θ with respect to an elastic principal axis of materials, is analyzed. The buckling direction varies with θ and the critical amount of shear. The numerical results show that only the flexural mode can indeed exist. Project supported by the National Natural Science Foundation of China (No. 19772032).     

The effect of transverse shear deformation on the buckling of two-layer composite columns with interlayer slip

This paper presents an efficient mathematical model for studying the buckling behavior of geometrically perfect elastic two-layer composite columns with interlayer slip between the layers. The present analytical model is based on the linearized stability theory and is capable of predicting exact critical buckling loads. Based on the parametric analysis, the critical buckling loads are compared to those in the literature. It is shown that the discrepancy between the different methods can be up to approximately 22%. In addition, a combined and an individual effect of pre-buckling shortening and transverse shear deformation on the critical buckling loads is studied in detail. A comprehensive parametric analysis reveals that generally the effect of pre-buckling shortening can be neglected, while, on the other hand, the effect of transverse shear deformation can be significant. This effect can be up to 20% for timber composite columns, 40% for composite columns very flexible in shear (pyrolytic graphite), while for metal composite columns it is insignificant.     

Torsional buckling of spherical shells under circumferential shear loads

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Analysis of Plane Phase Strains of Rods and Plates

The problem of a rod and a plate subjected to plane strain in the interval of the direct phase transformation is formulated as a nonlinear boundary-value thermoelastic problem with an implicit dependence on temperature (through a phase parameter simulating the volume fraction of new-phase crystals). An analytical solution of the problem of a rod bent into a ring and a plate bent into a tube as a result of phase strains under the action of a small end bending moment is given. A numerical analysis of the buckling problem of a titanium nickelide (alloy) rod (plate) under longitudinal compression in the interval of the direct phase transformation shows that buckling becomes possible if the compressive load is much lower than the Euler critical load calculated before the transformation. Branches of buckled equilibrium states corresponding to loads lower than the Euler load are plotted as functions of the phase parameter. In all cases considered, the deflections increase abruptly in the neighborhood of the critical points. The evolution of buckling modes is studied, and the phase-strain distributions along the rod (plate) are shown. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 156–164, March–April, 2006.     

Exact solution of the multi-cracked Euler–Bernoulli column

The use of distributions (generalized functions) is a powerful tool to treat singularities in structural mechanics and, besides providing a mathematical modelling, their capability of leading to closed form exact solutions is shown in this paper. In particular, the problem of stability of the uniform Euler–Bernoulli column in presence of multiple concentrated cracks, subjected to an axial compression load, under general boundary conditions is tackled. Concentrated cracks are modelled by means of Dirac’s delta distributions. An integration procedure of the fourth order differential governing equation, which is not allowed by the classical distribution theory, is proposed. The exact buckling mode solution of the column, as functions of four integration constants, and the corresponding exact buckling load equation for any number, position and intensity of the cracks are presented. As an example a parametric study of the multi-cracked simply supported and clamped–clamped Euler–Bernoulli columns is presented.     

Exact buckling solution for two-layer Timoshenko beams with interlayer slip

This paper deals with the buckling behavior of two-layer shear-deformable beams with partial interaction. The Timoshenko kinematic hypotheses are considered for both layers and the shear connection (no uplift is permitted) is represented by a continuous relationship between the interface shear flow and the corresponding slip. A set of differential equations is obtained from a general 3D bifurcation analysis, using the above assumptions. Original closed-form analytical solutions of the buckling load and mode of the composite beam under axial compression are derived for various boundary conditions. The new expressions of the critical loads are shown to be consistent with the ones corresponding to the Euler–Bernoulli beam theory, when transverse shear stiffnesses go to infinity. The proposed analytical formulae are validated using 2D finite element computations. Parametric analyses are performed, especially including the limiting cases of perfect bond and no bond. The effect of shear flexibility is particularly emphasized.     

Nonlinear in-plane thermal buckling of rotationally restrained functionally graded carbon nanotube reinforced composite shallow arches under uniform radial loading

The nonlinear in-plane instability of functionally graded carbon nanotube reinforced composite (FG-CNTRC) shallow circular arches with rotational constraints subject to a uniform radial load in a thermal environment is investigated. Assuming arches with thickness-graded material properties, four different distribution patterns of carbon nanotubes (CNTs) are considered. The classical arch theory and Donnell’s shallow shell theory assumptions are used to evaluate the arch displacement field, and the analytical solutions of buckling equilibrium equations and buckling loads are obtained by using the principle of virtual work. The critical geometric parameters are introduced to determine the criteria for buckling mode switching. Parametric studies are carried out to demonstrate the effects of temperature variations, material parameters, geometric parameters, and elastic constraints on the stability of the arch. It is found that increasing the volume fraction of CNTs and distributing CNTs away from the neutral axis significantly enhance the bending stiffness of the arch. In addition, the pretension and initial displacement caused by the temperature field have significant effects on the buckling behavior.     

