Lay-up Optimization of Laminated Composites Using a Modified Branch and Bound Method

Giacomo Canale1, Paul M. Weaver2, Felice Rubino3, *, Angelo Maligno3, Roberto Citarella4
1 Rolls-Royce plc, Moor Lane, Derby, UK
2 Department of Aerospace Engineering, University of Bristol, BS8 1TR, UK
3 Institute for Innovation in Sustainable Engineering, University of Derby, Quaker Way, DerbyDE1 3EE, UK
4 Department of Industrial Engineering, University of Salerno, via Giovanni Paolo II 132, 84084 Fisciano (SA), Italy

© 2018 Canale et al.

open-access license: This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International Public License (CC-BY 4.0), a copy of which is available at: ( This license permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

* Address correspondence to this author at the Institute for Innovation in Sustainable Engineering, University of Derby, Quaker Way, Derby DE1 3EE, UK, Tel: 01332593554; E-mails:;



Composite materials are widely used in the aerospace, marine and automotive industries. One of their main advantages is that their stacking sequence can be tailored to maximise/minimise a specific structural performance. Efficient and non-computational-expensive algorithms are always needed to find the optimum stacking sequence of a composite laminate whose thickness is either to be minimised or may be kept constant (i.e. the thickness and the plies orientation percentages are pre-determined; the problem of the optimisation is therefore permutational).


A modified branch and bound algorithm is proposed here and used to determine the stacking sequence for single and multi-objective optimisation problems. Laminate thickness and orientation percentages are either variables or determined a priori (the optimisation problem is therefore permutational). Computational time is drastically reduced when compared with other meta-heuristic techniques.


The proposed method is a branch and bound algorithm, modified from the original work proposed by Kim and Hwang [10]. The main novelty is the starting point of the optimisation sequence: a laminate formed by “Ideal” layers, described in this paper.


The modified branch and bound has been first tested with a laminate having fixed thickness and a fixed percentage of layer orientation. Three different problems have been investigated: maximisation of natural frequencies, minimisation of tip deflection and maximisation of buckling critical load. The algorithm has been also tested, secondly, for a problem of weight minimisation subjected to buckling and strength constraints.


The MBB has been shown to give good fidelity and significant computational advantages compared with a GA. Despite the simplicity of the structures in the numerical examples, it is anticipated that the MBB can be used to determine lay-ups in multi-part structures. The method was used to determine stacking sequences for several problems. The modified branch and bound method was shown to determine good laminate designs and offer significant efficiency savings.

A “Good Design” is here defined as a solution producing “Near Global Optima” fitness values by minimising the computational effort. It was shown that for a single objective without ply competition, global optima were obtained.

Keywords: Composite laminates, Stacking sequence optimisation, permutative GA, Buckling critical load, Constrained optimisation, Lamination parameters.