AIAA-2002-1513

BUCKLING LOAD CALCULATIONS OF THE ISOTROPIC SHELL A-8

USING A HIGH-FIDELITY HIERARCHICAL APPROACH

Johann Arbocz*

Delft University of Technology, The Netherlandsand

James H. Statues**

NASA Langley Research Center, Hampton, VA 23681-001

ABSTRACT

As a step towards developing a new design

philosophy, one that moves away from thetraditional empirical approach used today in

design towards a science-based designtechnology approach, a test series of 7 iso-tropic shells carried out by Arbocz andBabcock [1] at Caltech is used. It is shown howthe hierarchical approach to buckling load

calculations proposed by Arbocz et al [2] canbe used to perform an approach often called"high fidelity analysis", where the uncertaintiesinvolved in a design are simulated by refinedand accurate numerical methods. The Delft

Interactive Shell DEsign COde (short,

DISDECO} is employed for this hierarchical

analysis to provide an accurate prediction ofthe critical buckling load of the given shellstructure. This value is used later as areference to establish the accuracy of the

Level-3 buckling load predictions. As a finalstep in the hierarchical analysis approach, thecritical buckling load and the estimatedimperfection sensitivity of the shell are verified

by conducting an analysis using a sufficientlyrefined finite element model with one of the

current generation two-dimensional shellanalysis codes with the advanced capabilitiesneeded to represent both geometric andmaterial nonlinearities.

t_

Professor, Faculty of AerospaceEngineering, Fellow AIAASenior Engineer, Structures and MaterialsCompetency, Fellow AIAACopyright O 2002 by J. Arbocz andJ.H. StarnesPublished by AIAA with permission

INTRODUCTION

It is generally agreed that, in order to makethe development of the Advanced SpaceTransportation System a success and toachieve the very ambitious performance goals(like every generation of vehicles 10x safer and10x cheaper than the previous one), one mustmake full and efficient use of the technical

expertise accumulated in the past 50 years orso, and combine it with the tremendouscomputational power now available. It isobvious that with the strict weight constraintsused in space applications these performancegoals can only be achieved with an approachoften called "high fidelity analysis', where theuncertainties involved in a design are

simulated by refined and accurate numericalmodels. In the end the use of =high fidelity _numerical simulation will also lead to overallcost reduction, since the analysis and design

phase will be completed faster and onty thereliability of the final configuration needs to beverified by structural testing.

The light-weight shell structures used inaerospace applications are often bucklingcritical. The buckling load calculations are

usually carried out by one of the manycurrently available finite element basedcomputer codes [e.g., 3,4]. In order to reducecomputer execution time, buckling analysesare often done using only the small

displacement stiffness matrix K o . This

approach is used, despite the fact that the"initial stability problem" so formulated can onlygive physically meaningful answers if theelastic solutions based on K o (at least

approximately) are identically equal to zero [5].When the qualitative nature of the expected

behavior is completely unknown, the stability ofthe structure must be investigated using the full

tangent stiffness matrix KT in order to

guarantee accurate and reliable buckling loadand buckling mode predictions. In order todiscover the load level at which K T ceases to

be positive definite (that is, the load level whenCopyright C, 2002 by Johann Arbocz. Published by the &rnerican Institute of Aeronautics and Astronautics, Inc., with permission.

2396

buckling occurs), a step-by-step analysisprocedure is needed.

In addition, it is imperative (though oftencompletely neglected), that at the beginning ofany stability investigation, the accuracy of thediscrete model used should be checked

against available analytical or semi-analyticalresults. This step is part of a mandatory studyneeded in order to establish the dependence ofthe buckling load predictions on the meshdistribution used. Furthermore, as has beenpointed out in the past by Byskov [6], if onecarries out imperfection sensitivityinvestigations, which involve an extension ofthe solution into the postbuckling responseregion, further mesh refinement may beneeded since the wavelength of the dominantlarge deformation pattern may often decreasesignificantly.

Finally, whenever one is engaged in shellstability analysis it is especially important thatone is aware of the possible detrimental effectsof a whole series of factors, that have beeninvestigated extensively in the late 1960s andthe early 1970s. Thus for an accurate andreliable prediction of the critical buckling loadof a real structure, one must account not onlyfor the influence of initial imperfections [e.g.,7,8] and of the boundary conditions [e.g. 9], butone must also consider the effects of stiffener

and load eccentricity [e.g., 10] and theprebuckling deformations caused by the edgerestraints [e.g. 11,12].

A test series of 7 isotropic shells carriedout by Arbocz and Babcock [1] at Caltech isused to illustrate how such a hierarchical

approach to buckling load calculations can becarried out. The platform for the multi-levelcomputations, needed for an accurate

prediction of the critical buckling loads and areliable estimation of their imperfectionsensitivity, is provided by DISDECO [13]. Withthis open ended, hierarchical, interactivecomputer code the user can access from his

workstation a succession of programs ofincreasing complexity.

SOLUTION OF THE BUCKLING PROBLEM

In the following it will be shown that with thehelp of DISDECO, the Delft Interactive ShellDEsign COde, the shell designer can study thebuckling behavior of a specified shell, calculateits critical buckling load quite accurately andmake a reliable prediction of the expecteddegree of imperfection sensitivity of the criticalbuckling load. The proposed procedureconsists of a hierarchical approach, where theanalyst proceeds step-by-step from the simpler(Level-I) methods used by the early

investigators to the more sophisticatedanalytical and numerical (Level-2 and Level-3)methods used presently.

Level-1 Perfect Shell Buckling Analysis

The geometric and material properties ofthe isotropic shell A-8 of Ref. [1] are listed inTable 1.

Table 1.

Geometric properties of Caitech isotropic shell A-8

ttotal(= h) =0.00464 in (=0.117856 mm )

L =8.0 in (= 203.2 mm )

R =4.0 in (= 101.6 mrn )

E = 15.2x106 psi (= 1.0480x105 N/mrn 2 )

v =0.3

D

Assuming a perfect shell(W=0) and the

following membrane prebuckling state

t

W (°) = hWv = hA12 Xc

F(O)= Eh 2 1;Ly 2cR 2

where

(1)

o _ Nx . Eh

O'cg Nc_ ' ¢rct _', Ncg = Gcgh

and c = 43(1 -v 2)

then the nonlinear equations governing theprebuckling state are identically satisfied andthe linearlzed stability equations reduce to aset of equations with constant coefficients. Ithas been shown in Ref. [14] that by assumingan asymmetric bifurcation mode of the form

W(1) x y= hsinm_Ecosn _

where

(2)

m = k = number of axial haft wavesn = e = number of circumferential full waves

one can reduce the solution of the linearizedstability equations to an algebraic eigenvalue

problem. Notice that the eigenvalue Zmn

depends on the wave numbers m and n. The

critical load parameter Zc is the lowest of all

possible eigenvalues. Thus finding ;Lc

involves a search over the integer valued wavenumbers m and n. Using the Level-1

2397

computational module AXBIF [14] a searchover integer valued axial half-wave numbers myielded the lowest eigenvalues listed in Table 2for the specified circumferential wave numbersn.

Table 2.

Buckling loads of the Caltech isotropic shell A-8Buckling load map for the perfect shellusing AXBIF [14] (Nc£ = - 49.51517 Ib/in)

n = 4 _m = 1.000 (m = 34)

n = 5 Zm = 1.000 (m = 34)

n = 6 _m = 1.000 (m = 34)

n = 7 _.m = 1.000 (m = 33)

n = 8 ;Lm = 1.000 (m = 33)

n : 9 zm: 1.ooo (m = I)

_.cm = 1.000 (m = 33)

n = 10 X m : 1.000 (rn = 33)

n = 11 _m = 1.000 (m = 33)

n = 12 Xcm = 1.000 (m = 32)

n : 13 ;Lcm = 1.008 (rn = 2)

;Lcm = 1.000 (m = 32)

n = 14 _m : 1.000 (m = 32)

n = 15 Z.crn = 1.001 (m : 3)

_n = 1.000 (m = 31 )

As is known the lowest eigenvalue is arepeated eigenvalue with a high degree ofmultiplicity. The family of modes belonging to

the lowest eigenvalue ;Lc = 1.000 lie on the so-

called Koiter circle [7].To facilitate the interpretation of the

numerical results obtained, DISDECO providesthe user with various graphical interfaces.

