On single and bimodal optimum buckling loads of clamped columns

We study the problem of determining the optimum shape of a thin, elastic, clamped column of given length and volume, such that the fundamental buckling load is a maximum. The column cross-sections are assumed to be geometrically similar, and a minimum allowable value is specified for the cross-sectional area.Investigating the optimization problem parametrically in terms of this minimum constraint, we reveal a significant feature. There exists a threshold value for the constraint, beyond which the optimum columns are all associated with single mode optimum buckling loads, whereas, for any value of the constraint less than the threshold value, the optimum columns are associated with bimodal fundamental buckling loads.This bimodal behaviour necessitates an extension and a mathematical reformulation of the current optimization problem, which is outlined and solved in the paper. In particular, we revise the result hitherto considered to be the optimum solution for an unconstrained column with clamped ends.     

Lower bounds to fundamental frequencies and buckling loads of columns and plates

A general derivation of expressions for lower bounds to fundamental frequencies and buckling loads is given for the class of structures governed by linear elastic theory in the prebuckling state. These expressions involve two Rayleigh quotients both of which are upper bounds for the fundamental frequency under a prescribed load. The displacement trial functions must satisfy force and kinematic continuity but no other conditions are required. Thus, if appropriate high order base functions are used, the finite element procedure can be used to systematically narrow the difference between the upper and lower bounds.The theory is illustrated with several column and plate problems. The finite element method is applied to uniform and nonuniform columns with a representative set of boundary conditions. Elementary trial functions are used to show that reasonable bounds can also be obtained for plates subjected to known states of stress. Since the lower bound is obtained with a variation of the classical technique of Rayleigh, these results indicate that the method may be suitable for conservatively estimating buckling loads and fundamental frequencies of engineering structures.     

Local buckling of composite channel columns

Continuum Mechanics and Thermodynamics - The investigation concerns local buckling of compressed flanges of axially compressed composite channel columns. Cooperation of the member flange and web is...     

Influence of splices on the buckling of columns

The paper presents a procedure for the analysis of stability and initial post-buckling behaviour of spliced columns in sway and non-sway steel frames. The main assumptions are linear elasticity and geometrically perfect columns that are loaded by a compressive force which retains its direction as the column deflects. An energy-based formulation that includes a polynomial Rayleigh–Ritz approximation into the potential energy function, in combination with the Lagrange’s method of undetermined multipliers, has been found very convenient for this type of problem. The system is thus described by a set of kinematically admissible generalized coordinates and a single loading parameter. First, the critical state is characterized by means of linear eigenvalue analysis. A parametric study is implemented to assess the critical load. The numerical results are used to develop a relatively simple yet reasonably accurate engineering method for predicting the critical behaviour of spliced columns in sway and non-sway steel frames. The energy formulation is then applied to the search of post-buckling branches of bifurcation points. The approach embraces path-following methods based on perturbation schemes built on a Newton type iterative procedure. This is illustrated in the application to post-bifurcation in columns with different splice mechanical characteristics. The findings suggest that the splice tangent stiffness has a major influence on the overall column behaviour.     

Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

Upper and lower bounds of buckling loads

Upper and lower bounds of buckling load for a nonuniform elastic column under conservative loading are considered. Compatible admissible moment and displacement functions are expressed in terms of a compatible coordinate system. The generalized Timoshenko Quotient and the modified Schreyer and Shih formula are the proposed upper and lower bounds. Both bounds when iterated converge to the exact buckling load. The method described here is simple and convenient and applies to all self-adjoint problems without exception.     

A method for the estimation of plastic buckling loads

Certain elastic-plastic buckling problems require the solution of an appropriate incremental or “rate” boundary-value problem in order that physically meaningful results may be obtained. In this paper, it is shown that a recent general variational theorem by Neale[8] may be advantageous for the approximate solution of such problems. As an example, the buckling of elastic-plastic cylindrical shells under torsion is analyzed, wherein the material is assumed to obey the incremental theory of plasticity and the effects of initial imperfections in geometry are taken into account.     

Thermoelastic buckling of steel columns with load-dependent supports

The in-plane elastic buckling of a steel column with load-dependent supports under thermal loading is investigated. Two elastic rotational springs at the column ends are used to model the restraints which are provided by adjacent structural members or elastic foundations. The temperature is assumed to be linearly distributed across the section. Based on a nonlinear strain–displacement relationship, both the equilibrium and buckling equations are obtained by using the energy method. Then the limits for different buckling modes and the critical temperature of columns with different cases are studied. The results show that the proposed analytical solution can be used to predict the critical temperature for elastic buckling. The effect of thermal loading on the buckling of steel columns is significant. Furthermore, the thermal gradient plays a positive role in improving the stability of columns, and the effect of thermal gradients decreases while decreasing the modified slenderness ratios of columns. It can also be found that rotational restraints can significantly affect the column elastic buckling loads. Increasing the initial stiffness coefficient α or the stiffening rate β of thermal restraints will increase the critical temperature.     

