**6. Optimization of FGM pipes conveying fluid**

The determination of the critical flow velocity at which static or dynamic instability can be encountered is an important consideration in the design of slender pipelines containing

**Figure 13.** Isodiverts of for a two-panel wing model

constant value in order not to violate other performance requirements. Figure 12 shows a rectangular composite wing model constructed from uniform piecewise panels, where the design variables are defined to be the fiber volume fraction (Vf) and length (L) of each panel.

The isodiverts (lines of constant divergence speed) for a wing composed from two panels made of carbon-AS4/epoxy-3501-6 composite are shown in Figure 13. The selected design variables are (Vf1,L1) and (Vf2,L2). However, one of the panel lengths can be eliminated, because of the equality constraint imposed on the wing span. Another variable can also be discarded by applying the mass equality constraint, which further reduces the number of variables to only any two of the whole set of variables. Actually the depicted level curves represent the dimensionless critical flight speed augmented with the imposed equality mass constraint. It is seen that the function is well behaved, except in the empty regions of the first and third quadrants, where the equality mass constraint is violated. The final constrained optima was found to be (Vf1,L1)= (0.75, 0.5) and (Vf2,L2)= (0.25, 0.5), which corresponds to the maximum critical speed of 1.81, representing an optimization gain of about 15% above the reference value /2. The functional behavior of the critical flight speed *div Vˆ* of a three-panel model is shown in Figure 14, indicating conspicuous design trends for configurations with improved aeroelastic performance. As seen, the developed isodiverts have a pyramidal shape with its vertex at the design point (Vf2, L2)=(0.5, 1.0) having *div Vˆ* =/2. The feasible domain is bounded from above by the two lines representing cases of twopanel wing, with Vf1=0.75 for the line to the left and Vf3=0.25 for the right line. The contours near these two lines are asymptotical to them in order not to violate the mass equality constraint. The final global optimal solution, lying in the bottom of the pyramid, was calculated using the MATLAB optimization toolbox routines as follows: (Vfk, Lk)k=1,2,3 = (0.75, 0.43125), (0.5, 0.1375), (0.25, 0.43125) with *div Vˆ* =1.82, which represents an optimization gain of about 16%. Actually, the given exact mathematical approach ensured the attainment of global optimality of the proposed optimization model. A more general case would include

**Figure 12.** Composite wing model with material grading in spanwise direction

material grading in both spanwise and airfoil thickness directions.

The determination of the critical flow velocity at which static or dynamic instability can be encountered is an important consideration in the design of slender pipelines containing

**6. Optimization of FGM pipes conveying fluid** 

**Figure 14.** Isodiverts of *div Vˆ* in (Vf2-L2) design space for a three-panel wing model

flowing fluid. At sufficiently high flow velocities, the transverse displacement can be too high so that the pipe bends beyond its ultimate strength leading to catastrophic instabilities. In fact maximization of the critical flow velocity can be regarded as a major aspect in designing an efficient piping system with enhanced flexural stability. It can also have other desirable effects on the overall structural design and helps in avoiding the occurrence of large displacements, distortions and excessive vibrations, and may also reduce fretting among structural parts, which is a major cause of fatigue failure. The dynamic characteristics of fluid-conveying functionally graded materials cylindrical shells were investigated in [16]. A power-law was implemented to model the grading of material properties across the shell thickness and the analysis was performed using modal superposition and Newmark's direct time integration method. Reference [17] presented an analytical approach for maximizing the critical flow velocity, also known as divergence velocity, through multi-module pipelines for a specified total mass. Optimum solutions

