Resolution of the Twentieth Century Conundrum in Elastic Stability

The following sections are included:

• Dedication
• Preface
• Acknowledgements
• Contents

In this chapter we describe how the author got involved with the imperfection sensitivity concept, specifically how he got to be excited by the paradoxical result reported by Fraser and Budiansky (1969). Then rather brief historical excursion is made into the history of elastic stability that originated by the experiments of Pieter van Musschenbroek (1692–1761) and the theory of Leonhard Euler (1707–1783).

This chapter deals with the probabilistic resolution of the paradox created by the work of Fraser and Budiansky (1965) and other studies performed at Harvard. Bolotin's (1962) contribution is described in some detail. In it Bolotin utilized a single-term Bubnov–Galerkin method to derive reliability of the compressed structure. It is shown that multimode analysis was needed to find realistic reliability of structures.

This chapter investigates the combined effect of randomness in initial geometric imperfections and the applied loading on the reliability of axially compressed cylindrical shells. In order to gain an insight, we consider simplest possible case when both the initial imperfections and the applied loads are uniformly distributed. It is shown that hybrid randomness may either increase or decrease the reliability of the shell if the latter is treated as experiencing the sole randomness in initial imperfections.

This chapter deals with an alternative to probabilistic modeling. As if the probabilistic resolution of the shell stability was not sufficient, the author was blessed by yet another methodology that is not based on probabilistic modeling. The initial imperfection is that Fourier coefficients were treated as being bounded by an ellipsoidal set. Then the minimum buckling load is determined in an analytic form when the numerical code is available for buckling load evaluation. Thus, semianalytic and seminumeric methodology was proposed to bridge the gap between theory and practice. This methodology is known as the convex modeling of uncertainty (Ben-Haim and Elishakoff, 1990), guaranteed approach (Chernousko, 1990), worst-case design (Hlavaček et al., 2004), info-gap modeling (Ben-Haim, 2006), or max-min analysis (Wald, 1945, 1950; Sniedovich, 2007, 2008, 2010, 2012).

This chapter investigates the combined effect of three quantities, namely, randomness in initial geometric imperfections, thickness variations, and the applied loading on the reliability of axially compressed cylindrical shells. Some cooperative work with Warner T. Koiter and James H. Starnes, Jr. are first described. In order to gain insight we consider the simplest possible case when the initial imperfections, thickness variations, and the applied loads are uniformly distributed. It is shown that hybrid randomness may increase or decrease the reliability of the shell in contrast to the case when the latter is treated as experiencing the sole randomness in initial imperfections.

This chapter deals with hybrid optimization and antioptimization of the buckling load of composite cylindrical shells. The methodology, which has been developed in the previous works, is applied to a set of cylindrical composite shells, tested at German Aerospace Center (DLR). Furthermore, the existing approach is enhanced to fit within the design optimization scheme. The shells possess traditional imperfections in the form of Fourier series coefficients of their initial imperfection profile. Additionally, two nontraditional imperfections are included in the analysis. The available experimental data is enclosed by either 11-dimensional hyper-rectangle or hyper-ellipsoid. The minimum buckling load of the ensemble of such shells is determined by the antioptimization procedure. Then, this minimum load is maximized by varying the laminate angle. It is shown that the proposed method is a viable and relatively simple alternative to probabilistic approaches and successfully supplements them. It is shown that the proposed method is a successful supplement to probabilistic methods and the deterministic single buckle approach (SBA), since it is deterministic in nature and thus could appeal the engineers and investigators alike, and it takes into account the actual scatter of input data.

This chapter is devoted to various topics that appear to be outside of the main thrust of this book, namely of resolution of the conundrum created by the “curse of imperfection sensitivity.” Some cooperative work with Warner T. Koiter and James H. Starnes, Jr., are first described. Then we follow with a brief story of establishment of the ASME Medal in honor of Professor Koiter. Finally, the latest works are briefly reviewed along with the discussion of the ever-present question of priority.

The following section is included:

• Appendix A: Analytical Analysis of the Nonsymmetric Version of the Budiansky–Hutchinson Model
• Appendix B: Elastic Stability: From Musschenbroek & Euler to Koiter–There Was None Like Koiter
• Appendix C:
  • Data Basis
  • Minimum Volume Enclosing Hyper-Rectangle
  • Minimum Volume Enclosing Hyper-Ellipsoid
  • Numerical Determination of Buckling Load
  • Numerical Derivatives of Buckling Load
  • Experimentally Determined Buckling Loads
• References
• Author Index
• Subject Index