3. Frequency optimization of FGM thin-walled box beams

This section presents a mathematical model for optimizing the dynamic performance of thinwalled FGM box beams with closed cross sections. The objective function is to maximize the natural frequencies and place them at their target values to avoid the occurrence of large amplitudes of vibration. The variables considered include fiber volume fraction, fiber orientation angle, and ply thickness distributions. Various power-law expressions describing the distribution of the fiber volume fraction have been implemented, where the power exponent was taken as the main optimization variable [25]. The mass of the beam is kept equal to that of a known reference beam. Side constraints are also imposed on the design variables in order to avoid having unacceptable optimal solutions. A case study on the optimization of a cantilevered, single-cell spar beam made of carbon/epoxy composite is considered. The results for the basic case of uncoupled bending motion are given.

NL(j) = number of layers in the j-th segment. k = subscript for the k-th layer, k = 1, 2,…NL(j).

PNs

� � <sup>=</sup> <sup>H</sup>^ <sup>j</sup> <sup>¼</sup> <sup>P</sup>NLð Þ<sup>j</sup>

� ) = normalized length of the j-th segment.

<sup>j</sup>¼<sup>1</sup> <sup>L</sup>^<sup>j</sup> = normalized total beam length.

θkj = fiber orientation angle in the k-th layer in the j-th segment.

r^kj ¼ rkj=r<sup>o</sup> = normalized density of the k-th layer in the j-th segment.

Vf , kj = fiber volume fraction in the k-th layer in the j-th segment.

Γ<sup>j</sup> = πDj for circular C.S., Γ<sup>j</sup> = 2(aj + bj) for rectangular C.S.

<sup>r</sup>kj <sup>=</sup> <sup>r</sup><sup>f</sup> Vf , kj <sup>þ</sup> <sup>r</sup><sup>m</sup> <sup>1</sup> � Vf , kj � �, <sup>r</sup><sup>o</sup> <sup>¼</sup> 0.5(r<sup>f</sup> <sup>+</sup> <sup>r</sup>m).

� � = normalized thickness of the k-th layer in the j-th segment.

<sup>k</sup>¼<sup>1</sup> ^hkj= normalized total wall thickness of the j-th segment.

ds=∮ <sup>o</sup>ds = normalized circumference of the j-th segment cross section.

Figure 4. General configuration of a multisegment, composite box beam [25]. (a) Circular cross section, (b) rectangular

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165

<sup>L</sup>^<sup>j</sup> <sup>¼</sup> Lj=Lo

cross section.

<sup>L</sup>^ <sup>¼</sup> ð Þ¼ <sup>L</sup>=Lo

H^ <sup>j</sup> = Hj=H<sup>0</sup>

^hkj <sup>¼</sup> hkj=H<sup>0</sup>

<sup>Γ</sup>^<sup>j</sup> <sup>¼</sup> <sup>Γ</sup>j=Γ<sup>o</sup> <sup>¼</sup> <sup>∮</sup> <sup>J</sup>

#### 3.1. Structural dynamic analysis

Figure 4 shows a slender, composite thin-walled beam constructed from uniform segments, each of which has different cross-sectional dimensions, material properties, and length. Tapered shapes of an actual blade or wing spar can be adequately approximated by such a piecewise structural model with a sufficient number of segments. The various parameters and variables are normalized with respect to a reference beam, which is constructed from just one segment with single unidirectional lamina having equal fiber and matrix volume fractions, that is, Vfo = Vmo = 50%. The different quantities are defined in the following:

Ns = number of segments (panels).

j = subscript for the j-th segment, j = 1, 2,…….Ns.

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suggested the range 1 < ξ < 2 depending on the fiber array type, for example, hexagonal, square, etc. Usually, ξ is taken equal to 1.0 for theoretical analysis procedures in the case of carbon or

This section presents a mathematical model for optimizing the dynamic performance of thinwalled FGM box beams with closed cross sections. The objective function is to maximize the natural frequencies and place them at their target values to avoid the occurrence of large amplitudes of vibration. The variables considered include fiber volume fraction, fiber orientation angle, and ply thickness distributions. Various power-law expressions describing the distribution of the fiber volume fraction have been implemented, where the power exponent was taken as the main optimization variable [25]. The mass of the beam is kept equal to that of a known reference beam. Side constraints are also imposed on the design variables in order to avoid having unacceptable optimal solutions. A case study on the optimization of a cantilevered, single-cell spar beam made of carbon/epoxy composite is considered. The results for the basic case of

Figure 4 shows a slender, composite thin-walled beam constructed from uniform segments, each of which has different cross-sectional dimensions, material properties, and length. Tapered shapes of an actual blade or wing spar can be adequately approximated by such a piecewise structural model with a sufficient number of segments. The various parameters and variables are normalized with respect to a reference beam, which is constructed from just one segment with single unidirectional lamina having equal fiber and matrix volume fractions, that

3. Frequency optimization of FGM thin-walled box beams

Property Mathematical formula\*

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

Table 1. Halpin-Tsai semiempirical relations for calculating composite properties [23].

Young's modulus in direction (2) E22 Em (1 + ξηVf)/(1-ηVf); η = (E2f –Em)/(E2f + ξEm) Shear modulus G12 Gm (1 + ξηVf)/(1-ηVf); η = (G12f –Gm)/(G12f + ξGm)

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

Poisson's ratio ϑ<sup>12</sup> ϑmVm þ ϑ12<sup>f</sup> Vf Mass density r r<sup>m</sup> Vm + r<sup>f</sup> Vf

is, Vfo = Vmo = 50%. The different quantities are defined in the following:

glass fibrous composite laminates.

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

\*

164 Optimum Composite Structures

uncoupled bending motion are given.