Thus the results of the search for the critical

(lowest) buckling load ;Lc can be displayed in

a contour map as shown in Fig. 1. In order toprovide a quick overview of the distribution ofeigenvalues, the values displayed in thecontour plot are normalized Thus in Fig. 1 thefollowing normalized eigenvalues are plotted

_.m : Nm-49.51517

Notice that the critical buckling load can becalculated using a simple multiplication

= =1.000 (-49.51517) = -49.51517 Ib/in

Level-2 Perfect Shell Buckling Analysls

To investigate the effects of edge constraintand of different boundary conditions on the

critical buckling load of the perfect shell

(0=0) one has to switch to the Level-2

computational module ANILISA [15]. In thismodule the axisymmetric prebuckling state isrepresented by

W (°) = hW v + hw o (x)

F(O) Eh 2. 14 2+R2fo(X)]

(3)

It has been shown in Ref. [15] that with theseassumptions the prebuckling problem isreduced to the solution of a single fourth order

ordinary differential equation with constantcoefficients, which always admits exponentialsolutions. Closed form solutions for simply

supported and clamped boundary conditionshave been published in the literature [16].

For isotropic shells the linearized stability

equations admit separable solutions of theform

W (1) = hWl(X)cosne

F(1) = E Rh2 fl(x)cosneC

where e = yR

(4)

Using a generalization of Stodola's method[17] first published by Cohen [18] the resultingnonlinear eigenvalue problem is reduced to asequence of linearized eigenvalue problems.The resulting ordinary differential equations aresolved numerically by a technique known as"parallel shooting over N-intervals" [19]. Noticethat by this approach the effect of edgerestraint and the specific boundary conditionsare satisfied rigorously. To find the critical load

parameter Zc an n-search must be carried

out, whereby one must be careful to find not alocal minimum but the absolute minimum. Ascan be seen from the results presented inTable 3 the n-search using membrane

prebuckling and a rigorous satisfaction of SS-3(N x = v = w = M x = 0) boundary conditions for

the stability problem yields indeed the repeated

eigenvalues Zcm -1.0 located on the "Koiter"

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circle. The slight variations in the eigenvaluesare due to the fact that both the axial half-wavenumbers m and the circumferential full wave

numbers n are integers.

Table 3

Buckling loads of the Caltech isotropic shell A-8

(Nc£ =-49.51517 Ib/in)

Buckling load map for the perfect shell using

ANILISA [15] (B.C. N x = v = w = M x = 0)

Prebuckling: Membrane Nonlinear

n =8 _Lm= 1.10977(m=1) _,n1=0.951155

zm = 1.00001(m=33) _nl= 0.966181*

n = 9 ;Lm = 1.00004(m=1) _nl= 0.934990

_._n = 1.00000(m=33) _Lcnl=0.957035*

n = 10 zm = 1.08160(m=1) _Lnl= 0.940209

= 1.00011(m=33) _,cnl= 0.946124"

n = 11 _ = 1.12539(m--2) _,_1= 0.931492*

_m = 1.00035(m=32) _nl= 0.939878

n = 12 _m= 1.01481(m=2) zcnl=0.915200 *

_,crn = 1.00004(m=32) _,nl= 0.931464

n = 13 _,m = 1.00836(m=2) _L_I= 0.908528*

Z,m = 1.00004(m=32) _.cnl= 0.918482

n = 14 _,cm= 1.04064(m=3) _,_1= 0.903256

_,cm= 1.00030(m=31) _,nl= 0.906963"

n = 15 Z,m = 1.00060(m=3) zcnl= 0.891368

;Lm = 1.00000(m=31) _nl= 0.898837*

n = 16 ;Lm = 1.02041(m=3) _,cnl= 0.885185

_,crn= 1.00023(m=31) ;Lnl= 0.886606"

n = 17 _,_-- 1.00086(m=4) _nl= 0.875881*

xm = 1.00001(m=30) _,nl= 0.878312

n = 16 _m = 1.01300(m=4) _,nl= 0.868406*

= 1.00027(m=30) _Lnl= 0.869200

n = 19 _,cm = 1.00014(m=5) z,nl= 0.861087

_m = 1.00001(m=29) _,cnl= 0.861626*

n = 20 ;Lm = 1.00164(m=6) xnl= 0.854891

Z,m = 1.00006(m=28) _,cnl= 0.855033*

Table 3 - continuation

Prebuckllng: Membrane

n = 21 )_m = 1.00378(m=7)

n = 22

n=23

n=24

n = 25

n =26

n=27

Nonlinear

2_nl = 0.849839*

;Lm = 1.00023(m=27) _,cnl= 0.849927

= 1.00183(m=7) _.cnl-- 0.846336"

2,,m= 1.00014(m=27) 2_,1= 0.846344

_m = 1.00114(m=8) _,_1= 0.844502

_L_ = 1.00012(m:26) _nl= 0.844511"

Z,_ = 1.00071(m=10) Z_I= 0.844480"

_.cm = 1.00011(m=24) _1= 0.844481

Z,m = 1.00000(m=11) 2,,nl= 0.846279*

_,crn = 1.00000(m=23) z,nl= 0.846280

;Lm= 1.00002(m=13) _,1=0.849833"

= 1.00001(m=21) _nl= 0.849833

2.,m = 1.00027(m=17) 2,,_1= 0.854990"

_,m = 1,00032(m=18) _,_I= 0.854990

* anti-sym at x = 1_/2.

The most accurate Level-2 solutions are

obtained when one employs a rigorous

nonlinear prebuckling analysis. As can be seenfrom the results listed in Table 3, for this

particular shell the critical buckling loads with

nonlinear prebuckling are always lower thanthe corresponding results obtained using a

membrane prebuckling analysis. Notice that

the critical load N c can be calculated easily bymultiplying the lowest eigenvalue ;Lc by thenormalizing factor Nc£ = -49.51517 Ib/in

yielding

Nc = 2,,cNc_ = -41.815 Ib/in (n = 24)

In fig. 2 the critical buckling modes usingmembrane and rigorous nonlinear prebucklingare depicted. Notice that the solutions withnonlinear prebuckling differ significantly fromthe ones obtained using membraneprebuckling, especially at n = 24 where oneobserves a typical edge buckling typebehavior.

Level-3 Perfect Shell Buckllng Analysis

To verify the earlier predictions the finitedifference version [20] of the well known shell

2399

analysis code STAGS [21] will be used. Inorder to be able to represent the measuredinitial imperfections accurately, the whole shellwill be modeled.

Initially a convergence study must becarried out in order to establish the mesh sizeneeded for accurate modeling of the bucklingbehavior of the shell in question. For this

purpose the asymmetric bifurcation from anonlinear prebuckling path option was used,whereby the earlier results obtained with theLevel-2 module ANILISA listed in Table 3serve as a reference.

In the convergence study, at first, for a fixednumber of mesh points in the axial direction(NR = 161) the number of mesh points in thecircumferential direction (NC) was increaseduntil the bifurcation load approached a

horizontal tangent. As can be seen from Fig. 3the results converge to a limiting value frombelow at about NC = 261. Next, for a fixed

number of mesh points in the circumferentialdirection (NC = 161) the number of rows (NR)was varied. This time convergence is fromabove and as can be seen from Fig. 3 the

horizontal tangent is reached at about NR =261.

To illustrate the difference between usingcoarser meshes to speed up the computationsand the finer meshes which produce (nearly)

converged solutions, initially the first threeeigenvalues and eigenvectors are computed.

