**1. Introduction**

Numerous applications of numerical optimization to various structural design problems have been addressed in the literature. A comprehensive survey on this issue was given in [1], presenting a historical review and demonstrating the future needs to assimilate this technology into the practicing design environment. Different approaches were applied successfully by several investigators for treating stress, displacement, buckling and frequency optimization problems. In general, design optimization seeks the best values of a set of *n* design variables represented by the vector, Xnx1, to achieve, within certain *m* constraints, Gmx1(X), its goal of optimality defined by a set of *k* objective functions, Fkx1(X), for specified environmental conditions (see Figure 1). Mathematically, design optimization may be cast in the following standard form: Find the design variables Xnx1 that minimize

$$F(\underline{X}) = \sum\_{i=1}^{k} \mathcal{W}\_{fi} F\_i(\underline{X}) \tag{1a}$$

subject to

$$\mathsf{G}(\underline{\mathsf{X}}) \le 0 \text{ , } \mathsf{j} \text{=} \mathsf{1}, \mathsf{2}, \dots, \mathsf{ldots} \text{ } \mathsf{I} \tag{1b}$$

$$\mathbf{G}(\underline{\mathbf{X}}) = \mathbf{0} \text{ / } \mathbf{j} \text{=} \mathbf{I} + \mathbf{1}, \mathbf{I} + \mathbf{2}, \dots \text{m} \tag{1c}$$

$$0 \le \mathcal{W}\_{f\bar{t}} \le 1$$

$$\sum\_{i=1}^{k} \mathcal{W}\_{f\bar{t}} = 1$$

© 2012 Maalawi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 19

capable of withstanding high temperature gradients. *FGMs* may be defined as advanced composite materials that fabricated to have graded variation of the relative volume fractions of the constituent materials. Commonly, these materials are made from particulate composites where the volume fraction of particles varies in one direction, as shown in Figure 2, or several directions for certain applications. *FGMs* may also be developed using fiber reinforced layers with a volume fractions of fibers changing, rather than constant,

Table 1 summarizes the mathematical formulas for determining the equivalent mechanical and physical properties for known type and volume fractions of the fiber and matrix

experimentally for specified types of fiber and matrix materials. Experimental results fall

especially in case of glass and carbon composites. The 1 and 2 subscripts denote the principal directions of an orthotropic lamina, defined as follows: direction (1): principal fiber direction, also called fiber longitudinal direction; direction (2): In-plane direction

> *Vf)/(1-Vf)*;

*Vf)/(1-Vf)*; 

*m Vm+ f Vf*

\*Subscripts *"m"* and *"f"* refer to properties of matrix and fiber materials, respectively.

**Table 1.** Halpin-Tsai semi-empirical relations for calculating composite properties [4].

is called the reinforcing efficiency and can be determined

is taken as 100% for theoretical analysis procedures,

*=(E2f –Em)/(E2f +* 

*=(G12f –Gm)/(G12f +* 

*Em)*

> *Gm)*

producing grading of the material with favorable properties.

Property Mathematical formula\*

Young's modulus in direction (1) *E11 Em Vm+ E1f Vf*

Young's modulus in direction (2) *E22 Em (1 +*

Shear modulus *G12 Gm (1 +*

Assuming no voids are present, then *Vm+Vf =1.*

Poisson's ratioߴଵଶ ߴ ܸ ߴଵଶܸ

**Figure 2.** The concept of material grading

within a band of 1<<2. Usually,

perpendicular to fibers, transversal direction.