3.1. Structural dynamic analysis

Ns = number of segments (panels).

j = subscript for the j-th segment, j = 1, 2,…….Ns.

Figure 4. General configuration of a multisegment, composite box beam [25]. (a) Circular cross section, (b) rectangular cross section.

k = subscript for the k-th layer, k = 1, 2,…NL(j). <sup>L</sup>^<sup>j</sup> <sup>¼</sup> Lj=Lo � ) = normalized length of the j-th segment. <sup>L</sup>^ <sup>¼</sup> ð Þ¼ <sup>L</sup>=Lo PNs <sup>j</sup>¼<sup>1</sup> <sup>L</sup>^<sup>j</sup> = normalized total beam length. H^ <sup>j</sup> = Hj=H<sup>0</sup> � � <sup>=</sup> <sup>H</sup>^ <sup>j</sup> <sup>¼</sup> <sup>P</sup>NLð Þ<sup>j</sup> <sup>k</sup>¼<sup>1</sup> ^hkj= normalized total wall thickness of the j-th segment. ^hkj <sup>¼</sup> hkj=H<sup>0</sup> � � = normalized thickness of the k-th layer in the j-th segment. θkj = fiber orientation angle in the k-th layer in the j-th segment. <sup>Γ</sup>^<sup>j</sup> <sup>¼</sup> <sup>Γ</sup>j=Γ<sup>o</sup> <sup>¼</sup> <sup>∮</sup> <sup>J</sup> ds=∮ <sup>o</sup>ds = normalized circumference of the j-th segment cross section. Γ<sup>j</sup> = πDj for circular C.S., Γ<sup>j</sup> = 2(aj + bj) for rectangular C.S. r^kj ¼ rkj=r<sup>o</sup> = normalized density of the k-th layer in the j-th segment. Vf , kj = fiber volume fraction in the k-th layer in the j-th segment. <sup>r</sup>kj <sup>=</sup> <sup>r</sup><sup>f</sup> Vf , kj <sup>þ</sup> <sup>r</sup><sup>m</sup> <sup>1</sup> � Vf , kj � �, <sup>r</sup><sup>o</sup> <sup>¼</sup> 0.5(r<sup>f</sup> <sup>+</sup> <sup>r</sup>m).

NL(j) = number of layers in the j-th segment.

r<sup>f</sup> = fiber mass density, r<sup>m</sup> = matrix density.

m^ <sup>j</sup> ¼ mj=mo � � <sup>¼</sup> normalized mass per unit length of the j-th segment.

$$m\_{\rangle} = \Gamma\_{\rangle} \sum\_{k=1}^{N\_{\mathbb{L}}(j)} \rho\_{k\uparrow} h\_{k\uparrow} = \text{mass per unit length of the } j\text{-th segment, } m\_{\circ} = \Gamma\_{\circ} \rho\_{\circ} H\_{\circ}.$$

Ij = mass polar moment of inertia per unit length of the j-th segment.

$$\rho = \sum\_{k}^{N\_L(j)} \oint \rho\_{kj} h\_{kj} \left( y^2 + z^2 \right) \, ds.$$

The normalized total structural mass is given by the expression:

$$
\hat{M}\_s = M\_s / M\_0 = \sum\_{j=1}^{N\_s} \hat{M}\_j = \sum\_{j=1}^{N\_s} \hat{m}\_j \hat{L}\_j = \sum\_{j=1}^{N\_s} \hat{\Gamma}\_j \hat{L}\_j \sum\_{k=1}^{N\_L(j)} \hat{\rho}\_{kj} \hat{h}\_{kj} \tag{6}
$$

the extension displacement (U1) is uncoupled, as well as the edgewise bending (U2), while the flapping displacement (U3) is coupled with twist (φ). The general solution can be obtained by separating the space and time variables, where the time dependence is assumed to be harmonic with circular frequency, ω. The solutions for the uncoupled axial and bending equations are

The basic important case to be considered first is the uncoupled bending response, which exists in both CUS and CAS layup configurations. Using the multisegment model depicted in Figure 4 and considering flapping motion (U3), the associated eigenvalue problem can be written directly

<sup>3</sup> � <sup>ω</sup><sup>2</sup>

which must be satisfied over the length Lj of any segment composing the beam structure.

in the interval 0 <sup>≤</sup> <sup>x</sup> <sup>≤</sup> <sup>L</sup>^j, where <sup>x</sup> <sup>¼</sup> <sup>x</sup>^ � <sup>x</sup>^<sup>j</sup> is a local coordinate of the j-th segment and

j

Expressing the constants ai, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, 4 in terms of the state variables vector {S}<sup>T</sup> <sup>=</sup>

where [T(j)] is called the transfer matrix of the j-th segment with its elements given in detail in Ref. [25]. The state variable vectors can be computed progressively along the length of the beam by applying continuity among the interconnecting joints of the different segments composing the beam structure. An overall transfer matrix denoted by [T], which relates the state variables at both ends of the beam, can be obtained from the following matrix

The required frequency equation for determining the natural frequencies can then be obtained by applying the associated boundary conditions and considering only the nontrivial solution

<sup>x</sup> <sup>þ</sup> <sup>a</sup><sup>3</sup> sinh <sup>β</sup>^

<sup>3</sup> <sup>g</sup><sup>T</sup> at both ends of the j-th segment, we get

mjU<sup>3</sup> ¼ 0 (8)

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167

<sup>j</sup> <sup>U</sup>^ <sup>3</sup> <sup>¼</sup> <sup>0</sup> (9)

j

. Eq. (9) must be satisfied

x (10)

0 m<sup>0</sup> C33, <sup>0</sup> � �<sup>1</sup>=<sup>2</sup>

j

½ �¼ <sup>T</sup> <sup>T</sup>ð Þ Ns h i <sup>T</sup>ð Þ Ns�<sup>1</sup> h i……:: <sup>T</sup>ð Þ<sup>2</sup> h i <sup>T</sup>ð Þ<sup>1</sup> h i (12)

<sup>x</sup> <sup>þ</sup> <sup>a</sup><sup>4</sup> cosh <sup>β</sup>^

f g<sup>S</sup> <sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>T</sup>ð Þ<sup>j</sup> h if g<sup>S</sup> <sup>j</sup> (11)

straightforward, while those for the coupled equations involve much mathematics [26].

