Hydrostatic Buckling of Shells

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Journal of Constructional Steel Research 56 (2000) 1–16

www.elsevier.com/locate/jcsr

Hydrostatic buckling of shells with variousboundary conditions

Rodney Pinna a,*, Beverley F. Ronalds b

a Centre for Offshore Foundation Systems, The University of Western Australia, Nedlands, Perth, W.A.6907, Australia

b Centre for Oil & Gas Engineering, The University of Western Australia, Nedlands, Perth, W.A. 6907,

Australia

Received 3 June 1999; accepted 24 November 1999

Abstract

Eigenvalue buckling of cylindrical shells with various boundary conditions under hydrostatic

load is examined, using an energy method. Results are compared to known solutions, wherethese solutions exist. It is found that, for shells of intermediate length, buckling loads fordifferent end conditions may be determined by applying a simple, scalar multiplier to the pin-ended case. This does not apply to long shells, where the circumferential wave number n3.For n=2, the ring equation may be applied to all cases, as the boundary conditions no longerinfluence the solution. It is seen for the case of a shell with one end pinned and the other endfree that the buckling solution collapses to the long shell solution, for geometries of practicalinterest. The effect of radial elastic restraint at the open end is also examined, as an intermedi-ate case between pinned and free ends. The work has application to the design of suctioncaissons, where cylinder dimensions are usually in the range of intermediate length shells. ©2000 Elsevier Science Ltd. All rights reserved.

Keywords: Boundary conditions; Buckling; Elastic restraint; Shells; Suction caissons

1. Introduction

A novel application for cylindrical shells which has found increasing use is thatof the suction caisson. These have been used as a foundation system for a numberof offshore petroleum production facilities [1,2]. Such foundations are essentiallycylinders, with one end closed by an end cap, and the other end open. The upper

* Corresponding author.

0143-974X/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 1 4 3 - 9 7 4 X ( 9 9 ) 0 0 1 0 4 - 2



2 R. Pinna, B.F. Ronalds / Journal of Constructional Steel Research 56 (2000) 1–16

Nomenclature

a buckling multipler for different boundary conditions z1,q1, zq1 shell curvaturese z1,eq1,g zq1 shell strains

Ai undetermined coefficient

C Eh

1−v2, shell membrane stiffness

D Eh3

12(1−v2), shell bending stiffness

j number of terms in series solution

k ∗

f non-dimensionalised spring constantk f spring foundation modulusm, n longitudinal and circumferential wave numbers

N z0, N q0, N zq0 pre-buckle membrane force per unit lengthP∗

cr non-dimensionalised shell buckling loadPcr shell buckling pressureu1, v1, w1 displacements in the r , q and z directionsV potential energy function

Z Batdorf parameter, L2

ah√1−v2

closed end may range from a light steel plate to a heavily stiffened structure, which

may be idealised as providing either a pinned or clamped end condition (for example,a caisson attached to a jacket leg, as in Fig. 1), respectively, to the shell. The openend of the cylinder is effectively sealed by the sea bed. During installation, the

cylinder is subjected to varying amounts of lateral restraint from the surroundingsoil, and suction pressures which are increasing so as to continue the installationprocess against rising soil resistance. Thus, while the buckling load of the shellincreases, the applied load on the shell is also increasing. To enable the economicdesign of the shell structure, use may be made of the increase in buckling loadprovided by the surrounding soil mass.

To assess the increase in buckling load that may be possible, this paper examinesthe buckling load of a cylindrical shell with various end conditions. Lateral endrestraint is provided by means of radial Winkler springs in the case of an open-ended shell, with free and pinned extreme cases examined. Buckling loads are found

using energy functions, solved using a variational approach. Comparisons are alsomade with existing results, and a number of finite element solutions.




Fig. 1. Example of an effectively clamped suction caisson top.

2. Buckling analysis

The properties of the shell being considered are shown in Fig. 2. This shows anopen-ended cylinder, with elastic restraint around its base. The elastic restraint isprovided by a set of uniform Winkler springs, with modulus k f . Buckling analysisof the shell/spring system is performed using an eigenvalue solution method, basedon the method of variations [3]. To do this, the second variation in the energy func-tional is required. This may be divided into two parts: the contribution from the

shell, and that from the elastic support. Details for deriving these expressions maybe found in a number of texts [4,5]. The second variation of the potential energy of the shell is given by:




Fig. 2. Suction caisson parameters.

