Bend-free design of ellipsoids of revolution using variable stiffness composites

Journal Pre-proofs Bend-free Design of Ellipsoids of Revolution using Variable Stiffness Composites Shahrzad Daghighi, Mohammad Rouhi, Giovanni Zucco, Paul M. Weaver PII: DOI: Reference:

S0263-8223(19)32669-8 https://doi.org/10.1016/j.compstruct.2019.111630 COST 111630

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

16 July 2019 23 October 2019 30 October 2019

Please cite this article as: Daghighi, S., Rouhi, M., Zucco, G., Weaver, P.M., Bend-free Design of Ellipsoids of Revolution using Variable Stiffness Composites, Composite Structures (2019), doi: https://doi.org/10.1016/ j.compstruct.2019.111630

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Bend-free Design of Ellipsoids of Revolution using Variable Stiffness Composites Shahrzad Daghighia , Mohammad Rouhia , Giovanni Zuccoa , Paul M. Weavera a Bernal

Institute, School of Engineering, University of Limerick, Limerick, Ireland

Abstract Shells are commonly used in many structural applications due to their high specific load carrying capabilities. One of the most interesting features of shell structures is that they can resist external transverse loads by developing membrane stresses in the small deformation regime yet, in general, also generate inefficient bending deformations and stresses. In this study, a composite ellipsoid shell of revolution, under internal pressure, is designed for zero bending and curvature change. To this end, the stiffness properties of elliptical composite shell structures are tailored by fibre steering. A new definition for a bend-free state, independent of internal pressure, is presented. Based on this definition, the internal pressure-induced bending state of an isotropic ellipsoidal shell of revolution is compared with its tailored composite counterpart. Results show that up to a specific level of ellipticity, a bend-free state is achievable by fibre steering in elliptical composite shells of revolution. Finally, a failure study is performed to assess the potential improvement of the maximum allowable internal pressure by bend-free design.

Nomenclature

gϕ , gθ

vector tangent to the respective coordinate line (ϕ, θ)

(σ1T )ult , (σ1C )ult

tensile and compressive strengths of carbon/epoxy along fibre direc- R tion

(σ2T )ult , (σ2C )ult

tensile and compressive strengths of carbon/epoxy transverse to fibre direction

r

position vector of the points on the middle surface

χ

M ax(α, β)

(τ12 )ult

shear strength along fibres for carbon/epoxy

∆

rθ rϕ

α

moment-less factor

bot top ij , ij

strain at top and bottom of each individual ply of composite

αϕ , αθ

metric tensor coefficients

γ

β

curvature-less factor

angle used to change the direction of stresses and strains

A

extensional stiffness matrix

γϕθ

in-plane shear strain

Aani

orientation dependent part of ex- κϕ , κθ , κϕθ tensional stiffness matrix ν orientation independent part of 12 extensional stiffness matrix νxy , νyx

Aiso B D

bending-extensional stiffness matrix bending stiffness matrix

coupling

position vector of the points on the arbitrary surface

curvature changes in respective direction (ϕ, θ) poisson’s ratio for carbon/epoxy in-plane Poisson’s ratio in their respective directions

σ1 , σ2 , τ12

in-plane stresses for aluminium

σν

equivalent von Mises stress

top bot σij , σij

θ θ1 εϕ , εθ ϕ

stress at top and bottom of each H1 , H11 , H2 , H22 , coefficients of Tsai-Wu individual ply of composite H6 , H66 , H12 failure criteria curvilinear coordinate in latitudiMϕ , Mθ , Mϕθ bending moment resultant in renal direction spective direction (ϕ, θ) fibre direction in composite M L% percentage of moment-less area normal strain in respective direcNϕ , Nθ stress resultant in respective dition (ϕ, θ) rection (ϕ, θ) curvilinear coordinate in longituNϕθ shear stress resultant dinal direction

a

half major axes in ellipsoid of revolution

nelmθ

number of finite elements in ϕ direction

Aα

area of the structure where α is nelmϕ less than 0.01

number of finite elements in θ direction

Aβ

area of the structure where β is P less than 0.01 Qθ , Qϕ

internal pressure

Awhole

whole area of the structure

qθ , qϕ

in-plane tractions

b

radii of curvature

BF%

half minor axes in ellipsoid of rev- R olution rθ , rϕ percentage of bend-free area

CL%

percentage of curvature-less area

thickness

e

eccentricity of ellipsoid of revolu- U , U , U , U , U material invariants 1 2 3 4 5 tion V1A , V2A , V3A , V4A in-plane lamination parameters young modulus in respective direction for carbon/epoxy Z coordinate in vertical direction

E11 , E22 G12

t

shear modulus for carbon/epoxy

z

out of plane shears

radii of curvature in ϕ and θ direction

coordinate in thickness direction

1. Introduction Shell structures are human-made bio-inspired structures which can be found in nature in the form of eggs, leaves, skulls and so on [1]. Due to their excellent structural performance, shells are commonly used in engineering applications such as pressure vessels, submarine hulls, liquid storage tanks and aeroplane fuselages. In structural mechanics, generally a shell is defined as the volume between two surfaces where their distance from each other, the thickness, is small compared with the other two characteristic lengths. One of the most advantageous characteristics of shell structures that make them interesting is their ability to resist external transverse loads by developing membrane stresses, yet in general, can also generate inefficient bending deformations and stresses. An attractive research proposition is to control how the structure reacts under applied loads and boundary conditions. Based on Newton’s third law, for each action there is an equal and opposite reaction. These reactions can be in forms of forces or moments. When a structure is under bending moment, there is a plane, i.e. the so-called neutral plane, that does not carry in-plane normal stresses. Simply put, there is a waste of material around the neutral plane and this brings forth the notion of designing a bend-free shell structure. Bend-free design can be obtained by suppressing both bending moments and curvature changes in a structure. 2

