composite dome shape and pressure vessels optimization

Head Shape and Winding Angle Optimizationof Composite Pressure Vessels Basedon a Multi-level Strategy

A. Vafaeesefat & A. Khani

Received: 30 August 2007 /Accepted: 30 January 2008 /Published online: 19 February 2008# Springer Science + Business Media B.V. 2008

Abstract This paper presents a multi-level strategy for the optimization of compositepressure vessels with nonmetallic liners. The design variables for composite vessels includethe head shape, the winding angle, the layer thickness, the number of layers, and thestacking sequence. A parameter called “modified shape factor” is introduced as an objectivefunction. This parameter takes into account the effects of the internal pressure and volume,the vessel weight, and the composite material properties. The proposed algorithm usesgenetic algorithm and finite element analysis to optimize the design parameters. As a fewexamples, this procedure is implemented on geodesic and ellipsoidal heads. The resultsshow that for the given vessel conditions, the geodesic head shape with helical windingangle of nine degrees has the better performance.

Keywords Filament wound vessel . Optimization . Genetic algorithm .Multi-level strategy .

Stacking sequence . Shape factor . Geodesic

1 Introduction

High-pressure vessels are widely used in commercial and aerospace applications as well astransportation vehicles. Filament-wound composite pressure vessels, which utilize afabrication technique of filament winding to form high strength and light weight reinforcedplastic parts, are a major type of high pressure vessels. Pressure vessels normally consist oftwo distinct parts: cylindrical portion and heads, domes or caps. Heads are usually the mostimportant part in the pressure vessel design. The desired parameters for a good head shapeare higher burst pressure and internal volume and lower weight.

Appl Compos Mater (2007) 14:379–391DOI 10.1007/s10443-008-9052-8

A. Vafaeesefat (*)Mechanical Engineering Department, Imam Hussein University, Tehran, Irane-mail: [email protected]

A. KhaniFaculty of Energy and New Technologies, Aerospace Engineering, Shahid Behesti University, Tehran, Iran

Many works have been done to optimize the design parameters of filament woundpressure vessels. Fukunaga and Chou [1] presented a laminate optimization procedure forfilament wound cylindrical pressure vessels under stiffness and strength constraints. Adaliet al. [2] presented an optimum design algorithm for symmetrically laminated cylindricalpressure vessels. Krikanov [3] and Jaunky et al. [4] introduced an analytical laminateoptimization approach for composite pressure vessels under stiffness and strengthconstraints. The reported works are mainly based on simple analyzing or experimentalmethods and the head shape effect has not been considered.

Optimum design of dome contours for composite pressure vessels has been thesubject of many researches [5–8]. Hofeditz [5] applied the netting and orthotropicanalysis to solve dome design problems. Hojjati et al. [6] used the orthotropic plate theoryfor dome design of the polymeric composite pressure vessels. Lin and Hwang [7] used aparameter called performance factor to evaluate the structural efficiency of the vesseldomes. They introduced an optimum dome design method based on the Tsai–Hillfailure criterion and orthotropic plate theory. Liang et al. [8, 9] investigated the optimumdesign of dome contour for filament wound composite pressure vessels subjected togeometrical limitations, winding conditions and the Tsai–Wu failure criterion. They usedthe feasible direction method for maximizing the shape factor. However, the stackingsequences are not simultaneously considered in their optimization procedure.

In this paper, a multi-level strategy is introduced for the optimization of compositepressure vessels with nonmetallic liners. The multi-level optimization strategy is apowerful method for the problems whose plurality of design variables is relatively large.The benefit is that by reducing the number of design variables in each level,convergence in genetic algorithm occurs much faster. The multi-level procedure isseldom reported for the optimization of complex structures such as filament woundvessels. As an example, this strategy is implemented on composite vessels withgeodesic and ellipsoidal heads. However, the presented method could be applied forevery symmetric vessel with any kind of head shape. This strategy is subjected to theTsai–Wu failure criterion and problem of maximizing the modified shape factor usinggenetic algorithm. This study is limited to the symmetric composite vessels which havetwo same domes and opening radii. The relation between internal pressure and Tsai–Wufailure criterion is modeled through the finite element analysis and the geodesiccondition is considered to prevent fiber slipping.