Using a mesh of 161 rows and 181 columns(a model with 89488 D.O.F.'s and a maximumsemi-bandwidth of 713) and SS-3 boundary

conditions (N x = -N o, v = W = Mx = 0) the

following 3 lowest eigenvalues were obtained:

_,(c1) = 0.833361 (n = 26) --->

N(c1) = Z(cl)Nc£ = -41.264 Ib/in

Z(2) = 0.833374 (n = 26) -->

N (2) = _,(2)Nct =- 41,265 Ib/in

_(3) = 0.834238 (n = 25) --->

N (3) = _,(3)Nct= -41.307 Ib/in

Z(c1) = 0.844993 (n = 24) -->

N(c1 : =-41.840Ib n

;L(2) = 0.845006 (n = 24) --_

N(c2) = L(c2)Nc_, =- 41.84lib/in

_.(3) = 0.845100 (n = 25)

N(3) = Z(c3)Nct = -41.845 Ib/in

Details of the critical mode are displayed in

Figures 4-5 for the coarser 161x181 model andin Figures 6-7 for the finer 261-261 model.Notice that using the more refined 261x261model the sequence of the 3 lowest bucklingloads and the corresponding buckling modes

agree closely with the predictions obtained withthe Level-2 module ANILISA (see also Table 3)

IMPERFECTION SENSITIVITY STUDY

That initial imperfections may decrease theload carrying capacity of thin-walled shellstructures is by now widely known and

accepted. However, in order to calculate theeffect of initial imperfections one must knowtheir shape and amplitude, an information that

is rarely available.

In the absence of initial imperfection

measurements, as a first step one mustestablish whether a given shell-loading

combination is imperfection sensitive, and ifthe answer is positive to estimate how

damaging certain characteristic imperfection

shapes are.

Single Axisymmetric Imperfection

Based on Koiter's pioneering work on the

effect of initial imperfections [7,22] the simplestimperfection model consists of a singleaxisymmetric imperfection

_' = h_l cosi_x (5)L

Next these computations were repeated usinga mesh of 261 rows and 261 columns (a modelwith 207508 D.O.F.'s and a maximum semi-bandwidth of 1037) and the same SS-3

boundary conditions yielding the followingresults

where i is an integer denoting the number of

half-waves in the axial direction and _1 is the

amplitude of the axisymmetric imperfectionnormalized by the shell waU-thickness h.

If one assumes that both the axial load and the

boundary conditions are independent of thecircumferential coordinate, then the prebuck-

2400

ling solution will also be axisymmetric, a factthat simplifies the solution considerably.

Level-1 Analysis of AxisymmetricImperfection

Neglecting the effect of the prebucklingboundary conditions the nonlinear equationsgoverning the prebuckling state admit thefollowing axisymmetric solutions

W (°) =hW v +hwo(x)

F(o)= Eh 2 l_,y2+fo(x )cR 2

where

(6)

w , Xwo(x) = hE1cos i_

fo (x) =Z,Ci - _,

(1+ B210r'2) Eh 3 - x

2C_2_2 C F_lCOSi_L

(7)

1 -- (1+ B21°_2)2

Notice that the linearized stability equationsbecome now a set of equations with variablecoefficients. The reduced wave number oci

and the normalized stiffness coefficients A22,

B21 and 5_ 1 are all listed in Ref. [14].

It has been shown in Ref. [14] that byassuming an asymmetric bifurcation mode ofthe form

W (1)= hslnm=LcosnY (8)

whe re

m = k = number of axial half wavesn = e -- number of circumferential full waves

a Galerkin type approximate solution yields for

the eigenvalues (read, buckling loads) _. of theproblem a characteristic equation in the form ofa cubic polynomial

_3 _ (Zmn.c + 2_.c i _ (_1_18i=2m);L2 (9)

+{2Zmn'c + _'c i + (C2 -C1)_lai=2m}Zci _"

^ -2 2-{Zmn'c + C2_18i=2m + ((_3 - C4_im )_1 }Zci = 0

where

2401

5i=2m = 1

=0

if i=2m

otherwise

5im=1 ifi=m

= 0 otherwise

and the constants C1,(_2 .... are listed in

Appendix C of Ref. [14].Here it must be remembered that one will

only get any noticeable degrading influence of

the assumed axisymmetric imperfection if _1

is negative and if the coupling condition i=2mis satisfied. The physical explanation for thiscan be found in Koiter's 1963 paper [22].Furthermore, in order to obtain the smallest

real root of Eq. (9), for a given axisymmetric

imperfection _1 an n-search must be carried

out. It should also be noticed that the terms

involving the Kronecker delta _i=2m are all

linear in _1, and thus they dominate the buck-

ling behavior of the shell with axisymmetricimperfection.

Assuming that the most likely axisymmetdcimperfection of the silver pointed wax mandrelused to electroplate the isotropic shell A-8 isgiven by

= I_ 1 COS2_ L (10)

the Level-1 DISDECO computational moduleAXBIF generated the solid curve shown in Fig.8. Notice that the curve is normalized by

Nct =-49.515171b/in, the classical buckling

load of an isotropic shell, obtained around1910 by Lorenz [23], Timoshenko [24] andSouthwell [25]. Notice also that an initialimperfection amplitude equal to the wall

thickness of the shell (_1 = -1.0) generates a

"knockdown factor" of Zc = 0.706, resulting in

the following buckling load

N c = ;LcNct =

0.706(-49.51517) = -34.958 Ib/in

However, if one assumes an imperfection inthe form of the classic axisymmetric bucklingmode of Koiter [7]

_/= h_l cosict_ L

where

L 2/ C-. 2)

then for the isotropic shell A-8 ict - 34.0, and

the Level-1 DISDECO computational moduleAXBIF found that the minimum buckling load

1occurred when m = -_ic,e for different values of

n (which depended on the value of _1 ). These

results are shown in Fig. 9.For this case an initial imperfection amplitudeequal to the wall thickness of the shell

(_1 =-1.0) generates a "knockdown factor" of

;Lc = 0.196 resulting in the following rather low

buckling load

Nc= =0.196(-49.51517) = -9.705 Ib/in

The last two buckling loads clearlydemonstrate the importance of the shape ofthe initial imperfections involved and the role ofthe corresponding eigenvalues. Using the long

wave axisymmetric imperfection (i=2) the

corresponding normalized axisymmetricbuckling load (eigenvalue)of the perfect shell is

;Lci =144.321 (a very high value) yielding a

low imperfection sensitivity. For the short waveimperfection (i = 34) the corresponding

normalized axisymmetric buckling toad

(eigenvalue) of the perfect shell is

;Lci =1.000. Thus it lies on Koiter's circle

yielding a high imperfection sensitivity.

Level-2 Analysla of AxlsymmetricImperfection

Since the external loading, the boundaryconditions and the assumed initial imperfection

are axisymmetric, therefore the prebucklingsolution will also be axisymmetric. It has beenshown in Ref. [26] that by assuming

W (°) = hWv + hwo(X)

F(o) Eh 2. 1_ 2+R2fo(x)}

(11)

the solution of the nonlinear partial differential

equations governing the prebuckling state canbe reduced to the solution of a single fourthorder ordinary differential equation withconstant coefficients, which can be solved

routinely.For isotropic shells the resulting linearized

stability equations admit separable solutions ofthe form

W (1) = h Wl(X)cosne

F(I)= ERh2 f1(x)cosnOc

Ywhere e =--,

R

(12)

Solution proceeds as outlined on Ref. [26].Using an updated version of the Level-2computational module MANILISA [27] and SS-3 (N x -- v = w = M x = 0) boundary conditions

one obtains the results presented in Table 4.Notice that a rigorous nonlinear prebuckling

analysis was used and an n-search wascarried out for each specified axisymmetric

imperfection amplitude _1"

The values of Table 4 are plotted as the

dashed curve in Fig. 9. A comparison of theresults obtained via the Level-1 module AXBIF

(solid curve) and the Level-2 moduleMANILISA (dashed curve) shows that also inthe case of axisymmetric imperfections arigorous pre-buckling analysis should be used,Especially for very small initial imperfection

(r_ll < 0.1) the Level-1 predictionsamplitudes/ w

are inaccurate and overestimate the critical

buckling load. Notice further that both curveshave been normalized by classical buckling

load Nct = -49.51517 Ib/in. This way the effect

of using a rigorous prebuckling analysisbecomes easily discernible.

