On the optimal structural design for a nonconservative,elastic stability problem

The design of a cantilever column under a follower load is considered with the aim of maximizing the critical value of the load. The optimality condition is derived, and a modified Ritz method is used to determine an approximate solution for the bending stiffness. Results are obtained numerically for the case of a sandwich column with constant bending stiffness in each of two segments. It is found that, for the same structural weight, the optimal design yields a critical load significantly higher than that for a uniform column.This research was supported in part by the US Army Research Office–Durham and in part by the United States Navy under Grant No. NONR N00014-67-A-0191-0009.     

Optimum shape of columns with general conservative end loading

In this paper, the optimal shape of a column with the most general conservative state of loading is determined. The boundary conditions of a column contain, in general, 16 parameters; it is shown that the boundary values of a column with general conservative loading may, at most, depend on nine constants. The influence of these parameters on the optimal shape and corresponding buckling load is also obtained.     

On the integral equation method for buckling loads

An initial value method for the integral equation of the column is presented for determining the buckling load of columns. The differential equation of the column is reduced to a Fredholm integral equation. An initial value problem is derived for this integral equation, which is reduced to a set of ordinary differential equations with prescribed initial conditions in order to find the Fredholm resolvent. The singularities of the resolvent occur at the eigenvalues. Integration of the equations proceeds until the integrals become excessively large, indicating that a critical load has been reached. To check this method, numerical results are given for two examples, for which the critical load is well known. One is the Euler load of a simply supported beam, and the other case is the buckling load of a cantilever beam under its own weight. The advantage of this initial value method is that it can be applied easily to solve other nonlinear problems for which the critical loads are unknown. This approach will be illustrated in future papers.     

Experimental study on the behaviour of steel columns under seismic-induced axial impact load

It has been observed during major earthquakes that the so called soft-storey failure of an upper floor of a structure results in large impact load acting on structural members of the lower storeys. It may further lead to progressive collapse of the whole structure substantially intensifying human and material losses. Therefore, the aim of this paper is to investigate experimentally the behaviour of columns under bending (observed during an earthquake) that are additionally subjected to an axial plastic impact load. Steel columns with high slenderness ratio were considered in the study. In the experiment, impact load was created by a weight that was dropped onto the top of the column and stayed on it. The weight with the shape of a ball was made of clay so as to simulate more plastic impact. The results of the experiment show that the value of the permanent horizontal deformation of the column observed after impact depends much on the initial relative displacement between the base and the top of the column. As the initial displacement increases the values of the permanent deformation increases significantly indicating the significant reduction in the critical buckling force. The results obtained in the study can be utilized at the design stage of the structures in order to enhance their seismic resistance. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Détermination et maximisation de la charge critique d'une colonne de Hauger en présence d'amortissement

Résumé On détermine la charge critique de flottement d'une colonne soumise à une charge répartie non conservative de type suiveuse. La formulation tient compte de l'amortissement interne dû à la nature dissipative du matériau, et de l'amortissement externe dû à la résistance visqueuse du fluide ambiant. On effectue également l'optimisation numérique de la répartition massique de la colonne pour une charge critique maximum. Dans le cas de sections droites toutes semblables entre elles, la répartition optimale obtenue représente une amélioration de 144% par rapport à la charge critique de la colonne prismatique de même masse totale.

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The critical flutter load analysis of a column subjected to a distributed nonconservative follower loading is performed. The formulation includes the effects of internal material damping and external viscous damping. The optimum design of such a column for a maximum value of the critical loading is also determined numerically. If all cross-sections have a similar form, the optimum shape obtained represents an increase of 144% over the critical load of the uniform structure with the same overall mass.

     

Novel closed – form solutions in buckling of inhomogeneous columns under distributed variable loading

In this study we consider buckling of columns with variable stiffness, under axially distributed loading varying polynomially. The objective is to obtain closed – form solutions for the buckling load. The problem is posed in inverse setting: determine the column’s stiffness, so that it has the given, polynomial, buckling mode. Four sets of boundary conditions are investigated. Some perplexing results are obtained, namely, that irrespective of boundary conditions, the critical load of the column is the same; this occurs in conjunction with the fact that the obtained distribution for stiffness is different for each set of boundary conditions.     

