1. Appl Compos Mater (2007) 14:379–391
DOI 10.1007/s10443-008-9052-8
Head Shape and Winding Angle Optimization
of Composite Pressure Vessels Based
on 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 composite
pressure vessels with nonmetallic liners. The design variables for composite vessels include
the head shape, the winding angle, the layer thickness, the number of layers, and the
stacking sequence. A parameter called “modified shape factor” is introduced as an objective
function. This parameter takes into account the effects of the internal pressure and volume,
the vessel weight, and the composite material properties. The proposed algorithm uses
genetic algorithm and finite element analysis to optimize the design parameters. As a few
examples, this procedure is implemented on geodesic and ellipsoidal heads. The results
show that for the given vessel conditions, the geodesic head shape with helical winding
angle 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 as
transportation vehicles. Filament-wound composite pressure vessels, which utilize a
fabrication technique of filament winding to form high strength and light weight reinforced
plastic parts, are a major type of high pressure vessels. Pressure vessels normally consist of
two distinct parts: cylindrical portion and heads, domes or caps. Heads are usually the most
important part in the pressure vessel design. The desired parameters for a good head shape
are higher burst pressure and internal volume and lower weight.
A. Vafaeesefat (*)
Mechanical Engineering Department, Imam Hussein University, Tehran, Iran
e-mail: Abbas_v@yahoo.com
A. Khani
Faculty of Energy and New Technologies, Aerospace Engineering, Shahid Behesti University, Tehran, Iran

2. 380 Appl Compos Mater (2007) 14:379–391
Many works have been done to optimize the design parameters of filament wound
pressure vessels. Fukunaga and Chou [1] presented a laminate optimization procedure for
filament wound cylindrical pressure vessels under stiffness and strength constraints. Adali
et al. [2] presented an optimum design algorithm for symmetrically laminated cylindrical
pressure vessels. Krikanov [3] and Jaunky et al. [4] introduced an analytical laminate
optimization approach for composite pressure vessels under stiffness and strength
constraints. The reported works are mainly based on simple analyzing or experimental
methods and the head shape effect has not been considered.
Optimum design of dome contours for composite pressure vessels has been the
subject of many researches [5–8]. Hofeditz [5] applied the netting and orthotropic
analysis to solve dome design problems. Hojjati et al. [6] used the orthotropic plate theory
for dome design of the polymeric composite pressure vessels. Lin and Hwang [7] used a
parameter called performance factor to evaluate the structural efficiency of the vessel
domes. They introduced an optimum dome design method based on the Tsai–Hill
failure criterion and orthotropic plate theory. Liang et al. [8, 9] investigated the optimum
design of dome contour for filament wound composite pressure vessels subjected to
geometrical limitations, winding conditions and the Tsai–Wu failure criterion. They used
the feasible direction method for maximizing the shape factor. However, the stacking
sequences are not simultaneously considered in their optimization procedure.
In this paper, a multi-level strategy is introduced for the optimization of composite
pressure vessels with nonmetallic liners. The multi-level optimization strategy is a
powerful 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 is
seldom reported for the optimization of complex structures such as filament wound
vessels. As an example, this strategy is implemented on composite vessels with
geodesic and ellipsoidal heads. However, the presented method could be applied for
every symmetric vessel with any kind of head shape. This strategy is subjected to the
Tsai–Wu failure criterion and problem of maximizing the modified shape factor using
genetic algorithm. This study is limited to the symmetric composite vessels which have
two same domes and opening radii. The relation between internal pressure and Tsai–Wu
failure criterion is modeled through the finite element analysis and the geodesic
condition is considered to prevent fiber slipping.
2 Winding Pattern
Filament wound vessel design includes the design of the mandrel shape and the
calculation of the fiber path. In general, the mandrel shape can be determined by
imposed design requirements such as internal pressure, volume and manufacturing
convenience.
Finding the possible winding patterns on an arbitrary shape is one of the first
necessaries in order to introduce the optimization strategy for composite vessels. Since
the accuracy of finite element analysis is directly influenced by the winding

3. Appl Compos Mater (2007) 14:379–391 381
information, there is a need for the winding pattern to be actually modeled. In this
paper, the semi-geodesic path method is proposed, in which the slippage tendency
between the fiber and the mandrel is considered. The semi-geodesic path for general
filament wound structures is defined as follows [9]:
À Á
da l A2 sin2 a À rr'' cos2 a À r'A2 sin a
¼ ð1Þ
dz r'A2 cos a
A ¼ 1 þ r2
Equation 1 is defined on an arbitrary surface where α, z, θ, ρ and 1 are the winding
angle, the axial, circumferential and radial coordinate parameters, and the slippage tendency
between the fiber and the mandrel, respectively.
By setting the slippage tendency equal to zero in Eq. 1, the geodesic path equation is
obtained:
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 the
slippage 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 sin a 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 is
limited 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:
r0 cos a0
t¼ t0 ð4Þ
r cos a
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 the
vessel. The internal volume of the vessel is determined by the internal volume of the domes
and the drum. The outer area of the domes affects the whole weight of the filament wound

