Composite spherical shells at large deflections. Asymptotic analysis and applications

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Composite spherical shells at large deflections. Asymptotic analysis and applications Alexander Yu. Evkin Research and Development Department, Software for Structures, 81-1004, Townsgate Dr., Toronto, Canada

A R T I C LE I N FO

A B S T R A C T

Keywords: Composite spherical shell Asymptotic method Large deflections Energy barrier Design buckling load

The asymptotic method for spherical shells at large deflections was developed in the case of composite shell subject to external pressure. Reissner’s equations describing axially symmetric deformation of deep shells with arbitrary deflections and rotation angles were considered as initial ones. They were significantly simplified and asymptotic formulae for deformation energy, external pressure and stresses depending on deflection amplitude were obtained. The energy barrier criterion was applied for the composite shell for estimation of the shell metastability. Formula for design buckling pressure was derived for composite shells. It improves the significantly more conservative NASA SP-8032 recommendations. The full inversion of the shell was studied as well. Asymptotic formulae were obtained for evaluation of stresses and load deflection dependencies. They can be useful for design of diaphragms for positive expulsion propellant spherical tanks of launch vehicles and for other applications.

1. Introduction Our study is based on research published in papers [1–4]. In [1] a new small parameter proportional to the ratio of structure thickness to deflection amplitude was introduced by Evkin for asymptotic analysis of smooth isotropic spherical shell at large deflections. The nonlinear theory of shallow shells with small or moderate rotation angles of tangent to the shell surface was applied. The deflections were considered as large compared to shell thickness but still small compared to its radius. The obtained solution coincided with results of Pogorelov’s geometrical method [5,6]. In papers [2,3] large deflections of shells of revolution were studied by Evkin and Kalamkarov using asymptotic method with small parameter which was proportional to ratio of shell thickness to its curvature radius. Reissner’s equations [7,8], describing axially symmetric shell deformation with arbitrary deflections and rotation angles, were used. In the present paper we combine these two asymptotic approaches using a unified small parameter. We are focused on practical applications of composite spherical shells experiencing large deflections. Two very important practical scenarios are considered. The first one concerns the stability of the spherical shells under external pressure and estimation of knockdown factor (KDF) for design buckling load. Because of extremely high sensitivity of the structure to its geometrical imperfections, the obtained experimental data were spread in a wide interval. They were much smaller than buckling load

predicted for perfect shells and therefore the NASA design recommendation [9] suggests formula for KDF

ρ = 0.14 +

3.2 , λ>2 λ2

(1)

where λ is shell thickness parameter

φ R 12 λ = [12(1 − ν 2)]1 4 ⎛ ⎞ 2sin 2 ⎝h⎠

(2)

Here h and R are shell thickness and its radius, φ is half of the included angle of the spherical cap. Formula (1) was obtained as a lower bound estimation of experimental data collected at that time (1969). The design buckling pressure is a fraction of classical buckling load obtained by Zoelly [10] and Leibenzon [11] for perfect isotropic spherical shell:

qc =

2Eh ⌈3(1 − ν 2) ⌉1 2 R

(3)

where E and ν are Young’s modulus and Poisson’s ratio of shell material respectively. It is recognised now that formula (1) is overly conservative [12–15] mainly because structure manufacturing technology improved significantly allowing to produce shells of high quality. The formula (1) was improved by Evkin and Lykhachova [12,13] using energy barrier concept in combination with asymptotic method and by Wagner et al. [15] using the so called localized reduced stiffness method.

E-mail address: [email protected]. https://doi.org/10.1016/j.compstruct.2019.111577 Received 23 July 2019; Received in revised form 10 October 2019; Accepted 21 October 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Alexander Yu. Evkin, Composite Structures, https://doi.org/10.1016/j.compstruct.2019.111577

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authors are valid in the scope of shallow shell theory. Our asymptotic solution obtained in Section 5 extends the results to the case of deep (including complete) shells. 2. Reissner’s equations for axially symmetric deformation of spherical shells We apply Reissner’s equations [7,8] describing axially symmetric deformation of thin shell of revolution with large deflections and rotation angles. The relations between displacements and deformations of arbitrary point on the reference surface can be presented in the case of spherical shell of radius R as follows:

ε2 =

χ1 =

u , r0

du = (1 + ε1 ) cosψ − cosψ0 , ds

dv = (1 + ε1 ) sinψ − sinψ0 ds

sinψ0 ⎞ sinψ dψ (1 + ε1 ) , χ2 = ⎛ (1 + ε1) − − r r0 ⎠ ds R ⎝ ⎜

(4)

⎟

(5)

where ε1 and ε2 are the deformations of reference surface element in the meridional and circumferential directions, u and v are the radial (horizontal) and axial (vertical) displacements respectively, ψ0 and ψ are angles of inclination of the tangent to the meridian before and after shell deformation (Fig. 2), χ1 and χ2 curvature changes in the meridional and circumferential directions, s is the independent variable, which means the distance from the shell pole along meridian arc to the shell point being considered; whereas r0 and r are distances from this point to symmetry axis z before and after deformation. We have the obvious relations:

Fig. 1. Different positions of hemispherical expulsion diaphragm.

