**Acknowledgements**

better results at the previous stage of the study (see **Figure 9**), where Roman numeral indi-

**Figure 10** illustrates two examples of objective and other responses evolution during optimization process. These plots demonstrate very fast convergence to the quasi-optimal solutions for both optimized lay-up stacking. These solutions are not global optimum because they

The final features of all optimized "candidate" structures are present in **Figure 11**. All optimized lay-ups were ranked according to calculated responses. The optimized lay-ups (I *φin* <sup>=</sup> <sup>30</sup><sup>0</sup>

) provide minimal values of objective, but (II *φin* <sup>=</sup> <sup>30</sup><sup>0</sup>

its flexural rigidities are downscale. These considerations substantiate obligatoriness of multiobjective optimization at the design of load-carrying multilayered composite structures with orthotropic symmetry of the materials. The final decision can be made taking into account some constraints and requirements, for example, complexity of manufacturing, weight of ready structure, the natural vibration modes and eigenfrequencies and importance of a particular rigidity for the operability of the structure. An additional study of the strength of the composite layers according to the Tsai-Wu criterion [25] should be carried out at critical loads. Meanwhile, the used approach requires many tedious calculations, but allows us to obtain the visual substantial representations about chosen composite lay-ups for optimizing structures, and can be expanded to the composite structures with arbitrary number of arbitrarily

This chapter studies a problem of lay-up optimization for a cantilevered long tube-like composite structure with varied cross-section that is manufactured by winding of glass fiber unidirectional tape. The optimized composite structure is the tube-like cantilever slender beam experiencing distributed bending and torsion forces. The multilayered composite material assumed and modeled as a single phase anisotropic elastic homogeneous continuum. We determine the elastic properties of laminates, which used in the modeled tube, taking as input data the mechanical properties of reinforcing fibers and epoxy resin to determine initially the elastic properties of the unidirectional lamina. For each accepted lay-up scheme and unidirectional prepreg orientation of the symmetric balanced laminate formation, the elastic moduli were determined independently by two methods: by the finite element method and on the

The first stage of used optimization approach is based on the analysis of the angular distribution of all engineering constants of laminates. This analysis allows us to choose the small enough set of "candidate" lay-ups, which should be used at the modeling of the mechanical response of the studied structure at three different load scenarios. The higher level "candidates" were appointed for the final dynamic test, which includes applying full load to the selected structures and gives us the possibility to make the expert decision about final choice

; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup>

cates the type of lay-up (see **Figure 4**) and the value of angle is its start value.

characterized by the different values of all responses.

and (V *ψ* = 30<sup>0</sup>

50 Optimum Composite Structures

; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup>

oriented unidirectional layers.

base of the classical laminates theory.

**7. Conclusion**

mal total strain energy. Lay-up (V *ψ* = 45<sup>0</sup>

This work is implemented in the framework of the state mission for the South Scientific Center of Russian Academy of Science, project No. AAAA-A16-116012610052-3 and it was supported by the Russian Foundation for the Basic Research (Grant 15-08-99849 and Russian-Taiwanese project 16-58-52013 MNT\_a).