Small-scale effect on torsional buckling of multi-walled carbon nanotubes

The small-scale effect on the torsional buckling of multi-walled carbon nanotubes coupled with temperature change is investigated in this paper. A nonlocal multiple-shell model for the multi-walled carbon nanotubes surrounded an elastic medium under torsional and thermal loads is established, and then general solutions are obtained from the governing equations. The influence of the nonlocal effect on critical shear force and change in temperature is investigated. It is demonstrated that the critical shear force could be overestimated by the classical continuum theory and the nonlocal effect on critical buckling force decreases as the change in temperature increases at room or low temperature but increases as the change in temperature increases at higher temperature. Meanwhile, the effect of small size-scale is dependent on the buckling mode under different thermal environments. It is also shown that the innermost radius and the number of layer can affect the small-scale effect on critical change in temperature and buckling shear force. When the ratio of tube length and outmost radius are given, the critical shear force in each layer decreases and the nonlocal effect on the critical shear force becomes weaker as the innermost radius and the layer number increase.     

Effect of In-Plane Deformation on Lateral Buckling

ABSTRACT

In the classical analysis of the flexural-torsional buckling of beams, beam columns and rigid-jointed plane frames, it is assumed that the major axis rigidity is very large, so that the small in-plane deformations can be neglected. The effects of the in-plane deformations on lateral buckling are investigated in this paper for determinate beams and cantilevers, beam columns, continuous beams, and portal frames. This is done by deriving more accurate governing differential equations, and by obtaining closed form or numerical solutions of these. The results obtained indicate that the classical critical loads or moments are generally conservative, except for the members which are highly restrained laterally. The sources of error in the classical analysis are also studied, and their effects are demonstrated. The results of experiments on small scale beams, which are in close agreement with the theoretical predictions, are reported.     

Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

In the traditional continuum mechanics, the effects of surface free energy are generally ignored. However, this cannot be the case for nanostructures because of their high surface to volume ratio; surface energy plays an important role in the mechanical responses. In the present study, the nonlinear buckling and postbuckling characteristics of cylindrical nanoshells subjected to combined axial and radial compressions are investigated in the presence of surface energy effects. To this end, Gurtin–Murdoch elasticity theory is implemented into the classical first-order shear deformation shell theory to develop an efficient size-dependent shell model incorporating surface free energy effects. Subsequently, a boundary layer theory is employed including surface effects in conjunction with the nonlinear prebuckling deformations, the large postbuckling deflections and the initial geometric imperfection. Finally, a solution methodology based on a two-stepped singular perturbation technique is utilized to obtain the size-dependent critical buckling loads and equilibrium postbuckling paths corresponding to the both axial dominated and radial dominated loading cases. It is observed that for the both axial dominated and radial dominated loading cases, surface free energy effects cause to increase the both critical buckling load and critical end-shortening of shear deformable nanoshell made of silicon.     

In-plane stability of arches

Classical buckling theory is mostly used to investigate the in-plane stability of arches, which assumes that the pre-buckling behaviour is linear and that the effects of pre-buckling deformations on buckling can be ignored. However, the behaviour of shallow arches becomes non-linear and the deformations are substantial prior to buckling, so that their effects on the buckling of shallow arches need to be considered. Classical buckling theory which does not consider these effects cannot correctly predict the in-plane buckling load of shallow arches. This paper investigates the in-plane buckling of circular arches with an arbitrary cross-section and subjected to a radial load uniformly distributed around the arch axis. An energy method is used to establish both non-linear equilibrium equations and buckling equilibrium equations for shallow arches. Analytical solutions for the in-plane buckling loads of shallow arches subjected to this loading regime are obtained. Approximations to the symmetric buckling of shallow arches and formulae for the in-plane anti-symmetric bifurcation buckling load of non-shallow arches are proposed, and criteria that define shallow and non-shallow arches are also stated. Comparisons with finite element results demonstrate that the solutions and indeed approximations are accurate, and that classical buckling theory can correctly predict the in-plane anti-symmetric bifurcation buckling load of non-shallow arches, but overestimates the in-plane anti-symmetric bifurcation buckling load of shallow arches significantly.     

Stability of slender beams and frames resting on 2D elastic half-space

Making use of a mixed variational formulation based on the Green function of the substrate, which assumes as independent fields the structure displacements and the contact pressure, a simple and efficient finite element-boundary integral equation coupling method is derived and applied to the stability analysis of beams and frames resting on an elastic half-plane. Slender Euler–Bernoulli beams with different combinations of end constraints are considered. The examples illustrate the convergence to the existing exact solutions and provide new estimates of the buckling loads for different boundary conditions. Finally, nonlinear incremental analyses of rectangular pipes with compressed columns and free or pinned foundation ends are performed, showing that pipes stiffer than the soil may exhibit snap-through instability.     

Bending and stability analysis of gradient elastic beams

The problems of bending and stability of Bernoulli–Euler beams are solved analytically on the basis of a simple linear theory of gradient elasticity with surface energy. The governing equations of equilibrium are obtained by both a combination of the basic equations and a variational statement. The additional boundary conditions are obtained by both variational and weighted residual approaches. Two boundary value problems (one for bending and one for stability) are solved and the gradient elasticity effect on the beam bending response and its critical (buckling) load is assessed for both cases. It is found that beam deflections decrease and buckling load increases for increasing values of the gradient coefficient, while the surface energy effect is small and insignificant for bending and buckling, respectively.