4. Application: buckling optimization of anisotropic cylindrical shells

the hoop strain (ε<sup>0</sup>

4.1. Analytical buckling model

sure are cast in the following:

angular position φ and β ¼ vo � w<sup>0</sup>

sion [14]:

form of the stress-strain relationships in matrix form is

tangential direction s, w in the radial direction z).

Nss

extensional, coupling, and bending stiffness coefficients, respectively [1].

M<sup>0</sup>

M<sup>00</sup>

ss þ R N<sup>0</sup>

ss � R Nss <sup>þ</sup> <sup>β</sup>Nss � �<sup>0</sup>

o

Mss � � <sup>¼</sup> <sup>A</sup><sup>22</sup> B22

ss) and the circumferential curvature (Kss) of the mid-surface. The reduced

ss

Introductory Chapter: An Introduction to the Optimization of Composite Structures

http://dx.doi.org/10.5772/intechopen.81165

5

<sup>κ</sup>ss � � (5)

ss � <sup>β</sup>Nss � � <sup>¼</sup> <sup>β</sup> pR<sup>2</sup> (6.1)

<sup>B</sup><sup>22</sup> D22 � � <sup>ε</sup><sup>o</sup>

where Nss and Mss are the resultant distributed force and moment and (Aij, Bij, Dij) are the

Figure 2. Laminated composite cylindrical shell under external pressure (u displacement in the axial direction x, v in the

The governing differential equations of anisotropic long cylinders subjected to external pres-

where uo, vo, and wo are the displacements of a generic point (x, s) on the shell middle surface (z=0) in x, s, and z directions, respectively. The prime denotes differentiation with respect to the

ratio (h/R) ≤ 0.1, the critical buckling pressure can be determined using the mathematical expres-

þ p wo þ v<sup>0</sup>

o � � h i <sup>¼</sup> pR<sup>2</sup> (6.2)

� �=R: For the case of thin cylinders with thickness-to-radius

Structural buckling failure due to high external hydrostatic pressure is a major consideration in designing cylindrical shell-type structures. This section presents a direct approach for enhancing buckling stability limits of thin-walled long cylinders that are fabricated from multi-angle fibrous laminated composite lay-ups. The mathematical formulation employs the classical lamination theory for calculating the critical buckling pressure, where an analytical solution that accounts for the effective axial and flexural stiffness separately as well as the inclusion of the coupling stiffness terms is presented. The associated design optimization problem of maximizing the critical buckling pressure has been formulated in a standard nonlinear mathematical programming problem with the design variables encompassing the fiber orientation angles and the ply thicknesses as well. The physical and mechanical properties of the composite material are taken as preassigned parameters. The proposed model deals with dimensionless quantities in order to be valid for thin shells having different thickness-to-radius ratios. Results have been obtained for cases of filament wound cylinders fabricated from different types of composite materials.

The basic analysis and analytical formulation presented in this chapter are based on the work given by Maalawi [14], which provides good sensitivity to lamination parameters and allows the search for the needed optimal stacking sequences in a reasonable computational time. Referring to the structural model depicted in Figure 2, the significant strain components are Introductory Chapter: An Introduction to the Optimization of Composite Structures http://dx.doi.org/10.5772/intechopen.81165 5

Figure 2. Laminated composite cylindrical shell under external pressure (u displacement in the axial direction x, v in the tangential direction s, w in the radial direction z).

the hoop strain (ε<sup>0</sup> ss) and the circumferential curvature (Kss) of the mid-surface. The reduced form of the stress-strain relationships in matrix form is

$$
\begin{Bmatrix} N\_{\rm ss} \\ M\_{\rm ss} \end{Bmatrix} = \begin{bmatrix} A\_{22} & B\_{22} \\ B\_{22} & D\_{22} \end{bmatrix} \begin{Bmatrix} \varepsilon\_{\rm ss}^{\rho} \\ \kappa\_{\rm ss} \end{Bmatrix} \tag{5}
$$

where Nss and Mss are the resultant distributed force and moment and (Aij, Bij, Dij) are the extensional, coupling, and bending stiffness coefficients, respectively [1].