The approach and formula for design buckling pressure obtained in [12,13] for isotropic thin spherical caps is generalized for the case of composite structure in Section 4 of present paper. For now we have to mention that the main idea of energy barrier criterion is to define the load level when the structure turns out to be not sensitive to different perturbations. It happens when the pressure is relatively low. For example, according to formula (1) and [12,13], the design buckling pressure is about 15–30% of classical buckling load. The energy barrier is defined as full potential energy of the system at an intermediate unstable post-buckling equilibrium state (saddle in the potential energy landscape) which can be reached at relatively large deflections. It was shown in [12] that asymptotic method was appropriate for estimation of energy barrier at this load level, therefore it can be used for estimation of shell metastability for practical purposes. In the second practical scenario the shell deformations with large deflections are realizable by design, for example using diaphragms for positive expulsion propellant spherical tanks of launch vehicles [16–19]. The diaphragms separate fuel and pressure gas to avoid their mixing and to guarantee stable combustion process. They are designed to be fully inverted (Fig. 1). The load deflection diagrams as well as stress analysis are the main goals for the structural engineer in this case. This information is used for selecting material of the structure and its geometrical and stiffness parameters. The analysis of structure inversion is performed in Section 5 of our paper. Simple asymptotic formulae are obtained for pressure parameter and for maximum stresses depending on deflection amplitude. Gleich [16] and Woodruff [17] considered spherical metallic diaphragms reinforced in circumferential direction by wire rings to avoid random buckling. Elastomeric diaphragms are also used nowadays. More detailed information about diaphragm design requirements can be found in the recently published review [19] by Tam et al. New and interesting applications of shell theory were considered by Vella et al. [20] and by Knoche and Kierfeld [21] modeling microscopic polymeric capsules that may be used in drug delivery and can be fabricated by various methods. Elastic complete spherical shells are commonly found in nature as well: red blood cells, virus capsules, and pollen grains. The authors of the research [20,21] studied the shell behaviour at large deflections to determine both the internal pressure and elastic properties of capsules. These results are also relevant for the control of buckled shapes in the applications. The solutions obtained by

r0 = Rsinψ0 , r = r0 + u,

dψ0 1 = R ds

(6)

Here subscript 0 indicates properties of undeformed shell. The differential equilibrium equations of a shell element are the following:

cosψ dV V + qv ⎞ = −(1 + ε1 ) ⎛ ds ⎝ r ⎠

(7)

cosψ N dU U − 2 + qu⎞ = −(1 + ε1 ) ⎛ r ds ⎝ r ⎠

(8)

cosψ dM1 (M1 − M2) − U sinψ + V cosψ⎞ = −(1 + ε1 ) ⎛ ds ⎝ r ⎠

(9)

Here qu and qv , U and V are horizontal and vertical components of the external load q and stress resultant of the deformed shell; N1, N2 and M1, M2 are stress resultants and bending moments which are defined as being per unit length. Subscripts 1 and 2 in our notation correspond to meridional and circumferential directions respectively. There is an obvious relationship between internal forces:

N1 = Ucosψ + Vsinψ

(10)

Fig. 2. Axially symmetric deformation of spherical shell at large deflections: initial F1 and mirror- inverted F2 parts of the shell are connected by rounding inner boundary layer F3 at meridian coordinate ψ0 = ξ0 . 2

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The constitutive equations for orthotropic shell in case of axially symmetric deformation are the following:

Ni = Bij εj + Kij χj , Mi = Dij χj + Kij εj, i, j = 1, 2

ε∗2 =

In present research we introduce a new parameter

(11)

There is summation here with respect to index j . The Eq. (11) allow further simplification in the case of axially symmetric deformation of the shell with large deflections even though the shell is not symmetric with respect to its middle surface, because the bending of the shell in meridional direction is dominating. If the reference surface is selected as a neutral one with respect to meridional bending, the Eq. (11) have the simplified form

Ni = Bij εj, Mi = Dij χj

ε2 =

1 Rξ02

D11 a∗ B22

(16)

Here angle ξ0 defines position of inner boundary layer (Fig. 2) and therefore deflection at the shell pole

w0 = 2R (1 − cosξ0)

(17)

If ξ0 ~1, formula (16) corresponds to the definition of parameter (15) which was introduced in [2]. In the case of shallow shells ξ02 ≪ 1, w0 = Rξ02 and the introduced parameter (16) differs from (14) only by a factor of 0.25. We will apply parameter (16) as a small one to asymptotic analysis of spherical shells at large deflections.

(12)

Relationships (12) cover structures composed from laminates, orthogrid and waffle grid shells with small enough cells which can be modeled by shells with several layers neglecting discreteness of stiffeners (smeared stiffener approach). The stiffness coefficients Bij and Dij in particular cases can be calculated using methods described in [22–26]. In addition, we consider large deflections caused mostly by bending while the shell is subject to inwardly directed load. In this case we assume that ε1, ε2 ≪ 1 and the factor (1 + ε1) in Eqs. (7)–(9) can be omitted. To finalize the formulation of the initial boundary value problem, we introduce the three types of boundary conditions at ψ0 = φ

ψ = φ , u = 0, v = 0 ψ = φ , U = 0, v = 0 M0 = 0, u = 0, v = 0

h ⌈12(1 − ν 2) ⌉1 2 R

3.1. Asymptotic analysis of Reissner’s equation Let us introduce external pressure parametrization −

qv =

qv − , q = qu qc , qc = 4 a∗ D11 B22 qc u

R2

(18)

For the case of isotropic sphere qc is the classical buckling pressure. Then, as mentioned above, we consider the case of small membrane deformations: −

−

(20)

ε1 = εε1, ε2 = εε2 For independent variable we put

(13) −

ψ0 = ψ0 ξ0, s = Rψ0

The first one corresponds to the shell clamped along the shell edge ψ0 = φ , the second condition corresponds to the shell resting on a frictionless surface, and the last one is the simply supported edge condition.

The position of inner boundary layer is defined now by simple −

equation ψ0 = 1, which does not depend on the inner boundary layer position defined by ξ0 . Our goal is to obtain a fast changing solution in this vicinity, therefore we assume in the asymptotic analysis that

3. Asymptotic solutions

dF −1 ~ε F dψ0

The asymptotic analysis is based on the fact that inwardly directed large deflections are caused mostly by bending deformation of the shell. Pogorelov [5,6] in this case suggested considering the so called isometric transformation of the shell middle surface without its membrane deformation. This type of deformation requires much less energy than the membrane deformations, especially if deflections are large. However, the pure bending of the shell structures usually is not allowed because of their shapes or boundary conditions. Pogorelov suggested to consider isometric mapping of initial shell surface in the set of not smooth surfaces. For spherical shell it is a spherical segment which is mirror-inverted with respect to horizontal plane A-A (Fig. 2). The main deformation energy is concentered at the ridge (inner boundary layer) which is a junction of two parts of the shell (initial F1 and inverted F2 ). Pogorelov obtained a simple formula for this deformation energy taking into account bending in meridional direction and membrane deformation in circumferential direction. It was shown by Evkin [1] that Pogorelov’s geometrical method is a leading order asymptotic solution with the following small parameter

