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Buckling of cross-ply cylinders under hydrostatic pressure considering pressure stiffness
Ocean Engineering 38 (2011) 559–569
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Buckling of cross-ply cylinders under hydrostatic pressure considering pressure stiffness Izzet U. Cagdas a,n,1, Sarp Adali b a b
Civil Engineering Department, Faculty of Engineering, Akdeniz University, 07058 Antalya, Turkey School of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa
a r t i c l e in f o
abstract
Article history: Received 19 January 2010 Accepted 6 December 2010 Editor-in-Chief: A.I. Incecik Available online 22 December 2010
Buckling behavior of cross-ply cylinders under hydrostatic pressure is investigated using a semianalytical finite element based on a consistent first order shear deformable shell theory. Potential loss due to external pressure, also called pressure stiffness (PS) is taken into account by making use of Koiter’s related energy expression. A number of verification problems are solved and the numerical results are compared with the analytical results available in the literature and excellent agreement is observed. New numerical results are presented to assess the effect of PS on buckling due to hydrostatic pressure. It is shown that PS causes a decrease in the buckling load and this decrease depends on the size of the cylinder and the material. Also, issues related to thickness optimization are examined and optimal lamina thicknesses are determined for a number of cases with and without PS taken into account. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Stability Cross-ply cylinder Hydrostatic pressure Finite elements Pressure stiffness
1. Introduction Cross-ply laminated composite cylinders are prone to buckling and therefore should be designed accordingly to prevent catastrophic failure. In the present study the influence of loss of potential due to external hydrostatic pressure on the stability of cross-ply cylinders is investigated. This issue is often neglected by researchers in the field, leading to an overestimation of the buckling pressure. A semi-analytical finite element, which is based on a consistent shell theory, is developed and employed to obtain the numerical solution of the problem. The consistent shell theory employed here was developed by Leissa and Chang (1996) and Qatu (1999) and was used by Qatu (1999, 2004) to solve shell vibration problems. The present study is the first application of this shell theory to hydrostatic stability problems. Comparisons with published results are given by solving a number of verification problems and it is observed that pressure stiffness negatively affects the buckling pressure. In some cases the numerical results indicate up to 30% reduction in the buckling pressure. This shows the importance of taking this effect into account in stability calculations to obtain conservative buckling loads. Pressure stiffness was first considered for circular cylindrical isotropic shells under hydrostatic pressure by Koiter (1967). Brush and Almroth (1975) have explained the phenomenon in their n
Corresponding author. Tel.: + 90 5323101166; fax: + 90 2423106306. E-mail addresses: [email protected] (I.U. Cagdas), [email protected] (S. Adali). 1 Previous address: School of Mechanical Engineering, University of KwaZuluNatal, Durban, South Africa. 0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.12.005
textbook on the stability of structures. Hibbitt (1979) described pressure as a follower force and named the related stiffness terms as ‘‘pressure stiffness’’ terms. Loganathan et al. (1979) examined pressure stiffness in shell stability analysis and reported reduction in the buckling pressure. Subbiah and Natarajan (1980) developed an axisymmetric finite element for the instability analysis of ring stiffened isotropic shells of revolution and reported substantial reduction in buckling pressure due to the follower load effect. Pressure stiffness was considered in the analytical studies of Kardomateas (1996, 1997) and Kardomateas and Chung (1994) where only orthotropic cylinders were considered. Han et al. (2004) investigated buckling of long sandwich cylindrical shells under external pressure. Kasagi and Srinivasan (1995), Schokker et al. (1996), Sridharan and Kasagi (1997) and Kardomateas (1997) have also considered this effect by using the work of Koiter (1967). Kardomateas (2000) has improved the elasticity solution further by considering the effect of normal strains. Sridharan and Kasagi (1997) investigated buckling and associated non-linear response and collapse of moderately thick laminated composite cylindrical shells. Tanov et al. (1999) examined the effect of preloading on the dynamic buckling of laminated cylinders under sudden follower pressure. As can be seen from this survey so far, the literature on the influence of pressure stiffness on shell stability is not extensive. In previous studies on the optimization of composite cylinders under hydrostatic pressure, the effect of pressure stiffness was neglected. Diaconu et al. (2002), Messager et al. (2002), Smerdov (2000), Liang et al. (2003), and Ostwald (1990) studied optimization of composite cylindrical shells under combined loads including external pressure. Ross and Little (2001) investigated the buckling of
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Nomenclature i, j, k I GP L, H Le m
element nodes (letters i and j are also used as indices) the unit matrix Gauss point length and thickness of the cylinder length of a finite element total number of harmonics in the circumferential direction n number of half sine waves in the circumferential direction ncr critical value of n NL number of layers PS pressure stiffness q0 initially applied pressure qcr ,qcr dimensional and non-dimensional buckling pressures Rint, R, Rext internal, mean, and external radii of the cylinder x coordinate of a point in the curvilinear coordinate system lcr eigenvalue m1, m2 percentage errors E1, E2, E3 moduli of elasticity in 1, 2, 3 directions G12, G13, G23 moduli of shear n12, n13, n23 Poisson’s ratios Q ij transformed reduced stiffness coefficients Aij, Bij, Dij, Eij stiffness coefficients c0 coefficient ( ¼ 1/R) ^ ij modified stiffness coefficients Aij , A^ ij ,Bij , B^ ij ,Dij , D r, y, z radial, circumferential, and axial coordinates
er, ey, ez unit vectors in r, y, z directions ur, uy, uz radial, circumferential, and axial displacement components in the global c.s. a, y, z0 local coordinates of a Gauss point ea ,ezu unit vectors in a, z0 directions u, v, w displacements parallel to ea, ey, ezu directions. ca, cy :rotations of the transverse normal about y and a axes V1i, y, V3i nodal coordinates of node i V1i, V3i unit vectors in V1i,V3i directions Vai, Vyi nodal rotations of the transverse normal about unit vectors ey and V1i e1, e2, e3 unit vectors in 1, 2, 3 directions of the fiber c.s. ea, ey, gay, gyzu , gazu linear strain components in the local c.s. e0a, e0y, e0ya, e0ay, g0yzu linear (mid-surface) strain components in the local c.s. wa, wy, way, wya curvatures in the local c.s. nl nl nl enl 0a , e0y , e0ay , e0ya non-linear strain components in the local c.s. Be, Bw, Bf strain–displacement matrices Na, Ny, Nay, Nya force resultants in the local c.s. Ma, My, May, Mya moments in the local c.s. Qa, Qy shear forces in the local c.s. N, M, Q vectors of forces, moments, and shear forces Ue element strain energy Ve element potential energy of the applied stresses S a matrix of membrane stresses G a vector of derivatives of in-plane deformations at a Gauss point Kn, KnG, Knq global stiffness, geometric stiffness, and pressure stiffness matrices corresponding to nth harmonic
a corrugated carbon fiber cylinder under external hydrostatic pressure. Blachut and Smith (2008) examined stability of a multisegment underwater pressure hull. Kruzelecki and Trzeciak (2000) and Liang et al. (2004) optimized axially symmetrical shells and spheres under hydrostatic pressure. The effect of pressure stiffness on optimal thickness design is studied here for the first time. In the present study, the layer thicknesses are taken as design variables for the optimal design of cross-ply cylinders. The objective of the optimization is to maximize the external buckling pressure taking the pressure stiffness into account. It is shown that optimal lamina thickness differs when pressure stiffness is considered as compared to the case when PS is neglected. However the difference in the optimal layer thicknesses for these cases is found to be marginal even though the buckling loads can differ substantially when PS is taken into account. The details of the finite element method employed in the present study are given in Section 2. To assess the accuracy of the FEM, two verification problems are solved in Section 3. The solutions of these problems also serve to elucidate the effect of PS on the buckling load as a function of the stiffness of the material and the thickness of the cylinders. Results on the optimal thicknesses of the cross-ply cylinders are given in Section 4, which provides a detailed study of the title problem. Several conclusions are summarized in Section 5 based on the numerical results obtained in the present study.
thickness H. The position of a point on the cylinder is defined by the cylindrical coordinates r, y, z where (r, y, z) denote the radial, circumferential, and axial coordinates and (ur, uy, uz) denote the radial, circumferential, and axial displacement components, respectively. A local coordinate system (a, y, z0 ) is defined at a Gauss point on the mid-surface of the cylinder where u, v, and w denote the displacements parallel to a, y, and z0 coordinates and ca and cy are the rotations of the transverse normal about a and y axes. The coordinate of a point in the curvilinear element (shape function) coordinate system is denoted by (x). A three noded shell finite element of length ‘Le’ is schematically shown in Fig. 1. The global, nodal, and local displacement components and unit vectors are shown in Fig. 2. The model is based on the following displacement field: 9 9 8 2 38 ur > cn 0 0 0 0 > unr > > > > > > > > > n> > > > > 6 7 > > 0 sn 0 0 0 7> > > > uy > = < uy > m 6 X 6 7< n = 6 0 0 cn 0 0 7 uz uz ¼ ð1Þ 6 7> > > > > > 7> n¼06 0 > > > > Vani > V 0 0 0 c > > > > 4 5 n a i > > > > > > ; ; :V > : Vn > 0 0 0 0 sn yi yi
2. Finite element formulation
2.1. Strain–displacement relations
In the present section formulation of an axisymmetric (cylindrical) shell element is given based on a thick shell theory developed by Leissa and Chang (1996) and Qatu (1999). A laminated composite cylinder under uniform hydrostatic loading is considered, which is of length L, mean radius R, and
The local displacement field corresponding to the first order shear deformation theory is given as
where cn, sn, and m denote cos(ny), sin(ny), and the total number of harmonics in the circumferential direction, respectively, and n stands for the nth harmonic in the circumferential direction. Vai and Vyi are the nodal rotations.
