Surrogate-based optimization of stiffened shells including load-carrying capacity and imperfection sensitivity Peng Hao a , Bo Wang a,n , Gang Li a , Kuo Tian a , Kaifan Du a , Xiaojun Wang b , Xiaohan Tang b a State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116023, PR China b Beijing Institute of Astronautical Systems Engineering, Beijing 100076, PR China article info Article history: Received 19 May 2012 Received in revised form 8 June 2013 Accepted 8 June 2013 Keywords: Stiffened shell Load-carrying capacity Imperfection sensitivity Hyperbolic generatrix shape Surrogate-based optimization abstract Stiffened shells are very sensitive to initial imperfections, especially for geometrical imperfections. In this study, based on a 3-m-diameter orthogrid stiffened shell, post-buckling and imperfection sensitivity analyses are performed by use of explicit dynamic method. Result indicates that stiffened shell with outward hyperbolic generatrix shape shows a lower imperfection sensitivity as compared to the cylindrical one. Then, a framework of surrogate-based optimization including load-carrying capacity and imperfection sensitivity for stiffened shells is introduced, and the illustrative example demonstrates that the proposed framework has the potential to provide robust optimum design for realistic stiffened shells. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction Stiffened shells constitute one of the most common compo- nents in aerospace structures, such as interstages, fuel tanks of launch vehicle and missile, as shown in Fig. 1. For these applica- tions, axial inertia load caused by acceleration during ascent is the main design condition of stiffened shells [1]. Therefore, global buckling is one of the main failure modes, and collapse load is the governing assessment for this type of thin-walled structures, which is specified by that point of the load vs. end-shortening curve where a sharp decrease occurs and thus limiting the load- carrying capacity [2]. In the past, stiffened shells were usually made of metallic material, such as aluminum alloys [3]. With the advent of new composite material in aerospace applications, stiffened shells made of composite material were developed rapidly in the last few decades [4]. However, due to the deep insight into the metallic behavior, metallic stiffened shells are even potentially able to work in the post-buckling field. From the point- of-view of reliability and economy, metallic stiffened shells are considered as an efficient design choice for application, which are still used in the interstages and fuel tanks of CZ, Titan, Atlas and Delta launch vehicles [5]. However, initial geometrical imperfections were proved to reduce the load-carrying capacity of stiffened shell drastically compared to that of the geometrically perfect structure [6]. With the advent of China's newly developing and future heavy-lift launch vehicles, the diameter of core stage increases from 3.35 m to 5 m, then to 9 m, and the influence of R/t (ratio of shell radius to equivalent wall thickness) becomes an important issue [7]. Obviously, launch vehicle would be overweight eventually with an unchanged R/t, while with regard to a largely increased R/t, the effect of initial imperfections on the load-carrying capacity of stiffened shell would become more significant, which results in an extremely high imperfection sensitivity. When investigating the effect of initial imperfections, it has been standard practice to predict the collapse load of geometri- cally perfect structure and then to reduce this collapse load by a knockdown factor (KDF) [8,9]. Based on a semi-empirical method, KDFs of stiffened shells with varying R/t were proposed by NASA SP-8007 [10]. Also, in China's current design code, KDFs for tank designs were almost derived from the early experimental results. However, with the development of manufacturing technology and material system, these KDFs were proved to be overly conservative [7,9,11–13]. The excessive conservatism usually corresponds to an increase of structural weight, which is particularly important in the development of large-diameter launch vehicle. The determi- nation of KDFs for large-diameter stiffened shells cannot depend entirely on buckling experiments due to the expensive cost. Furthermore, accounting for imperfections requires more detailed Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/tws Thin-Walled Structures 0263-8231/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2013.06.004 n Corresponding author. Tel.: +86 411 84706382; fax: +86 411 84708390. E-mail address: wangbo@dlut.edu.cn (B. Wang). Thin-Walled Structures 72 (2013) 164–174