Application of variable slippage coefficients to the design of filament wound toroidal pressure vessels

Accepted Manuscript Application of variable slippage coefficients to the design of filament wound toroidal pressure vessels Lei Zu, Weidong Zhu, Huiyue Dong, Yinglin Ke PII: DOI: Reference:

S0263-8223(17)30939-X http://dx.doi.org/10.1016/j.compstruct.2017.03.094 COST 8415

To appear in:

Composite Structures

Received Date: Accepted Date:

22 March 2017 27 March 2017

Please cite this article as: Zu, L., Zhu, W., Dong, H., Ke, Y., Application of variable slippage coefficients to the design of filament wound toroidal pressure vessels, Composite Structures (2017), doi: http://dx.doi.org/10.1016/ j.compstruct.2017.03.094

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Application of variable slippage coefficients to the design of filament wound toroidal pressure vessels Lei Zu, Weidong Zhu*, Huiyue Dong, Yinglin Ke College of Mechanical Engineering, Zhejiang University, Zhejiang 310027, China

Abstract: The non-geodesics using variable slippage coefficients along the winding trajectories are gaining increasing attention for composite pressure vessels. The aim of this paper is to evaluate the influence of several slippage coefficient distributions on fiber trajectories and the resulting mass of toroidal pressure vessels. The distributions of slippage coefficients are represented in terms of three functions satisfying C1 continuity. The governing equations for designing non-geodesics on a toroid are formulated using the given functions of slippage coefficients. For each function, the initial fiber angle and the shell thickness are considered as the design variables, and the minimum mass is taken as the objective. With the aid of the classical lamination theory and the SQP algorithm, the optimal fiber trajectories are respectively determined for the given three slippage coefficient distributions, and the corresponding masses of the toroids are then obtained and compared to each other. The results indicate that toroids produced using variable slippage coefficients show improved performance than toroids using constant slippage coefficients; this is mainly due to the maximum utilization of the laminate strength. It is also revealed that the weight efficiency of toroidal vessels can be significantly improved using the quadratic distribution function of fiber slippage coefficients. Keywords: Composite Materials; Filament winding; Toroidal pressure vessel; Non-geodesic; Slippage

coefficient

*Corresponding author. Tel.: +86 571 87953929; Fax: +86 571 87953929 E-mail address: [email protected] (W. Zhu).

distribution

1. Introduction Composite pressure vessels have been extensively used in aerospace, nuclear, underwater, transportation and storage industries due to their significant advantages as compared to traditional metal cylinders. The filament winding process is a manufacturing technique which involves continuous fibers being accurately positioned according to a predetermined trajectory on a rotating mandrel to shape a closed shell. The process is normally CNC controlled to permit highly automated production. Impregnated fibers are oriented to match the principal stress direction in a laminate, allowing for optimal reinforcement loading. More recently, filament wound toroidal pressure vessels show great potential for applications in a variety of industries due to their high structural performance, high efficient configuration, and low aspect ratio. A toroid is doubly curved and provides a promising alternative shape for spaces that have limited height and volume. The geometry of a toroid can be considered as a bent, endless cylindrical shell. The use of a toroid saves materials in the end caps and, when considering its shape, it is more weight-effective than a cylinder. Filament-wound convex bodies have so far been studied for many years and research on standard axisymmetrical parts such as cylinders, spheres, cones and domes has became relatively simple and mature [1-4]. When winding such bodies, a predetermined fiber path is repeated continuously by indexing the axis. In contrast, a torus has a doubly-closed body with a convex-concave surface. Many researchers have been devoted to determine the optimal fiber trajectories for filament-wound toroidal pressure vessels that simultaneously satisfy structural and manufacturable requirements. Their design methods were mainly based on the nettingdictated paths, geodesics or non-geodesics. Zu et al. [5-7] suggested a number of design methods for acheiving optimal fiber trajectories, geometries, and winding patterns of filament-wound toroidal pressure vessels. They developed a CAD system for designing and manufacturing these vessels. Hu et al. [8] reported an innovative approach to the design in