were given for simply supported pipes with the design variables taken to be the wall thickness and length of each module composing the pipeline. A recent work [18] considered stability optimization of FGM pipelines conveying fluid, where a general multimodal model was formulated and applied to cases with different boundary conditions. A more spacious optimization model was given and extending the analysis to cover both effects of material, thickness grading and type of support boundary conditions. The model incorporated the effect of changing the volume fractions of the constituent materials for maximizing the critical flow velocity while maintaining the total mass at a constant value. Additional constraints were added to the optimization model by imposing upper limits on the fundamental eigenvalue to overcome the produced multiplicity near the optimum solution. Figure 15 shows the pipe model under consideration consisting of rigidly connected thinwalled tubes, each of which has different material properties, cross-sectional dimensions and length. The tube thickness, *h*, is assumed to be very small as compared with the mean diameter, *D*. The pipe conveys an incompressible fluid flowing steadily with an axial velocity *Uk* through the kth module. The variation in the velocity across the cross section was neglected, and the pipe was assumed to be long and slender so that the classical engineering theory of bending can be applicable. The effects of structural damping, damping of surroundings and gravity were not considered. Practical designs ignoring small damping, which has stabilizing effect on the system motion, are always conservatives. The model axis in its un-deformed state coincides with the horizontal x-axis, and the free small motion of the pipe takes place in a two dimensional plane with transverse displacement, *w.*

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 37

*k=1,2,…,Nm* (20)

In equation (16), *U* stands for the flow velocity through the pipe module having the maximum cross sectional area *Amax* and *Nm* is the total number of modules composing the pipeline. It is noted that consideration of the continuity equation provides that *UkAk=UAmax ,k=1,2,…Nm* Possible boundary conditions at the end supports of the pipeline are stated in

For a cantilevered pipeline, static instability caused by divergence is unlikely to happen. The non-trivial solution of the associated characteristic equation results in a vanishing bending displacement over the entire span of the pipeline. For such pipe configuration, dynamic instability (flutter) may only be considered. The state variable are defined by the vector

At two successive joints *(k)* and *(k+1)* the state vectors are related to each other by the matrix

 Zk+1 = [ Tk ] Zk (22) where *[Tk]* is a square matrix of order 4x4 known as the transmission or transfer matrix of the kth pipe module. For a pipeline built from *Nm* - uniform modules, Eq.(18) can be applied

where *[T]* is called the overall transmission matrix formed by taking the products of all the intermediate matrices of the individual modules. Therefore, applying the boundary conditions and considering only the non-trivial solution, the resulting characteristic equation can be solved numerically for the critical flow velocity, *U*. Extensive computer experimentation for obtaining the non-trivial solution of Eq.(19), for various pipe configurations, has demonstrated that the critical velocity can be multiple in some zones in the design space. This means that the eigenvalues cross each other, indicating multi-modal

*<sup>k</sup> k k Z w M F w w EIw EIw* (21)

*ZNm+1 = [ T ] Z1* (23)

where *<sup>k</sup> max k k*

(a) Hinged-Hinged *(H/H): w(0)=w*

 *w(1)=w*

(b) Clamped-Hinged *(C/H): w(0)=w*

 *w(1)=w*

(c) Clamped-Clamped *(C/C): w(0)=w*

 *w(1)=w*

*U*

the following:

equation

at successive joints to obtain

*k k kk*

which is valid over the length of any kth module of the pipe, i.e. *0* 

*(0)=0* 

*(1)=0* 

*(0)=0* 

 *(1)=0*

 *(0)=0* 

*T*

*(1)=0*

*A A U*

*EkI A E I*

<sup>2</sup> 0 *w wk* (19)

 *x* 

 *Lk***,** where *x =x-xk*

**Figure 15.** General configuration of a piecewise axially graded pipe conveying fluid.

The various parameters are normalized by their corresponding values of a baseline pipe having the same total mass and length, material and fluid properties, and boundary conditions as well. The baseline pipe has uniform mass and stiffness distributions along its length and is made of two different materials denoted by *(A)* and *(B)* with equal volume fractions *(V)*, i.e. *VA=VB= 50%*. The governing differential equation in dimensionless form:

