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Stability of fiber trajectories for winding toroidal pressure vessels
Composite Structures 94 (2012) 1855–1860
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Stability of fiber trajectories for winding toroidal pressure vessels Lei Zu ⇑ Department of Engineering Mechanics, School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, PR China Design and Production of Composite Structures, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
a r t i c l e
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Article history: Available online 25 November 2011 Keywords: Filament-wound pressure vessel Trajectory stability Torus Slippage Bridging
a b s t r a c t In this paper the fiber trajectory stability of filament wound toroidal pressure vessels is evaluated for two most frequently used patterns: single helical winding, helical and hoop winding. The basic equations of equilibrium for fibers on a torus are given based on netting analysis. The governing equations that determine helical winding angles along the meridional direction are derived for the two winding patterns. The slippage coefficients of the obtained fiber trajectories are calculated using the non-geodesic law and differential geometry. The condition between the hoop-to-helical thickness ratio and the relative bend radius of the toroid is also formulated to prevent fiber bridging on the concave surface. The fiber slippage and bridging tendencies are outlined and compared, corresponding to various relative bend radii and hoop-to-helical thickness ratios. The results show that the single helical winding process provides better stability of fiber trajectories, in terms of both fiber slippage and bridging, than the helical and hoop winding. The toroidal vessel with larger relative bend radius requires lower coefficient of friction between the fiber bundle and the supporting surface. The present analysis for fiber trajectory stability affords a useful reference tool for designing filament-wound toroidal pressure vessels. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Filament winding have so far been extensively used in manufacturing processes in composites industry. Continuous fibers can be oriented to match the direction and magnitude of stresses in a laminated structure, leading to optimal structural performance. Filament-wound toroidal pressure vessels have recently gained wide attention in the storage of pressurized liquids and gases. Remarkable applications involve fuel tanks that store highpressure gaseous hydrogen, compressed natural gas, and propellant. These applications have great potential to facilitate the development of fuel cell and LPG vehicles, nuclear reactors, underwater applications, spacecrafts, etc. [1–5]. Compared to other shapes of pressure vessels, a toroid-shaped vessel exhibits several advantages in often providing better utilization of space, more stable center of gravity of the storage system, and greater protection for the pressure regulator. By neglecting several practical problems like the connection of devices for filling it, the toroid is an attractive alternative for spaces having limited height/length, for example toroidal pressure vessels in the spare wheel position of LPG vehicles able to reduce space consumption within the vehicle, and toroidal breathing apparatus that are ⇑ Address: Design and Production of Composite Structures, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands. Tel.: +31 15 2789616; fax: +31 15 2781151. E-mail address: [email protected] 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.11.027
largely ergonomic in nature as compared to cylinders [6]. Furthermore, a toroid-shaped vessel has a better structural efficiency than the conventional cylindrical shape, due to its relatively homogeneous thickness distribution as a result of the absence of end enclosure. However, determining the optimal winding trajectories of an internally pressurized toroid poses several challenges not present with other shells of revolution such as cylinders. These challenges arise partly from the instability of fiber trajectories on the mandrel due to the unique geometry of the toroid. During the past decades, the design and analysis for filament-wound toroidal pressure vessels have been developed in a variety of directions. Li and Cook [7] considered a toroid consisting of an inner metal liner and an outer fiber overwrap, and developed a full mathematical approach to the design of meridionally overwound toroids using a membrane shell theory, taking into account the load-bearing capability of the overwrap and its interaction with the liner. Cook et al. [6] employed a metal–aramid fiber hybrid construction into the design of lightweight toroidal breathing apparatus, taking into account stress rupture and stress relaxation effects in aramid fibers and the fatigue arising from repeated filling–discharge cycles. Zu et al. [8– 10] obtained the optimal geodesic and non-geodesic trajectories for winding circular toroids and further determined the adapted cross sectional shapes for filament-wound toroidal pressure vessels, using both the netting and orthotropic plate theory. Marketos [11] only considered geodesic winding patterns and assumed the toroidal shell undergoes in-plane membrane forces. The circular
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cross section proved to be not structurally optimal because the change in the meridional stress around the cross-section is not compatible with the change of the geodesic winding angle. Maksimyuk and Chernyshenko [12] examined the distribution of the displacements, strains, and stresses in orthotropic toroidal shells made of nonlinearly elastic composites. Blachut [13] presented a numerical study into strength and static stability of externally pressurized filament wound CFRP toroids with closed circular and non-circular cross-sections. Previous studies have concentrated on the use of geodesic or semi-geodesic trajectories; however, the design freedom of geodesics is severely limited because once initial winding conditions of fiber position and orientation have been chosen, the entire fiber trajectory and corresponding lay-up thickness distribution are determined. Semi-geodesic winding offers more design freedom but still remains limited to a constant coefficient of slippage during the winding process. In general neither geodesic nor semi-geodesic winding can place fibers in the principal stress direction of the toroid-shaped shell under internal pressure. It is thus expected to design the fiber trajectories that are aligned in the direction of the maximum principal stress along their entire length [14–16] and that consequently lead to the optimal lay-up structure. Designing optimal laminate lay-up is not the only issue; the fibers must be stable on the mandrel and be exactly placed along trajectories as predetermined by strength calculation. The fiber trajectories and corresponding winding angles cannot be chosen arbitrarily because of the requirements of windability. Any deviation from geodesics will require a lateral force of friction to prevent the fiber sliding from its original trajectory. In addition, since a half torus is concave the condition which avoids fiber bridging should also be respected when designing winding trajectories. Therefore an analysis for the stability of fiber trajectories is imperative which allows accurate fiber placement to achieve the optimal vessel performance dictated by structural design. The overlook of fiber trajectory stability will make the optimally designed lay-up parameters difficult to be realized using filament winding technology and hence invalidate the structural design and optimization. In this paper we emphasize the importance of the stability analysis for fiber trajectories and accordingly derive the general criteria and equations that determine the distributions of the fiber slippage and bridging tendencies. The laminate lay-up design of toroidal pressure vessels are based on the netting theory and the resulting fiber trajectories have the form of non-geodesics with non-constant slippage coefficients. The distributions of slippage coefficients along the meridional direction and the minimum winding angles that avoid fiber bridging are outlined to evaluate the fiber trajectory stability of toroidal pressure vessels. The primary goal of this paper is to provide an important reference with respect to the fiber stability for designing toroidal pressure vessels. 2. Stability of fiber trajectories on a surface The vector parameterization of an arbitrary mandrel surface in generalized curvilinear orthogonal coordinates can be represented as:
r ¼ rðu; v Þ
ð1Þ
where r is a vector-valued function of the parameters (u, v) and the parameters vary within a certain domain D in the parametric uvplane. The u, v-curves are lines of constant and denote meridians and parallels for shells of revolution. An arbitrary fiber curve on the surface r(u, v) can be given by its arclength parametrization:
rðsÞ ¼ rðuðsÞ; v ðsÞÞ
ð2Þ
An infinitesimal elementary piece P0P1 (|P0P1| = ds) of a fiber curve is here considered, as depicted in Fig. 1. P0P1 is subjected
Fig. 1. An infinitesimal element of a tensioned fiber trajectory on an arbitrary surface.
