5. Optimization of FGM wings against divergence

The use of the in-plane grading in aeroelastic design was first exploited by Librescu and Maalawi [6], who introduced the underlying concepts of using material grading in optimizing subsonic rectangular wings against torsional instability. Exact mathematical models were developed allowing the material physical and mechanical properties to change in the wing spanwise direction, where both continuous and piecewise structural models were successfully implemented. In this section, analytical solutions are developed for slender tapered composite wings through optimal grading of the material volume fraction in the spanwise direction. The enhancement of the wing torsional stability is measured by maximization of the critical flight speed at which aeroelastic divergence occurs. The total structural mass is maintained at a value equals to that of a known baseline design in order not to violate other performance requirements. Figure 11 depicts a slender wing constructed from Np panels with trapezoidal planform and known airfoil cross section. The wing is considered to be made of unidirectional fiber-reinforced composites with variable fiber volume fraction in the spanwise direction. The flow is taken to be steady and incompressible, and the aspect ratio is assumed to be sufficiently large so that the classical engineering theory of torsion can be applicable and the state of deformation described in terms of one space coordinate.

The chord distribution is assumed to have the form:

given by Eq. (3). The corresponding design variable vector is defined by X

= (0.3, 0.3, 0, 0.75) and XU

Figure 10. Normalized critical speed <sup>Ω</sup>^ cr augmented with the mass constraint (M^ <sup>¼</sup> <sup>1</sup>:0) in (p-Δf) design space.

maximum critical speed increased by 14% above that of the baseline design with active mass

A last optimization strategy to be addressed here is to combine the two criteria in a single

Minimize <sup>F</sup> ¼ � <sup>Ω</sup>^ cr <sup>þ</sup> <sup>T</sup>^cr

τmax τallow 

Eq. (22) assumes that whirling and torsional buckling instabilities are of equal relative importance. This model resulted in a balanced improvement in both stabilities with active mass constraint. The attained optimal solution was found to have a uniform distribution of the fiber volume fraction with its upper limiting value of 70% and wall thickness = 0.935. The corresponding optimal values of the design objectives were <sup>Ω</sup>^ cr <sup>¼</sup> <sup>1</sup>:135 and <sup>T</sup>^cr <sup>¼</sup> <sup>1</sup>:161, representing optimization gains 13.5 and 16.1%, respectively, as measured from the baseline

Subject to <sup>M</sup>^ � <sup>1</sup> <sup>≤</sup> <sup>0</sup>

!

!

� 1:0 ≤ 0

!

objective function subject to the mass, strength, and side constraints.

optimal design variable vector was calculated to be Xopt

with lower and upper limits XL

178 Optimum Composite Structures

constraint.

design.

!

<sup>¼</sup> Vfð Þ<sup>0</sup> ; Vf <sup>1</sup>

= (0.7, 0.7, ∞, 1.25). The attained

¼ ð Þ 0:7; 0:3; 5:61; 0:955 at which the

2 ; <sup>p</sup>; <sup>H</sup>^

(32)

$$\mathbf{C}(\mathbf{x}) = \mathsf{C}\_{\mathsf{r}} \left( 1 - \beta\_{\mathsf{c}} \mathbf{x} \right), \beta\_{\mathsf{c}} = (1 - \Delta\_{\mathsf{c}}) \tag{33}$$

The symbol Δ<sup>c</sup> denotes the chord taper ratio (= tip chord Ct/root chord Cr) and x (= x1/L) denotes the dimensionless spanwise coordinate. The equivalent shear modulus G of a unidirectional reinforced composite, thin-walled cross section can be determined from the relation [35]:

$$\mathbf{G} = f\_1 \,\, \mathbf{G}\_{12} \tag{34}$$

where f1 is a function that depends on the geometry and thickness ratio of the cross section (h/ C) and the ratio (G12/ G13), where G12 and G13 are the in-plane and out-of-plane shear moduli,

Figure 11. Trapezoidal wing planform and cross section geometry. (a) Multipanel, piecewise wing model, (b) airfoil section and applied airloads.

respectively (refer to Table 1). C is the chord length and h is the maximum thickness of the cross section. For many types of fibrous composites that are commonly utilized in aerospace industry [23], such as carbon/epoxy and graphite epoxy, both moduli are approximately equal, G12 ≈ G13.

