6.1.1. Solutions for cantilevered two-module pipe with uniform thickness (h1 = h2 = 1.0)

Considering the case of two-module pipe, a direct and fast way for checking out system stability for any desired set of the dimensionless design variables (Vf1, L)k = 1,2 is given here. Lower and upper bounds are imposed on the design variables in order not to violate other strength and manufacturing requirements. The fiber volume fraction is constrained to be within the range 30% up to 70%, while the dimensionless length is between 0.0 and 1.0. The mass ratio MRo is taken to be 2.0.

volume fraction of the material of the second segment does not fall in range between 0.3 and 0.7. The maximum flutter velocity (Uf) and its corresponding flutter frequency (ωf) occur in the region colored with dark brown L1 = 0.36 and Vf1 = 0.3. The maximum Uf and its corresponding ω<sup>f</sup> are 13.67 and 58.4, respectively. Table 5 gives several standard optimal solutions for the two-module case study. The global optimum design point is seen to be (Vf, L)k = 1,2 = (0.35, 0.40), (0.60, 0.60), at which the normalized flutter velocity reached a value of 13.31 corresponding to

Optimization of Functionally Graded Material Structures: Some Case Studies

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As a major concern in producing efficient structures with enhanced properties and tailored response, this chapter presents appropriate design optimization models for improving performance and operational efficiency of different types of composite structural members. The concept of material grading has been successfully applied by incorporating the distribution of the volume fractions of the composite material constituents in the mathematical model formulation. Various scenarios in modeling the spatial variation of material properties of functionally graded structures are addressed. The associated optimization strategies include frequency maximization of thin-walled composite beams, optimization of drive shafts against torsional buckling and whirling instability, and maximization of the critical flight speed of subsonic aircraft wings. Design variables encompass the distribution of volume fraction, ply angle, and wall thickness as well. Detailed optimization models have been formulated and presented for improving the dynamic performance and increasing the overall stiffness-to-mass level of thin-walled composite beams. The objective functions have been measured by maximizing the natural frequencies and place them far away from the excitation frequencies, while maintaining the total structural mass at a constant value. For discrete models, the optimized beams can be constructed from any arbitrary number of uniform segments where the length of each segment has shown to be an important variable in the optimization process. It has also been proved that expressing all parameters in dimensionless forms results in naturally scaled design variables, constraints, and objective functions, which are favored by a variety of optimization algorithms. The attained optimal solutions using continuous grading depend entirely upon the prescribed power-law expression, which represents additional constraint on the optimization problem. Results show that material grading in the spanwise direction is much more better than grading through the wall thickness of the cross section. Regarding optimization of FGM drive shafts, it was shown that the best model is to combine torsional buckling and whirling in a single objective function subject to mass constraint. This has produced a balanced improvement in both stabilities with

In the context of aeroelastic stability of aircraft structures, an analytical model has been formulated to optimize subsonic trapezoidal wings against divergence. It was shown that by using material and thickness grading simultaneously, the aeroelastic stability boundary can be broaden by more than 50% above that of a known baseline design having the same total structural mass. Other stability problems concerning fluid-structure interaction have also been addressed. Both flutter and divergence optimization have been considered, and several design

active mass constraint at the attained optimal design point.

23.47% optimization gain.

7. Conclusions

Dimensionless flutter velocity and flutter frequency are obtained from the frequency and velocity branches at the four modes. The lowest frequency and velocity among the four modes at which Imag(ω) = 0.0 are considered the flutter velocity and frequency. These computed values at different conditions are employed in constructing the flutter velocity and frequency contours as shown in Figure 16. The white regions shown in both figures indicate that the fiber

Figure 16. Contour plots of flutter velocity and frequency in (Vf1-L1) design space.


Table 5. Standard solutions for a cantilevered two-module pipeline with uniform thickness (h1 = h2 = 1.0, M = 1.0).

volume fraction of the material of the second segment does not fall in range between 0.3 and 0.7. The maximum flutter velocity (Uf) and its corresponding flutter frequency (ωf) occur in the region colored with dark brown L1 = 0.36 and Vf1 = 0.3. The maximum Uf and its corresponding ω<sup>f</sup> are 13.67 and 58.4, respectively. Table 5 gives several standard optimal solutions for the two-module case study. The global optimum design point is seen to be (Vf, L)k = 1,2 = (0.35, 0.40), (0.60, 0.60), at which the normalized flutter velocity reached a value of 13.31 corresponding to 23.47% optimization gain.
