Numerical Methods for Determining the Dynamic Buckling Critical Load of Thin Shells. State of the Art (original) (raw)
Related papers
Buckling of thin shells: Recent advances and trends
This paper provides a review of recent research advances and trends in the area of thin shell buckling. Only the more important and interesting aspects of recent research, judged from a personal view point, are discussed. In particular, the following topics are given emphasis: (a) imperfections in real structures and their influence; (b) buckling of shells under local/non-uniform loads and localized compressive stresses; and (c) the use of computer buckling analysis in the stability design of complex thin shell structures.
Comments on effective use of numerical modelling and extended classical shell buckling theory
Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 2023
2. Extended classical shell buckling theory A number of reviews [31-34] have covered some of the background and important applications of the RS extension to classical shell buckling theory. These have demonstrated that for wide ranges of shell forms and loading conditions, the lower limits to the reductions in buckling loads caused by increasing levels of geometric (and loading) imperfections are provided by RS analysis, including the treatment of the development of material failure [6,31-33]. The philosophical basis of the RS method can be formulated [34] in terms of the following lemmas: Lemma 2.1. Significant geometric nonlinearity in structural behaviour results from changes in incremental membrane energy (or equivalently membrane stiffness). Corollary 2.2. Significant nonlinearity in the buckling of shell and plate structures derives from changes in incremental membrane energy. Corollary 2.3. Nonlinearity arising from changes of incremental bending energy is practically negligible. Lemma 2.4. Loss of stiffness (or load carrying capacity) in the buckling 1 and postbuckling (see footnote 1) of shells is the result of loss of incremental membrane energy. Corollary 2.5. Loss of stiffness (or load carrying capacity) in the postbuckling of shells can only occur if the initial buckling mode (critical bifurcation mode) contains membrane energy. Corollary 2.6. Loss of stiffness (or load carrying capacity) resulting from changes of incremental bending energy in the postbuckling of shells is practically negligible.
BUCKLING ANALYSIS OF SHELLS SUBJECTED TO COMBINTED LOADS
IAEME Publication, 2014
A semi-analytical isoparametric finite element with three nodes per element and five degrees of freedom per node has been used for the solution. Moderately thick shell theory has been used for the analysis. Second order strains with the in plane and transverse non-linear terms are used for the derivation of geometric matrix. Full Fourier expansion is used in the circumferential direction to overcome the coupling that arises due to material anisotropy and torque prestress. Comparison of the results obtained due to finite element is made with simplified solutions using two thin shell theories with and without shear deformation. The effects of combined load (axial compression and external pressure) on pre-buckling characteristics of composite circular cylindrical and conical shells of various geometric properties have been presented.
Thin-Walled Structures, 2012
In the present work buckling stresses of prismatic flat and stiffened shell structures are derived within the framework of the Kantorovich approach, making reference to both Von Karman and Koiter-Sanders theories, the latter exploiting the Green-Lagrange strain tensor which is needed if the expected buckling modes involve comparable in-plane and out-of-plane displacements. Additionally, in order to highlight the contribution of each nonlinear term of the Koiter-Sanders model, two further intermediate choices are also considered, namely an enhanced Von Karman model and a spurious model which collects selected terms from different theoretical approaches, generally adopted in literature with the aim of simplifying the numerical analyses. The obtained results show how in buckling problems where the weight of in-plane displacements cannot be neglected the Von Karman model tends to overestimate the critical load, while the three considered alternative models result substantially equivalent, at least from the practical standpoint.