to longitudinal tension forces F0 and F1, a normal reaction force Fv perpendicular to the surface, a friction force Ff tangential to the mandrel surface. The Darboux frame [17], which takes into account the fact that the curve r(s) lies on the surface r(u, v), is here considered (Fig. 1). Using the right-handed orthonormal basis (n, T, n T) the principal curvature vector j is in the plane spanned by n T and n, and can thus be decomposed into two orthogonal vectors [17]:
j ¼ jg ðn TÞ þ jn n
ð3Þ
where n, T denote the unit normal and tangent vector, respectively; jn gives the component of the principal curvature vector normal to the surface at a point and is called the normal curvature; jg gives the component of the curvature vector tangential to the surface and is called the geodesic curvature. The principal curvature j points towards the center of the curvature of the fiber curve and can be expressed by:
j ¼ jN
ð4Þ
where j is the curvature of the fiber trajectory at a point (P0 or P1); N is the principal normal vector in the Frenet Frame [17]. For an infinitesimal arclength the magnitudes of the tension at both sides of P0P1 are considered equal:
jF0 j ¼ jF1 j ¼ ‘
ð5Þ
Thus we have:
F0 ¼ ‘ T0
ð6Þ
F1 ¼ ‘ T1
ð7Þ
The equilibrium equations of the forces exerting on P0P1 in directions normal and tangent to the surface, can respectively be formulated as:
ðF0 þ F1 Þnn þ Fn ¼ 0
ð8Þ
ðF0 þ F1 Þðn TÞðn TÞ þ Ff ¼ 0
ð9Þ
Substituting Eqs. (6) and (7) into (8) and (9), leads to:
Fn ¼ ‘ðT1 T0 Þnn
ð10Þ
Ff ¼ ‘ðT1 T0 Þðn TÞðn TÞ
ð11Þ
Since |P0P1| = ds, according to differential geometry we have [17]
T1 T0 ¼ dsT0 ¼ jdsN
ð12Þ
With the aid of Eq. (12), Eqs. (10) and (11) reduce to:
Fn ¼ ‘jdsNnn
ð13Þ
Ff ¼ ‘jdsNðn TÞðn TÞ
ð14Þ
Substituting Eq. (4) into (3) followed by multiplying both sides of Eq. (3) by the normal vector n, leads to:
Nn ¼ jn
ð15Þ
Nðn TÞ ¼ jg :
ð16Þ
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Substitution of Eqs. (15) and (16) into (13) and (14) yields:
Fn ¼ jn ‘jdsn
ð17Þ
Ff ¼ jg ‘jdsðn TÞ
ð18Þ
To prevent fiber sliding on the supporting surface, the friction force Ff should always be less than the maximum static friction between the supporting surface and the fiber bundle:
jFf j lmax jFn j
ð19Þ
where lmax is the coefficient of maximum static friction between the fiber and the mandrel surface or between the fiber and the previously overwound layer. Note that lmax can be affected by the surface quality, fiber morphology, and the resin viscosity, etc. [18]. Substitution of Eqs. (17) and (18) into (19) gives the nonslippage criterion of fiber trajectories on an arbitrary surface:
jjg =jn j lmax
ð20Þ Fig. 2. Geometry of a torus with its fiber trajectory.
The slippage coefficient k is defined as the ratio of the geodesic curvature to the normal curvature [19,20]:
k ¼ jjg =jn j
ð21Þ
The non-slippage criterion Eq. (20) can then be rewritten as:
k lmax
ð22Þ
The slippage coefficient k represents the slippage tendency between the fiber bundle and the supporting surface. The region of possible winding patterns can also be found according to k. To avoid fiber sliding, k should always be less than lmax which ranges from 0.2 (between the fiber and the previously wound layer) to 0.35 (between the fiber and the mandrel surface) [18,21]. In addition, the fibers may bridge on the concave surface and lose contact with the mandrel surface unless the normal reaction force Fn is acting in the same direction to the normal vector n (if pointing outward), or in the opposite direction to the normal vector n (if pointing inward):
Fn n 0 ðoutward pointing normalÞ Fn n 0 ðinward pointing normalÞ
ð23Þ ð24Þ
Please note that the normal vector is here considered always outward-pointing. Substituting Eq. (17) into (23) and taking into account that the product ‘jds is constantly greater than 0, leads to:
jn 0
ð25Þ
Eq. (25) provides the non-bridging criterion of fiber trajectories on an arbitrary surface. Please note that the fiber bridging can generally be eliminated by changing the winding angles. 3. Fiber trajectories based on netting analysis
We consider here an infinitesimal elementary piece of a surface being part of the thin toroidal vessel (Fig. 3). The normal reaction force is acting as a counterbalance to the internal pressure [9]:
F n ¼ p Ru Du Rv Dv
where p is the internal pressure. For a thin-walled pressure vessel, the membrane approximation theory for the stress analysis is here considered. According to the force equilibrium in the direction normal to the surface element, Fn can be expressed in terms of the membrane forces in two principal directions:
F n ¼ Nu Rv Dv sin Du þ Nv Ru Du sin Dv
ð28Þ
where Nu, Nv are the in-plane forces per unit length in the meridional and parallel directions. The equality relation for the membrane forces can be obtained by equalizing the right-hand expression of Eq. (27) with that of Eq. (28):
Nu Nv þ ¼p Ru Rv
ð29Þ
The axial equilibrium of a toroidal shell can be formulated as shown in Fig. 4:
pp½ðR0 þ r0 cos uÞ2 R20 2pðR0 þ r0 cos uÞNu cos u ¼ 0
ð30Þ
Hence
Nu ¼
pr 0 2R0 þ r0 cos u 2 R0 þ r 0 cos u
A toroidal pressure vessel is an axisymmetric shell of revolution with a circular cross-section that does not intersect the axis of revolution. The torus (surface of a closed doughnut) is formed by revolving a circle of radius r0 about a circle of radius R0 > r0 lying in an orthogonal plane, as pictured in Fig. 2. R0 is the distance between the center of the cross-section and the axis of rotation, being referred to as the major radius; r0 is the radius of the tube, being called the minor radius. The meridional direction u for a circular toroid is tangent to the cross-section and the parallel direction v follows the circle of revolution for the toroid. The regular parameterization of the torus is given by:
8 9 > < ðR0 þ r 0 cos v Þ cos u > = Sðu; v Þ ¼ ðR0 þ r 0 cos v Þ sin u ð0 u 2p; 0 v 2pÞ > > : ; r 0 sin v ð26Þ
ð27Þ
Fig. 3. An elementary surface belonging to a pressure vessel.
ð31Þ
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Fig. 4. Part of a toroidal shell loaded by an internal pressure p.