4. 382 Appl Compos Mater (2007) 14:379–391
structure. The geometry of the domes has influence on the burst pressure of the vessel and
affects the failure criterion.
3.1 Geodesic Dome Contours
Using netting analysis simultaneously with the Eq. 3, the geodesic dome shape can be
derived [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 r
r0 cos a0 t 3
z ¼ Àr0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi à dt ð5Þ
1 ð1 À t3 Þ cos2 a0 Â t2 ð1 þ t2 Þ À sin2 a
In this research, the integral Eq. 5 is numerically calculated using parameters α0 and ρ0
which are related to the cylindrical portion. Accordingly, for different radial coordinate
parameter ( ρ), axial coordinate parameter (z) are calculated and the dome shape is
determined. The winding angle values (α) are calculated for each radial distance from the
central axis (ρ) using the Eq. 4. Figure 1 shows different geodesic head geometries and their
relationship 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 is
the 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 opening
radius variation (Eq. 3).
4 Finite Element Model
Finite element is used to analyze the three dimensional model of the vessel. Stress
concentration is high at the junction of the cylinder and the head due to sudden change of
Fig. 1 Different geodesic
head shape for different winding
angle

5. Appl Compos Mater (2007) 14:379–391 383
Fig. 2 Different ellipsoidal head
shapes
the curvature in this area. For this reason, the smaller shell elements are modeled in this
area compared to other locations. Moreover, since the structure is symmetric, only half of
vessel is modeled.
There are several failure criteria to predict failure in the composite materials. In this
work, the 3D Tsai–Wu criterion is utilized which is defined by:
x3 ¼ A þ B ð6Þ
where
À Á2 À Á2 À Á2
ðσx Þ2 σy ð σz Þ 2 σxy σyz ðσxz Þ2 Cxy σx σy
A¼À À À þ 2 þ 2 þ 2 þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
σ fxt σ fxc σ fyt σ fyc σ fzt σ fzc σ fxy σ fyz σ fxz σ fxt σ fxc σ fyt σ ftc
Cyz σy σz Cxz σx σz
þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f f f f
σ yt σ yc σ zt σ zc σ fxt σ fxc σ fxt σ fzc
and
# # #
1 1 1 1 1 1
B¼ þ σx þ þ σy þ þ σz
σ fxt σ fxc σ fyt σ fyc σ fzt σ fzc
and ξ3, σ, σ ft and σ fc are the three dimensional Tsai–Wu criterion, the applied stresses,
tension strength and compression strength, respectively. Cxz, Cyz and Cxy are coupling

6. 384 Appl Compos Mater (2007) 14:379–391
coefficients for Tsai–Wu theory. Indices x, y and z show the fiber direction, it’s
perpendicular direction in and out of the element plane, respectively. Mechanical properties
of the carbon-epoxy composite material used in this research are detailed in Table 1. Mass
density 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-metallic
liner. 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 pressure
470 bar is applied. The finite element analysis results are presented versus longitudinal
distance 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 helical
layers (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 is
usually selected a constant value or an integer factor of it. Optimization procedure for
composite pressure vessels is divided into two levels; namely, head shape and laminate
weight optimization. The schematic diagram of the multi-level optimization strategy for
different head shapes is presented in Fig. 4.
In the first level, known head shapes with different winding angles are compared based
on the maximum shape factor. In this level, the procedure is done for an initial stacking
sequence and number of layers that may not be optimum. In the second level, the optimum
stacking sequence and number of layers will be selected. This two-level procedure is
repeated until no clear progress is seen. Using the multi-level optimization procedure, the
number of required finite element analysis is significantly deceased.
Table 1 Materials properties of Materials properties
composite material (Strengths
are in MPa and elastic modulus Exx 110.3
are in GPa)
Eyy 15.2
Ezz 8.97
Gxy 4.9
Gxz 4.9
Gyz 3.28
νxy 0.25
νxz 0.25
νyz 0.365
σ fxt 1,918
σ fxc 1,569
σ fyt 247
σ fyc 1,245
σ fzt 247
σ fzc 1,245
σxy 68.9
σyz 34.5
σxz 34.5