Stability of Pflüger's column with imperfections

The stability of a column loaded with continuously distributed tangential load is analyzed. It is assumed that imperfections in the shape (initial deformation) and loading (a small concentrated force) are present. It is shown that the influence of the imperfections is modelled by a Whithney cusp.     

The stability of the shape of an unattached elastic rod with stiff flanges on its ends

The stability of an unattached column consisting of an elastic rod with stiff flanges on its ends under longitudinal compression is investigated. The load under which the plane of the flange surface is tilted from the plane of the support surface is found. This tilting is accompanied by considerable rotation (reversing) of the flanges and corresponding bending of the rod axis. Abrupt replacement of the rectilinear or bent equilibrium shape by an equilibrium shape that is non-contiguous to it occurs. It is established that the columns behave differently when this a change in the equilibrium shape occurs, depending on the ratio of the length of the rod to the length of the flanges.     

Buckling analysis of non-prismatic columns based on modified vibration modes

In this paper, a new procedure is formulated for the buckling analysis of tapered column members. The calculation of the buckling loads was carried out by using modified vibrational mode shape (MVM) and energy method. The change of stiffness within a column is characterized by introducing a tapering index. It is shown that, the changes in the vibrational mode shapes of a tapered column can be represented by considering a linear combination of various modes of uniform-section columns. As a result, by making use of these modified mode shapes (MVM) and applying the principle of stationary total potential energy, the buckling load of tapered columns can be obtained. Several numerical examples on tapered columns demonstrate the accuracy and efficiency of the proposed analytical method.     

轴向弹塑性应力波作用下直杆中的分叉问题

考虑了一个弹塑性直杆的动力屈曲问题,将其归结为轴向阶跃应力波的传播导致的分叉问题,分析了横向惯性效应的影响,并考虑了应力波反射的作用,给出了相应的屈曲条件,最后进行了数值分析,从中得到了一些有益的结论。     

Optimal design of multi-purpose beam-columns

The paper deals with the optimal design of an elastic pinended member of given volume that is to serve as a beam for a part of its design life and as a column for the rest. The optimal design can be interpreted in two ways. Firstly, it is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection as a beam. Secondly, it is the design that has the least deflection as a beam under a midspan concentrated load, subject to a minimum permissible Euler buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.     

The von Mises girder

The von Mises girder is the simplest example of an elastic system that exhibits nonlinear behavior, bifurcates, and has nonunique equilibrium states. The classical von Mises girder consists of two joined elastic compressible rods with small slopes. A typical manifestation of stability loss under vertical forces is buckling into the mirror-reflected shape. The critical load is a limiting point on the force-deflection curve.     

The Reduced Basis Method for an Elastic Buckling Problem

In this work, we apply the Reduced Basis (RB) Method to the field of nonlinear elasticity. In this first stage of research, we analyze a buckling problem for a compressed 2D column: Here, the trivial linear solution is computed for an arbitrary load; the critical load, marking the transition to nonlinearity, is then identified through an eigenvalue problem. The linear problem satisfies the Lax-Milgram conditions, allowing the implementation of both a Successive Constraint Method for an inexpensive lower bound of the coercivity constant and of a rigorous and efficient a posteriori error estimator for the RB approximation. Even though only a non-rigorous estimator is available for the buckling problem, the actual RB approximation of the output is more than satisfactory, and the gain in computational efficiency significant. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Subcritical and critical states of a crack with failure zones

The problem on the stress–strain state near a mode I crack in an infinite plate is solved in the frame of a cohesive zone model. The complex variable method of Muskhelishvili is used to obtain the crack opening displacements caused by the cohesive traction, which models the failure zone at the crack tip, as well as by the external load. The finite stress condition and logarithmic singularity of the derivative of the separation with respect to the coordinate at the tip of a physical crack are taken into account.The cohesive traction distribution is sought in a piecewise linear form, nodal values of which are being numerically chosen to satisfy the traction-separation law. According to this law, the cohesive traction is coupled with the corresponding separation and fracture toughness. The tips of the physical crack and cohesive zone (geometric variables) along with the discrete cohesive traction are used as the problem parameters determining the stress-strain state. If the crack length is included in the set, then the critical crack size can be found for the given loading intensity.The obtained determining system of equations is solved numerically. To find the initial point for a standard numerical algorithm, the asymptotic determining system is derived. In this system, the geometric variables can be easily eliminated, which make it possible to linearize the system.In the numerical examples, the one-parameter traction-separation laws are used. Influence of the shape parameters of the law on the critical crack size and the corresponding cohesive length is studied. The possibility of using asymptotic solutions for determining the critical parameters is analysed. It is established that the critical crack length slightly depends on the shape parameter, while the cohesive length shows a strong dependence on the shape of cohesive laws.     