materials [4]. The factor

Mass density

**Figure 1.** Design optimization process

The weighting factors Wfi measure the relative importance of the individual objectives with respect to the overall design goal. Several computer program packages are available now for solving a variety of design optimization models. Advanced procedures are carried out by using large-scale, general purpose, finite element-based multidisciplinary computer programs [2], such as *ASTROS*, *MSC/NASTRAN* and *ANSYS*. The *MATLAB* optimization toolbox is also a powerful tool that includes many routines for different types of optimization encompassing both unconstrained and constrained minimization algorithms [3]. Design optimization of sophisticated structural systems involves many objectives, constraints and variables. Therefore, creation of a detailed optimization model incorporating, simultaneously, all the relevant design features is virtually impossible. Researchers and engineers rely on simplified models which provide a fairly accurate approximation of the real structure behaviour. This chapter presents some of the underlying concepts of applying optimization theory for enhancing the stability, dynamic and aeroelastic performance of functionally graded material *(FGM)* structural members. Such concept of *FGM*, in which the properties vary spatially within a structure, was originated in Japan in 1984 during the space project, in the form of proposed thermal barrier material capable of withstanding high temperature gradients. *FGMs* may be defined as advanced composite materials that fabricated to have graded variation of the relative volume fractions of the constituent materials. Commonly, these materials are made from particulate composites where the volume fraction of particles varies in one direction, as shown in Figure 2, or several directions for certain applications. *FGMs* may also be developed using fiber reinforced layers with a volume fractions of fibers changing, rather than constant, producing grading of the material with favorable properties.

**Figure 2.** The concept of material grading

18 Advances in Computational Stability Analysis

**Figure 1.** Design optimization process

The weighting factors Wfi measure the relative importance of the individual objectives with respect to the overall design goal. Several computer program packages are available now for solving a variety of design optimization models. Advanced procedures are carried out by using large-scale, general purpose, finite element-based multidisciplinary computer programs [2], such as *ASTROS*, *MSC/NASTRAN* and *ANSYS*. The *MATLAB* optimization toolbox is also a powerful tool that includes many routines for different types of optimization encompassing both unconstrained and constrained minimization algorithms [3]. Design optimization of sophisticated structural systems involves many objectives, constraints and variables. Therefore, creation of a detailed optimization model incorporating, simultaneously, all the relevant design features is virtually impossible. Researchers and engineers rely on simplified models which provide a fairly accurate approximation of the real structure behaviour. This chapter presents some of the underlying concepts of applying optimization theory for enhancing the stability, dynamic and aeroelastic performance of functionally graded material *(FGM)* structural members. Such concept of *FGM*, in which the properties vary spatially within a structure, was originated in Japan in 1984 during the space project, in the form of proposed thermal barrier material Table 1 summarizes the mathematical formulas for determining the equivalent mechanical and physical properties for known type and volume fractions of the fiber and matrix materials [4]. The factor is called the reinforcing efficiency and can be determined experimentally for specified types of fiber and matrix materials. Experimental results fall within a band of 1<<2. Usually, is taken as 100% for theoretical analysis procedures, especially in case of glass and carbon composites. The 1 and 2 subscripts denote the principal directions of an orthotropic lamina, defined as follows: direction (1): principal fiber direction, also called fiber longitudinal direction; direction (2): In-plane direction perpendicular to fibers, transversal direction.