Elastoplastic buckling and post-buckling analysis of sandwich columns

Sandwich structures are widely used in many industrial applications thanks to their interesting compromise between lightweight and high mechanical properties. This compromise is realized thanks to the presence of different parts in the composite material, namely the skins which are particularly thin and stiff relative to the homogeneous core material and possibly core reinforcements. Owing to these geometric and material features, sandwich structures are subject to global but also local buckling phenomena which are mainly responsible for their collapse. The buckling analysis of sandwich materials is therefore an important issue for their mechanical design. In this respect, this paper is devoted to the theoretical study of the local/global buckling and post-buckling behavior of sandwich columns under axial compression. Only symmetric sandwich materials are considered with homogeneous and isotropic core/skin layers. First, the buckling problem is analytically addressed, by solving the so-called bifurcation equation in a 3D framework. The bifurcation analysis is performed using an hybrid model (the two faces are represented by Euler–Bernoulli beams, whereas the core material is considered as a 2D continuous solid), considering both an elastic and elastoplastic core material. Closed-form expressions are derived for the critical loadings and the associated bifurcation modes. Then, the post-buckling response is numerically investigated using a 2D finite element bespoke program, including finite plasticity, arc-length methods and branch-switching procedures. The numerical computations enable us to validate the previous analytical solutions and describe several kinds of post-critical responses up to advanced states, depending on geometric and material parameters. In most cases, secondary bifurcations occur during the post-critical stage. These secondary modes are mainly due to the modal interaction phenomenon and give rise to unstable post-buckled solutions which lead to final collapse.     

Response of an elastic half-space with power-law nonhomogeneity to static loads

In this paper a series of problems for an isotropic elastic half-space with power-law nonhomogeneity are considered. The action of surface vertical and horizontal forces applied to the half-space is studied. A part of the paper deals with the case of zero-valued surface shear modulus (for positive values of the power determining the nonhomogeneity). This condition leads to simple solutions for two-dimensional (2D) case when radial distribution of stresses exists for surface loads concentrated along an infinite line. Corresponding results for the three-dimensional (3D) case are constructed on the basis of the relationships between 2D and 3D solutions developed in the paper. A more complicated case, in which the shear modulus at the surface of the half-space differs from zero, is treated using fundamental solutions of the differential equations for Fourier–Bessel transformations of displacements. In the paper the fundamental solutions are built in the following two forms: (a) a combination of functions expressing displacements of the half-space under the action of vertical and horizontal forces in the case of zero surface shear modulus, and (b) a representation of the fundamental solutions using confluent hypergeometric functions. The results of numerical calculation given in the paper relate to Green functions for the surface vertical and horizontal point forces.     

Torsional buckling of spherical shells under circumferential shear loads

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Dynamic buckling of shallow curved structures under stochastic loads

In this paper, the dynamic stability of shallow structures such as arches and curved panels under stochastically fluctuating loads is studied. Sufficient conditions guaranteeing the almost-sure stability in both symmetric modes and unsymmetric modes of deformation are obtained first by Infante's method. Necessary and sufficient conditions are determined by evaluating the largest Lyapunov exponent of the perturbed solution.     

Analytic upper-bound estimates for the critical loads of perfect ribbed shells

A new approximate approach is proposed to find upper-bound estimates for the critical loads of ribbed shells. Seventeen cases are considered, and the minimum critical load parameter is determined __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 74–83, December 2005.     

Finding the buckling load of non-uniform columns using the iteration perturbation method

The aim of this study is to calculate the critical load of variable inertia columns. The example studied in this paper can be used as a paradigm for other non-uniform columns. The wavelength of equivalent vibratory system is used to calculate the critical load of the trigonometrically varied inertia column. In doing so, the equilibrium equation of the column is theoretically studied using the perturbation method. Accuracy of the calculated results is evaluated by comparing the solution with numerical results. Effect of improving the initial guess on the solution accuracy is investigated. Effects of varying parameters of the trigonometrically varied inertia and the uniformly tapered columns on their stability behavior are studied. Finally, using the so-called “perfectibility” parameter, two design goals, i.e., being lightweight and being strong, are studied for the discussed columns.     

Generalized sensitivity and probabilistic analysis of buckling loads of structures

The imperfection sensitivity law by Koiter played a pivotal role in the early stage of research on initial post-buckling behaviors of structures, but seems somewhat overshadowed by numerical approaches in the computer age. In this paper, to make this law consistent with practical application, the law is extended to implement the influence of a number of imperfections, and the second-order (minor) imperfections are considered, in addition to the first-order (major) imperfections considered in the Koiter law. Explicit formulas are presented to be readily applicable to the numerical evaluation of imperfection sensitivity. A procedure to describe the probabilistic variation of critical loads is presented for the case where initial imperfections of structures are subject to a multivariate normal distribution; the formula for the probability density function of critical loads is derived by considering up to the second-order imperfections. The validity and usefulness of the present procedure are demonstrated through the application to truss structures.     

Effects of lateral loads and constraints on buckling of cracked thin plates under compressive edge loads

Critical buckling load is important in investigation of behaviors of thin plates or shells. The presence of cracks in these structures can affects their safety factor with respect to the common modes of failure such as buckling. Some analytical solutions were obtained for un-cracked plates with different boundary conditions. Their numerical results have good agreement with these solutions. In this paper, we also studied the effects of lateral loads and constraints on the critical buckling load of cracked plates under axial compression, experimentally and numerically. Effects of length and orientation of cracks are investigated in presence of the lateral loads. Finally, tests data are compared with the results of numerical calculations.     

Critical loads of shells with high-modulus reinforcement

The analytical method developed to determine the upper and lower critical stresses is applied to cylindrical shells reinforced with stringers and rings. Buckling modes typical of shells reinforced with discrete ribs are considered. The minimum critical loads are determined and compared with available experimental data. Perfect and imperfect ribbed shells with high-modulus reinforcement are studied. It is proved that the effect of small axisymmetric imperfections in such shells is not so drastic as in isotropic shells. Ribs made of a high-modulus material can enhance the stability of shells by a factor of tens