Composite Materials Research Progress

Optimization of Laminated Composite Structures… 71
problem including integer variables, genetic algorithms have been used (Haftka and Gurdal,
1992, Le Riche and Haftka, 1993). The topology optimization formulation of Figure 4.4 was
used by Beckers (1999) and (Stegmann and Lund, 2005) to solve this problem with discrete
and continuous design variables, respectively. Another approach, still based on the discrete
character of the problem, is proposed by Carpentier et al. (2006). It consists in using a lay-up
table defined based on buckling, geometric and industrial rules considerations. This table,
which satisfies the ply drop-off continuity restrictions is determined numerically. Once it is
obtained a given laminate total thickness corresponds to a stacking sequence (via a column of
the table). The optimization process then consists in optimizing the local thickness of a set of
contiguous laminates defining the structure. Each laminate has equivalent homogenized
properties with 0, ±45 and 90° plies. Based on the lay-up table, the stacking sequence is
therefore known everywhere in the structure for different local optimal thicknesses and the
composite material can be drapped.
Figure 4.7. Illustration of a lay-up table for 0, ±45 and 90° plies.
5. Problems Solved in the Literature
5.1. Structural Responses
When designing laminated composite structures the functions entering the optimization
problem (2.1) are classically the stiffness, the vibration frequencies, the structural stability
and the plies’ strength. (see Abrate, 1994, for a detailed review of the literature). It is
interesting to note that for orthotropic laminates maximizing the stiffness, the frequency or
the first buckling load will provide the same solution (Pedersen, 1987 and Grenestedt, 1990).
On top of that, it should be noted that optimizing a laminated structure against plies strength
or stiffness will result in different designs. It results that the local (stress) effects are very
important in the optimal design of composite structures (Tauchert and Adibhatla, 1985,
Fukunaga and Sekine, 1993, and Hammer, 1997).