Results on best theories for metallic and laminated shells including Layer-Wise models E. Carrera a,b,⇑ , M. Cinefra a , A. Lamberti a , M. Petrolo b a Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy b School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083, Australia article info Article history: Available online 18 February 2015 Keywords: Carrera Unified Formulation Shell Best Theory Diagram Advanced models abstract This paper deals with Best Theory Diagrams (BTDs) for metallic and laminated shells. The BTD is a curve that is defined over a 2D reference frame in which the horizontal axis indicates the error of a shell model with respect to a reference solution whereas the vertical axis indicates the number of displacement vari- ables of the model. The best reduced model is a refined model that offers the lowest possible error for a given number of variables. The relevant terms of a model are detected by means of the axiomatic/asymp- totic method (AAM), and the error is related to a given variable with respect to an exact or quasi-exact solution. In this work, a genetic algorithm has been used to obtain the BTD. The Carrera Unified Formulation (CUF) has been employed to build the refined models. The CUF makes it possible to generate automatically, and in a unified manner, any plate or shell models. Equivalent Single Layer (ESL) and Layer Wise (LW) refined models have been considered. The governing equations for shells have been obtained through the Principle of Virtual Displacements (PVD), and Navier-type closed form solutions have been considered. BTDs have been constructed by considering the influence of several parameters, such as vari- ous geometries, material properties, layouts, different displacement/stress components and loadings. The accuracies of some well-known theories have been evaluated and compared with BTD reduced models. The results suggest that, since the BTD depends on the problem characteristics to a great extent, the sys- tematic adoption of the CUF and the AAM can be considered as a powerful tool to evaluate the accuracy of any structural theory against a reference solution for any structural problem. Ó 2015 Elsevier Ltd. All rights reserved. 1. Introduction Laminated composite and metallic shells are widely employed in several structural engineering applications, and many mathe- matical models have been developed over the last decades for the structural analysis of plates and shells. The solution of the 3D elasticity equations can be computationally prohibitive and valid only for a few geometries, material characteristics and boundary conditions. Computationally cheaper 2D structural models are commonly used to analyse shells and plates. The first model that was developed was the Kirchoff–Love ([1,2]). According to this model, transverse shear and normal strains are assumed to be negligible with respect to the other stress and strain components. An extension of this model to multilayered structures is referred to as the Classical Lamination Theory (CLT). Further details on shell theories can be found in [3]. The inclusion of shear effects can be carried out according to the Reissner–Mindlin model [4,5] that leads to the First Order Shear Deformation Theory (FSDT). Further refinements of the FSDT can be obtained through the Vlasov model and the Reddy–Vlasov model [6,7] to account for the homogeneous conditions for the transverse shear stresses at the top and bottom shell/plate surfaces. A refined model that accounts for both the transverse shear and normal stress effects, i.e. that fulfills Koiter’s recommendation [8], was developed by Hildebrand, Reissner and Thomas [9]. Other sig- nificant contributions on laminated shell models can be found in [10–17]. The number of unknown variables in the theories that were mentioned above is independent of the number of layers. These theories are commonly referred to as Equivalent Single Layer mod- els (ESL). An alternative method is the Layer-Wise (LW) approach [18–23] in which each layer is seen as an independent plate and http://dx.doi.org/10.1016/j.compstruct.2015.02.027 0263-8223/Ó 2015 Elsevier Ltd. All rights reserved. ⇑ Corresponding author at: Department of Mechanical and Aerospace Engineer- ing, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy. Tel.: +39 011 090 6871; fax: +39 011 090 6899. E-mail addresses: erasmo.carrera@polito.it (E. Carrera), maria.cinefra@polito.it (M. Cinefra), alessandro.lamberti@polito.it (A. Lamberti), marco.petrolo@rmit.edu. au (M. Petrolo). URL: https://www.mul2.com (E. Carrera). Composite Structures 126 (2015) 285–298 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct