Buckling Of Elastic Column Research Articles

Dynamic Analysis for Overturning of Pile Driving Machine, etc., on Soft Ground

There were quite a few overturning accidents of heavy machinery which have a high center of gravity such as pile driving machines, cranes, jacks, or working vehicles with aerial platform. Researches to study the background of the accidents have taken place from the structural stability point of view by the authors. The structural instability, as can be seen typically in the elastic buckling of columns, may be a blind spot because the structures deflect in different directions to that of the load, which makes it difficult to predict. According to the past investigations, the overturning mechanisms can be categorized into the three: (1) Type of overturning moment, (2) Type of structural instability and (3) Type of equilibrium transition, in which Type of equilibrium transition plays important role on the overturning on soft ground. The past investigations were based on the static analysis. Dynamic analysis is essential to include the effect of inertial forces which will significantly affect the overturning accidents on the soft ground. In the static analysis, the displacement inclination is considered to stop at the equilibrium position, but in the dynamic analysis it will continue increasing due to the inertial force. Then, it is possible the displacement may go beyond the equilibrium and even the stability limit to overturn. In this paper, the equations of motion will be first introduced, then behaviors of the machinery from time to time, i.e., the inclination-time curves, will be obtained by solving the equations. It is shown that the dynamic analysis can explain more precisely the overturning behaviors of the machinery on the soft foundation than the static analysis.

Elastic buckling of columns with a discrete elastic torsional restraint

Cold-formed steel haunched portal frames are popular structures in industrial and housing applications. They are mostly used as sheds, garages, and shelters, and are common in rural areas. Cold-formed steel portal frames with spans of up to 30 m are now being constructed in Australia. As they are relatively new to the market, current design recommendations are fairly limited. In the specific frame system analyzed herein, the column is partially restrained against twist rotation at an intermediate point where the knee brace joining the column and rafter is connected. An experimental program was carried out on a series of portal frame systems composed of back-to-back channels for the columns and rafters. It was found that changing the knee brace and knee brace-to-column connection bracket significantly affected the buckling capacity of the column, however this was not captured in design calculations. In order to correctly predict frame behavior and ultimate loads for design purposes, the column buckling capacity must be accurately calculated. This paper presents an energy method approach to calculate the buckling load of a column with an intermediate elastic torsional restraint. Various end conditions of the column are considered including column base semi-rigidity, as well as multiple loading conditions. Displacement functions are determined based on measured experimental data. The Southwell and Meck plot methods to determine column buckling loads are discussed. The column buckling loads determined from the plot methods and calculated by the energy analysis are compared to the experimental column buckling loads. It is shown that the energy method outlined herein predicts the buckling load within 6% for columns with an intermediate elastic torsional restraint.

Modelagem computacional aplicada à flambagem elástica de colunas

<p>Flambagem é um fenômeno de instabilidade que ocorre em elementos estruturais esbeltos submetidos à carregamento de compressão. A flambagem elástica é caracterizada por uma repentina deflexão lateral do membro estrutural esbelto, quando a carga crítica de flambagem é atingida. Em colunas a flambagem elástica é considerada o colapso da estrutura. O objetivo desse trabalho é apresentar três diferentes modelos computacionais para tratar a flambagem elástica de colunas. O software ANSYS, que é baseado no Método dos Elementos Finitos (MEF), foi adotado para desenvolver esses modelos, empregando os elementos BEAM189, SHELL93 e SOLID45. As condições de contorno e a carga compressiva devem ser aplicadas de maneira específica em cada modelo, dependendo das características do elemento finito. As seguintes condições de contorno para a coluna foram consideradas: (a) birrotulada, (b) engastada-livre, (c) biengastada e (d) engastada-rotulada. As soluções analíticas, para as condições de contorno consideradas, foram usadas para a verificação dos modelos computacionais empregados. Todos os resultados numéricos obtidos estão em concordância com os analíticos, sendo a maior diferença encontrada de 2,83 %. Além disso, as formas flambadas das colunas obtidas numericamente também estão de acordo com as configurações teóricas. A vantagem para o uso dos modelos computacionais desenvolvidos com os elementos finitos SHELL93 e SOLID45 é a possibilidade de analisar colunas com furos ou com geometrias complexas. </p><div> </div>

