Reduced stiffness buckling of sandwich cylindrical shells under uniform external pressure

Thin-Walled Structures 43 (2005) 1188–1201 www.elsevier.com/locate/tws

Reduced stiffness buckling of sandwich cylindrical shells under uniform external pressure Mitao Ohgaa,*, Aruna Sanjeewa Wijenayakaa, James G.A. Crollb a

Department of Civil and Environmental Engineering, Ehime University, 3, Bunkyo-cho, Matsuyama 790-8577, Japan b Department of Civil and Environmental Engineering, University Collage of London, London WC1E 6BT, UK Received 7 May 2004; accepted 30 March 2005 Available online 21 June 2005

Abstract A reduced stiffness lower bound method for the buckling of laterally pressure loaded sandwich cylindrical shell is proposed. Also, an attempt is made to assess the validity of the proposed reduced stiffness lower bound with FEM numerical examples. In addition, the proposed method is compared with classical and Plantema’s approaches of the buckling of the laterally pressure loaded sandwich cylindrical shell. Comparison of the proposed reduced stiffness lower bound with that obtained from non-linear FEM analysis verifies that it indeed provides a safe lower bound to the buckling of laterally pressure loaded sandwich cylindrical shells. The attractive feature of the proposed reduced stiffness method is that it can be readily used in designing laterally pressure loaded sandwich cylindrical shells without being concerned about geometrical imperfections. q 2005 Elsevier Ltd. All rights reserved. Keywords: Sandwich shell; Reduced stiffness; Buckling; Lower bound; Lateral pressure; Geometrical imperfections

1. Introduction Derived from high flexural strength-to-weight ratio compared to that of single material architecture, sandwich construction, though relatively new, is playing an increasingly important role in structures. In addition, the sandwich construction results in lower lateral * Corresponding author. Tel./fax: C81 89 927 9816. E-mail address: [email protected] (M. Ohga).

0263-8231/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2005.03.006

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

1189

deformations, higher buckling resistance, and better performance under dynamic conditions than do other single-material configurations [1]. Vinson JR [1] in his recent publications predicts that “it’s many advantages, development of new materials, and the need for high-performance and low-weight structures will insure that the sandwich construction will continue to be in demand”. Even with these advantages, initial studies has shown that the presence of geometrical imperfections on the laterally pressure loaded sandwich cylindrical shell can drastically reduce its buckling strength. Under these circumstances, a lower bound to the buckling of laterally pressure loaded sandwich cylindrical shell would not only provide safe design parameters but also make it simple to design by eliminating parameters such as geometrical imperfections. Therefore, a reduced stiffness lower bound method for the buckling of laterally pressure loaded sandwich cylindrical shell is proposed in this paper. Its characteristics that may be useful to the designer are listed and explained with numerical examples. Also, an attempt is made to assess the validity of the proposed reduced stiffness lower bound with extensive numerical examples. In addition, it is compared with classical and Plantema’s [2] approaches for the buckling of the laterally pressure loaded sandwich cylindrical shell. This approach is based on a simple linearized classical buckling method based on energy methods and variational principles [3–5]. This allows the analysis of variation of energies during buckling by considering different axial and circumferential wave numbers. On account of the imperfection sensitivity associated with loss of strength at large initial imperfections of the sandwich cylindrical shell, the membrane component of the total potential energy cannot be expected to provide stability of the shell. Therefore, this component is eliminated from the total potential energy to obtain the reduced energy from which the reduced strength is derived. Finally, the reduced stiffness lower bound is obtained as a function of the classical bucking strength. After that, FEM analysis of the geometrically imperfect sandwich cylindrical shell is used to verify the proposed reduced stiffness method. Comparison of the FEM and reduced stiffness lower bounds shows that the proposed method provides a safe lower bound to the buckling of laterally pressure loaded sandwich cylindrical shells. In addition, the proposed classical buckling method and the Plantema’s [2] approach agree very well that they are almost the same. The attractive feature of the proposed reduced stiffness method is that designers can readily use this method to design laterally pressure loaded sandwich cylindrical shells without being worried about the geometric imperfection that might arise during fabrication and construction. It is also simple to use and could be highly useful during the initial estimations of the design of laterally pressure loaded sandwich cylindrical shell.

