**3. Stability of FGM long cylinders under external pressure**

A common application of composites is the design of cylindrical shells under the action of external hydrostatic pressure, which might cause collapse by buckling instability. Examples are the underground and underwater pipelines, rocket motor casing, boiler tubes subjected to external steam pressure, and reinforced submarine structures. The composite cylindrical vessels for underwater applications are intended to operate at high external hydrostatic pressure (sometimes up to 60 MPa). For deep- submersible long-unstiffened vessels, the hulls are generally realized using multilayered, cross-ply, composite cylinders obtained following the filament winding process. Previous numerical and experimental studies have shown that failure due to structural buckling is a major risk factor for thin laminated cylindrical shells. Figure 7 shows the structural model used in reference [13], where the effect of changing the fiber volume fraction in each lamina was taken in the formulation of the structural model.

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 27

22 22 22 22 B D

21 2

(9)

2

1

2 2

3 3

*B*

(11)

(12)

(13)

*h ˆ ˆ z z* is the dimensionless

*k)* of the individual *k-th* ply,

*A B* 

ani 22 22 1

11 16 11 16 12 12 16 66 16 66 26 26

ABB B

11 16 11 16 12 12 16 66 16 66 26 26

BB D D D

ABB B and S BDD D

*A A [ [ [] SS S ] ]*

*A A A A B B*

ani ani

(7)

 *0.1*. The

(10)

(8)

ani ani

shear forces *(Nxx, Nxs)* must vanish along the free edges. The bending and twisting moments *(Mxx, Mxs)* may also be neglected. The final closed form solution for the critical buckling

3 2

<sup>1</sup> 1 1 B D 3 , 1 2 A A

*p ( )( ), ( )( ) R R R*

which is only valid for thin rings/cylinders with thickness-to-radius ratio (*h/R)*

ani ani

 B B D D

*cylinder ani*

ani 22 22

1 *n*

*k*

2

3

*z z / h* is a dimensionless coordinate, and *<sup>k</sup> k k* <sup>1</sup> *ˆ*

volume fraction (*Vfk*), thickness *(hk)* and fiber orientation angle (

*n ij ij kk k*

*k*

*n ij ij kk k k*

*ij ij k k k*

*A h (Q ) ( ) ˆ ˆ z z*

<sup>1</sup> 2 <sup>1</sup>

<sup>1</sup> 3 <sup>1</sup>

*<sup>h</sup> D (Q ) (z z ) ˆ ˆ*

thickness of the *kth* lamina. The associated optimization problem shall seek maximization of the critical buckling pressure *pcr* while maintaining the total structural mass constant at a value equals to that of a reference baseline design. Optimization variables include the fiber

*<sup>h</sup> B (Q ) (z z ) ˆ ˆ*

*ani T*

 B D *ani*

 

stiffness coefficients *Aani, Bani* and *Dani* are calculated, for the case of long cylinders from:

pressure is given by the following mathematical expression:

*D ( /)*

*A B*

*ani ring*

*Aij* are called the extensional stiffnesses given by:

*Dij* are called the bending stiffnesses:

where *k k ˆ*

*Bij* are called the bending-extensional stiffnesses given by:

*B B*

*ani*

*cr*

And for circular rings:

where *S*<sup>1</sup>

2

**Figure 7.** Laminated composite shell under external pressure (*u* displacement in the axial direction *x, v* in the tangential direction *s, w* in the radial direction *z*)

The governing differential equations of anisotropic rings/long cylinders subjected to external pressure are cast in the following [13]:

$$
\mathbf{M}'\_{ss} + \mathbf{R} (\mathbf{N}'\_{ss} - \mathbf{B} \mathbf{N}\_{ss}) = \mathbf{B} \ p \mathbf{R}^2
$$

$$
\mathbf{M}''\_{ss} - \mathbf{R} \{ \mathbf{N}\_{ss} + (\mathbf{B} \mathbf{N}\_{ss})' + p(\mathbf{r}\_{\overline{\mathbf{u}} \mathbf{o}\_o + \mathbf{r}\_o' \mathbf{o}\_o) \} = p \mathbf{R}^2 \tag{6}
$$