The stability of a circular plate of shape memory alloy during a direct martensite transformation

In different formulations, analytical solutions are obtained of the problem of the axisymmetrical loss of stability of a circular plate of shape memory alloy undergoing a direct martensite transformation under a compression load. It is established that the solution based on the “fixed phase composition” concept gives an upper estimate of the critical load, while the solution based on the “continuing loading” concept gives a lower estimate. For a clamped plate, an intermediate solution is found using the “elastic unloading” concept that satisfies all the equations of the problem with zero variations in external loads and corresponds to a thickness of the additional phase transformation layer that is independent of the radial coordinate.     

Exploration of the encased nanocomposites functionally graded porous arches: Nonlinear analysis and stability behavior

This paper explores the nonlinear stability mechanism of the functionally graded porous (FGP) arch reinforced by graphene nanocomposites. Both the pores and the nanocomposites are distributed symmetrically to the mid-surface of the arch but not uniformly in the cross-section so that the bending stiffness can be best improved. The arch is confined in an elastic medium with a radially-pointed concentrated load at the crown position. The confinement of the medium results in a symmetrical deformed shape of the arch, which can be described by an admissible displacement function. Associated with the thin-walled arch theory and the principle of minimum potential energy, analytical predictions are obtained to express the critical buckling load, as well as the hoop force and bending moment. Subsequently, a numerical model is developed to simulate the medium and the arch in ABAQUS software. By introducing the modified arc-length method, the equilibrium paths of the encased arch are traced. After comparison in terms of the critical buckling load and the equilibrium paths, it is found the numerical results are in good accordance with the analytical solutions. Finally, particular attention is paid to the parameters that may impact the buckling load, such as the porosity coefficient, the weight fraction, the central angle, the geometry of the Graphene platelets (GPLs) et al.     

Optimal design of multi-purpose structures

The paper presents the optimal (maximum transverse stiffness) design of an elastic, simply supported member of given volume that is to serve as a beam or as a column at different times during its design life. The optimal design can be interpreted in two ways. It is the design that has the maximum Euler buckling load in column action, subject to a prescribed maximum deflection in beam action under a uniformly distributed load; it is also the design that has the least deflection at midspan under a uniformly distributed load, subject to a lower limit on its buckling load in column action. The effectiveness of the optimal design is judged by comparing it with a prismatic bar of the same volume.The author is grateful to Professor W. Prager for suggesting several improvements to an earlier version of the paper.     

Free vibration and buckling of tapered columns made of axially functionally graded materials

This paper presents a unified model to analyze the free vibration and buckling of axially functionally graded Euler-Bernoulli columns subjected to an axial compressive force. The material properties vary linearly along the longitudinal direction, and column with circular and square cross sections is linearly tapered. The governing differential equations of the problem are derived and solved using the direct integral method combined with the determinant search technique. The computed results are compared with those reported in the literature and obtained from the finite element software ADINA. Numerical examples for natural frequency, buckling load and their corresponding mode shapes are given to highlight the effects of modular ratio, taper ratio and cross sectional shape as well as the end condition.     

The perturbation method applied to tube drawing with floating plug

The determination of the load on the machine tool is crucial in the mechanics of metal forming processes as it has a critical influence over the deformation of the material into a new shape and the optimal conditions for the process. The problems of strip, wire, tube drawing without plug, tube drawing with floating plug [1-4] viscoplastic deformation has no exact solutions, but this paper addresses them using an approximate method. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Application of the Perturbation Method in the Buckling Optimization of a Heavy Elastic Column of Constant Aspect Ratio

This paper addresses the problem of maximizing the first Euler load of a column of constant aspect ratio (moment of inertia varies with the square of the cross-sectional area), considering the column self-weight, subject to the constraint of fixed volume of column material and height. The coupled nonlinear integro-differential equations of optimality and stability, generated through variational principles, have been solved using the method of parameter perturbation. The optimal column has cross-sectional area that follows a perturbed two-thirds power law; the first Euler load is up to 87% larger than that of the corresponding column of uniform cross-section having the same volume and height.