In this way, it is possible to use the full load carrying potential of structures and as a result have less weight. Currently, weight reduction of structures is of great interest especially in transportation systems due to its effects on fuel consumption, material cost and environmental issues. In composite shell structures, a bend-free state can be achieved in several ways. For symmetric laminates a moment-less state requires zero curvature change of the shell, while for asymmetric laminates a linear dependency of in-plane and out-ofplane deformations can result in a moment-less state [2]. Design of moment-less isotropic shells under internal pressure can be achieved for surfaces with constant radius of curvature such as circular cylinders or spherical domes [3]. For isotropic structures one of the most common methods to prevent bending is to locally increase the thickness. Many researchers have worked on shells with variable thickness [4–7]. The formulation for shells with variable thickness was first introduced by Reissner [4] in 1908. Reissner considered that thickness varies linearly in a shell. This concept later was used by several researchers in a variety of applications. In 1954, Horne [8] described shells with zero bending stresses under uniform axial loading. In 1966, Murthy and Kiusalaas [9] found the condition for an isotropic thin shell of revolution under axisymmetric loading in order to suppress bending. In 1970, Chicurel and Wu [10] studied closed shells of revolution under uniform pressure load. They varied shell thickness in order to suppress bending and as a result they presented a closed-form expression for the thickness as a function of polar angle. They also found a geometric q requirement for the moment-less condition of orthotropic shells ν under internal pressure as a function of νxy in which νxy and νyx are in-plane Poisson’s ratios. Chicurel yx later in 1972 [11], studied isotropic shells of revolution under uniform axial loading and he found an exact solution for shell configurations in a zero bending state. In the same year, Kulkarni and Fredrick [? ] used thickness variation to suppress bending in orthotropic shells of revolution with a given meridian geometry. In 1977, Farshad [12] obtained tension-less and moment-less domes using variable thickness. Using the variation of thickness method to suppress bending results in changing the geometry of the structure. But once the geometry and the shape of the shell is prescribed, the material properties of the shell are the only design parameters that can be tailored throughout the structure to make it bend-free. Another drawback of using thickness variation arises in manufacturing, in fact sometimes it is difficult to manufacture shell structures with variable thickness. Moreover, suppressing bending in a structure by this method, means that the thickness should be increased locally and as a result the weight also increases, which is not of interest. In 1969, Pao [3] studied the design of moment-less shells of revolution with axisymmetric loading condition such as in pressure vessels and liquid containers. He used the general form of the equilibrium equations to show that composites with variable elastic properties are needed to obtain moment-less vessels. As an illustrative example he presented the moment-less requirement by tailoring the elastic constants of the material for the pressure vessels with ellipsoidal bulkheads. In 1975, Nemirovskii and Starostin [13] designed general shells with bend-free states by neglecting the change in curvature in the compatibility equations. As such, they obtained necessary conditions on strains to make the structure bend-free. Then they satisfied these conditions in two ways. Firstly, by tailoring the material properties via changing the reinforcement volume fractions and secondly, by introducing secondary loadings to the structure. Another way to tailor the stiffness throughout composite structures is by using variable angle tow (VAT) laminates. In VAT laminates, the fibre orientation is tailored spatially in the surface of the structure which results in varying stiffness properties. Many researchers used VAT techniques to achieve several desired behaviours [14–18]. In 2012, White and Weaver [2] extended Nemirovskii and Starostin’s theory [13] by tailoring the material properties using fibre steering. They presented the general equations for bend-free shells that can be used for a wide variety of problems. They used stiffness tailoring techniques to steer fibre tows and tailor local properties. In their study, they represented the ply angles in composite laminates as functions of required stresses and strains. In order to evaluate the drawbacks and benefits of this method, they applied this technique to elliptical cylinders and ellipsoids of revolution, both under uniform internal pressure. By using bend-free theory, they showed that both bending moments and curvature changes can be completely suppressed for doubly curved shell structures. However, for elliptical cylinders, this theory fails. In such a structure, to suppress bending everywhere, additional secondary loading should also be applied. In this study, an ellipsoidal shell of revolution subjected to uniform internal pressure is designed for the bend-free state using the method described in [2]. Firstly, a governing equation for bend-free conditions is

3

obtained. In the governing expression, the stiffness is expressed using lamination parameters and material invariants [19]. This governing equation is then used for tailoring properties throughout the structure. Then, the lamination parameters that make the structure bend-free are extracted and the transformation from the lamination parameter design space to fibre direction design space is performed using a least-squares optimisation procedure. It is worth noting that tow-steering by automated fibre placement (AFP), in reality, generates gaps and/or overlaps, and consequently, thickness variation in laminates. In doubly-curved shells, even unidirectional tows cannot cover the entire surface without generating gaps or overlaps. In this study, however, the effects of gaps/overlaps are not considered, partly due to the anticipation of new manufacturing technology such as 3D printing that is under development. However, for the time being, to fully capture the real behavior of such structures, these effects should be considered in practical designs. To verify the bend-free state of the design, the structure has been modelled using the commercial Finite Element Analysis (FEA) software ABAQUS. After performing mesh sensitivity analysis and validating the model using analytical solutions, the bend-free designed structure is analysed. A new criterion for verifying the bend-free state of the structure that is independent of the amount of internal pressure is suggested. It is the first time in the literature that such a definition is proposed for bend-free states of the structure. Since the bend-free state is both moment-less and curvature-less, this criterion includes two separate parts that control both curvature changes and bending moments upon loading. The proposed criterion defines bending moment and curvature change based on the stresses and strains in the structure. Stresses and strains are direction dependent parameters and so as to ensure no bending in the structure occurs the proposed criterion is assessed in all directions. To check the validity of the bend-free design, the states of bending moment and curvature change due to internal pressure in the baseline (isotropic case) and tailored structures are assessed using FEA and compared with each other. Finally, in order to extend our knowledge of bend-free design and its advantages, a failure analysis is performed to examine the performance of this new design. This paper is organised as follows. In Section 2 , the methodology for designing bend-free ellipsoids of revolution is presented. Subsection 3.1 presents the criteria for verifying the bend-free state. The modelling procedure is described in Subsection 3.2. Numerical results and discussions are presented in Subsection 3.3. Section 4 deals with the failure study and provides a comparison of maximum allowable pressure for structures with bend-free design and conventional materials and finally, in Section 5 a summary and conclusions are presented. 2. Methodology An ellipsoid of revolution is a quadratic surface generated by revolving an ellipse about one of its principal axes. An oblate ellipsoid of revolution with major and minor axes as 2a and 2b is considered and shown in Fig. 1.

4

z θ ϕ

R r

Z

t

Y X (a) Geometry of ellipsoid of revolution.

(b) Position vectors in differential element of the shell.

Fig. 1 Fibre direction designed for bend-free state.

Each arbitrary point of the shell is defined by (ϕ, θ, z) where ϕ and θ are orthogonal curvilinear coordinates in longitudinal and latitudinal directions that span the middle surface and z is the coordinate in the thickness direction. Assume r and R are position vectors of the points on the middle surface and arbitrary surface respectively as shown in Fig. 1. The distance between 2 points (ϕ, θ, 0) and (ϕ + dϕ, θ + dθ, 0) is determined by 2 (ds)2 = dr · dr = αϕ (dϕ)2 + αθ2 (dθ)2 (1) √ √ where αϕ = gϕϕ and αθ = gθθ are the metric tensor coefficients. In this definition, gϕϕ = g ϕ · g ϕ and gθθ = g θ · g θ where g ϕ and g θ are vectors tangent to the ϕ and θ coordinate lines respectively and are determined by g ϕ = ∂r/∂ϕ = r ϕ and g θ = ∂r/∂θ = r θ . In this study, the shell is considered to be thin and shallow. Therefore, the kinematic term (z/R) is assumed to be zero. By assuming a linear strain field and Kirchhoff’s hypothesis [1], equilibrium equations can be expressed as  ∂(αϕ Nϕθ ) ∂(αθ Nϕ ) ∂α αϕ αθ θ  + ∂θϕ Nϕθ − ∂α  ∂ϕ + ∂θ ∂ϕ Nθ + rϕ Qϕ = αϕ αθ qϕ  ∂(αθ Nϕθ ) ∂(αϕ Nθ ) ∂αϕ αϕ αθ θ (2) + ∂θ + ∂α ∂ϕ ∂ϕ Nϕθ − ∂θ Nϕ + rθ Qθ = αϕ αθ qθ  2 2 2  α α ( Nϕ + Nθ ) = ∂ Mϕ + ∂ Mϕθ + ∂ Mθ + α α P ϕ θ