order to determine the winding parameters of filament wound toroidal pressure vessels using differential geometry and a finite element method (FEM). Li et al. [9] presented a full mathematical method for the optimal design of a filament-wound toroid that resulted in 46% of the weight of a monolithic metal toroidal vessel of uniform wall thickness. Marketos [10] reported an optimal winding angle in combination with an optimal meridian shape of the toroidal mandrel. Using a similar method, Mitkevich et al. [11] clarified the winding trajectory and meridian shape of the mandrel and its transformation into given surface elements, using the example of a toroidal membrane. Cook et al. [12] argued that a toroidal shaped vessel for compressed-air storage has several improved qualities that make it better than a conventional cylindrical shape in a design for a breathing apparatus. Koussios [13, 14] and Zu [15] reported the use of cross-sectional shapes of isotensoid toroids and compared the structural efficiency of such toroids with circular ones. A previous report on use of the netting theory had a severe deficiency due to the designs, which completely neglected the matrix effect and were only based on fiber strength. Pressure vessels based on netting-based designs are built on the condition of equal shell strains, and they therefore have high-level performance. However, netting-based fiber trajectories belong to the class of non-geodesics; therefore, the rigorous requirements of friction coefficients (≤0.3~0.5 in general) during the winding process will significantly confine the available design space [16]. An example of this restriction is there is a limit on the smaller relative bending radius that a non-geodesic-isotensoid toroid can obtain [17]. Previous research has mostly concentrated on geodesic or semi-geodesic trajectories with a constant slippage coefficient throughout the entire structure, in addition to the netting-based designs; however, a design utilizing the variable slippage coefficients along the fiber paths has been overlooked. If one is using either geodesics or semi-geodesics, it generally follows that the fibers are not properly placed along the principal stress direction. Accordingly, this does

not result in maximum structural performance. Thus, it is desirable to determine the fiber trajectories with non-constant (variable) slippage coefficient distributions to produce a more efficient toroid. In this report, a design method is written out in detail for the determination of fiber trajectories with variable slippage coefficients for filament-wound toroids that are subjected to uniform internal pressure. Utilizing the classical lamination theory and the fiber slippage law, the differential equations for obtaining non-geodesics with variable slippage coefficients were formulated for a torus while simultaneously keeping the layer thickness build-up along the meridional direction in mind. Three continuous functions were deliberately selected to define the fiber slippage coefficient distributions along the meridional coordinate as a preliminary attempt to investigate non-geodesics using non-constant slippage coefficients and to make the evaluation convenient and explicit. The present design method is demonstrated by filamentwound carbon-epoxy toroidal pressure vessels with relative bending radii of 3~6, reflecting on the an incredibly general design approach. The resulting fiber trajectories and vessel mass are then determined and compared to each other, in order to find the best distribution type of fiber slippage coefficients for winding a toroidal pressure vessel.

2. Non-geodesics for winding a torus The geometry of a toroid is described as a surface of revolution with circular cross sections surrounding the axis of rotation (Z-axis, see Fig. 1). A toroid has a bending radius, R, and a radius of the tube, r, and it is a doubly-closed shell in the meridional (φ) and parallel (θ) directions. The vector of a torus is represented as: ( R + r cos ϕ ) cos θ    S (θ , ϕ ) = ( R + r cos ϕ ) sin θ    r sin ϕ  

(1)

where φ and θ respectively denote the angular coordinates along the meridional and parallel direction of the torus, as shown in Fig.1. Let λ represent the coefficient of slippage tendency between the fiber bands and the mandrel surface. It can thus be defined as the slippage coefficient (the ratio of the geodesic curvature, κg, to the normal curvature, κn [18]):

λ = κ g κn

(2)

According to the Liouville formula [19], κg for the torus can be expressed as: (3) where α is the winding angle, as shown in Fig. 1; l is the fiber length. κn can be given by the Euler formula [20]:

κn = κθ cos2 α + κϕ sin2 α

(4)

where κn is the normal curvature, κφ and κθ are the meridional and parallel curvatures, respectively. According to the definition in differential geometry [19], κφ and κθ are given by:

κϕ＝ −

cos ϕ R + r cos ϕ

, κθ = −

1 r

(5)

Substituting Eq. (5) into (4) and plugging the result and Eq. (3) into Eq. (2) leads to: dα cos ϕ 1 sin ϕ cos 2 α + sin 2 α )λ − cos α = −( dl R + r cos ϕ r R + r cos ϕ

(6)

Since the fiber trajectory has an angle α as related to the parallel circle of the torus, the geometric relations dθ/dl, dφ/dl and α, can be given by (see Fig. 2): dθ cos α dϕ sin α , = = r dl R + r cos ϕ dl

(7)