#### 4.1. Analytical buckling model

selecting the search direction [9]. The most commonly used approaches are the random search, conjugate directions, and conjugate gradients methods. Other algorithms for solving global optimization problems may be classified into heuristic methods that find the global optimum only with high probability and methods that guarantee to find a global optimum with some accuracy. The simulated annealing technique and the genetic algorithms (GAs) belong to the former type, where analogies to physics and biology to approach the global optimum are utilized. The simulated annealing technique is an iterative search method based on the simulation of thermal annealing of critically heated solids. Hasancebi et al. [10] applied it to find the optimum design of fiber composite structures as an efficient method to solve multi-objective optimization models. On the other hand, the GAs [11, 12] are based on the principles of natural genetics and natural selection. GAs do not utilize any gradient information during the searching process. Narayana Naik et al. [12] used GA and various failure mechanisms based on different failure criteria to reach an optimal composite structure. Another robust algorithm in solving complex problems of optimal structural design is named particle swarm optimization algorithm (PSOA). This algorithm is based on the behavior of a colony of living things, such as a swarm of insects like ants, bees, and wasps, a folk of birds, or a school of fish. Omkar et al. [13] applied PSOA to achieve a specified strength with minimizing weight and total cost of a composite structure under different failure criteria. To the author's knowledge, GA has been the most efficient stochastic method for obtaining the global optimum design of compos-

4. Application: buckling optimization of anisotropic cylindrical shells

Structural buckling failure due to high external hydrostatic pressure is a major consideration in designing cylindrical shell-type structures. This section presents a direct approach for enhancing buckling stability limits of thin-walled long cylinders that are fabricated from multi-angle fibrous laminated composite lay-ups. The mathematical formulation employs the classical lamination theory for calculating the critical buckling pressure, where an analytical solution that accounts for the effective axial and flexural stiffness separately as well as the inclusion of the coupling stiffness terms is presented. The associated design optimization problem of maximizing the critical buckling pressure has been formulated in a standard nonlinear mathematical programming problem with the design variables encompassing the fiber orientation angles and the ply thicknesses as well. The physical and mechanical properties of the composite material are taken as preassigned parameters. The proposed model deals with dimensionless quantities in order to be valid for thin shells having different thickness-to-radius ratios. Results have been obtained for cases of filament wound cylinders fabricated from different

The basic analysis and analytical formulation presented in this chapter are based on the work given by Maalawi [14], which provides good sensitivity to lamination parameters and allows the search for the needed optimal stacking sequences in a reasonable computational time. Referring to the structural model depicted in Figure 2, the significant strain components are

ite structures.

4 Optimum Composite Structures

types of composite materials.

The governing differential equations of anisotropic long cylinders subjected to external pressure are cast in the following:

$$\mathcal{M}'\_{ss} + \mathcal{R} \{ \mathcal{N}'\_{ss} - \beta \mathcal{N}\_{ss} \} = \beta \,\mathrm{p} \mathbb{R}^2 \tag{6.1}$$

$$\mathbf{M}''\_{ss} - \mathbf{R} \left[ \mathbf{N}\_{ss} + \left( \beta \mathbf{N}\_{ss} \right)' + p \left( w\_o + v'\_o \right) \right] = p \mathbf{R}^2 \tag{6.2}$$

where uo, vo, and wo are the displacements of a generic point (x, s) on the shell middle surface (z=0) in x, s, and z directions, respectively. The prime denotes differentiation with respect to the angular position φ and β ¼ vo � w<sup>0</sup> o � �=R: For the case of thin cylinders with thickness-to-radius ratio (h/R) ≤ 0.1, the critical buckling pressure can be determined using the mathematical expression [14]:

$$p\_{cr} = 3\left[\frac{D\_{22}}{R^3}\right]\left[\frac{1-\left(\psi^2/a\right)}{1+\alpha+2\psi}\right] \tag{7.1}$$

$$
\psi = \left(\frac{1}{R}\right) \left(\frac{B\_{22}}{A\_{22}}\right) \tag{7.2}
$$