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 37

$$
\pi \varpi^{\prime\prime\prime\prime} + \chi^2\_{\cdot k} \overline{w}^{\prime\prime} = 0 \tag{19}
$$

where *<sup>k</sup> max k k k k kk A A U U EkI A E I* 

36 Advances in Computational Stability Analysis

were given for simply supported pipes with the design variables taken to be the wall thickness and length of each module composing the pipeline. A recent work [18] considered stability optimization of FGM pipelines conveying fluid, where a general multimodal model was formulated and applied to cases with different boundary conditions. A more spacious optimization model was given and extending the analysis to cover both effects of material, thickness grading and type of support boundary conditions. The model incorporated the effect of changing the volume fractions of the constituent materials for maximizing the critical flow velocity while maintaining the total mass at a constant value. Additional constraints were added to the optimization model by imposing upper limits on the fundamental eigenvalue to overcome the produced multiplicity near the optimum solution. Figure 15 shows the pipe model under consideration consisting of rigidly connected thinwalled tubes, each of which has different material properties, cross-sectional dimensions and length. The tube thickness, *h*, is assumed to be very small as compared with the mean diameter, *D*. The pipe conveys an incompressible fluid flowing steadily with an axial velocity *Uk* through the kth module. The variation in the velocity across the cross section was neglected, and the pipe was assumed to be long and slender so that the classical engineering theory of bending can be applicable. The effects of structural damping, damping of surroundings and gravity were not considered. Practical designs ignoring small damping, which has stabilizing effect on the system motion, are always conservatives. The model axis in its un-deformed state coincides with the horizontal x-axis, and the free small motion of the pipe takes place in a two dimensional plane with transverse displacement, *w.*

**Figure 15.** General configuration of a piecewise axially graded pipe conveying fluid.

The various parameters are normalized by their corresponding values of a baseline pipe having the same total mass and length, material and fluid properties, and boundary conditions as well. The baseline pipe has uniform mass and stiffness distributions along its length and is made of two different materials denoted by *(A)* and *(B)* with equal volume fractions *(V)*, i.e. *VA=VB= 50%*. The governing differential equation in dimensionless form:

$$k = 1, 2, \ldots, \mathbf{N}\_m \tag{20}$$

which is valid over the length of any kth module of the pipe, i.e. *0 x Lk***,** where *x =x-xk*

In equation (16), *U* stands for the flow velocity through the pipe module having the maximum cross sectional area *Amax* and *Nm* is the total number of modules composing the pipeline. It is noted that consideration of the continuity equation provides that *UkAk=UAmax ,k=1,2,…Nm* Possible boundary conditions at the end supports of the pipeline are stated in the following:

(a) Hinged-Hinged *(H/H): w(0)=w(0)=0 w(1)=w(1)=0*  (b) Clamped-Hinged *(C/H): w(0)=w(0)=0 w(1)=w (1)=0* (c) Clamped-Clamped *(C/C): w(0)=w (0)=0 w(1)=w(1)=0*

For a cantilevered pipeline, static instability caused by divergence is unlikely to happen. The non-trivial solution of the associated characteristic equation results in a vanishing bending displacement over the entire span of the pipeline. For such pipe configuration, dynamic instability (flutter) may only be considered. The state variable are defined by the vector

$$\underline{\mathbf{Z}}\_{k}^{T} = \left[ \underline{\mathbf{w}} \,\upvarphi \,\mathbf{M} \,\boldsymbol{F} \right]\_{k} = \left[ \underline{\mathbf{w}} - \underline{\mathbf{w}}' - EI\underline{\mathbf{w}}'' - EI\underline{\mathbf{w}}''' \right]\_{k} \tag{21}$$