C33,jU<sup>0000</sup>

U^ <sup>0000</sup> <sup>3</sup> � <sup>β</sup>^<sup>4</sup>

, <sup>C</sup>^33,j <sup>¼</sup> <sup>C</sup>33,j=C33,0, and <sup>ω</sup>^ <sup>¼</sup> <sup>ω</sup>L<sup>2</sup>

<sup>x</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup> cos <sup>β</sup>^

3.1.3. Solution procedure of uncoupled bending motion

Normalizing with respect to the reference beam, we get:

x^ ¼ ð Þ x=Lo : The general solution is well known to be:

<sup>3</sup> � C33U<sup>000</sup>

<sup>U</sup>^ <sup>3</sup>ð Þ¼ <sup>x</sup> <sup>a</sup><sup>1</sup> sin <sup>β</sup>^

j

in the form:

where β^ j = ffiffiffiffi

{U<sup>3</sup> � U<sup>0</sup>

multiplication:

<sup>ω</sup>^ <sup>p</sup> <sup>m</sup>^ <sup>j</sup>=C^33,j � �<sup>1</sup>=<sup>4</sup>

<sup>3</sup> � C33U<sup>00</sup>

of the resulting matrix equation.

where Mo ¼ moLo ¼ ΓoroHoLo is the total mass of the uniform baseline design. A quantity with subscript "o" refers to a reference beam parameter.

#### 3.1.1. Constitutive relationships

The displacement field of anisotropic thin-walled closed cross-sectional beams was derived by Dancila and Armanios [26], who used a variational asymptotic approach to obtain the following constitutive equations:

$$\begin{Bmatrix} F\_x\\ M\_x\\ M\_y\\ M\_z \end{Bmatrix} = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} & \mathbf{C}\_{14} \\\\ \mathbf{C}\_{12} & \mathbf{C}\_{22} & \mathbf{C}\_{23} & \mathbf{C}\_{24} \\\\ \mathbf{C}\_{13} & \mathbf{C}\_{23} & \mathbf{C}\_{33} & \mathbf{C}\_{34} \\\\ \mathbf{C}\_{14} & \mathbf{C}\_{24} & \mathbf{C}\_{34} & \mathbf{C}\_{44} \end{bmatrix} \begin{Bmatrix} \boldsymbol{U}'\_1\\ \boldsymbol{\phi}' \\ \boldsymbol{\upmu}'\\ \boldsymbol{\upmu}''\\ \boldsymbol{\upmu}''' \end{Bmatrix} \tag{7}$$

where Fx, Mx, My, and Mz stand for the axial force, torsional, and bending moments, respectively, and Cmn are the cross-sectional stiffness coefficients derived in terms of closed-form integrals of the geometry and material constants. The notations U1, U2, U3, and ϕ are the kinematic variables representing the average displacements and rotation of the cross section. The primes denote differentiation with respect to x.

#### 3.1.2. Equations of motion

The general equations of motion for the free vibration analysis are derived using Hamilton's principle and expressed in terms of the kinematic variables, where it was shown that a closed form solution is not available [25]. However, particular choices of cross-sectional shape and layup can produce zero coupling coefficients in the equations of motion. Two special layup configurations can be considered, namely circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS). The equations of the CUS type consist of two coupled equations for extension-twist and two uncoupled bending equations. For the CAS type, the extension displacement (U1) is uncoupled, as well as the edgewise bending (U2), while the flapping displacement (U3) is coupled with twist (φ). The general solution can be obtained by separating the space and time variables, where the time dependence is assumed to be harmonic with circular frequency, ω. The solutions for the uncoupled axial and bending equations are straightforward, while those for the coupled equations involve much mathematics [26].

#### 3.1.3. Solution procedure of uncoupled bending motion

r<sup>f</sup> = fiber mass density, r<sup>m</sup> = matrix density.

<sup>k</sup> <sup>∮</sup> <sup>r</sup>kjhkj <sup>y</sup><sup>2</sup> <sup>þ</sup> <sup>z</sup><sup>2</sup> � � ds.

3.1.1. Constitutive relationships

ing constitutive equations:

3.1.2. Equations of motion

� � <sup>¼</sup> normalized mass per unit length of the j-th segment.

Ns

j¼1

Ij = mass polar moment of inertia per unit length of the j-th segment.

The normalized total structural mass is given by the expression:

<sup>M</sup><sup>b</sup> <sup>s</sup> <sup>¼</sup> Ms=M<sup>0</sup> <sup>¼</sup> <sup>X</sup>

subscript "o" refers to a reference beam parameter.

Fx Mx My Mz 9 >>>>>=

>>>>>; ¼

8 >>>>><

>>>>>:

The primes denote differentiation with respect to x.

<sup>k</sup>¼<sup>1</sup> <sup>r</sup>kjhkj = mass per unit length of the j-th segment, mo <sup>¼</sup> <sup>Γ</sup>oroHo.