2 Winding Pattern

Filament wound vessel design includes the design of the mandrel shape and thecalculation of the fiber path. In general, the mandrel shape can be determined byimposed design requirements such as internal pressure, volume and manufacturingconvenience.

Finding the possible winding patterns on an arbitrary shape is one of the firstnecessaries in order to introduce the optimization strategy for composite vessels. Sincethe accuracy of finite element analysis is directly influenced by the winding

380 Appl Compos Mater (2007) 14:379–391

information, there is a need for the winding pattern to be actually modeled. In thispaper, the semi-geodesic path method is proposed, in which the slippage tendencybetween the fiber and the mandrel is considered. The semi-geodesic path for generalfilament wound structures is defined as follows [9]:

dadz

¼ l A2 sin2 a � rr'' cos2 a� �� r'A2 sin a

r'A2 cosað1Þ

A ¼ 1þ r2

Equation 1 is defined on an arbitrary surface where α, z, θ, ρ and 1 are the windingangle, the axial, circumferential and radial coordinate parameters, and the slippage tendencybetween the fiber and the mandrel, respectively.

By setting the slippage tendency equal to zero in Eq. 1, the geodesic path equation isobtained:

r sin a ¼ cte ð2Þ

The geodesic path introduces the shortest path between two points on a surface.Therefore, geodesic fiber path is a special kind of semi-geodesic fiber path for which theslippage tendency is zero. This kind of winding is named “isotensoid winding” in literature.In order to obtain the winding angle at each point, we have:

r sina ¼ r0 sin a0 ¼ rb1 sin 90 ¼ rb2 sin 90 ð3Þ

where α0 and ρ0 are the winding angle and the radius of the vessel in the cylindrical part,respectively. Also ρb1 and ρb2 denote left and right dome opening radii, respectively.Therefore, for geodesic winding, two domes must completely have similar opening radii.When the two openings are not the same, semi-geodesic path must be used. This study islimited to symmetric vessels and, therefore, the geodesic winding pattern is applied.Thickness of the helical layers at each point of the head is obtained by:

t ¼ r0r

cos a0

cosat0 ð4Þ

where t and t0 are the helical layer thickness on the head and cylinder, respectively.

3 Different Head Shapes

The head shape has certain effects on the internal volume, weight and burst pressure of thevessel. The internal volume of the vessel is determined by the internal volume of the domesand the drum. The outer area of the domes affects the whole weight of the filament wound

Appl Compos Mater (2007) 14:379–391 381

structure. The geometry of the domes has influence on the burst pressure of the vessel andaffects the failure criterion.

3.1 Geodesic Dome Contours

Using netting analysis simultaneously with the Eq. 3, the geodesic dome shape can bederived [10]. The coordinates of the constructive points of the geodesic dome profile( ρ and z) are obtained from the numerical integration of the below equation:

z ¼ �r0

Z rr0

1

cosa0t3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� t3ð Þp

cos2 a0 � t2 1þ t2ð Þ � sin2 a� � dt ð5Þ

In this research, the integral Eq. 5 is numerically calculated using parameters α0 and ρ0which are related to the cylindrical portion. Accordingly, for different radial coordinateparameter ( ρ), axial coordinate parameter (z) are calculated and the dome shape isdetermined. The winding angle values (α) are calculated for each radial distance from thecentral axis (ρ) using the Eq. 4. Figure 1 shows different geodesic head geometries and theirrelationship to the fiber-winding angle in the cylindrical portion (α0).