Nv can be solved by substitution of Eq. (31) into (29):
Nv ¼
pr 2
ð32Þ
For thin shells, the membrane forces are considered to be constant through the thickness. The meridional force, Nu, increases as the parameter u decreases while the parallel force, Nv, remains constant throughout the toroid. The netting theory that considers only the loading of fibers and neglects any contribution by the resin matrix and any interaction between the fibers, is here employed [22–24]. These assumptions do not cause any significant error in the analysis, as long as the fibers are primarily loaded in tension and the transverse and shear stresses in the composite laminate are low compared to the ultimate tensile strength of the fibers. The netting theory gives a very simple result in design and evaluation of composite pressure vessels, but it is well suited for filament-wound pressure vessels [25]. Fig. 5 shows a basic element of helically and hoop wound fibers, intersecting a unit area of the torus in the meridional and parallel directions, respectively. The force equilibrium equations in the u and v directions can respectively be given by:
pr 0 2 pr 2R þ r cos u ra ta sin2 a þ rh th sin2 90 ¼ 0 0 0 2 R0 þ r 0 cos u
ra ta cos2 a þ rh th cos2 90 ¼
ð33Þ ð34Þ
ð35Þ
The relative bend radius, which denotes the tube slenderness, is defined as:
k ¼ R0 =r 0
ð36Þ
Substituting Eqs. (35) and (36) into (33) and (34) and solving for a, results in:
a ¼ tan
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 g þ kþcos u ð0 u 2pÞ 1þg
where g is the hoop-to-helical layer thickness ratio, defined as:
g ¼ th =ta
ð38Þ
Eq. (37) provides the periodically fiber trajectories on the torus based on the netting analysis. The winding angles of the fiber trajectories depend on the relative bend radius and the hoop-to-helical thickness ratio. For the given k and g, the initial winding conditions of fiber position and angle are able to completely determine the fiber trajectories. Note that for the single helical winding g = 0 and Eq. (37) can be rewritten as: 1
where ra, rh, ta, th are the helical and the hoop fiber stress and thickness, respectively; a is the winding angle between the fiber curve and the corresponding parallel curve. The condition of equal shell strains [14,26] is always achieved with the netting solution, given by:
ea ¼ eh ) ra ¼ rh
Fig. 6. Winding angle distributions for the single helical winding (g = 0) and the helical and hoop winding (g = 0.5).
a ¼ tan
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1þ ð0 u 2pÞ k þ cos u
Fig. 6 presents the netting-based winding angle developments for the single helical winding and the helical and hoop winding (k = 3–6, g = 0.5), respectively. The results show that at each position along the meridional direction the single helical winding provides a bigger fiber angle than the helical and hoop winding. 4. Stability-ensuring criteria for fiber trajectories on a torus According to Liouville formula [17], kg is obtained by:
kg ¼
da sin u cos a þ dl R þ r cos u
ð40Þ
where l is the fiber length; kn is related to the principal curvatures in the respective meridional and parallel directions, given by [17]: 2
kn ¼ ku sin a þ kv cos2 a ð37Þ
ð39Þ
ð41Þ
For a torus, ku and kv are given by:
1 ku ¼ ; r
kv ¼
cos u R þ r cos u
ð42Þ
Substituting Eq. (42) into (41) leads to:
Kn ¼
cos u 1 2 cos2 a sin a R þ r cos u r
ð43Þ
In addition, the following relation holds:
du sin a ¼ dl r Fig. 5. A netting-based element per unit area of helically and hoop wound fibers.
ð44Þ
Substituting Eqs. (40), (43), and (44) into (21), the slippage coefficient k can be expressed as a function of u:
L. Zu / Composite Structures 94 (2012) 1855–1860
da sin a þ sin u cos a du kðuÞ ¼ cos u k þ cos u 2 2 cos a þ sin a k þ cos u
ð45Þ
where da/du can be obtained by the first derivative of Eq. (37) with respect to u:
pffiffiffiffiffiffiffiffiffiffiffiffi k sin u 1 þ g da pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ du 2ð3k þ 2 cos uÞ ðk þ cos uÞ½ð2 gÞk þ ð1 gÞ cos u
ð46Þ
Substitution of Eq. (46) into (45) gives: ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin uð3k þ 2 cos uÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ ðk þ cos uÞð1 gÞ þ k ð1 þ gÞð2k þ cos uÞ k¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4ðk þ cos uÞ3=2 ð3k þ 2 cos uÞ½ðk þ cos uÞð1 gÞ þ k
ð47Þ The fiber bridging tendency should also be examined since the torus has a convex–concave surface. Substitution of Eq. (43) into (25) leads to the condition for preventing fiber bridging over the torus: k þ1g cos u tan2 a ¼ kþcos u k þ cos u 1þg
ð48Þ
Eq. (48) can further reduce to:
g ¼ th =ta < 2ð1 1=kÞ
1859
critical angles for avoiding fiber bridging; the obtained fiber trajectories will therefore not bridge over the entire surface. In addition, for a given k there is a maximum value for g able to guarantee the validity of the non-bridging criterion. The result is depicted in Fig. 8, where the non-bridging-ensuring feasible {k, g}-field is dashed. It should be noted here that the non-bridging criterion is consistently valid for the convex surface because Eq. (48) always holds true for the range u = [90°, 90°]. Fig. 9 shows the distributions of the slippage coefficient k along the meridional coordinate u, corresponding to various k (k = 3–6, g = 0.5). The results reveal that the fiber trajectories exhibit an increasing slippage tendency from the inner/outer equator towards the crest of the toroid; however, the maximum slippage tendency is not present exactly at the crest but somewhere between the crest and the inner equator, depending on the k-value. It is thus suggested that more attention should be paid to the winding of the concave surface of the toroid. Fig. 10 illustrates the maximum slippage coefficients of the obtained fiber trajectories corresponding to the most common combinations of k and g (k = 3–6, g = 0–1.5). It is shown that the fiber slippage tendency significantly decreases with the increase of k, and slightly increases with the increase of g. The relative bend radius proves to be the most influential factor on fiber slippage tendency, while the thickness ratio has less influence.
ð49Þ
Eq. (49) provides the non-bridging criterion for fiber trajectories on the torus. It should be noted here that for the single helical winding Eq. (49) is always satisfied since g = 0. Accordingly the single helical winding trajectories determined using the netting theory will never bridge over the torus. 5. Results and discussion Eqs. (47) and (49) give the criteria for assessing the stability of fiber trajectories designed using the netting-based approach. It is shown that for the helical and hoop winding the fiber stability is influenced by the relative band radius k and the hoop-to-helical layer thickness ratio g while for the single helical winding k is the only influential factor. Fig. 7 outlines the distribution of the minimum required winding angles as dictated by the non-bridging criterion, as compared to the winding angles of fiber trajectories obtained using the netting analysis. It can be seen that for the most common k-range [3,6], the netting-based winding angles are much bigger than the
Fig. 8. Feasible field of {k, g}-combinations for ensuring non-bridging (shaded area).