Respectively substituting the two expressions of Eq. (7) into Eq. (6) leads to a system of differential equations for determining the non-geodesic trajectories on a torus:

 dα  r cos ϕ cos α  r sin ϕ =± + sin α  λ −  ( R + r cos ϕ )tgα  dϕ  ( R + r cos ϕ )tgα    dα = ± cos ϕ cos α + ( R + r cos ϕ )tgα sin α  λ − sin ϕ    dθ r

(8)

The slippage coefficient λ must not necessarily be constant along the fiber paths for winding a toroidal pressure vessel. Counting on the allowable friction coefficient µ, λ could be varied in the range [-µ, µ]. The derivative of the fiber angle at the outer and inner peripheries must be the same as the derivative of a geodesic to ensure C1 continuity of the winding trajectories when passing the upper-lower half junction. Hence, λ should be zero for φ = π and φ = 0. The three explicit functions for formulating slippage coefficient distributions in terms of φ-coordinate are a constant function, a linear function and a quadratic function. The three functions are shown here to create non-geodesic trajectories (see Fig. 3):

µ   f1 (ϕ ) = 2  π  2µ  (0 ≤ ϕ ≤ ) ϕ    2 λ =  f 2 (ϕ ) =  π   2 µ (π − ϕ ) ( π ≤ ϕ ≤ π )   π 2   f 3 (ϕ ) = 4 µ2 ϕ (π − ϕ ) π 

(9)

where µ indicates the maximal allowable friction coefficient between the fiber band and the mandrel surface or between the composite layer and the previously wound layer. Note that µ strongly relies on the smoothness of the mandrel surface, the resin viscosity, the material system, the fiber tension, and other factors [20-22]. Substituting Eq. (9) into (8) gives:

 dα  r cos ϕ cos α  r sin ϕ = ± f i (ϕ )  + sin α  −   dϕ  ( R + r cos ϕ )tgα  ( R + r cos ϕ )tgα   d α = ± f (ϕ ) cos ϕ cos α + ( R + r cos ϕ )tgα sin α  − sin ϕ i    dθ r

(10)

Let K be the relative bending radius. We have: K = R/r

(11)

Substitution of Eq. (11) into (10) leads to:  dα  cos ϕ cos α  sin ϕ = ± fi (ϕ )  + sin α  −   dϕ  ( K + cos ϕ )tgα  ( K + cos ϕ )tgα   dα = ± f (ϕ ) cos ϕ cos α + ( K + cos ϕ )tgα sin α − sin ϕ [ ] i  dθ

(12)

For λ=0, the solution of Eq. (12) is the Clairaut equation for determining geodesics. The fiber trajectories will deviate from the geodesics in every other case (λ≠0) and is therefore regarded as non-geodesics. No analytical solution of Eq. (12) exists for λ≠0. The nongeodesic trajectories can be calculated step by step using the Runge-Kutta method [23] with the aid of the initial conditions of the winding process, as demonstrated in Figure 4. The absolute value of λ is less than 0.2 for wet winding, whereas it can exceed 0.5 [24] for dry winding. It should be noted that the slippage coefficient λ could also be negative [25].

3. Minimum mass design of toroidal pressure vessels

3.1 Design variables and objective function The goal of this design is to search the minimum mass of non-geodescally overwound toroidal pressure vessels. The wall thickness t0 at the equator and the initial winding angle α0 are considered as the design variables, as stated by:

X = (α 0 , t0 ) The mass of the composite shell of a toroid is considered as the objective function of the present problem: 2π

M = 2π rγ ( R + r ) sin α 0t0 ∫ csc α dϕ 0

(13)

where γ is the density of the used materials, α is a function of φ, given by Eq. (12). The following dimensionless variables are here used for analysis: M =M⋅

2Y YT , t =t⋅ T 3 pr 2γ pπ r

(14)

where p is the internal pressure and YT is the transverse tensile strength of a unidirectional lamina. Substituting Eq. (14) into (13), the dimensionless mass of a toroid can be represented by: 2π

M = ( K + 1) sin α 0 t0 ∫ csc α dϕ

(15)

0

3.2 Design constraints The toroidal surface has a concave section. Therefore, the winding may produce fiber bridging that fibers may not closely attach to the mandrel surface, unless the normal curvature κn meets the below requirement [18]:

κn < 0

(16)

Based on Eqs. (4) and (5), fibers on the convex part of a torus will not bridge since the normal curvature κn is constantly less than 0. However, the winding angle α on the concave section has to fulfill the following non-bridging condition for the satisfaction of Eq. (16): − cos ϕ − tg 2α < 0 K + cos ϕ

(17)