**Table 1.** Halpin-Tsai semi-empirical relations for calculating composite properties [4].

An excellent review paper dealing with the basic knowledge and various aspects on the use of *FGMs* and their wide applications was given in [5]. It was shown that *FGMs* can be promising in several applications such as, spacecraft heat shields, high performance structural elements and critical engine components. A few studies have addressed the dynamics and stability of *FGM* structures. Closed-form expressions for calculating the natural frequencies of an axially graded beam were derived in [6]. The modulus of elasticity was taken as a polynomial of the axial coordinate along the beam's length, and an inverse problem was solved to find the stiffness and mass distributions so that the chosen polynomial serve as an exact mode shape. Another work [7] considered stability of FGMstructures and derived closed form solution for the mode shape and the buckling load of an axially graded cantilevered column. A semi-inverse method was employed to obtain the spatial distribution of the elastic modulus in the axial direction. In reference [8], the buckling of simply supported three-layer circular cylindrical shell under axial compressive load was considered. The middle layer sandwiched with two isotropic layers was made of an isotropic *FGM* whose Young's modulus varies parabolically in the thickness direction. Classical shell theory was implemented under the assumption of very small thickness/radius and very large length/radius ratios. Numerical results showed that the buckling load increases with an increase in the average value of Young's modulus of the middle layer. In the field of structural optimization, reference [9] considered frequency optimization of a cantilevered plate with variable volume fraction according to simple power-laws. Genetic algorithms was implemented to find the optimum values of the power exponents, which maximize the natural frequencies, and concluded that the volume fraction needs to be varied in the longitudinal direction of the plate rather than in the thickness direction. A direct method was proposed in [10] to optimize the natural frequencies of functionally graded beam with variable volume fraction of the constituent materials in the beam's length and height directions. A piecewise bi-cubic interpolation of volume fraction values specified at a finite number of grid points was used, and a genetic algorithm code was applied to find the needed optimum designs. It is the main aim of this chapter to present some fundamental issues concerning design optimization of different types of functionally graded composite structures. Practical realistic optimization models using different strategies for enhancing stability, structural dynamics, and aeroelastic performance are presented and discussed. Design variables represent material type, structure geometry as well as cross sectional parameters. The mathematical formulation is based on dimensionless quantities; therefore the analysis can be valid for different configurations and sizes. Such normalization has led to a naturally scaled optimization models, which is favorable for most optimization techniques. Case studies concerning optimization of FGM composite structures include buckling of flexible columns, stability of thin-walled cylinders subject to external pressure, frequency optimization of FGM bars in axial motion, and critical velocity maximization in pipe flow as a measure of raising the stability boundary. The use of the concept of material grading for enhancing the aeroelastic stability of composite wings have been also addressed. Several design charts that are useful for direct Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 21

k k P 0P 1 2 *<sup>ˆ</sup> <sup>ˆ</sup> w w , P ˆ , k , ,....Ns ˆ ˆ <sup>E</sup> Ik <sup>k</sup>* (2)

1 2 34 *k k w(x) sin x cos x x ˆ a a aa P P* (3)

determination of the optimal values of the design variables are introduced. In all, the given mathematical models can be regarded as useful design tools which may save designers from

The consideration of buckling stability of elastic columns can be crucial factor in designing efficient structural components. In references [11, 12], optimization models of the strongest columns were developed for maximizing the critical buckling load under equality mass constraint. Emphasizes were given to thin-walled tubular sections, which are more economical than solid sections in resisting compressive loads. The given formulation considered columns made of uniform segments with different material properties, crosssectional parameters and length, as shown in Figure 3. The simplest problem of equilibrium of a column compressed by an axial force, *P*, was first formulated and solved by the great mathematician L. Euler in the middle of the eighteenth century. The associated 4th-order

where *( )'* means differentiation with respect to the dimensionless coordinate *xˆ* and *Ns* is the total number of segments. The various dimensionless quantities denoted by (**^**) are defined in Table 2. Equation (2) must be satisfied in the interval 0 *x Lk*, where *x <sup>k</sup> x x ˆ ˆ* **.** 

having to choose the values of some of their variables arbitrarily.

**2. Buckling optimization of elastic columns**

governing differential equation in dimensionless form:

Its general solution is:

2

**Figure 3.** General configuration of a piecewise axially graded thin-walled column

nodes of the *Kth* segment, which results in the following matrix relation:

The coefficients *ai'*s in Equation (3) can be expressed in terms of the state variables at both

determination of the optimal values of the design variables are introduced. In all, the given mathematical models can be regarded as useful design tools which may save designers from having to choose the values of some of their variables arbitrarily.