7. Appl Compos Mater (2007) 14:379–391 385
Fig. 3 Vessel dimensions (in mm) when the helical winding angle of fiber is ±9 in cylindrical section (left).
Longitudinal distance (s) (right)
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, vessel
characteristics 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:
V
K' ¼ ð7Þ
W= ðMax: Tsai WuÞ
+
where W, V, and + are, respectively, the dome weight, the internal volume, and the vessel
density. In the mentioned two-level strategy, the modified shape factor is compared for
different heads and winding angles. Since selection is done through a comparative
approach, failure criterion is put in the denominator and values greater than 1 do not make
Fig. 4 Schematic diagram of
multi-level optimization strategy Selection of initial approximate
for different heads number of layers and thicknesses
plus an arbitrary stacking
Selection of the best head shape
and winding angle based on Level 1
maximum modified shape factor
Stacking sequence and
number of layers optimization Level 2
No _
K jjä1 ä K jj ≤ ε
_1
Yes
Optimum design

8. 386 Appl Compos Mater (2007) 14:379–391
any difficulties due to relatively selection. On the other hand, after the second level is
carried out, this problem is solved. By applying this evaluation factor, there is no need to
impose any constraints.
5.2 Laminate Weight Optimization Using GA
The procedure for the optimization of the number of layers and the stacking sequence via
genetic algorithm is applied on the best head shape and the winding angle selected in the
first level. The laminate optimization procedure is configured to find the stacking sequence
and 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 obtained
using genetic algorithm for a constant number of layers, and the number of layers is
decreased or increased to reach the minimum number of layers with Tsai–Wu below one
through an iterative approach. For the first estimation of the number of layers, netting
analysis [10] or any other appropriate method can be used.
In fact, two design variables of number of layers and stacking sequence do not have the
same effect on the weight and strength. To be more precise, the number of layers has a
larger effect than stacking sequence. On the other hand, some variables such as the number
of layers have a linear and predictable effect on the structural behavior; whereas, some like
stacking sequence have a nonlinear effect. Therefore, using one objective function to find
the minimum number of layers (minimum weight) while its Tsai–Wu criterion is below one
may not converge to the optimum solution. For these reasons, these two variables are
separated and optimized via different approaches.
6 Genetic Algorithm Implementation
The optimal stacking sequence is obtained using genetic algorithm for minimum number of
layers. As mention before, stacking sequence optimization is a sub-problem in the weight
minimization. This problem is formulated for the strength maximization of a minimum
number of layers is defined by:
MinSS MaxLN;S Tsai À WuðSS; LN; S Þ
subject to:
Number of layers NL
Layer thickness 1 mm
Layer angles 90, α, −α
where SS is the stacking sequence of fiber angles which is the design variable; LN is the
layer number with maximum failure criterion; S is the longitudinal distance (Fig. 3) where
the maximum failure criterion takes place along the vessel mane and NL is the number of
layers.
Design variables in stacking sequence optimization (winding angle of fiber) are not
continuous. Therefore, in each model the winding angle design variables are restricted to
the discrete values of +α and −α for the helical layers and 90 for the hoop ones. For

9. Appl Compos Mater (2007) 14:379–391 387
example, if the helical winding angle is 9° and a geodesic head based on this angle is
implemented, the sequences of angles are limited to +9°, −9° and 90°. To apply this
discontinuity and prevent the selection of other angles, the stacking sequence of the
layer angle is coded to a binary set. Every two digit in the binary set are represented as
an angle. Based on our definition, different combinations of zero and one are translated
as follow:
½0; 0Š ¼ 90; ½1; 1Š ¼ À90; ½1; 0Š ¼ 9; ½0; 1Š ¼ À9:
The GA program works with the set of binary values. For example, the stacking
sequence for a multi-layer composite composed of nine layers is defined by:
½0; 1; 1; 1; 0; 0; 1; 0; 0; 1; 1; 0; 0; 0; 1; 0Š:
For applying to FE code, this stacking sequence is translated as (angles +90° and −90°
are equal for the fiber):
½À9; 90; 90; 9; À9; 9; 90; 9Š:
7 Results
As two application examples, the two-level optimization strategy is applied for vessels with
geodesic and ellipsoidal heads. For geodesic heads, the winding angle should be only
optimized in first level (Fig. 1); whereas for ellipsoidal ones the aspect ratio is optimized in
addition to winding angle (Fig. 2).
7.1 Geodesic Domes
Numerical analyses are performed on different geodesic head shapes with various winding
angles (Fig. 2), and similar stacking sequences and numbers of layers. Nine layers are
selected 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 cylindrical
portion as the first point in the optimization procedure. As shown in Fig. 5, the winding
angle of 9° has the best performance in the first level. In the second level, stacking
sequence optimization is carried out for the best winding angle of 9° to find the optimal
stacking sequence. The optimum stacking sequence [90, 9, −9, 90, 9, −9, 90, 9, 90] is
obtained.
As shown in the Table 2, the maximum Tsai–Wu failure criterion is less than one for the
optimal stacking sequence. Therefore, the number of layers is decreased to eight layers to
see if it is possible to obtain a lighter laminate. As shown in Table 2, the failure criterion for
the optimum stacking sequence of eight layers is greater than one; therefore, nine is the
least 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 to
obtain a lighter laminate for the minimum number of layers. When the numbers of helical
and 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 to
replace helical layers with hoop ones.
The whole described two-level strategy should be repeated until no considerable
improvement is seen in the objective function. As shown in Fig. 5, the optimum stacking