ε02 =

2h ⌈3(1 − ν 2) ⌉1 2 w0

2 B12 D11 , a∗ = 1 − B11 B22 a∗ B22

(22)

where F is any function describing stress-strain condition of the shell at the inner boundary layer. However, we assume that geometrical and stiffness parameters of the structure are slowly changing functions, therefore we can treat them as constant calculating them at ψ0 ≈ ξ0 . Equilibrium Eqs. (7)–(9) can be simplified as the following:

dV = −qv , ds

dU N = 2, ds r

dM1 = Usinψ − Vcosψ ds

(23)

In the second equation we put qu = 0 . This component of the external load does not do any work, because there is only vertical displacement at mirror reflection of the shell segment. Obtained Eq. (23) together with (10) allow us to define the leading order terms of internal membrane forces: −

−

−

N2 = ε −1qc Rsinψ0 N2, U = 0.25qc RU , V = εqc RV

(24)

Here, in the previous and the following formulae the functions with bar above have order of unity in the asymptotic expansion. One can see from (24) that internal membrane force N2 acting in circumferential direction is the most important one. The curvature change in meridional direction is dominating in the boundary layer, therefore

(14)

where w0 is deflection amplitude of the shell. In [2,3] asymptotic method was applied to the case of shells of revolution with small parameter

1 ε∗2 = R

(21)

−

−

χ1 = ε −1χ1 R, χ2 = χ2 R

(25)

Now we have simplified constitutive equations in the form

M1 = D11 χ1 , M2 = D21 M1 D11, N2 = a∗ B22 ε2, ε1 = −B12 ε2 B11 (15)

(26)

Using first equations of (5), (23) and (26), we obtain the asymptotic representation of equilibrium Eq. (9) in the following form:

which in case of isotropic smooth spherical shell takes the form 3

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ε2

d 2ψ −2

−

stress and strain of the shell, we obtain for the shell deformation energy

= Usinψ + qv̂ cosψ

dψ0

(27)

W=

where qv̂ is defined according to the first equation of (23) as

qv̂ = 4

ψ0 − qv dψ0 0

∫

(28)

W = a∗ πε∗3 B22 R2ξ02 sinξ0 J (ξ0)

−2

J (ξ0) =

= cosψ − cosψ0

dψ0

(36)

where

−

d 2U

(35)

which is calculated at inner boundary layer F3 . According to above obtained relations we have

Joining two first equations of (4) and using (26) and (23), we obtain the second equation

ε 2sin2 ψ0

∬ (M1 χ1 + N2 ε2 ) dF3

(29)

ξ∗

∫−∞ ((f ' )2 + (ϑ')2) dξ

(37)

and

If ε = 0 we have two trivial solutions: ψ = ψ0 and ψ = −ψ0 . The first solution corresponds to the initial pre-compressed part of the spherical shell, the second one is the mirror-inverted part of the shell (Fig. 2). To obtain fast changing boundary value functions connecting these two

The work of external load is defined by vertical displacement v (ψ0) of the shell surface at the mirror-inverted area F2

solutions, we introduce a new variable ξ = (ψ0 − 1) ε . Finally, we have

v (ψ0) = 2R (cosψ0 − cosξ0)

− d 2ψ = Usinψ + qv̂ (ξ0) cosψ dξ 2

therefore

−

f = sinξ0 U ξ0, ϑ = ψ ξ0

−

− d 2U

dξ 2

(30)

A (ξ0) = 4πR3

cosψ − cosξ0 = sin2 ξ0

−'

(32)

These boundary conditions should be taken into account when inner boundary layer is close to the shell edge (ξ∗ ~1). If it is far from the shell edge (ξ∗ ~1 ε ) , we have to satisfy the condition of transition to the initial shell equilibrium state −

(33)

−

cosψ − cosψ0 d 2U = dξ 2 sin2 ξ0

−

where U0 = −qv̂ (ξ0) cotξ0 is the solution corresponding to spherical shell in initial bendingless equilibrium state. If we consider the case of large deflections when the inner boundary layer is far from pole (1 ε ≫ 1) , then the inner boundary layer should tend to the solution corresponding to the inverted part of the sphere

J (ξ0) = 2.223 + 0.112ξ02

R

sinξ0 U (ξ ), M1 =

'

The corresponding solutions for functions f and solution ϑ are sketched in Fig. 3 in the case when ξ02 ≪ 1. These graphs show the distribution of internal forces in inner boundary layer. We calculated their maximum values for different ξ0 and obtained the approximations

We completed the formulation of the boundary value problem as a leading order problem of the asymptotic analysis. It is remarkable that the problem does not depend on geometrical and stiffness parameters of − the shell. If it is solved for functions ψ (ξ ) and U (ξ ) , the stresses of the structure can be estimated using the formulae for internal forces

N1 = 0, N2 =

(44) '

ψ = −ξ0, U = −U0 as ξ∗ → −∞

−'

(43)

We solved this problem numerically for different values of ξ0 and obtained the following formula for the minimum of the functional

−

3 D11 (a∗ B22 ) 4

(42)

defining the equilibrium state of the shell at large deflections. There are two possible cases: when inner boundary layer is close or far from the shell edge. In the first case the deformation energy changes rapidly if ξ0 → φ and therefore dJ dξ0 ~1 ε . When the inner boundary layer is far enough from the shell edge (ξ∗ ~1 ε ), dJ dξ0 ~1 and we obtain qv ~ε . In this case the load parameter qv̂ can be dropped from Eq. (30) and we have a system of homogenous differential equations. It was shown in [2] that they deliver minimum of functional (37) under condition (31), which has the form

−

ψ = φ, U = 0

1 4

(41)

Finally we obtain the equation

−'