2. RS buckling strength of the laterally pressure loaded sandwich cylindrical shell 2.1. Classical buckling strength A convenient way of examining the various possible equilibrium paths described by the stationarity of the total potential energy of sandwich cylindrical shell of length L, mean radius a, face thickness hf and core thickness hc (Fig. 1) is to first define the membrane

1190

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

ζ (w)

q s(v) βx

h=hf +hc+hf

βs

a L

x(u)

(a)

(b)

Fig. 1. Geometry and applied stresses of the laterally loaded sandwich cylindrical shell.

fundamental state. For the present problem this has been approximated as given below F ðNxF ; NsF ; Nxs Þ Z ð0; Kqa; 0Þ

(1a)

F Þ Z ð0; 0; 0Þ ðMxF ; MxF ; Mxs

(1b)

where q is the externally applied radial lateral pressure, Ef the Young’s module of the face and, nf the Poison’s ratio of the face. NF and MF are the fundamental stresses and moments in subscripted planes and directions. At the increment of the applied load, the total potential energy P of the sandwich cylindrical shell can be divided into its fundamental (P0), linear (P1) and quadratic (P2) components as given below P Z P0 C P1 C P2 C P3 C/

(2)

The quadratic component is in concern as it controls the stability of the laterally pressure loaded sandwich cylindrical shell. It can be written as ððð 1 0 0 0 0 0 0 ½sx0 3x0 C ss0 3s0 C txs P2 Z gxs C tx2 gx2 C ts2 gs2 ds dx d2 2 ðð 1 ðNsF 3s00 Þds dx Z U C V s ð3Þ C 2 Where U is the strain energies. And, Vs is the non-linear membrane energy term. Using the sign convention and notations given in Fig. 2, the linear incremental stress–strain relations can be written as sx0 Z Df ð3x0 C nf 3s0 Þ

(4a)

ss0 Z Df ð3s0 C nf 3x0 Þ

(4b)

0 0 Z Gf gxs txs

(4c)

0 0 Z Gc gxz txz

(4d)

0 0 Z Gc gsz tsz

(4e)

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

ζ (w)

nx

τ xs

ns

Outer face x(u)

τsζ τxs

1191

nx

Core Inner face

s(v) τxζ

τxs τxs

ns

Fig. 2. Element of a sandwich shell, stresses in the core and faces.

Here, Df Z Ef =ð1K n2f Þ. And, Gc and Gf are the core and face material shear strength, respectively. Also the non-linear stress–strain relations can be written as sx00 Z Df ð3x00 C nf 3s00 Þ

(5a)

ss00 Z Df ð3s00 C nf 3x00 Þ

(5b)

The linear strain–displacement relations defined by the orthogonal curvilinear coordinate system (x, s, z) coinciding with lines of principle curvatures (Fig. 1) are given below 3x0 Z

vu vb C z x Z 3x0 C zkx vx vx

vv w vb C C z s Z 3s0 C zks vs a vs   vv vu vbs vbx 0 gxs Z C C z C Z gxs0 C zkxs vx vs vx vs 3s0 Z

(6a) (6b)

(6c)

0 gxz Z

vw C bx vx

(6d)

0 Z gsz

vw v K C bs vs a

(6e)

The non-linear strain components that associate with the non-linear stress resultants s 00 x, and s 00 s can be written as   1 vw 2 1 0 2 Z ðwx Þ (7a) 3x00 Z 2 vx 2 3s00

  1 vw 2 1 0 2 Z Z ðws Þ 2 vs 2

(7b)

1192

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

Assuming that the cylindrical sandwich shell is simply supported at the ends, the displacements in Eqs. (6) and (7) can be formulated into functions as given below u Z A1 cos as cos rx

(8a)

v Z A2 sin as sin rx

(8b)

w Z A3 cos as sin rx

(8c)

bx Z A4 cos as cos rx

(8d)

bs Z A5 sin as sin rx

(8e)