Table 4Buckling loads of the Caltech isotropic shell A-8

(Nc£ = -49.51517 Ib/in)

Short wave axisyrnmetric imperfection

4 x_//h =_1cos3 lz_ using MANILISA [27]

(B.C.: N x = v = w =M x = 0)

0.

- 0.01

- 0.02

- 0.03

- 0.04

- 0.05

- 0.06

- 0.07

- 0.08

- 0.09

0.844480 (n=24)

0.847663 (n=21)

0.786079 (n=21)

0.740449 (n=21)

0.703421 (n=21)

0.671958 (n=21)

0.644481 (n=21)

0.620047 (n=21)

0.598039 (n=21)

0.578030 (n=21)

_1 _nl

- 0.1 0.549168 (n=17)

- 0.2 0.420091 (n=17)

- 0.3 0.354645 (n=19)

- 0.4 0.305484 (n=18)

- 0.5 0.271507 (n=17)

- 0.6 0.247002 (n=16)

- 0.7 0.229213 (n=15)

- 0.8 0.214064 (n=20)

- 0.9 0.200103 (n=19)

- 1.0 0.189011 (n=18)

2402

Level-3 Analysis of AxlsymmetrlcImperfection

Recalling that since both the axial load andthe boundary conditions are independent of thecircumferential coordinate, therefore theprebuckling solution will also be axisymmetric,one can use once again the asymmetricbifurcation from a nonlinear prebuckling pathoption. By modeling the full shell the code canchoose itself the critical number of full waves inthe circumferential direction. No n-search must

be carried out. Using a uniformly spaced meshof 161 rows and 181 columns and the user

written subroutine option WlMP to introducethe following axisymmetric imperfection

= 1.0hcos 2_ xL

(remember STAGS defines W positiveoutward) the following critical bifurcation loadwas found

Nc = -32.582 Ib/in

As can be seen from Fig. 10 the criticalbuckling mode has n = 9 full waves in the

circumferential direction and a single half wavein the axial direction. The nondimensional

bifurcation load of the shell with axisymmetric

imperfection is for _1 = 1.0

Lc _ Nc _ -32.582 _0.65802Nc£ -49.51517

Notice also that the Level-2 MANILISA

prediction (_,nl = 0.683, n = 9) agrees closely

with the Level-3 STAGS-A prediction

(_.nl =0.658, n = 9). The slight difference is

partly due to the fact that MANILISA uses the

Donnell type nonlinear shell equations,

whereas STAGS-A employs the higher order

Marlowe-FI0gge equations.

Single Asymmetric Imperfectlon

The effect of a single asymmetric initialimperfection can be investigated either bysolving the full nonlinear response problem orby employing the well known Lyapunov-Schmidt-Koiter [7] reduction technique. Wheninvestigating the degrading effect of a singlemode asymmetric imperfection

• x yW = t_2 slnmnEcosn _- (13)

where m and n are integers denoting thenumber of axial half-waves and the number of

2403

circumferential full waves, respectively,instability occurs at the limit point of theprebuckling state in the generalized load-deformation space. Assuming that theeigenvalue problem for the critical (lowest)buckling load Ac will yield a unique asym-

metric buckling mode W (1), then for an

imperfect shell (_2 #0) the shape of the

generalized load-deflection curve in the vicinityof the bifurcation point A = Ac is given by the

following asymptotic expansion

(A - A c )_ = AcaP_2 + Acb_ 3 + ...

- -(^ - ^c + o( 2)(14)

Expressions for the postbuckling coefficients=a" and =b" and the imperfection forms factors"(x" and "13" are derived in References [28,29].If the limit point is close to the bifurcation point,

then the maximum load A s that the structurecan carry prior to buckling can be evaluatedfrom Eq. (14) by maximizing A with respect toF_. For cases where the first postbucklingcoefficient "a" is zero, this analysis yields themodified Koiter formula [29]

(1-Ps) 3'2 = 3jE Yb[1-- (1-p_ )l 21(15)

where Ps = As/Ac.

Notice that, if the second postbucklingcoefficient "b" is positive, Eq. (15) has no realsolutions. Thus the buckling load of thespecified shell-loading combination is not

sensitive to small asymmetric initialimperfections of the shape given by Eq. (13).If, however, the second postbucklingcoefficient "b" is negative, the equilibrium loadA decreases following buckling and thebuckling load of the real structure As issensitive to the asymmetric initial imperfectionspecified by Eq. (13).

Level-1 Analysis of AsymmetricImperfection

For the isotropic shell under investigation,

as can be seen from the partial results listed inTable 2, there are many identical eigenvalues.

Hence, strictly speaking, the proposed form of

the perturbation expansion given by Eqs. (14)is not applicable, since the nonlinear

interaction between the many nearlysimultaneous eigenmodes is not accounted for.

Thus the following results, where one

considers the eigenfunctions corresponding tocertain critical eigenvalues chosen one at the

time,canat bestgivean indicationas to theseverity of the expected imperfectionsensitivity.

Initially, we assume a long waveasymmetricimperfectionaffineto oneof thecriticalbucklingmodesof theperfectCaltechisotropicshellA-8,computedby the Level-1computationalmoduleAXBIF(seealso Table

3)

= hE 2 sinTtXc°s9YL R

(16)

Next using the Level-1 computational moduleBFACT to carry out the initial postbucklinganalysis yields the following results

Kc = Km = 1.00004 (m = 1, n = 9)

b = - 0.010636 a = [_= 1.0

Substituting these values into Eq. (15), onecan plot the degrading effect of an asymmetricimperfection of the shape given by Eq. (16) as

a function of its amplitude _2. As can be seen

from Fig. 11 an initial imperfection amplitudeequal to the wall thickness of the shell

(_2 = 1.0) generates a "knockdown factor" of

Xs =0.679, resulting in the following low

buckling load

Ns = XsNct =

0.679 (-49.51517) = -33.621 Ib/in

Notice that the imperfection form factors

"a" and "13" are identical equal to 1.0 because

BFACT uses membrane prebuckling to

calculate the necessary first and second orderfields and the assumed asymmetric

imperfection shape of Eq. (16) is affine to thebuckling mode. If, however, one assumes thefollowing short-wave asymmetric imperfection

V y_/= hT.,2sin 17_" cos 27L R

which lies also on the Koiter circle, then the

initial postbuckling analysis, done once againby the Level-1 computational module BFACT,yields the following results

Xc= X_ = 1.00025(m =17,n = 27)

b = -220.84 a = 13= 1.0

Using these values in Eq. (15) one obtains Fig.12. As can be seen, an initial imperfection

amplitude equal to the shell wall thickness

(T..,2= 1.0) this time generates a =knockdown

factor" of Ks =0.0249, resulting in an

unrealistically low buckling load of

Ns = KsNct =

0.0249(-49.51517) = -1.233 Ib/in

Here one obviously cannot rely on the

prediction of the Level-1 computational moduleBFACT, which uses membrane prebuckling

analysis. In such a situation, as has beenshown by Hutchinson and Frauenthal [30], onemust repeat the postbuckling analysis using arigorous prebuckling solution which includesthe nonlinear effects due to the edge restraints.