Fig. 7. Minimum non-bridging-ensuring angles as compared to winding angles of the netting-based fiber trajectories.
Fig. 9. Slippage tendency distributions along the meridional direction.
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Despite the fact that the small relative bend radii (k < 3) result in the instability of fiber trajectories on the torus, the present netting-based winding trajectories do satisfy the stability-ensuring requirements for the most commonly used toroids (k P 3) in today’s industries. Since this paper only consider the fiber slippage and bridging tendencies for analyzing the stability of winding trajectories, in the future the criteria of uniform and full coverage will also be considered in order to enable the fiber bands equally distributed, without leaving any gaps or creating excessive overlaps. However, the present analysis proves very useful in the design phase of filament-wound toroidal pressure vessels. References
Fig. 10. Maximum slippage coefficients of the obtained fiber trajectories for the most common k and g.
As seen in the results of the stability analysis, it is shown that for toroidal pressure vessels with k > 4 the fiber trajectories determined using the netting theory can well satisfy the stability-ensuring criteria; for toroidal vessels with 3 6 k 6 4, the maximum coefficient of friction between the supporting surface and the placed fiber bundle should be guaranteed to reach 0.4–0.5, in order to prevent fiber trajectories deviating from the netting-dictated trajectories. However, for toroids with k < 3 and g > 1.5, it is strongly suggested to employ geodesic or semi-geodesic trajectories instead of the present netting-based trajectories, with a compromise between structural performance and windability. 6. Conclusions In this paper an analysis for fiber trajectory stability is elaborated for providing a design reference of filament-wound toroidal pressure vessels. The fiber trajectories are determined using the netting theory in combination with the special geometry of the torus. The netting analysis establishes the simple and straightforward relationship among the fiber trajectories, the internal pressure, material properties and processing parameters. Compared to the geodesic or semi-geodesic winding, the present nettingbased approach can place the fibers according to the principal stress direction of the toroidal shell under internal pressure and thus lead to a maximum utilization of fiber strength as well as a maximum structural performance. To evaluate the effect of the geometry and the lay-up on the stability of the netting-based trajectories, the fiber slippage and bridging tendencies of the obtained fiber trajectories are illustrated for various relative bend radii and hoop-to-helical thickness ratios. It is revealed that toroidal pressure vessels with larger relative bend radius and lower hoopto-helical thickness ratio lead to better fiber stability in terms of either slippage or bridging tendency. The results also conclude that for the most commonly used toroids, the winding trajectories designed by the netting theory satisfy both the non-slippage and non-bridging criteria, and show good fiber stability for the winding process. In addition, it is worth noting that the netting-based fiber trajectories will in general fulfill the non-bridging condition if they satisfy the non-slippage criterion.
[1] Ross CTF. A conceptual design of an underwater missile launcher. Ocean Eng 2005;32:85–99. [2] Ruggiero EJ, Jha A, Park G, Inman DJ. A literature review of ultra-light and inflated toroidal satellite components. Shock Vib Digest 2003;35:171–81. [3] Weisberg AH, Myers B, Berry G. Hydrogen storage using lightweight tanks. In: Proceedings of the 2002 US DOE hydrogen program review. NREL/CP-61032405; 2002. [4] Baker WR, Bello SD, Marcuzzi D, Sonato P, Zaccaria P. Design of a new toroidal shell and support for RFX. Fusion Eng Des 2002;64:461–6. [5] Kisioglu Y. Burst Pressure Determination of vehicle toroidal oval cross-section LPG fuel tanks. J Press Ves Technol 2011;133:0312021–25. [6] Cook J, Chambers JK, Richards BJ. Toroidal pressure vessels for breathing apparatus. In: Proc of international SAMPE Europe conference, Paris, France; April 1998. [7] Li SG, Cook J. An analysis of filament overwound toroidal pressure vessels and optimum design of such structures. J Press Ves Technol 2002;124:215–22. [8] Zu L, Koussios S, Beukers A. Design of filament-wound circular toroidal hydrogen storage vessels based on non-geodesic fiber trajectories. Int J Hydrogen Energy 2010;35:660–70. [9] Zu L, Koussios S, Beukers A. Optimal cross sections of filament-wound toroidal hydrogen storage vessels based on continuum lamination theory. Int J Hydrogen Energy 2010;35:10419–29. [10] Zu L, Zhang DH, Xu YQ, Xiao DJ. Integral design and simulation of composite toroidal hydrogen storage tanks. Int J Hydrogen Energy 2012;37:1027–36. [11] Marketos JD. Optimum toroidal pressure vessel filament wound along geodesic lines. AIAA J 1963;1:1942–5. [12] Maksimyuk V A, Chernyshenko IS. Nonlinear elastic state of thin-walled toroidal shells made of orthotropic composites. Int Appl Mech 1999;35:1238–45. [13] Blachut J. Buckling and first ply failure of a composite toroidal pressure hull. Comput Struct 2004;82:1981–92. [14] Vasiliev VV, Krikanov AA. New generation of filament-wound composite pressure vessels for commercial applications. Compos Struct 2003;62:449–59. [15] Zu L, Koussios S, Beukers A. Design of filament-wound domes based on continuum theory and non-geodesic roving trajectories. Compos Part A: Appl Sci Manuf 2010;41:1312–20. [16] Zu L, Koussios S, Beukers A. Integral design for filament-wound composite pressure vessels. Polym Polym Compos 2011;19:413–20. [17] Gray A. Modern differential geometry of curves and surfaces. Boca Ration: CRC Press; 1993. [18] Wang R, Jiao W, Liu W, Yang F, He X. Slippage coefficient measurement for non-geodesic filament-winding process. Comp Part A: Appl Sci Manuf 2011;42:303–9. [19] De Carvalho J, Lossie M, Vandepitte D, Van Brussel H. Optimization of filament wound parts based on non-geodesic winding. Compos Manuf 1995;6:79–84. [20] Zu L, Koussios S, Beukers A. Shape optimization of filament wound articulated pressure vessels based on non-geodesic trajectories. Compos Struct 2010;92:339–46. [21] Park JS, Hong CS, Kim CG, Kim CU. Analysis of filament wound composite structures considering the change of winding angles through the thickness direction. Compos Struct 2002;55:63–71. [22] Krikanov AA. Composite pressure vessels with higher stiffness. Compos Struct 2000;48:119–27. [23] Zu L, Koussios S, Beukers A. Design of filament-wound isotensoid pressure vessels with unequal polar openings. Compos Struct 2010;92:2307–13. [24] Evans JT, Gibson AG. Composite angle ply laminates and netting analysis. Proc R Soc Lond A 2002;458:3079–88. [25] Rosato DV, Grove CS. Filament winding: its development, manufacture, applications, and design. New York: Interscience; 1964. [26] Hojjati M, Safavi AV, Hoa SV. Design of domes for polymeric composite pressure vessels. Compos Eng 1995;5:51–9.