Using the classical elasticity and aerodynamic strip theories, the governing differential equation of torsional stability in dimensionless form is [35]:

$$\left(\hat{\mathbf{G}}\hat{\mathbf{J}}\boldsymbol{\alpha}'\right)' + \hat{\boldsymbol{V}}^2 \boldsymbol{\alpha}(\mathbf{x}) = \mathbf{0} \tag{35}$$

Minimize � <sup>V</sup>^ div

� �

<sup>P</sup>^bk <sup>¼</sup> <sup>1</sup>:<sup>0</sup>

The preassigned parameters that do not change during the optimization process include the wing semispan (b), the chord taper ration (Δc), airfoil type and geometry, and fiber and resin material types. This model has been applied to obtain wing designs with improved torsional stability by maximizing the divergence speed (Vdiv) without weight penalty. The selected material is carbon-AS4/epoxy-3501-6 composite (see Table 2), which has favorable characteristics and is highly desirable in manufacturing aircraft structures. The baseline design has uniform mass and stiffness distributions and is made of uniform unidirectional fibrous composite with equal volume fraction of the matrix and fiber materials, that is, Vfo = 50%. Figures 12 and 13 show the developed level curves of constant divergence speed (also named isodiverts) for two-panel wings with chord tapering ratio, Δ<sup>c</sup> = 0.5. Actually, these curves represent the dimensionless critical speed, augmented with the equality mass constraint. Examining Figure 12, it is seen that the Vdiv function is well behaved and continuous everywhere in the selected design space except in the empty regions to the upper left and right regions, where the equality mass constraint is violated. The feasible domain is bounded from

k¼1,2,::Np

≤ ð Þ 0:75; 1:25; 1:0

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

Optimization of Functionally Graded Material Structures: Some Case Studies

(39)

181

Subject to <sup>M</sup>^ <sup>s</sup> � <sup>1</sup>:<sup>0</sup> <sup>≤</sup> <sup>0</sup>

ð Þ <sup>0</sup>:25; <sup>0</sup>:5; <sup>0</sup>:<sup>0</sup> <sup>≤</sup> Vf ; ^h; ^<sup>b</sup>

Figure 12. Isodivert in (Vf1- b1) design space for a two-panel wing model (h1 = h2 = 1.0, <sup>Δ</sup><sup>c</sup> = 0.5, <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1).

The associated boundary conditions of the elastic angle of attack, α, are α(0) = 0 and α<sup>0</sup> ð Þ¼ 1 0. The symbol <sup>G</sup>^ <sup>¼</sup> <sup>G</sup>12=G12, <sup>o</sup> denotes the dimensionless shear modulus, ^<sup>J</sup> <sup>¼</sup> <sup>J</sup>=Jr denotes the dimensionless torsion constant, and the prime denotes differentiation with respect to the dimensionless coordinate x=x1/L. The dimensionless flight speed is defined by <sup>V</sup>^ <sup>¼</sup> VCrb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rae=2GJr p , where (GJ)r is the torsional stiffness of the baseline design at root. The shear modulus G12,0 of the baseline design can be calculated by taking Vfo = 50%.

Considering the K-th panel of the wing as shown in Figure 11a, and using the transformation y = (1-βx), Eq. (25) takes the form:

$$z\alpha'' + 3\alpha' + a\_k^2 \alpha = 0, \quad \text{(1-\beta x\_{k+1})} \le y \le \left(1 - \beta \mathbf{x}\_k\right) \tag{36}$$

where ak <sup>¼</sup> <sup>V</sup>^ <sup>=</sup><sup>β</sup> ffiffiffiffiffiffiffiffiffiffiffi ^ hkG^ <sup>k</sup> q , ^ hk and G^ <sup>k</sup> are the normalized wall thickness and shear modulus of the kth wing panel, respectively. The general solution of Eq. (26) is:

$$\alpha(y) = A\_1 \frac{J\_2\left(2\sqrt{a\_k y}\right)}{y} - A\_2 \frac{Y\_2\left(2\sqrt{a\_k y}\right)}{y} \tag{37}$$

where J2 and Y2 are Bessel's function of the first and second kind with order 2, respectively [35], and A1 and A2 are the constants of integration. The dimensionless internal torsional moment, T, can be obtained by differentiating Eq. (27) and multiplying by the dimensionless shear rigidity. Applying the boundary conditions at stations (k) and (k + 1), the constants A1 and A2 can be expressed in terms of the state variables at station (k), which can be related to those at station (k+1) by the transfer matrix relation:

$$\begin{Bmatrix} \alpha\_{\mathbf{k}+1} \\ \mathbf{T}\_{\mathbf{k}+1} \end{Bmatrix} = \begin{bmatrix} \mathbf{E}^{(\mathbf{k})} \end{bmatrix} \begin{Bmatrix} \alpha\_{\mathbf{k}} \\ \mathbf{T}\_{\mathbf{k}} \end{Bmatrix} \tag{38}$$