The filament-wound shell are multi-layers with helically overwound fibers comprising ±α layers, each of which forms an angle ply. The stress components in an individual layer with respect to the material axes are represented in terms of the biaxial internal forces [26]: 1  σ 1 = t ( m11 Nϕ + m12 Nθ )  1  σ 2 = ( m21 Nϕ + m22 Nθ ) t  1  τ 12 = t ( m31 Nϕ + m32 Nθ ) 

(18)

where constants mij can be calculated by: m11 = c 2 + 2cs

Q 22 Q16 − Q12 Q 26 2

Q11 Q 22 − Q12

, m12 = s 2 + 2cs

Q11 Q 26 − Q12 Q16 2

Q11 Q 22 − Q12

,

m21 = s 2 − 2cs

Q 22 Q16 − Q12 Q 26 2

Q11 Q 22 − Q12

m31 = − sc + (c 2 − s 2 )

, m22 = c 2 − 2cs

Q 22 Q16 − Q12 Q 26 2

Q11 Q 22 − Q12

Q11 Q 26 − Q12 Q16 2

,

Q11 Q 22 − Q12

, m32 = sc + (c 2 − s 2 )

Q11 Q 26 − Q12 Q16 2

. (19)

Q11 Q 22 − Q12

where c=sinα, s=cosα. Qij are the elements of the transformed reduced stiffness matrix, given by: Q11 = Q11c 4 + 2(Q12 + 2Q66 )c 2 s 2 + Q22 s 4  2 2 4 4 Q12 = (Q11 + Q12 − 4Q66 )c s + Q12 (c + s )  4 2 2 4 Q 22 = Q11s + 2(Q12 + 2Q66 )c s + Q22 c  3 3 2 2 Q16 = −Q22 cs + Q11c s − (Q12 + 2Q66 )(c − s )cs  3 3 2 2 Q 26 = −Q22 c s + Q11cs + (Q12 + 2Q66 )(c − s )cs Q = (Q + Q − 2Q )c 2 s 2 + Q (c 2 − s 2 ) 2  66 11 12 12 66

(20)

where [Q] is the reduced stiffness matrix and its elements can be stated as: Q11 =

E1 1 −ν 12 v21

, Q22 =

E2 1 −ν 12 v21

, Q12 =

ν 12 E2 , Q66 = G12 . 1 −ν 12 v21

(21)

where E1, E2, G12 and ν12 are the longitudinal elastic modulus, the transverse elastic modulus, the shear modulus and the major Poisson's ratio of the unidirectional layer. The internal forces of the toroidal shell can be given by [18]: Nϕ =

pr pr 2 K + cos ϕ ⋅ , Nθ = 2 K + cos ϕ 2

(22)

The failure load is determined by the Tsai-Wu failure criterion [27]:

Fiσ i + Fijσ iσ j ≤ 1 (i, j = 1, 2, 6)

(23)

where F11, F22, F12, F66, F1, F2 and F6 are given by: F11 =

1 1 1 1 1 1 , F22 = , F66 = 2 , F12 = − , F1 = , − XT XC YT YC S XT XC 2 X T X C YT YC F2 =

1 1 − , F6 = 0 YT YC

(24)

where XT, XC, YC, and YT are the tensile and compressive strengths of the unidirectional layer in the fiber and transverse directions, and S is the in-plane shear failure strength. Substituting Eqs. (19) ~ (22) into Eq. (18) and plugging the results into Eq. (23) yields the strength constraint that the fiber lay-up of the toroidal pressure vessel should satisfy. 3.3 Design approach The optimization involves the minimization of the dimensionless mass M , constrained by non-bridging condition Eq. (17) and failure criterion Eq. (23). The optimization is related to the nonlinear constrained optimization. The sequential quadratic programming (SQP) method is regarded as effective for solving this class of optimization problems. Since the number of the fibers located in any parallel circle of the composite shell is constant, the shell thickness at any meridional coordinate can be formulated on the basis of the thickness at the equator, given by: t=

( K + 1) sin α 0 ⋅ t0 ( K + cos ϕ ) sin α

(25)

When a non-geodesic path that complies with the constraints, is assigned as a set of design variables in the ith iteration step of the optimization procedure, the wall thickness distribution is determined by Eq. (25) and then the constrained equations (17) and (23) are evaluated at all points along the fiber trajectory. The points used for evaluation are determined by solving the non-geodesic equation (12) at each iteration step. The objective function M is calculated using the Gaussian quadrature rule at the end of each iteration step. The current design point is then updated as a new point in this procedure and the above steps are repeated until a termination criterion is satisfied.