3.2 Ellipsoidal Dome Contours

There may be different ellipsoidal shapes according to the value of aspect ratio (e) which isthe ratio of the ellipse diameter along the vessel axis (H) to the diameter perpendicular to it(D) (Fig. 2). The dome geometry is also affected by the winding angle through openingradius variation (Eq. 3).

4 Finite Element Model

Finite element is used to analyze the three dimensional model of the vessel. Stressconcentration is high at the junction of the cylinder and the head due to sudden change of

Fig. 1 Different geodesichead shape for different windingangle


the curvature in this area. For this reason, the smaller shell elements are modeled in thisarea compared to other locations. Moreover, since the structure is symmetric, only half ofvessel is modeled.

There are several failure criteria to predict failure in the composite materials. In thiswork, the 3D Tsai–Wu criterion is utilized which is defined by:

x3 ¼ Aþ B ð6Þwhere

A ¼ � σxð Þ2σ f

xtσfxc

� σy

� �2σ f

ytσfyc

� σzð Þ2σ f

ztσfzc

þ σxy

� �2σ f

xy

� �2 þσyz

� �2σ f

yz

� �2 þσxzð Þ2

σ fxz

� �2 þCxyσxσyffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

σ fxtσ

fxcσ

fytσ

ftc

q

þ Cyzσyσzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ f

ytσfycσ

fztσ

fzc

q þ Cxzσxσzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ f

xtσfxcσ

fxtσ

fzc

qand

B ¼ 1

σ fxt

þ 1

σ fxc

" #σx þ 1

σ fyt

þ 1

σ fyc

" #σy þ 1

σ fzt

þ 1

σ fzc

" #σz

and ξ3, σ, σft and σ f

c are the three dimensional Tsai–Wu criterion, the applied stresses,tension strength and compression strength, respectively. Cxz, Cyz and Cxy are coupling

Fig. 2 Different ellipsoidal headshapes


coefficients for Tsai–Wu theory. Indices x, y and z show the fiber direction, it’sperpendicular direction in and out of the element plane, respectively. Mechanical propertiesof the carbon-epoxy composite material used in this research are detailed in Table 1. Massdensity of the composite material is 1,565 kg/m3.

The outside diameter of the cylindrical section is 330 mm and its length is 650 mm(Fig. 3). The defined vessel is a compressed natural gas pressure vessel with a non-metallicliner. The working pressure, test pressure, and burst pressure based on ISO 1439 [10, 11]are 200, 300, and 470 bar, respectively. In the finite element model, a uniform pressure470 bar is applied. The finite element analysis results are presented versus longitudinaldistance on the vessel mane from the middle cross section of the vessel (Fig. 3).

5 Multi-level Optimization Strategy

The design variables of composite vessel consist of the head shape, the angle of helicallayers (α), the layers thickness (t), the number of layers (NL) and the stacking sequence.Due to the continuous nature of filament winding process, the thickness of all layers isusually selected a constant value or an integer factor of it. Optimization procedure forcomposite pressure vessels is divided into two levels; namely, head shape and laminateweight optimization. The schematic diagram of the multi-level optimization strategy fordifferent head shapes is presented in Fig. 4.

In the first level, known head shapes with different winding angles are compared basedon the maximum shape factor. In this level, the procedure is done for an initial stackingsequence and number of layers that may not be optimum. In the second level, the optimumstacking sequence and number of layers will be selected. This two-level procedure isrepeated until no clear progress is seen. Using the multi-level optimization procedure, thenumber of required finite element analysis is significantly deceased.