## **2. Buckling optimization of elastic columns**

20 Advances in Computational Stability Analysis

An excellent review paper dealing with the basic knowledge and various aspects on the use of *FGMs* and their wide applications was given in [5]. It was shown that *FGMs* can be promising in several applications such as, spacecraft heat shields, high performance structural elements and critical engine components. A few studies have addressed the dynamics and stability of *FGM* structures. Closed-form expressions for calculating the natural frequencies of an axially graded beam were derived in [6]. The modulus of elasticity was taken as a polynomial of the axial coordinate along the beam's length, and an inverse problem was solved to find the stiffness and mass distributions so that the chosen polynomial serve as an exact mode shape. Another work [7] considered stability of FGMstructures and derived closed form solution for the mode shape and the buckling load of an axially graded cantilevered column. A semi-inverse method was employed to obtain the spatial distribution of the elastic modulus in the axial direction. In reference [8], the buckling of simply supported three-layer circular cylindrical shell under axial compressive load was considered. The middle layer sandwiched with two isotropic layers was made of an isotropic *FGM* whose Young's modulus varies parabolically in the thickness direction. Classical shell theory was implemented under the assumption of very small thickness/radius and very large length/radius ratios. Numerical results showed that the buckling load increases with an increase in the average value of Young's modulus of the middle layer. In the field of structural optimization, reference [9] considered frequency optimization of a cantilevered plate with variable volume fraction according to simple power-laws. Genetic algorithms was implemented to find the optimum values of the power exponents, which maximize the natural frequencies, and concluded that the volume fraction needs to be varied in the longitudinal direction of the plate rather than in the thickness direction. A direct method was proposed in [10] to optimize the natural frequencies of functionally graded beam with variable volume fraction of the constituent materials in the beam's length and height directions. A piecewise bi-cubic interpolation of volume fraction values specified at a finite number of grid points was used, and a genetic algorithm code was applied to find the needed optimum designs. It is the main aim of this chapter to present some fundamental issues concerning design optimization of different types of functionally graded composite structures. Practical realistic optimization models using different strategies for enhancing stability, structural dynamics, and aeroelastic performance are presented and discussed. Design variables represent material type, structure geometry as well as cross sectional parameters. The mathematical formulation is based on dimensionless quantities; therefore the analysis can be valid for different configurations and sizes. Such normalization has led to a naturally scaled optimization models, which is favorable for most optimization techniques. Case studies concerning optimization of FGM composite structures include buckling of flexible columns, stability of thin-walled cylinders subject to external pressure, frequency optimization of FGM bars in axial motion, and critical velocity maximization in pipe flow as a measure of raising the stability boundary. The use of the concept of material grading for enhancing the aeroelastic stability of composite wings have been also addressed. Several design charts that are useful for direct

The consideration of buckling stability of elastic columns can be crucial factor in designing efficient structural components. In references [11, 12], optimization models of the strongest columns were developed for maximizing the critical buckling load under equality mass constraint. Emphasizes were given to thin-walled tubular sections, which are more economical than solid sections in resisting compressive loads. The given formulation considered columns made of uniform segments with different material properties, crosssectional parameters and length, as shown in Figure 3. The simplest problem of equilibrium of a column compressed by an axial force, *P*, was first formulated and solved by the great mathematician L. Euler in the middle of the eighteenth century. The associated 4th-order governing differential equation in dimensionless form:

$$\|\hat{w}\prime\prime + \|\mathbf{P}\_{\mathbf{k}}\prime\hat{w} = 0,\qquad \mathbf{P}\_{\mathbf{k}} = \sqrt{\hat{\mathbf{P}}\!\!/ \hat{\mathbf{E}}\_{\mathbf{k}} \hat{\mathbf{I}}\_{\mathbf{k}}}\prime,\ k = 1, 2, \dots \text{Ns} \tag{2}$$

where *( )'* means differentiation with respect to the dimensionless coordinate *xˆ* and *Ns* is the total number of segments. The various dimensionless quantities denoted by (**^**) are defined in Table 2. Equation (2) must be satisfied in the interval 0 *x Lk*, where *x <sup>k</sup> x x ˆ ˆ* **.**  Its general solution is:

$$\hat{u}(\overline{\mathbf{x}}) = \_{\mathcal{A}1}\text{sin }\;\_{1}\text{p}\_{k}\overline{\mathbf{x}} + \_{\mathcal{A}2}\text{cos }\;\_{1}\text{p}\_{k}\overline{\mathbf{x}} + \_{\mathcal{A}3}\overline{\mathbf{x}} + a\_{4} \tag{3}$$