10. 388 Appl Compos Mater (2007) 14:379–391
Fig. 5 Modified shape factor for
different winding angles
sequence is examined for different winding angles in the second iteration and 9° has again
the 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 consecutive
generations is set to 1e-6. If this difference becomes less than this defined value, the
program will stop. The mutation and crossover operators have been selected binary and
arithmetic, respectively.
Figure 6 shows the Tsai–Wu failure criterion for the optimum stacking sequence with
geodesic 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. After
this point, the cylindrical section has a higher Tsai–Wu failure criterion than other areas on
the dome.
To show that the obtained winding angle is optimum, the optimum stacking sequence for
three models of nine-layered shells with helical winding angles 9°, 20° and 30° are
compared in Table 3. The results show that the optimum stacking sequences of composite
shells 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, the
number of layers for the helical angle 9° is suitable whereas for 20°, 30°, this number
should be increased to obtain the acceptable Tsai–Wu. These results show the fact that
helical angle 9° has better performance compared to 20° and 30° as shown in Fig. 5.
Table 2 The optimal stacking sequence for composite shells with helical winding angle ±9°
Number Optimal stacking sequence Maximum Tsai–Wu Vessel weight (kg)
of layers failure criterion
8 [90, 9, −9, 90, 90, −9, 9, 90] 1.145 5.977
9 [90, 9, −9, 90, 9, −9, 90, 9, 90] 0.955 6.833

11. Appl Compos Mater (2007) 14:379–391 389
Fig. 6 Tsai–Wu failure criterion 1.2
for the optimized composite
nine-layered shell with geodesic 1
head and helical winding
Tsai-Wu
angle ±9° 0.8
0.6
0.4
0.2
0
0 0.1 0. 2 0. 3 0.4 0.5 0. 6
Longitudinal Distance (m)
7.2 Ellipsoidal Heads
The proposed two-level optimization procedure was also implemented on ellipsoidal head
shape with different aspect ratios. The first level is done in two stages itself. First, only the
effect of head geometry is studied and all other variables are considered constant. The angle
stacking sequence [90, 20, −20, 90, 20, −20, 90, 20, 90] and layer thickness of 1 mm in the
cylindrical portion is used as the first point in the optimization procedure. This is the
optimum stacking sequence obtained for geodesic heads. Note that the procedure is not
sensitive to the start point; therefore, any appropriate method can be used for the first guess
of layer thickness and number of layers.
From Fig. 7, it is seen that for different aspect ratios (0.2e2) 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 the
previous selection is studied and all other variables are considered constant.
As shown in Fig. 8, the maximum modified shape factor occurred for the angle of
21°. If the domain of helical layer orientation angles between 5° and 40° is divided by the
pre-assigned increment of five degrees, the winding angle of 20° will be selected in this
level.
The second level belongs to the stacking sequence and number of layers optimization
while the head shape aspect ratio and the winding angle values obtained from the last two
levels 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 improvement
is seen in the objective function. It is seen from Figs. 7 and 8 that previous results (aspect
ratio 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] 0.955
20 [20, 90, −20, 20, 20, −20, −20, 20, 20] 6.871
30 [90, 30, −30, 30, −30, 30, −30, 90, 90] 11.105

12. 390 Appl Compos Mater (2007) 14:379–391
Fig. 7 Modified shape factor
for different ellipsoidal
aspect ratios
be seen in the value of modified shape factor and the procedure is ended after two
iterations.
8 Conclusion
In this paper, a multi-level optimization strategy is introduced which can be utilized in
combination with genetic algorithm to optimize composite pressure vessels with different
head shapes such as geodesic, and ellipsoidal. The main advantage of this process is that a
minimum 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 ellipsoidal
heads. Among different winding angles for geodesic head shape, the winding angle of nine
Fig. 8 Modified shape factor for
different winding angles

13. Appl Compos Mater (2007) 14:379–391 391
degrees has the best performance. Moreover, the results show that ellipsoidal heads have
generally weaker performance compared to geodesic head.
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