Some Graphical Interpretations of Melan’s Theorem for Shakedown Design

Bree Interaction Diagrams have long been one of the major visual design guides for employing and evaluating shakedown in engineering applications. These diagrams provide representations of the realms in which elastoplastic behaviors, including shakedown, are found for a material and structure under variable loads. The creation of these diagrams often relies upon some combination of upper or lower bound shakedown theorems and numerical shakedown limit determination techniques. Part of the utility of these diagrams is that, for a given structure and loading conditions, inspecting them is sufficient to determine whether shakedown will occur or not. The diagrams cannot however, give the designer insight into how the conditions for shakedown are met. This chapter presents some graphical interpretations of one of the common methods for shakedown determination: the use of Melan’s Lower Bound Theorem. The intent is to provide additional insight for designers regarding how shakedown conditions are satisfied. In this way, additional directions for modifying designs to recover shakedown behavior may also be identified. Revisiting this well-established theorem from a graphical and pedagogical approach, also provides a foundation for interdisciplinary innovation. The particular focus is on simple examples that highlight ways in which Melan’s theorem may be applied to shakedown design problems. N. Vermaak (✉) ⋅ M. Boissier Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA e-mail: [email protected] M. Boissier e-mail: [email protected] L. Valdevit Department of Mechanical and Aerospace Engineering, University of California, Irvine, CA 92697, USA e-mail: [email protected] R.M. McMeeking Mechanical Engineering Department, University of California, Santa Barbara, CA 93106, USA e-mail: [email protected] © Springer International Publishing AG 2018 O. Barrera et al. (eds.), Advances in Direct Methods for Materials and Structures, DOI 10.1007/978-3-319-59810-9_11 179 180 N. Vermaak et al.