According to the filament-winding manufacturing process, each ply is characterized by its angle θ<sup>k</sup> with respect to the cylinder axis x. The stacking sequence is denoted by [θ1/θ2/…/θn], where the angles are given in degrees, starting from the outer surface of the shell. In addition, in a real-world manufacturing process, the filament-winding angles θ<sup>k</sup> must be chosen from a limited range of allowable lower (θL) and upper (θU) values according to technology references. It is important to mention here that the volume fractions of the constituent materials of the composite structure is assumed to not significantly change during optimization, so that the

Introductory Chapter: An Introduction to the Optimization of Composite Structures

http://dx.doi.org/10.5772/intechopen.81165

The functional behavior of the candidate objective function, as represented by maximization of the dimensionless buckling pressure ^pcr, is thoroughly investigated in order to see how it is changed with the optimization variables in the selected design space. The final optimum designs recommended by the model will directly depend on the mathematical form and

The first case study to be considered herein is a long thin-walled cylindrical shell fabricated from E-glass/vinyl ester composites with the lay-up composed of only two plies (n=2) having equal thicknesses ( ^h<sup>1</sup> <sup>¼</sup> ^h<sup>2</sup> <sup>¼</sup> <sup>0</sup>:5) and different fiber orientation angles. Considering the case of �63� angle ply, the present model gives ^pcr= 4.23, i.e., Pcr = 4.23 � 1.708 � (h/R)

depending on the shell thickness-to-radius ratio. The actual dimensional values of the critical buckling pressure for the different thickness ratios are given in Table 2 for the cases of baseline design [0�], helically wound [�63�], and [�90�] hoop layers. The unconstrained maximum

For a two-ply long cylinder fabricated from graphite/epoxy composites, Figure 3 shows the developed level curves of the dimensionless buckling pressure, ^pcr (also named isomerits or isobars) in the (θ<sup>1</sup> � θ2) design space. As seen in the figure, the maximum value of ^pcr reaches a value of 18.57 for a hoop wound construction. Table 3 presents the solutions for the [�45�] angleply and the [90�] cross-ply constructions for different thickness-to-radius ratios. These solutions

^pcr = 1.00 4.23 6.10

(1/15) 506.07 2140.69 3087.05 (1/20) 213.50 903.11 1302.35 (1/25) 109.31 462.39 666.80 (1/50) 13.66 57.80 83.35

Table 2. Critical buckling pressure for E-glass/vinyl ester cylinder with different lay-ups.

Baseline [0�] Helically wound [�63�] Hoop plies [�90�]

value of ^pcr = 6.1 occurs at the design points [θ1/θ2] = [�90, �90].

<sup>3</sup> GPa,

7

total structural mass remains constant at its reference value of the baseline design.

4.4. Optimal solutions

(h/R)

[Pcr <sup>=</sup> ^pcr � 1.708 � <sup>10</sup><sup>6</sup> (h/R)

<sup>3</sup> KPa].

behavior of the objective function.

4.4.1. Two-layer anisotropic long cylinder

$$a = \left(\frac{1}{R^2}\right) \left(\frac{D\_{22}}{A\_{22}}\right) \tag{7.3}$$

#### 4.2. Definition of the baseline design

It is convenient first to normalize all variables and parameters with respect to a baseline design, which has been selected to be a unidirectional orthotropic laminated cylinder with the fibers parallel to the shell axis x. Optimized designs shall have the same material properties, mean radius R, and total shell thickness h of the baseline design. Expressions for calculating the critical buckling pressure (Pcro) of the baseline design are defined in Table 1, which depend upon the type of composite material utilized and the shell thickness-to-radius ratio (h/R) as well.

#### 4.3. Optimization model

The search for the optimized lamination can be performed by coupling the analytical buckling shell model to a standard nonlinear mathematical programming procedure. The resulting optimization problem may be cast in the following standard form to

$$\text{minimize} \qquad -\hat{p}\_{cr} \tag{8.1}$$

$$\text{subject to} \qquad h\_L \le \hat{h}\_k \le h\_{\text{l}\prime} \tag{8.2}$$

$$
\Theta\_L \mathfrak{S} \Theta\_k \mathfrak{S} \Theta\_{\mathcal{U}} \qquad k = 1, 2, \dots \\
n \tag{8.3}
$$

$$\sum\_{k=1}^{n} \hat{h}\_k = 1\tag{8.4}$$

where ^pcr ¼ pcr=pcro is the dimensionless critical buckling pressure, and (hL, hU) are the lower and upper bounds imposed on the individual dimensionless ply thicknesses ^hk <sup>¼</sup> hk=h.