It is now possible to compute the state variables progressively along the wing span by applying continuity requirements of the variables (α, T) among the interconnecting boundaries of the various wing panels. The divergence speed can be calculated by applying the boundary conditions and considering the nontrivial solution of the resulting equations (similar to the procedure outlined in Section 3.1.3). The associated optimization problem may be cast in the following:

$$\begin{aligned} \text{Minimize} &- \hat{V}\_{\text{div}}\\ \text{Subject to} & \hat{M}\_s - 1.0 \le 0 \end{aligned}$$

$$(0.25, 0.5, 0.0) \le \left( V\_f, \hat{h}, \hat{b} \right)\_{k=1, 2, \dots \text{Np}} \le (0.75, 1.25, 1.0) \tag{39}$$

$$\sum \hat{b}\_k = 1.0$$

The preassigned parameters that do not change during the optimization process include the wing semispan (b), the chord taper ration (Δc), airfoil type and geometry, and fiber and resin material types. This model has been applied to obtain wing designs with improved torsional stability by maximizing the divergence speed (Vdiv) without weight penalty. The selected material is carbon-AS4/epoxy-3501-6 composite (see Table 2), which has favorable characteristics and is highly desirable in manufacturing aircraft structures. The baseline design has uniform mass and stiffness distributions and is made of uniform unidirectional fibrous composite with equal volume fraction of the matrix and fiber materials, that is, Vfo = 50%. Figures 12 and 13 show the developed level curves of constant divergence speed (also named isodiverts) for two-panel wings with chord tapering ratio, Δ<sup>c</sup> = 0.5. Actually, these curves represent the dimensionless critical speed, augmented with the equality mass constraint. Examining Figure 12, it is seen that the Vdiv function is well behaved and continuous everywhere in the selected design space except in the empty regions to the upper left and right regions, where the equality mass constraint is violated. The feasible domain is bounded from

respectively (refer to Table 1). C is the chord length and h is the maximum thickness of the cross section. For many types of fibrous composites that are commonly utilized in aerospace industry [23], such as carbon/epoxy and graphite epoxy, both moduli are approximately equal,

Using the classical elasticity and aerodynamic strip theories, the governing differential equa-

<sup>þ</sup> <sup>V</sup>^ <sup>2</sup>

The symbol <sup>G</sup>^ <sup>¼</sup> <sup>G</sup>12=G12, <sup>o</sup> denotes the dimensionless shear modulus, ^<sup>J</sup> <sup>¼</sup> <sup>J</sup>=Jr denotes the dimensionless torsion constant, and the prime denotes differentiation with respect to the dimensionless coordinate x=x1/L. The dimensionless flight speed is defined by <sup>V</sup>^ <sup>¼</sup> VCrb ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

where (GJ)r is the torsional stiffness of the baseline design at root. The shear modulus G12,0 of the

Considering the K-th panel of the wing as shown in Figure 11a, and using the transformation

� � <sup>≤</sup> <sup>y</sup> <sup>≤</sup> <sup>1</sup> � <sup>β</sup>xk

hk and G^ <sup>k</sup> are the normalized wall thickness and shear modulus of

Y<sup>2</sup> 2 ffiffiffiffiffiffi aky � � p

<sup>k</sup>α ¼ 0, 1-βxkþ<sup>1</sup>

J<sup>2</sup> 2 ffiffiffiffiffiffi aky � � p <sup>y</sup> � <sup>A</sup><sup>2</sup>

α<sup>k</sup>þ<sup>1</sup> Tkþ<sup>1</sup>

( )

where J2 and Y2 are Bessel's function of the first and second kind with order 2, respectively [35], and A1 and A2 are the constants of integration. The dimensionless internal torsional moment, T, can be obtained by differentiating Eq. (27) and multiplying by the dimensionless shear rigidity. Applying the boundary conditions at stations (k) and (k + 1), the constants A1 and A2 can be expressed in terms of the state variables at station (k), which can be related to

<sup>¼</sup> <sup>E</sup>ð Þ <sup>k</sup> h i <sup>α</sup><sup>k</sup>

It is now possible to compute the state variables progressively along the wing span by applying continuity requirements of the variables (α, T) among the interconnecting boundaries of the various wing panels. The divergence speed can be calculated by applying the boundary conditions and considering the nontrivial solution of the resulting equations (similar to the procedure outlined in Section 3.1.3). The associated optimization problem may be cast in the