Level-2 Analysis of AsymmetricImperfection

To investigate the effects of edge-constraintand/or different boundary conditions on the

imperfection sensitivity of the critical bucklingload of the Caltech isotropic shell A-8 one hasto switch to the Level-2 module ANILISA [15]

and run its postbuckling analysis option. In thismodule, as described eartier, the axisymmetric

prebuckling state is represented by Eqs. (3),the buckling modes by Eqs. (4) and the

postbuckling state by

W (2) = h[wcx(x)+ wf_(x)cosne]

ERh 2F(2) = _ [fez(x) + f13( x ) cos 2n0]

c

(18)

where 8 = y / R. Details of the computational

procedures used are reported in Refs. [15,26].Initially, let us assume that the specified

asymmetric imperfection is given by Eq. (16)

_ = h_2 sin=Xcos9eL

where (_= y / R . Running ANILISA with rigo-

rous prebuckling and SS-3 boundaryconditions (Nx = -No, v = w = Mx = 0) yields

the following results

Xc = Knl = 0.934990(n = 9)

b =-0.016202; _= 0.20830; 6 =-0.38232

Using Eq. (15) to plot the degrading effectof the asymmetric imperfection specified by

Eq. (16) as a function of its amplitude _2 one

2404

obtains the results displayed in Fig. 13 as asolid line.

Here one must remember that Koiter's

Sensitivity Theory is asymptotically exact, thatis, it yields accurate predictions for sufficientlysmall imperfections, whereby what issufficiently small may vary from case to case.Also, Eq. (15) was obtained by using theperturbation expansion given by Eq. (14),

where terms of order (_) are neglected. As

can be seen from the dotted curve plotted inFig. 13, by using more advancedcomputational modules such as COLLAPSE[31], where a full nonlinear solution is used and

terms up to and including order (_2) are

kept, one obtains slightly lower predictions.

Notice that up to about _2 =0.1 the

asymptotic predictions from ANILISA and thenonlinear results of COLLAPSE agree veryclosely. Thus one can say that in this case therange of validity of the asymptotic solution is

0>_2 _0.1.

For an initial imperfection amplitude equal

to the wall thickness of the shell (T_.,2= 1.0) the

asymptotic solution yields a =knockdown factor"

of Zs =0.806, which corresponds to a

buckling load of

N s = ZsNct =

0.806(-49.51517) = -39.909 Ib/in

For the same initial imperfection amplitude

of _2 =1.0 the nonlinear solution using

COLLAPSE yields a =knockdown factor" of

Xs = 0.68, which corresponds to a bucklingload of

Ns = ;LsNct =

0.68(-49.51517) = -33.670 Ib/in

Let us next consider the short wave

asymmetric imperfection given by

_/= hT.._2sin 17_Xcos 240 (19)L

where e= y/R. The rigorous asymptotic

solution obtained by running ANILISA with SS-3 boundary conditions (N x = -N o,

v=w=M x =0) yields now the followingresults

Zc = _nl = 0.844480 (n = 24)

b = --0.50329; a = 0.20113; _ = -0.012022

Using these values with Eq. (15) yields thesolid curve plotted in Fig. 14. Rerunning thesame case with the nonlinear module

COLLAPSE yields the dotted curve plotted inFig. 14.

Notice that in this case the asymptoticpredictions from ANILISA and the nonlinearresults from COLLAPSE agree closely up to

about _ = 0.03. Thus the range of validity of

the asymptotic solution is only 0 > _ ___0.03.

For an initial imperfection amplitude of

T__=1.0 the asymptotic solution yields a

"knockdown factor" of _s = 0.400, which

corresponds to a buckling load of

Ns = ZsNcp =

0.400(-49.51517) = -19.806 Ib/in

For the same initial imperfection amplitude of

F._ =1.0 the nonlinear solution yields a

=knockdown factor" of Xs =0.30, which

corresponds to a buckling load of

N s = ;LsNcf =

0.30(-49.51517) = -14.855 Ib/in

For an initial imperfection amplitude equal

to the wall thickness of the shell (T.,2= 1.0) the

asymptotic solution yields a =knockdown factor"

of _s =0.806, which corresponds to a

buckling load of

Ns = ZsNct =

0.806(-49.51517) = -39.909 Ib/in

MEASURED INITIAL IMPERFECTIONS

In order to apply the theory of imperfectionsensitivity with confidence, one must know thetype of imperfections that occur in practice. In1969 Arbocz and Babcock [1] published theresults of buckling experiments where, for thefirst time, the actual initial imperfections andthe prebuckling growth of the midsurface ofelectroplated isotropic shells were carefullymeasured and recorded by means of anautomated scanning mechanism.

Mideurface Initial Imperfections

As can be seen from Fig. 15, the measuredinitial midsurface imperfections of shell A-8 [1]show a rather general distribution dominated

24O5

by an n=2 mode. One can use the followingdouble Fourier series

_/ 7 _ . x 18-= EW,ocO,,,c+E. o,CO ,-hi=1 I=1

(20)

19 99

_=1 k,_=l

115x y

+k,t=l

to represent the measured initial imperfectionsaccurately, where Fourier coefficients withabsolute values less than 0.001 are neglected.

A convenient measure of the size of the

initial imperfections is their root-mean-square(RMS) value. By definition

2_R L

A2 s = 12 R---CJ" (x'y12dxdy 121 o o

Thus

h =_.z_ ioI k,_

2 2 (22)= Aax i + Aasy

For the A-8 shell the RMS values are

Z_axi = 0.5077 ; Aasy = 0.6882

Using these measured RMS values with thepreviously discussed initial imperfectionmodels yields the following buckling loadpredictions:

For the long wave

_/= h_l cos2_ Laxisymmetric imperfection

if _1 = Aaxi = --0.5077 then from

MANILISA [27] Xnl = 0.824 (n = 9)

pcnl = 2_R2_nlNct = -1025.428 Ibs

For the short wave

_./= h_l cos 34_ Laxisymmetric imperfection

if _1 = Aaxi = -0.5077 then from

MANILISA [27] _1 = 0.266 (n = 27)

pnl= 27¢R;_lNct = -331.024 Ibs

Turning now to the long wave asymmetric

imperfection _/= hT--.2sin_Xcos9 yL R

if _ = Aasy = 0.6882 then from

COLLAPSE [31] X_ I = 0.740 (n = 9)

pnl = 2_RZ_.INcl = _920.894 Ibs

For the short wave asymmetricx y

imperfection _/= h_2 sin17_cos24_

if _ = Aasy = 0.6882 then from

COLLAPSE [31] _1 = 0.324 (n = 24)

p;'= =403202bs

Comparing these buckling load predictions withthe experimental buckling load of

Pexp = -782"71bs it is clear that the long wave

imperfection models yield an upper bound,whereas the short wave imperfection models

yield a lower bound.For a more accurate estimate of the

buckling load one must use the measuredinitial imperfections in codes like STAGS

[20,21] to carry out 2-dimensional nonlinearcollapse analysis. Employing a user wdttensubroutine WlMP to input the double Fourierseries of Eq. (20), the 161x181 STAGS modelyielded with SS-3 boundary condition

(N x = -N o, v = w = M x = 0) a collapse load of

Ps =-900-1031bs' whereas the same model

with C-4 boundary condition (u = uo ,

v=w=w, x=0) yielded a collapse load of

Ps = -976.4671bs" Both of these values are

significantly higher than the experimental

buckling load of Pexp = -782-71bs •

Boundary Initial Imperfections

In References [32,33] it was shown that

boundary imperfections can have a significantdegrading effect on the buckling loads ofaxially compressed cylindrical shells. Theflatness of the end ring, attached to the load-cell of the Caltech test set-up shown in Fig. 16,

was measured about 20 years after it was

2406

used to test shell A-8. As can be seen in Fig.17 there is a very large amplitude waviness ofthe end support, with maximum deviations ofabout + 3 wall thicknesses.

The measured boundary imperfections aredecomposed in a 1-dimensional Fourier series

Uo = ub(Y) = (23)

_bhflao + n___l(ancosnY + bn sinnY) t

This series then is used in STAGS [20] tomodel the effect of boundary initialimperfections using a modified C-4 boundarycondition

U=Ub(Y) , v=w=W,x=0 (24)

and taking advantage of the dual loadingsystems provided by the program.