uða, y,zuÞ ¼ u0 ða þ yÞ þzuca ða, yÞ
ð2:1Þ
vða, y,zuÞ ¼ v0 ða þ yÞ þzucy ða, yÞ
ð2:2Þ
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
wða, y,zuÞ ¼ w0 ða, yÞ
ð2:3Þ
Accordingly, the linear and non-linear strain–displacement relations are
ea ¼ e0a þzuwa ey ¼
ð3:1Þ
1 ðe0y þ zuwy Þ 1 þ zu=R
enl 0a ¼
i 1h ðu0, a Þ2 þ ðv0, a Þ2 þ ðw0, a Þ2 2
enl 0y ¼
i 1 h ðu0, y Þ2 þðv0, y Þ2 þ ðw0, y Þ2 þ 2w0 v0, y 2v0 w0, y þ v20 þ w20 2 2R ð3:7Þ
gay ¼ e0ay þ gyzu ¼
1 1 e0ya þzu wya þ way 1 þ zu=R 1 þ zu=R
1 R
ð3:8Þ
1 R
ð3:9Þ
ðv0 w0, a þ w0 v0, a Þ enl 0ay ¼
ð3:3Þ
ðu0, a u0, y þ v0, a v0, y þ w0, a w0, y Þ enl 0ya ¼ where 9 9 8 8 ðu0, a Þ e0a > > > > > > > > > > > > > > > 1 = < e0y = < ðw0 þ v0, y Þ > R , ¼ e0 ¼ Þ ðv e > > > > 0, a 0ay > > > > > > > > > > > > 1 :e ; : ðu0, y Þ ;
ð3:4Þ
gazu ¼ ca þ w0, a
ð3:5Þ
0ya
z, uz
(
z, uz
u¼
Rext
¼
ya
R
a, y
)
ca þ w0, a : vR0 þ cy
w0, y R
2.2. The force and moment resultants
j
The force and moment resultants, schematically shown in Fig. 3, are given below; see Qatu (2004). 9 2 9 8 38 Na > e0a > > A11 A12 A16 A16 > > > > > > > > > > 6 > > 7> < Ny = 6 A12 A^ 22 A26 A^ 26 7< e0y = 7 ¼6 6 7 Na y > e0ay > > > > > > 4 A16 A26 A66 A66 5> > > > > > > > > : Nya ; A16 A^ 26 A66 A^ 66 : e0ya ; 9 2 38 wa > B11 B12 B16 B16 > > > > > > 6 7> 6 B12 B^ 22 B26 B^ 26 7< wy = 6 7 þ6 ð4Þ 7 way > > > 4 B16 B26 B66 B66 5> > > > > ; : ^ ^ wya B16 B26 B66 B66
Le
Le
Rint j
z', w
Gauss Point k
(
i
Rint
R
g0azu g0yz
R
)
9 8 ðca, a Þ > > > > > > > 1 = < ðcy, y Þ > y R , ¼ v¼ c Þ ð w > > > y, a > ay > > > > > > > > > > > > : w ; : 1 ðc Þ ; 9 8 wa > > > > > > > =
The linear terms of the strain–displacement relations used here were also derived by Qatu (1999) and Toorani and Lakis (2000).
H i
ð3:6Þ
ð3:2Þ
1 ðg0yzu Þ 1þ zu=R
561
k H
, u
r, ur
r, ur
Fig. 1. Details of the element geometry and the coordinate systems. (a) Global coordinate system and (b) local coordinate system.
z, uz mid-surface
mid-surface uzi
Vαi uθi
i
eθ i
uri
V3i V1i
Vθi θ
v0
GP
eθ,e2 w
u0
R
ez
ez', e3
GP
eα,e1 eθ
uθ r, ur
er
Fig. 2. Global, nodal and local variables: (a) displacement components and (b) unit vectors.
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Nα
Mαθ
Qα Nαθ
Mα
Nθα Qθ
eθ
Nθ
ez' eα
Nθ Qθ Nθα
Mθ eθ
Mθα
ez'
eα
Mθ
Mθα
Mα
Nαθ Qα Mαθ
Nα
Fig. 3. Force resultants and moments in the local coordinate system: (a) force resultant and (b) moments.
9 8 Ma > > > > > > > = < My >
9 38 > > e0a > B11 B12 B16 B16 > > > > > 6 7 6 B12 B^ 22 B26 B^ 26 7< e0y = 7 ¼6 6 7> e0ay > May > > > > > > 4 B16 B26 B66 B66 5> > > > > > > ; :M > ^ ^ B16 B26 B66 B66 : e0ya ; ya 9 2 38 wa > D11 D12 D16 D16 > > > > > > > 6 ^ 22 D26 D ^ 26 7 6 D12 D 7< wy = 6 7 þ6 7 w > > 4 D16 D26 D66 D66 5> > > ay > ; :w > ^ 26 D66 D ^ 66 > D16 D ya (
Qa Qy
) ¼
2
" 5 A55 6 A45
A45 A^ 44
#(
gazu g0yzu
Minimization of the total potential energy and consecutive integration in circumferential direction yield the element stiffness matrix given below in Eq. (9) Z 1 n Kne ¼ kp ðBTw DBw þ BTe BBw þBTf CBf þ BTe ABe þ BTw BBe Þ ðz, x ÞR dx 1
ð9Þ ð5Þ where, k ¼2 for n ¼0, and k¼1 for n ¼1, y, m. Be, Bw, Bf are the strain–displacement matrices.
)
2.4. Stability analysis ð6Þ
The rigidity terms are given below (see Leissa and Chang, 1996) Aij ¼ Aij c0 Bij
A^ ij ¼ Aij þc0 Bij
ð7:1Þ
Bij ¼ Bij c0 Dij
B^ ij ¼ Bij þc0 Dij
ð7:2Þ
Dij ¼ Dij c0 Eij
^ ij ¼ Dij þ c0 Eij D
ð7:3Þ
where c0 ¼ 1=R, i,j ¼ 1,2,4,5,6 and the matrices A, B, D, and E are given by Aij ¼
NL X
ðQ ij Þk ðzku zuk1 Þ
The potential energy of the applied stresses, Ve arising from the action of the in-plane stresses on the corresponding second order strains is given by Eq. (10) Z Ve ¼ ðNT e0NL Þ dA ð10Þ A
where N is the vector of in-plane forces at the Gauss points that are calculated by a primary static analysis. Minimization of the potential energy of the in-plane forces yields the element geometric stiffness matrix given by: Z 1 Ke,n ðGT SGÞn ðz, x ÞR dx ð11Þ G ¼ kp 1
k¼1
Bij ¼
NL 1X ðQ Þ ðzu2 zu2k1 Þ 2 k ¼ 1 ij k k
Dij ¼
NL 1X ðQ Þ ðzu3 zu2k1 Þ 3 k ¼ 1 ij k k
Eij ¼
NL 1X ðQ Þ ðzu4 zu4k1 Þ 4 k ¼ 1 ij k k
where S is a matrix of membrane stresses and G is a vector of derivatives of in-plane deformations at a Gauss point. The buckling load parameter lcr, which is defined as the ratio of actual buckling pressure to the initially applied pressure (q0) at which the shell buckles, is obtained by solving the eigenvalue problem detðKn lcr KnG Þ ¼ 0 n
where z0 k, z0 k 1 are the coordinates of the lower and upper surfaces of kth layer in z0 direction, and NL is the number of layers. Q ij are the transformed reduced stiffness coefficients defined by Whitney (1987). These modified stress resultant equations are a direct consequence of the ð1 þ z=RÞ term in the kinematic relations, as stated by Bert (1967) and Qatu (1999, 2004). 2.3. Element stiffness matrix The strain energy Ue of element e can be written as: Z 1 ðNT e0 þ MT v þ Q T uÞ dA Ue ¼ 2 A where N ¼ Ae0 þ Bv, M ¼ Be0 þDv, and Q ¼ Cu.
ð8Þ
ð12Þ KnG
is the global geometric where K is the global stiffness matrix and stiffness matrix corresponding to the nth harmonic. This eigenvalue problem can be modified by multiplying ðKn lcr KnG Þ by the inverse of Kn and then dividing by lcr. This operation yields the modified eigenvalue problem 1 I ¼0 ð13Þ det ðKn Þ1 KnG
lcr
1 where I denotes the unit matrix. The highest eigenvalue (lcr ) obtained by the solution of this eigenvalue problem corresponds to the lowest value of the buckling load parameter lcr. The corresponding buckling pressure is equal to qcr ¼ q0 lcr . The critical value of n, and the corresponding buckling parameter are determined by trial. The numerical results presented below are obtained by four point Gauss integration and using Matlabs built-in function ‘eigs’.
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
2.5. Pressure stiffness
Table 1 Critical values of pressure (Pa 10 6) for clamped [90/90/90]s cylinders. ˚ ˚ ˚
The expression giving the potential loss due to external pressure used in this study is given below Z ^ p ¼ 1 lp½wðe0a þ e0y Þ þðuc þ vc Þ dS P ð14Þ a y 2 where u, v, e0a, e0y, ca, and cy denote displacement in a direction, displacement in y direction, membrane strain component in a direction, membrane strain component in y direction, and rotations, respectively. Note that a similar expression was also used by Jha and Inman (2004) who studied vibration of a gossamer structure. The buckling load parameter for the externally pressurized case can be obtained by solving the modified eigenvalue problem n
det½K
lcr ðKnG þ Knq Þ
563
¼0
ð15Þ
where Knq is the global pressure stiffness matrix corresponding to the nth harmonic.
3. Verification problems
H (mm)
L R
qcra
3.175
1 2 5
3.450 (5) 1.940 (4) 0.951 (3)
3.4173 (5) 1.9465 (4) 0.9414 (3)
3.3015 (5) 1.8380 (4) 0.8437 (3)
3.5 5.9 11.6
6.35
1 2 5
18.61 (5) 11.03 (3) 5.929 (3)
18.5211 (5) 11.0876 (3) 5.9902 (3)
17.9577 (5) 10.0370 (3) 5.3888 (3)
3.1 10.5 11.2
12.7
1 2 5
94.45 (4) 52.60 (3) 28.10 (2)
92.2937 (4) 52.5812 (3) 28.2758 (2)
88.6725 (4) 48.2255 (3) 22.2208 (2)
4.1 9.0 27.3
a b c
qcrb
qcrc
m1 (%)
Anastasiadis et al. (1994). Current study, without PS. Current study, with PS, m1 ¼ 9% error9.