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Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Stability of fiber trajectories for winding toroidal pressure vessels Lei Zu ⇑ Department of Engineering Mechanics, School of Civil Engineering and Transportation, South China University of Technology, Guangzhou 510641, PR China Design and Production of Composite Structures, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands
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Article history: Available online 25 November 2011 Keywords: Filament-wound pressure vessel Trajectory stability Torus Slippage Bridging
a b s t r a c t In this paper the fiber trajectory stability of filament wound toroidal pressure vessels is evaluated for two most frequently used patterns: single helical winding, helical and hoop winding. The basic equations of equilibrium for fibers on a torus are given based on netting analysis. The governing equations that determine helical winding angles along the meridional direction are derived for the two winding patterns. The slippage coefficients of the obtained fiber trajectories are calculated using the non-geodesic law and differential geometry. The condition between the hoop-to-helical thickness ratio and the relative bend radius of the toroid is also formulated to prevent fiber bridging on the concave surface. The fiber slippage and bridging tendencies are outlined and compared, corresponding to various relative bend radii and hoop-to-helical thickness ratios. The results show that the single helical winding process provides better stability of fiber trajectories, in terms of both fiber slippage and bridging, than the helical and hoop winding. The toroidal vessel with larger relative bend radius requires lower coefficient of friction between the fiber bundle and the supporting surface. The present analysis for fiber trajectory stability affords a useful reference tool for designing filament-wound toroidal pressure vessels. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Filament winding have so far been extensively used in manufacturing processes in composites industry. Continuous fibers can be oriented to match the direction and magnitude of stresses in a laminated structure, leading to optimal structural performance. Filament-wound toroidal pressure vessels have recently gained wide attention in the storage of pressurized liquids and gases. Remarkable applications involve fuel tanks that store highpressure gaseous hydrogen, compressed natural gas, and propellant. These applications have great potential to facilitate the development of fuel cell and LPG vehicles, nuclear reactors, underwater applications, spacecrafts, etc. [1–5]. Compared to other shapes of pressure vessels, a toroid-shaped vessel exhibits several advantages in often providing better utilization of space, more stable center of gravity of the storage system, and greater protection for the pressure regulator. By neglecting several practical problems like the connection of devices for filling it, the toroid is an attractive alternative for spaces having limited height/length, for example toroidal pressure vessels in the spare wheel position of LPG vehicles able to reduce space consumption within the vehicle, and toroidal breathing apparatus that are ⇑ Address: Design and Production of Composite Structures, Faculty of Aerospace Engineering, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands. Tel.: +31 15 2789616; fax: +31 15 2781151. E-mail address: [email protected] 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.11.027
largely ergonomic in nature as compared to cylinders [6]. Furthermore, a toroid-shaped vessel has a better structural efficiency than the conventional cylindrical shape, due to its relatively homogeneous thickness distribution as a result of the absence of end enclosure. However, determining the optimal winding trajectories of an internally pressurized toroid poses several challenges not present with other shells of revolution such as cylinders. These challenges arise partly from the instability of fiber trajectories on the mandrel due to the unique geometry of the toroid. During the past decades, the design and analysis for filament-wound toroidal pressure vessels have been developed in a variety of directions. Li and Cook [7] considered a toroid consisting of an inner metal liner and an outer fiber overwrap, and developed a full mathematical approach to the design of meridionally overwound toroids using a membrane shell theory, taking into account the load-bearing capability of the overwrap and its interaction with the liner. Cook et al. [6] employed a metal–aramid fiber hybrid construction into the design of lightweight toroidal breathing apparatus, taking into account stress rupture and stress relaxation effects in aramid fibers and the fatigue arising from repeated filling–discharge cycles. Zu et al. [8– 10] obtained the optimal geodesic and non-geodesic trajectories for winding circular toroids and further determined the adapted cross sectional shapes for filament-wound toroidal pressure vessels, using both the netting and orthotropic plate theory. Marketos [11] only considered geodesic winding patterns and assumed the toroidal shell undergoes in-plane membrane forces. The circular
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cross section proved to be not structurally optimal because the change in the meridional stress around the cross-section is not compatible with the change of the geodesic winding angle. Maksimyuk and Chernyshenko [12] examined the distribution of the displacements, strains, and stresses in orthotropic toroidal shells made of nonlinearly elastic composites. Blachut [13] presented a numerical study into strength and static stability of externally pressurized filament wound CFRP toroids with closed circular and non-circular cross-sections. Previous studies have concentrated on the use of geodesic or semi-geodesic trajectories; however, the design freedom of geodesics is severely limited because once initial winding conditions of fiber position and orientation have been chosen, the entire fiber trajectory and corresponding lay-up thickness distribution are determined. Semi-geodesic winding offers more design freedom but still remains limited to a constant coefficient of slippage during the winding process. In general neither geodesic nor semi-geodesic winding can place fibers in the principal stress direction of the toroid-shaped shell under internal pressure. It is thus expected to design the fiber trajectories that are aligned in the direction of the maximum principal stress along their entire length [14–16] and that consequently lead to the optimal lay-up structure. Designing optimal laminate lay-up is not the only issue; the fibers must be stable on the mandrel and be exactly placed along trajectories as predetermined by strength calculation. The fiber trajectories and corresponding winding angles cannot be chosen arbitrarily because of the requirements of windability. Any deviation from geodesics will require a lateral force of friction to prevent the fiber sliding from its original trajectory. In addition, since a half torus is concave the condition which avoids fiber bridging should also be respected when designing winding trajectories. Therefore an analysis for the stability of fiber trajectories is imperative which allows accurate fiber placement to achieve the optimal vessel performance dictated by structural design. The overlook of fiber trajectory stability will make the optimally designed lay-up parameters difficult to be realized using filament winding technology and hence invalidate the structural design and optimization. In this paper we emphasize the importance of the stability analysis for fiber trajectories and accordingly derive the general criteria and equations that determine the distributions of the fiber slippage and bridging tendencies. The laminate lay-up design of toroidal pressure vessels are based on the netting theory and the resulting fiber trajectories have the form of non-geodesics with non-constant slippage coefficients. The distributions of slippage coefficients along the meridional direction and the minimum winding angles that avoid fiber bridging are outlined to evaluate the fiber trajectory stability of toroidal pressure vessels. The primary goal of this paper is to provide an important reference with respect to the fiber stability for designing toroidal pressure vessels. 2. Stability of fiber trajectories on a surface The vector parameterization of an arbitrary mandrel surface in generalized curvilinear orthogonal coordinates can be represented as:
r ¼ rðu; v Þ
ð1Þ
where r is a vector-valued function of the parameters (u, v) and the parameters vary within a certain domain D in the parametric uvplane. The u, v-curves are lines of constant and denote meridians and parallels for shells of revolution. An arbitrary fiber curve on the surface r(u, v) can be given by its arclength parametrization:
rðsÞ ¼ rðuðsÞ; v ðsÞÞ
ð2Þ
An infinitesimal elementary piece P0P1 (|P0P1| = ds) of a fiber curve is here considered, as depicted in Fig. 1. P0P1 is subjected
Fig. 1. An infinitesimal element of a tensioned fiber trajectory on an arbitrary surface.
