**6. Pseudo-optimal choice of the preferred lay-ups**

In the final stage of investigation, the optimizing structure made of the preferred lay-ups "candidates" has been loaded simultaneously by the systems of three forces: bending by the

**Figure 9.** Dependencies of maximum deflections and torsion angles of the loaded beams.

values of effective elastic moduli for the laminate with given lay-up. Note that according to the homogenization hypothesis this laminate is considered as the solid body with

In the next stage of investigation, the angular dependencies of the effective moduli were investigated. In order to avoid the significant reduction of the structural rigidity in some directions, these dependencies should not have sharp drops, corresponding to these directions. Some examples of such angular dependencies are shown in **Figure 8**. These dependencies have been used for the preliminary selections of the "candidates" lay-ups, which have been further used

The mechanical testing of the structure with different lay-ups and ply stacking angles has been implemented using FE model (see **Figures 1**–**3**) by using three abovementioned types of static bending and torsion loading. These loads were applied separately; for each load

deflections max(*vb*), max(*wb*) and torsion angles *θ* on the free face of the beam tip were calculated. At the solving of these subtasks a linear parametric solver was used. Results of these testing were used for the refinement of the "candidates," that is, select lay-ups that provide the minimum strain energy, bending and torsion deformations. Our study demonstrated a strong coupling of total strain energy of deformed structure with the values of maximum beam deflection (bending load cases) and maximum torsion angles (twisting load case). Moreover, two very interesting and important facts were established. First, we found that sensitivity of both maximum and averaged von Mises stress to the structural symmetry of orthotropic material is noticeably less comparing to the total elastic strain energy. Thereby we do not give here any von Mises dependencies. Second result is the practically independent response of the structure on the twisting load of the lay-ups II, III and IV. The reason for this is the practical identity of the shear modulus of these lay-up schemes at the same

The calculated dependencies of maximum deflections and torsion angles of the loaded beams on the lay-up parameters are present in **Figure 9**. These results together with the angular dependencies of elastic moduli that are partially presented in **Figure 8** allowed to select eight "candidates" for further optimization. These candidates had to have acceptable values for the

In the final stage of investigation, the optimizing structure made of the preferred lay-ups "candidates" has been loaded simultaneously by the systems of three forces: bending by the

*tot*, the maximum and averaged von Mises stress, two end

to test them in mechanical testing of the optimized slender structure.

**5. Mechanical testing of structure with different laminate lay-ups**

uniform structure.

46 Optimum Composite Structures

scenario the total strain energy *Eel*

lamina angular orientation (see **Figure 7b**–**d**).

total strain energy and rigidity under all loading scenarios.

**6. Pseudo-optimal choice of the preferred lay-ups**

distributed forces oriented along y and z axis and twisting torque applied to the external surface and given by Eqs. (1) and (2). The studied responses included total strain energy, the maximum bending deflections and objective, which was accepted in the normalized dimensionless form.

$$Obj = \mathcal{O}/\mathfrak{G} + \max\{\upsilon\_i\} + \max\{\upsilon\_i\}/1.5\tag{5}$$

Where, *θ* is the torsion angle and *vb*

function determined by Eq. (5).

*Eei*

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

tion procedures, we used such "candidates": (I *φin* <sup>=</sup> <sup>30</sup><sup>0</sup>

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

); (V *ψ* = 45<sup>0</sup>

imposed.

(V *ψ* = 30<sup>0</sup>

the preferred lay-ups).

, *wb*

tively. Naturally, definition of such objective function and weights for its components has a significant element of arbitrariness. Therefore, the total strain energy, the beam deflections and torsion angle were also monitored during optimization process along with the objective

For the definiteness of the optimization problem statement the following constraints were

At the optimization workflow, the forward structural mechanics problem has been called by the built-in gradient-free Nelder-Mead optimizer. In order to start the iterative optimiza-

**Figure 11.** Comparison among the best lay-ups of the optimized structure (stems with the filled symbols correspond to

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

*tot* ≤ 160*J* ; *θ* ≤ 4.50 ;

); (V *ψ* = 60<sup>0</sup>

are the beam tip deflections along the *y* and *z*, respec-

http://dx.doi.org/10.5772/intechopen.76566

Optimization of Lay-Up Stacking for a Loaded-Carrying Slender Composite Beam

max(*vb*) <sup>≤</sup> 0.46*<sup>m</sup>* ; max(*wb*) <sup>≤</sup> 0.86*<sup>m</sup>* (6)

); (II *φin* <sup>=</sup> <sup>30</sup><sup>0</sup>

) and (V *ψ* = 75<sup>0</sup>

); (III *φin* <sup>=</sup> <sup>30</sup><sup>0</sup>

; *<sup>φ</sup>in* <sup>=</sup> <sup>30</sup><sup>0</sup>

); (IV *φin* <sup>=</sup> <sup>30</sup><sup>0</sup>

), which had

);

49

**Figure 10.** Evolution of lay-up angle, objective and mechanical properties of structure during optimization for two "candidate" lay-ups: (I *φ in* = 30 0) and (V *ψ in* = 30 0; *φ in* = 15 0).