Revisiting finite difference and finite element methods applied to structural mechanics within enriched continua

In this paper, we revisit the capability of numerical approaches such as finite difference methods and finite element methods, in approximating exact one-dimensional continuous eigenvalue problems (such as lateral vibrations of a string, the axial or the torsional vibrations of a bar, and the buckling of elastic columns). The numerical methods analysed in this paper are converted into difference equations. Following a continualization procedure or the method of differential approximation, the difference operators are then expanded in differential operators via Taylor expansion or Pade approximants. Analogies between the finite numerical approaches and some equivalent enriched continuum are shown, using this continualization procedure. The finite difference methods (first-order or higher-order finite difference methods) are shown to behave as integral-based nonlocal media (or stress gradient media), while the finite element method is found to behave as gradient elasticity media (or strain gradient media). The length scale identification of each equivalent enriched continuum strongly depends on the order of the numerical method considered. For the finite difference methods, the length scale identification of the equivalent nonlocal medium depends on the static versus dynamic analysis, whereas this length scale appears to be independent of inertia effects for the finite element method. Some comparisons between the exact discrete eigenvalue problems and the approximated continuous ones show the efficiency of the continualization procedure. An equivalent enriched Rayleigh quotient can be defined for each numerical method: the integral-based nonlocal method gives a lower bound solution to the exact eigenvalue multiplier, whereas the gradient elasticity method furnishes an upper bound solution.

Comments and discussion: “Elastic buckling of columns with end restraint effects”

Comments and discussion: “Elastic buckling of columns with end restraint effects”

Response to discussion on “Elastic buckling of columns with end restraint effects”

Response Paper to S.E. Djellab's Comments and discussion concerning: R. Adman,M. Saidani, “Elastic buckling of columnswith end restraint effects”, Journal of Constructional Steel Research 87 (2013) 1–5. In what follows is a response to the points of substance made in the discussion paper. The discusser raised issues relating to the rigour and accuracy of the results reported in the original paper. The response below has been presented following the same format adopted by the discusser.

Discussion of “Buckling of a Weakened Column” by C. Y. Wang, C. M. Wang, and Tun Myint Aung

The paper is a welcome addition to the literature dealing with the analytical solutions of the elastic buckling of columns having a weakened cross section or junction. This paper is a very good benchmark for the numerical approaches. The discusser wants to show how, using two numerical approaches—the contact element approach and the parallel connecting beams approach—the same results could be obtained.

Flexural–Torsional Buckling of Columns with Oblique Eccentric Restraints

This paper is concerned with the elastic flexural.torsional buckling of concentrically loaded columns with oblique eccentric restraints. Oblique flexural restraints may resist deflections as well as flexural rotations, while torsional restraints may resist twist rotations and warping. Flexural restraints may act at the shear center or may be eccentric. Restraints may be concentrated at points along a column or distributed along portions of its length, and may be rigid or elastic. Oblique flexural restraints cause coupling between the principal axis deflections and rotations, while eccentric flexural restraints cause coupling between the principal axis deflections and rotations and the twist rotations and warping displacements. The general buckling mode involves simultaneous bending about both principal axes and torsion. This paper discusses the nature of oblique eccentric restraints, summarizes their finite element analysis, presents examples of their effects on the elastic buckling of columns, and demonstrates the design of columns with oblique eccentric restraints.

Finite-Element Analysis of the Flexural Buckling of Columns with Oblique Restraints

This paper is concerned with the elastic flexural buckling of doubly symmetric columns with oblique restraints under concentric loading. Oblique restraints cause coupling between the principal axis deflections and rotations, and the flexural buckling mode involves simultaneous bending about both principal axes. Oblique restraints may resist deflections as well as rotations, and may be rigid or elastic. They may be concentrated at points along a column, or distributed along portions of its length. This paper discusses the nature of oblique end restraints, summarizes their finite element analysis, presents examples of their effects on the elastic buckling of columns, and demonstrates the design of columns with oblique restraints.