 where the prime denotes differentiation with respect to angular position , and *o o ( ) / R. v w* Two possible solutions for Eq. (6) can be obtained; one for the pre-buckled state and the other termed as the bifurcation solution obtained by perturbing the displacements about the pre-buckling solution. For laminated composite rings and long cylindrical shells the only significant strain components are the hoop strain *( <sup>o</sup> ss )* and the circumferential curvature ( *ss )* of the mid-surface. In the case of thin rings the axial and shear forces *(Nxx, Nxs)* must vanish along the free edges. The bending and twisting moments *(Mxx, Mxs)* may also be neglected. The final closed form solution for the critical buckling pressure is given by the following mathematical expression:

$$p\_{cr} = 3\left[\frac{D\_{\rm ani}}{R^3}\right]\left[\frac{1 - (\boldsymbol{\eta}^2/\alpha)}{1 + \alpha + 2\boldsymbol{\eta}}\right], \quad \boldsymbol{\eta} = (\frac{1}{R})(\frac{\mathbf{B}\_{\rm ani}}{\mathbf{A}\_{\rm ani}}), \quad \alpha = (\frac{1}{R})(\frac{\mathbf{D}\_{\rm ani}}{\mathbf{A}\_{\rm ani}}) \tag{7}$$

which is only valid for thin rings/cylinders with thickness-to-radius ratio (*h/R) 0.1*. The stiffness coefficients *Aani, Bani* and *Dani* are calculated, for the case of long cylinders from:

$$
\begin{bmatrix}
\mathbf{A}\_{ani} & \mathbf{B}\_{ani} \\
\mathbf{B}\_{ani} & \mathbf{D}\_{ani}
\end{bmatrix}\_{\text{cylinder}} = \begin{bmatrix}
\mathbf{A}\_{22} & \mathbf{B}\_{22} \\
\mathbf{B}\_{22} & \mathbf{D}\_{22}
\end{bmatrix} \tag{8}
$$

And for circular rings:

$$
\begin{bmatrix} A\_{ani} & \mathbf{B}\_{ani} \\ \mathbf{B}\_{ani} & \mathbf{D}\_{ani} \end{bmatrix}\_{ring} = \begin{bmatrix} A\_{22} & \mathbf{B}\_{22} \\ \mathbf{B}\_{22} & \mathbf{D}\_{22} \end{bmatrix} - \begin{bmatrix} \mathbf{S}\_{21} \end{bmatrix}^{T} \begin{bmatrix} \mathbf{S}\_{1} \end{bmatrix}^{-1} \begin{bmatrix} \mathbf{S}\_{2} \end{bmatrix} \tag{9}
$$

$$\text{where}$$

26 Advances in Computational Stability Analysis

the structural model.

**3. Stability of FGM long cylinders under external pressure** 

A common application of composites is the design of cylindrical shells under the action of external hydrostatic pressure, which might cause collapse by buckling instability. Examples are the underground and underwater pipelines, rocket motor casing, boiler tubes subjected to external steam pressure, and reinforced submarine structures. The composite cylindrical vessels for underwater applications are intended to operate at high external hydrostatic pressure (sometimes up to 60 MPa). For deep- submersible long-unstiffened vessels, the hulls are generally realized using multilayered, cross-ply, composite cylinders obtained following the filament winding process. Previous numerical and experimental studies have shown that failure due to structural buckling is a major risk factor for thin laminated cylindrical shells. Figure 7 shows the structural model used in reference [13], where the effect of changing the fiber volume fraction in each lamina was taken in the formulation of

**Figure 7.** Laminated composite shell under external pressure (*u* displacement in the axial direction *x, v*

The governing differential equations of anisotropic rings/long cylinders subjected to

<sup>2</sup> *M N ss R )p (Nss ss R*

*o o ( ) / R. v w* Two possible solutions for Eq. (6) can be obtained; one for the pre-buckled state and the other termed as the bifurcation solution obtained by perturbing the displacements about the pre-buckling solution. For laminated composite rings and long

circumferential curvature ( *ss )* of the mid-surface. In the case of thin rings the axial and

where the prime denotes differentiation with respect to angular position

cylindrical shells the only significant strain components are the hoop strain *( <sup>o</sup>*

<sup>2</sup> *M N wv ss R ( ) p( )] p [Nss ss o o R* (6)

, and

*ss )* and the

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

external pressure are cast in the following [13]:

where *S*<sup>1</sup> 11 16 11 16 12 12 16 66 16 66 26 26 2 11 16 11 16 12 12 16 66 16 66 26 26 ABB B ABB B and S BDD D BB D D D *A A A A B B B* (10)