At two successive joints *(k)* and *(k+1)* the state vectors are related to each other by the matrix equation

$$
\underline{\nabla}^{k+1} = \begin{bmatrix} \Gamma \underline{r} \end{bmatrix} \underline{\nabla} \tag{22}
$$

where *[Tk]* is a square matrix of order 4x4 known as the transmission or transfer matrix of the kth pipe module. For a pipeline built from *Nm* - uniform modules, Eq.(18) can be applied at successive joints to obtain

$$
\underline{\mathbf{Z}}^{\rm Nm+1} = \begin{bmatrix} \ \mathbf{T} \ \end{bmatrix} \ \underline{\mathbf{Z}}^{\rm I} \tag{23}
$$

where *[T]* is called the overall transmission matrix formed by taking the products of all the intermediate matrices of the individual modules. Therefore, applying the boundary conditions and considering only the non-trivial solution, the resulting characteristic equation can be solved numerically for the critical flow velocity, *U*. Extensive computer experimentation for obtaining the non-trivial solution of Eq.(19), for various pipe configurations, has demonstrated that the critical velocity can be multiple in some zones in the design space. This means that the eigenvalues cross each other, indicating multi-modal solutions (i.e. Bi- Tri- Quadri- modal solutions). Such a multiplicity introduces singularity of the eigenvalue derivatives with respect to the design variables, which does not allow the use of gradient methods. Therefore, it is necessary to formulate the optimization problem with respect to the critical velocity connected with two, three, or four simultaneous divergence modes. The present formulation employs multi-dimensional, non-gradient search techniques to find the required optimum solutions [2, 3]. This formulation requires only simple function evaluations without computing any derivatives for either the objective function or the design constraints. The additional constraints, which ought to be added to the optimization problem, are [18]:

$$\mathsf{U}\mathsf{U}\mathsf{i}\leq\mathsf{U}\mathsf{j},\quad\mathsf{j}\neq\mathsf{2},\mathsf{3}...\mathsf{m}.\tag{24}$$

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 39

*(1/2)=0*. For three-module *H/H* pipeline, the attained

3.2235 4.5645 6.3325

3.6235 5.1355 7.0965

become *w(0)=w*

H/H C/H C/C

H/H C/H C/C

constant total mass.

**Table 6.** Optimal solutions for two-module pipelines

*(0)=0* and *w*

20.81% optimization gain above the baseline value

*(1/2)=w* maximum value of the critical velocity was found to be 3.7955, occurring at the design point *(Vf, h, L)k= (0.625,0.5,0.15625), (0.7,1.1375, 0.6875), (0.625, 0.5, 0.15625)*. This represents about

Support (Vf, h, L)k=1,2 Ucr,max

**Material grading only**  (0.550, 1.0, 0.800), (0.300, 1.0, 0.200) (0.525, 1.0, 0.875), (0.325, 1.0, 0.125) (0.675, 1.0, 0.125), (0.475, 1.0, 0.875)

**Combined material & thickness grading** (0.70, 1.0, 0.75), (0.65, 0.75, 0.25) (0.70, 0.95, 0.9), (0.50, 0.85, 0.10) (0.70, 1.0, 0.60), (0.65, 0.85, 0.40)

**Figure 16.** Effect of material grading on the critical flow velocity for a two-module, *H/H* pipe with

.