Because the grooves of the end rings werefilled with liquid Cerrolow at the time shell A-8was installed in the test set-up, it is to beexpected that after cooling off the hardenedliquid metal has filled-in all the gaps. The effectof this unknown end support was modeled byvarying the amplitude of the boundary

imperfection T_ between 0 and 0.1. As can be

seen from the results displayed in Fig. 18,including both the measured initial midsurfaceimperfections of Eq. (20) and the measuredboundary imperfections of Eq. (23) with an

amplitude of T.,b-0.035, the calculated

collapse load agrees well with the experimentalbuckling load.

DISCUSSION OF THE RESULTS

When comparing and analyzing the resultsobtained sofar it is important to keep in mindthat all Level-1 and Level-2 solutions are

based on approximate representations of theunknown functions. As pointed out in theprevious sections Level-1 solutions use asingle term double Fourier seriesapproximation to reduce the solution of thestability problem, formulated in terms of partialdifferential equations, to algebraic eigenvalueproblems. The effect of edge restraint isneglected (one uses a membrane prebucklingsolution) and the assumed field functions

satisfy approximately SS-3 (N x = -N o,

v = w = M x = 0) boundary conditions.

Level-2 solutions eliminate the y-dependence by a truncated Fourier

decomposition in the circumferential direction.

The resulting system of nonlinear ordinarydifferential equations are solved numerically,whereby both the specified boundaryconditions and the effect of edge restraint arerigorously satisfied. Thus by this approach theonly approximation is that one represents thevariation of the solution in the circumferential

direction by a single harmonic with n fullwaves, whereby an n-search is used toestablish which wave number is the criticalone. The Level-2 module ANILISA can also be

used to investigate the effect of using differentboundary conditions. In Table 5, the results forfour different boundary conditions are

presented. As expected the fully clamped C4boundary conditions has the highest criticalbuckling load. The increase in load carryingcapacity with respect to the weaker SS3boundary conditions is about the same as foran isotropic shell of similar characteristicdimensions (same L/R and PJt ratios) of Ref.[34].

The Level-3 solutions are based either on a2-dimensional finite difference or finite element

formulation. In both cases, if one uses theappropriate meshes, one can obtain rigoroussolutions where all nonlinear effects are

properly accounted for. The only real problemwith Level-3 type solutions is that for eachproblem one must establish the appropriatemesh size. Coarse meshes yield inaccuratesolutions. What is coarse depends on theparticular problem under investigation. Thus,for a general nonlinear solution a convergencestudy must always be carried out.

Using a hierarchical simulation platform

such as DISDECO (Delft Interactive Shell

DEsign Code), where the analyst has at hisdisposal computational modules of differentlevel of sophistication, such a convergence

study can be carried out relatively quickly andaccurately. In Table 5 a summary of the results

obtained in this study is presented usingnormalized variables.

Table 5

Summary of buckling load calculations of theCaltech isotropic shell A-8

(Nct= -49.51517 Ib/in)

Perfect shell analysis

Level-1 Membrane Prebuckting -Repeated eigenvalues with high multiplicity

Lm = 1.000 ---> Pc = -1244.5 Ibs

2407

Level-2 Nonlinear Prebuckling

SS-3 2_,I = 0.844480(n = 24)

--> Pc = -1050.9 Ibs

SS-4 _1 = 0.866974(n = 25)

"-> Pc = -1078.9 Ibs

C-3 Z,nl = 0.910305(n = 24)

Pc = -1132.8 Ibs

C-4 _1 = 0.926871(n = 25)

--> Pc = -1153.4 Ibs

Level-3 SS-3 B.C.

161x181 Model

Z_ I = 0.833361(n = 26)

-> Pc =-1037'11bs

261 x261 Model

2_,I = 0.844993(n = 24)

--) Pc = -1051.6 Ibs

Imperfect Shell Analysis

Long wave axisymmetric imperfection -

_1 = Aaxi = -0.5077

Level-2 ;Lnl = 0.824(n = 9)

---> Pc = -1025.4 Ibs

Short wave axisymmetric imperfection -

_1 = Aaxi = -0.5077

Level-2 _1 = 0.266(n = 27)

--> Pc = -331.0 Ibs

Long wave asymmetric imperfection -

_2 = Aasy = 0.6882

Level-2 Knl = 0.740(n = 9)

"--> Ps = -920-91bs

Short wave asymmetric imperfection -

_2 = Aasy = 0.6882

Level-2 ;k,nl = 0.324(n -- 24)

---> Ps = -403.21bs

Measured midsurface initial imperfection -see Eq. (20)

Level-3 SS-3 B.C. --> Ps = -900-11bs

Level-3 C-4 B.C. '-> Ps = -976"51bs

Measured midsurface + boundary (T--=b = 0.035)

initial imperfection - see Eqs. (20) and (23)

Level-3 Modified C-4 B.C.

(see Eq. (24)) --_ Ps =-780.01bs

Experimental Buckling Load

--> Pexp = -782.7 Ibs

Notice that by including both the midsurfaceand the boundary initial imperfections oneobtained a very good agreement indeed.

The study of the buckling behavior of theisotropic shell A-8 was undertaken with theobjective of establishing a rational analysisprocedure for incorporation of initialimperfections into the design of buckling criticalstructures. For a shell of revolution under

axially symmetric loading there are thefollowing four items necessary for theapplication of the proposed procedure:

a. Buckling analysis capability for shells ofrevolution under axially symmetric loading.

(A computer code such as SRA [18] orBOSOR [35] will satisfy this requirement.)

b. Imperfection sensitivity analysis capabilityfor shells of revolution under axially

symmetric loading. (A computer code suchas SRA [18] or STASOR [36] wilt satisfy thisrequirement.)

c. Geometrically nonlinear analysis capabilityfor a shell of revolution, which can handle

the asymmetry introduced by the generalasymmetric initial imperfections. (TheSTAGS [20,21] computer code will satisfythis requirement.)

d. A design imperfection.At the present time this is the most elusiveitem.

The imperfection model introduced byImbert [37], following an idea by Donnell andWan [38], is one approach, but has yet to beextended to larger cylindrical shell structuresand other shells of revolution.

coN_q _C.ML gt

By relying on a series of theoretical results ofvarious degree of sophistication published inthe literature, the hierarchical approach used inthis paper has resulted in a series of bucklingload predictions of increasing accuracy. It wasshown that in order to be able to arrive at areliable prediction of the critical buckling load

2408

and to make an estimate of its imperfectionsensitivity which can be used with confidence,

one must proceed step by step from simple tomore complex models and solution proce-dures.

In particular one can state, that in order topredict the critical buckling load accurately andto make a reliable estimate of its imperfectionsensitivity, the nonlinear effects caused by theedge restraint conditions must be included inthe analysis. Any solution procedure whichfails to account for these effects, should be

suspect of having provided incorrect results.The most approximate of the here

described analyses, the Level-1 solutionswhich neglect the effects caused by the edgerestraints, can still be used to great advantageto establish the approximate behavior of a shellsubjected to the specified external loading.However, depending on the value of theprebuckling stiffness, resulting from thedifferent types of wall constructions used, thesolutions may be either conservative ornonconservative.

As can be seen from the results shown in

Table 5, the buckling load of the isotropic shellA-8 is sensitive to all the initial imperfectionshapes investigated. For a more specificprediction of the final collapse load, the finalgoal of a "High Fidelity Buckling LoadAnalysis", one has to carry out a refined Level-3 analysis including measured values of all thesignificant generalized imperfections such asthe traditional shell-wall imperfections,variations in the shell-end or loading surfacegeometry and especially for composite shellsvariations in the shell-wall thicknessdistribution. It has been shown in Ref. 33 that

such an approach yields very good agreementbetween the predicted collapse toad and theexperimental buckling load.

ACKNOWLEDGEMENT

The research reported in this paper wassupported in part by NASA Grant NAG 1-2129.This aid is gratefully acknowledged.

1.

2.