Tk

( )

αð Þ¼ x 0 (35)

� � (36)

<sup>y</sup> (37)

ð Þ¼ 1 0.

rae=2GJr p ,

(38)

<sup>G</sup>^^Jα<sup>0</sup> � �<sup>0</sup>

The associated boundary conditions of the elastic angle of attack, α, are α(0) = 0 and α<sup>0</sup>

tion of torsional stability in dimensionless form is [35]:

baseline design can be calculated by taking Vfo = 50%.

<sup>z</sup>α<sup>00</sup> <sup>þ</sup> <sup>3</sup>α<sup>0</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

the kth wing panel, respectively. The general solution of Eq. (26) is:

αð Þ¼ y A<sup>1</sup>

y = (1-βx), Eq. (25) takes the form:

ffiffiffiffiffiffiffiffiffiffiffi ^ hkG^ <sup>k</sup> q

, ^

those at station (k+1) by the transfer matrix relation:

where ak <sup>¼</sup> <sup>V</sup>^ <sup>=</sup><sup>β</sup>

following:

G12 ≈ G13.

180 Optimum Composite Structures

Figure 12. Isodivert in (Vf1- b1) design space for a two-panel wing model (h1 = h2 = 1.0, <sup>Δ</sup><sup>c</sup> = 0.5, <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1).

above by the two curved lines representing the upper and lower limiting constraints imposed on the volume fraction of the outboard blade panel. The contours inside the feasible domain are not allowed to penetrate these borderlines and obliged to turn sharply to be asymptotes to them, in order not to violate the mass constraint. The final attained optimal solutions are summarized in Table 4. It can be observed that good wing patterns shall have the lower limit of the fiber volume fraction at the tip and the upper limit at root. Using material and wall thickness grading together results in a considerable enhancement of the wing torsional stability.

6. Optimization of composite thin-walled pipes conveying fluid

modulus of elasticity (E) can be determined using the formulas of Table 1.

<sup>þ</sup> <sup>U</sup><sup>2</sup> V00

ð Þ EI <sup>k</sup>V<sup>0000</sup>

Figure 14. Multimodule composite pipe conveying flowing fluid.

equation [39]:

The associated eigenvalue problem is described by the fourth-order ordinary differential

ffiffiffiffi βo q

<sup>V</sup><sup>0</sup> � mkω<sup>2</sup>

V ¼ 0 (40)

þ 2iωU

where V(x) is the dimensionless mode shape satisfying boundary conditions, and ω is the corresponding dimensionless frequency of oscillation, which will be, in general, a complex number to be determined by the requirement of nontrivial solutions, V(x) 6¼ 0. More details for

The subject of vibration and stability of thin pipes conveying flowing fluids is of a considerable practical interest. An advanced textbook by Paїdoussis [36] gives an excellent review of the several developments made in this research area. Practical models for enhancing static and dynamic stability characteristics of pipelines constructed from uniform modules were addressed by Maalawi et al. [37, 38], where the relevant design variables were selected to be the mean diameter, wall thickness, and length of each module composing the pipeline. The general case of an elastically supported pipe, covering a variety of boundary conditions, was also investigated. Distinct domains of the flutter instability boundaries were presented for different ratios of the fluid-to-pipe mass, and the variation of the critical flow velocity with support flexibility was examined and discussed. Concerning pipelines made of advanced FGMs, this section presents a mathematical model for enhancing the overall system stability against flutter and/or divergence under mass constraint. Figure 14 shows a FGM pipe conveying flowing fluid with the coordinate system chosen such that the x-axis coincides with the longitudinal centroidal axis in its undeformed position, while the y- and z-axes coincide with the cross section principal axes. The pipe model consists of rigidly connected thin-walled circular tubes made of unidirectional fibrous composite material. Each pipe module has different material properties, wall thickness, and length. Such a configuration results in a piecewise axial grading of either the material of construction or the wall thickness in the direction of the pipe axis. Assuming no voids are present, the distributions of the mass density (r) and

Optimization of Functionally Graded Material Structures: Some Case Studies

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

183

Figure 13. Isodivert in (Vf1- b1) design space for a two-panel wing model (h2 = 0.5, Vf2 = 0.3, <sup>Δ</sup><sup>c</sup> = 0.5, <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1).