Table 1. Material properties and critical buckling pressure of the baseline design (Pcro).

According to the filament-winding manufacturing process, each ply is characterized by its angle θ<sup>k</sup> with respect to the cylinder axis x. The stacking sequence is denoted by [θ1/θ2/…/θn], where the angles are given in degrees, starting from the outer surface of the shell. In addition, in a real-world manufacturing process, the filament-winding angles θ<sup>k</sup> must be chosen from a limited range of allowable lower (θL) and upper (θU) values according to technology references. It is important to mention here that the volume fractions of the constituent materials of the composite structure is assumed to not significantly change during optimization, so that the total structural mass remains constant at its reference value of the baseline design.

#### 4.4. Optimal solutions

pcr <sup>¼</sup> <sup>3</sup> <sup>D</sup><sup>22</sup> R3

> <sup>ψ</sup> <sup>¼</sup> <sup>1</sup> R � � B<sup>22</sup> A<sup>22</sup> � �

<sup>α</sup> <sup>¼</sup> <sup>1</sup> R2 � � D<sup>22</sup>

type of composite material utilized and the shell thickness-to-radius ratio (h/R) as well.

optimization problem may be cast in the following standard form to

It is convenient first to normalize all variables and parameters with respect to a baseline design, which has been selected to be a unidirectional orthotropic laminated cylinder with the fibers parallel to the shell axis x. Optimized designs shall have the same material properties, mean radius R, and total shell thickness h of the baseline design. Expressions for calculating the critical buckling pressure (Pcro) of the baseline design are defined in Table 1, which depend upon the

The search for the optimized lamination can be performed by coupling the analytical buckling shell model to a standard nonlinear mathematical programming procedure. The resulting

> Xn k¼1

and upper bounds imposed on the individual dimensionless ply thicknesses ^hk <sup>¼</sup> hk=h.

E11 E22 G12 ν<sup>12</sup>

Material type Orthotropic mechanical properties\* (GPa) Pcro � (h/R)

E-Glass/vinyl ester 41.06 6.73 2.5 0.299 1.708 Graphite/epoxy 130.0 7.0 6.0 0.28 1.757 S-Glass/epoxy 57.0 14.0 5.7 0.277 3.567 \*E11 = longitudinal modulus, E22 = hoop modulus, ν<sup>12</sup> = Poisson's ratio for axial load, ν<sup>21</sup> = ν12E22/E11.

Table 1. Material properties and critical buckling pressure of the baseline design (Pcro).

where ^pcr ¼ pcr=pcro is the dimensionless critical buckling pressure, and (hL, hU) are the lower

minimize —^pcr (8.1)

subject to hL ≤ ^hk ≤ hU, (8.2)

θ<sup>L</sup> ≤ θ<sup>k</sup> ≤ θ<sup>U</sup> k ¼ 1, 2,…:n (8.3)

^hk <sup>¼</sup> <sup>1</sup> (8.4)

<sup>3</sup> (GPa)

4.2. Definition of the baseline design

4.3. Optimization model

6 Optimum Composite Structures

� � <sup>1</sup> � <sup>ψ</sup><sup>2</sup>

A<sup>22</sup> � �

=α � � 1 þ α þ 2ψ " #

(7.1)

(7.2)

(7.3)

The functional behavior of the candidate objective function, as represented by maximization of the dimensionless buckling pressure ^pcr, is thoroughly investigated in order to see how it is changed with the optimization variables in the selected design space. The final optimum designs recommended by the model will directly depend on the mathematical form and behavior of the objective function.

#### 4.4.1. Two-layer anisotropic long cylinder

The first case study to be considered herein is a long thin-walled cylindrical shell fabricated from E-glass/vinyl ester composites with the lay-up composed of only two plies (n=2) having equal thicknesses ( ^h<sup>1</sup> <sup>¼</sup> ^h<sup>2</sup> <sup>¼</sup> <sup>0</sup>:5) and different fiber orientation angles. Considering the case of �63� angle ply, the present model gives ^pcr= 4.23, i.e., Pcr = 4.23 � 1.708 � (h/R) <sup>3</sup> GPa, depending on the shell thickness-to-radius ratio. The actual dimensional values of the critical buckling pressure for the different thickness ratios are given in Table 2 for the cases of baseline design [0�], helically wound [�63�], and [�90�] hoop layers. The unconstrained maximum value of ^pcr = 6.1 occurs at the design points [θ1/θ2] = [�90, �90].