rϕ

rθ

∂ϕ2

∂ϕ∂θ

∂θ 2

ϕ θ

where rϕ and rθ are the radii of curvature in ϕ and θ directions, Qϕ and Qθ are the out-of-plane resultants, qϕ and qθ are the in-plane tractions and P is the internal pressure. Compatibility equations can also be expressed as      1 ∂(αθ εθ ) ∂(αϕ γϕθ ) ∂αθ ∂αϕ αϕ ∂ 1 1   − + + ε + γ + − γϕθ + ψ1 = 0  ϕ ϕθ   rϕ ∂ϕ ∂θ ∂ϕ ∂θ 2 ∂θ rϕ rθ         1 ∂(αϕ εϕ ) ∂(αθ γϕθ ) ∂αϕ ∂αθ αθ ∂ 1 1    − + + ε + γ + − γϕθ + ψ2 = 0 ϕ ϕθ r ∂θ ∂ϕ ∂θ ∂ϕ 2 ∂ϕ rθ rϕ θ     ∂ 1 ∂(αθ εθ ) ∂(αϕ γϕθ ) ∂αθ ∂αϕ   − + + ε + γ + ...  ϕ ϕθ  ∂ϕ αϕ ∂ϕ ∂θ ∂ϕ ∂θ         ∂ 1 ∂(αϕ εϕ ) ∂(αθ γϕθ ) ∂αϕ ∂αθ   − + + εϕ + γϕθ + ψ3 = 0  ∂θ αθ ∂θ ∂ϕ ∂θ ∂ϕ

5

shear

(3a) (3b) (3c)

where ψ1 , ψ2 and ψ3 are defined as   ∂κ ∂κ ∂αθ ∂αϕ  ψ1 = αθ θ − αϕ ϕθ + (κθ − κϕ ) − 2 κϕθ   ∂ϕ ∂θ ∂ϕ ∂θ    ∂κϕ ∂κϕθ ∂αϕ ∂αθ ψ2 = αϕ − αθ + (κϕ − κθ ) − 2 κϕθ  ∂θ ∂ϕ ∂θ ∂ϕ    κϕ κθ    ψ3 = αϕ αθ ( r + r ) θ

(4a) (4b) (4c)

ϕ

In this study, it is assumed that the curvature changes of the shell’s mid-plane are zero. κϕ = κθ = κϕθ = 0

(5)

Based on these assumptions and according to the linear constitutive equations for a laminated composite, if the changes in curvatures are all zero then a moment-less state demands that Bε is zero as satisfied (in a sufficient way) by symmetric laminates. N = Aε + Bκ M = Bε + Dκ

(6)

Substituting Eq. (5) into compatibility equations Eq. (3) and considering bend-free states, results in d(rθ εθ ) + (rθ εϕ − rϕ εϕ ) cot ϕ = 0 dϕ

(7)

for moment-less and curvature-less states. A simple solution of Eq. (7) is rθ εϕ = rϕ εθ

(8)

Using the equilibrium equations shown in Eq. (2) for this shell with a moment-less state, which only has inplane stress resultants and is a classical application of the membrane hypothesis, the in-plane stress resultants can be found as  P a2  Nϕ = 2b∆ 12 (9) Nθ = Nϕ (2 − ∆)   Nϕθ = 0 where ∆= rθ =

rθ a = ( )2 sin2 ϕ + cos2 ϕ rϕ b a2

(10)

1 2

b∆ For a moment-less and curvature-less state (M = 0, κ = 0) Eq. (6) can be rewritten as N = Aε = (Aiso + Aani )ε where Aiso and Aani are isotropic and anisotropic components of the A matrix where   U1 U4 0 Aiso = t U4 U1 0  0 0 U5  A  1 A A A V1 U2 + V2 U3 −V2A U3 2 V3 U2 + V4 U3 −V2A U3 −V1A U2 + V2A U3 12 V3A U2 + V4A U3  Aani = t  1 A 1 A A A −V2A U3 2 V3 U2 + V4 U3 2 V3 U2 + V4 U3 6

(11)

(12)

and Ui and ViA are the material invariants (independent of orientation angles) and in-plane lamination parameters (dependent on orientation angles), respectively as introduced by Tsai et al. [19]. Eq. (11) can also be written as     Nϕ εϕ  Nθ  = A  ε θ  (13) Nϕθ γϕθ Because of the symmetry of loading and geometry, the structure is shear-free, and therefore, Nϕθ and γϕθ vanish. Substituting Eq. (12) into Eq. (13) gives      A   Nϕ U1 U4 V1 U2 + V2A U3 −V2A U3 εϕ =t + (14) Nθ U4 U1 εθ −V2A U3 −V1A U2 + V2A U3 Using Eq. (8) and (9), Eq. (14) can be expressed as      A 1 U1 U4 V U + V AU Nϕ =t + 1 2 A 2 3 2−∆ U4 U1 −V2 U3

−V2A U3 A −V1 U2 + V2A U3



  ∆ εθ 1

(15)

which can be rewritten as Nϕ = (U1 ∆ + U4 + V1A U2 ∆ + V2A U3 ∆ − V2A U3 )εθ t

(16a)

Nϕ (U4 ∆ + U1 − V2A U3 ∆ − V1A U2 + V2A U3 )εθ = t (2 − ∆)

(16b)

As the left-hand sides of Eqs. (16) are identical to each other, by equating the right-hand sides and after some manipulations the governing equation for a bend-free ellipsoid of revolution under internal pressure can be obtained as:     U4 − U3 V2A 2(U4 − U3 V2A ) U1 − U2 V1A + U3 V2A − ∆2 + 2∆ 1 − + − =0 (17) U1 + U2 V1A + U3 V2A U1 + U2 V1A + U3 V2A U1 + U2 V1A + U3 V2A In the following discussion, design of ellipsoids of revolution under uniform internal pressure is described. This approach is divided into two main steps. Firstly, the optimum stiffness distribution for the bend-free structure is obtained using Eq. (17), which has been found for a bend-free state in terms of lamination parameters. Secondly, the optimum fibre direction, corresponding to the stiffness obtained from the previous step is found. More details on how to obtain the tow trajectory using lamination parameters can be found in [20, 21]. For a prescribed geometry, i.e. known (a, b, t) and a given ϕ, Eq. (17) becomes the equation of a straight line in the V1A − V2A plane. Fig. 2 shows example lines (ϕ = 10◦ , 60◦ and 90◦ ). It is also worth noting that since the lamination parameters are trigonometric functions, they depend on each other. This dependency is introduced with ( Constraint1 : 2(V1A )2 − (V2A ) − 1 ≤ 0 (18a) Constraint2 : V2A − 1 ≤ 0

(18b)

which limit the feasible region for V1A and V2A [22, 23]. This feasible region for V1A and V2A that makes the structure moment-less and curvature-less, is also shown in Fig. 2.