Table 2 Critical values of pressure (Pa 10 6) for clamped [01/901/01]s cylinders. H (mm)
L R
qcra
qcrb
qcrc
3.175
1 2 5
2.344 (7) 1.069 (5) 0.517 (4)
2.2834 (7) 1.0857 (5) 0.5321 (4)
2.2409 (7) 1.0443 (5) 0.4997 (4)
1.9 4.0 6.5
6.35
1 2 5
15.03 (6) 6.205 (4) 2.758 (3)
14.8788 (6) 6.1752 (4) 2.7941 (3)
14.5172 (6) 5.8202 (4) 2.4956 (3)
2.5 6.1 12.0
12.7
1 2 5
89.62 (5) 35.16 (4) 15.99 (3)
88.2243 (6) 34.7196 (4) 16.2949 (3)
86.3514 (6) 32.9215 (4) 14.6529 (3)
2.2 5.5 11.2
m1 (%)
Next two verification problems are solved in order to compare the results obtained by the finite element formulation developed above with available results in the literature. This enables us to study the effect of pressure stiffness on the buckling loads. Problem I: First, results presented by Anastasiadis et al. (1994) are duplicated, even though pressure stiffness was not taken into account in this study. In general, excellent agreement is observed with the reference results when pressure stiffness is neglected. Same problems are also solved by taking PS into account and the results obtained here indicate considerable reductions in buckling loads. Problem II: As stated before, stability results involving pressure stiffness are scarce. However, the studies of Kardomateas (1996) and Kardomateas and Chung (1994), where results were obtained by a 3D analytical solution, provide reliable results. Even though their results are valid only for a specific boundary condition and loading, and only applicable to orthotropic cylinders, they provide valuable information to verify the numerical procedure used here.
The percentage error m1 is defined as q q m1 ¼ crðwith PSÞ crðwithout PSÞ 100: qcrðwith PSÞ
3.1. Problem I: laminated circular tube under lateral pressure
3.2. Problem II: graphite/epoxy (orthotropic) cylinder under hydrostatic pressure
A clamped–clamped laminated cylinder under uniform lateral pressure (i.e. the pressure loading is not applied on the top and bottom of the cylinder) is analyzed. The results presented by Anastasiadis et al. (1994) for this problem, which are based on a first order shear deformable shell theory, are given in Tables 1 and 2 together with the numerical results obtained here using 50 equal length elements. The results are given for length to radius ratios L/R¼1, 2 ,5 and for various shell thicknesses, H. The critical wave numbers (ncr) are also given in parenthesis. The cylinder is clamped at both ends and the radius is R¼190.5 mm. Anastasiadis et al. (1994) seem to have applied the pressure at the mid-surface. Therefore, it is assumed that hydrostatic pressure acts at the midsurface. (Normally, outer surface of the cylinders should be loaded.) The material properties are given as follows
Circumferentially reinforced, i.e. [901/901], graphite/epoxy and glass/epoxy cylinders of length L, external radius 1 m constant thickness H, and L/Rext ¼10 under hydrostatic pressure are analyzed. At the ends of the shells, only radial and circumferential displacements are set equal to zero. In the present study, axial displacement was also set equal to zero at one of the ends to be able to numerically solve the problem. Results obtained using 50 equal length elements are presented in Tables 3 and 4 in comparison with the analytical results based on 3D elasticity obtained by Kardomateas (1996). In Table 3, comparison was also made with the numerical results presented by Santos et al. (2005) who formulated a 3D semi-analytical finite element, which was not based on a shell theory. The material properties are given below
E1 ¼ 206:844 109 Pa,
E2 ¼ 18:6159 109 Pa,
G12 ¼ G13 ¼ 4:48162 109 Pa G23 ¼ 2:55107 109 Pa,
n12 ¼ n13 ¼ 0:21, n23 ¼ 0:25:
a b c
Anastasiadis et al. (1994). Current study—without PS. Current study—with PS, m1 ¼ 9% error9.
ð16Þ
Graphite/epoxy: E1 ¼ 140 GPa, E2 ¼ 9:1GPa, E3 ¼ 9:9 GPa, G12 ¼ 4:3 GPa, G13 ¼ 4:7 GPa G23 ¼ 5:9 GPa, n12 ¼0.3, n13 ¼0.020, n23 ¼0.49.
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
Glass/epoxy: E1 ¼ 57 GPa, E2 ¼ E3 ¼ 14 GPa, G12 ¼ G13 ¼ 5:7 GPa, G23 ¼ 4:3 GPa, n12 ¼0.277, n13 ¼0.068, n23 ¼ 0.4.
In Tables 3–5, qcr , Rint, and Rext denote the non-dimensional buckling pressure, internal radius, and external radius of the cylinder, respectively. The axial compressive force acting on top of the cylinder is calculated as f ¼ q0 pðRint þHÞ2 . The non-dimensional buckling pressure is defined as qcr ¼
qcr R3ext E1 H3
ð17Þ
As can be seen from Tables 3 and 4, the numerical results obtained here are in very good agreement with the reference results. In order to have a better understanding, the same problems are solved again for L=Rext ¼ 5 and the results are presented in Table 5, which shows the effect of pressure stiffness on the buckling pressure for thick cylinders. As the ratio Rext/Rint increases, the difference between the cases with and without PS decreases; however, this decrease is more pronounced for the stiffer material, i.e. for graphite epoxy. The critical wave number for the results presented in Tables 3–5 is n ¼2 in all cases.
3.3. Discussion of the verification results It is noted that the numerical results given in Tables 1 and 2 are in excellent agreement with the analytical results available in the literature when PS is neglected. The same problems are also solved taking PS into account and the reduction in buckling pressures is observed. It is also observed that this reduction is more pronounced for higher L/R ratios, that is, the results become less accurate as L/R increases when PS is neglected. The results presented in Tables 3–5 reveal the significance of pressure stiffness for orthotropic cylinders. It is observed that the negative influence of PS increases with decrease in thickness. This decrease is due to the fact that the pre-buckling deformations are higher for the thinner cylinders. Santos et al. (2005) have also examined
50
qcr/h (Pax10-6/mm)
564
Table 3 Graphite/epoxy (orthotropic) cylinder under hydrostatic pressure, L/Rext ¼ 10. Rext Rint
qcr elasticitya
1.05 1.10 1.15 1.20 1.25 1.30
0.2576 0.2513 0.2347 0.2166 0.1978 0.1808
b
0.3706 0.3508 0.3417 0.3322 0.3217 0.3108
qcr current study (without PS)
qcr current study (with PS)
m2 (%)
0.3447 0.3211 0.2878 0.2535 0.2227 0.1967
0.2687 0.2581 0.2467 0.2308 0.2128 0.1948
28.3 24.4 16.7 9.8 4.7 1.0
1.05 1.10 1.15 1.20 1.25 1.30
0.2813 0.2744 0.2758 0.2764 0.2755 0.2733
20
2
3
4
5
6 n
7
qcr current study (with PS)
m2 (%)
0.3992 0.3648 0.3574 0.3501 0.3399 0.3268
0.3092 0.2859 0.2841 0.2830 0.2804 0.2764
29.1 27.6 25.8 23.7 21.2 18.2
9
10
H = 3.175 mm. H = 6.35 mm. H = 12.7 mm.
40
qcr current study (without PS)
8
50
Table 4 Glass/epoxy (orthotropic) cylinder under hydrostatic pressure, L/Rext ¼10. qcr elasticitya
30
0
Kardomateas (1996). Santos et al. (2005).
Rext Rint
40
10
current study
qcr/h (Pax10-6/mm)
a
qcr FEMb
H = 3.175 mm. H = 6.35 mm. H = 12.7 mm.
current study
30 20 10
a
0 2
3
4
5
6 n
7
8
9
10
Fig. 4. qcr/H vs. n graphs for [0/90/0]s, L/R¼ 1 and H ¼3.175, 6.35, 12.7 mm: (a) without PS and (b) with PS.
Kardomateas and Chung (1994).
Table 5 Orthotropic cylinder under hydrostatic pressure, L/Rext ¼ 5. Rext Rint
1.05 1.10 1.15 1.20 1.25 1.30
Graphite/epoxy
Glass/epoxy
qcr (without PS)
qcr (with PS)
m2 (%)
qcr (without PS)
qcr (with PS)
m2 (%)
0.4615 0.3358 0.2936 0.2585 0.2259 0.1991
0.3777 0.2810 0.2535 0.2332 0.2145 0.1964
22.2 19.5 15.8 10.8 5.3 1.4
0.8920 0.4802 0.4019 0.3698 0.3494 0.3329
0.7238 0.3932 0.3331 0.3107 0.2978 0.2879
23.2 22.1 20.7 19.0 17.3 15.6
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0.40 0.35 0.30
qcr
0.25 0.20 0.15
Rext/Rint = 1.04
0.10
Rext/Rint = 1.08 Rext/Rint = 1.16
Rext/Rint = 1.05
0.05 0
0.05
0.1
0.15
0.2
0.25 h90
0.3
0.35
0.4
0.45
0.5
Fig. 5. Curves of qcr vs. h90 for a cross-ply laminate (901/01/01/901) with L/Rext ¼10, (ncr ¼ 2).
565
the stability of orthotropic shells using an axisymmetric solid element without considering PS and have presented numerical results in comparison with the results presented by Kardomateas (1996). The discrepancies in the results presented by Santos et al. (2005) also support the significance of PS issue investigated in the present study. As expected, the influence of PS is generally less for the case L/Rext ¼5 compared to the case L/Rext ¼ 10. This suggests that the influence of PS increases with increasing L/Rext ratio, i.e. the influence is more pronounced for longer cylinders. The reason can be explained as follows; when L/Rext increases the ratio of loss of potential to the total potential increases because a greater surface area is under hydrostatic pressure. The results presented in Table 5 show that the influence of PS is closely related to the material used. The PS effect is higher for cylinders made of glass epoxy material compared to the cylinders made of carbon epoxy. The reason for this is the fact that glass epoxy is less stiff and hence will be affected more by PS. This is an important finding because glass epoxy may be preferable to carbon epoxy for underwater applications due to its lower cost; see Swanson (1997).
Fig. 6. Contour graphs for glass-epoxy, L/Rext ¼ 5, (901/01/01/901), (ncr ¼ 2 3). (a) qcr without PS, (b) qcr with PS, and (c) error m (%).
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An important observation can be made when the results in Tables 1 and 2 and Tables 3–5 are compared. The influence of PS is expected to be higher for thinner cylinders and the results presented in Tables 3–5 confirm this expectation. However, examining the error percentages in Tables 1 and 2, an opposite conclusion may be reached. In order to investigate this observation more closely, qcr/H vs. n graphs are plotted for the stacking sequence [01/901/01]s, L/R¼1 and H¼3.175, 6.35, 12.7 mm as shown in Fig. 4(a) (without PS) and 4(b) (with PS) where the critical qcr/H values corresponding to the critical wave number are shown on the graphs. Note that the critical wave numbers can be easily obtained by trial because there are no local minima. Note also that according to Sridharan and Kasagi (1997) the effect of PS is especially important in the analysis of overall buckling with n o4. The graphical results presented in Fig. 4(a) and (b) are in agreement with this statement. Comparison of Fig. 4(a) and (b) shows that the reduction in critical pressure is higher for lower modes. Returning back to the results presented in Tables 1 and 2 it can be observed that the critical wave number, ncr, decreases with increase in L/R ratio. The explanation of the contradiction lies in this observation.