In which, aZn/a, rZnP/L, where m is the axial half-wave number and n, the circumferential full-wave number. Ai is the amplitude of the displacement function. Of the present concern is the quadratic component P2 of the total potential energy for it is that controls the critical behavior, and from which the homogeneous equations yielding the critically stable state is derived. The first variation of the quadratic component P2 of Eq. (3) can be written as ÐÐÐ 0 0 0 0 0 0 0 0 dP2 Z ½sx d3x C ss0 d3s0 C txs dgxs C tx2 dgx2 C ts2 dgs2 ds dx d2 Ð Ð F 00 s C Ns d3s ds dx Z dU C dV ð9Þ Application of the variational principles to Eq. (9), with stress–stain relations from Eqs. (4) and (5), strain–displacement functions from Eqs. (6) and (7), and displacement functions from Eq. (8) results in a set of hamogeneous equations 3 2 3 2 C11 C12 C13 C14 C15 A1 7 6 7 6 6 C21 C22 C23 C24 C25 7 6 A2 7 7 6 7 6 7 6 7 6 (10) 6 C31 C32 ðC33 K lÞ C34 C35 7 6 A3 7 Z 0 7 6 7 6 7 7 6C 6 C44 C45 5 4 A4 5 4 41 C42 C43 A5 C51 C52 C53 C54 C55 where the coefficients, CijZCji are obtained as C11 Z r2 DM1 C a2 DM2 ; C12 Z Kðnf arDM1 C arDM2 Þ; nf S C13 Z K rDM1 ; C14 Z C15 Z 0 C21 Z a2 DM1 C r2 DM2 C 2s ; a a a a Ss C23 Z DM1 C SS ; C24 Z 0; C25 Z K a a a 1 2 2 C33 Z 2 DM1 C r Sx C a Ss ; C34 Z rSx ; C35 Z KaSs ; a C44 Z r2 DB1 C a2 DB2 C Sx C45 Z KðnF arDB1 C arDB2 Þ; C55 Z a2 DB1 C r2 DB2 C Ss

l Z qaa2

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

1193

Here membrane (DM), bending (DB) and shear (S) stiffness are obtained as; DM1 Z

2Ef h; ð1 K nf Þ f

DB2 Z

Gf fðh C 2hf Þ3 K h3c g; 12 c

DM2 Z 2hf Gf ;

DB1 Z

Sx Z Gc hc ;

Ef fðhc C 2hf Þ3 K h3c g 12ð1 K n2f Þ Ss Z G c hc

The classical buckling load, qc and modes of sandwich cylindrical shell are obtained as solutions of the homogeneous equations. By solving Eq. (10), amplitudes A1, A2, A4, and A5 of the displacement functions can be obtained as a function of A3. A1 Z K

A2 Z

(11a)

2 ðC12 C13 K C11 C23 ÞðC45 K C44 C55 Þ C ðC11 C25 ÞðC34 C45 K C35 C44 Þ A3 Z A20 A3 2 2 K C C ÞðC 2 K C C Þ C11 C25 C44 K ðC12 11 22 44 55 45 (11b)

A4 Z K

A5 Z

C12 A20 C C13 A3 Z A10 A3 C11

ðC34 C C45 A50 Þ 0 A 4 A3 C44

(11c)

2 ðC12 C13 KC11 C23 ÞðKC25 C44 ÞKðC34 C45 KC35 C44 ÞðC12 KC11 C22 Þ A3 Z A50 A3 2 2 2 ðC12 KC11 C22 ÞðC45 KC44 C55 ÞKC11 C25 C44 (11d)

By back-substituting Eq. (11) into (10), the classical lateral buckling strength is derived as qc Z

1 ðC31 A10 CC32 A20 CC33 CC34 A40 CC35 A50 Þ aa2

(12)

The classical critical buckling coefficient kc can be written as kc Z

qc 2Ef hf

(13)

2.2. Reduced stiffness strength The derivation of the classical buckling strength allows analysis of each energy component for their behavior during buckling of the laterally pressure loaded sandwich cylindrical shell. In this way those energy components that affect the stability of the shell are identified. Elimination of those energy components from the total potential energy results in the reduced stiffness model [2,3,5].

