6. Optimization of composite thin-walled pipes conveying fluid

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

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

opt <sup>¼</sup> Vf ; ^

(0.2500, 1.0, 0.3406)2

(0.25, 0.5250, 0.3)2

(0.4125, 1.0, 0.1375)2 (0.2500, 1.0, 0.2875)3

(0.30, 0.65, 0.05)2 (0.25, 0.50, 0.3)3

h; ^ b 

k¼1, 2,::Np

V^ div,max Optimization gain %

2.0523 3.93%

3.2275 63.44%

2.05625 4.13%

3.3475 69.52%

!

Table 4. Optimal piecewise wing designs with chord taper = 0.5 (baseline value = 1.9747).

Np Type of grading <sup>X</sup>

2 Material (0.5906, 1.0, 0.6594)1

3 Material (0.6000, 1.0, 0.5750)1

Material and thickness (0.75, 1.0625, 0.7)1

Material and thickness (0.75, 1.10, 0.65)1

stability.

182 Optimum Composite Structures

The subject of vibration and stability of thin pipes conveying flowing fluids is of a considerable practical interest. An advanced textbook by Paїdoussis [36] gives an excellent review of the several developments made in this research area. Practical models for enhancing static and dynamic stability characteristics of pipelines constructed from uniform modules were addressed by Maalawi et al. [37, 38], where the relevant design variables were selected to be the mean diameter, wall thickness, and length of each module composing the pipeline. The general case of an elastically supported pipe, covering a variety of boundary conditions, was also investigated. Distinct domains of the flutter instability boundaries were presented for different ratios of the fluid-to-pipe mass, and the variation of the critical flow velocity with support flexibility was examined and discussed. Concerning pipelines made of advanced FGMs, this section presents a mathematical model for enhancing the overall system stability against flutter and/or divergence under mass constraint. Figure 14 shows a FGM pipe conveying flowing fluid with the coordinate system chosen such that the x-axis coincides with the longitudinal centroidal axis in its undeformed position, while the y- and z-axes coincide with the cross section principal axes. The pipe model consists of rigidly connected thin-walled circular tubes made of unidirectional fibrous composite material. Each pipe module has different material properties, wall thickness, and length. Such a configuration results in a piecewise axial grading of either the material of construction or the wall thickness in the direction of the pipe axis. Assuming no voids are present, the distributions of the mass density (r) and modulus of elasticity (E) can be determined using the formulas of Table 1.

The associated eigenvalue problem is described by the fourth-order ordinary differential equation [39]:

$$\dot{\phi}(EI)\_k V^{\prime\prime\prime} + \mathcal{U}^2 V^{\prime} + 2i\omega\mathcal{U}\sqrt{\beta\_o}V^{\prime} - m\_k\omega^2 V = 0\tag{40}$$

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

Figure 14. Multimodule composite pipe conveying flowing fluid.

the definition of the various parameters and dimensionless design variables are given in Ref. [39]. Both static and dynamic instability phenomena can be involved for the physical model described by Eq. (30), depending on the type of boundary conditions at the pipe ends. The notation ()<sup>0</sup> means total spatial derivative. The general solution can be obtained using standard power series methods:

$$V(\mathbf{x}) = \sum\_{j=1}^{4} A\_j e^{j p\_j \mathbf{x}} \tag{41}$$

nonlinear equations derived from the consideration of nontrivial solution of characteristic equation. The effect of flow for small velocity is to damp the system in all modes. At higher velocities, some of the modes become less damped and the corresponding branches cross the Re(ω)-axis, indicating the existence of unstable oscillations of the system. If a branch passes through the

Optimization of Functionally Graded Material Structures: Some Case Studies

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

185

To verify the developed formulation, the classical problem dealing with one-module cantilevered pipe is considered first. The dimensionless wall thickness and length of the pipe are assigned at a value of 1.0, while the volume fraction at 50%. Figure 15 depicts variation of the critical flutter velocity and frequency with the mass density ratio MRo, covering a wide range of pipe and fluid mass densities. It is seen that there are four subdomains with the associated flutter modes defined in the specified intervals of the mass ratio. The upper and lower bounds determine the critical values of the mass ratios at which some of the frequency branches cross each other at the same value of the flutter velocity. The overall flutter mode may be regarded as composed of different quasimodes separated at the shown "jumps" in the Uf-MRo curve. The calculated mass density ratios at the three indicated frequency jumps are 0.4225, 2.29, and 12.33, respectively. They correspond to multiple points of neutral stability, where for a finite incremental increase in the flow velocity, the system becomes unstable, then regains stability, and then once again becomes unstable with a noticeable abrupt increase in the flutter frequency. Next, we consider a baseline design made of carbon/epoxy composites (see Table 2) with mass ratio MRo = 2.0. The calculated values of the dimensionless flutter velocity and frequency are found to be Uf = 10.78 and ω<sup>f</sup> = 26, respectively. Keeping the total dimensionless mass constant at the value corresponding to the baseline design, the best solution having the greatest flutter velocity was found to be (Vf, h1) = (0.70, 0.9345), which corresponds