The reduction in critical pressure with increase in thickness occurs because critical wave number decreases, which is more effective than the change in thickness. Therefore, it cannot be stated that PS will have less influence when thickness of the cylinder increases.
4. Optimal thickness design The effect of PS on optimal thickness design is studied next. 4.1. Problem formulation The design objective is to maximize the buckling load of the cross-ply laminated cylinder of total thickness H by optimizing the layer thicknesses, Hi, where i¼1, 2, y, NL. At the ends of the shells, only radial and circumferential displacements are set equal to zero, i.e. the cylinder is simply supported. The related optimization problem is defined below in Eq. (18). Optimization problem: determines the ply thicknesses Hi, where i¼1, 2, y, NL of a symmetric cross-ply laminate of total thickness H,
Fig. 7. Contour graphs for glass-epoxy, L/Rext ¼10, (901/01/01/901), (ncr ¼ 2 3). (a) qcr without PS, (b) qcr with PS, and (c) error m (%).
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such that the buckling parameter qcr is maximized, viz. maxqcr Hi
such that
NL X
Hi ¼ H
ð18Þ
i¼1
The ‘Golden Section Method’, which is a powerful one dimen¨ sional minimization algorithm defined in Gurdal et al. (1999), is used to obtain the optimal values.
4.2. Optimization results Next, we investigate the effect of various problem parameters on the optimal values of the design variables and the buckling pressure. The numerical results are obtained using 50 elements and material used is specified as glass-epoxy the properties of which are given in Section 3.2. A 4-layered symmetric cross-ply cylinder with a stacking sequence (901/01/01/901) is considered. The nondimensional thicknesses of the 01 and 901 plies are given by h0 ¼ H0 =Hand h90 ¼ H90 =H, respectively, where H0 and H90 are the thicknesses of 01 and 901 plies. h0 can be written in terms of h90 as h0 ¼0.5 h90 so that the only unknown is h90. The optimal value
567
of h90 will be denoted by h90, opt. The results are presented for simply supported boundary conditions. Fig. 5 shows the curves of the non-dimensional buckling parameter qcr plotted against h90 for L/Rext ¼10 and Rext/Rint ¼1.04, 1.05, 1.08, and 1.16 considering the effect of PS. It can be observed from Fig. 5 that the optimal value of h90, h90, opt, is roughly equal to 0.27 for Rext/Rint ¼1.04 and rapidly increases with increasing Rext/Rint and finally becomes equal to 0.5 for Rext/Rint ¼1.16. For Rext/Rint 41.16, h90, opt ¼0.5, that is, the 01 layer thickness vanishes. Next the effects of PS and the layer thicknesses h90 on the buckling load are investigated. Contour graphs of qcr and the percentage error m2 are shown in Figs. 6–8 for L/Rext ¼5, 10, 20 respectively, with respect to the ratio Rext/Rint and h90. Here the error m2 denotes the percentage difference in the results without and with PS effect included in the calculation of the buckling loads. The percentage error m2 is defined as qcrðwith PSÞ qcrðwithout PSÞ 100 qcrðwith PSÞ
m2 ¼
ð19Þ
Contour graphs of qcr without and with taking PS into account are shown in Fig. 6(a) and (b), respectively, for L/Rext ¼5.
Fig. 8. Contour graphs for glass-epoxy, L/Rext ¼20, (901/01/01/901), (ncr ¼ 2). (a) qcr without PS, (b) qcr with PS, (c) error m (%)
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Corresponding graphs for L/Rext ¼10 and L/Rext ¼20 are shown in Figs. 7(a) and (b), and 8(a) and (b). The critical wave number is either 2 or 3 for L/Rext ¼5 and L/Rext ¼ 10 and is equal to 2 for L/Rext ¼20. Comparison of Figs. 6(a), 7(a), and 8(a) with Figs. 6(b), 7(b), and 8(b), respectively, indicates the differences in the buckling loads with respect to Rext/Rint and h90 when PS is taken into account. Figs. 6(c) and 7(c) show that there exists a transition region in error where the critical wave number ncr changes. The critical wave number is found to be equal to 3 below and equal to 2 above this transition region. The sharp increase in m2 at the transition region with increase in h90 and Rext/Rint ratio shows that the influence of PS is higher for n ¼2 compared to n ¼3. Fig. 8(c) shows that for L/Rext ¼ 20, the error m2 varies between 27% and 30% and is highest for the lowest values of h90 and Rext/Rint ratio and gradually decreases with increasing h90 and Rext/Rint ratio and as such it is not similar to the cases for the shorter cylinders with L/Rext ¼5 and 10 shown in Figs. 6(c) and 7(c). This is due to the fact that for L/Rext ¼20, the buckling mode n¼2 while for L/Rext ¼5, Table 6 Optimization results for glass-epoxy cylinder under hydrostatic pressure with stacking sequence (901/01/01/901). L Rext
Rext Rint
With PS h90,
opt
ncr
qcr,max
h90,
5. Conclusions
m2 (%)
Without PS
qcr,max
and 10 it is n ¼2 or n ¼3 depending on h90 and Rrxt/Rint. The increase in rigidity with increasing h90 and Rext/Rint reduces the pre-buckling deformations and therefore the influence of PS on the buckling load also decreases. Fig. 8(a) and (b) shows that qcr increases with increase in h90 and is not significantly affected by Rext/Rint ratio. Thus, the Rext/Rint ratio does not affect qcr substantially for relatively long cylinders. Fig. 8(c) shows that the influence of PS is similar to the L/Rex ¼10 case with the exception that m2 depends heavily on Rext/Rint ratio rather than on h90. Optimal values of h90 for glass-epoxy cylinder under hydrostatic pressure are given in Table 6 with L/Rext ¼5 and 10 and Rext/Rint ¼ 1.04, 1.10, and 1.16. It is observed that h90, opt increases with increasing Rext/Rint ratio for L/Rext ¼10, but this behavior is not monotonous for L/Rext ¼5. h90, opt values are marginally smaller for the case without PS as compared to the case with PS. It is noted that qcr,max values corresponding to h90, opt are higher for the case without PS. The buckling mode shapes of optimal glass-epoxy cylinders with L/Rext ¼5, 10 and Rext/Rint ¼ 1.04 are shown in Fig. 9(a) and (b).
opt
ncr
5
1.04 1.10 1.16
0.7964 0.4402 0.3409
0.3631 0.2931 0.3558
3 2 2
0.8825 0.5439 0.4138
0.3614 0.2872 0.3475
3 2 2
10.8 23.6 21.4
10
1.04 1.10 1.16
0.3782 0.2892 0.2845
0.2730 0.4023 0.4480
2 2 2
0.4906 0.3695 0.3569
0.2712 0.3995 0.4441
2 2 2
29.7 27.8 25.4
An axisymmetric finite element formulation for moderately thick cross-ply cylinders accounting for Koiter’s pressure stiffness expression is presented. Several verification problems are solved and the accuracy of the element is assessed. Some new results are presented concerning the effect of pressure stiffness on the buckling load. Moreover, an optimization study is performed in order to determine the optimal thicknesses of laminated cross-ply cylinders under hydrostatic pressure. It is observed that the numerical results obtained are very close to the 3D analytical results presented by Kardomateas (1996) when pressure stiffness is taken into account. It is also observed that the influence of pressure stiffness depends on the material properties and materials with lower stiffness are influenced more by pressure stiffness. It is shown that the influence of pressure stiffness may be higher for thicker cylinders. ‘The large discrepancy between the experimental and theoretical buckling loads of thin cylindrical shells is generally attributed to geometric imperfections and there is a vast literature devoted to the analysis of geometrically imperfect cylindrical shells’, as stated by Gusic et al., 2000. The results obtained here show that the influence of pressure stiffness may also contribute to the accuracy of the results and can be as important as the influence of imperfections, and should be included in stability analysis of composite cylinders under hydrostatic pressure. Not taking the PS factor into account in the design of composite cylinders under hydrostatic pressure results in the use of higher knock-down factors which increases the production cost and the weight. It is shown here that optimal thicknesses differ only marginally when PS is considered.
Acknowledgements The financial assistance of NRF (National Research Foundation of South Africa) is gratefully acknowledged. References
Fig. 9. Buckling mode shapes of optimal glass-epoxy (901/01/01/901) cylinders, Rext/Rint ¼ 1.04, with PS, (a) L/Rext ¼5, ncr ¼3, qcr ¼ 0:7907 and (b) L/Rext ¼ 10, ncr ¼2, qcr ¼ 0:3774.
Anastasiadis, J.S., Tabiei, A., Simitses, G.J., 1994. Instability of moderately thick, laminated, cylindrical shells under combined axial compression and pressure. Composite Structures 27, 367–378. Blachut, J., Smith, P., 2008. Buckling of multi-segment underwater pressure hull. Ocean Engineering 35, 247–260.