**Figure 3.** General configuration of a piecewise axially graded thin-walled column

The coefficients *ai'*s in Equation (3) can be expressed in terms of the state variables at both nodes of the *Kth* segment, which results in the following matrix relation:

$$\begin{bmatrix} \hat{w}\_{k+1} \\\\ \Phi\_{k+1} \\\\ \hat{\mathbf{M}}\_{k+1} \\\\ \hat{\mathbf{P}}\_{k+1} \\ \hat{\mathbf{P}}\_{k+1} \end{bmatrix} = \begin{bmatrix} 1 & \frac{\mathbf{\cdot}\mathbf{S}\_{\mathbf{k}}}{\mathbf{P}\_{\mathbf{k}}} & \frac{\mathbf{\cdot}(1 - \mathbf{C}\_{\mathbf{k}})}{\hat{\mathbf{P}}} & (\frac{\mathbf{S}\_{\mathbf{k}}}{\hat{\mathbf{P}}\_{\mathbf{P}}} - \frac{\hat{L}\_{k}}{\hat{\mathbf{P}}} \\\\ \mathbf{0} & \mathbf{C}\_{\mathbf{k}} & \frac{\mathbf{P}\_{k}S\_{k}}{\hat{\mathbf{P}}} & \frac{(\mathbf{I} - \mathbf{C}\_{\mathbf{k}})}{\hat{\mathbf{P}}} \\\\ \mathbf{0} & \frac{\mathbf{\cdot}\mathbf{\hat{P}}\_{\mathbf{k}}}{\mathbf{P}\_{\mathbf{k}}} & \mathbf{C}\_{\mathbf{k}} & \frac{\mathbf{S}\_{\mathbf{k}}}{\mathbf{P}\_{\mathbf{k}}} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix} \begin{matrix} \hat{w}\_{k} \\\\ \Phi\_{k} \\\\ \hat{\mathbf{M}}\_{k} \\\\ \hat{\mathbf{P}}\_{k} \end{matrix} \tag{4}$$

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 23

Therefore, the strongest column design problem may be cast in the following:

ܲcr

1 *<sup>s</sup> Mˆ*

1

(5)

*L )k=1,2,3 = (0.095, 0.0875), (0.38125, 0.140625), (0.69177,0.271875)*

*B(g/cm3) EB(GPa)*

*2)*. Other types of materials were also addressed in [12] including,

1

Side constraints are always present by imposing lower and upper limits on the design variables to avoid having odd-shaped unrealistic column design in the final optimum solutions. Reference [12] presented optimum patterns for cases of simply supported and cantilevered FGM columns constructed from unidirectional fibrous composites with properties given in Table 3. The case of a symmetrical simply supported (*S.S.*) column made of E-glass/epoxy and constructed from different number of segments is depicted in Figure 4. For the case of 6-segments, the optimum zone of the dimensionless critical buckling load augmented with the mass equality constraint was determined and found to be well behaved in the selected (*VA – L)3* design space. Referring to Figure 5, three distinct regions can be observed: two empty regions to the left and right violating the mass equality constraint and the middle feasible region containing the global optimum solution. It is also seen that the optimal feasible domain is bounded from left and right by two heavy zigzagged lines owing to the fact that many contours are stuck to these borderlines and are not allowed to penetrate them for not violating the imposed mass constraint. The final optimum design

where *(Pcr)max =11.649375*. This represents an optimization gain of about *18.0%* relative to the

carbon/epoxy, S-glass/epoxy, E-glass/Vinyl-ester and S-glass/Vinyl-ester. In all cases the

Composite material material *(A)* = Fibers material *(B)* = matrix

*A(g/cm3) EA (GPa)*

E-glass/epoxy 2.54 73.0 1.27 4.3

E-glass/Vinylester 2.54 73.0 1.15 3.5

S-glass/epoxy 2.49 86.0 Carbon/epoxy 1.81 235.0

S-glass/Vinylester 2.49 86.0

**Table 3.** Material properties of selected fiber-reinforced composites [12]

*Ns k k ˆ L* 

Maximize

Subject to

point was found to be *(VA, ˆ*

baseline value *(*

where *Sk=SinPkLk* and *Ck=CosPkLk*. Applying Equation (4) successively to all the segments composing the column and taking the products of all the resulting matrices, the state variables at both ends of the column can be related to each other through an overall transfer matrix. Therefore, by the application of the appropriate boundary conditions and consideration of the non-trivial solution, the associated characteristic equation for determining the critical buckling load can be accurately obtained. The exact buckling analysis outlined above can be coupled with a standard nonlinear mathematical programming algorithm for the search of columns designs with the largest possible resistance against buckling. It is important to bear in mind that design optimization is only as meaningful as its core structural analysis model. Any deficiencies therein will certainly be reflected in the optimization process.