Where, *θ* is the torsion angle and *vb* , *wb* are the beam tip deflections along the *y* and *z*, respectively. Naturally, definition of such objective function and weights for its components has a significant element of arbitrariness. Therefore, the total strain energy, the beam deflections and torsion angle were also monitored during optimization process along with the objective function determined by Eq. (5).

For the definiteness of the optimization problem statement the following constraints were imposed.

 *Eei tot* ≤ 160*J* ; *θ* ≤ 4.50 ; max(*vb*) <sup>≤</sup> 0.46*<sup>m</sup>* ; max(*wb*) <sup>≤</sup> 0.86*<sup>m</sup>* (6)

At the optimization workflow, the forward structural mechanics problem has been called by the built-in gradient-free Nelder-Mead optimizer. In order to start the iterative optimization procedures, we used such "candidates": (I *φin* <sup>=</sup> <sup>30</sup><sup>0</sup> ); (II *φin* <sup>=</sup> <sup>30</sup><sup>0</sup> ); (III *φin* <sup>=</sup> <sup>30</sup><sup>0</sup> ); (IV *φin* <sup>=</sup> <sup>30</sup><sup>0</sup> ); (V *ψ* = 30<sup>0</sup> ; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup> ); (V *ψ* = 45<sup>0</sup> ; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup> ); (V *ψ* = 60<sup>0</sup> ; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup> ) and (V *ψ* = 75<sup>0</sup> ; *<sup>φ</sup>in* <sup>=</sup> <sup>30</sup><sup>0</sup> ), which had

**Figure 11.** Comparison among the best lay-ups of the optimized structure (stems with the filled symbols correspond to the preferred lay-ups).

**Figure 10.** Evolution of lay-up angle, objective and mechanical properties of structure during optimization for two

"candidate" lay-ups: (I *φ in* = 30 0) and (V *ψ in* = 30 0; *φ in* = 15 0).

48 Optimum Composite Structures

better results at the previous stage of the study (see **Figure 9**), where Roman numeral indicates the type of lay-up (see **Figure 4**) and the value of angle is its start value.

of quasi-optimal structure. The short discussion of the obtained results confirms necessity of multiobjective approach to the studied optimization problem, taking into account many requirements and constraints that allows to make the final choice of the best lay-up parameters.

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

, Natalia Snezhina<sup>2</sup>

3 National Kaohsiung University of Science and Technology, Kaohsiung City, Taiwan R.O.C.

[1] Baker A, Dutton S, Kelly D. Composite Materials for Aircraft Structures. 2nd ed. Reston:

[2] Ostergaard MG, Ibbotson AR, Le Roux O, Prior AM. Virtual testing of aircraft structures.

[3] Taylor JE. A formulation for optimal structural design with optimal materials. In: Rozvany GIN and Olhoff N, editors. Topology Optimization of Structures and Composite Continua. Dordrecht/Boston: Kluver Academic Publishers; 2000. pp. 49-55.

[4] Rao JS. Advances in aero structures. In: Proceedings of 12th International Conference on

[5] Gillet A, Francescato P, Saffre P. Single- and multi-objective optimization of composite structures: The influence of design variables. Journal of Composite Materials.

[6] Optimization of Composite: Recent Advances and Application [Internet]. 2011. Available

CEAS Aeronautical Journal. 2011;**1**:83-103. DOI: 10.1007/s13272-011-0004-x

Vibration Problems (ICOVP-2015); December 14-17, 2015; Guwahati. p. 20

from: www.altairproductdesign.com [Accessed: February 01, 2018]

and WU Jiing-Kae3

Optimization of Lay-Up Stacking for a Loaded-Carrying Slender Composite Beam

http://dx.doi.org/10.5772/intechopen.76566

51

**Acknowledgements**

project 16-58-52013 MNT\_a).

\*, Igor Zhilyaev1

\*Address all correspondence to: sergnshevtsov@gmail.com 1 South Center of Russian Academy, Rostov on Don, Russia

2 Don State Technical University, Rostov on Don, Russia

AIAA Inc.; 2004. p. 597. ISBN 1-56347-540-5

2010;**44**(4):457-480. DOI: 10.1177/0021998309344931

**Author details**

Sergey Shevtsov<sup>1</sup>

**References**

ISBN 0-7923-6806-1

**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 characterized by the different values of all responses.

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> ) and (V *ψ* = 30<sup>0</sup> ; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup> ) provide minimal values of objective, but (II *φin* <sup>=</sup> <sup>30</sup><sup>0</sup> ) provides the minimal total strain energy. Lay-up (V *ψ* = 45<sup>0</sup> ; *<sup>φ</sup>in* <sup>=</sup> <sup>15</sup><sup>0</sup> ) provides the greatest torsional stiffness, but 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 oriented unidirectional layers.