Figure 15. Variation of flutter speed and frequency with mass ratio for a uniform one-module cantilevered pipe (Vf1 = 50%,

origin, that i, ω = 0, the case of static instability (called divergence) is reached.

6.1. Flutter solutions

to Uf = 12.517 and ω<sup>f</sup> = 31.8615.

h1 = 1, L1 = 1).

pj, j = 1,…4 are the four roots of the fourth-order polynomial:

$$p^4 - (a\_k \mathcal{U}^2)p^2 - \left(2\omega \mathcal{U}\sqrt{\beta\_o}\alpha\_k\right)p - m\_k\omega^2 a\_k = 0 \quad , a\_k = 1/(\mathcal{E}\_k h\_k) \tag{42}$$

An efficient method [38] is successfully implemented to find the complex roots of Eq. (32) by formulating a special companion matrix and finding the associated eigenvalues for any desired values of the variables αk, βo, ω, and U. The transfer matrix [Tk] of the kth pipe module can be obtained by performing the matrix multiplications

$$\begin{bmatrix} T\_k \end{bmatrix} = \begin{bmatrix} P^{(k)} \end{bmatrix} \begin{bmatrix} E^{(k)} \end{bmatrix} \begin{bmatrix} P^{(k)} \end{bmatrix}^{-1} \tag{43}$$

$$
\begin{bmatrix} p^{(k)} \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & 1 \\ -ip\_1 & -ip\_2 & -ip\_3 & -ip\_4 \\ (Eh)\_k p\_1^2 & (Eh)\_k p\_2^2 & (Eh)\_k p\_3^2 & (Eh)\_k p\_4^2 \\ i(Eh)\_k p\_1^3 & i(Eh)\_k p\_2^3 & i(Eh)\_k p\_3^3 & i(Eh)\_k p\_4^3 \end{bmatrix} \tag{44}$$

and [E] is a diagonal matrix with elements:

$$E\_{\vec{\eta}}^{(k)} = e^{i\vec{\eta}\_{\vec{\eta}}^{(k)}L\_k} \tag{45}$$

Applying the appropriate boundary conditions and considering only the nontrivial solution, the resulting characteristic equation can be solved numerically for the frequency and the critical flow velocity. The system is stable or unstable according to whether the imaginary component of the frequency, ω, is positive or negative, respectively. In case of neutral stability, ω is wholly real. As the flow velocity increases, the system may become unstable in one or more of its normal modes. The critical flow velocity is the greatest velocity for which the system is stable in all its modes. The characteristic equation for cantilevered pipe is:

$$T\_{33}T\_{44} - T\_{34}T\_{43} = 0,\tag{46}$$

where Tij are the elements of the overall transfer matrix. For a specific mode number, the proper starting frequency is determined by solving a subsidiary eigenvalue problem corresponding to the case of stationary fluid inside the vibrating pipe. A globally convergent optimization algorithm, known as Levenberg-Marquardt algorithm [38], has been applied to solve the resulting nonlinear equations derived from the consideration of nontrivial solution of characteristic equation. The effect of flow for small velocity is to damp the system in all modes. At higher velocities, some of the modes become less damped and the corresponding branches cross the Re(ω)-axis, indicating the existence of unstable oscillations of the system. If a branch passes through the origin, that i, ω = 0, the case of static instability (called divergence) is reached.