Materials properties

Exx 110.3Eyy 15.2Ezz 8.97Gxy 4.9Gxz 4.9Gyz 3.28νxy 0.25νxz 0.25νyz 0.365σ f

xt 1,918σ f

xc 1,569σ f

yt 247σ f

yc 1,245σ f

zt 247σ f

zc 1,245σxy 68.9σyz 34.5σxz 34.5

Table 1 Materials properties ofcomposite material (Strengthsare in MPa and elastic modulusare in GPa)


5.1 Head Shape and Winding Angle Selection

The head shape factor is used to compare the performance of the different head shapes.Since the Tsai–Wu failure criterion takes into account the effects of internal pressure, vesselcharacteristics and material properties, the pressure and strength terms in the shape factor[9] can be replaced by Tsai–Wu. Consequently, a new factor named “modified shape factor”is introduced as below:

K' ¼ V

W=+

� �Max:Tsai Wuð Þ

ð7Þ

where W, V, and + are, respectively, the dome weight, the internal volume, and the vesseldensity. In the mentioned two-level strategy, the modified shape factor is compared fordifferent heads and winding angles. Since selection is done through a comparativeapproach, failure criterion is put in the denominator and values greater than 1 do not make

Fig. 3 Vessel dimensions (in mm) when the helical winding angle of fiber is ±9 in cylindrical section (left).Longitudinal distance (s) (right)

Level 1

Level 2

Selection of initial approximate number of layers and thicknesses

plus an arbitrary stacking

Selection of the best head shape and winding angle based on

maximum modified shape factor

Stacking sequence and number of layers optimization

ää jj KK 1

Optimum design

No

Yes

Level 1

Level 2

Selection of initial approximate number of layers and thicknesses

plus an arbitrary stacking

Selection of the best head shape and winding angle based on

maximum modified shape factor

Stacking sequence and number of layers optimization

≤__ jj KK 1

Optimum design

ε

Fig. 4 Schematic diagram ofmulti-level optimization strategyfor different heads


any difficulties due to relatively selection. On the other hand, after the second level iscarried out, this problem is solved. By applying this evaluation factor, there is no need toimpose any constraints.

5.2 Laminate Weight Optimization Using GA

The procedure for the optimization of the number of layers and the stacking sequence viagenetic algorithm is applied on the best head shape and the winding angle selected in thefirst level. The laminate optimization procedure is configured to find the stacking sequenceand number of layers which have the minimum weight with a failure criterion below one.Therefore, the problem is formulated as below:

Min W subject to Tsai Wu < 1:

In this procedure, the optimal stacking sequence with maximum strength is obtainedusing genetic algorithm for a constant number of layers, and the number of layers isdecreased or increased to reach the minimum number of layers with Tsai–Wu below onethrough an iterative approach. For the first estimation of the number of layers, nettinganalysis [10] or any other appropriate method can be used.

In fact, two design variables of number of layers and stacking sequence do not have thesame effect on the weight and strength. To be more precise, the number of layers has alarger effect than stacking sequence. On the other hand, some variables such as the numberof layers have a linear and predictable effect on the structural behavior; whereas, some likestacking sequence have a nonlinear effect. Therefore, using one objective function to findthe minimum number of layers (minimum weight) while its Tsai–Wu criterion is below onemay not converge to the optimum solution. For these reasons, these two variables areseparated and optimized via different approaches.

6 Genetic Algorithm Implementation

The optimal stacking sequence is obtained using genetic algorithm for minimum number oflayers. As mention before, stacking sequence optimization is a sub-problem in the weightminimization. This problem is formulated for the strength maximization of a minimumnumber of layers is defined by:

MinSS MaxLN;S Tsai�Wu SS; LN; Sð Þsubject to:

Number of layers NLLayer thickness 1 mmLayer angles 90, α, −α

where SS is the stacking sequence of fiber angles which is the design variable; LN is thelayer number with maximum failure criterion; S is the longitudinal distance (Fig. 3) wherethe maximum failure criterion takes place along the vessel mane and NL is the number oflayers.