to longitudinal tension forces F0 and F1, a normal reaction force Fv perpendicular to the surface, a friction force Ff tangential to the mandrel surface. The Darboux frame [17], which takes into account the fact that the curve r(s) lies on the surface r(u, v), is here considered (Fig. 1). Using the right-handed orthonormal basis (n, T, n T) the principal curvature vector j is in the plane spanned by n T and n, and can thus be decomposed into two orthogonal vectors [17]:
j ¼ jg ðn TÞ þ jn n
ð3Þ
where n, T denote the unit normal and tangent vector, respectively; jn gives the component of the principal curvature vector normal to the surface at a point and is called the normal curvature; jg gives the component of the curvature vector tangential to the surface and is called the geodesic curvature. The principal curvature j points towards the center of the curvature of the fiber curve and can be expressed by:
j ¼ jN
ð4Þ
where j is the curvature of the fiber trajectory at a point (P0 or P1); N is the principal normal vector in the Frenet Frame [17]. For an infinitesimal arclength the magnitudes of the tension at both sides of P0P1 are considered equal:
jF0 j ¼ jF1 j ¼ ‘
ð5Þ
Thus we have:
F0 ¼ ‘ T0
ð6Þ
F1 ¼ ‘ T1
ð7Þ
The equilibrium equations of the forces exerting on P0P1 in directions normal and tangent to the surface, can respectively be formulated as:
ðF0 þ F1 Þnn þ Fn ¼ 0
ð8Þ
ðF0 þ F1 Þðn TÞðn TÞ þ Ff ¼ 0
ð9Þ
Substituting Eqs. (6) and (7) into (8) and (9), leads to:
Fn ¼ ‘ðT1 T0 Þnn
ð10Þ
Ff ¼ ‘ðT1 T0 Þðn TÞðn TÞ
ð11Þ
Since |P0P1| = ds, according to differential geometry we have [17]
T1 T0 ¼ dsT0 ¼ jdsN
ð12Þ
With the aid of Eq. (12), Eqs. (10) and (11) reduce to:
Fn ¼ ‘jdsNnn
ð13Þ
Ff ¼ ‘jdsNðn TÞðn TÞ
ð14Þ
Substituting Eq. (4) into (3) followed by multiplying both sides of Eq. (3) by the normal vector n, leads to:
Nn ¼ jn
ð15Þ
Nðn TÞ ¼ jg :
ð16Þ
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Substitution of Eqs. (15) and (16) into (13) and (14) yields:
Fn ¼ jn ‘jdsn
ð17Þ
Ff ¼ jg ‘jdsðn TÞ
ð18Þ
To prevent fiber sliding on the supporting surface, the friction force Ff should always be less than the maximum static friction between the supporting surface and the fiber bundle:
jFf j lmax jFn j
ð19Þ
where lmax is the coefficient of maximum static friction between the fiber and the mandrel surface or between the fiber and the previously overwound layer. Note that lmax can be affected by the surface quality, fiber morphology, and the resin viscosity, etc. [18]. Substitution of Eqs. (17) and (18) into (19) gives the nonslippage criterion of fiber trajectories on an arbitrary surface:
jjg =jn j lmax
ð20Þ Fig. 2. Geometry of a torus with its fiber trajectory.
The slippage coefficient k is defined as the ratio of the geodesic curvature to the normal curvature [19,20]:
k ¼ jjg =jn j
ð21Þ
The non-slippage criterion Eq. (20) can then be rewritten as:
k lmax
ð22Þ
The slippage coefficient k represents the slippage tendency between the fiber bundle and the supporting surface. The region of possible winding patterns can also be found according to k. To avoid fiber sliding, k should always be less than lmax which ranges from 0.2 (between the fiber and the previously wound layer) to 0.35 (between the fiber and the mandrel surface) [18,21]. In addition, the fibers may bridge on the concave surface and lose contact with the mandrel surface unless the normal reaction force Fn is acting in the same direction to the normal vector n (if pointing outward), or in the opposite direction to the normal vector n (if pointing inward):
Fn n 0 ðoutward pointing normalÞ Fn n 0 ðinward pointing normalÞ
ð23Þ ð24Þ
Please note that the normal vector is here considered always outward-pointing. Substituting Eq. (17) into (23) and taking into account that the product ‘jds is constantly greater than 0, leads to:
jn 0
ð25Þ
Eq. (25) provides the non-bridging criterion of fiber trajectories on an arbitrary surface. Please note that the fiber bridging can generally be eliminated by changing the winding angles. 3. Fiber trajectories based on netting analysis
We consider here an infinitesimal elementary piece of a surface being part of the thin toroidal vessel (Fig. 3). The normal reaction force is acting as a counterbalance to the internal pressure [9]:
F n ¼ p Ru Du Rv Dv
where p is the internal pressure. For a thin-walled pressure vessel, the membrane approximation theory for the stress analysis is here considered. According to the force equilibrium in the direction normal to the surface element, Fn can be expressed in terms of the membrane forces in two principal directions:
F n ¼ Nu Rv Dv sin Du þ Nv Ru Du sin Dv
ð28Þ
where Nu, Nv are the in-plane forces per unit length in the meridional and parallel directions. The equality relation for the membrane forces can be obtained by equalizing the right-hand expression of Eq. (27) with that of Eq. (28):
Nu Nv þ ¼p Ru Rv
ð29Þ
The axial equilibrium of a toroidal shell can be formulated as shown in Fig. 4:
pp½ðR0 þ r0 cos uÞ2 R20 2pðR0 þ r0 cos uÞNu cos u ¼ 0
ð30Þ
Hence
Nu ¼
pr 0 2R0 þ r0 cos u 2 R0 þ r 0 cos u
A toroidal pressure vessel is an axisymmetric shell of revolution with a circular cross-section that does not intersect the axis of revolution. The torus (surface of a closed doughnut) is formed by revolving a circle of radius r0 about a circle of radius R0 > r0 lying in an orthogonal plane, as pictured in Fig. 2. R0 is the distance between the center of the cross-section and the axis of rotation, being referred to as the major radius; r0 is the radius of the tube, being called the minor radius. The meridional direction u for a circular toroid is tangent to the cross-section and the parallel direction v follows the circle of revolution for the toroid. The regular parameterization of the torus is given by:
8 9 > < ðR0 þ r 0 cos v Þ cos u > = Sðu; v Þ ¼ ðR0 þ r 0 cos v Þ sin u ð0 u 2p; 0 v 2pÞ > > : ; r 0 sin v ð26Þ
ð27Þ
Fig. 3. An elementary surface belonging to a pressure vessel.
ð31Þ
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Fig. 4. Part of a toroidal shell loaded by an internal pressure p.