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Liang, C.C., Chen, H.W., Jen, C.Y., 2003. Optimum design of filament–wound multilayer-sandwich submersible pressure hulls. Ocean Engineering 30, 1941–1967. Liang, C.C., Shiah, S.W., Jen, C.Y., Chen, H.W., 2004. Optimum design of multiple intersecting spheres deep-submerged pressure hull. Ocean Engineering 31, 177–199. Loganathan, K., Chang, S.C., Gallagher, R.H., Abel, J.F., 1979. Finite element representation and pressure stiffness in shell stability analysis. International Journal for Numerical Methods in Engineering 14, 1413–1429. Messager, T., Pyrz, M., Gineste, B., Chauchot, P., 2002. Optimal laminations of thin underwater composite cylindrical vessels. Composite Structures 58, 529–537. Ostwald, M., 1990. Optimum weight design of sandwich cylindrical shells under combined loads. Computers & Structures 37 (3), 247–257. Qatu, M.S., 1999. Accurate equations for laminated composite deep thick shells. International Journal of Solids and Structures 36 (19), 2917–2941. Qatu, M.S., 2004. Vibration of Laminated Plates and Shells. Elsevier, UK. Ross, C.T.F., Little, A.P.F., 2001. The buckling of a corrugated carbon fibre cylinder under external hydrostatic pressure. Ocean Engineering 28, 1247–1264. Santos, H., Mota Soares, C.M., Mota Soares, C.A., Reddy, J.N., 2005. A semi-analytical finite element model for the analysis of laminated 3D axisymmetric shells: bending, free vibration and buckling. Composite Structures 71, 273–281. Schokker, A., Sridharan, S., Kasagi, A., 1996. Dynamic buckling of composite shells. Computers & Structures 59 (1), 43–53. Smerdov, A., 2000. A computational study in optimum formulations of optimization problems on laminated cylindrical shells for buckling II. shells under external pressure. Composites Science and Technology 60, 2067–2076. Sridharan, S., Kasagi, A., 1997. On the buckling and collapse of moderately thick composite cylinders under hydrostatic pressure. Composites Part B 28B, 583–596. Subbiah, J., Natarajan, R., 1980. Stability analysis of ring stiffened shells of revolution. Computers & Structures 14 (5–6), 479–490. Swanson, S.R., 1997. Introduction to Design and Analysis with Advanced Composite Materials. Prentice-Hall, New Jersey. Tanov, R., Tabiei, A., Simitses, G.J., 1999. Effect of static preloading on the dynamic buckling of laminated cylinders under sudden pressure. Mechanics of Composite Materials and Structures 6, 195–206. Toorani, M.H., Lakis, A.A., 2000. General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects. Journal of Sound and Vibration 237 (4), 561–615. Whitney, J.M., 1987. Structural Analysis of Laminated Anisotropic Plates. Technomic, Lancaster, PA.
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Buckling of cross-ply cylinders under hydrostatic pressure considering pressure stiffness Izzet U. Cagdas a,n,1, Sarp Adali b a b
Civil Engineering Department, Faculty of Engineering, Akdeniz University, 07058 Antalya, Turkey School of Mechanical Engineering, University of KwaZulu-Natal, Durban, South Africa
a r t i c l e in f o
abstract
Article history: Received 19 January 2010 Accepted 6 December 2010 Editor-in-Chief: A.I. Incecik Available online 22 December 2010
Buckling behavior of cross-ply cylinders under hydrostatic pressure is investigated using a semianalytical finite element based on a consistent first order shear deformable shell theory. Potential loss due to external pressure, also called pressure stiffness (PS) is taken into account by making use of Koiter’s related energy expression. A number of verification problems are solved and the numerical results are compared with the analytical results available in the literature and excellent agreement is observed. New numerical results are presented to assess the effect of PS on buckling due to hydrostatic pressure. It is shown that PS causes a decrease in the buckling load and this decrease depends on the size of the cylinder and the material. Also, issues related to thickness optimization are examined and optimal lamina thicknesses are determined for a number of cases with and without PS taken into account. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Stability Cross-ply cylinder Hydrostatic pressure Finite elements Pressure stiffness
1. Introduction Cross-ply laminated composite cylinders are prone to buckling and therefore should be designed accordingly to prevent catastrophic failure. In the present study the influence of loss of potential due to external hydrostatic pressure on the stability of cross-ply cylinders is investigated. This issue is often neglected by researchers in the field, leading to an overestimation of the buckling pressure. A semi-analytical finite element, which is based on a consistent shell theory, is developed and employed to obtain the numerical solution of the problem. The consistent shell theory employed here was developed by Leissa and Chang (1996) and Qatu (1999) and was used by Qatu (1999, 2004) to solve shell vibration problems. The present study is the first application of this shell theory to hydrostatic stability problems. Comparisons with published results are given by solving a number of verification problems and it is observed that pressure stiffness negatively affects the buckling pressure. In some cases the numerical results indicate up to 30% reduction in the buckling pressure. This shows the importance of taking this effect into account in stability calculations to obtain conservative buckling loads. Pressure stiffness was first considered for circular cylindrical isotropic shells under hydrostatic pressure by Koiter (1967). Brush and Almroth (1975) have explained the phenomenon in their n
Corresponding author. Tel.: + 90 5323101166; fax: + 90 2423106306. E-mail addresses: [email protected] (I.U. Cagdas), [email protected] (S. Adali). 1 Previous address: School of Mechanical Engineering, University of KwaZuluNatal, Durban, South Africa. 0029-8018/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2010.12.005
textbook on the stability of structures. Hibbitt (1979) described pressure as a follower force and named the related stiffness terms as ‘‘pressure stiffness’’ terms. Loganathan et al. (1979) examined pressure stiffness in shell stability analysis and reported reduction in the buckling pressure. Subbiah and Natarajan (1980) developed an axisymmetric finite element for the instability analysis of ring stiffened isotropic shells of revolution and reported substantial reduction in buckling pressure due to the follower load effect. Pressure stiffness was considered in the analytical studies of Kardomateas (1996, 1997) and Kardomateas and Chung (1994) where only orthotropic cylinders were considered. Han et al. (2004) investigated buckling of long sandwich cylindrical shells under external pressure. Kasagi and Srinivasan (1995), Schokker et al. (1996), Sridharan and Kasagi (1997) and Kardomateas (1997) have also considered this effect by using the work of Koiter (1967). Kardomateas (2000) has improved the elasticity solution further by considering the effect of normal strains. Sridharan and Kasagi (1997) investigated buckling and associated non-linear response and collapse of moderately thick laminated composite cylindrical shells. Tanov et al. (1999) examined the effect of preloading on the dynamic buckling of laminated cylinders under sudden follower pressure. As can be seen from this survey so far, the literature on the influence of pressure stiffness on shell stability is not extensive. In previous studies on the optimization of composite cylinders under hydrostatic pressure, the effect of pressure stiffness was neglected. Diaconu et al. (2002), Messager et al. (2002), Smerdov (2000), Liang et al. (2003), and Ostwald (1990) studied optimization of composite cylindrical shells under combined loads including external pressure. Ross and Little (2001) investigated the buckling of
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Nomenclature i, j, k I GP L, H Le m
element nodes (letters i and j are also used as indices) the unit matrix Gauss point length and thickness of the cylinder length of a finite element total number of harmonics in the circumferential direction n number of half sine waves in the circumferential direction ncr critical value of n NL number of layers PS pressure stiffness q0 initially applied pressure qcr ,qcr dimensional and non-dimensional buckling pressures Rint, R, Rext internal, mean, and external radii of the cylinder x coordinate of a point in the curvilinear coordinate system lcr eigenvalue m1, m2 percentage errors E1, E2, E3 moduli of elasticity in 1, 2, 3 directions G12, G13, G23 moduli of shear n12, n13, n23 Poisson’s ratios Q ij transformed reduced stiffness coefficients Aij, Bij, Dij, Eij stiffness coefficients c0 coefficient ( ¼ 1/R) ^ ij modified stiffness coefficients Aij , A^ ij ,Bij , B^ ij ,Dij , D r, y, z radial, circumferential, and axial coordinates
er, ey, ez unit vectors in r, y, z directions ur, uy, uz radial, circumferential, and axial displacement components in the global c.s. a, y, z0 local coordinates of a Gauss point ea ,ezu unit vectors in a, z0 directions u, v, w displacements parallel to ea, ey, ezu directions. ca, cy :rotations of the transverse normal about y and a axes V1i, y, V3i nodal coordinates of node i V1i, V3i unit vectors in V1i,V3i directions Vai, Vyi nodal rotations of the transverse normal about unit vectors ey and V1i e1, e2, e3 unit vectors in 1, 2, 3 directions of the fiber c.s. ea, ey, gay, gyzu , gazu linear strain components in the local c.s. e0a, e0y, e0ya, e0ay, g0yzu linear (mid-surface) strain components in the local c.s. wa, wy, way, wya curvatures in the local c.s. nl nl nl enl 0a , e0y , e0ay , e0ya non-linear strain components in the local c.s. Be, Bw, Bf strain–displacement matrices Na, Ny, Nay, Nya force resultants in the local c.s. Ma, My, May, Mya moments in the local c.s. Qa, Qy shear forces in the local c.s. N, M, Q vectors of forces, moments, and shear forces Ue element strain energy Ve element potential energy of the applied stresses S a matrix of membrane stresses G a vector of derivatives of in-plane deformations at a Gauss point Kn, KnG, Knq global stiffness, geometric stiffness, and pressure stiffness matrices corresponding to nth harmonic
a corrugated carbon fiber cylinder under external hydrostatic pressure. Blachut and Smith (2008) examined stability of a multisegment underwater pressure hull. Kruzelecki and Trzeciak (2000) and Liang et al. (2004) optimized axially symmetrical shells and spheres under hydrostatic pressure. The effect of pressure stiffness on optimal thickness design is studied here for the first time. In the present study, the layer thicknesses are taken as design variables for the optimal design of cross-ply cylinders. The objective of the optimization is to maximize the external buckling pressure taking the pressure stiffness into account. It is shown that optimal lamina thickness differs when pressure stiffness is considered as compared to the case when PS is neglected. However the difference in the optimal layer thicknesses for these cases is found to be marginal even though the buckling loads can differ substantially when PS is taken into account. The details of the finite element method employed in the present study are given in Section 2. To assess the accuracy of the FEM, two verification problems are solved in Section 3. The solutions of these problems also serve to elucidate the effect of PS on the buckling load as a function of the stiffness of the material and the thickness of the cylinders. Results on the optimal thicknesses of the cross-ply cylinders are given in Section 4, which provides a detailed study of the title problem. Several conclusions are summarized in Section 5 based on the numerical results obtained in the present study.
thickness H. The position of a point on the cylinder is defined by the cylindrical coordinates r, y, z where (r, y, z) denote the radial, circumferential, and axial coordinates and (ur, uy, uz) denote the radial, circumferential, and axial displacement components, respectively. A local coordinate system (a, y, z0 ) is defined at a Gauss point on the mid-surface of the cylinder where u, v, and w denote the displacements parallel to a, y, and z0 coordinates and ca and cy are the rotations of the transverse normal about a and y axes. The coordinate of a point in the curvilinear element (shape function) coordinate system is denoted by (x). A three noded shell finite element of length ‘Le’ is schematically shown in Fig. 1. The global, nodal, and local displacement components and unit vectors are shown in Fig. 2. The model is based on the following displacement field: 9 9 8 2 38 ur > cn 0 0 0 0 > unr > > > > > > > > > n> > > > > 6 7 > > 0 sn 0 0 0 7> > > > uy > = < uy > m 6 X 6 7< n = 6 0 0 cn 0 0 7 uz uz ¼ ð1Þ 6 7> > > > > > 7> n¼06 0 > > > > Vani > V 0 0 0 c > > > > 4 5 n a i > > > > > > ; ; :V > : Vn > 0 0 0 0 sn yi yi
2. Finite element formulation
2.1. Strain–displacement relations
In the present section formulation of an axisymmetric (cylindrical) shell element is given based on a thick shell theory developed by Leissa and Chang (1996) and Qatu (1999). A laminated composite cylinder under uniform hydrostatic loading is considered, which is of length L, mean radius R, and
The local displacement field corresponding to the first order shear deformation theory is given as
where cn, sn, and m denote cos(ny), sin(ny), and the total number of harmonics in the circumferential direction, respectively, and n stands for the nth harmonic in the circumferential direction. Vai and Vyi are the nodal rotations.