For a two-ply long cylinder fabricated from graphite/epoxy composites, Figure 3 shows the developed level curves of the dimensionless buckling pressure, ^pcr (also named isomerits or isobars) in the (θ<sup>1</sup> � θ2) design space. As seen in the figure, the maximum value of ^pcr reaches a value of 18.57 for a hoop wound construction. Table 3 presents the solutions for the [�45�] angleply and the [90�] cross-ply constructions for different thickness-to-radius ratios. These solutions


Table 2. Critical buckling pressure for E-glass/vinyl ester cylinder with different lay-ups.

4.4.2. Three-layer anisotropic long cylinder

[Pcr = ^pcr � 1.757 � <sup>10</sup><sup>6</sup> (h/R)

solutions were obtained for the stacking sequences [0�

Baseline [0�

<sup>3</sup> KPa].

Table 4. Critical buckling pressure, Pcr for graphite/epoxy cylinder [θ1/θ2/θ1].

4.4.3. Four-layer sandwiched anisotropic cylinder

increase of (17.92 � 16.43)/16.43 = 9.1%.

constraint P<sup>4</sup>

(h/R)

Results for a cylinder constructed from three, equal-thickness layers with stacking sequence denoted by [θ1/θ2/θ1] are given in Table 4. The same behavior can be observed as before but with slight change in the attained values. It was found that for the range �30� > θ<sup>1</sup> > 30� the critical buckling pressure is not much affected by variation in the ply angle θ2. A substantial increase in the critical buckling pressure by changing the ply angles can be observed. Similar

(1/15) 520.59 859.57 9331.19 (1/50) 14.06 23.21 251.94 (1/120) 1.02 1.68 18.23

The same graphite/epoxy cylinder is reconsidered here with changing the stacking sequence to become �20� equal-thickness layers sandwiched in between outer and inner 90� hoop layers with unequal thicknesses, i.e., ( ^h<sup>2</sup> <sup>¼</sup> ^h3) and ( ^h<sup>1</sup> 6¼ ^h4), such that the thickness equality

( ^h1, ^h2) design space. The contours inside the feasible domain, which is bounded by the three lines ^h<sup>1</sup> <sup>¼</sup> 0 and ^h<sup>2</sup> <sup>¼</sup> 0 and ^h<sup>1</sup> <sup>þ</sup> <sup>2</sup> ^h<sup>2</sup> <sup>¼</sup> 1 (i.e., ^h<sup>4</sup> <sup>¼</sup> <sup>0</sup>), are obliged to turn sharply to be asymptotes to the line ^h<sup>4</sup> <sup>¼</sup> 0, in order not to violate the thickness equality constraint. This is why they appear in the figure as zigzagged lines. At the design point ( ^h1, ^h2) = (0.25, 0.25), the dimensionless buckling pressure ^pcr = 16.43 (see Figure 4 and Table 5). As a general observation, as the thickness of the hoop layers increases, a substantial increase in the critical buckling pressure will be achieved, e.g., at ( ^h1, ^h2) = (0.33, 0.17), ^pcr= 17.92 representing a percentage

Finally, the obtained results have indicated that the optimized laminations induce significant increases, always exceeding several tens of percent, of the buckling pressures with respect to the reference or baseline design. It is assumed that the volume fractions of the composite material constituents do not significantly change during optimization, so that the total structural mass remains constant. It has been shown that the overall stability level of the laminated composite shell structures under considerations can be substantially improved by finding the optimal stacking sequence without violating any imposed side constraints. The stability limits

<sup>k</sup>¼<sup>1</sup> ^hk¼1 is always satisfied. Figure 4 shows the developed <sup>p</sup>^cr -isomerits in the

2/90�]s and [90�

<sup>3</sup>] [0�/90�/0�] [90�/0�/90�]

Introductory Chapter: An Introduction to the Optimization of Composite Structures

http://dx.doi.org/10.5772/intechopen.81165

9

^pcr = 1.00 1.651 17.92

2/0�]s.