1194

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

From Eq. (3), quadratic component P2 can be written as s P2 Z UM C UB C US C VM

(14)

Where UM, UB and US are the membrane, bending and shear energies, respectively. Those can be divided into their components in respective directions and planes; x s sx UM Z UM C UM C UM

(15a)

UB Z UBx C UBs C UBsx

(15b)

US Z USxz C USsz

(15c)

The energy components in Eq. (15) can be written as ðð 1 x fDM1 ð3x0 C nf 3s0 Þ3x0 gds dx Z UM 2 xs Z UM

1 2

s Z UM

1 2

UBx Z

1 2

UBs Z

1 2

UBxs Z

1 2

USxz Z

1 2

USsz Z

1 2

ðð

fDM1 gxs0 gxs0 gds dx

(16b)

fDM1 ð3s0 C nf 3x0 Þ3s0 gds dx

(16c)

fDB1 ðkx C nf ks Þkx gds dx

(16d)

fDB1 ðks C nf kx Þks gds dx

(16e)

ðð ðð ðð

(16a)

ðð

fDB2 kxs0 kxs0 gds dx

ðð ðð

(16f)

fSx gxz gxz gds dx

(16g)

fSs gsz gsz gds dx

(16h)

s VM , the contribution arising from the non-linear stresses and strains can be written as ðð 1 s ðaðws0 Þ2 Þds dx Z qUE Zq (17) VM 2

Then, by considering the energy components of quadratic component of the total potential energy the classical buckling strength (qc) can be written as UM C UB C US K qc UE Z 0

(18)

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

1195

On account of the imperfection sensitivity associated with loss of strength at large initial imperfections of the sandwich cylindrical shell, the membrane component of the total potential energy cannot be expected to provide stability of the shell. Therefore, this component is eliminated from the total potential energy to obtain the reduced energy from which the reduced strength (qrs) is derived. UB C US K qrs UE Z 0

(19)

By substituting UE from Eq. (18) into (19), the reduced stiffness buckling strength can be written as function of the classical buckling strength; qrs Z

UB C US q UM C UB C US c

(20)

3. Non-linear analysis of the geometrically imperfect sandwich cylindrical shell 3.1. Non-linear FEM analysis The validity of the proposed reduced stiffness methods is ascertained by comparing it with results obtained with non-linear FEM analysis of the geometrically imperfect sandwich cylindrical shell; the lower bound obtained in this way is compared with the reduced stiffness lower bound. The FEM program developed for this purpose uses so called 9-node isoparametric shell element with five degrees of freedom at each nodal point corresponding to three displacements and two rotations. The use of the isoparametric shell element allows the layered analysis of the sandwich cylindrical shell and thus allows different material properties through the thickness of the cylinder. 3.2. Initial geometrical imperfection shape FEM analysis of the perfect laterally pressure loaded sandwich cylindrical shell shows bi-mode shape consisting one axial half-wave (m) and number of circumferential full waves (n) as can be seen in Fig. 3. In order to induce buckling of the shell, a very small point imperfection (w0/hZ1.0) is applied at the middle of the shell (Fig. 3b). Without this small point imperfection, the shell only shrink due to the applied lateral load as can be seed in Fig. 3a. When L/aZ2.0, the critical mode shape consists of one axial half-wave and five circumferential full waves (Fig. 3c). In addition, sandwich cylindrical shells with L/a equals 3.0, 4.0 and 5.0 were also analyzed. The critical mode shapes of all these models are exactly similar to those of the classical buckling method as can be seen in Fig. 3(c–f). Therefore, the critical classical buckling mode shape was introduced as the initial geometrical imperfection shape of the sandwich cylindrical shell in FEM analysis. The analysis was repeated by varying the magnitude of the initial imperfection until a lower bound from the FEM analysis was obtained. This lower bound is compared with the reduced stiffness lower bound in order to verify its validity.

1196

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

Fig. 3. FEM buckling analysis of the perfect shell (EfZ2.05!105 MPa, vfZ0.3, a/hfZ500, aZ1.0 m, VZ0.01).