uða, y,zuÞ ¼ u0 ða þ yÞ þzuca ða, yÞ
ð2:1Þ
vða, y,zuÞ ¼ v0 ða þ yÞ þzucy ða, yÞ
ð2:2Þ
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
wða, y,zuÞ ¼ w0 ða, yÞ
ð2:3Þ
Accordingly, the linear and non-linear strain–displacement relations are
ea ¼ e0a þzuwa ey ¼
ð3:1Þ
1 ðe0y þ zuwy Þ 1 þ zu=R
enl 0a ¼
i 1h ðu0, a Þ2 þ ðv0, a Þ2 þ ðw0, a Þ2 2
enl 0y ¼
i 1 h ðu0, y Þ2 þðv0, y Þ2 þ ðw0, y Þ2 þ 2w0 v0, y 2v0 w0, y þ v20 þ w20 2 2R ð3:7Þ
gay ¼ e0ay þ gyzu ¼
1 1 e0ya þzu wya þ way 1 þ zu=R 1 þ zu=R
1 R
ð3:8Þ
1 R
ð3:9Þ
ðv0 w0, a þ w0 v0, a Þ enl 0ay ¼
ð3:3Þ
ðu0, a u0, y þ v0, a v0, y þ w0, a w0, y Þ enl 0ya ¼ where 9 9 8 8 ðu0, a Þ e0a > > > > > > > > > > > > > > > 1 = < e0y = < ðw0 þ v0, y Þ > R , ¼ e0 ¼ Þ ðv e > > > > 0, a 0ay > > > > > > > > > > > > 1 :e ; : ðu0, y Þ ;
ð3:4Þ
gazu ¼ ca þ w0, a
ð3:5Þ
0ya
z, uz
(
z, uz
u¼
Rext
¼
ya
R
a, y
)
ca þ w0, a : vR0 þ cy
w0, y R
2.2. The force and moment resultants
j
The force and moment resultants, schematically shown in Fig. 3, are given below; see Qatu (2004). 9 2 9 8 38 Na > e0a > > A11 A12 A16 A16 > > > > > > > > > > 6 > > 7> < Ny = 6 A12 A^ 22 A26 A^ 26 7< e0y = 7 ¼6 6 7 Na y > e0ay > > > > > > 4 A16 A26 A66 A66 5> > > > > > > > > : Nya ; A16 A^ 26 A66 A^ 66 : e0ya ; 9 2 38 wa > B11 B12 B16 B16 > > > > > > 6 7> 6 B12 B^ 22 B26 B^ 26 7< wy = 6 7 þ6 ð4Þ 7 way > > > 4 B16 B26 B66 B66 5> > > > > ; : ^ ^ wya B16 B26 B66 B66
Le
Le
Rint j
z', w
Gauss Point k
(
i
Rint
R
g0azu g0yz
R
)
9 8 ðca, a Þ > > > > > > > 1 = < ðcy, y Þ > y R , ¼ v¼ c Þ ð w > > > y, a > ay > > > > > > > > > > > > : w ; : 1 ðc Þ ; 9 8 wa > > > > > > > =
The linear terms of the strain–displacement relations used here were also derived by Qatu (1999) and Toorani and Lakis (2000).
H i
ð3:6Þ
ð3:2Þ
1 ðg0yzu Þ 1þ zu=R
561
k H
, u
r, ur
r, ur
Fig. 1. Details of the element geometry and the coordinate systems. (a) Global coordinate system and (b) local coordinate system.
z, uz mid-surface
mid-surface uzi
Vαi uθi
i
eθ i
uri
V3i V1i
Vθi θ
v0
GP
eθ,e2 w
u0
R
ez
ez', e3
GP
eα,e1 eθ
uθ r, ur
er
Fig. 2. Global, nodal and local variables: (a) displacement components and (b) unit vectors.
562
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
Nα
Mαθ
Qα Nαθ
Mα
Nθα Qθ
eθ
Nθ
ez' eα
Nθ Qθ Nθα
Mθ eθ
Mθα
ez'
eα
Mθ
Mθα
Mα
Nαθ Qα Mαθ
Nα
Fig. 3. Force resultants and moments in the local coordinate system: (a) force resultant and (b) moments.
9 8 Ma > > > > > > > = < My >
9 38 > > e0a > B11 B12 B16 B16 > > > > > 6 7 6 B12 B^ 22 B26 B^ 26 7< e0y = 7 ¼6 6 7> e0ay > May > > > > > > 4 B16 B26 B66 B66 5> > > > > > > ; :M > ^ ^ B16 B26 B66 B66 : e0ya ; ya 9 2 38 wa > D11 D12 D16 D16 > > > > > > > 6 ^ 22 D26 D ^ 26 7 6 D12 D 7< wy = 6 7 þ6 7 w > > 4 D16 D26 D66 D66 5> > > ay > ; :w > ^ 26 D66 D ^ 66 > D16 D ya (
Qa Qy
) ¼
2
" 5 A55 6 A45
A45 A^ 44
#(
gazu g0yzu
Minimization of the total potential energy and consecutive integration in circumferential direction yield the element stiffness matrix given below in Eq. (9) Z 1 n Kne ¼ kp ðBTw DBw þ BTe BBw þBTf CBf þ BTe ABe þ BTw BBe Þ ðz, x ÞR dx 1
ð9Þ ð5Þ where, k ¼2 for n ¼0, and k¼1 for n ¼1, y, m. Be, Bw, Bf are the strain–displacement matrices.
)
2.4. Stability analysis ð6Þ
The rigidity terms are given below (see Leissa and Chang, 1996) Aij ¼ Aij c0 Bij
A^ ij ¼ Aij þc0 Bij
ð7:1Þ
Bij ¼ Bij c0 Dij
B^ ij ¼ Bij þc0 Dij
ð7:2Þ
Dij ¼ Dij c0 Eij
^ ij ¼ Dij þ c0 Eij D
ð7:3Þ
where c0 ¼ 1=R, i,j ¼ 1,2,4,5,6 and the matrices A, B, D, and E are given by Aij ¼
NL X
ðQ ij Þk ðzku zuk1 Þ
The potential energy of the applied stresses, Ve arising from the action of the in-plane stresses on the corresponding second order strains is given by Eq. (10) Z Ve ¼ ðNT e0NL Þ dA ð10Þ A
where N is the vector of in-plane forces at the Gauss points that are calculated by a primary static analysis. Minimization of the potential energy of the in-plane forces yields the element geometric stiffness matrix given by: Z 1 Ke,n ðGT SGÞn ðz, x ÞR dx ð11Þ G ¼ kp 1
k¼1
Bij ¼
NL 1X ðQ Þ ðzu2 zu2k1 Þ 2 k ¼ 1 ij k k
Dij ¼
NL 1X ðQ Þ ðzu3 zu2k1 Þ 3 k ¼ 1 ij k k
Eij ¼
NL 1X ðQ Þ ðzu4 zu4k1 Þ 4 k ¼ 1 ij k k
where S is a matrix of membrane stresses and G is a vector of derivatives of in-plane deformations at a Gauss point. The buckling load parameter lcr, which is defined as the ratio of actual buckling pressure to the initially applied pressure (q0) at which the shell buckles, is obtained by solving the eigenvalue problem detðKn lcr KnG Þ ¼ 0 n
where z0 k, z0 k 1 are the coordinates of the lower and upper surfaces of kth layer in z0 direction, and NL is the number of layers. Q ij are the transformed reduced stiffness coefficients defined by Whitney (1987). These modified stress resultant equations are a direct consequence of the ð1 þ z=RÞ term in the kinematic relations, as stated by Bert (1967) and Qatu (1999, 2004). 2.3. Element stiffness matrix The strain energy Ue of element e can be written as: Z 1 ðNT e0 þ MT v þ Q T uÞ dA Ue ¼ 2 A where N ¼ Ae0 þ Bv, M ¼ Be0 þDv, and Q ¼ Cu.
ð8Þ
ð12Þ KnG
is the global geometric where K is the global stiffness matrix and stiffness matrix corresponding to the nth harmonic. This eigenvalue problem can be modified by multiplying ðKn lcr KnG Þ by the inverse of Kn and then dividing by lcr. This operation yields the modified eigenvalue problem 1 I ¼0 ð13Þ det ðKn Þ1 KnG
lcr
1 where I denotes the unit matrix. The highest eigenvalue (lcr ) obtained by the solution of this eigenvalue problem corresponds to the lowest value of the buckling load parameter lcr. The corresponding buckling pressure is equal to qcr ¼ q0 lcr . The critical value of n, and the corresponding buckling parameter are determined by trial. The numerical results presented below are obtained by four point Gauss integration and using Matlabs built-in function ‘eigs’.
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
2.5. Pressure stiffness
Table 1 Critical values of pressure (Pa 10 6) for clamped [90/90/90]s cylinders. ˚ ˚ ˚
The expression giving the potential loss due to external pressure used in this study is given below Z ^ p ¼ 1 lp½wðe0a þ e0y Þ þðuc þ vc Þ dS P ð14Þ a y 2 where u, v, e0a, e0y, ca, and cy denote displacement in a direction, displacement in y direction, membrane strain component in a direction, membrane strain component in y direction, and rotations, respectively. Note that a similar expression was also used by Jha and Inman (2004) who studied vibration of a gossamer structure. The buckling load parameter for the externally pressurized case can be obtained by solving the modified eigenvalue problem n
det½K
lcr ðKnG þ Knq Þ
563
¼0
ð15Þ
where Knq is the global pressure stiffness matrix corresponding to the nth harmonic.