−

(40)

dA dW = dξ0 dξ0

ψ = φ, U = 0

−

(cosψ0 − cosξ0 ) qv sinψ0 dψ0

dA = 2πR3qsin3 ξ0 dξ0

In the first order of asymptotic solution we have to calculate all slowly changing functions (including geometrical, stiffness parameters of the structure and qv̂ in (28)) at ψ0 = ξ0 . The boundary conditions (13) are defined at ψ0 = φ or at ξ∗ = (φ ξ0 − 1) ε . They have the following asymptotic form

ψ = ξ0, U = U0 as ξ∗ → +∞

ξ0

(39)

For the case of uniformly distributed external pressure q we have

(31)

ψ' = 0, U = 0

∫0

(38)

maxf ' = 0.402 + 0.0094ξ02, max ϑ' = 0.947 + 0.058ξ02

(45)

The maximum relative error in approximations (44),(45) is less than 0.5%. Having formulae (45) one can calculate maximum values of internal forces at leading order of asymptotic

D11 ' D ψ (ξ ), M2 = 21 M1 εR D11

maxN2 = a∗ B22 ε∗ ξ0 maxf ′ , N1 ~εN2

(34)

D11 ξ0 D maxM1 max ϑ', maxM2 = 21 Rε∗ D11

(46)

3.2. Variational approach to asymptotic analysis

maxM1 =

According to Lagrange principle, the full potential energy Π = W − A attains a stationary value at an equilibrium state in comparison to all kinematically possible displacements. Here W is deformation energy of the shell and A is work of external forces done on the feasible displacements. Using only leading order components of the

For the case of smooth isotropic shell we have the following formulae for maximum membrane and bending stresses

maxσ2m = Eε∗ ξ0 maxf ' , maxσ1b = Eε∗ ξ0 max ϑ' 3 (1 − ν 2)

(47)

(48)

One can see that the maximum bending stress is much greater than 4

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author of the present paper tried to do this years ago but did not succeed. Having two asymptotical solutions at large and small deflections, it is possible to obtain a joint solution suitable for the entire range of the deflection and external pressure. However, the asymptotic solution obtained in [29] for small deflections is valid in the very small load − range 0.95 ≤ q ≤ 1, therefore intermediate points for construction of joint solution were needed in [29] as well as in our research [28]. Another very important result was obtained by Baumgarten and Kierfeld [29]. They have shown by analytical and numerical analysis that there is a very strong dependence between buckling load caused by local external force and initial localized geometrical imperfection of the shell. They suggested a formula for point load which is equivalent to initial deflections. We also noted about this dependence and correlation between internal and external local perturbations in [12] using qualitative analysis of shell equations. This important conclusion allows applying energy barrier criterion not only for estimation of metastability and buckling load of the shell under different external perturbations but also use it for evaluation of design buckling load of the structure having geometrical imperfections. It was done in [12,13] for the case of smooth spherical shell. In the next Section of the present paper we will generalize the approach for the case of composite spherical shell.

Fig. 3. Bending and membrane stresses distribution at the inner boundary layer.

the membrane one. According to Fig. 3 they are reached at ξ = 0 when membrane stresses are equal to 0. In general case shell stiffness parameters depend on coordinate ξ0 . Let us consider the case when these parameters change slowly, therefore we can treat them as constants taking derivative of deformation energy (36) with respect to ξ0 . Using (42) for the case of uniform external pressure we obtain the final formulae

4. Design buckling pressure for composite shells Composite spherical shells are significantly more efficient compared to smooth isotropic structures. However, they are still very sensitive to imperfections [15,30], therefore the influence of different perturbations on buckling load should be investigated to estimate the design buckling pressure. The main idea of our approach is to find the level of pressure which splits up the area of excessive sensitivity of the structure to possible perturbations. For this purpose we use the energy barrier criterion for shell metastability evaluation. The reader can find the history of developing of energy barrier concept in [12]. Here we just briefly describe the main idea. Typically, spherical shell under external pressure has several equilibrium states − for the given load q < 1. They are represented by points A, B, D on load volume change diagram sketched in Fig. 4a. Point C corresponds to − buckling load of perfect shell (q =1) . Straight line in Fig. 4a represents initial pre-buckling equilibrium states. Descending part of the curve CBD corresponds to unstable post-buckling equilibrium path. Increasing equilibrium branch shows stable post-buckling states. They are usually located very far from the initial states and are reached at large deflections. The increasing part of the diagram is caused by the effect of plastic deformation or shell edge conditions. In Fig. 4b the full potential energy of the system is sketched depending on the shell deflection for the fixed load value. Point A corresponds to initial stable equilibrium state of the shell, point B indicates maximum of the potential at the intermediate unstable state. Corresponding energy ΠAB is the energy barrier which should be overcome by the system to snap through to post-buckling stable equilibrium state. It is used as a measure of metastability of the structure. For that reason it is reasonable to compare the energy barrier with deformation energy accumulated by structure at its initial state A, which can be calculated as triangle dashed area in Fig. 4a. The energy barrier itself is illustrated in Fig. 4a as dashed area ACB. Analysis of post-buckling behaviour of the shell shows that minimum pressure on load volume change diagram is reached at very large deflections with corresponding large values of energy barrier (see for example [13] and Section 5 of the present paper). However, it is − shown in [12] that load qL corresponding to minimum volume change

−

q = q qc = ε∗ T (ξ0) 8, w0 = 2R (1 − cosξ0)

(49)

T (ξ0)=[J (ξ0 )(2ξ0 sinξ0 + ξ02 cosξ0) + ξ02 sinξ0 dJ dξ0] sin3 ξ0

(50)

which yield relationship between deflection amplitude w0 at the shell − pole and pressure parameter q . 3.3. The case of shallow shells The nonlinear theory of shallow shells with moderate rotation angles corresponds to the following assumptions

sinξ0 ≈ ξ0, cosξ0 ≈ 1 − 0.5ξ02

(51)

In this case we obtain

J = J0 = 2.223, w0 = Rξ02, maxf ' = 0.402, max ϑ' = 0.947

(52)

and finally −

q =

3ε∗ J0 8

R w0

(53)