<sup>M</sup><sup>b</sup> <sup>j</sup> <sup>¼</sup> <sup>X</sup> Ns

j¼1

where Mo ¼ moLo ¼ ΓoroHoLo is the total mass of the uniform baseline design. A quantity with

The displacement field of anisotropic thin-walled closed cross-sectional beams was derived by Dancila and Armanios [26], who used a variational asymptotic approach to obtain the follow-

C<sup>11</sup> C<sup>12</sup>

C<sup>12</sup> C<sup>22</sup>

C<sup>13</sup> C<sup>23</sup>

C<sup>14</sup> C<sup>24</sup>

where Fx, Mx, My, and Mz stand for the axial force, torsional, and bending moments, respectively, and Cmn are the cross-sectional stiffness coefficients derived in terms of closed-form integrals of the geometry and material constants. The notations U1, U2, U3, and ϕ are the kinematic variables representing the average displacements and rotation of the cross section.

The general equations of motion for the free vibration analysis are derived using Hamilton's principle and expressed in terms of the kinematic variables, where it was shown that a closed form solution is not available [25]. However, particular choices of cross-sectional shape and layup can produce zero coupling coefficients in the equations of motion. Two special layup configurations can be considered, namely circumferentially uniform stiffness (CUS) and circumferentially asymmetric stiffness (CAS). The equations of the CUS type consist of two coupled equations for extension-twist and two uncoupled bending equations. For the CAS type,

<sup>m</sup>^ jL^<sup>j</sup> <sup>¼</sup> <sup>X</sup> Ns

C<sup>13</sup> C<sup>14</sup>

8 >>>>><

>>>>>:

U0 1 φ0 U<sup>00</sup> 3 U<sup>00</sup> 2 9 >>>>>=

>>>>>;

C<sup>23</sup> C<sup>24</sup>

C<sup>33</sup> C<sup>34</sup>

C<sup>34</sup> C<sup>44</sup>

j¼1

Γ^jL^<sup>j</sup> N XLð Þj k¼1

<sup>r</sup>^kj^hkj (6)

(7)

m^ <sup>j</sup> ¼ mj=mo

= PNLð Þ<sup>j</sup>

PNLð Þ<sup>j</sup>

166 Optimum Composite Structures

mj ¼ Γ<sup>j</sup>

The basic important case to be considered first is the uncoupled bending response, which exists in both CUS and CAS layup configurations. Using the multisegment model depicted in Figure 4 and considering flapping motion (U3), the associated eigenvalue problem can be written directly in the form:

$$\mathbf{C}\_{33,j}\mathbf{U}\_3^{\prime\prime\prime} - \omega^2 \mathbf{m}\_j \mathbf{U}\_3 = \mathbf{0} \tag{8}$$

which must be satisfied over the length Lj of any segment composing the beam structure. Normalizing with respect to the reference beam, we get:

$$
\hat{\mathbf{U}}\_3^{\prime\prime\prime} - \hat{\boldsymbol{\beta}}\_j^4 \hat{\mathbf{U}}\_3 = \mathbf{0} \tag{9}
$$

where β^ j = ffiffiffiffi <sup>ω</sup>^ <sup>p</sup> <sup>m</sup>^ <sup>j</sup>=C^33,j � �<sup>1</sup>=<sup>4</sup> , <sup>C</sup>^33,j <sup>¼</sup> <sup>C</sup>33,j=C33,0, and <sup>ω</sup>^ <sup>¼</sup> <sup>ω</sup>L<sup>2</sup> 0 m<sup>0</sup> C33, <sup>0</sup> � �<sup>1</sup>=<sup>2</sup> . Eq. (9) must be satisfied in the interval 0 <sup>≤</sup> <sup>x</sup> <sup>≤</sup> <sup>L</sup>^j, where <sup>x</sup> <sup>¼</sup> <sup>x</sup>^ � <sup>x</sup>^<sup>j</sup> is a local coordinate of the j-th segment and x^ ¼ ð Þ x=Lo : The general solution is well known to be:

$$\hat{L}\_3(\overline{\mathbf{x}}) = a\_1 \sin \hat{\boldsymbol{\beta}}\_{\uparrow} \overline{\mathbf{x}} + a\_2 \cos \hat{\boldsymbol{\beta}}\_{\uparrow} \overline{\mathbf{x}} + a\_3 \sinh \hat{\boldsymbol{\beta}}\_{\uparrow} \overline{\mathbf{x}} + a\_4 \cosh \hat{\boldsymbol{\beta}}\_{\uparrow} \overline{\mathbf{x}} \tag{10}$$

Expressing the constants ai, i <sup>¼</sup> <sup>1</sup>, <sup>2</sup>, <sup>3</sup>, 4 in terms of the state variables vector {S}<sup>T</sup> <sup>=</sup> {U<sup>3</sup> � U<sup>0</sup> <sup>3</sup> � C33U<sup>00</sup> <sup>3</sup> � C33U<sup>000</sup> <sup>3</sup> <sup>g</sup><sup>T</sup> at both ends of the j-th segment, we get

$$\{\mathcal{S}\}\_{j+1} = \left[T^{(j)}\right] \{\mathcal{S}\}\_{j} \tag{11}$$

where [T(j)] is called the transfer matrix of the j-th segment with its elements given in detail in Ref. [25]. The state variable vectors can be computed progressively along the length of the beam by applying continuity among the interconnecting joints of the different segments composing the beam structure. An overall transfer matrix denoted by [T], which relates the state variables at both ends of the beam, can be obtained from the following matrix multiplication:

$$\mathbb{E}\left[T\right] = \left[T^{(\text{Ns})}\right] \left[T^{(\text{Ns}-1)}\right] \dots \dots \dots \left[T^{(2)}\right] \left[T^{(1)}\right] \tag{12}$$

The required frequency equation for determining the natural frequencies can then be obtained by applying the associated boundary conditions and considering only the nontrivial solution of the resulting matrix equation.