Design variables in stacking sequence optimization (winding angle of fiber) are notcontinuous. Therefore, in each model the winding angle design variables are restricted tothe discrete values of +α and −α for the helical layers and 90 for the hoop ones. For


example, if the helical winding angle is 9° and a geodesic head based on this angle isimplemented, the sequences of angles are limited to +9°, −9° and 90°. To apply thisdiscontinuity and prevent the selection of other angles, the stacking sequence of thelayer angle is coded to a binary set. Every two digit in the binary set are represented asan angle. Based on our definition, different combinations of zero and one are translatedas follow:

0; 0½ � ¼ 90; 1; 1½ � ¼ �90; 1; 0½ � ¼ 9; 0; 1½ � ¼ �9:

The GA program works with the set of binary values. For example, the stackingsequence for a multi-layer composite composed of nine layers is defined by:

0; 1; 1; 1; 0; 0; 1; 0; 0; 1; 1; 0; 0; 0; 1; 0½ �:For applying to FE code, this stacking sequence is translated as (angles +90° and −90°

are equal for the fiber):

�9; 90; 90; 9;�9; 9; 90; 9½ �:

7 Results

As two application examples, the two-level optimization strategy is applied for vessels withgeodesic and ellipsoidal heads. For geodesic heads, the winding angle should be onlyoptimized in first level (Fig. 1); whereas for ellipsoidal ones the aspect ratio is optimized inaddition to winding angle (Fig. 2).

7.1 Geodesic Domes

Numerical analyses are performed on different geodesic head shapes with various windingangles (Fig. 2), and similar stacking sequences and numbers of layers. Nine layers areselected using netting analysis with the arbitrary angle and thickness stacking sequences of[90, 90, α, −α, α, −α, α, 90, 90] and [1, 1, 0.5, 0.5, 0.5, 0.5, 0.5, 1, 1] in the cylindricalportion as the first point in the optimization procedure. As shown in Fig. 5, the windingangle of 9° has the best performance in the first level. In the second level, stackingsequence optimization is carried out for the best winding angle of 9° to find the optimalstacking sequence. The optimum stacking sequence [90, 9, −9, 90, 9, −9, 90, 9, 90] isobtained.

As shown in the Table 2, the maximum Tsai–Wu failure criterion is less than one for theoptimal stacking sequence. Therefore, the number of layers is decreased to eight layers tosee if it is possible to obtain a lighter laminate. As shown in Table 2, the failure criterion forthe optimum stacking sequence of eight layers is greater than one; therefore, nine is theleast acceptable number of layers. Since the hoop layers are a little lighter than helical ones,the possibility of replacing helical layers with hoop ones should be checked in order toobtain a lighter laminate for the minimum number of layers. When the numbers of helicaland hoop layers are limited to four and five, the optimal stacking sequence [90, 9, −9, 9, 90,90, 9, 90, 90] with failure criterion value of 1.924 is obtained; therefore, it is not possible toreplace helical layers with hoop ones.

The whole described two-level strategy should be repeated until no considerableimprovement is seen in the objective function. As shown in Fig. 5, the optimum stacking


sequence is examined for different winding angles in the second iteration and 9° has againthe best performance. Therefore, the procedure is finished.

In this approach, the genetic algorithm parameters are selected through trial and error.The initial population and the maximum number of generations are 50 and 100,respectively. The allowable difference between population individuals for two consecutivegenerations is set to 1e-6. If this difference becomes less than this defined value, theprogram will stop. The mutation and crossover operators have been selected binary andarithmetic, respectively.

Figure 6 shows the Tsai–Wu failure criterion for the optimum stacking sequence withgeodesic head and helical winding angle 9°. This figure shows that the maximum Tsai–Wu(that is 0.955 in Table 2) is located at the junction of cylinder section and the head. Afterthis point, the cylindrical section has a higher Tsai–Wu failure criterion than other areas onthe dome.

