Contents lists available at ScienceDirect Thin-Walled Structures journal homepage: www.elsevier.com/locate/tws Full length article Rapid identification of pre-buckling states: A case of cylindrical shell Natalia I. Obodan, Victor J. Adlucky, Vasilii A. Gromov ⁎ Oles Honchar Dnepropetrovsk National University, Gagarina av., 72, Dnepropetrovsk 49010, Ukraine ARTICLE INFO Keywords: Rapid assessment Buckling of thin-walled structures Non-linear boundary problem for von Karman equations The inverse bifurcation problem Identification of pre-buckling state ABSTRACT The problem to identify pre-buckling states for thin-walled shell corresponds to the problem to identify pre- bifurcation solutions (the inverse bifurcation problem) for von Karman equations that govern the structure. Typical solution sequences similar to those of post-bifurcation solutions observed along the bifurcation paths of the nonlinear boundary problem for von Karman equations are extracted to serve as precursors of bifurcation (tools to solve the problem). The method allows one to divide all operations required to solve the problem under study into two non-equal parts. The most time-consuming part (to trace bifurcation paths and cluster the re- spective solution) is performed off-line, while the part of the algorithm that is carried out on-line (the identi- fication algorithm) requires a relatively small number of arithmetic operations. This allows development of the efficient system of rapid identification of pre-buckling states. 1. Introduction The bifurcation theory employed to investigate a thin-shell structure makes it possible to consider both direct and inverse bifurcation pro- blems. The direct bifurcation problem, the most conventional one, implies that one estimates buckling (bifurcation) loads for various ex- ternal loadings, boundary conditions and so on. As far as dependences of buckling loads on problem parameters are strongly non-monotonous it is necessary (in order to solve the problem) to trace all its bifurcation paths and ascertain its complete bifurcation set (for example, [1]). The term ‘inverse bifurcation problem’ is conventionally used in two distinct senses. The first statement suggests that one seeks for such values of problem parameters that the respective buckling load satisfies certain demands – by way of illustration we may point to the problems to find the worst initial imperfection or the infavourable load [2] (that is the imperfection/load corresponding to the lowest possible buckling load). Papers [3–5] deal with an approach (to solve this problem) based upon specific perturbation functions with single or multiple localized dents (dimple-shape imperfections); to find the worst imperfection, authors propose to find minimum buckling load among those corre- sponding the perturbations typical for experimental studies [3,5–7]; another approach employs nonlinear buckling modes as perturbations [5]; Schenk and Schuëller [8] propose to analyze statistically experi- mental post-buckling shapes [7]. In our view, this problem can be solved by tracing all its post- buckling (bifurcation) paths, since nonlinear buckling modes associated with the lowest buckling loads correspond to (secondary and tertiary) bifurcation paths with relatively small low boundaries of existence domains [1]. The deformed shapes corresponding to these paths, usually, a single dent, a group of dents, or a ‘belt’ of dents [1], are similar to those employed in single or multiple perturbation load ap- proach [3,5,6]. The second statement of the inverse bifurcation problem implies that one attempts to predict buckling (or to put it differently, to identify pre-buckling state) provided a sequence of deformed shapes is ob- served. This statement is a subject of much current interest as far as it manifests itself in actual practice as the problem of rapid sustainability assessment of a damaged thin-walled structure. On the other hand, robust design [9], which is growing more popular in engineering, im- plies that one is able rapidly identify every possible buckling state. The present paper concerns with a novel approach to predict thin- walled shell buckling that it is the second statement of the inverse bi- furcation problem for thin-shell structures. The approach utilizes knowledge about post-buckling (bifurcation) paths traced for the re- spective static nonlinear elastic problems – namely, typical sequences of solutions (deformed shapes) associated with post-buckling bifurcation paths serve as bifurcation precursors; the observed sequences of de- formed shapes may correspond to processes unfolding in time. It is worth stressing that in the frameworks of dynamical analysis for this type of partial differential equations (PDEs) it is possible to solve the inverse bifurcation problem for a particular right-hand member (load function) only; while the proposed approach can be employed to identify pre-buckling state for any right-hand member. Rapid assessment is associated with the concept of progressive https://doi.org/10.1016/j.tws.2017.12.034 Received 3 August 2017; Received in revised form 1 December 2017; Accepted 27 December 2017 ⁎ Corresponding author. E-mail addresses: obodann@gmail.com (N.I. Obodan), adluckyv@rambler.ru (V.J. Adlucky), stroller@rambler.ru (V.A. Gromov). Thin-Walled Structures 124 (2018) 449–457 0263-8231/ © 2017 Elsevier Ltd. All rights reserved. T