#### 3.2. Formulation of the optimization problem

Several design objectives can exist in structural optimization including minimum mass, maximum natural frequencies, minimum manufacturing cost, etc. [17]. Considering the reduction of vibration level, two optimization alternatives can be formulated, namely, frequency placement by separating the natural frequencies from the harmonics of the excitations or direct maximization of the natural frequencies. The latter can ensure a simultaneous balanced improvement in both stiffness and mass distributions of the vibrating structure. The related optimization problems are usually formulated as nonlinear mathematical programming problem where iterative techniques are implemented for finding the optimal solution in the selected design space. Numerous computer programs [18] are available to solve nonlinear optimization models, which can be interacted with structural and eigenvalue analyses routines. The MATLAB toolbox optimization routines can be useful in solving some types of unconstrained and constrained optimization problems. One of the most commonly applied routines that find the constrained optima of a nonlinear merit function of many variables is named "fmincon" [19].

Figure 5 depicts the functional behavior of the dimensionless fundamental frequency param-

It is remarked that the function is continuous and well behaved everywhere in (Vf – θ) design space. The contours are symmetric about the horizontal line θ = 0 where the constrained global maxima occurs when the fiber volume fraction reaches its upper limiting value. It can then be concluded that the unidirectional lamina is favorable when considering beam designs with

Property Fiber: Carbon-AS4 Matrix: Epoxy-3501-6

E2f = 15.0

Modulus of rigidity (GPa) G12f = 27.0 Gm = 1.60 Poisson's ratio ν12f = 0.20 ν<sup>m</sup> = 0.35

) r<sup>f</sup> = 1.81 r<sup>m</sup> = 1.27

<sup>p</sup> function augmented with the constraintM^ <sup>s</sup> <sup>¼</sup> 1 in Vf � <sup>θ</sup> � � design space (Ns = 1, NL = 1, <sup>L</sup>^ <sup>¼</sup> <sup>1</sup>Þ.

ð Þ <sup>0</sup>:25; <sup>0</sup>:75; �π=<sup>2</sup> <sup>≤</sup> Vf ; <sup>H</sup>^ ; <sup>θ</sup>

Modulus of elasticity (GPa) E1f = 235.0

Table 2. Material properties of fiber and matrix materials [23].

<sup>p</sup> combined with the structural mass constraint (M^ <sup>s</sup> <sup>¼</sup> 1). The imposed side constraints

� � <sup>≤</sup> ð Þ <sup>0</sup>:75; <sup>1</sup>:25; <sup>π</sup>=<sup>2</sup> (15)

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Optimization of Functionally Graded Material Structures: Some Case Studies

Em = 4.30

eter ffiffiffiffiffiffi ω^ <sup>1</sup>

Density (g/cm<sup>3</sup>

Figure 5. Level curves of ffiffiffiffiffiffi

ω^ <sup>1</sup>

are:

#### 3.2.1. Basic optimization problem

Before performing the necessary mathematics, it is essential to recognize that design optimization is only as meaningful as its core model of structural analysis. Any deficiencies therein will absolutely be affected in the optimization process. Consider the basic problem of a uniform cantilevered, thin-walled, single-cell spar constructed from just one segment with one unidirectional lamina (Ns = 1, NL = 1). The total length and outer cross-sectional dimensions are given preassigned values equal to those of the baseline design. The remaining set of variables is, therefore, X = Vf ; H^ ; θ � �. The associated frequency equation for such a basic case is:

$$\cos\hat{\beta}\hat{L}\cosh\hat{\beta}\hat{L} = -1,\\
or\ \cos\sqrt{\hat{\alpha}}\left(\hat{m}/\hat{\mathsf{C}}\_{33}\right)^{\frac{1}{4}}\hat{L}\cosh\sqrt{\hat{\alpha}}\left(\hat{m}/\hat{\mathsf{C}}\_{33}\right)^{\frac{1}{4}}\hat{L} = -1\tag{13}$$

It is seen that ffiffiffiffi <sup>ω</sup>^ <sup>p</sup> is an implicit function of the design variables and can be calculated numerically by any suitable method such as Newton-Raphson or the Bisection method. However, the frequency equation can be solved directly for the whole term ffiffiffiffi <sup>ω</sup>^ <sup>p</sup> <sup>m</sup>^ <sup>=</sup>C^<sup>33</sup> � �<sup>1</sup> 4 L^ without regard to the specific values of the design variables. The computed roots are:

$$\sqrt{\hat{\omega}\_i} = \left(\hat{\mathbb{C}}\_{33}/\hat{m}\right)^{\dagger} \begin{pmatrix} 1\\ \hat{L} \end{pmatrix} \begin{pmatrix} 1.8751, 4.6941, 7.8548, \dots \\ \end{pmatrix} \begin{pmatrix} -0.5\\ \end{pmatrix} \quad i \ge 4 \tag{14}$$

In Eq. (14), the frequency parameter ffiffiffiffiffi ω^ i <sup>p</sup> can be imagined as an explicit function of the design variables. So, for prescribed values of the design variables within the domain of side constraints, ffiffiffiffiffi ω^ i <sup>p</sup> can be obtained directly from the above equation. Therefore, it is possible to place the frequency at its desired value and obtain the corresponding value of any one of the design variables directly from Eq. (14). The selected composite material of construction is made of epoxy-3501-6 and carbon-AS4 (see Table 2), which has favorable properties and is highly recommended in many applications of civil, aerospace, and mechanical engineering [23].