*Aij* are called the extensional stiffnesses given by:

$$A\_{i\bar{j}} = \hbar \sum\_{k=1}^{n} (\overline{\mathcal{Q}}\_{i\bar{j}})\_k (\hat{z}\_k - \hat{z}\_{k-1}) \tag{11}$$

*Bij* are called the bending-extensional stiffnesses given by:

$$B\_{ij} = \frac{\hbar^2}{2} \sum\_{k=1}^{n} (\overline{\mathbb{Q}}\_{ij})\_k (\hat{\boldsymbol{z}}\_k^2 - \hat{\boldsymbol{z}}\_{k-1}^2) \tag{12}$$

*Dij* are called the bending stiffnesses:

$$D\_{ij} = \frac{h^3}{\mathfrak{D}} \sum\_{k=1}^n (\overline{\mathcal{Q}}\_{ij})\_k (\hat{z}\_k^3 - \hat{z}\_{k-1}^3) \tag{13}$$

where *k k ˆ z z / h* is a dimensionless coordinate, and *<sup>k</sup> k k* <sup>1</sup> *ˆ h ˆ ˆ z z* is the dimensionless thickness of the *kth* lamina. The associated optimization problem shall seek maximization of the critical buckling pressure *pcr* while maintaining the total structural mass constant at a value equals to that of a reference baseline design. Optimization variables include the fiber volume fraction (*Vfk*), thickness *(hk)* and fiber orientation angle (*k)* of the individual *k-th* ply, *k=1, 2,…..n* (total number of plies). Side constraints are always imposed on the design variables for geometrical, manufacturing or logical reasons to avoid having unrealistic odd shaped optimum designs. The first case study to be examined herein is a long thin-walled cylindrical shell fabricated from E-glass/epoxy composites with the lay-up made of only two plies *(n=2)* having fibers parallel to the x-axis (i.e. *1=2=0*). Considering the case with no side inequality constraints imposed on the design variables, Figure 8 shows the developed ܲcr level curves, augmented with the mass equality constraint, in *(Vf1-* <sup>1</sup> *ˆ h )* design space.

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 29

*)* with equal thicknesses and same material properties of E-

**S-glass/Epoxy**

**E-glass/Vinylester**

*) = (0.375, 0.0)* with *p*cr*=0.9985*,

*) with (pcr*)*max=3.45766*, which means

*)* and *(-*

*)* layers. A

*)* plies

*o)*

attained solution is *(pcr)max* =1.2105 at the design point (0.75, 0.215), (0.4315, 0.785), showing that good shell designs with higher stability level ought to have a thinner inner layer with higher fiber volume fraction and a thicker outer layer with less volume fraction. To see the effect of the ply angle, another case of study has been considered for a cylinder constructed

glass/epoxy composites. This type of stacking sequence is widely used in filament wound

indicating a degradation in the stability level below the baseline design. The absolute

*)=(0.5,90o*

that the dimensional critical pressure*, pcr=3.45766x2.865=9.906 x(h/R)3 GPa*. Figure 9 depicts

for the different types of the selected composite materials. All shall have the same optimal solution (*Vfk*,*hk*)*k=1,2* = (0.75, 0.215), (0.4315, 0.785), independent upon the shell thickness-to

circular shells since such a manufacturing process inherently dictates adjacent *(*

the final global optimum designs of cylinders constructed from adjacent *(+*

**Figure 9.** Variation of the absolute maximum buckling pressure with ply angle for balanced *(*

**E-glass/Epoxy**

Other cases of study include optimization of two different constructions of multi-layered cylinders made of *AS-4* carbon/epoxy composites. The first one is called a lumped-layup construction with the inner half of its wall composed of 90o hoop layers and the outer half

solutions are given in Table 4, indicating that good designs shall have thicker hoop wound layers with higher volume fraction of the fibers near the upper limiting values imposed by the manufacturers. On the other hand, the sandwiched helically wound layers are seen to be

helically wound layers. The second type has different stacking sequence where

**0 10 20 30 40 50 60 70 80 90 P ly a ngle , de gre e s.**

**Carbon/Epoxy**

layers are sandwiched in between outer and inner *90o* hoop layers. Optimum

radius ratio *(h/R)*, a major contribution of the given formulation.