1

2d 2V a

C

2e2 z1e2q12ve z1eq11−v

2 g zq1 d z dq

a

2 N z0w21, z N q0

w21,q

a2

2 N zq0w1, z

w1,q

a d z dqa

D

22 z1

2q12v z1q12(1v) zq1 d z dq (1)

1

2Pcr

v21v1w1,qv1,qw1w2

1 d z dq

where the subscript 1 denotes the post-buckled state, while 0 denotes the pre-buckledshell state. The expression for the second variation in potential energy of the springrestraint is:

1

2d 2V s

ak f

2 w21( z L,q) d z dq (2)

Adding Eqs. (1) and (2) leads to the final variational expression. With this equationit is then necessary to substitute expressions for the shell displacements, strains andcurvatures after buckling. The required expressions are [5]:

e z1∂ zu1( z,q) (3)




eq1∂qv1( z,q)+w1( z,q)

a (4)

g zq1∂ zv1( z,q)∂qu1( z,q)

a (5)

z1∂ z, zw1( z,q) (6)

q1∂qv1( z,q)−∂q,qw1( z,q)

a2 (7)

zq11

2

∂ zv1( z,q)−2∂ z,qw1( z,q)

a (8)

These expressions are valid for both shallow and non-shallow shells. Substitutingthese into the variational energy expressions results in an equation for the secondvariation in terms of the pre-buckling membrane stresses and the post-buckled shelldisplacements. Hence, to determine eigenvalues of the problem, it is necessary tohave expressions for both of these. For the following analysis, it is assumed that thepre-buckled shell stresses are adequately described by shell membrane theory. Thisleads to the membrane forces per unit length of:

N z0Pcr a

2 (9)

N q0Pcr a (10)

N zq00 (11)

The displacement functions that describe the post-buckled shape of the shelldepend on the boundary conditions that are present (see Table 1). In this paper six

Table 1Lateral displacement boundary conditions

Designation Boundary conditions w1( z=0) w1( z= L) ∂w1( z=0)

∂ z

∂w1( z= L)

∂ z

P-P Both ends pinned 0 0 Free Free

C-C Both ends clamped 0 0 0 0

P-F Upper edge pinned, lower edge free 0 Free Free Free

C-F Upper edge clamped, lower edge free 0 Free 0 Free

P-E Upper edge pinned, lower edge elastically 0 Elastic Free Free

restrained

C-E Upper edge clamped, lower edge elastically 0 Elastic 0 Free

restrained




cases are examined, with the displacement functions shown in Table 2. With one of these sets of functions in place, the buckling load is determined using the Rayleigh–Ritz method. The total expression for the second variation in potential energy of the

system is minimised against each of the undetermined coefficients Ai, i.e.

∂ Ai1

2d 2V

1

2d 2V s0 for i1…2 j (12)

This produces a set of j simultaneous equations, [C 1]{ A1…A2 j}+Pcr [C 2]{A1…A2 j}=0. The eigenvalues of this equation then give the critical buckling loadin terms of n and, where applicable, m. Where m is present, the lowest bucklingload is found with m=1. Further details may be found in Pinna and Ronalds [6].

The cases of a cylinder with an open, unrestrained end (C-F and P-F) are special

cases of a cylinder with an elastically restrained end (C-E and P-E). The displacementfunctions for these two end conditions are the same, with the buckling mode shapesdependent on the values of the coefficients A j found by minimising the energy func-tion, which in turn depend on the value of the spring constant k f . As the end restraintbecomes stiffer, the change in curvature along the length of the shell also becomesgreater. To accurately model this change in curvature, it is necessary to ensure thata sufficient number of terms, j, are included in the displacement functions.

It should be noted that Eq. (1) and Eqs. (3)–(8) are valid for all circumferentialwave numbers n. Omission of the last line of Eq. (1) would result in a set of equationsvalid for shells under dead-loading. These equations would produce results that are

accurate for short to intermediate length shells under hydrostatic pressure loading,but inaccurate for long shells (n3). The point where a shell becomes long is afunction of both its geometry and end conditions, as discussed in the next section.

The lateral spring stiffness k f may be non-dimensionalised by

k ∗ f k f La

Eh2 (13)

The use of this factor is also discussed below.

Table 2

Displacement functions for various boundary conditions

Designation u1 v1 w1

P-P A1 sin(nq)cosmp z

L A2 cos(nq)cosmp z

L A3 sin(nq)sinmp z

L

C-C A1 sin(nq)sin2mp z

L A2 cos(nq)cos2mp z

L 1 A3 sin(nq)cos2mp z

L 1

P-E, P-Esin(nq)

i

Ai z

Li−1

cos(nq)i

Ai+ j z

Li

sin(nq)i

Ai+2 j z

Li

C-F, C-E sin(nq)i

Ai z L

i

cos(nq)i

Ai+ j z L

i+1

sin(nq)i

Ai+2 j z L

i+1




3. Results

Fig. 3 shows the effect of increasing end restraint on the buckling load of both

P-E and C-E shells, where the buckling load is non-dimensionalised by:

Fig. 3. Shell buckling load as k f varies ( Z =500, L / a=1.61889).




