Review Probabilistic resolution of the twentieth century conundrum in elastic stability Isaac Elishakoff Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, United States article info Article history: Received 23 January 2012 Received in revised form 21 March 2012 Accepted 4 April 2012 Available online 2 June 2012 Keywords: Shell buckling Imperfection sensitivity Probabilistic approach Monte Carlo method First-order second moment method abstract This paper overviews the efforts that led to resolution of the 20th century conundrum in elastic stability of shells. In particular, the dramatic disagreement between theoretical and experimental results and the subsequent introduction of the empirical knockdown factor, is discussed in detail. The mismatch between theory and experiment was qualitatively explained by Warner Tjrdus Koiter, in his now-famous thesis, as well as in the paper by Lloyd H. Donnell and C.C. Wan. However, these studies did not offer means for rigorous, theoretical derivation of the knockdown factor for the shells with generic imperfection patterns encountered in practice. Numerous attempts to resolve the conundrum via deterministic theoretical, experimental and probabilistic analyses remained unsuccessful. The concendrum consists in two facts. On one hand, it consists of impossibility of using of hundreds and perhaps thousands of deterministic studies in predicting the rigorous knockdown factors. On the other hand, it lies in the fact that Wynstone Barrie Fraser and Bernard Budiansky (1969) [157] and numerous other investigators, although recognized the need to utilize probabilistic approach to resolve the above concendrum asserted that the buckling load of stochastic structures was a deterministic quantity. Some investigators suggested to use that result as the design load. In 1979, this author lucked out on reliability-based theoretical means for derivation of the knockdown factor and its judicious allocation. & 2012 Elsevier Ltd. All rights reserved. 1. How I got involved with the imperfect world of imperfection sensitivity In the spring semester of 1977, Professor Bernard Budiansky of Harvard University was expected at the Department of Aeronau- tical Engineering at the Technion—Israel Institute of Technology, as part of his sabbatical. Professor Joseph Singer, head of the Structures Group, asked me to read some of Professor Budiansky’s works, interested in some of them and discuss my ideas with him. I spent considerable time in the library on this assignment and eventually found the paper [157] on imperfection sensitivity of an infinite column resting on a nonlinear elastic foundation. What attracted me most in it was the conclusion from which the following excerpt is reproduced word for word, to avoid any impression of selective quoting: ‘‘We consider the buckling of an ensemble of infinitely long columns, with initial deflection, resting on nonlinear elastic foundations. The initial deflections are assumed to be Gaussian, stationary random functions of known autocorrelations and the problem is solved by the method of equivalent linearization. We find that each column in the ensemble has the same buckling load that depends only on autocorrelation of the initial deflection function [boldface mine—I.E.]. Results are presented for columns whose initial deflection functions have an exponential—cosine autocorrelation.’’ My first reaction was: ‘‘Wow!’’, since I could not possibly visualize all columns having the same buckling load even though each of them necessarily possess a different imperfection profile! The authors stated further: ‘‘y we show that each column in the ensemble has the same buckling load which depends only on the autocorrelation of the initial deflection and not on a particular realization of one of these functions.’’ This brought some relief: at least, the buckling load depends on the autocorrelation function of the initial imperfections, if not on a particular realization of them. The authors themselves had some reservations. Choosing a exponential-cosine autocorrelation function, and remarked: ‘‘Whether or not structural imperfection can be validly repre- sented by this autocorrelation function and spectral density Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures 0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.04.002 E-mail address: elishako@fau.edu Thin-Walled Structures 59 (2012) 35–57