from two balanced plies *(*

local minimum was found near the design point *(Vf1,*

maximum occurred at the design points *(Vf1,*

cylinders with structural mass preserved constant

**0**

**5**

**10**

**15**

**20**

**Max. buckling pressure x (h/R)3, GPa.**

**25**

**30**

**35**

thinner and have less fiber volume fractions.

made of

the *20o* *20o*

**Figure 8.** Optimum design space containing *pcr*-isobars augmented with the mass equality constraint 1 0 *Mˆ .* . Case of two-layer, E-glass/epoxy cylinder with fibers parallel to cylinder axis (*1=2=0*).

It is seen that such a constrained objective function is well behaved in the selected design space having the shape of a tent with its ceiling formed by two curved lines, above which the mass equality constraint is violated. Their zigzagged pattern is due to the obliged turning of many contours, which are not allowed to penetrate the tent's ceiling and violate the mass equality constraint. The curve to the left represent a 100% fiber volume fraction of the outer ply, *Vf2*, while the other curve to the right represents zero volume fraction, that is *Vf2=0%.* Two local minima with *pcr* near a value of 0.90 can be observed: one to the lower left zone near the design point (*Vfk*,*hk*)*k=1,2* = (0.15, 0.25), (0.6165, 0.75) while the other lies at the upper right zone close to the point (0.625, 0.745), (0.135, 0.255). This represents degradation in the stability level by about 10.6% below the baseline value. On the other hand, the unconstrained absolute optimum value of the dimensionless critical buckling pressure was found to be 1.7874 at the design point (1.0, 0.145), (0.415, 0.855). A more realistic optimum design has been obtained by imposing the side constraints: *0.25≤ Vfk≤ 0.75, k=1, 2*. The attained solution is *(pcr)max* =1.2105 at the design point (0.75, 0.215), (0.4315, 0.785), showing that good shell designs with higher stability level ought to have a thinner inner layer with higher fiber volume fraction and a thicker outer layer with less volume fraction. To see the effect of the ply angle, another case of study has been considered for a cylinder constructed from two balanced plies *()* with equal thicknesses and same material properties of Eglass/epoxy composites. This type of stacking sequence is widely used in filament wound circular shells since such a manufacturing process inherently dictates adjacent *()* layers. A local minimum was found near the design point *(Vf1,) = (0.375, 0.0)* with *p*cr*=0.9985*, indicating a degradation in the stability level below the baseline design. The absolute maximum occurred at the design points *(Vf1,)=(0.5,90o ) with (pcr*)*max=3.45766*, which means that the dimensional critical pressure*, pcr=3.45766x2.865=9.906 x(h/R)3 GPa*. Figure 9 depicts the final global optimum designs of cylinders constructed from adjacent *(+)* and *(-)* plies for the different types of the selected composite materials. All shall have the same optimal solution (*Vfk*,*hk*)*k=1,2* = (0.75, 0.215), (0.4315, 0.785), independent upon the shell thickness-to radius ratio *(h/R)*, a major contribution of the given formulation.

28 Advances in Computational Stability Analysis

plies *(n=2)* having fibers parallel to the x-axis (i.e.

level curves, augmented with the mass equality constraint, in *(Vf1-* <sup>1</sup> *ˆ*

*k=1, 2,…..n* (total number of plies). Side constraints are always imposed on the design variables for geometrical, manufacturing or logical reasons to avoid having unrealistic odd shaped optimum designs. The first case study to be examined herein is a long thin-walled cylindrical shell fabricated from E-glass/epoxy composites with the lay-up made of only two

inequality constraints imposed on the design variables, Figure 8 shows the developed ܲcr -

**Figure 8.** Optimum design space containing *pcr*-isobars augmented with the mass equality constraint

It is seen that such a constrained objective function is well behaved in the selected design space having the shape of a tent with its ceiling formed by two curved lines, above which the mass equality constraint is violated. Their zigzagged pattern is due to the obliged turning of many contours, which are not allowed to penetrate the tent's ceiling and violate the mass equality constraint. The curve to the left represent a 100% fiber volume fraction of the outer ply, *Vf2*, while the other curve to the right represents zero volume fraction, that is *Vf2=0%.* Two local minima with *pcr* near a value of 0.90 can be observed: one to the lower left zone near the design point (*Vfk*,*hk*)*k=1,2* = (0.15, 0.25), (0.6165, 0.75) while the other lies at the upper right zone close to the point (0.625, 0.745), (0.135, 0.255). This represents degradation in the stability level by about 10.6% below the baseline value. On the other hand, the unconstrained absolute optimum value of the dimensionless critical buckling pressure was found to be 1.7874 at the design point (1.0, 0.145), (0.415, 0.855). A more realistic optimum design has been obtained by imposing the side constraints: *0.25≤ Vfk≤ 0.75, k=1, 2*. The