For the case of smooth isotropic shells we have a simple formula −

q =

3J0 8[12(1 − ν 2)]1

4

h w0

(54)

which was obtained by Evkin [1] applying asymptotic method and earlier by Pogorelov [5,6] using geometrical method. The last one differs from (54) only by factor 1/[(1 − ν 2)]1 4 . Formula (54) was confirmed for the case of large deflections (small load parameter) by comparison with experimental and numerical data in [1,27,28]. The same result was obtained by Baumgarten and Kierfeld in the recently published paper [29]. The authors calculated more precise value 2.223 for the factor J0 . We obtained J0 = 2.225 in our calculations before and then rounded it to 2.23 in our previous publications. In the present paper we use a more precise value. In addition, Baumgarten and Kierfeld [29] made another very important step in the asymptotic analysis of post-buckling behaviour of smooth isotropic spherical shell. − They obtained the asymptotic solution for small deflections (q 1) . The

−

ΔVmin (Fig. 4a) splits up the area of intensive sensitivity of the shell to possible perturbations. Based on that, the design buckling pressure formula was suggested in [12] for smooth isotropic shell. We will generalize this approach to the case of composite shells. 5

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Fig. 4. Equilibrium states A, B, D on diagrams: a) load-volume change; b) full potential energy versus deflection. Energy barrier ΠAB corresponds to intermediate unstable equilibrium state B.

the initial pre-compressed shell which is calculated as 0.5qΔVA . It is striking that energy barrier does not depend on any geometrical or − − stiffness shell parameters if q = qL . After substitution of (61) in (62) we

We consider composite shell whose membrane and bending stiffness components are constant and equal in any direction. This corresponds to the case of optimal stiffness distribution regarding to possible local shell buckling. Waffled, orthogrid and some laminate structures can be modeled by isotropic homogenous shell:

−

have E = 1 6. Energy barrier is a relatively fast changing function (62) − depending on load parameter q to the power of 5 in the denominator. For example, if load is changed by about 20%, the energy barrier changes 2.5 times. Therefore we can conclude that the structure is very − sensitive to perturbations if the pressure is above qL . On the other hand, if we decrease the load parameters by 20%, the energy barrier will be − relatively high. We suggested in [12] the knockdown factor 0.8q L for − estimation of design buckling load parameter qD . Exactly the same formula is valid for composite spherical shells:

(55)

B11 = B22, B12 = B21, D11 = D22 , D12 = D21

Following the methodology described in [12], we calculate the volume change in different equilibrium shell stages. In initial momentless stage A with constant deflection wA = 0.5qR2 (B22 + B12) −

ΔVA = 4πq R2 (1 − cosφ) a∗ D11 B22 (B22 + B12)

(56) −

For the buckling stage C of perfect shell we put q = 1 to calculate corresponding volume change ΔVC . The volume change of mirror-inverted shell with deflection w0 at the shell pole is the following

ΔVB = 0.5πw02 R2

−

qD =

ΔV =

− w02 (1 + ν ) VA + VB =q + 8R (1 − cosφ) VC

B22 (1 − ν 2) D11

(58)

where ν = B12 B22 . Using (54) and introducing new variables

hef =

hef 4Rsin2 (φ 2) 12D11 , ε2 = , λ2 = 12(1 − ν 2) 2 hef B22 w0 12(1 − ν ) (59)

we finally obtain the asymptotic formula for volume change −

ΔV =

3J0 ε 1 + 8 4(1 − ν ) λ2ε 4

(60)

which coincides with the result obtained in [12] for smooth isotropic shell. For this case hef = h and structure slenderness λ is defined −

by (2). Minimizing ΔV with respect to ε , we obtain formula for corresponding pressure parameter − 3J qL = ⎛ 0 ⎞ ⎝ 8 ⎠

4 5

1 λ2 5 (1 − ν )1

5

−

qD =

0.69 (1 − ν )1 5λ2

5

+ 0.05e−2(λ − 3), λ ≥ 3

(64)

The Eq. (64) is plotted in Fig. 6 by solid line. The NASA recommendation [9] is presented here by another solid line. Estimations suggested in the present research yield significantly more optimistic results for design buckling pressure parameter. All formulae are valid in the case of elastic shell buckling. The influence of geometric imperfections on energy barrier and therefore on shell metastability is studied in [13]. This influence becomes important for 4 ≤ λ ≤ 18 and for shells having relatively large initial deflections (with signature of

(61)

which is the same as in [12]. We also obtain asymptotic formula for energy barrier which coincides with result [12] − J4 3 3 E = ⎛ ⎞ −5 0 16 ⎝ ⎠ 2q (1 − ν ) λ2

(63)

with new generalized definition of shell stiffness parameter λ according to (59). One can see from obtained formula that by increasing bending stiffness in meridional direction, we can increase not only buckling load of perfect shells qc (18), but also decrease stiffness parameter λ and therefore increase knockdown factor (63). The knock down factor is increasing proportionally to effective thickness hef to the power of 0.2. This result reflects the fact that reinforced structures are somewhat less sensitive to imperfections (similar to thicker shells). The graph of energy barrier (62) is sketched in Fig. 5 for λ = 36.5, ν = 0.3 by dashed curve. Horizontal dashed lines correspond − − to load parameters qD = 0.176 , qL = 0.22 . The solid line corresponds to result obtained in [12] using numerical simulation of isotropic clamped semi-spherical shell with ratio R h = 200 . One can see that asymptotic solution yields conservative estimation of energy barrier with small difference compared to the numerical result. The formula (63) is plotted in Fig. 6 by dashed curve. The influence of boundary conditions is studied in [13] for the case of clamped spherical shells with stiffness parameter λ < 5. The formula (63) was slightly corrected by additional term:

(57)

Relative total volume change −

0.69 , λ>5 (1 − ν )1 5λ2 5

(62)

It is normalized with respect to deformation energy of the shell in 6

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Fig. 7. Pressure parameter versus deflection amplitude: dashed lines correspond to asymptotic solutions (49),(53) obtained for complete spherical shell; another dashed line is asymptotic solution obtained in Section 5.2 for clamped semi-spherical cap; it almost coincides with solid curve representing numerical solution [27] obtained for isotropic clamped semisphere (R h = 100 , ν = 0.3).