7

1.5

𝐕𝟐𝐀 Constraint1

Feasible reagion 1

Constraint2 0.5

φ=90

𝐕𝟏𝐀

0 -1.5

-1

-0.5

0

0.5

1

1.5

-0.5

φ=60 -1

φ=10 -1.5

Fig. 2 Feasible bend-free region for current study.

As finding a unique solution is of interest, the intersection of the green lines with the lower boundary of the feasible region was considered to obtain the values of the lamination parameters. As shown in Fig. 2, when ϕ varies from 0◦ at the pole (top and bottom of structure) to 90◦ at the equator, the fibre orientation angle changes from 45◦ (V1A = 0 and V2A = −1) to 0◦ (V1A = 1 and V2A = 1). Up until now, the optimum lamination parameters are found for the whole structure. These lamination parameters should be transformed to fibre orientation design space which means that the fibre orientation corresponding to optimum stiffness should be determined for the whole domain in the structure. Lamination parameters are functions of the fibre direction as defined by Z 1 1 A [cos 2θ cos 4θ] dˆ z (19) V = 2 −1 where zˆ = 2 zt is the normalised thickness coordinate. A least squares optimisation process is used to find the angles corresponding to the designed lamination parameters. The objective function for this optimisation is defined as F = (V1A − V1∗ )2 + (V2A − V2∗ )2

(20)

In particular, the objective function is the least-square distance between the optimum lamination parameters (V1∗ and V2∗ found in the previous step) and the lamination parameters defined as a function of fibre orientation. It is assumed that the laminate is balanced, symmetric with 4 layers, i.e. [θ1 , −θ1 ]s , the fibre direction is obtained and shown in Fig. 3. It is worth noting that the fibre directions are the same at locations with the same distance from the equator of the ellipsoid of revolution (e.g. θ1 has the same value for ϕ = 85◦ and ϕ = 95◦ ).

8

50 45

Fibre direction

40 35

φ= 0°

30

25 φ= 90°

20 15

φ= 180°

10 θ1

5 0 0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 φ

(a) θ1 based on the ϕ.

(b) 3D schematic of θ1 over the structure.

Fig. 3 Fibre direction designed for bend-free state.

The constraints defined in Eq. (18), bound the feasible region of the eccentricity (e =

q 1 − ( ab )2 ). Fig.

4 shows the bend-free locus of V1A and V2A for different ratios of (a/b) at three different locations, (ϕ = 10◦ , 45◦ and 90◦ ). As observed, ϕ = 90◦ (equator) has the highest sensitivity to variation of a/b in keeping the bend-free lamination parameters within the feasible region.

1≤

𝑎 𝑏

≤1.5

(b) ϕ = 45◦ .

1

1

(a) ϕ = 10◦ .

(c) ϕ = 90◦ .

Fig. 4 Bend-free locus of V1A and V2A for different ratios of (a/b).

Considering the critical value for lamination parameter which is V1A = V2A = 1 when ϕ is 90◦ as shown in Fig. 2, for a given material, the maximum value of eccentricity, which guarantees a bend-free state, is obtained from Eq. (17) as s U1 + U2 + U3 p eM ax = 1 − (21) U1 + U2 + 2U3 − U4 + 2U1 U2 + 2U22 + 2U3 U2 + U32 − 2U3 U4 + U42 The material used in this study is carbon/epoxy with E11 = 140 GP a, E22 = 10 GP a, G12 = 5 GP a and υ12 = 0.3. For this material, the maximum value of eccentricity is equal to 0.696. In other words, for this specific material this approach can make the structure bend-free only if the eccentricity does not exceed 0.696. At the equator—which has the maximum difference of curvature in the ϕ and the θ directions— ε eccentricities beyond the eM ax = 0.696 generate strain states in which εϕθ exceeds a threshold (see Eqs. (8) and (10)) that cannot be maintained with any material system [2]. In order to observe a noticeable improvement in the bend-free state of the structure compared with an isotropic case, the maximum value for eccentricity (0.696) is chosen. This is due to the fact that the isotropic sphere is bend-free by its nature and bending increases in the structure by increasing the eccentricity.

9

3. Numerical verification of bend-free state 3.1. Bend-free criteria In this section a criterion for the bend-free state of a pressurised ellipsoid of revolution independent of the magnitude of internal pressure is proposed. An arbitrary point of the structure is considered to be bend-free if and only if at that point the bending moments and curvature changes are both negligible in all directions. Therefore, it is necessary to define two quantities describing the amount of bending moment and curvature change independent from the amount of internal pressure. To this end, two factors (α and β) are defined as top σ − σ bot ij ij α = top with i, j = ϕ, θ (22) bot σij + σij top  − bot ij ij β = top ij + bot ij

with i, j = ϕ, θ

(23)

top These expressions are valid at the ply level in which σij and top ij are stress and strain at the top of each bot bot individual ply, while σij and ij are stress and strain at its bottom. The term α is defined as a measure of the bending moment in the structure, while β measures the curvature change. By these definitions, some extreme cases can be illustrated as shown in Table 1.

10

Table 1 Extreme cases of α and β factors.

top bot σij = σij >0

α=0

bot top ij = ij >0

β=0

top bot σij = σij <0

α=0

bot top ij = ij <0

β=0

top top bot bot | σij |=| σij | and σij σij <0

α = Inf

top bot bot | top ij |=| ij | and ij ij <0

β = Inf

top bot σij = 0 or σij =0

α=1

bot top ij = 0 or ij = 0

β=1

Pure tension

Pure compression

Pure bending

Combined bending and tension or compression

To assess the bend-free state using these two factors, a maximum (or threshold) limit was prescribed for each factor, below which the structure is considered to be bend-free. Here it is considered that when α and β are less than 0.01 for all layers, the structure is moment-less and curvature-less, respectively. Therefore, when both factors are less than 0.01 the structure is bend-free. Up until now, α and β have been expressed as functions of stresses and strains evaluated in ϕ and θ directions. However, both stress and strain values change in different directions and consequently α and β depend on ϕ and θ directions. In order to ensure that there is no curvature change and bending moment in any direction of the structure, α and β have to be evaluated in all directions. Therefore, α and β defined in Eqs. (22) and (23) are calculated using the stresses and strains evaluated in all directions by γ changing from 0◦ to 360◦ in 1◦ steps as shown in Fig. 5.