1 0 *Mˆ .* . Case of two-layer, E-glass/epoxy cylinder with fibers parallel to cylinder axis (

*1=*

*2=0*). Considering the case with no side

*h )* design space.

*1=2=0*).

**Figure 9.** Variation of the absolute maximum buckling pressure with ply angle for balanced *(o)* cylinders with structural mass preserved constant

Other cases of study include optimization of two different constructions of multi-layered cylinders made of *AS-4* carbon/epoxy composites. The first one is called a lumped-layup construction with the inner half of its wall composed of 90o hoop layers and the outer half made of *20o* helically wound layers. The second type has different stacking sequence where the *20o* layers are sandwiched in between outer and inner *90o* hoop layers. Optimum solutions are given in Table 4, indicating that good designs shall have thicker hoop wound layers with higher volume fraction of the fibers near the upper limiting values imposed by the manufacturers. On the other hand, the sandwiched helically wound layers are seen to be thinner and have less fiber volume fractions.

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 31

to be within close ranges (called frequency windows) of those corresponding to a reference or baseline design, which are adjusted to be far away from the critical exciting frequencies. The main idea is to tailor the mass and stiffness distributions in such a way to make the objective function a minimum under the imposed mass constraint. The second alternative for reducing vibration is the direct maximization of the system natural frequencies as expressed by equation (11). Maximization of the natural frequencies can ensure a simultaneous balanced improvement in both of stiffness and mass of the vibrating structure. It is a much better design criterion than minimization of the mass alone or maximization of the stiffness alone. The latter can result in optimum solutions that are strongly dependent on the limits imposed on either the upper values of the allowable deflections or the acceptable values of the total structural mass, which are rather arbitrarily chosen. The proper determination of the weighting factors Wfi should be based on the fact that each frequency ought to be maximized from its initial value corresponding to a baseline design having uniform mass and stiffness properties. Reference [14] applied the concept of material grading for enhancing the dynamic performance of bars in

2

where ܷ ൌ ܷȀܮ is the dimensionless amplitude and ߱ෝ =߱ܮඥߩȀܧ dimensionless frequency. Both continuous and discrete distributions of the volume fractions of the selected composite material were analyzed in [14]. The general solution of Eq. (12), where the modulus of elasticity and mass density vary in the axial direction, can be expressed by the following

2

*m*

*(x) <sup>ˆ</sup> a xˆ*

*ˆ*

1 *m m*

The unknown coefficients *am,n* can be determined by substitution into the differential equation (12) and equating coefficients of like powers of *xˆ* . Table 5 summarizes the appropriate mathematical expressions of the frequency equation for any desired case, which can be obtained by application of the associated boundary conditions and consideration of

Variation of the volume fractions in *FGM* structures is usually described by power-law distributions. Figure 10 shows both linear and parabolic models for material grading along the bar span. Results given in [14] showed that, for Fixed-Fixed and Fixed-Free boundary conditions, patterns with higher fiber volume fraction near the fixed ends are always

*U(x) (x) ˆ ˆ C* 

. Appropriate values of the target frequencies are usually chosen

<sup>2</sup> 0 0 1 *U dE dU ˆ ˆˆ ˆ ˆ <sup>d</sup> E . U , x ˆ ˆˆ <sup>d</sup> dx dx ˆ ˆ xˆ* (16)

(17)

(18)

*m's* are two linearly independent solutions

desired (target) frequency <sup>i</sup>

power series:

that have the form:

nontrivial solutions.

\*

axial motion. The associated eigenvalue problem is cast in the following:

2

<sup>1</sup> m (n m) *<sup>n</sup> m,n n m*

where *Cm's* are the constants of integration and

**Table 4.** Optimum buckling design of multi-layered, AS-4, FGM composite cylinders