Fig. 5. Normalized energy barrier versus pressure for shell with stiffness λ = 36.5: dashed line corresponds to asymptotic formula (62); solid curve re− presents numerical solution [13] (R h = 200) . Load parameter qL = 0.22 corresponds to equilibrium state with minimum volume change with energy bar−

−

−

rier E ≈ 17% ; qD = 0.8qL = 0.176 is design buckling load parameter for this shell with λ = 36.5.

the shell. 5. Analysis of deep shells at large deflections In this Section we will study the inversion of spherical shell under external pressure. 5.1. The complete spherical shell First of all, we will estimate the error of shallow shell theory at large deflections compared to Reissner’s equations. Estimating influence of second terms in (44) and (45) with factor ξ02 we can conclude that maximum error can reach approximately 10% in (44) and about 5% in formula (45) at ξ0 = π 2 . But the most important factor is sin3 ξ0 in the denominator of expression (50). It is related to work of external pressure and it can yield significant error if we put sinξ0 ≈ ξ0 according to shallow shell theory. The approximation (52) for deflection amplitude w0 instead of (49) also leads to weighty error if ξ0 = π 2 . In Fig. 7 the load deflection diagrams corresponding to asymptotic solutions are plotted by dashed lines. The visible difference between the two graphs in Fig. 7 starts approximately at w0 R = 0.2 . The line corresponding to formula (54) yields decreasing dependence with minimal pressure − value qmin = 0.589ε∗ at the end of the interval w0 R = 2 . In contrast, the more precise theory (49) and (50) yields minimum

Fig. 6. Design buckling pressure parameter versus shell stiffness λ : asymptotic formulae (63), (64) and NASA recommendation [9].

about shell thickness). Our approach to knock down factor estimation is based on considering post-buckling equilibrium states of the shell at relatively large axially symmetric deflections. Non-axially symmetric post-buckling states can be of two types: corresponding to modes obtained using linear buckling theory (buckling caused by additional stresses at the edge of the shell) and modes corresponding to isometric transformation of the shell surface. The first type exists only at high level of pressure (close to classical one) and therefore cannot be accepted when estimating knockdown factor for design buckling load which is much lower according to experimental data. The second type of post-buckling modes is characterised by extremely large deflections, therefore they are hardly reached by reasonable level of perturbations. For much more details see [12] Section 6.2 as well as [13,31,32]. All conclusions obtained in these researches are applicable to the case of considered types of composite shells [31,32] except the case when the shell is reinforced in meridian direction only, which is not a common practical case. We also have to note that for reinforced shells we used “smeared stiffener approach” which is valid if the shell has small enough cells. The model does not allow to catch effects of local buckling of skin or stiffeners of the structure which may occur before general buckling of

−

qmin = 1.063ε∗

(65)

on the curve at w0 R = 1.51. We see significant difference in results. However, the load parameter at this range is very small especially for thin shells. For the case of isotropic shell −

qmin = 0.575

h R 1 − ν2

(66)

The same result was obtained numerically for complete isotropic spherical shell in [23–35]. Minimum load value is sometimes called an unbuckle pressure [29]. The numerical solution is represented in Fig. 7 by solid line. It is obtained by Korovaitsev [27] for isotropic clamped semi-spherical shell with ratio R h = 100 . There is small difference between asymptotic and numerical solutions at w0 R ≤ 1.5. The sharply increasing difference at w0 R > 1.6 is caused by the influence of boundary conditions because in this case the inner boundary layer approaches the shell edge. It will be considered in the next Section. 7

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ϑ+ (0) = b − 1, ϑ− (0) = b + 1

(73)

Eqs. (72), (73) guarantee continuity of horizontal displacement and rotation angle functions. The values of parameters a and b are selected during the numerical analysis as delivering minimum of potential energy (37). In addition, the continuity condition of displacement in vertical direction should be satisfied. It has the form ξ∗

∫ sin 0

ξ (2 + ϑ+) ξ0 ϑ+ cos 0 dξ = 2 2

0

∫ ⎛sin ⎜

−∞

⎝

ξ0 (ϑ−−2) ϑ ) cos − + sinξ0 ⎞⎟ dξ 2 2 ⎠

(74)

for deep shells. In the case of shallow caps it can be simplified ξ∗

0

0

−∞

∫ ϑ+dξ = ∫ ϑ−dξ

If the inner boundary layer is far enough from the shell edge, a = b = 0 and Eqs. (74), (75) are satisfied automatically because of deformation symmetry. Asymptotic formulation of edge conditions can be found in Appendix A. For example, clamped edge conditions are the following

Fig. 8. Maximum membrane and bending stresses versus deflection amplitude: dashed lines correspond to asymptotic solution (48) obtained for complete spherical shell; another dashed lines represent asymptotic solution obtained in Section 5.2 for clamped semi-spherical cap; it is compared with numerical solution [27] (solid curve) obtained for isotropic clamped semisphere (R h = 100 , ν = 0.3).