11

Fig. 5 Directions where stress and strain are evaluated.

Finally, to have a better idea of the moment-less, curvature-less and bend-free state of the structure, results are presented in forms of area percentage. BF% is the percentage of bend-free area to the total area of the structure. Likewise, M L% and CL% are the percentage of moment-less area and the percentage of curvature-less area, respectively, defined as BF% =

Aα ∩ Aβ · 100 Awhole

M L% = CL% =

Aα

(24)

· 100

(25)

Aβ · 100 Awhole

(26)

Awhole

where Aα and Aβ are the respective areas of the structure where α and β are less than 0.01, Awhole is the whole area of the structure. 3.2. Numerical Modelling To verify the bend-free states resulting from our analytical approach, a FEA was performed for the pressurised ellipsoids. In particular, an isotropic ellipsoid of revolution with a = 500 mm and b = 358 mm, subjected to uniform internal pressure P = 100 kP a, was investigated. Aluminium with E = 70 GP a and υ = 0.33 was chosen. Shell elements S8R5 and ST RI65 were used in the commercial software ABAQUS [24]. A mesh convergence study was also performed to ensure that FEA results were not mesh size dependent. In order to minimise distortional locking, during the mesh refinement process, the aspect ratio of the finite elements was optimised to be as close as possible to a value of 1. The mesh convergence study was done by studying the strain variation for three different mesh sizes. In particular, the isotropic ellipsoid of revolution described above has been meshed with three different mesh sizes. Details of these mesh are reported in Table 2, in which nelmϕ and nelmθ are the number of elements in the ϕ and θ directions, respectively. It should be noted that the number of elements along ϕ varies with the geometry of the ellipsoid for keeping the aspect ratio of the FEA equal to 1, as described above. As shown in Fig. 6, the strain has the smoothest trend, in both ϕ and θ directions, for Model3.

12

Table 2 Details of the mesh used for investigating the strain trend. Model 1 2 3

nelmθ 72 144 360

nelmϕ 158 312 716

370

400 ε𝜑-Model1 εϕ-Model1 ε𝜑-Model2 εϕ-Model2 εϕ-Model3 ε𝜑-Model3

360

350

εθ-Model1 εθ-Model1

300

εθ-Model2 εθ-Model2

εθ-Model3 εθ-Model3

250

200

εθ (µε)

340

εϕ (µε)

350

330 320

150 100

50 0

310

-50 300

-100 -150

290 0

15

30

45

60

75

90

φ (deg)

105

120

135

150

165

0

180

15

30

45

60

75

90

φ (deg)

105

120

135

150

165

180

Fig. 6 Strain variation in the structure.

40

40

35

35

30

30

25

25

Nθ (KN/m)

Nϕ (KN/m)

Fig. 7 shows the numerical results in terms of normal stress resultants compared with analytical solutions found using Eq. (9). As observed the maximum error between numerical and analytical results is 0.02% in ϕ direction and 1.02% in θ direction for Model3 as described in Table 2.

20 15

10

Numerical Nθ Numerical N𝜽

20 15 10

Analytical N𝜑 Numerical N𝜑

5

Analytical N𝜽 Analytical Nθ

5 0

0 0

15

30

45

60

75

90 105 φ (deg)

120

135

150

165

180

0

15

30

45

60

75

90 105 φ (deg)

120

135

150

165

Fig. 7 Comparison of analytical and numerical results for Nϕ and Nθ .

Based on the results shown here, Model3 was selected for FEA in this study. 3.3. Results and Discussion In this section, FEA results for three different ellipsoids of revolution, designed for exhibiting bend-free states, are presented. In order to gain more insight regarding the validity of the proposed design method, the numerical results from these three ellipsoids of revolution were compared with their corresponding isotropic baseline models. The ellipsoids of revolution analysed in this study are listed in Table 3. These structures were modelled in ABAQUS and through-thickness stresses and strains at an example location are plotted and shown in Figs. 8 to 9. Here, the results are plotted for ϕ = 31.41◦ which is approximately located at one third of the distance from the pole. As shown, the rate of through-thickness variation of stress and strain is reduced by stiffness tailoring which indicates that structures with optimally designed VAT designs have less 13

180

bending moment and curvature changes compared with their isotropic counterparts. It is also observed that the change of slope increases by increasing the thickness. Table 3 Details of the ellipsoids of revolution. Model 1 2 3 4 5 6

Material Isotropic VAT for Bend-free Isotropic VAT for Bend-free Isotropic VAT for Bend-free

14

Thickness (mm) 0.1 0.1 1 1 10 10

t [mm]

Stress in ϕ direction

Strain in ϕ direction 0.5

0.5

VAT

VAT

0.1

ISO

ISO

0.25

z/t

z/t

0.25

0

0

-0.25

-0.25

-0.5

-0.5

300

305

310 σϕ (MPa)

315

3000

320

0.5

3200

3400 3600 εϕ (µε)

3800

0.5 VAT

1

VAT

ISO

ISO

0.25

0

z/t

z/t

0.25

-0.25

0

-0.25

-0.5

-0.5 30

30.5

31 σϕ (MPa)

31.5

32

300

0.5

320

340 360 εϕ (µε)

380

400

0.5

10

VAT

VAT

ISO

ISO

0.25

0.25

z/t

z/t

4000

0

-0.25

0

-0.25

-0.5

-0.5 3

3.05

3.1 σϕ (MPa)

3.15

3.2

30

32

34 36 εϕ (µε)

38

Fig. 8 Through-thickness σϕ and ϕ for isotropic and bend-free (VAT) design at ϕ = 31.41◦ .

15

40

t [mm]

Stress in θ direction

Strain in θ direction 0.5

0.5

VAT

VAT

0.1

ISO

ISO

0.25

z/t

z/t

0.25

0

-0.25

-0.25

-0.5

-0.5 220

225

230 235 σθ (MPa)

1500

240

0.5

2000

2500 εθ (µε)

3000

VAT

VAT

ISO

ISO

0.25

0.25

z/t

0

-0.25

0

-0.25

-0.5

-0.5 22

22.5

23

23.5

24

150

200

250

300

350

εθ (µε)

σθ (MPa)

0.5

0.5 VAT

VAT

10

ISO

ISO

0.25

z/t

0.25

z/t

3500

0.5

1

z/t

0

0

0

-0.25

-0.25

-0.5

-0.5 2.2

2.25

2.3 2.35 σθ (MPa)

15

2.4

20

25 εθ (µε)

30

Fig. 9 Through-thickness σθ and θ for isotropic and bend-free (VAT) design at ϕ = 31.41◦ .

