The results o f experim ents on axial load ing o f cylindrical shells (th in enough to buckle below th e elastic lim it and too short to buckle as Euler colum ns) are n o t in good agree­ m en t w ith previous theories, w hich have been based on th e assum ptions o f perfect in itia l shape and in fin itesim al de­ flections. Experim ental failure stresses range from 0.6 to 0.15 of th e theoretical. T he discrepancy is apparently considerably greater for brass and m ild -stee l specim ens th an for duralum in and increases w ith th e rad ius-th ick ness ratio. There is an equally great discrepancy betw een observed and predicted shapes o f buckling deflections. In th is paper an approxim ate large-deflection theory is developed, w hich perm its in itia l eccen tricities or devia­ tions from cylindrical shape to be considered. True in ­ stab ility is, o f course, im possib le under such con d itions; th e stress d istrib ution is no longer un iform , and it is assum ed th a t final failure takes place w hen th e m axim um stress reaches th e y ield point. T he effect o f in itia l eccen ­ tricities and o f large deflections is m u ch greater th a n for th e case o f sim ple stru ts. M easurem ents o f in itia l ec ­ centricities in actual cylinders have n o t been m ade; h ow ­ ever, it is show n th a t m ost o f th ese discrepancies can be explained i f th e in itia l deviations from cylindrical form are assum ed to be resolved in to a double harm onic series, and i f certain reasonable assu m p tion s are m ade as to th e m ag­ n itu d es o f th ese com p on en ts o f th e deviations. W ith th ese assu m p tion s th e fa ilin g stress is found to be a fu n c ­ tio n o f th e y ield p o in t as w ell as o f th e m odu lu s o f e la stic ­ ity and th e rad ius-th ick ness ratio . O n th e basis o f th is a ten ta tive design form ula [5] is proposed, w hich involves relations suggested by th e theory b u t is based on experi­ m en ta l data. It is show n th a t sim ilar discrepancies betw een experi­ m en ts and previous theories on th e b uck ling o f th in cy lin ­ ders in pure bending can be reasonably explained on th e sam e basis, and th a t th e m axim u m bend ing stress can be tak en as abou t 1.4 tim es th e values given by E quation [5]. It is a lso show n th a t puzzling features in m an y other buckling problem s can probably be explained by sim ilar considerations, and it is hoped th a t th is d iscussion m ay help to open a new field in th e stu d y o f buckling problem s. T he large-deflection theory developed in th e paper should be usefu l in exploring th is field, and m ay be used in other applications as well. T he paper presents th e resu lts o f abou t a hundred new te s ts o f thin, cylinders in axial com pression and bending, w hich, togeth er w ith num erous te s ts by L u n d q u ist,3 form th e experim ental evidence for th e conclusions arrived at.

A new theory for the buckling of thin cylinders under axial compression and ending

A review of recent researches on FGM cylindrical structures under coupled physical interactions, 2000–2015

Summary
The paper deals with two main advantages in the analysis of slender elastic structures both achieved through the mixed (stress and displacement) format with respect to the more commonly used displacement one: (i) the smaller error in the extrapolations usually employed in the solution strategies of nonlinear problems and (ii) the lower polynomial dependence of the problem equations on the finite element degrees of freedom when solid finite elements are used. The smaller extrapolation error produces a lower number of iterations and larger step length in path-following analysis and a greater accuracy in Koiter asymptotic method. To focus on the origin of the phenomenon, the two formats are derived for the same finite element interpolation. The reduced polynomial dependence improves the Koiter asymptotic strategy in terms of both computational efficiency, accuracy and simplicity. Copyright © 2016 John Wiley & Sons, Ltd.

Advantages of the mixed format in geometrically nonlinear analysis of beams and shells using solid finite elements

Decision tree-based machine learning to optimize the laminate stacking of composite cylinders for maximum buckling load and minimum imperfection sensitivity

The design of lightweight structures is often driven by buckling phenomena. Increasing demands for fuel-efficient aircraft structures makes post-buckled designs attractive from a structural weight perspective. However, the need for reliable and efficient design tools that accurately model the emerging nonlinear post-buckled landscape, potentially one containing multiple branches, remains. With this aim, a previously derived flat shell element, MISS-4, is extended to the geometrically nonlinear analysis of variable-angle tow (VAT) composite plates using Koiter's asymptotic approach. The curvilinear fiber paths in VAT lamina open the design space for tailoring the buckling and post-buckling capability of plates and shells. A finite element implementation of Koiter's asymptotic approach allows the pre-critical and post-critical behavior of slender elastic structures to be evaluated in a computationally efficient manner. Its implementation uses a fourth-order expansion of the strain energy, and requires both the structural modeling and finite element discretization procedures to be, at least, of fourth order. The corotational approach adopted in the MISS-4 element readily fulfills this requirement by starting from a linear finite element discretization. VAT plates with prismatic fiber variations and different loading conditions are analyzed using the MISS-4 element and numerical results of the post-buckling paths are presented. The computed equilibrium paths are compared to benchmark results using the commercial finite element package ABAQUS, and strong asymptotic solutions of the differential equations. The results document the good accuracy and reliability of the proposed modeling approach, and also highlight the importance of multi-modal analysis when multiple buckling modes coincide as is the case in long plates, shells and other optimized thin-walled structures.

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Ke Liang

Papers

The Koiter-Newton approach using von Kármán kinematics for buckling analyses of imperfection sensitive structures

A Koiter‐Newton approach for nonlinear structural analysis

Stacking sequence influence on imperfection sensitivity of cylindrical composite shells under axial compression

Co-rotational finite element formulation used in the Koiter–Newton method for nonlinear buckling analyses

Nonlinear buckling analysis of the conical and cylindrical shells using the SGL strain based reduced order model and the PHC method