3.3. FEM numerical examples Geometrically imperfect sandwich cylindrical shells having L/aZ1.0–5.0 were analyzed with the FEM program. Each model was reanalyzed by varying the transverse shear flexibility parameter V Z Ef hf =ð4aGc ð1K y2f ÞÞ in order to asses its effects. First, the equilibrium paths were obtained by plotting the stress parameter (qfem/qc) with the respective lateral deformation parameter (w/h). The same for L/a equals five is given in Fig. 4(a) (VZ0.1), (b) (VZ0.01) and (c) (VZ0.001). The variation of a typical equilibrium path is such that the stress peaks up at buckling and then reduces as the deformation develops as it can be seen in Fig. 4. As the magnitude of the initial imperfection (w0/h) increases, the maximum stress on the equilibrium path ðqmax fem =qc Þ reduces. However, beyond a certain initial imperfection, buckling of the sandwich cylindrical shell does not occur. The limiting curve on which the bucking can be observed is named as ‘the limiting equilibrium path’. The maximum stress on this ‘limiting equilibrium path’ is taken as the lower bound from the FEM analysis ðqlow fem =qc Þ. 3.4. Plantema strength Strength curves obtained with the simplified formulas proposed by F. J. Plantema [2] with that from classical and reduced stiffness methods for the laterally pressure

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

(a) qfem 1.2 qc

q max

qc

fem

1.0

Plantema Classical

0.8

RS low q fem

0.6

qc w0 /h=0.4 w0 /h=0.8 w0 /h=1.2 w0 /h=1.4 w0 /h=1.5

0.4 0.2

0

1197

5

10

15

w/h

20

V=0.1 (b) qfem

max

1.2

q fem

qc Plantema Classical

qc 1.0 0.8

RS low q fem

0.6

w0 /h=0.4 w0 /h=0.8 w0 /h=1.2 w0 /h=1.3 w0 /h=1.4

0.4 0.2

0

qc

5

10

15 w/h 20

V=0.01 (c) max

qfem 1.2

q fem

qc Plantema Classsical

qc 1.0 0.8

RS q low

0.6

fem

w0 /h=0.4 w0 /h=0.8 w0 /h=1.2 w0 /h=1.3 w0 /h=1.4

0.4 0.2

0

qc

5

10

15 w/h 20

V=0.001 Fig. 4. Non-linear Buckling analysis of laterally pressure loaded sandwich cylinder with L/aZ5.0, VZ0.1, EfZ 2.05!105 MPa, vfZ0.3, a/hfZ500, hc/hfZ6, aZ1.0 m.

1198

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

loaded sandwich cylindrical shell are also given in Fig. 4. The formulas proposed by Plantema [2] have been derived by assuming that radial displacement consists of single longitudinal half-sine wave and n circumferential full waves. The classical bucking strength and Plantema strength [2] are in good agreement that they are almost the same. 3.5. Imperfection sensitivity plot The imperfection sensitivity plot is obtained by plotting the maximum stress on each equilibrium path against the respective initial imperfection parameter (w0/h). The same for the L/a equals five is given in Fig. 5. As it can be seen in this figure, it is clearly evident that as the magnitude of the initial imperfection increases the maximum stress on the equilibrium path reach from and above the reduced stiffness lower bound. Also, the FEM lower bound occurs just above the reduce stiffness strength. 3.6. Comparison of FEM, classical, Plantema and RS methods Variation of classical, Plantema [2], reduced stiffness and the FEM lower bound buckling strengths with L/a are given in Fig. 6 (a) (VZ0.1), (b) (VZ0.01), and (c) (VZ 0.001). All of them seem to reduce as L/a increases. In fact, there is sharp descending in the strengths. The classical buckling and Plantema strengths [2] are almost the same. The points corresponding FEM lower bound occur just above the reduced stiffness lower bound. This verifies the validity of the proposed reduced stiffness method that it provides a close and safe lower bound for the buckling of laterally pressure loaded sandwich cylindrical shells.

max

V = 0.1 V = 0.01 V = 0.001 Classical

q fem 1.1 qc 1.0

low

q fem qc

0.9

0.8

V = 0.001 V = 0.01 V = 0.1 RS Lower bound

0.2

0.4

0.6

0.8

1.0

1.2 1.4 w0 / h

Fig. 5. Imperfection sensitivity plot of laterally pressure loaded sandwich cylindrical shell with L/aZ5.0, EfZ 2.05!105 MPa, vfZ0.3, a/hfZ500, hc/hfZ6, aZ1.0 m.