16

35

Numerical results found by ABAQUS were subsequently used for verifying the bend-free state using the criteria proposed in Section 3.1. The factors α and β were evaluated for all layers of VAT ellipsoids of revolution as well as for isotropic ones. Fig. 10 shows the contour plot of maximum amount of α, while Fig. 11 shows the contour plot of maximum amount of β in the structures. Considering those results shown in Figs. 10 and 11 for the isotropic ellipsoids with t = 1 mm (Model3 ) and t = 10 mm (Model5 ), a relatively large amount of bending moment and curvature changes are generated at the equator. The explanation for this behaviour reflects the largest difference between two geometrical radii of curvature in this area. For the same structures, this value decreases by reducing the difference between the two radii of curvature. It can also be observed that by increasing the thickness, the bending moment and curvature changes increase, especially at the equator. Based on these results, it is possible to totally suppress bending moment and curvature changes for the thickness of 0.1 mm and 1 mm using VAT design. Although it is not possible to fully suppress the bending moment and curvature change in all cases, in comparison with the isotropic base-line case, the tailored structures always exhibit less bending moment and curvature changes. As an example, for an ellipsoid of revolution with thickness of 10 mm, the structure is not fully bend-free, however, the maximum amount of αM ax and βM ax factors for a tailored VAT structure representing a bend-free state (χM ax = M ax(αM ax , βM ax ) = 0.021) is approximately four orders of magnitude lower than its corresponding isotropic base-line case (χM ax = M ax(αM ax , βM ax ) = 142.474). Fig. 12 shows the contour plots for the intersection of maximum α and β for each model. Considering results shown in Figs. 10 and 11, it is observed that the curvature-less area is smaller than the moment-less area. Therefore, the contour plot for the intersection of α and β is similar to the contour plot of β.

17

t [mm]

Isotropic

VAT designed for bend-free state

0.1

1

10

Fig. 10 Contour plots of maximum α factor for each model (Dimensions are in metre).

18

t [mm]

Isotropic

VAT designed for bend-free state

0.1

1

10

Fig. 11 Contour plots of maximum β factors for each model (Dimensions are in metre).

19

t [mm]

Isotropic

VAT designed for bend-free state

0.1

1

10

Fig. 12 Contour plots of χ=M ax(α, β) for each model (Dimensions are in metre).

For each model, moment-less, curvature-less and bend-free percentage areas, defined with Eqs. (24) to (26), are shown as bar charts in Fig. 13. A summary of results is also shown in Tables 4 to 5. Results show the potential of tailored composite structures in suppressing bending deformations and stresses compared with their isotropic counterparts with the same geometry and thickness. However, by increasing the thickness to 10 mm, it is not possible to achieve a fully bend-free state even by tailoring the stiffness using VAT. The possible explanation for this behaviour is the non-satisfaction of one of the assumptions used in this method: 20

Moment-less Percentage

100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

Isotropic layer1

layer2 layer3 layer4

Curvature-less Percentage

(a) Moment-less area percentage. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

Isotropic Layer1

Layer2 Layer3 Layer4

Bend-free Percentage

(b) Curvature-less area percentage. 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

Isotropic Layer1

Layer2 Layer3 Layer4

(c) Bend-free area percentage.

Fig. 13 Moment-less, Curvature-less and Bend-free area percentage.

21

the shell has to be thin and shallow. Simply put, the ratio of t/R should be sufficiently small. For thicknesses of 0.1 mm, 1 mm and 10 mm, the ratio t/R are 1.435e-4, 1.435e-3 and 1.435e-2, respectively. It can be concluded from these results that t/R = 1.435e-2 is not sufficiently small for this specific doubly-curved structure and the structure can be considered to be a thick shell. Table 4 Moment-less, Curvature-less and Bend-free area percentages for VAT structures. Layer 1

2

3

4

Area % Moment-less Curvature-less Bend-free Moment-less Curvature-less Bend-free Moment-less Curvature-less Bend-free Moment-less Curvature-less Bend-free

Model2 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%

Model4 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100% 100%

Model6 61.9% 56.8% 56.8% 61.9% 56.8% 56.8% 63.6% 56.8% 56.8% 63.7% 58.4% 58.4%

Table 5 Moment-less, Curvature-less and Bend-free area percentages for isotropic structures. Area % Moment-less Curvature-less Bend-free

Model1 100% 64.0% 64.0%

Model3 77.0% 47.6% 47.6%

Model5 0% 0% 0%

In order to have a better insight, in-plane stress and strains at the shell’s poles are plotted in terms of ϕ in Fig. 14. It should be noted that results are symmetric with respect to the equator and therefore they are only plotted for ϕ between 0◦ and 90◦ . As shown in Fig. 14, when the thickness is 0.1 mm, stresses in ϕ and θ directions for VAT structures are similar to isotropic structures. This is due to the fact that for a thickness of 0.1 mm, then the isotropic structure is already moment-less because of its relative thinness. However, for this thickness, the strain variation is different for VAT and isotropic structures. In the ϕ direction, the strain increases at the pole (ϕ = 0◦ ) and decreases at the equator (ϕ = 90◦ ). In the θ direction, the strain increases over the entire domain. For a thickness of 1 mm, the stress in the ϕ direction slightly changes for VAT, while the strain variation is totally different for VAT and isotropic structures. Strain in the θ direction increases for VAT in comparison with the isotropic case. For a thickness of 10 mm, differences between VAT and isotropic structures are more significant. The difference between σ top and σ bot decreases in both ϕ and θ directions for VAT structures compared to their isotropic counterparts. Strain in the ϕ direction increases at the pole while decreases at the equator. The difference between strains at the top and bottom decreases for VAT compared with the isotropic case. Strain in the θ direction, increases for VAT compared with the isotropic case when the thickness is 10 mm.

22

σϕ (MPa)

0.1

σϕ-bot-ISO σϕ_bot_ISO σϕ-top-ISO σϕ_top_ISO σϕ-bot-VAT σϕ_bot_VAT σϕ-top-VAT σϕ_top_VAT

330 310 290

4350 3800 εϕ (µε)

350

270

1600 ϕ (deg)

350

75

90

σθ-bot-ISO σθ_bot_ISO σθ-top-ISO σθ_top_ISO σθ-bot-VAT σθ_bot_VAT σθ-top-VAT σθ_top_VAT

280 σθ (MPa)

60

210 140 70 0

σϕ (MPa)

0

30

45

ϕ (deg)

36 34 32 30 28 26 24 15

30

45

ϕ (deg)

35 30 25 20 15 10 5 0

75

60

75

15

30

45

ϕ (deg)

3.7

60

75

3.1

ϕ (deg)

3.5

60

75

30

2.8

2.1 1.4

0.7 0 0

15

30

45

ϕ (deg)

60

75

90

30

15

30

60

75

90

45

60

75

90

ϕ (deg)

15

30

45

60

75

90

45

60

75

90

ϕ (deg)

0

15

30

ϕ (deg)

49 39 29 19 9 -1 -11

εεθ_bot_ISO θ-bot-ISO εεθ_top_ISO θ-top-ISO εεθ_bot_VAT θ-bot-VAT εεθ_top_VAT θ-top-VAT

0

15

30

45

ϕ (deg)

Fig. 14 Stress and strain plots in ϕ direction.