Nv can be solved by substitution of Eq. (31) into (29):
Nv ¼
pr 2
ð32Þ
For thin shells, the membrane forces are considered to be constant through the thickness. The meridional force, Nu, increases as the parameter u decreases while the parallel force, Nv, remains constant throughout the toroid. The netting theory that considers only the loading of fibers and neglects any contribution by the resin matrix and any interaction between the fibers, is here employed [22–24]. These assumptions do not cause any significant error in the analysis, as long as the fibers are primarily loaded in tension and the transverse and shear stresses in the composite laminate are low compared to the ultimate tensile strength of the fibers. The netting theory gives a very simple result in design and evaluation of composite pressure vessels, but it is well suited for filament-wound pressure vessels [25]. Fig. 5 shows a basic element of helically and hoop wound fibers, intersecting a unit area of the torus in the meridional and parallel directions, respectively. The force equilibrium equations in the u and v directions can respectively be given by:
pr 0 2 pr 2R þ r cos u ra ta sin2 a þ rh th sin2 90 ¼ 0 0 0 2 R0 þ r 0 cos u
ra ta cos2 a þ rh th cos2 90 ¼
ð33Þ ð34Þ
ð35Þ
The relative bend radius, which denotes the tube slenderness, is defined as:
k ¼ R0 =r 0
ð36Þ
Substituting Eqs. (35) and (36) into (33) and (34) and solving for a, results in:
a ¼ tan
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 g þ kþcos u ð0 u 2pÞ 1þg
where g is the hoop-to-helical layer thickness ratio, defined as:
g ¼ th =ta
ð38Þ
Eq. (37) provides the periodically fiber trajectories on the torus based on the netting analysis. The winding angles of the fiber trajectories depend on the relative bend radius and the hoop-to-helical thickness ratio. For the given k and g, the initial winding conditions of fiber position and angle are able to completely determine the fiber trajectories. Note that for the single helical winding g = 0 and Eq. (37) can be rewritten as: 1
where ra, rh, ta, th are the helical and the hoop fiber stress and thickness, respectively; a is the winding angle between the fiber curve and the corresponding parallel curve. The condition of equal shell strains [14,26] is always achieved with the netting solution, given by:
ea ¼ eh ) ra ¼ rh
Fig. 6. Winding angle distributions for the single helical winding (g = 0) and the helical and hoop winding (g = 0.5).
a ¼ tan
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1þ ð0 u 2pÞ k þ cos u
Fig. 6 presents the netting-based winding angle developments for the single helical winding and the helical and hoop winding (k = 3–6, g = 0.5), respectively. The results show that at each position along the meridional direction the single helical winding provides a bigger fiber angle than the helical and hoop winding. 4. Stability-ensuring criteria for fiber trajectories on a torus According to Liouville formula [17], kg is obtained by:
kg ¼
da sin u cos a þ dl R þ r cos u
ð40Þ
where l is the fiber length; kn is related to the principal curvatures in the respective meridional and parallel directions, given by [17]: 2
kn ¼ ku sin a þ kv cos2 a ð37Þ
ð39Þ
ð41Þ
For a torus, ku and kv are given by:
1 ku ¼ ; r
kv ¼
cos u R þ r cos u
ð42Þ
Substituting Eq. (42) into (41) leads to:
Kn ¼
cos u 1 2 cos2 a sin a R þ r cos u r
ð43Þ
In addition, the following relation holds:
du sin a ¼ dl r Fig. 5. A netting-based element per unit area of helically and hoop wound fibers.
ð44Þ
Substituting Eqs. (40), (43), and (44) into (21), the slippage coefficient k can be expressed as a function of u:
L. Zu / Composite Structures 94 (2012) 1855–1860
da sin a þ sin u cos a du kðuÞ ¼ cos u k þ cos u 2 2 cos a þ sin a k þ cos u
ð45Þ
where da/du can be obtained by the first derivative of Eq. (37) with respect to u:
pffiffiffiffiffiffiffiffiffiffiffiffi k sin u 1 þ g da pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ du 2ð3k þ 2 cos uÞ ðk þ cos uÞ½ð2 gÞk þ ð1 gÞ cos u
ð46Þ
Substitution of Eq. (46) into (45) gives: ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 sin uð3k þ 2 cos uÞpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k þ ðk þ cos uÞð1 gÞ þ k ð1 þ gÞð2k þ cos uÞ k¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi 4ðk þ cos uÞ3=2 ð3k þ 2 cos uÞ½ðk þ cos uÞð1 gÞ þ k
ð47Þ The fiber bridging tendency should also be examined since the torus has a convex–concave surface. Substitution of Eq. (43) into (25) leads to the condition for preventing fiber bridging over the torus: k þ1g cos u tan2 a ¼ kþcos u k þ cos u 1þg
ð48Þ
Eq. (48) can further reduce to:
g ¼ th =ta < 2ð1 1=kÞ
1859
critical angles for avoiding fiber bridging; the obtained fiber trajectories will therefore not bridge over the entire surface. In addition, for a given k there is a maximum value for g able to guarantee the validity of the non-bridging criterion. The result is depicted in Fig. 8, where the non-bridging-ensuring feasible {k, g}-field is dashed. It should be noted here that the non-bridging criterion is consistently valid for the convex surface because Eq. (48) always holds true for the range u = [90°, 90°]. Fig. 9 shows the distributions of the slippage coefficient k along the meridional coordinate u, corresponding to various k (k = 3–6, g = 0.5). The results reveal that the fiber trajectories exhibit an increasing slippage tendency from the inner/outer equator towards the crest of the toroid; however, the maximum slippage tendency is not present exactly at the crest but somewhere between the crest and the inner equator, depending on the k-value. It is thus suggested that more attention should be paid to the winding of the concave surface of the toroid. Fig. 10 illustrates the maximum slippage coefficients of the obtained fiber trajectories corresponding to the most common combinations of k and g (k = 3–6, g = 0–1.5). It is shown that the fiber slippage tendency significantly decreases with the increase of k, and slightly increases with the increase of g. The relative bend radius proves to be the most influential factor on fiber slippage tendency, while the thickness ratio has less influence.
ð49Þ
Eq. (49) provides the non-bridging criterion for fiber trajectories on the torus. It should be noted here that for the single helical winding Eq. (49) is always satisfied since g = 0. Accordingly the single helical winding trajectories determined using the netting theory will never bridge over the torus. 5. Results and discussion Eqs. (47) and (49) give the criteria for assessing the stability of fiber trajectories designed using the netting-based approach. It is shown that for the helical and hoop winding the fiber stability is influenced by the relative band radius k and the hoop-to-helical layer thickness ratio g while for the single helical winding k is the only influential factor. Fig. 7 outlines the distribution of the minimum required winding angles as dictated by the non-bridging criterion, as compared to the winding angles of fiber trajectories obtained using the netting analysis. It can be seen that for the most common k-range [3,6], the netting-based winding angles are much bigger than the
Fig. 8. Feasible field of {k, g}-combinations for ensuring non-bridging (shaded area).
