Asymptotic analyses of the buckling of imperfect columns on nonlinear elastic foundations

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Abstract.Approximate asymptotic expressions are obtained for the buckling stresses and autocorrelation of the lateral displacement of infinitely long imperfect columns resting on nonlinear elastic foundations. The imperfections are assumed to be homogeneous Gaussian random functions with known autocorrelation. The formulas are discussed and compared with previous results obtained by means of truncated hierarchy and equivalent linearization techniques.Introduction. In this paper a perturbation scheme is used to study a model imperfection-sensitive structure. We consider an ensemble of infinitely long imperfect columns resting on nonlinear elastic foundations. The stress-free initial displacements of the columns are assumed to be homogeneous zero-mean Gaussian random functions of positions along the column. In the analysis, approximate asymptotic expressions that are applicable for small mean square of the imperfections are sought for the buckling stresses and the autocorrelation of the displacements.In an earlier study of this problem [1] Fraser and Budiansky used an equivalent linearization technique to obtain the buckling stresses for random imperfections with two-parameter exponential-cosine autocorrelation functions. In a subsequent study [2] in which other types of imperfections were considered, Amazigo, Budiansky and Carrier obtained asymptotic expressions for the buckling load by means of both equivalent linearization and truncated-hierarchy methods.The perturbation scheme used in this paper appears more satisfactory and yields slightly different asymptotic expressions for the buckling load. In addition an asymptotic expression is obtained for the buckling displacement. Differential equation.The nondimensional form of the differential equation governing the lateral displacement of an infinite column on a "softening" nonlinear elastic foundation is Lw -w3 = -2\ew'0', -00 < x < <*>,where L( ) = ( )"" + 2X( )" + ( ) with ( )' = {d/dx){ ). The nondimensional axial coordinate x, lateral deflection w, axial load parameter X, and stress-free initial displacement w0 are related to the physical quantities by x = (kJEiy^X, w = {h/lh)1/2W, X = P/2(EIk1

“…As in [2], we observe that asymptotically the buckling load depends only on the value of the spectral density at u = 1. Eq.…”

Section: The Solution Of Eq (8) Issupporting

confidence: 73%

“…In a subsequent study [2] in which other types of imperfections were considered, Amazigo, Budiansky and Carrier obtained asymptotic expressions for the buckling load by means of both equivalent linearization and truncated-hierarchy methods.…”

Section: Introductionmentioning

confidence: 99%

Buckling of stochastically imperfect columns on nonlinear elastic foundations

Amazigo¹

1971

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“…The perturbation scheme used here was developed in [1J. It is found that the range of values of Z for imperfection-sensitivity remains the same and the loss in the buckling load for the three types of imperfections parallels that obtained for columns on nonlinear foundations [1,2], Kdrmdn-Donnell equations.…”

Section: Introductionmentioning

confidence: 90%

Asymptotic analysis of the buckling of externally pressurized cylinders with random imperfections

Amazigo¹

1974

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Abstract.The buckling of finite circular cylindrical shells with random stress-free initial displacements which are subjected to lateral or hydrostatic pressure is studied using a perturbation scheme developed in an earlier paper [1], A simple approximate asymptotic expression is obtained for the buckling load for small magnitudes of the imperfection. This result is compared with earlier results obtained for localized imperfections and imperfections in the shape of the linear buckling mode. Introduction.It is generally recognized that the buckling loads of some elastic structures are substantially reduced by the presence of nonuniformities in these structures. These nonuniformities or imperfections may be in the elastic or geometric properties of the structure. In [7,8], Koiter developed a general theory of post-buckling behavior and derived simple asymptotic formulae for the buckling load of a class of elastic structures with imperfections in the shape of their classical (linear) buckling

Section: Introductionmentioning

confidence: 99%

“…It is found that the effect of one Fourier coefficient in the expansion of the imperfection dominates in the asymptotic expression for the deflection and the dynamic buckling load. The static buckling of this structure has been studied extensively [3,4,5].A number of related dynamic response and buckling analyses are noteworthy. Budiansky and Hutchinson [6,7,8] formulated a general approximate theory of dynamic buckling by an extension of Koiter's static theory of post-buckling behavior [9].…”

mentioning

confidence: 99%

Untitled

1973

Introduction.In this paper a formal two-time perturbation expansion (see, e.g., [1,2]) is used to study the dynamic response and buckling of a model imperfectionsensitive structure. We consider a finite imperfect column resting on nonlinear elastic foundation subject to step loading. A simple expression is obtained for the dynamic buckling load for small imperfections and small initial conditions. It is found that the effect of one Fourier coefficient in the expansion of the imperfection dominates in the asymptotic expression for the deflection and the dynamic buckling load. The static buckling of this structure has been studied extensively [3,4,5].A number of related dynamic response and buckling analyses are noteworthy. Budiansky and Hutchinson [6,7,8] formulated a general approximate theory of dynamic buckling by an extension of Koiter's static theory of post-buckling behavior [9]. In these theories the imperfections are assumed to be in the shape of the classical buckling mode. With such a restriction this problem can be handled by the theory of Budiansky and Hutchinson. In this analysis this restriction on the imperfection need not be made.