where *U1* is the first eigenvalue representing the dimensionless critical flow velocity, *Uj's* are the subsequent higher eigenvalues and *m* is the assumed modality of the final optimum solution. All constraints are augmented with the objective function through penalty multiplier terms, and the number of active constraints at the optimum design point can automatically detect the actual modality of the problem. In the case of single mode optimization, none of the constraints become active at the optimal solution. It is noted that the total mass and length equality constraints can be used to eliminate some of the design variables, which help reducing the dimensionality of the optimization problem. The *MATLAB* optimization toolbox is a powerful tool that includes many routines for different types of optimization encompassing both unconstrained and constrained minimization algorithms [3]. One of its useful routines is named *"fmincon"* which finds the constrained minimum of an objective function of several variables. Figure 16 depicts the functional behavior of the dimensionless critical flow velocity, *Ucr,1* augmented with the equality mass constraint, *Ms=1*. It is seen that the function is well behaved and continuous everywhere in the design space *(Vf -L)1*, except in the empty region located at the upper right of the whole domain, where the mass equality constraint is violated. The feasible domain is seen to be split by the baseline contours (*Ucr=*) into two distinct zones. The one to the right encompasses the constrained global maxima, which is calculated to be *Ucr=3.2235* at the optimal design point *(Vf ,L)k*=1,2 =(0.550, 0.80), (0.30, 0.20). Actually, each design point inside the feasible domain corresponds to different material properties as well as different stiffness and mass distributions, while maintaining the total structural mass constant. Figure 17 shows the developed isodiverts (lines of constant divergence velocity, *Ucr,1*) in the (*Vf1-Vf2)*  design space. The equality mass constraint is violated in the first and third quadrants and the cross lines *Vf1=50%* and *Vf2=50%* represent the isodiverts of the baseline value . For the case of a clamped-hinged (C/H), two-module pipe, the global maxima was calculated to be *(Vf ,L)k*=1,2 =(0.525, 0.875), (0.325, 0.125) at which *Ucr,1*=4.5645. Table 6 summarizes the attained optimal solutions for the different types of boundary conditions. Cases of combined material and thickness grading are also included, showing a truly and significant optimization gain for the different pipe configurations. More results indicated that for the case of *H/H* pipelines, good patterns must be symmetrical about the mid-span point. Therefore, it can be easier to cope with symmetrical configurations, which reduce computational efforts significantly, and the total number of variables to half. In this case, the boundary conditions become *w(0)=w(0)=0* and *w(1/2)=w(1/2)=0*. For three-module *H/H* pipeline, the attained maximum value of the critical velocity was found to be 3.7955, occurring at the design point *(Vf, h, L)k= (0.625,0.5,0.15625), (0.7,1.1375, 0.6875), (0.625, 0.5, 0.15625)*. This represents about 20.81% optimization gain above the baseline value .


**Table 6.** Optimal solutions for two-module pipelines

38 Advances in Computational Stability Analysis

the optimization problem, are [18]:

split by the baseline contours (*Ucr=*

solutions (i.e. Bi- Tri- Quadri- modal solutions). Such a multiplicity introduces singularity of the eigenvalue derivatives with respect to the design variables, which does not allow the use of gradient methods. Therefore, it is necessary to formulate the optimization problem with respect to the critical velocity connected with two, three, or four simultaneous divergence modes. The present formulation employs multi-dimensional, non-gradient search techniques to find the required optimum solutions [2, 3]. This formulation requires only simple function evaluations without computing any derivatives for either the objective function or the design constraints. The additional constraints, which ought to be added to

where *U1* is the first eigenvalue representing the dimensionless critical flow velocity, *Uj's* are the subsequent higher eigenvalues and *m* is the assumed modality of the final optimum solution. All constraints are augmented with the objective function through penalty multiplier terms, and the number of active constraints at the optimum design point can automatically detect the actual modality of the problem. In the case of single mode optimization, none of the constraints become active at the optimal solution. It is noted that the total mass and length equality constraints can be used to eliminate some of the design variables, which help reducing the dimensionality of the optimization problem. The *MATLAB* optimization toolbox is a powerful tool that includes many routines for different types of optimization encompassing both unconstrained and constrained minimization algorithms [3]. One of its useful routines is named *"fmincon"* which finds the constrained minimum of an objective function of several variables. Figure 16 depicts the functional behavior of the dimensionless critical flow velocity, *Ucr,1* augmented with the equality mass constraint, *Ms=1*. It is seen that the function is well behaved and continuous everywhere in the design space *(Vf -L)1*, except in the empty region located at the upper right of the whole domain, where the mass equality constraint is violated. The feasible domain is seen to be

 *Uj , j=2,3…m.* (24)