F__=E.EBF=B.C_E_

Arbocz, J. and Babcock, C.D. Jr., '_l'he

Effect of General Imperfections on theBuckling of Cylindrical Shells," J. Appl.Mech., Vol. 36, 1969, pp. 28-38.Arbocz, J., Starnes, J.H. and Nemeth,M.P., "A Hierarchical Approach toBuckling Load Calculations," Proceedings40th AIAAJASM F_JASCE/AH S/ASC

Structures, Structural Dynamics andMaterials Conference, April 12-15, 1999,St. Louis, MO, pp. 289-299.

3. Anonymous, NASTRAN, The MacNeal-Schwendler Corporation, Los Angeles,California.

4. Anonymous, ABAQUS, Hibbitt, Karlsson& Sorensen, Inc., Providence, RhodeIsland.

5. Zienkiewicz, O.C., "The Finite ElementMethod," (The third, expanded andrevised edition of The Finite Element

Method in Engineering Science),McGraw-Hill Book Company (UK) Limited,London, 1977, p. 504.

6. Byskov, E., "Smooth PostbucklingStresses by a Modified Finite ElementMethod," DCAMM Report No. 380,Technical University of Denmark, Lyngby,1988.

7. Koiter, W.T., "On the Stability of ElasticEquilibrium," Ph.D. Thesis (in Dutch), TH-Delft, The Netherlands, H.J. Paris,Amsterdam, 1945, English translationNASA TTF-10, 1967, pp. 1-833.

8. Budiansky, B. and Hutchinson, J.W.,"Dynamic Buckling of ImperfectionSensitive Structures," Proceedings 11thIUTAM Congress, Munich, 1964, JuliusSpringer Verlag, Berlin, 1966, pp. 636-651.

9. Hoff, N.J., "Buckling of Thin Shells,"Proceedings of an Aerospace Symposiumof Distinguished Lecturers in Honor ofTheodore von Karm._n on his 80th

Anniversary, Institute of AerospaceSciences, New York, 1961, pp. 1-42.

10. Stuhlman, C.E., De Luzio, A. and Almroth,B., =Influence of Stiffener Eccentricity andEnd Moment on Stability of Cylinders inCompression," AIAA Journal, Vol. 4, No.5, May 1966, pp. 872-877.

11. Fischer, G., "0ber den Einfluss dergelenkingen Lagerung auf die Stabilit_td0nnwandiger Kreiszylinderschalen unterAxiallast und Innendruck," Z.f. Flug-wissenschaften, Vol. 11, 1963, pp. 111-119.

12. Stein, M., The Influence of PrebucklingDeformations and Stresses on the

Buckling of Perfect Cylinders," NASA TR-190, February 1964.

13. Arbocz, J. and Hol, J.M.A.M., =ShellStability Analysis in a Computer-AidedEngineering (CAE) Environment,"Proceedings 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamicsand Materials Conference, April 19-22,1993, La Jolla, California, pp. 300-314.

14. Arbocz, J., '_he Effect of Initial

Imperfections on Shell Stability AnUpdated Review," Report LR-695, DelftUniversity of Technology, Faculty ofAerospace Engineering, The Netherlands,September 1992.

2409

15. Arbocz, J. and Hol, J.M.A.M., "Koiter's

Stability Theory in a Computer AidedEngineering (CAE) Environment," Int. J.Solids and Structures, Vol. 26, No. 9/10,

pp. 945-973, 1990.16. Booton, M., "Buckling of Imperfect

Anisotropic Cylinders under CombinedLoading," UTIAS Report No. 203,University of Toronto, 1976.

17. Von K&rm,_n, T. and Blot, M.A.,"Mathematical Methods in Engineering,"McGraw-Hill Book Co., New York, London,

1940, p. 317.18. Cohen, G.A., "Computer Analysis of

Asymmetric Buckling of Ring-stiffenedOrthotropic Shells of Revolution," AIAAJournal, Vol. 6, No. 1, January 1968, pp.141-149.

19. Keller, H., "Numerical Methods of Two-Point Boundary Value Problems," BlaisdellPublishing Co., Waltham, Mass., 1968.

20. Almroth, B.O., Brogan, F.A., Miller, E.,Zele, F. and Peterson, H.T., "CollapseAnalysis for Shells of General Shape; I1.User's Manual for the STAGS-A Computer

Code," Technical Report AFFDL-TR-71-8,Air Force Flight Dynamics Laboratory,Wright-Patterson Air Force Base, Ohio,March 1973.

21. Brogan, F.A., Rankin, C C. and Cabiness,H.D., "STags Access Routines-STARReference Manual," Report LMSCP032595, Version 2.0, Lockheed Palo AltoResearch Laboratory, Palo Alto, California,June 1994.

22. Koiter, W.T., '`The Effect of AxisymmetricImperfections on the Buckling of CylindricalShells under Axial Compression," Koninkl.Ned. Akad. Wetenschap. Proc. B66, 1963,pp. 265-279.

23. Lorenz, R., "AchsensymmetrischeVerzerrungen in d0nnwandigenHohlzylindern," Z..VDI, Vol. 52, 1908, pp.1706-1713.

24. Timoshenko, S., "Einige Stabilit&ts-probleme der Elastizit&tstheorie," Z. Math.Phys., Vol. 58, 1910, pp. 337-385.

25. Southwell, R.V., =On the General Theoryof Elastic Stability," Phil Trans. Roy. Soc.London, Series A, Vol. 213, 1914, pp.187-244.

26. Arbocz, J. and Hol, J.M.A.M., "ANILISA -Computational Module for Koiter'sImperfection Sensitivity Theory,, ReportLR-582, Delft University of Technology,

Faculty of Aerospace Engineering, TheNetherlands, January 1989.

27. Romkes, A., "Stability and ImperfectionSensitivity of Anisotropic Cylindrical Shellsunder General Boundary Conditions,"Memorandum M-809, Delft University of

Technology, Faculty of AerospaceEngineering, The Netherlands, July 1997.

28. Hutchinson, J.W. and Amazigo, J.C.,

"Imperfection Sensitivity of EccentricallyStiffened Cylindrical Shells," AIAAJournal, Vol. 5, No. 3, March 1967, pp.392-401.

29. Cohen, G.A., "Effect of a NonlinearPrebuckling State on the PostbucklingBehavior and Imperfection Sensitivity ofElastic Structures," AIAA Journal, Vol. 6,

No. 8, August 1968, pp. 1616-1619.30. Hutchinson, J.W. and Frauenthal, J.C.,

"Elastic Postbuckling Behaviour ofStiffened and Barreled Cylindrical Shells,"

J. Appl. Mech., Vol. 36, 1969, pp. 784-790.

31. Arbocz, J., de Vries, J. and Hol, J.M.A.M.,

"On the Buckling of Imperfect AnisotropicShells with Elastic Edge Supports underCombined Loading - Part I: Theory andNumerical Analysis", Memorandum M-849,Delft University of Technology, Faculty ofAerospace Engineering, The Netherlands,

May 1997.32. Arbocz, J., The Effect of Imperfect

Boundary Conditions on the CollapseBehavior of Anisotropic Shells," Int. J.Solids Structures, Vol. 27, Numbers 46-47,

2000, pp. 6891-6915.33. Hilburger, M.W. and Stames, J.H. Jr.,

"Effects of Imperfections on the Buckling

Response of Compression-LoadedComposite Shells," Proceedings 41stAIANASME/ASCF-JAHS/ASC Structures,Structural Dynamics and MaterialsConference, 3-6 April 2000, Atlanta,Georgia, (Paper AIAA-2000-1382).

34. Almroth, B.O., "Influence of EdgeConditions on the Stability of AxiallyCompressed Cylindrical Shells", NASACR-161, February 1965.

35. Bushnell, D., "Stress, Stability andVibration of Complex Branched Shells ofRevolution: Analysis and User's Manual forBOSOR-4," NASA CR-2116, 1972.

36. Weustink, A.P.D., =Stability Analysis of

Anisotropic Shells of Revolution underAxisymmetric Load with General BoundaryConditions", Memorandum M-881, DelftUniversity of Technology, Faculty ofAerospace Engineering, The Netherlands,November 2000.