P∗

cr L2a

p 2 D Pcr (14)

It can be seen that there is a substantial increase in the buckling load in each case.For the geometry shown in the figure, the fully restrained buckling loads are 19.8and 2.10 times the unrestrained buckling loads for the P-E and C-E cases, respect-ively. Also shown in this figure are a number of results produced by finite elementanalysis using the ABAQUS [7] program. Details of this analysis can be found inPinna and Ronalds [6]. It may be seen that for both cases, there is very good agree-ment between the results found using the method outlined above, and the finiteelement analysis.

For the case of a shell with one end pinned and the other free, the analysis assumesthat buckling occurs with n2. It is found in both the variational and finite element

analyses that other very low buckling loads may occur for this combination of endconditions, where these modes correspond to rigid body eigenmodes. These modescan be restricted so that buckling occurs in a circumferential mode, which reflectsthe situation for a suction caisson. In the finite element analysis, this is done byfixing the shell at four nodes, around the top of the shell, against axial displacement.Alternatively, an n=2 solution may be found by ignoring the rigid body modes. Forthe installation of a suction caisson, it would be expected that the rigid body modeswould be restricted, and buckling would therefore occur with n2. This assumptionis carried through this analysis. In contrast, Koga and Morimatsu [8] give a bucklingload for a P-F cylinder (S3-FR in their notation) of zero, using an asymptotic method,

and with n2.The variation in buckling load, as the geometry of the shell is changed, is shown

for various elastic end restraint conditions for P-E and C-E shells in Figs. 4 and 5.These graphs show that the increase in buckling load shown in Fig. 3 is valid for arange of Z values. One immediately apparent difference between these two figuresis that, for the case of a shell with a clamped top, there is a relatively large number of circumferential waves in the buckling solution. In comparison, P-F shells of practicalgeometries (that is, of intermediate length) have a buckling mode with n2. Apply-ing the assumption that n=2 results in a solution identical to that for a long shell,that is [5]:

P∗

cr 3

p 2 1−v2 Z

h

a (15)

Any sufficiently long shell’s critical load will converge on Eq. (15), which isindependent of the shell’s end conditions. For all types of boundary conditions exam-ined, this occurs after the value of Z where n first equals 2. A distinguishing propertyof P-F shells is that they enter this long shell mode immediately for practical geo-metries. However, as may be seen in Fig. 4, with the presence of some lateral restraintat the base of the shell, buckling occurs with n2, and thus the series form of the

solution is required.Fig. 6 demonstrates this, showing the eigenmode for a P-F shell with no elastic

restraint. It can be seen that there are only two circumferential waves around the




Fig. 4. P-E shells with various elastic end restraint conditions (a / h=200).

shell. Also, the lack of rotational restraint at the top of the shell is evident. Thiscircumferential mode is similar to that which would be obtained for a much longershell with other boundary conditions. These results are also reflected in Fig. 3. Themuch larger increase in buckling load for the P-E shell as k ∗ f increases, comparedwith the C-E shell, is caused by the change in buckling mode from a long to anintermediate shell. The C-E shell does not undergo this change, as it remains in the

intermediate shell solution regime.Bounds for these cases are found by setting the value of k ∗ f to either zero or

infinity. The result of doing this is shown in Fig. 7. In this figure, solid lines are




Fig. 5. C-E shells with various elastic end restraint conditions (a / h=200).

those determined from the variational analysis, while dotted lines are found usingthe multiplier a . The critical buckling load is then given by:

P∗

cr ab (16)

where

b2 1+8 Z

3p 2 (17)

where values for a are given in Table 3, and b [9] is a lower bound approximationfor the buckling value of a P-P shell. It can be seen that, for all cylinders with




Fig. 6. Eigenmode for a P-E shell.

solutions in the intermediate shell range, the buckling load can be arrived at byapplying a multiplier to the pin supported case. As the P-F case falls immediatelyinto the long shell range, it is not proportional to b, and it does not follow thismultiplier rule. Table 3 also compares the multipliers obtained here with those pre-viously published for these bounding cases. It can be seen that there is no difference

between published results, and the results found here using exact trigonometric dis-placement functions, that is, the P-P and C-C cases. Further, for the cases wheretruncated series solutions are used (C-F and C-P), the agreement with publishedresults is also good. It should be noted that the solution arrived at by Malik et al. [10]was by an approximate method, using a similar starting equation to that employed byKoga and Morimatsu [8]. The a multiplier can also be used for intermediate restraint.This is shown in Figs. 4 and 5, where the lateral restraint at the elastically supportedend varies from none to fully effective in each case. It can be seen that for n4,the buckling load can be expressed in terms of a .

