7. Conclusions

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

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

Design variables: (Vf, L)k = 1,2 Dimensionless Uflutter Stability improvement %

21.99 11.97

20.96 23.47 (max.)

17.81 18.46

18.92 8.81

13.15 12.07

13.04 13.31

12.70 12.77

12.82 11.73

(0.50, 0.50), (0.50, 0.50) 10.78 0.00 (baseline)

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

mass ratio MRo is taken to be 2.0.

186 Optimum Composite Structures

(0.30, 0.20), (0.55, 0.80) (0.30, 0.50), (0.70, 0.50)

(0.35, 0.25), (0.55, 0.75) (0.35, 0.40), (0.60, 0.60)

(0.40, 0.50), (0.60, 0.50) (0.40, 0.60), (0.65, 0.40)

(0.45, 0.50), (0.55, 0.50) (0.45, 0.75), (0.65, 0.25)

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

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

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 active mass constraint at the attained optimal design point.

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 charts that are useful for direct determination of the optimal values of the design variables are given. It has been confirmed that the segment length is the most significant design variable in the whole optimization process. Some investigators who apply finite elements have not recognized that the length of each element can be taken as a main design variable in the whole set of optimization variables. The results from the present approach reveal that piecewise grading of the material can be promising in producing truly efficient structural designs with enhanced stability, dynamic, and aeroelastic performance. It is the author's wish that the results presented in this chapter will be compared and validated through other optimization techniques such as genetic algorithms or any appropriate global optimization algorithm.

[4] Cho JR, Shin SW. Material composition optimization for heat-resisting FGM by artificial neural network. Composites Part A Applied Science and Manufacturing. 2004;35:585-594

Optimization of Functionally Graded Material Structures: Some Case Studies

http://dx.doi.org/10.5772/intechopen.82411

189

[5] Goupee AJ, Vel SS. Optimization of natural frequencies of bidirectional functionally graded beams. Journal of Structural and Multidisciplinary Optimization. 2006;32(6):73-484

[6] Librescu L, Maalawi K. Material grading for improved aeroelastic stability of composite

[7] Birman V, Byrd WL. Modeling and analysis of functionally graded materials and struc-

[8] Elishakoff I, Guede Z. Analytical polynomial solutions for vibrating axially graded beams. Journal of Mechanics and Advanced Materials and Structures. 2004;11:517-533

[9] Qian L, Batra R. Design of bidirectional functionally graded plate for optimal natural

[10] Maalawi KY. Functionally graded bars with enhanced dynamic performance. Journal of

[11] Elishakoff I, Endres J. Extension of Euler's problem to axially graded columns. Journal of

[12] Li S-R, Batra R. Buckling of axially compressed thin cylindrical shells with functionally

[13] Maalawi KY. Optimization of elastic columns using axial grading concept. Engineering

[14] Maalawi KY. Use of material grading for enhanced buckling design of thin-walled composite rings/long cylinders under external pressure. Composite Structures. 2011;93(2):

[15] Shin-Yao K. Flutter of rectangular composite plates with variable fiber spacing. Compos-

[16] Maalawi KY. Stability dynamic and aeroelastic optimization of functionally graded composite structures. In: Coskun SB, editor. Advances in Computational Stability Analysis.

[17] Rao S. Engineering Optimization: Theory and Practice. 4th ed. New York: John Wiley &

[18] Maalawi K, Badr M. Design optimization of mechanical elements and structures: A review

[19] Venkataraman P. Applied Optimization with MATLAB Programming. 2nd ed. New

[20] Chen WH, Gibson RF. Property distribution determination of non-uniform composite beams from vibration response measurements and Galerkin's method. Journal of Applied

with application. Journal of Applied Sciences Research. 2009;5(2):221-231

graded middle layer. Journal of Thin-Walled Structures. 2006;44:1039-1047

wings. Journal of Mechanics of Materials and Structures. 2007;2(7):1381-1394

tures. Applied Mechanics Reviews. 2007;60(5):195-216

frequencies. Journal of Sound and Vibration. 2005;280:415-424

Mechanics of Materials and Structures. 2011;6(1-4):377-393

Intelligent Material Systems and Structures. 2005;16(1):77-83

Rijeka: InTech; 2012. pp. 17-42. DOI: 10.5772/45878.ch2

York: John Wiley & Sons; 2009. ISBN: 978-0470084885

Structures. 2009;31(12):2922-2929

ite Structures. 2011;93:2533-2540

Sons; 2009. ISBN: 978-0470183526

Mechanics. 1998;65:127-133

351-359

Actually, the most economic structural design that will perform its intended function with adequate safety and durability requires much more than the procedures that have been described in this chapter. Further optimization studies must depend on a more accurate analysis of constructional cost. This combined with probability studies of load applications and materials variations should contribute to further efficiency achievement. Much improved and economical designs for the main structural components may be obtained by considering multidisciplinary design optimization, which allows designers to incorporate all relevant design objectives simultaneously. Finally, it is important to mention that, while FGM may serve as an excellent optimization and material tailoring tool, the ability to incorporate optimization techniques and solutions in practical design depend on the capacity to manufacture these materials to required specifications. Conventional techniques are often incapable of adequately addressing this issue. In conclusion, FGMs represent a rapidly developing area of science and engineering with numerous practical applications. The research needs in this area are uniquely numerous and diverse, but FGMs promise significant potential benefits that fully justify the necessary effort.