23

45

ϕ (deg)

εϕ-bot-ISO εϕ_bot_ISO εϕ-top-ISO εϕ_top_ISO εϕ-bot-VAT εϕ_bot_VAT εϕ-top-VAT εϕ_top_VAT

20

90

σθ-bot-ISO σθ_bot_ISO σθ-top-ISO σθ_top_ISO σθ-bot-VAT σθ_bot_VAT σθ-top-VAT σθ_top_VAT

90

40

εθ (µε)

45

75

50

0 30

60

εεθ_bot_ISO θ-bot-ISO εεθ_top_ISO θ-top-ISO εεθ_bot_VAT θ-bot-VAT εεθ_top_VAT θ-top-VAT

10

15

15

480 380 280 180 80 -20 -120 0

2.2 0

45

ϕ (deg)

εϕ-bot-ISO εϕ_bot_ISO εϕ-top-ISO εϕ_top_ISO εϕ-bot-VAT εϕ_bot_VAT εϕ-top-VAT εϕ_top_VAT

0

2.5

σθ (MPa)

450 400 350 300 250 200 150

90

2.8

30

εεθ_bot_ISO θ-bot-ISO εεθ_top_ISO θ-top-ISO εεθ_bot_VAT θ-bot-VAT εεθ_top_VAT θ-top-VAT

0

90

σϕ-bot-ISO σϕ_bot_ISO σϕ-top-ISO σϕ_top_ISO σϕ-bot-VAT σϕ_bot_VAT σϕ-top-VAT σϕ_top_VAT

3.4

15

4500 3550 2600 1650 700 -250 -1200

90

σθ-bot-ISO σθ_bot_ISO σθ-top-ISO σθ_top_ISO σθ-bot-VAT σθ_bot_VAT σθ-top-VAT σθ_top_VAT

0

σϕ (MPa)

60

σϕ_bot_ISO σϕ-bot-ISO σϕ_top_ISO σϕ-top-ISO σϕ_bot_VAT σϕ-bot-VAT σϕ-top-VAT σϕ_top_VAT

0

σθ (MPa)

15

0

εθ (µε)

45

εϕ (µε)

30

εθ (µε)

15

εϕ (µε)

0

10

εϕ-bot-ISO εϕ_bot_ISO εϕ-top-ISO εϕ_top_ISO εϕ-bot-VAT εϕ_bot_VAT εϕ-top-VAT εϕ_top_VAT

2700

2150

250

1

3250

60

75

90

4. Failure analysis and parametric study To gain more insight into the bend-free design and its advantages compared to conventional material systems such as isotropic or laminated composites with constant fibre orientations, a failure study was performed. With this aim, the ellipsoid of revolution with semi-major and semi-minor axes as a = 500 mm, b = 358 mm and t = 0.1 mm and a balanced symmetric laminate with 4 layers, i.e. [θ1 , −θ1 ]s was considered. The maximum uniform internal pressure for the bend-free structure was found using a first-ply failure based on the Tsai-Wu failure criterion. Based on this criterion, material is considered to fail, when H1 σ1 + H2 σ2 + H6 τ12 + H11 σ1 2 + H22 σ2 2 + H66 τ12 2 + 2H12 σ1 σ2 <1

(27)

is not satisfied, where the coefficients of Tsai-Wu failure criterion can be determined based on the material properties given by 1 1 H1 = T − C (28) (σ1 )ult (σ1 )ult H11 = H2 =

1 (σ1T )ult (σ1C )ult 1

(σ2T )ult

H22 =

−

1 (σ2C )ult

(29) (30)

1 (σ2T )ult (σ2C )ult

(31)

H6 = 0

(32)

H66 =

1 (τ12 )2ult

H12 = −

1 2(σ1T )2ult

(33) (34)

where (σ1T )ult and (σ1C )ult are tensile and compressive strengths along the fibre direction, (σ2T )ult and (σ2C )ult are tensile and compressive strengths transverse to the fibre direction and (τ12 )ult is the shear strength for carbon/epoxy. This criterion was assessed by evaluating the stress state in each layer of composite found numerically by ABAQUS. To have a fair comparison, maximum internal pressure was also found for structures made out of unidirectional composite. In order to cover all possible constant fibre orientations, 19 structures were studied with different constant fibre orientations varying from 0◦ to 90◦ in steps of 5◦ . The composite material considered in this study is carbon/epoxy with the material properties shown in Table 6. Table 6 Material properties of Carbon/Epoxy. Property Density (σ1T )ult (σ1C )ult (σ2T )ult (σ2C )ult (τ12 )ult

Unit kg/m3 GPa GPa MPa MPa MPa

Value 1600 1.54 1.12 50 250 0.5(σ2T )ult

Maximum uniform internal pressure was also found for the structure that is made from isotropic material. To this end, von Mises failure theory was considered for aluminium material. The yield strength was considered to be 110 M P a for aluminium with the density of 2700 kg/m3 . The von Mises failure criterion

24

is calculated for two extreme cases, at both inner and outer surfaces of the structure. The von Mises failure criterion considers the material to fail when σν (equivalent von Mises stress) q 2 σν = σ12 − σ1 σ2 + σ22 + 3τ12 (35) is greater than the material’s yield strength. The maximum allowable internal pressure found based on the considered failure theories are shown in Table 7 as well as Fig. 15 in which the CompX represents the structure made from unidirectional composite with θ1 = X ◦ . Table 7 Maximum allowable uniform pressure per weight for each model. Model Comp0 Comp5 Comp10 Comp15 Comp20 Comp25 Comp30 Comp35 Comp40 Comp45 Comp50 Comp55 Comp60 Comp65 Comp70 Comp75 Comp80 Comp85 Comp90 VAT ISO

Max Pressure (kP a) 7.3 7.4 7.5 7.8 8.4 9.7 12.5 18.7 26.9 20.5 17.2 15.7 15.3 14.1 13.5 13.4 13.7 14 14.2 81.8 31.5

Max Pressure/Weight (kP a/kg) 20.24 20.52 20.80 21.63 23.30 26.90 34.67 51.86 74.60 56.85 47.70 43.54 42.43 39.10 37.44 37.16 37.99 38.83 39.38 226.85 51.77

25

Pressure/Density (kPa/kg)

250

200

150

100

50

0

Fig. 15 Maximum allowable uniform pressure per weight for each model.