Figure 5 depicts the functional behavior of the dimensionless fundamental frequency parameter ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> combined with the structural mass constraint (M^ <sup>s</sup> <sup>¼</sup> 1). The imposed side constraints are:

$$(0.25, 0.75, -\pi/2) \le \left(V\_f, \hat{H}, \theta\right) \le (0.75, 1.25, \pi/2) \tag{15}$$

It is remarked that the function is continuous and well behaved everywhere in (Vf – θ) design space. The contours are symmetric about the horizontal line θ = 0 where the constrained global maxima occurs when the fiber volume fraction reaches its upper limiting value. It can then be concluded that the unidirectional lamina is favorable when considering beam designs with


Table 2. Material properties of fiber and matrix materials [23].

3.2. Formulation of the optimization problem

168 Optimum Composite Structures

3.2.1. Basic optimization problem

is, therefore, X = Vf ; H^ ; θ

It is seen that ffiffiffiffi

straints, ffiffiffiffiffi

ω^ i

� �

^ <sup>β</sup>^L^ ¼ �1, or cos ffiffiffiffi

frequency equation can be solved directly for the whole term ffiffiffiffi

� �<sup>1</sup>

the specific values of the design variables. The computed roots are:

<sup>4</sup> 1 L^ � �

ω^ i

cos β^L cosh

ffiffiffiffiffi ω^ i <sup>p</sup> <sup>¼</sup> <sup>C</sup>^33=m^

In Eq. (14), the frequency parameter ffiffiffiffiffi

a nonlinear merit function of many variables is named "fmincon" [19].

Several design objectives can exist in structural optimization including minimum mass, maximum natural frequencies, minimum manufacturing cost, etc. [17]. Considering the reduction of vibration level, two optimization alternatives can be formulated, namely, frequency placement by separating the natural frequencies from the harmonics of the excitations or direct maximization of the natural frequencies. The latter can ensure a simultaneous balanced improvement in both stiffness and mass distributions of the vibrating structure. The related optimization problems are usually formulated as nonlinear mathematical programming problem where iterative techniques are implemented for finding the optimal solution in the selected design space. Numerous computer programs [18] are available to solve nonlinear optimization models, which can be interacted with structural and eigenvalue analyses routines. The MATLAB toolbox optimization routines can be useful in solving some types of unconstrained and constrained optimization problems. One of the most commonly applied routines that find the constrained optima of

Before performing the necessary mathematics, it is essential to recognize that design optimization is only as meaningful as its core model of structural analysis. Any deficiencies therein will absolutely be affected in the optimization process. Consider the basic problem of a uniform cantilevered, thin-walled, single-cell spar constructed from just one segment with one unidirectional lamina (Ns = 1, NL = 1). The total length and outer cross-sectional dimensions are given preassigned values equal to those of the baseline design. The remaining set of variables

> <sup>ω</sup>^ <sup>p</sup> <sup>m</sup>^ <sup>=</sup>C^<sup>33</sup> � �<sup>1</sup> 4 L cosh ^ ffiffiffiffi

ically by any suitable method such as Newton-Raphson or the Bisection method. However, the

variables. So, for prescribed values of the design variables within the domain of side con-

the frequency at its desired value and obtain the corresponding value of any one of the design variables directly from Eq. (14). The selected composite material of construction is made of epoxy-3501-6 and carbon-AS4 (see Table 2), which has favorable properties and is highly recommended in many applications of civil, aerospace, and mechanical engineering [23].

<sup>p</sup> can be obtained directly from the above equation. Therefore, it is possible to place

. The associated frequency equation for such a basic case is:

<sup>ω</sup>^ <sup>p</sup> is an implicit function of the design variables and can be calculated numer-

<sup>ω</sup>^ <sup>p</sup> <sup>m</sup>^ <sup>=</sup>C^<sup>33</sup> � �<sup>1</sup> 4

<sup>ω</sup>^ <sup>p</sup> <sup>m</sup>^ <sup>=</sup>C^<sup>33</sup> � �<sup>1</sup> 4

ð1:8751, 4:6941, 7:8548, …πð Þ i � 0:5 i ≥ 4 (14)

<sup>p</sup> can be imagined as an explicit function of the design

<sup>L</sup>^ ¼ �1 (13)

L^ without regard to

Figure 5. Level curves of ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> function augmented with the constraintM^ <sup>s</sup> <sup>¼</sup> 1 in Vf � <sup>θ</sup> � � design space (Ns = 1, NL = 1, <sup>L</sup>^ <sup>¼</sup> <sup>1</sup>Þ.

maximum bending frequency. The optimal design point was found to be (Vf, H, ^ θ)=(0.75, 0.92, 0) at which ffiffiffiffiffiffi ω1 � � p max ¼ 2:02589. This corresponds to an optimization gain of about 8.04% as measured from the reference value 1.8751. Before ending this section, it is interesting to address here the dual optimization problem of minimizing the total structural mass under preserved frequency ( ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> =1.8751). The optimal solution was calculated to be (Vf, H, ^ <sup>θ</sup>)=(0.50, 0.915, 0) and M^ s,min = 0.915, which corresponds to a mass saving of 8.5% as compared to the baseline design.

A couple of words are stated here regarding the side constraints in Eq. (15). First of all, it is reminded that the main focus of the present study is to optimize the fiber volume fraction in order to achieve higher values of the natural frequencies without mass penalty. The optimization is performed with respect to a known baseline design, which is considered to be conservative having reserve strength to withstand severe dynamic loads. The imposed side constraint on the total wall thickness, normalized with respect to that of the baseline design, is included for consideration of strength and stability requirements, which are not considered in the present study. So, the imposed limits with a percentage of 25% below or above that of the baseline can be practically accepted for the given model formulation. On the other hand, appropriate values of the upper and lower bounds imposed on the fiber volume fraction are chosen to avoid having unacceptable designs from the manufacture point of view. For example, the filament winding is usually associated with the highest fiber volume fractions. With careful control of fiber tension and resin content, values of around 75% would be reasonable [27].