Theoretical study of dynamic elastic buckling of columns subjected to intermediate velocity impact loads

Dynamic elastic buckling of simply supported columns subjected to intermediate velocity impact is theoretically studied in this paper. The dynamic buckling equation is set up. Theoretical solution of dynamic responses of columns subjected to a half-sine shape intermediate velocity impact load is derived. Based on the characteristics of the theoretical solution, a dynamic buckling criterion is proposed to determine the critical buckling condition and to estimate the dynamic buckling critical load. Theoretical results obtained in this study are compared with experimental data. They are also compared with numerical results obtained by other authors. Good agreements between them are observed.

On the eigenvalue of nuclear reaction and self-weight buckling

We give an instructive analogy between the buckling of elastic column under self-weight and fast neutrons chain reaction for a cylindrical geometrical configuration.

Research on elastic buckling of columns, beam and plates: Focussing on formulas and design charts

Although there are many numerical methods which may be used to evaluate critical elastic design parameters accurately, designers still prefer to sacrifice accuracy for simplicity. Approximate formulas or design charts from handbooks and standard texts are used, or conservative assumptions are made to simplify the problem so as to justify their use. Until powerful, user-friendly and inexpensive software becomes readily available to designers, approximate formulas and design charts continue to be used. In recent years, collaborative work between the authors and their research associates has yielded a number of approximate buckling formulas and design charts for critical loads for columns, beams and plates. A review of these research investigations is presented herein, together with some of the newly developed buckling formulas and design charts.

Use of Engineering Strain and Trefftz Theory in Buckling of Columns

The elastic buckling of columns with allowance for prebuckling axial shortening is analyzed using Trefftz's initial stress theory of 1933 and the engineering strain definition; the results are compared with earlier results based on the Green strain definition. When the latter definition is used, the buckling load can be obtained using the Euler solution after scaling the flexural rigidity by the ratio of the original column length to the prebuckling shortened length. It is found that when the engineering strain is used instead, the corresponding factor is equal to the square of the aforementioned ratio.

Stability of structures: elastic, inelastic, fracture, and damage theories

Buckling of elastic columns by equilibrium analysis Buckling of elastic frames by equilibrium analysis Dynamic analysis and general concept of stability Energy analysis of discrete or discretized elastic systems Energy analysis of continuous elastic structures and approximate methods Thin-wall elastic beams Elastic plates and shells Elastoplastic buckling Creep buckling Stability of inelastic structures and path bifurcation Stability of massive three-dimensional bodies Fracture as a stability problem Damage as a stability problem.

Buckling of columns: Allowance for axial shortening

This paper investigates the effect of axial shortening on (i) the elastic buckling of columns with a continuous elastic restraint, (ii) the elastic buckling of rotating columns and (iii) the free vibration of columns under a static axial load. These column problems can be solved in a unified approach because the resulting energy functional is similar. The field differential equation is derived by minimizing the energy functional with respect to the lateral displacement function via calculus of variations. The buckling load or fundamental frequency may be obtained by analytically solving the two-point boundary-value problem. It was found that the boundary conditions and the restraint parameter or angular velocity parameter affect the influence of axial shortening on the buckling load. In vibrating columns, tensile forces enhance the effect of axial stretching on the fundamental frequency.

Vibration and buckling of beam‐columns subjected to non‐uniform axial loads

AbstractThe small‐displacement free vibration of elastic Euler‐Bernoulli beams subjected to non‐uniform axial forces and the buckling of elastic columns are both analysed by means of various types of beam finite elements. The procedure incorporates beams and columns of varying cross‐sections, such as linear tapers, inhomogeneous beams and columns, distributed axial forces, elastic end and interior restraints and point masses with linear and rotary inertias. All of these topics individually or in some combinations have been analysed by others. The purpose of this paper is to bring together under a single umbrella the various problems studied by others, and to provide the solution to one problem apparently not yet solved—the buckling and vibration of tapered columns under non‐uniform axial thrust. The mass, stiffness and geometric stiffness matrices for the standard beam element are fully written out for direct incorporation into existing finite element programs. A FORTRAN subroutine for generating these matrices, and those for various higher‐order beam elements, is also provided.