**Table 2.** Definition of dimensionless quantities

Therefore, the strongest column design problem may be cast in the following:

ܲcr

Maximize

22 Advances in Computational Stability Analysis

1

*k*

*ˆ F*

reflected in the optimization process.

*k*

 

*<sup>ˆ</sup> Mˆ*

1 k kk

1 k k k


*k k*

*w ˆ <sup>ˆ</sup> ) <sup>L</sup> )*

1 k k k k k

0 0 0

<sup>1</sup>

*k k*

k k

<sup>P</sup> (1-C 0 C


where *Sk=SinPkLk* and *Ck=CosPkLk*. Applying Equation (4) successively to all the segments composing the column and taking the products of all the resulting matrices, the state variables at both ends of the column can be related to each other through an overall transfer matrix. Therefore, by the application of the appropriate boundary conditions and consideration of the non-trivial solution, the associated characteristic equation for determining the critical buckling load can be accurately obtained. The exact buckling analysis outlined above can be coupled with a standard nonlinear mathematical programming algorithm for the search of columns designs with the largest possible resistance against buckling. It is important to bear in mind that design optimization is only as meaningful as its core structural analysis model. Any deficiencies therein will certainly be

Quantity Non-dimensionalization\*

*ˆ L / L*

1

*f + m)/2*.

\*Baseline design parameters: *L*=total column's length, *h*=wall thickness, *I*= second moment of area, *E*=modulus

*=*mass density*= (*

*Ns s k k k k Mˆ ˆ ˆ ˆ h L* 

Axial coordinate *x x/L ˆ* Length of *Kth* segment *<sup>k</sup> <sup>k</sup> L*

Transverse deflection *w w/L ˆ* Wall thickness *<sup>k</sup> <sup>k</sup> ˆ h / h h*

Modulus of elasticity *<sup>k</sup> <sup>k</sup> Eˆ E / E*

Mass density *k k ˆ /*

Total structural mass

of elasticity *= (Ef+ Em)/2*,

**Table 2.** Definition of dimensionless quantities

Second moment of area *<sup>k</sup> <sup>k</sup> <sup>k</sup> ˆ /I ( ) ˆ I I h*

Bending moment *Mˆ M \* (L / EI)* Shearing force *<sup>ˆ</sup>* <sup>2</sup> *F F \* ( / EI) <sup>L</sup>* Axial force *<sup>ˆ</sup>* <sup>2</sup> *P P \* ( / EI) <sup>L</sup>*

P P P P

*Pˆ ˆˆ S ) P P ˆ ˆ*

*k*

*wˆ*

*k*

*Mˆ*

*k*

(4)

*k*

*ˆ F* Subject to

$$
\hat{M}\_s = 1
$$

$$
\sum\_{k=1}^{Ns} \hat{L}\_k = 1
$$

Side constraints are always present by imposing lower and upper limits on the design variables to avoid having odd-shaped unrealistic column design in the final optimum solutions. Reference [12] presented optimum patterns for cases of simply supported and cantilevered FGM columns constructed from unidirectional fibrous composites with properties given in Table 3. The case of a symmetrical simply supported (*S.S.*) column made of E-glass/epoxy and constructed from different number of segments is depicted in Figure 4. For the case of 6-segments, the optimum zone of the dimensionless critical buckling load augmented with the mass equality constraint was determined and found to be well behaved in the selected (*VA – L)3* design space. Referring to Figure 5, three distinct regions can be observed: two empty regions to the left and right violating the mass equality constraint and the middle feasible region containing the global optimum solution. It is also seen that the optimal feasible domain is bounded from left and right by two heavy zigzagged lines owing to the fact that many contours are stuck to these borderlines and are not allowed to penetrate them for not violating the imposed mass constraint. The final optimum design point was found to be *(VA, ˆ L )k=1,2,3 = (0.095, 0.0875), (0.38125, 0.140625), (0.69177,0.271875)* where *(Pcr)max =11.649375*. This represents an optimization gain of about *18.0%* relative to the baseline value *(2)*. Other types of materials were also addressed in [12] including, carbon/epoxy, S-glass/epoxy, E-glass/Vinyl-ester and S-glass/Vinyl-ester. In all cases the