The effect of non-dimensionalising k f in accordance with Eq. (13) is shown in

Figs. 8 and 9 for P-E and C-E shells, respectively. The buckling load is given interms of the multiplier a . These figures provide further evidence that the nondimen-sional restraint factor k ∗ f is appropriate over the complete range where the shell




Fig. 7. Buckling loads for various end conditions.

transitions from weak to near full lateral support. For large values of k ∗ f , the vari-ations in the nondimensional buckling loads in Fig. 8 are caused by restricting thevalue of n to an integer. The minimum buckling load that is predicted analyticallymay occur for a non-integer value of n. Physically, however, a complete number of

waves must form around the shell when buckling occurs. Thus, the minimum foundwith integer n, while corresponding to the physical minimum, may not correspondto the absolute minimum that could be found. The spread of curves at low k ∗ f in




Table 3

Multipliers for various boundary conditions

Designation a Published results

P-F Not applicable

C-F 0.58 0.58 [10]; 0.6 [8]

P-P 1

C-P 1.22 1.25 [10]; 1.25 [8]

C-C 1.5 1.5 [8]; 1.5 [12]

Fig. 8 is caused by the restriction that n=2, as discussed above. Fig. 9 also showsthe effect of allowing n to be a non-integer. It may be seen that this gives muchbetter convergence over a wide range of Z values.

4. Discussion

The set of graphs provided allow for the calculation of buckling loads for shellswith various end conditions. For the case where the shell has one of the limitingboundary conditions (C-C, C-P or C-F), the a multiplier given in Table 3 can beapplied to the P-P case to arrive at the buckling load. Where a cylinder has anelastically supported free end of stiffness k ∗ f , then Fig. 8 or Fig. 9 can be used todetermine the value of a , depending on the top end condition (pinned or clamped).The use of a multiplier factor a is advantageous as it allows existing design codes,which generally provide formulae for the P-P case only, to be applied to other bound-ary conditions representative of suction caissons. After applying the multiplier to theP-P case, knock-down factors for imperfections and elasto-plastic behaviour can be

included in design calculations. Such factors are provided in, for example, the “DnVBuckling Strength Analysis” code [11]. This applies to shells in the short to inter-mediate length range.

For long shells, the effects of boundary conditions may be ignored. Thus, the ringbuckling formula, Eq. (15), can be applied directly. It should be noted, however,that the point where a solution enters into the long shell solution depends on boththe shell geometry and the boundary conditions. For the case of a shell with P-Fend conditions, or with only small lateral restraint, then the ring solution can beapplied for all practical geometries. It is also found in this study that a P-F shellwill tend to buckle in a rigid body mode, if there is no axial restraint present. Only

a minimal axial restraint is required, however, to force the shell into a circumferentialbuckling mode. During the installation of a suction caisson, it is likely that the pres-ence of even a relatively flexible end cap would be sufficient to force this mode.




Fig. 8. Buckling load for P-E boundary conditions, Z varying (50 Z 5000).

5. Conclusions

The results presented in this paper allow the hydrostatic buckling load of cylindri-cal shells with various end conditions to be found. For cylinders with elastic lateralrestraint at one end, Figs. 8 and 9 give the non-dimensional buckling load, based




Fig. 9. Buckling load for C-E boundary conditions, Z varying (250 Z 5000).

on the boundary conditions at the other end. The non-dimensional equation for lateral

elastic restraint, Eq. (13), works well for shells with either pinned or clamped topends, over all geometries where the shell can be considered of intermediate length.For all cases, a simple multiplier can be applied to the buckling load for a pinned–




pinned shell, allowing the adaptation of existing buckling codes for offshore struc-tures to boundary conditions relevant to suction caisson design.

For long shells, boundary conditions have no effect on the buckling load, and Eq.

(15) can be applied directly to all cases. Where a shell becomes long depends onvarious parameters, and is indicated by a buckling solution with n=2. Guidance fordetermining where a long shell solution can be applied is offered by existing buck-ling codes.

Acknowledgements

The authors gratefully acknowledge the support of the Special Research Centrefor Offshore Foundation Systems, funded through the Australian Research Council’sResearch Centres Program.

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Hydrostatic Buckling of Shells

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