Figure 3. ^pcr-isomerits for a graphite/epoxy, two-layer cylinder in [θ<sup>1</sup> /θ2] design space (^h<sup>1</sup> <sup>¼</sup> ^h<sup>2</sup> <sup>¼</sup> <sup>0</sup>:5).

are also valid for lay-ups [03 �]s, [903 �]s, [452 �/�452 �]s, and [45�/�45�/45�/�45�]s. The case of a helically wound lay-up construction [+θ/�θ] with unequal play thicknesses ^h<sup>1</sup> and^h2, such that their sum is held fixed at a value of unity, has also been investigated. Computer solutions have shown that no significant change in the resulting values of the critical buckling pressure can be remarked in spite of the wide change in the ply thicknesses. This is a natural expected result since the stiffness coefficients A22, B22, and D22 remain unchanged for such lay-up construction.


Table 3. Critical buckling pressure, Pcr for graphite/epoxy cylinder with different lay-ups.


Table 4. Critical buckling pressure, Pcr for graphite/epoxy cylinder [θ1/θ2/θ1].

#### 4.4.2. Three-layer anisotropic long cylinder

Results for a cylinder constructed from three, equal-thickness layers with stacking sequence denoted by [θ1/θ2/θ1] are given in Table 4. The same behavior can be observed as before but with slight change in the attained values. It was found that for the range �30� > θ<sup>1</sup> > 30� the critical buckling pressure is not much affected by variation in the ply angle θ2. A substantial increase in the critical buckling pressure by changing the ply angles can be observed. Similar solutions were obtained for the stacking sequences [0� 2/90�]s and [90� 2/0�]s.

#### 4.4.3. Four-layer sandwiched anisotropic cylinder

are also valid for lay-ups [03

8 Optimum Composite Structures

(h/R)

[Pcr <sup>=</sup> ^pcr � 1.757 � <sup>10</sup><sup>6</sup> (h/R)

<sup>3</sup> KPa].

�]s, [903

�]s, [452

(1/15) 520.59 3071.50 9667.40 (1/50) 14.06 82.93 261.02 (1/120) 1.02 5.99 18.88

Table 3. Critical buckling pressure, Pcr for graphite/epoxy cylinder with different lay-ups.

Figure 3. ^pcr-isomerits for a graphite/epoxy, two-layer cylinder in [θ<sup>1</sup> /θ2] design space (^h<sup>1</sup> <sup>¼</sup> ^h<sup>2</sup> <sup>¼</sup> <sup>0</sup>:5).

�/�452

helically wound lay-up construction [+θ/�θ] with unequal play thicknesses ^h<sup>1</sup> and^h2, such that their sum is held fixed at a value of unity, has also been investigated. Computer solutions have shown that no significant change in the resulting values of the critical buckling pressure can be remarked in spite of the wide change in the ply thicknesses. This is a natural expected result since the stiffness coefficients A22, B22, and D22 remain unchanged for such lay-up construction.

Baseline [0�] Helically wound [�45�] Hoop plies [�90�]

^pcr = 1.00 5.9 18.57

�]s, and [45�/�45�/45�/�45�]s. The case of a

The same graphite/epoxy cylinder is reconsidered here with changing the stacking sequence to become �20� equal-thickness layers sandwiched in between outer and inner 90� hoop layers with unequal thicknesses, i.e., ( ^h<sup>2</sup> <sup>¼</sup> ^h3) and ( ^h<sup>1</sup> 6¼ ^h4), such that the thickness equality constraint P<sup>4</sup> <sup>k</sup>¼<sup>1</sup> ^hk¼1 is always satisfied. Figure 4 shows the developed <sup>p</sup>^cr -isomerits in the ( ^h1, ^h2) design space. The contours inside the feasible domain, which is bounded by the three lines ^h<sup>1</sup> <sup>¼</sup> 0 and ^h<sup>2</sup> <sup>¼</sup> 0 and ^h<sup>1</sup> <sup>þ</sup> <sup>2</sup> ^h<sup>2</sup> <sup>¼</sup> 1 (i.e., ^h<sup>4</sup> <sup>¼</sup> <sup>0</sup>), are obliged to turn sharply to be asymptotes to the line ^h<sup>4</sup> <sup>¼</sup> 0, in order not to violate the thickness equality constraint. This is why they appear in the figure as zigzagged lines. At the design point ( ^h1, ^h2) = (0.25, 0.25), the dimensionless buckling pressure ^pcr = 16.43 (see Figure 4 and Table 5). As a general observation, as the thickness of the hoop layers increases, a substantial increase in the critical buckling pressure will be achieved, e.g., at ( ^h1, ^h2) = (0.33, 0.17), ^pcr= 17.92 representing a percentage increase of (17.92 � 16.43)/16.43 = 9.1%.