ϑ+ = 0, f+' = 0 at ξ = ξ∗

ϑ− → 0, f−' → 0 if ξ → −∞

5.2. The influence of boundary conditions

W = a∗ πε∗3 B22 R2ξ02 sinξ0 J (ξ0 ) η (C )

(77)

For given values of ξ0 , ξ∗, a and b , the above boundary value problems were solved numerically. By changing a and b , we satisfied condition (75) for deep shell or (75) for shallow shell and minimum condition for potential energy (37). The result of calculation is presented in Appendix B for clamped semi-spherical and shallow shells. In general case instead of (36), for deformation energy we have (78)

Here the scale factor η (C ) is introduced which depends on variable

−

We divide the domain of variable ξ = (ψ0 − 1) ε into two intervals: (−∞, 0] and [0, ξ∗]. The first inner interval is the inverted part of the shell, the second outer interval is the initial part of the sphere. We use minus sign for the solution corresponding to the inner interval and plus sign for the second interval. For smooth parts of the solution we have

C = (φ − ξ0) ε∗

(79)

With relative error less than 0.5% the scale factor can be approximated as

η (C ) = 0.77 + C −0.8 − 0.658C −1.6 + 0.175C −2.4 +

−

(67)

0.025 cosh(4(2 − C ))

(80)

for the case of shallow shell and as

Considering the solution additional to (67) and introducing new variables according to (38), we have the following differential equations for fast changing boundary layer functions

η (C ) =

0.99 0.05 + , X = e−C 1.0 − 1.59X + 4.156X 2 + 4.141X 3 cosh(4(2 − C )) (81)

sin(ξ0 (ϑ ± 1)) d2ϑ =f sinξ0 dξ 2

(68)

cos (ξ0 (ϑ ± 1)) − cosξ0 d 2f = dξ 2 ξ0 sinξ0

(69)

for the case of clamped semisphere. These approximations, as well as obtained numerical data, are shown in Appendix B. They slightly depend on deepness of the shell φ . Variation of full potential energy yields dependence of pressure parameter on ξ0 . The deflection w0 at the shell pole should be calculated using (49). The result for clamped semi-spherical shell is shown in Fig. 7 by dashed curve which almost coincides with the corresponding numerical solution [27]. We have to note here that by taking derivative − of (78) with respect to ξ0 , we obtain the pressure parameter q 1 because derivative dη dξ0 = −dη dC ε . The graphs for maximum bending and membrane stresses are sketched using data from Tables 2 and 3 of Appendix B. They are shown in Fig. 8 and compared with numerical result. There is a good agreement between asymptotic and numerical solutions which can be even better for the case of thinner shells. We can estimate the zone of edge influence on the stress state of the shell using graphs plotted in Fig. 3 and data presented in Appendix B. The inner boundary stresses decay at ξ ≈ 4 and they are still small ξ ≈ 3, therefore the effect of boundaries should be taken into account if

connecting these two solutions. In the case of shallow shell theory with moderate rotation angles we obtain

d2ϑ = f (ϑ ± 1) dξ 2

(70)

d 2f = ∓ ϑ − 0.5ϑ2 dξ 2

(71)

Asymptotic Eqs. (68), (69) for deep shells, as well as Eqs. (70), (71) for shallow caps, are derived in Appendix A. The conditions of smooth connection of both solutions are the following:

f+' (0) = f−' (0) = a;

(76)

Finally we have the decay boundary conditions

Graphs corresponding to maximum values of stresses are plotted in Fig. 8. Here maximum membrane stresses are represented by function ξ0 maxf ' and maximum bending stresses by function ξ0 max ϑ' 3 (1 − ν 2) (dashed lines), therefore they are equal to maxσ (Eε∗) in both cases (where maxσ = maxσ2m, maxσ1b ) for isotropic shells (see formulae (48)). We put ν = 0.3 for comparison with numerical results (solid lines) obtained in [27] for R h = 100 . The difference in two results at w0 R > 1.6 is caused by the influence of clamped edges of the shell.

ψ = ± ψ0 , U = =∓ qv̂ (ψ0) cotψ0

(75)

(72) 8

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The suggested method allows estimating energy barrier required for the shell to snap through from initial equilibrium state to post-buckling one. This energy barrier is used as a measure of metastability of the shell for estimation of knockdown factor for design buckling load. This approach is applied and validated for the case of unstiffened shell in [12]. In the present research we generalized the obtained result for the case of composite spherical shell (64). The case of full inversion of complete spherical shell was considered as well. We obtained load deflection diagram and compared it with numerical solution. This comparison shows good agreement if w0 hef > 3. We discovered the minimum of load parameter − qmin = 1.063ε∗, ε∗2 = hef (R 12a∗ ) at w0 R = 1.51. The minimum is absent on load deflection diagram obtained using shallow shell theory which is valid if approximately w0 R ≤ 0.25. The obtained asymptotic formulae allowed estimating membrane and bending stresses of the shell. Their maximums are reached at inner boundary layer. Comparison with numerical solution shows good agreement in this case as well. Asymptotic formulae were obtained first for the case when inner boundary layer is far enough from the shell edge (82). In this case the − load parameter is small (q ε ) in leading order of asymptotic solution. However, the load is increasing sharply if the layer approaches the shell edge. The asymptotic formulation of the problem was obtained in this case, which does not depend on pressure, stiffness and geometrical parameters of the structure. It depends on ξ0 if post-buckling dimple is deep. In the case of shallow shells (ξ02 6 < < 1) it depends only on boundary conditions. The particular case of clamped shells was studied in the present paper. We plan to investigate the influence of other types of boundary conditions on shell behavior at large deflections including the question of bistability of spherical caps [36]. Morphing between equilibrium shell states is often desirable, for example in mechanical metamaterials and origami structures [37–39].

φ − ξ0 ≤ 3ε∗. This can be confirmed considering boundary zone in Fig. 7,8. For the given shell parameter R h = 100 and ν = 0.3 we have the inequality in the form ξ0 ≥ π 2 − 3ε∗ or in terms of deflection amplitude w0 R ≥ 2 − 6ε∗. For the presented in Fig. 7,8 numerical solution we have w0 R ≥ 1.67. In general case, we obtained the inequality

w0 R ≥ 2H R − 2Cε∗

(82)