To show that the obtained winding angle is optimum, the optimum stacking sequence forthree models of nine-layered shells with helical winding angles 9°, 20° and 30° arecompared in Table 3. The results show that the optimum stacking sequences of compositeshells with geodesic heads with different winding angles (and opening diameter) of 9°, 20°and 30°, have the lowest to the highest values of Tsai–Wu, respectively. Therefore, thenumber of layers for the helical angle 9° is suitable whereas for 20°, 30°, this numbershould be increased to obtain the acceptable Tsai–Wu. These results show the fact thathelical angle 9° has better performance compared to 20° and 30° as shown in Fig. 5.

Fig. 5 Modified shape factor fordifferent winding angles

Table 2 The optimal stacking sequence for composite shells with helical winding angle ±9°

Numberof layers

Optimal stacking sequence Maximum Tsai–Wufailure criterion

Vessel weight (kg)

8 [90, 9, −9, 90, 90, −9, 9, 90] 1.145 5.9779 [90, 9, −9, 90, 9, −9, 90, 9, 90] 0.955 6.833


7.2 Ellipsoidal Heads

The proposed two-level optimization procedure was also implemented on ellipsoidal headshape with different aspect ratios. The first level is done in two stages itself. First, only theeffect of head geometry is studied and all other variables are considered constant. The anglestacking sequence [90, 20, −20, 90, 20, −20, 90, 20, 90] and layer thickness of 1 mm in thecylindrical portion is used as the first point in the optimization procedure. This is theoptimum stacking sequence obtained for geodesic heads. Note that the procedure is notsensitive to the start point; therefore, any appropriate method can be used for the first guessof layer thickness and number of layers.

From Fig. 7, it is seen that for different aspect ratios (0.2<e<2) of the ellipsoidal heads,the maximum modified shape factor belongs to e=0.6.

Next, only the effect of winding angle for the selected aspect ratio (e=0.6) from theprevious selection is studied and all other variables are considered constant.

As shown in Fig. 8, the maximum modified shape factor occurred for the angle of21°. If the domain of helical layer orientation angles between 5° and 40° is divided by thepre-assigned increment of five degrees, the winding angle of 20° will be selected in thislevel.

The second level belongs to the stacking sequence and number of layers optimizationwhile the head shape aspect ratio and the winding angle values obtained from the last twolevels are constant. The stacking sequence [20, 20, 20, −20, 20, 20, −20, 20, −20, 20, 90,90, 90, −20, 90, 90, −20, 90] is obtained through using genetic algorithm.

The described two-level strategy should be repeated until no considerable improvementis seen in the objective function. It is seen from Figs. 7 and 8 that previous results (aspectratio of 0.6 and winding angle of 20°) are repeated. Therefore, no certain improvement can

Table 3 The optimal stacking sequence for nine-layered composite shells with different winding angles

Winding angle (°) Optimal stacking sequence Maximum Tsai–Wu failure criterion

9 [90, 9, −9, 90, 9, −9, 90, 9, 90] 0.95520 [20, 90, −20, 20, 20, −20, −20, 20, 20] 6.87130 [90, 30, −30, 30, −30, 30, −30, 90, 90] 11.105

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0. 2 0. 3 0.4 0.5 0. 6

Longitudinal Distance (m)T

sai-

Wu

Fig. 6 Tsai–Wu failure criterionfor the optimized compositenine-layered shell with geodesichead and helical windingangle ±9°


be seen in the value of modified shape factor and the procedure is ended after twoiterations.

8 Conclusion

In this paper, a multi-level optimization strategy is introduced which can be utilized incombination with genetic algorithm to optimize composite pressure vessels with differenthead shapes such as geodesic, and ellipsoidal. The main advantage of this process is that aminimum number of F.E. analysis is required compared to the other optimization approach.As two examples, two-level optimization strategy is applied for geodesic and ellipsoidalheads. Among different winding angles for geodesic head shape, the winding angle of nine

Fig. 8 Modified shape factor fordifferent winding angles

Fig. 7 Modified shape factorfor different ellipsoidalaspect ratios


degrees has the best performance. Moreover, the results show that ellipsoidal heads havegenerally weaker performance compared to geodesic head.

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