Finally, the obtained results have indicated that the optimized laminations induce significant increases, always exceeding several tens of percent, of the buckling pressures with respect to the reference or baseline design. It is assumed that the volume fractions of the composite material constituents do not significantly change during optimization, so that the total structural mass remains constant. It has been shown that the overall stability level of the laminated composite shell structures under considerations can be substantially improved by finding the optimal stacking sequence without violating any imposed side constraints. The stability limits

References

8262

tures. 2003;62:123-128

Computing. 2011;11(1):489-499

2008;83:354-367

[1] Daniel I, Ishai O. Engineering Mechanics of Composite Materials. 2nd ed. Oxford Univer-

Introductory Chapter: An Introduction to the Optimization of Composite Structures

http://dx.doi.org/10.5772/intechopen.81165

11

[2] Kollar LP, Springer GS. Mechanics of Composite Structures. 1st ed. United Kingdom:

[3] Niu MCY. Composite Airframe Structures. 3rd ed. Hong Kong: Hong Kong Conmilit Press

[4] Kassapoglou C. Design and Analysis of Composite Structures: With Applications to Aerospace Structures. 2nd ed. New Jersey: Wiley; 2013. 410p. ISBN: 978-1118401606 [5] Ye J. Laminated Composite Plates and Shells, 3D Modelling. 1st ed. London: Springer-

[6] Kumar SS. Smart Composite Structures. 1st ed. United States: Lambert Academic Publish-

[7] Sonmez FO. Optimum design of composite structures: A literature survey (1969–2009). Journal of Reinforced Plastics and Composites. 2017;36(1):3-39. DOI: 10.1177/073168441666

[8] Ganguli R. Optimal design of composite structures: A historical review. Journal of the

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

[10] Hasancebi O, Carba S, Saka M. Improving the performance of simulated annealing in structural optimization. Structural and Multidisciplinary Optimization. 2010;41:189-203

[11] Walker M, Smith RE. A technique for the multi-objective optimization of laminated composite structures using genetic algorithms and finite element analysis. Composite Struc-

[12] Narayana Naik G, Gopalakrishnan S, Ganguli R. Design optimization of composites using genetic algorithms and failure mechanism based failure criterion. Composite Structures.

[13] Omkar SN, Senthilnath J, Khandelwal R, Narayana Naik G, Gopalakrishnan S. Artificial bee colony (ABC) for multi-objective design optimization of composite structures. Applied Soft

[14] Maalawi KY. Optimal buckling design of anisotropic rings/long cylinders under external

pressure. Journal of Mechanics of Materials and Structures. 2008;3(4):775-793

Indian Institute of Science. 2013;93(4):557-570. ISSN: 0970-4140 Coden-JIISAD

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

sity Press, New York; 2006. 432p. ISBN: 978-0195150971

Ltd; 2011. 500p. ISBN: 978-9627128069

ing; 2013. 180p. ISBN: 978-3659322532

Cambridge University Press; 2009. 500p. ISBN: 978- 0521126908

Verlag; 2003. DOI: 10.1007/978-1447100959. 273p. ISBN: 978-1447100959

Figure 4. Design space for a sandwich lay-up graphite/epoxy cylinder [90/20/90].


Table 5. Critical buckling pressure, Pcr (KPa), for graphite/epoxy cylinder [90/20/90].

of the optimized shells have been substantially enhanced as compared with those of the reference or baseline designs.