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

(b) q(Kgf/cm2)

(a) q(kgf/cm2) qc qp low q fem qlow rs

30 20

0 1

(b) q(Kgf/cm2) qc qp low q fem

30

20

10

2

3

4

5

L/a

0

qc qp low qfem q rslow

30

q rslow

20

10

1199

10

1

2

4

3

L/a

5

0

1

V=0.01

V=0.1

2

3

4

L/a

5

V=0.001

Fig. 6. Variation of the classical and plantema strengths, FEM and reduced stiffness lower bounds with L/a (EfZ 2.05!105 MPa, vfZ0.3, a/hfZ500, hc/hfZ6, aZ1.0 m).

4. Characteristics of classical and reduced stiffness buckling methods The variation of the classical (qc) Eq. (12) and RS (qrs) Eq. (20) buckling strengths with L/a and transverse shear flexibility parameter v are given in Fig. 7. As can be seen from this figure, both reduced stiffness and classical buckling strengths reduces as the parameter L/a increases. Also, the three curves corresponding to V equals to 0.1, 0.01 and 0.001 of both reduced stiffness and classical buckling strengths merge separately to one line as the L/a increases. This implies that as the L/a ratio increase, the RS and classical critical buckling strengths become independent of the core material shear strength. In other words, the contribution of the core to the total strength of the shell becomes comparatively less. qc&qrs 40 V=0.1

V=0.01 30

V=0.001

20 Classical 10 RS 0

1

2

3

4

5 6 7 8 9 10

log(L/a) Fig. 7. Variation of classical and RS buckling strength with L/a (EfZ2.05!105 MPa, vfZ0.3, a/hfZ500, hc/hfZ 6, aZ1.0 m).

1200

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

qc&qrs 16 V=0.0

14

12

Classical RS

10 0

100

200

300

1/V Fig. 8. Variation of classical and RS buckling strengths with 1/V (EfZ2.05!105 MPa, vfZ0.3, L/aZ2.0, a/hfZ 500, aZ1.0 m).

Further, both classical and RS buckling coefficients corresponding to VZ0.01 and 0.001 are almost equal. This characteristic is described here. Variation of qc, and the qrs with 1/V is given in Fig. 8 (L/aZ2.0). Both qc and qrs seem to have a similar behavior with 1/V that the critical classical and RS buckling strengths asymptotically reach the value corresponding to v equals zero. When 1/VO200, the rate of increase of both qc and qrs are almost zero. This implies, when 1/VO200, the contributions of the core to the total strength of the shell are comparatively less in both classical and reduced stiffness methods.

5. Conclusions The reduced stiffness method provides safe and close lower bounds for the buckling of laterally pressure loaded sandwich cylindrical shells. One of the attractive features of the proposed reduced stiffness elastic lower method is that it provides safe estimates of buckling loads that do not depend on the specification of the precise magnitude of the imperfection spectra. And thus designers can readily apply this method without being worried about the credible geometric imperfections that might arise during fabrication and construction of the laterally pressure loaded sandwich cylindrical shell. One of the characteristics of the classical and reduced stiffness buckling strengths is that as L/a increase, they become independent of the core material shear strength. In other words, the contribution of the core to the total strength of the shell becomes comparatively less. Also, as core shear strength increases, the contributions of the core to the total strength of the shell become comparatively less in both classical and reduced stiffness methods.

References [1] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Lancaster, PA 17604, USA: Thechnomic Publishing Company; 1999. [2] Plantema FJ. Sandwich construction: the bending and buckling of sandwich beams, plates and shells. New York, London, Sydney: Wiley; 1966.

M. Ohga et al. / Thin-Walled Structures 43 (2005) 1188–1201

1201

[3] Ohga M, Masazumi U. RS buckling strength of cylindrical sandwich shells under axial pressures (Japanese). J Struct Eng 2001;47A:27–34. [4] Ohga M, Wijenayaka AS. Explicit lower bounds for imperfection sensitive buckling of axially loaded sandwich cylindrical shells. In: Sing-ping Chiew, editor. International symposium on new perspectives of shell and spatial structures. Taipei, Taiwan: National Taiwan University; 2003. p. 1561–8. [5] Croll JGA, Batista RC. Explicit lower bounds for the buckling of axially loaded cylinders. Int J Mech Sci 1981;23(6):331–43.