As shown in Fig. 15, the VAT structure designed to be bend-free exhibits the best performance by some considerable margin with respect to the best unidirectional composite laminates. It is worth noting in Fig. 15 that the trend of maximum allowable pressure to weight increases for unidirectional composites by increasing θ1 from 0◦ to 40◦ . A unidirectional composite structure with θ1 = 40◦ has the best performance amongst all composites with constant fibre orientations. Similar analysis for a quasi-isotropic (QI) laminate [±45/0/90]s with the same total thickness results in the maximum pressure to failure of about 74.8 kP a which is approximately 9% less than the bend-free VAT design. It is worth noting that not only is the QI laminate not necessarily bend-free, but also it belongs to a much larger design space in which [θ1 / − θ1 ]s layups are sub-optimal. Therefore, to have a fair comparison, this QI laminate should be compared with an optimum VAT laminate with a [±θ1 /θ2 /θ3 ]s stacking sequence of which the QI layup is a member and has more potential for improvement, but is out of scope of this study. The cross section of structures made from isotropic, composites with constant fibre orientations (θ1 changing from 0◦ to 90◦ with the step of 10◦ ) and also tailored VAT design under uniform internal pressure of 100 kP a are shown in Fig. 16.

26

0.5

θ1= 90°

Increasing θ1 in composites with constant fibre orientations from 0° to 90°

0.4

0.3

θ1= 0° 0.2

0.1

0 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

-0.1

-0.2

-0.3

VAT Isotropic Composite with constant fibre orientations

-0.4

-0.5

Fig. 16 Cross section after and before applying pressure, initial cross section is shown in black.

As shown in this figure, the tailored structure with VAT, expands self-similarly under internal pressure due to the fact that it is curvature-less. An interesting aspect shown in Fig. 16, is that the displacement at the two poles smoothly changes direction by increasing the θ1 , from 0◦ to 90◦ . To be more specific, for θ1 = 0◦ , the structure contracts locally at poles under internal pressure while for θ1 = 90◦ , the structure expands at poles. This behaviour lies in the fact that for the structure with θ1 = 0◦ , the structure is stiffer in the ϕ direction while for structure with θ1 = 90◦ , the structure is stiffer in the θ direction. Fig. 17 demonstrates the top view of the structure and shows how it is easier for the structure with θ1 = 0◦ to fail at its pole in comparison with the structure with θ1 = 90◦ . It also explains the deformation shape for structures with θ1 = 0◦ and θ1 = 90◦ as shown in Fig. 16 at the poles. This confirm the results shown in Table 7 which states the maximum allowable internal pressure for unidirectional composite structure with θ1 = 90◦ is about twice that of the maximum allowable pressure for unidirectional composite with θ1 = 0◦ .

27

θ1 = 90°

ϕ b

Y

a X

θ

θ1 = 0°

Z Y

Y

X X Fig. 17 Comparison of structures with constant fibre orientations at 0◦ and 90◦ .

Fig. 18 shows the variation of Tsai-Wu criterion with ϕ for tailored structure with VAT, unidirectional composites with θ1 = 90◦ and θ1 = 0◦ and also the variation of von Mises criterion with ϕ for the isotropic case. It should be noted that each structure is under its maximum allowable pressure based on its appropriate failure criterion. The most notable effect described in this table is the variation of normalised Tsai-Wu or von Mises stress values that smoothly decreases from poles to equator. This trend is independent of material properties and reaches the peak around the poles that can lead to failure at these areas and makes these areas critical for failure. Taken together, the findings of this section reveal the significant improvement in performance of tailored VAT compared with both isotropic material and conventional composites with constant fibre orientation.

28

Comp0

Comp90

1.2

1.2 Tsai-Wu Criterion-Layer1

1

Tsai-Wu Criterion-Layer2

0.8

Tsai-Wu Criterion-Layer3

0.6

Tsai-Wu Criterion-Layer4

0.8

Tsai-Wu

Tsai-Wu

1

0.4

0.6

Tsai-Wu Criterion-Layer1 Tsai-Wu Criterion-Layer2 Tsai-Wu Criterion-Layer3 Tsai-Wu Criterion-Layer4

0.4 0.2

0.2 0

0

-0.2

-0.2 0

20

40

60

80

ϕ

100

120

140

160

0

180

20

40

60

VAT

ϕ

100

120

140

160

180

ISO 1.2

1.2

Tsai-Wu Criterion-Layer1 Tsai-Wu Criterion-Layer2 Tsai-Wu Criterion-Layer3 Tsai-Wu Criterion-Layer4

0.8

1

0.8

Von Mises

1

Tsai-Wu

80

0.6

0.4

0.6 Von Mises Stress at inner Surface/Yield Strength Von Mises Stress at outer Surface/Yield Strength

0.4 0.2

0.2

0

0 0

20

40

60

80

ϕ

100

120

140

160

180

0

20

40

60

80

ϕ

100

120

140

160

Fig. 18 Failure criterion plot through ϕ direction.

5. Conclusion A pressurised ellipsoidal shell of revolution has been considered for designing bend-free states by tailoring material stiffness using fibre steering. To this end, an analytical model was used to achieve the bend-free state. A new definition of bend-free state independent of the amount of internal pressure has been introduced. Based on this definition, the bend-free state of isotropic and VAT ellipsoidal shells of revolution under internal pressure was assessed by performing finite element analysis. Results show that for pressurised ellipsoids of revolution, depending on the material properties, a bendfree state is achievable using fibre steered composite laminates up to a certain limit of eccentricity. Using carbon/epoxy material, bend-free states were found to be achievable up to an eccentricity of, e = 0.696. It is also worth noting that the proposed method to achieve fully bend-free states is valid for shells with sufficiently small thickness ratios (t/R ≤ 1.435e-3 in this study). Although it cannot fully suppress the internal pressure induced bending in thicker shells, stiffness tailoring can considerably reduce it compared with their isotropic counterparts. For instance, in this work approximately a 50% improvement of bend-free state was achieved for a shell with t/R =1.435e-2. Finally, the failure study reveals the exceptional performance of bend-free designs and provides a deeper insight in to these types of structure. The single most striking observation to emerge from the failure study is the improvement of 204% for maximum pressure to weight of the bend-free design compared with its best constant stiffness counterpart, i.e. θ1 = 40◦ . An improvement of 9% for maximum pressure to weight of a bend-free design was also found to be achievable in comparison with the quasi isotropic counterpart. Further research should be undertaken to consider the effect of gaps/overlaps and thickness variation that are inevitable in doubly-curved composite shells manufactured by AFP. Furthermore, this design method 29

180

can be generalized for obtaining bend-free states on thick doubly-curved shells. Finally, this method can be extended for designing bend-free thin walled structures with different complex geometries.

Acknowledgement The authors would like to thank Science Foundation Ireland (SFI) for funding Spatially and Temporally VARIable COMPosite Structures (VARICOMP) Grant No. (15/RP/2773) under its Research Professor programme.

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