Fig. 7. Minimum non-bridging-ensuring angles as compared to winding angles of the netting-based fiber trajectories.
Fig. 9. Slippage tendency distributions along the meridional direction.
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Despite the fact that the small relative bend radii (k < 3) result in the instability of fiber trajectories on the torus, the present netting-based winding trajectories do satisfy the stability-ensuring requirements for the most commonly used toroids (k P 3) in today’s industries. Since this paper only consider the fiber slippage and bridging tendencies for analyzing the stability of winding trajectories, in the future the criteria of uniform and full coverage will also be considered in order to enable the fiber bands equally distributed, without leaving any gaps or creating excessive overlaps. However, the present analysis proves very useful in the design phase of filament-wound toroidal pressure vessels. References
Fig. 10. Maximum slippage coefficients of the obtained fiber trajectories for the most common k and g.
As seen in the results of the stability analysis, it is shown that for toroidal pressure vessels with k > 4 the fiber trajectories determined using the netting theory can well satisfy the stability-ensuring criteria; for toroidal vessels with 3 6 k 6 4, the maximum coefficient of friction between the supporting surface and the placed fiber bundle should be guaranteed to reach 0.4–0.5, in order to prevent fiber trajectories deviating from the netting-dictated trajectories. However, for toroids with k < 3 and g > 1.5, it is strongly suggested to employ geodesic or semi-geodesic trajectories instead of the present netting-based trajectories, with a compromise between structural performance and windability. 6. Conclusions In this paper an analysis for fiber trajectory stability is elaborated for providing a design reference of filament-wound toroidal pressure vessels. The fiber trajectories are determined using the netting theory in combination with the special geometry of the torus. The netting analysis establishes the simple and straightforward relationship among the fiber trajectories, the internal pressure, material properties and processing parameters. Compared to the geodesic or semi-geodesic winding, the present nettingbased approach can place the fibers according to the principal stress direction of the toroidal shell under internal pressure and thus lead to a maximum utilization of fiber strength as well as a maximum structural performance. To evaluate the effect of the geometry and the lay-up on the stability of the netting-based trajectories, the fiber slippage and bridging tendencies of the obtained fiber trajectories are illustrated for various relative bend radii and hoop-to-helical thickness ratios. It is revealed that toroidal pressure vessels with larger relative bend radius and lower hoopto-helical thickness ratio lead to better fiber stability in terms of either slippage or bridging tendency. The results also conclude that for the most commonly used toroids, the winding trajectories designed by the netting theory satisfy both the non-slippage and non-bridging criteria, and show good fiber stability for the winding process. In addition, it is worth noting that the netting-based fiber trajectories will in general fulfill the non-bridging condition if they satisfy the non-slippage criterion.
[1] Ross CTF. A conceptual design of an underwater missile launcher. Ocean Eng 2005;32:85–99. [2] Ruggiero EJ, Jha A, Park G, Inman DJ. A literature review of ultra-light and inflated toroidal satellite components. Shock Vib Digest 2003;35:171–81. [3] Weisberg AH, Myers B, Berry G. Hydrogen storage using lightweight tanks. In: Proceedings of the 2002 US DOE hydrogen program review. NREL/CP-61032405; 2002. [4] Baker WR, Bello SD, Marcuzzi D, Sonato P, Zaccaria P. Design of a new toroidal shell and support for RFX. Fusion Eng Des 2002;64:461–6. [5] Kisioglu Y. Burst Pressure Determination of vehicle toroidal oval cross-section LPG fuel tanks. J Press Ves Technol 2011;133:0312021–25. [6] Cook J, Chambers JK, Richards BJ. Toroidal pressure vessels for breathing apparatus. In: Proc of international SAMPE Europe conference, Paris, France; April 1998. [7] Li SG, Cook J. An analysis of filament overwound toroidal pressure vessels and optimum design of such structures. J Press Ves Technol 2002;124:215–22. [8] Zu L, Koussios S, Beukers A. Design of filament-wound circular toroidal hydrogen storage vessels based on non-geodesic fiber trajectories. Int J Hydrogen Energy 2010;35:660–70. [9] Zu L, Koussios S, Beukers A. Optimal cross sections of filament-wound toroidal hydrogen storage vessels based on continuum lamination theory. Int J Hydrogen Energy 2010;35:10419–29. [10] Zu L, Zhang DH, Xu YQ, Xiao DJ. Integral design and simulation of composite toroidal hydrogen storage tanks. Int J Hydrogen Energy 2012;37:1027–36. [11] Marketos JD. Optimum toroidal pressure vessel filament wound along geodesic lines. AIAA J 1963;1:1942–5. [12] Maksimyuk V A, Chernyshenko IS. Nonlinear elastic state of thin-walled toroidal shells made of orthotropic composites. Int Appl Mech 1999;35:1238–45. [13] Blachut J. Buckling and first ply failure of a composite toroidal pressure hull. Comput Struct 2004;82:1981–92. [14] Vasiliev VV, Krikanov AA. New generation of filament-wound composite pressure vessels for commercial applications. Compos Struct 2003;62:449–59. [15] Zu L, Koussios S, Beukers A. Design of filament-wound domes based on continuum theory and non-geodesic roving trajectories. Compos Part A: Appl Sci Manuf 2010;41:1312–20. [16] Zu L, Koussios S, Beukers A. Integral design for filament-wound composite pressure vessels. Polym Polym Compos 2011;19:413–20. [17] Gray A. Modern differential geometry of curves and surfaces. Boca Ration: CRC Press; 1993. [18] Wang R, Jiao W, Liu W, Yang F, He X. Slippage coefficient measurement for non-geodesic filament-winding process. Comp Part A: Appl Sci Manuf 2011;42:303–9. [19] De Carvalho J, Lossie M, Vandepitte D, Van Brussel H. Optimization of filament wound parts based on non-geodesic winding. Compos Manuf 1995;6:79–84. [20] Zu L, Koussios S, Beukers A. Shape optimization of filament wound articulated pressure vessels based on non-geodesic trajectories. Compos Struct 2010;92:339–46. [21] Park JS, Hong CS, Kim CG, Kim CU. Analysis of filament wound composite structures considering the change of winding angles through the thickness direction. Compos Struct 2002;55:63–71. [22] Krikanov AA. Composite pressure vessels with higher stiffness. Compos Struct 2000;48:119–27. [23] Zu L, Koussios S, Beukers A. Design of filament-wound isotensoid pressure vessels with unequal polar openings. Compos Struct 2010;92:2307–13. [24] Evans JT, Gibson AG. Composite angle ply laminates and netting analysis. Proc R Soc Lond A 2002;458:3079–88. [25] Rosato DV, Grove CS. Filament winding: its development, manufacture, applications, and design. New York: Interscience; 1964. [26] Hojjati M, Safavi AV, Hoa SV. Design of domes for polymeric composite pressure vessels. Compos Eng 1995;5:51–9.