**Table 3.** Material properties of selected fiber-reinforced composites [12]

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 25

buckling load was found to be very sensitive to variation in the segment length. Investigators who use approximate methods, such as finite elements, have not recognized that the length of each element can be taken as a main optimization variable in addition to the cross-sectional properties. The increase in the number of segments would, naturally, result in higher values of the dimensionless critical buckling load. However, care ought to be taken for the corresponding increase in cost due to the resulting complications in the

Other cases of cantilevered columns were also investigated. The associated boundary conditions are: at 0 *xˆ wˆ* 0 , and at 1 0 *x . ˆ* 0 *M F ˆ ˆ* . Figure 6 shows the attained optimal solutions for cantilevered columns made of unidirectional E-glass/epoxy composites and constructed from different number of segments *(Ns).* For a three-segment column, the global optimal solution was found to be *(Pcr)max=2.90938* occurring at the design point *(VA,Lk)k=1,2,3 = (0.70, 0.514), (0.4125, 0.2785), (0.122,0.2075)*. This means that the strongest column made of only three segments can withstand a buckling load 18% higher than that with uniform mass and stiffness distributions, which represents a truly optimized column design. In fact, the exact buckling load can be obtained for any number of segments, type of cross section and type of boundary conditions. The given multi-segment model has the advantageous of achieving global optimality for the strongest columns shape that can be manufactured economically from any arbitrary number of segments. Sensitivity of the design variables on the buckling load should be included in a more general formulation.

**Figure 6.** Strongest cantilevered columns with axial material grading: Material (A)=E-glass fibers,

material (B)=epoxy matrix

associated assembling and manufacturing procedures.

**Figure 4.** Optimum simply supported columns with piecewise axial material grading

**Figure 5.** Optimum zone for a symmetrical 6-segment *S.S.* columns made of E-glass/epoxy composites.

buckling load was found to be very sensitive to variation in the segment length. Investigators who use approximate methods, such as finite elements, have not recognized that the length of each element can be taken as a main optimization variable in addition to the cross-sectional properties. The increase in the number of segments would, naturally, result in higher values of the dimensionless critical buckling load. However, care ought to be taken for the corresponding increase in cost due to the resulting complications in the associated assembling and manufacturing procedures.

24 Advances in Computational Stability Analysis

**Figure 4.** Optimum simply supported columns with piecewise axial material grading

**Figure 5.** Optimum zone for a symmetrical 6-segment *S.S.* columns made of E-glass/epoxy composites.

Other cases of cantilevered columns were also investigated. The associated boundary conditions are: at 0 *xˆ wˆ* 0 , and at 1 0 *x . ˆ* 0 *M F ˆ ˆ* . Figure 6 shows the attained optimal solutions for cantilevered columns made of unidirectional E-glass/epoxy composites and constructed from different number of segments *(Ns).* For a three-segment column, the global optimal solution was found to be *(Pcr)max=2.90938* occurring at the design point *(VA,Lk)k=1,2,3 = (0.70, 0.514), (0.4125, 0.2785), (0.122,0.2075)*. This means that the strongest column made of only three segments can withstand a buckling load 18% higher than that with uniform mass and stiffness distributions, which represents a truly optimized column design. In fact, the exact buckling load can be obtained for any number of segments, type of cross section and type of boundary conditions. The given multi-segment model has the advantageous of achieving global optimality for the strongest columns shape that can be manufactured economically from any arbitrary number of segments. Sensitivity of the design variables on the buckling load should be included in a more general formulation.

**Figure 6.** Strongest cantilevered columns with axial material grading: Material (A)=E-glass fibers, material (B)=epoxy matrix