#### 3.2.2. Optimization model for discrete grading

A comprehensive analysis and formulation of discrete optimization models for beam structures considering both stability and dynamic performance were formulated in [28], where mathematical programming coupled with finite element analysis procedures was implemented. For the case of a two-segment spar beam, (Ns = 2, NL = 1), the reduced optimization problem can be defined as follows:

$$\begin{aligned} \text{Minimize } \mathbf{F}(\underline{\mathbf{X}}) &= -\Big(\sqrt{\hat{\boldsymbol{\alpha}}\_{1}}\big) \\ \text{Subject to } \hat{\boldsymbol{M}}\_{s} &= 1 \\ \sum\_{1}^{\text{Ns}} \hat{L}\_{j} &= 1 \\ (0.25, 0.0) &\leq \Big(\text{Vf}\_{j}, \hat{L}\_{j}\Big)\_{j=1,2} \leq (0.75, 1.0) \end{aligned} \tag{16}$$

mass constraint is indicated. In the left one, the fiber volume fraction is equal to 100%, violating the imposed side constraint. The feasible domain is seen to be split into two distinct zones separated by the baseline contour, which is represented by the vertical line Vf1 = 50%. The constrained optimum is to found to be (Vfj, <sup>L</sup>^jÞ<sup>j</sup>¼1,<sup>2</sup> = (0.75, 0.50), (0.25, 0.50) corresponding

� � design space (Ns = 2).

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171

For continuous grading models, the associated optimization problem is cast as follows: find

<sup>F</sup>ð Þ¼� <sup>X</sup> ffiffiffiffiffiffi

<sup>M</sup>^ <sup>s</sup> <sup>¼</sup> <sup>1</sup> 0:33 ≤ Δ<sup>f</sup> ≤ 3:0 P ≥ 0

ω^ <sup>1</sup> p

(17)

the design variables vector X=(Δf, p), which minimizes the objective function:

<sup>p</sup> function augmented with <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1 in Vf <sup>1</sup>; <sup>L</sup>^<sup>1</sup>

<sup>p</sup> )max = 2.0645 with 10.10% optimization gain.

3.2.3. Optimization model for continuous grading

ω^ <sup>1</sup>

subject to the constraints:

Figure 6. Level curves of ffiffiffiffiffiffi

to ( ffiffiffiffiffiffi ω^ <sup>1</sup>

Using the equality constraints, two of the design variables can be expressed in terms of the other two variables. Figure 6 shows the functional behavior of the dimensionless frequency combined with the structural mass constraint. It is remarked that the function is well defined in the feasible domain of the selected design space (VA �L) ^ <sup>1</sup>. Two empty regions can be observed at the upper left and right parts of the design space, where violation of the equality

Figure 6. Level curves of ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> function augmented with <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1 in Vf <sup>1</sup>; <sup>L</sup>^<sup>1</sup> � � design space (Ns = 2).

mass constraint is indicated. In the left one, the fiber volume fraction is equal to 100%, violating the imposed side constraint. The feasible domain is seen to be split into two distinct zones separated by the baseline contour, which is represented by the vertical line Vf1 = 50%. The constrained optimum is to found to be (Vfj, <sup>L</sup>^jÞ<sup>j</sup>¼1,<sup>2</sup> = (0.75, 0.50), (0.25, 0.50) corresponding to ( ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> )max = 2.0645 with 10.10% optimization gain.

#### 3.2.3. Optimization model for continuous grading

For continuous grading models, the associated optimization problem is cast as follows: find the design variables vector X=(Δf, p), which minimizes the objective function:

$$F(\underline{X}) = -\sqrt{\hat{w}\_1}$$

subject to the constraints:

maximum bending frequency. The optimal design point was found to be (Vf, H,

and resin content, values of around 75% would be reasonable [27].

ð Þ <sup>0</sup>:25; <sup>0</sup>:<sup>0</sup> <sup>≤</sup> Vfj; <sup>L</sup>^<sup>j</sup>

in the feasible domain of the selected design space (VA �L)

� �

Using the equality constraints, two of the design variables can be expressed in terms of the other two variables. Figure 6 shows the functional behavior of the dimensionless frequency combined with the structural mass constraint. It is remarked that the function is well defined

observed at the upper left and right parts of the design space, where violation of the equality

3.2.2. Optimization model for discrete grading

defined as follows:

<sup>p</sup> =1.8751). The optimal solution was calculated to be (Vf, H,

max ¼ 2:02589. This corresponds to an optimization gain of about 8.04% as

measured from the reference value 1.8751. Before ending this section, it is interesting to address here the dual optimization problem of minimizing the total structural mass under preserved

M^ s,min = 0.915, which corresponds to a mass saving of 8.5% as compared to the baseline design.

A couple of words are stated here regarding the side constraints in Eq. (15). First of all, it is reminded that the main focus of the present study is to optimize the fiber volume fraction in order to achieve higher values of the natural frequencies without mass penalty. The optimization is performed with respect to a known baseline design, which is considered to be conservative having reserve strength to withstand severe dynamic loads. The imposed side constraint on the total wall thickness, normalized with respect to that of the baseline design, is included for consideration of strength and stability requirements, which are not considered in the present study. So, the imposed limits with a percentage of 25% below or above that of the baseline can be practically accepted for the given model formulation. On the other hand, appropriate values of the upper and lower bounds imposed on the fiber volume fraction are chosen to avoid having unacceptable designs from the manufacture point of view. For example, the filament winding is usually associated with the highest fiber volume fractions. With careful control of fiber tension