) into two distinct zones. The one to the right

. For the

*U1*

the cross lines *Vf1=50%* and *Vf2=50%* represent the isodiverts of the baseline value

encompasses the constrained global maxima, which is calculated to be *Ucr=3.2235* at the optimal design point *(Vf ,L)k*=1,2 =(0.550, 0.80), (0.30, 0.20). Actually, each design point inside the feasible domain corresponds to different material properties as well as different stiffness and mass distributions, while maintaining the total structural mass constant. Figure 17 shows the developed isodiverts (lines of constant divergence velocity, *Ucr,1*) in the (*Vf1-Vf2)*  design space. The equality mass constraint is violated in the first and third quadrants and

case of a clamped-hinged (C/H), two-module pipe, the global maxima was calculated to be *(Vf ,L)k*=1,2 =(0.525, 0.875), (0.325, 0.125) at which *Ucr,1*=4.5645. Table 6 summarizes the attained optimal solutions for the different types of boundary conditions. Cases of combined material and thickness grading are also included, showing a truly and significant optimization gain for the different pipe configurations. More results indicated that for the case of *H/H* pipelines, good patterns must be symmetrical about the mid-span point. Therefore, it can be easier to cope with symmetrical configurations, which reduce computational efforts significantly, and the total number of variables to half. In this case, the boundary conditions

**Figure 16.** Effect of material grading on the critical flow velocity for a two-module, *H/H* pipe with constant total mass.

Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 41

chapter will be compared and validated through other optimization techniques such as genetic algorithms or any appropriate global optimization algorithm. Further optimization studies must depend on a more accurate analysis of constructional cost. This combined with probability studies of load applications and materials variations, should contribute to further efficiency achievement. Much improved and economical designs for the main structural components may be obtained by considering multi-disciplinary design optimization, which allows designers to incorporate all relevant design objectives simultaneously. Finally, it is important to mention that, while FGM may serve as an excellent optimization and material tailoring tool, the ability to incorporate optimization techniques and solutions in practical design depend on the capacity to manufacture these materials to required specifications. Conventional techniques are often incapable of adequately addressing this issue. In conclusion, FGMs represent a rapidly developing area of science and engineering with numerous practical applications. The research needs in this area are uniquely numerous and diverse, but FGMs promise significant potential benefits

that fully justify the necessary effort.

Sons, ISBN: 978-0470183526, New York.

Univ. Press, New York.

Structures, 16(1): 77-83.

John Wiley & Sons, ISBN: 978-0470084885, New York.

Structures. Applied Mechanics Reviews, ASME 60: 195-216.

Natural Frequencies. Journal of Sound and Vibration, 280: 415-424.

*National Research Centre, Mechanical Engineering Department, Cairo, Egypt* 

[1] Maalawi K., Badr M. (2009) Design Optimization of Mechanical Elements and Structures: A Review with Application. Journal of Applied Sciences Research 5(2**):** 221–231. [2] Rao S. (2009) Engineering Optimization: Theory and Practice, 4th edition, John Wiley &

[3] Venkataraman P. (2009) Applied Optimization with MATLAB Programming, 2nd edition,

[4] Daniel I., Ishai O. (2006) Engineering Mechanics of Composite Materials, 2nd ed., Oxford

[5] Birman V., Byrd W. (2007) Modeling and Analysis of Functionally Graded Materials and

[6] Elishakoff I., Guede Z. (2004) Analytical Polynomial Solutions for Vibrating Axially Graded Beams. Journal of Mechanics and Advanced Materials and Structures, 11: 517-

[7] Elishakoff I., Endres J. (2005) Extension of Euler's Problem to Axially Graded Columns: Two Hundred and Sixty Years Later. Journal of Intelligent Material systems and

[8] Shi-Rong Li, Batra R. (2006) Buckling of Axially Compressed Thin Cylindrical Shells with Functionally Graded Middle Layer. Journal of Thin-Walled Structures, 44: 1039-1047. [9] Qian L., Batra R. (2005) Design of Bidirectional Functionally Graded Plate for Optimal

**Author details** 

Karam Maalawi

**8. References** 

533.

**Figure 17.** Isodiverts in the (*Vf1-Vf2*) design space for a two-module, *H/H* pipe.