Buckling Load of Thin Spherical Shells Based on the Theorem of Work and Energy
International Journal of Engineering and Technology, 2013
Thin spherical shells usually fail due to buckling. An empirical equation to predict their buckling load is derived based on the theorem of work done and energy released in the inversion of a section of a shell and nonlinear finite element (FE) modeling done using ABAQUS to determine their post-buckling behavior. It is observed that the initial buckling is sensitive to initial geometrical imperfections but the post-buckling load is little influenced. Therefore, the post-buckling load is used to predict a more realistic load as compared to classical buckling theory prediction
Dynamic elastic and plastic buckling of complete spherical shells
International Journal of Solids and Structures, 1974
A theoretical investigation is undertaken into the dynamic instability of complete spherical shells which are loaded impulsively and made from either linear elastic or elasticplastic materials. It is shown that certain harmonics grow quickly and cause a shell to exhibit a wrinkled shape which is characterized by a critical mode number. The critical mode numbers are similar for spherical and cylindrical elastic shells having the same R/h ratios and material parameters, but may be larger or smaller in an elastic-plastic spherical shell depending on the values of the various parameters. Threshold velocities are also determined in order to obtain the smallest velocity that a shell can tolerate without excessive deformation. The threshold velocities for the elastic and elastic-plastic spherical shells are larger than those which have been published previously for cylindrical shells having the same R/h ratios and material parameters. NOTATION shell thickness time Young's and tangent moduli, respectively mean radius of spherical shell initial impulsive velocity 2(1-v) h-1 1-2v equivalent yield strain and equivalent stress, respectively El& n(n + 1) Poisson's ratio and density, respectively yield stress a()/ar or a()/ci7 a()/ax, where x = 8, 4, 7 or y.
A comparison of techniques for computing the buckling loads of stiffened shells
Computers & Structures, 1993
Abatraet-Three different methods are employed to estimate the buckling loads of several ring stiffened and orthotropic cylindrical shells using finite elements. The methods used are a nonlinear bifurcation analysis and two linearized buckling analyses, one that ignores the initial displacement stiffness matrix, and one that includes it. Large differences are observed between the predictions made by the two linearized buckling analyses for a range of shell geometries. Detailed studies of a shell with six stiffeners demonstrate that them differences am caused by different versions of the linearized eigenvalue problem, rather than by the use of different numerical formations. These diserepaneies are also observed for orthotropic shells when L/R is small (< 1) and the degree of orthotropy (EJE,) is high (2 10). Investigations of the prebuckling behavior of some of the cylinders show that the problem is caused by significant nonlinear pmbuckling deformations. This means that a nonlinear bifurcation analysis must sometimes be used to accurately estimate the buckling load of stiffened shells.
Relaxation oscillations and buckling prognosis for shallow thin shells
Zeitschrift für angewandte Mathematik und Physik, 2020
This paper reports the possibilities of predicting buckling of thin shells with nondestructive techniques during operation. It examines shallow shells fabricated from high-strength materials. Such structures are known to exhibit surface displacements exceeding the thickness of the elements. In the explored shells, relaxation oscillations of significant amplitude could be generated even under relatively low internal stresses. The problem of cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, works by many authors who studied the geometry of the surface formed after buckling (postbuckling behavior) were used. The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. This circumstance enables a detailed analysis of the obtained equation to be performed and a description to be given of the physical phenomena-relaxation oscillations-that are unique to thin high-strength shells. It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence at which stable periodic motions are converted into dynamic chaos. The amplitude and period of the relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure. It is emphasized that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Because the amplitude and the frequency of the oscillations are known, this fact enables an experimental facility to be proposed for prediction of shell buckling with nondestructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding the specified limits. Based on the results of the experimental measurements, a formula is obtained to estimate the safety against buckling (safety factor) of the structure.
Journal of the International Association for Shell and Spatial Structures, 2017
This paper analyses the buckling shapes of spherical shells subjected to concentrated load. A theoretical investigation, based upon geometric considerations, shows a variety of possible buckling shapes, including polygonal ones. The results show that there exists a certain difference between the geometric behavior of shells characterized by different radius-thickness ratios. An analogy between the buckling edge of the shell and a compressed planar elastic ring is also shown, which gives a better view on the point of transformation of the buckling edge from a circle to a polygon. The results of numerical and experimental analyses are also exhibited to verify the theoretical achievements.