A comprehensive analysis and formulation of discrete optimization models for beam structures considering both stability and dynamic performance were formulated in [28], where mathematical programming coupled with finite element analysis procedures was implemented. For the case of a two-segment spar beam, (Ns = 2, NL = 1), the reduced optimization problem can be

Minimize Fð Þ¼ <sup>X</sup> – ffiffiffiffiffiffi

j¼1,2

Subject to <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> <sup>1</sup>

X Ns

1

≤ ð Þ 0:75; 1:0

ω^ <sup>1</sup> � � p

<sup>L</sup>^<sup>j</sup> <sup>¼</sup> <sup>1</sup>

at which ffiffiffiffiffiffi

170 Optimum Composite Structures

frequency ( ffiffiffiffiffiffi

ω1 � � p

ω^ <sup>1</sup>

^ θ)=(0.75, 0.92, 0)

^ θ)=(0.50, 0.915, 0) and

(16)

^ <sup>1</sup>. Two empty regions can be

$$\begin{aligned} \hat{M}\_s &= 1\\ 0.33 \le \Delta\_f \le 3.0\\ P &\ge 0 \end{aligned} \tag{17}$$

Solutions obtained by applying the power-law model of Eq. (3) have shown that no improvements can be achieved using grading of the fiber volume fraction in the thickness direction. On the other hand, grading in spanwise direction has shown some interesting results. Considering spanwise grading according to Eq. (4), Figure 7 depicts the level curves of the fundamental frequency parameter ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> combined with the mass constraint in the design space <sup>Δ</sup><sup>f</sup> ; <sup>p</sup> � �. It is observed that the feasible domain is bounded from below and above by the constraint curves corresponding to the upper and lower bounds imposed on the fiber volume fractions at tip and root. The horizontal line Δ<sup>f</sup> = 1.0 (i.e., Vf = 50% at root and tip) split the domain into two zones. The lower zone encompasses the constrained optimum solution: ( ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> )max <sup>=</sup> 2.01875 at the design point (Δf, P)opt. = (0.34, 1.01).

Table 3 summarizes the attained optimal solutions for the different grading patterns. It is seen that the highest optimization gain is obtained by using spanwise grading of Eq. (5) with the

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One of the important design issues in mechanical industries is the buckling and whirling instabilities that may arise from the loads applied to a power transmission shaft. These instabilities result in a reduced control of the vehicle, undesirable performance, and often cause damage, sometimes catastrophic, to the vehicle structure. Therefore, by incorporating such considerations into an early design optimization [29], the design space of a power transmission shaft will be reduced such that undesirable instability effects can be avoided during the range of the vehicle's mission profile. Figure 8 shows an idealized structural model of a long, slender composite shaft having circular thin-walled cross section. The main structure is constructed solely of functionally graded, fibrous composite materials. The laminate coordinates are defined by x parallel to the shaft axis, y points to the tangential direction, and z points to the radial direction. Predictions of both torsional buckling and whirling instabilities are based on simplified analytical solutions of equivalent beam and shell structures. The coupling between bending and torsional deformations, introduced by the composite construction, and its influ-

Bert and Kim [30] derived the governing differential equations of torsional buckling in the form:

Nx,x þ Nyx,y � 2Tu,xy ¼ 0 Nxy,x <sup>þ</sup> Ny, <sup>y</sup> <sup>þ</sup> Mxy, <sup>x</sup>=<sup>R</sup> <sup>þ</sup> My,y=<sup>R</sup> � <sup>2</sup>T v,y <sup>þ</sup> w, <sup>x</sup>=<sup>R</sup> <sup>¼</sup> <sup>0</sup>

where Nx and Ny are the normal forces, Nxy and Nyx are shear forces, Mx and My are bending moments, and Mxy and Myx are torsional moments. All are applied to the midsurface and measured per unit wall thickness of the shaft. T is the applied torque, R is the mean radius, and (u, v, w) are the displacements of a generic point on the middle surface of the shaft wall. An iterative process is outlined in Ref. [30] for calculating the buckling torque for specified boundary conditions. There are other simple empirical equations based on experimental studies that can give a reasonable estimate of the buckling torque. The most commonly used formula for the case

<sup>H</sup> ð Þ <sup>0</sup>:<sup>272</sup> ð Þ Ex <sup>0</sup>:<sup>25</sup> Ey

where Tcr is the critical buckling torque and H is the total wall thickness of the shaft. Expressions of the equivalent modulii of elasticity in the axial (Ex) and hoop (Ey) directions for

,xy <sup>þ</sup> My, yy � Ny=<sup>R</sup> <sup>þ</sup> <sup>2</sup>T v,x=<sup>R</sup> � w,xy <sup>¼</sup> <sup>0</sup>

<sup>0</sup>:<sup>75</sup>ð Þ <sup>H</sup>=<sup>R</sup> <sup>1</sup>:<sup>5</sup> (19)

(18)

4. Optimization of FGM drive shafts against torsional buckling and

coordinate exponent n=3.

ence on such instabilities is considered.

of simply supported shaft is [31]:

4.1. Torsional buckling optimization problem

Mx, xx <sup>þ</sup> Mxy <sup>þ</sup> Myx

Tcr <sup>¼</sup> <sup>2</sup>πR<sup>2</sup>

whirling

Figure 7. Level curves of ffiffiffiffiffiffi ω^ <sup>1</sup> <sup>p</sup> function augmented with <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1 in <sup>Δ</sup><sup>f</sup> ; <sup>p</sup> � � design space (Ns = NL = 1) with spanwise grading "Eq. 4."


Table 3. Optimal solutions using different grading patterns (M^ <sup>s</sup> = 1).

Table 3 summarizes the attained optimal solutions for the different grading patterns. It is seen that the highest optimization gain is obtained by using spanwise grading of Eq. (5) with the coordinate exponent n=3.