4. Results and discussion

A filament-wound carbon-epoxy toroid with relative bending radii K = 3~6 are considered to show the influence of the given three slippage coefficient distributions on the structural

mass of toroids and fiber trajectories. The shell is made of T300/5208 graphite/epoxy with elastic constants of E1=142 GPa, E2=10.8 GPa, G12=5.49 GPa and ν12=0.3. The strength values are XT=1568 MPa, YT=57 MPa, YC=212 MPa, XC=1341 MPa, and S=80 MPa [29]. Figure 5 presents a collection of the optimum non-geodesic paths that correspond to the given slippage coefficient distributions (Eq. [9]). The continuous lines denote the constantcoefficient-based non-geodesic trajectory (i.e. λ≡ 0.1 ). It is revealed that the winding angles determined using the quadratic function is distributing in the range of 50º-55º, which is almost same with the optimum fiber orientation for cylindrical vessels under biaxial stress ratio 2. On the other hand, the slippage coefficient distributions based on the constant and the linear functions result in fiber angle developments in a relatively large range, and the fiber angles at the inner periphery are below 50º. Therefore, the quadratic distribution of fiber slippage coefficients produces a more desirable distribution of the fiber stress in the composite shell of toroidal pressure vessels. Figure 6 shows the distributions of dimensionless longitudinal stress in the fiber direction for toroidal pressure vessels (K = 4) with the given slippage coefficient distributions. The results demonstrate that the toroidal shell obtained using the quadratic distribution of slippage coefficients has higher stresses than the toroid using a linear or constant function while the Tsai-Wu criterion Eq. (23) is still satisfied for the composite shell. Consequently, the strength of the composite shell can be maximally utilized with the aid of the quadratic distribution of slippage coefficients. As a result of this, the toroidal vessel determined using the quadratic distribution is considerably thinner than the others, as shown in Fig. 7. Figure 8 illustrates the optimal values of dimensionless structural mass for the given three slippage coefficient distributions, which correspond to the relative bending radii ranging from 3 to 6. The results show that filament-wound toroids determined using variable slippage coefficients is consistently lighter than those using a constant slippage coefficient. In addition,

compared to the linear distribution, the quadratic distribution of slippage coefficients gains lighter structures and thus better weight efficiency of toroidal pressure vessels.

5. Conclusions

The primary aim of this study was to outline a novel design approach to determine nongeodesic trajectories with variable slippage coefficients, and to assess the influence of the use of non-constant slippage coefficients on the fiber trajectories and the resulting structural performance of toroidal pressure vessels. An optimal design that minimizes the mass of nongeodesics-based toroidal vessels was presented using classical lamination theory, the Tsai-Wu criterion, and manufacturable constraints. The results show that the toroidal pressure vessel produced based on variable slippage coefficients along the fiber paths have an improved stress distribution and therefore a maximum utilization of the laminate strength under internal pressure, compared to the non-geodesics using a constant slippage coefficient. Hence, a reasonable distribution of the fiber stress in the parallel and meridional directions can be obtained and consequently increase the weight efficiency of filament-wound toroidal pressure vessels. It is also indicated that the best performance of toroidal pressure vessels can be achieved based on the quadratic distribution of fiber slippage coefficients. The non-constant slippage coefficient distributions were successfully applied to the design of composite toroidal pressure vessels. The present design method is thus an interesting alternative for improving the structural performance of filament-wound composite pressure vessels.

Acknowledgements

This research is supported by the National Natural Science Foundation of China (Grant No. 11302168) and the Fundamental Research Funds for the Central Universities (Grant No. 143101001).

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Figure Captions

Figure 1. Geometry of a toroidal pressure vessel and its fiber trajectories. Figure 2 . Geometrical relations for θ, φ, l and α. Figure 3. The three slippage coefficient distributions (µ = 0.2). Figure 4. Winding angle developments for the given three slippage coefficient distributions

(µ = 0.2, α0 = 60º). Figure 5. Optimal fiber trajectories for the given three slippage coefficient distributions (µ =

0.2, K = 4). Figure 6. Dimensionless longitudinal stress for the given three slippage coefficient

distributions (µ = 0.2, K = 4). Figure 7. Dimensionless shell thicknesses for the given three slippage coefficient

distributions (µ = 0.2, K = 4). Figure 8. Dimensionless vessel mass for the given three slippage coefficient distributions (µ =

0.2, K = 3~6).