Minimum cost design of hybrid cross-ply cylinders with uncertain material properties subject to external pressure Isaac Sfiso Radebe a , Sarp Adali b,n a Department of Mechanical Engineering, Durban University of Technology, Durban, South Africa b Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban, 4041, South Africa article info Article history: Received 2 January 2014 Accepted 22 June 2014 Available online 18 July 2014 Keywords: Hybrid design Material uncertainty Cross-ply cylinder External pressure Minimum cost Sensitivity abstract Minimum cost design of hybrid cross-ply cylinders is presented which employ high-stiffness and expensive materials in the surface layers and the low-stiffness and inexpensive layers in the middle layers to combine the advantages of the two materials. Hybrid construction takes advantage of the sandwich effect whereby most of the load is carried by the surface layers. The cylinder is subject to external pressure with the material properties displaying uncertain-but-bounded variations around their nominal values. For a given external pressure, the material cost is minimized by minimizing the thickness of the surface layers. Analysis to determine the worst-case combination of material uncertainties makes use of convex modeling to compute the least favorable solution defined here as the minimum buckling pressure. The minimum cost designs are investigated for various problem parameters such as the wall thickness and the level of uncertainty. The relative sensitivities of the buckling pressure to material properties are also studied by defining sensitivity indices. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Hybridization in composite structures can be used effectively to reduce the material costs by using a more expensive high-stiffness material in the outer layers and a less expensive low-stiffness material in the inner layers. A typical case would be to use a combination of CFRP (Carbon Fiber Reinforced Plastics) and GFRP (Glass Fiber Reinforced Plastics) noting that CFRP is about an order of magnitude more expensive than GFRP. A hybrid layup makes a better use of the expensive material by employing it in the outer layers where it can contribute most to the stiffness of the component (Rahul et al., 2006; Aiello and Ombres, 2007; Abachizadeh and Tahani, 2009; Karakaya and Soykasap, 2011; Montagnier and Hochard, 2013). The cost of the component can be further reduced by minimizing the thickness of the outer layers which is the objective of the present study for a hybrid cylinder subject to external pressure. Previous studies on the optimization of hybrid laminates include Adali and Duffy (1992, 1993), Adali et al. (1995), Adali and Verijenko (1997, 2001), and Aiello and Ombres (1996). Noting that variations in the elastic constants of composite materials are not unusual, the design of the composite cylinder is given subject to uncertainties in the material properties which may arise for a number of reasons such as manufacturing toler- ances, fiber misalignment, defects and voids in the fiber/matrix composition, and imperfect bonding between fibers and the matrix. In such situations material properties may be treated as uncertain variables, thereby taking the variations in the elastic constants into account in the design. For a robust design capable of operating under material uncertainty, the design should be able to take the so-called “worst-case” into account. Studies on robust designs include Du and Chen (2002), Lee and Park (2001), Lee et al. (2013), McDowell et al. (2010), Parkinson (2000) and Sandgren and Cameron (2002). Material uncertainties can be studied by employing prob- abilistic and stochastic models which require statistical data on problem parameters such as probability density functions of random variables. In many cases accurate estimates of prob- ability distributions are fairly difficult to obtain due to lack of sufficient information. On the other hand, information on the bounds on elastic constants could be available leading to bounded-but-uncertain properties. In such cases the problem can be analyzed using convex models of uncertainty which provide an effective alternative to probabilistic models. In the non-probabilistic modeling of uncertainties, variations about the average values of the uncertain variables can be expressed in terms of ellipsoidal sets which lead to the computation of the least favorable solution (worst-case) and consequently to a Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/oceaneng Ocean Engineering http://dx.doi.org/10.1016/j.oceaneng.2014.06.010 0029-8018/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. Tel.: þ27 31 2603203; fax: þ27 31 2603217. E-mail addresses: sfisor@dut.ac.za (I. Sfiso Radebe), adali@ukzn.ac.za (S. Adali). Ocean Engineering 88 (2014) 310–317