#### 6.1. Flutter solutions

the definition of the various parameters and dimensionless design variables are given in Ref. [39]. Both static and dynamic instability phenomena can be involved for the physical model described by Eq. (30), depending on the type of boundary conditions at the pipe ends. The notation ()<sup>0</sup> means total spatial derivative. The general solution can be obtained using standard power series

> V xð Þ¼ <sup>X</sup> 4

ffiffiffiffi βo q αk � �

Tk ½ �¼ <sup>P</sup>ð Þ<sup>k</sup> h i

1 1 �ip<sup>1</sup> �ip<sup>2</sup>

<sup>1</sup> ð Þ Eh <sup>k</sup>p<sup>2</sup>

<sup>1</sup> i Eh ð Þkp<sup>3</sup>

Eð Þ<sup>k</sup> jj ¼ e ipð Þ<sup>k</sup>

system is stable in all its modes. The characteristic equation for cantilevered pipe is:

Applying the appropriate boundary conditions and considering only the nontrivial solution, the resulting characteristic equation can be solved numerically for the frequency and the critical flow velocity. The system is stable or unstable according to whether the imaginary component of the frequency, ω, is positive or negative, respectively. In case of neutral stability, ω is wholly real. As the flow velocity increases, the system may become unstable in one or more of its normal modes. The critical flow velocity is the greatest velocity for which the

where Tij are the elements of the overall transfer matrix. For a specific mode number, the proper starting frequency is determined by solving a subsidiary eigenvalue problem corresponding to the case of stationary fluid inside the vibrating pipe. A globally convergent optimization algorithm, known as Levenberg-Marquardt algorithm [38], has been applied to solve the resulting

ð Þ Eh <sup>k</sup>p<sup>2</sup>

i Eh ð Þkp<sup>3</sup>

pj, j = 1,…4 are the four roots of the fourth-order polynomial:

<sup>p</sup><sup>4</sup> � <sup>α</sup>kU<sup>2</sup> � �p<sup>2</sup> � <sup>2</sup>ω<sup>U</sup>

can be obtained by performing the matrix multiplications

<sup>p</sup>ð Þ<sup>k</sup> h i ¼

and [E] is a diagonal matrix with elements:

j¼1 Aje ipj

An efficient method [38] is successfully implemented to find the complex roots of Eq. (32) by formulating a special companion matrix and finding the associated eigenvalues for any desired values of the variables αk, βo, ω, and U. The transfer matrix [Tk] of the kth pipe module

<sup>E</sup>ð Þ<sup>k</sup> h i

2

2

<sup>P</sup>ð Þ<sup>k</sup> h i�<sup>1</sup>

ð Þ Eh <sup>k</sup>p<sup>2</sup>

i Eh ð Þkp<sup>3</sup>

1 1 �ip<sup>3</sup> �ip<sup>4</sup>

<sup>3</sup> ð Þ Eh <sup>k</sup>p<sup>2</sup>

<sup>3</sup> i Eh ð Þkp<sup>3</sup>

T33T44 � T34T43 ¼ 0, (46)

4

4

<sup>j</sup> Lk (45)

<sup>p</sup> � mkω<sup>2</sup>

<sup>x</sup> (41)

α<sup>k</sup> ¼ 0 , α<sup>k</sup> ¼ 1=ð Þ Ekhk (42)

(43)

(44)

methods:

184 Optimum Composite Structures

To verify the developed formulation, the classical problem dealing with one-module cantilevered pipe is considered first. The dimensionless wall thickness and length of the pipe are assigned at a value of 1.0, while the volume fraction at 50%. Figure 15 depicts variation of the critical flutter velocity and frequency with the mass density ratio MRo, covering a wide range of pipe and fluid mass densities. It is seen that there are four subdomains with the associated flutter modes defined in the specified intervals of the mass ratio. The upper and lower bounds determine the critical values of the mass ratios at which some of the frequency branches cross each other at the same value of the flutter velocity. The overall flutter mode may be regarded as composed of different quasimodes separated at the shown "jumps" in the Uf-MRo curve. The calculated mass density ratios at the three indicated frequency jumps are 0.4225, 2.29, and 12.33, respectively. They correspond to multiple points of neutral stability, where for a finite incremental increase in the flow velocity, the system becomes unstable, then regains stability, and then once again becomes unstable with a noticeable abrupt increase in the flutter frequency. Next, we consider a baseline design made of carbon/epoxy composites (see Table 2) with mass ratio MRo = 2.0. The calculated values of the dimensionless flutter velocity and frequency are found to be Uf = 10.78 and ω<sup>f</sup> = 26, respectively. Keeping the total dimensionless mass constant at the value corresponding to the baseline design, the best solution having the greatest flutter velocity was found to be (Vf, h1) = (0.70, 0.9345), which corresponds to Uf = 12.517 and ω<sup>f</sup> = 31.8615.

Figure 15. Variation of flutter speed and frequency with mass ratio for a uniform one-module cantilevered pipe (Vf1 = 50%, h1 = 1, L1 = 1).