3. Verification problems
H (mm)
L R
qcra
3.175
1 2 5
3.450 (5) 1.940 (4) 0.951 (3)
3.4173 (5) 1.9465 (4) 0.9414 (3)
3.3015 (5) 1.8380 (4) 0.8437 (3)
3.5 5.9 11.6
6.35
1 2 5
18.61 (5) 11.03 (3) 5.929 (3)
18.5211 (5) 11.0876 (3) 5.9902 (3)
17.9577 (5) 10.0370 (3) 5.3888 (3)
3.1 10.5 11.2
12.7
1 2 5
94.45 (4) 52.60 (3) 28.10 (2)
92.2937 (4) 52.5812 (3) 28.2758 (2)
88.6725 (4) 48.2255 (3) 22.2208 (2)
4.1 9.0 27.3
a b c
qcrb
qcrc
m1 (%)
Anastasiadis et al. (1994). Current study, without PS. Current study, with PS, m1 ¼ 9% error9.
Table 2 Critical values of pressure (Pa 10 6) for clamped [01/901/01]s cylinders. H (mm)
L R
qcra
qcrb
qcrc
3.175
1 2 5
2.344 (7) 1.069 (5) 0.517 (4)
2.2834 (7) 1.0857 (5) 0.5321 (4)
2.2409 (7) 1.0443 (5) 0.4997 (4)
1.9 4.0 6.5
6.35
1 2 5
15.03 (6) 6.205 (4) 2.758 (3)
14.8788 (6) 6.1752 (4) 2.7941 (3)
14.5172 (6) 5.8202 (4) 2.4956 (3)
2.5 6.1 12.0
12.7
1 2 5
89.62 (5) 35.16 (4) 15.99 (3)
88.2243 (6) 34.7196 (4) 16.2949 (3)
86.3514 (6) 32.9215 (4) 14.6529 (3)
2.2 5.5 11.2
m1 (%)
Next two verification problems are solved in order to compare the results obtained by the finite element formulation developed above with available results in the literature. This enables us to study the effect of pressure stiffness on the buckling loads. Problem I: First, results presented by Anastasiadis et al. (1994) are duplicated, even though pressure stiffness was not taken into account in this study. In general, excellent agreement is observed with the reference results when pressure stiffness is neglected. Same problems are also solved by taking PS into account and the results obtained here indicate considerable reductions in buckling loads. Problem II: As stated before, stability results involving pressure stiffness are scarce. However, the studies of Kardomateas (1996) and Kardomateas and Chung (1994), where results were obtained by a 3D analytical solution, provide reliable results. Even though their results are valid only for a specific boundary condition and loading, and only applicable to orthotropic cylinders, they provide valuable information to verify the numerical procedure used here.
The percentage error m1 is defined as q q m1 ¼ crðwith PSÞ crðwithout PSÞ 100: qcrðwith PSÞ
3.1. Problem I: laminated circular tube under lateral pressure
3.2. Problem II: graphite/epoxy (orthotropic) cylinder under hydrostatic pressure
A clamped–clamped laminated cylinder under uniform lateral pressure (i.e. the pressure loading is not applied on the top and bottom of the cylinder) is analyzed. The results presented by Anastasiadis et al. (1994) for this problem, which are based on a first order shear deformable shell theory, are given in Tables 1 and 2 together with the numerical results obtained here using 50 equal length elements. The results are given for length to radius ratios L/R¼1, 2 ,5 and for various shell thicknesses, H. The critical wave numbers (ncr) are also given in parenthesis. The cylinder is clamped at both ends and the radius is R¼190.5 mm. Anastasiadis et al. (1994) seem to have applied the pressure at the mid-surface. Therefore, it is assumed that hydrostatic pressure acts at the midsurface. (Normally, outer surface of the cylinders should be loaded.) The material properties are given as follows
Circumferentially reinforced, i.e. [901/901], graphite/epoxy and glass/epoxy cylinders of length L, external radius 1 m constant thickness H, and L/Rext ¼10 under hydrostatic pressure are analyzed. At the ends of the shells, only radial and circumferential displacements are set equal to zero. In the present study, axial displacement was also set equal to zero at one of the ends to be able to numerically solve the problem. Results obtained using 50 equal length elements are presented in Tables 3 and 4 in comparison with the analytical results based on 3D elasticity obtained by Kardomateas (1996). In Table 3, comparison was also made with the numerical results presented by Santos et al. (2005) who formulated a 3D semi-analytical finite element, which was not based on a shell theory. The material properties are given below
E1 ¼ 206:844 109 Pa,
E2 ¼ 18:6159 109 Pa,
G12 ¼ G13 ¼ 4:48162 109 Pa G23 ¼ 2:55107 109 Pa,
n12 ¼ n13 ¼ 0:21, n23 ¼ 0:25:
a b c
Anastasiadis et al. (1994). Current study—without PS. Current study—with PS, m1 ¼ 9% error9.
ð16Þ
Graphite/epoxy: E1 ¼ 140 GPa, E2 ¼ 9:1GPa, E3 ¼ 9:9 GPa, G12 ¼ 4:3 GPa, G13 ¼ 4:7 GPa G23 ¼ 5:9 GPa, n12 ¼0.3, n13 ¼0.020, n23 ¼0.49.
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
Glass/epoxy: E1 ¼ 57 GPa, E2 ¼ E3 ¼ 14 GPa, G12 ¼ G13 ¼ 5:7 GPa, G23 ¼ 4:3 GPa, n12 ¼0.277, n13 ¼0.068, n23 ¼ 0.4.
In Tables 3–5, qcr , Rint, and Rext denote the non-dimensional buckling pressure, internal radius, and external radius of the cylinder, respectively. The axial compressive force acting on top of the cylinder is calculated as f ¼ q0 pðRint þHÞ2 . The non-dimensional buckling pressure is defined as qcr ¼
qcr R3ext E1 H3
ð17Þ
As can be seen from Tables 3 and 4, the numerical results obtained here are in very good agreement with the reference results. In order to have a better understanding, the same problems are solved again for L=Rext ¼ 5 and the results are presented in Table 5, which shows the effect of pressure stiffness on the buckling pressure for thick cylinders. As the ratio Rext/Rint increases, the difference between the cases with and without PS decreases; however, this decrease is more pronounced for the stiffer material, i.e. for graphite epoxy. The critical wave number for the results presented in Tables 3–5 is n ¼2 in all cases.
3.3. Discussion of the verification results It is noted that the numerical results given in Tables 1 and 2 are in excellent agreement with the analytical results available in the literature when PS is neglected. The same problems are also solved taking PS into account and the reduction in buckling pressures is observed. It is also observed that this reduction is more pronounced for higher L/R ratios, that is, the results become less accurate as L/R increases when PS is neglected. The results presented in Tables 3–5 reveal the significance of pressure stiffness for orthotropic cylinders. It is observed that the negative influence of PS increases with decrease in thickness. This decrease is due to the fact that the pre-buckling deformations are higher for the thinner cylinders. Santos et al. (2005) have also examined
50
qcr/h (Pax10-6/mm)
564
Table 3 Graphite/epoxy (orthotropic) cylinder under hydrostatic pressure, L/Rext ¼ 10. Rext Rint
qcr elasticitya
1.05 1.10 1.15 1.20 1.25 1.30
0.2576 0.2513 0.2347 0.2166 0.1978 0.1808
b
0.3706 0.3508 0.3417 0.3322 0.3217 0.3108
qcr current study (without PS)
qcr current study (with PS)
m2 (%)
0.3447 0.3211 0.2878 0.2535 0.2227 0.1967
0.2687 0.2581 0.2467 0.2308 0.2128 0.1948
28.3 24.4 16.7 9.8 4.7 1.0
1.05 1.10 1.15 1.20 1.25 1.30
0.2813 0.2744 0.2758 0.2764 0.2755 0.2733
20
2
3
4
5
6 n
7
qcr current study (with PS)
m2 (%)
0.3992 0.3648 0.3574 0.3501 0.3399 0.3268
0.3092 0.2859 0.2841 0.2830 0.2804 0.2764
29.1 27.6 25.8 23.7 21.2 18.2
9
10
H = 3.175 mm. H = 6.35 mm. H = 12.7 mm.
40
qcr current study (without PS)
8
50
Table 4 Glass/epoxy (orthotropic) cylinder under hydrostatic pressure, L/Rext ¼10. qcr elasticitya
30
0
Kardomateas (1996). Santos et al. (2005).
Rext Rint
40
10
current study
qcr/h (Pax10-6/mm)
a
qcr FEMb
H = 3.175 mm. H = 6.35 mm. H = 12.7 mm.
current study
30 20 10
a
0 2
3
4
5
6 n
7
8
9
10
Fig. 4. qcr/H vs. n graphs for [0/90/0]s, L/R¼ 1 and H ¼3.175, 6.35, 12.7 mm: (a) without PS and (b) with PS.
Kardomateas and Chung (1994).
Table 5 Orthotropic cylinder under hydrostatic pressure, L/Rext ¼ 5. Rext Rint
1.05 1.10 1.15 1.20 1.25 1.30
Graphite/epoxy
Glass/epoxy
qcr (without PS)
qcr (with PS)
m2 (%)
qcr (without PS)
qcr (with PS)
m2 (%)
0.4615 0.3358 0.2936 0.2585 0.2259 0.1991
0.3777 0.2810 0.2535 0.2332 0.2145 0.1964
22.2 19.5 15.8 10.8 5.3 1.4
0.8920 0.4802 0.4019 0.3698 0.3494 0.3329
0.7238 0.3932 0.3331 0.3107 0.2978 0.2879
23.2 22.1 20.7 19.0 17.3 15.6
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
0.40 0.35 0.30
qcr
0.25 0.20 0.15
Rext/Rint = 1.04
0.10
Rext/Rint = 1.08 Rext/Rint = 1.16
Rext/Rint = 1.05
0.05 0
0.05
0.1
0.15
0.2
0.25 h90
0.3
0.35
0.4
0.45
0.5
Fig. 5. Curves of qcr vs. h90 for a cross-ply laminate (901/01/01/901) with L/Rext ¼10, (ncr ¼ 2).