when the effect of boundary conditions can be neglected. Here H is the height of the spherical cap, 3 ≤ C ≤ 4 depending on boundary conditions. 6. Conclusion New parameter ε 2 = hef (Rξ02 12a∗ ), hef = 12D11 B22 is introduced for asymptotic analyses of composite shell behaviour at large deflections but still small membrane deformations described by Reissner’s equations. For unstiffened shell we have hef = h, a∗ = (1 − ν 2) . If the post-buckling dimple is shallow, then ε 2 = hef (w0 12a∗ ) , if w0 R then ε 2 hef (R 12a∗ ) . Obviously, the introduced parameter is small if deflections are large. It is used for asymptotic simplification of Reissner’s equations which were significantly reduced. We obtained boundary value problem which is independent from load level and its type, as well as from geometrical and stiffness shell parameter. For the deep dimple it depends slightly on angle ξ0 (meridional coordinate) defining position of inner boundary layer connecting two parts of the shell: the mirror-inverted and the initial one. The deformation energy of the structure is concentrated at this inner boundary layer. In contrast, work of external load is defined by the mirror-inverted part of the shell. For the shallow dimple (ξ02 6 < < 1) we obtained boundary value problem which is independent of all system parameters including load, shell geometrical and stiffness parameters and angle ξ0 . Solving it once, we obtained formula for deformation energy and work of external load. The connection between them was established by variation of full potential energy with respect to ξ0 (or deflection amplitude w0 ). For the case of deep dimple we solved the boundary value problem for different values of ξ0 and then obtained formulae (45)-(50) describing load, maximum bending and membrane internal forces depending on ξ0 and deflection amplitude. Using them we studied several practically important problems.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix. A asymptotic formulation of boundary value problem for inverted shell −

In the case when inner boundary layer is far enough from shell edge, the load parameter is small (q ε ) at large deflections and the asymptotic − boundary value problem does not depend on load. When the layer approaches the edge, the load parameter q 1. We will show that the boundary value problem in this case also does not depend on load parameter if we consider the solution as additional to two trivial solutions defining as ψ = ± ψ0 A.1 Differential equations First of all we will examine all Reissner’s equations against these two solutions and will obtain estimations of the internal forces. In both cases we can apply membrane (momentless) shell theory [40,41] which is well studied. The main condition of using this theory is a slow change of all geometrical, stiffness parameters of the structure and load. Mathematically this condition can be expressed as

dF dψ0 ~F

(A1)

where F is any geometrical, stiffness or load function. In addition we assume that ε1 ≪ 1. According to this theory we have to put M1 = M2 = 0 in (9) therefore

U = ± Vcotψ0

(A2)

From (7) we obtain

V=

R sinψ0

ψ0

∫ qv sinψ0 dψ0

(A3)

0

For the case of uniformly distributed pressure we have V = 0.5qRsinψ0 and U = ± 0.5qRcosψ0 . In general case we have the following estimation V qR and U qR . Therefore from (8) and (10) we obtain 9

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Table 1 Classical boundary conditions in terms of asymptotic analysis functions. ψ = ψ0

M1 = 0

U=0

u=0

ϑ=0

ϑ' = 0

f=0

f' = 0

Table 2 Numerical solution of asymptotic boundary value problem for clamped shallow spherical shell. C

a

b

J (C )

maxf '

maxϑ'

3.0 2.0 1.0 0.5 0.25

−0.05 −0.13 −0.21 −0.16 −0.09

0.014 0.135 0.29 0.160 −0.085

2.36 2.65 2.82 3.22 5.86

0.439 0.533 0.635 0.561 0.432

1.03 1.17 1.13 2.13 5.23

Fig. 9. Numerical results from Tables 2 and 3 and approximations of scale function η (C ) for deep and shallow shells.

N20 qR, N10 qR

(A4)

Comparing with membrane force (46) in inner boundary layer we have

N20

maxN2 ε

(A5)

N20

−

'

maxN2 = 2εq maxf . Therefore we can consider both initial and inverted parts of the shell as free from In the case of uniform pressure deformations in the leading order of asymptotic while constructing solution at the inner boundary layer. We have to note that condition (A1) should be satisfied only at the region of inner boundary layer. For example, the method can be used in the case of concentrated force applied at the shell pole if the layer is sufficiently far from the pole. A.2 Boundary conditions We consider classical boundary conditions at the shell edge. In all cases the vertical displacement is absent: v (φ) = 0 . In addition two pairs of conditions are introduced: M1 (φ) = 0 or ψ (φ) = ψ0 (φ) and U (φ) = 0 or u (φ) = 0 . They are shown in the Table 1 in terms of introduced functions of asymptotic analysis. Only last condition f ' = 0 needs some explanation. According to (4) equation u = 0 leads to ε2 = 0 . Deformation ε2 is proportional to N2 because of N1 = 0 in the leading order of asymptotic. Condition N2 = 0 yields f ' = 0 . More generally, instead of condition on horizontal displacement u we have to use function f ' (72). Finally, the formulation of boundary value problems considered in Section 5.2 can be derived using properties described in A1 and A2. Appendix B. Numerical solution of boundary value problem from Section 5.2 B.1 Shallow shell The result of the solution of boundary value problem for shallow shell is presented in Table 2. The potential energy parameter J (C ) reflects the boundary condition influence. It tends to J0 = 2.223, if C increases. The function η (C ) = J (C ) 2.223 is approximated by Eq. (80). It is sketched in Fig. 9 by solid line and compared with data from Table 2 which are represented in Fig. 9 by diamond signs. For C > 4 there is no influence of edge conditions therefore we have to put η (C ) = 1 in this case.

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Table 3 Numerical solution of asymptotic boundary value problem for clamped semi-spherical shell. C

a

b

J (C )

maxf '

maxϑ'

3.0 2.0 1.0 0.5 0.25

0.0 −0.2 −0.34 −0.22 −0.12

0.0554 −0.0084 −0.032 −0.125 −0.285

2.66 3.024 3.203 3.98 7.63

0.446 0.560 0.652 0.531 0.396

1.23 1.42 1.45 2.49 7.12

B.2 Inversion of clamped semi-spherical shell The result of the solution is presented in Table 3. The corresponding function η (C ) = J (C ) J0 (π 2) is approximated by Eq. (81). It is sketched in Fig. 9 by dashed curve and compared with corresponding data from Table 2 which are represented by circle signs. For C > 4 we have to put J (C ) = J0 (π 2) = 2.50 and η (C ) = 1. One can see that there is very small difference in function η (C ) in case of shallow and deep spherical shells.

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