565
the stability of orthotropic shells using an axisymmetric solid element without considering PS and have presented numerical results in comparison with the results presented by Kardomateas (1996). The discrepancies in the results presented by Santos et al. (2005) also support the significance of PS issue investigated in the present study. As expected, the influence of PS is generally less for the case L/Rext ¼5 compared to the case L/Rext ¼ 10. This suggests that the influence of PS increases with increasing L/Rext ratio, i.e. the influence is more pronounced for longer cylinders. The reason can be explained as follows; when L/Rext increases the ratio of loss of potential to the total potential increases because a greater surface area is under hydrostatic pressure. The results presented in Table 5 show that the influence of PS is closely related to the material used. The PS effect is higher for cylinders made of glass epoxy material compared to the cylinders made of carbon epoxy. The reason for this is the fact that glass epoxy is less stiff and hence will be affected more by PS. This is an important finding because glass epoxy may be preferable to carbon epoxy for underwater applications due to its lower cost; see Swanson (1997).
Fig. 6. Contour graphs for glass-epoxy, L/Rext ¼ 5, (901/01/01/901), (ncr ¼ 2 3). (a) qcr without PS, (b) qcr with PS, and (c) error m (%).
566
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
An important observation can be made when the results in Tables 1 and 2 and Tables 3–5 are compared. The influence of PS is expected to be higher for thinner cylinders and the results presented in Tables 3–5 confirm this expectation. However, examining the error percentages in Tables 1 and 2, an opposite conclusion may be reached. In order to investigate this observation more closely, qcr/H vs. n graphs are plotted for the stacking sequence [01/901/01]s, L/R¼1 and H¼3.175, 6.35, 12.7 mm as shown in Fig. 4(a) (without PS) and 4(b) (with PS) where the critical qcr/H values corresponding to the critical wave number are shown on the graphs. Note that the critical wave numbers can be easily obtained by trial because there are no local minima. Note also that according to Sridharan and Kasagi (1997) the effect of PS is especially important in the analysis of overall buckling with n o4. The graphical results presented in Fig. 4(a) and (b) are in agreement with this statement. Comparison of Fig. 4(a) and (b) shows that the reduction in critical pressure is higher for lower modes. Returning back to the results presented in Tables 1 and 2 it can be observed that the critical wave number, ncr, decreases with increase in L/R ratio. The explanation of the contradiction lies in this observation.
The reduction in critical pressure with increase in thickness occurs because critical wave number decreases, which is more effective than the change in thickness. Therefore, it cannot be stated that PS will have less influence when thickness of the cylinder increases.
4. Optimal thickness design The effect of PS on optimal thickness design is studied next. 4.1. Problem formulation The design objective is to maximize the buckling load of the cross-ply laminated cylinder of total thickness H by optimizing the layer thicknesses, Hi, where i¼1, 2, y, NL. At the ends of the shells, only radial and circumferential displacements are set equal to zero, i.e. the cylinder is simply supported. The related optimization problem is defined below in Eq. (18). Optimization problem: determines the ply thicknesses Hi, where i¼1, 2, y, NL of a symmetric cross-ply laminate of total thickness H,
Fig. 7. Contour graphs for glass-epoxy, L/Rext ¼10, (901/01/01/901), (ncr ¼ 2 3). (a) qcr without PS, (b) qcr with PS, and (c) error m (%).
I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
such that the buckling parameter qcr is maximized, viz. maxqcr Hi
such that
NL X
Hi ¼ H
ð18Þ
i¼1
The ‘Golden Section Method’, which is a powerful one dimen¨ sional minimization algorithm defined in Gurdal et al. (1999), is used to obtain the optimal values.
4.2. Optimization results Next, we investigate the effect of various problem parameters on the optimal values of the design variables and the buckling pressure. The numerical results are obtained using 50 elements and material used is specified as glass-epoxy the properties of which are given in Section 3.2. A 4-layered symmetric cross-ply cylinder with a stacking sequence (901/01/01/901) is considered. The nondimensional thicknesses of the 01 and 901 plies are given by h0 ¼ H0 =Hand h90 ¼ H90 =H, respectively, where H0 and H90 are the thicknesses of 01 and 901 plies. h0 can be written in terms of h90 as h0 ¼0.5 h90 so that the only unknown is h90. The optimal value
567
of h90 will be denoted by h90, opt. The results are presented for simply supported boundary conditions. Fig. 5 shows the curves of the non-dimensional buckling parameter qcr plotted against h90 for L/Rext ¼10 and Rext/Rint ¼1.04, 1.05, 1.08, and 1.16 considering the effect of PS. It can be observed from Fig. 5 that the optimal value of h90, h90, opt, is roughly equal to 0.27 for Rext/Rint ¼1.04 and rapidly increases with increasing Rext/Rint and finally becomes equal to 0.5 for Rext/Rint ¼1.16. For Rext/Rint 41.16, h90, opt ¼0.5, that is, the 01 layer thickness vanishes. Next the effects of PS and the layer thicknesses h90 on the buckling load are investigated. Contour graphs of qcr and the percentage error m2 are shown in Figs. 6–8 for L/Rext ¼5, 10, 20 respectively, with respect to the ratio Rext/Rint and h90. Here the error m2 denotes the percentage difference in the results without and with PS effect included in the calculation of the buckling loads. The percentage error m2 is defined as qcrðwith PSÞ qcrðwithout PSÞ 100 qcrðwith PSÞ
m2 ¼
ð19Þ
Contour graphs of qcr without and with taking PS into account are shown in Fig. 6(a) and (b), respectively, for L/Rext ¼5.
Fig. 8. Contour graphs for glass-epoxy, L/Rext ¼20, (901/01/01/901), (ncr ¼ 2). (a) qcr without PS, (b) qcr with PS, (c) error m (%)
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I.U. Cagdas, S. Adali / Ocean Engineering 38 (2011) 559–569
Corresponding graphs for L/Rext ¼10 and L/Rext ¼20 are shown in Figs. 7(a) and (b), and 8(a) and (b). The critical wave number is either 2 or 3 for L/Rext ¼5 and L/Rext ¼ 10 and is equal to 2 for L/Rext ¼20. Comparison of Figs. 6(a), 7(a), and 8(a) with Figs. 6(b), 7(b), and 8(b), respectively, indicates the differences in the buckling loads with respect to Rext/Rint and h90 when PS is taken into account. Figs. 6(c) and 7(c) show that there exists a transition region in error where the critical wave number ncr changes. The critical wave number is found to be equal to 3 below and equal to 2 above this transition region. The sharp increase in m2 at the transition region with increase in h90 and Rext/Rint ratio shows that the influence of PS is higher for n ¼2 compared to n ¼3. Fig. 8(c) shows that for L/Rext ¼ 20, the error m2 varies between 27% and 30% and is highest for the lowest values of h90 and Rext/Rint ratio and gradually decreases with increasing h90 and Rext/Rint ratio and as such it is not similar to the cases for the shorter cylinders with L/Rext ¼5 and 10 shown in Figs. 6(c) and 7(c). This is due to the fact that for L/Rext ¼20, the buckling mode n¼2 while for L/Rext ¼5, Table 6 Optimization results for glass-epoxy cylinder under hydrostatic pressure with stacking sequence (901/01/01/901). L Rext
Rext Rint
With PS h90,
opt
ncr
qcr,max
h90,
5. Conclusions
m2 (%)
Without PS
qcr,max
and 10 it is n ¼2 or n ¼3 depending on h90 and Rrxt/Rint. The increase in rigidity with increasing h90 and Rext/Rint reduces the pre-buckling deformations and therefore the influence of PS on the buckling load also decreases. Fig. 8(a) and (b) shows that qcr increases with increase in h90 and is not significantly affected by Rext/Rint ratio. Thus, the Rext/Rint ratio does not affect qcr substantially for relatively long cylinders. Fig. 8(c) shows that the influence of PS is similar to the L/Rex ¼10 case with the exception that m2 depends heavily on Rext/Rint ratio rather than on h90. Optimal values of h90 for glass-epoxy cylinder under hydrostatic pressure are given in Table 6 with L/Rext ¼5 and 10 and Rext/Rint ¼ 1.04, 1.10, and 1.16. It is observed that h90, opt increases with increasing Rext/Rint ratio for L/Rext ¼10, but this behavior is not monotonous for L/Rext ¼5. h90, opt values are marginally smaller for the case without PS as compared to the case with PS. It is noted that qcr,max values corresponding to h90, opt are higher for the case without PS. The buckling mode shapes of optimal glass-epoxy cylinders with L/Rext ¼5, 10 and Rext/Rint ¼ 1.04 are shown in Fig. 9(a) and (b).
opt
ncr
5
1.04 1.10 1.16
0.7964 0.4402 0.3409
0.3631 0.2931 0.3558
3 2 2
0.8825 0.5439 0.4138
0.3614 0.2872 0.3475
3 2 2
10.8 23.6 21.4
10
1.04 1.10 1.16
0.3782 0.2892 0.2845
0.2730 0.4023 0.4480
2 2 2
0.4906 0.3695 0.3569
0.2712 0.3995 0.4441
2 2 2
29.7 27.8 25.4
An axisymmetric finite element formulation for moderately thick cross-ply cylinders accounting for Koiter’s pressure stiffness expression is presented. Several verification problems are solved and the accuracy of the element is assessed. Some new results are presented concerning the effect of pressure stiffness on the buckling load. Moreover, an optimization study is performed in order to determine the optimal thicknesses of laminated cross-ply cylinders under hydrostatic pressure. It is observed that the numerical results obtained are very close to the 3D analytical results presented by Kardomateas (1996) when pressure stiffness is taken into account. It is also observed that the influence of pressure stiffness depends on the material properties and materials with lower stiffness are influenced more by pressure stiffness. It is shown that the influence of pressure stiffness may be higher for thicker cylinders. ‘The large discrepancy between the experimental and theoretical buckling loads of thin cylindrical shells is generally attributed to geometric imperfections and there is a vast literature devoted to the analysis of geometrically imperfect cylindrical shells’, as stated by Gusic et al., 2000. The results obtained here show that the influence of pressure stiffness may also contribute to the accuracy of the results and can be as important as the influence of imperfections, and should be included in stability analysis of composite cylinders under hydrostatic pressure. Not taking the PS factor into account in the design of composite cylinders under hydrostatic pressure results in the use of higher knock-down factors which increases the production cost and the weight. It is shown here that optimal thicknesses differ only marginally when PS is considered.
Acknowledgements The financial assistance of NRF (National Research Foundation of South Africa) is gratefully acknowledged. References
Fig. 9. Buckling mode shapes of optimal glass-epoxy (901/01/01/901) cylinders, Rext/Rint ¼ 1.04, with PS, (a) L/Rext ¼5, ncr ¼3, qcr ¼ 0:7907 and (b) L/Rext ¼ 10, ncr ¼2, qcr ¼ 0:3774.
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