37. Imbert, J., "The Effect of Imperfections onthe Buckling of Cylindrical Shells,"Aeronautical Engineer Thesis, CaliforniaInstitute of Technology, 1971.

38. Donnell, L.H. and Wan, C.C., "Effect of

Imperfections on Buckling of ThinCylinders and Columns under AxialCompression," J. App. Mech., Vol. 17,1950, pp. 73-83.

2410

Buckling toads contour mop for the Coltech isotropic shell A--8

normol[zed buckling Iood lombdoc =_ Uc // --49.51 51 7

o0

EoI

_o°>

0_o°

6

x0

_o

E3 0E 0

0

0o

I ] [

......1.1

1.0

....""1.1'

2

• _............</.-i = 1.0 i i

).00 _.o0 8.00 12.0o 16.00 20.00

number of c[rcumferent[ol full wove5 -- n

contou_

levels1 1 .00

2 1.10

._% 1 .20

4 1.50

5 2.00

Fig. 1 Distribution of buckling loads based on Level-1 membrane prebucklinganalysis - Caltech isotropic shell A-8.

2411

w v +w o

i

a. Prebuckling Shape

-I,O

0.0

t.o

! I IolidW 2 dalhad

b. Buckting Component. Uodes

a. Membrane prebuckling, n=9

;_ = 1.00006

i wv +w O

-,3

-.2 __

-.11.

.Or

"_(max)-L ---="

O. Prebuckling Shape

-t.o

o.o

J i.o

W 1 IOlldW 2 dolhld

_meK)-L

b. Buckling Component Modes

b. Nonlinear prebuckling, n=24

_,nl = 0.844480

w v +w o

-1.o

o.o

317-.2

-.I

.o _=(mox)-L

o. Prebuckling Shape

W 1 IOJld

W2 delhld

t 1.0 ;(mol)-t.

b. Buckling Component Modes

c. Nonlinear prebuckling, n=24

;_nl = 0.844481

Fig. 2 Buckling modes of the axially compressed isotropic shell A-8

SS-3 B.C. -N x = -N o, v = W = M x = 0 : Nct = -49.51517 Ib/in.

2412

-11(Q

(AO

(n.-¢

Q

00

,,3-,j

m

r-e'J

!

0

N"

Zc _] •

_o

E

v

]

7 _

Zfl

II

bo

0.70Normoir|od OxIOI iooa Iornbdoa m Nx / --4Q.51517

0.74 0.78 0,82 0.80

I " i - ! . w - I ' • - I0.80

II

_J

9,.

I

"" II

9 _..I

Ii N,

3f |I =I

I

I

!

I

I

?I

I

I

| _'_1, i , 1

tl

,.. -,",, n

i =", I|

, , - _t, ,

¢

Z;0I

®

Z0

I

ood

t'-t

00

m

V

X

IIV

Oo4

×

®

C°_

"o

ooooo=

Perfect shell A--8 -- 1 61 rows x 181 Columns --

w--field, eige_vatue 1 , Nc=--41 .264 Ib/in,

O00000UOOOOO_ O OOOO@@@OOOOOOOUOOO0000000000@O00@00(_,O00000000000000000000000000000000000000OO0000000000

0 000000000000000000000000000000000(O00000000C,O0000

I?000 O )O 0000 00000000 0000000000GOD000 O0"O ' 00(-6

,/eo.oo 18o.oo aTo.oo

c;rc. coordinate y [0. <= y <-- 360.00]

S$--3 B.C. -- u =0 at told--plane

deltc-- tetc,= _*_ 60 degrees

0.00_60.00

conto_Jr-

levels

1 --0.802 --0.622223 --0.444444 --0.266675 --0.08889

6 0.088897 0.26667

8 0.44,4-449 0.62222

1 0 0.80

a. Contour-map of the critical (lowest) buckling mode

I=grfllct lib4111 A--8 -- 161 r'o,_| x 181 ¢olu_"_ll -- S$ ---_ 8,C. -- _=0 ot mi'4--plar_m

w--f_el¢l, oi_or_vol_ _ , N¢----_1 .264. I1_/i_° CloIta--tetom3150 Clo_roew

g!

30

g o

q

I

To._ .3_33 2 (sol? _.oo • 33_ • gloo'_"

#xisl ilto_.i o r_

/

Fig. 4

b. Axial trace of the critical (lowest) buckling mode

Buckling mode of the axially compressed isotropic shell A-8

SS-3 B.C. N x -- -N o, v -- W = M x = 0; Ncg = -49.51517 Ib/in; STAGS-A results.

2414

Perfect ilhell A--IB -- 161 rows x 181 ¢oi_._mr_ll -- $S -Ii B.¢- -- umO Ot r-r_icl--plOl-le

w--f;oJd. ; m -- .o ge_vc_lLJo 1 . NC 41 2(5.4- Ibf;r% ¢loJto--tetom_O de_lrHII

g

_zj Jl II It II II ]1 II II It Jl II II It II II II II JI [[ II II If II II ill

i_l I/I/I/I/I/I/1t I/I/II I/II I1II tl I/JlI/II I/II I1JlIIIII

o+oo oo.¢m0 t JIO.OO t Iio.oo ll4.o.oo llOO.OO _llo.oo

(=|reumferent;ol Iltotion (degrooll)

c. Circumferential trace of the critical (lowest) buckling mode

Perfect shell A--8 _ 161 fowl x 1 a 1 ¢olumnlll -- SS--3 B.C. -- u--O ot tutti--plane

w--f;eld, i;glr_vol_o 1. N¢_--41 .26_- I_J;r_. _oltc=wtlt_R.RSO degr_ell

:: :: :: :: :: :: : :; :_) : :: i " :: ;: : :0 •

o "_!Ti(_ !! i .i _ _ _! i _ _ !_-_ _ !i i _i _ _! _ _I I_:®. :(_i : :. _i _: i : _ :i :: _: : :o: .,_,.: .: :: _ :: :_ :: :: :i : i

-"r_! i iiio __!ii

o :: :: : _ :: : :: !: : _ :. :_ :! : :: : : :_ : ! :: _) _:

_ .-: _ :" _ :; _# :' _" ._ i _ " _ ,_ i: _: _,_ .-" _ "

c|rc_J_'_ferllnt[ol stotior_ (d

d. Circumferential trace of the critical (lowest) buckling mode

Fig. 5 Buckling mode of the axially compressed isotropic shell A-8

SS-3 B.C. N x = -N o, v = W = M x = 0; Nct = -49.51517 Ib/in; STAGS-A results.

24]5

Perfect shell A--8 -- 261 rows x 261 columns -- SS--3 B.C. -- u--O o± mid--ploNe

w--f;elcI, e;ger_value 1 , Nc----4-1 .83997 Ib/in, delta--tetom360 degrees

6

)00000000000000000}0000000 0000000000000000000VN

X

IV

doL..Jq

,¢X

o

oE

"oL

t00000000000000000000000000000000000000000000000._60. Oo

0.00 gO.O_ 180.00 270.00

cTrc. coord[nc:te y [O. <-- y <-- 360.00]

Cor_toLJr

levels

1 --0.80

2 --0.62222

3 -- O. ,4--4-4-4-4

4 --0.26667

5 --0.08889

6 0.08889

7 0.26667

8 0.44444

90. 6222210 0.80

a. Contour.-map of the critical (lowest) buckling mode

Perfect shill! A--8 -- :Z61 rows x 2.61 colu_nnll _ SS--3 B.C. -- _ I 0 O t rnr_--pl_ne

w--f_llll{:_, O[gCI_V{]ILII! 1. NC----41.8--_g87 Ib/;n, _lalta--tlltom360 de_lrells

g

o.0@ ' 1.0o 2.,)o 3.¢>0 4.00 I1.00 i.o0 .O(: -aMlll mt@tFo_

b. Axial trace of the critical (lowest) buckling mode

Fig. 6 Buckling mode o! the axially compressed isotropic shell A-8

SS-3 B.C. N x = -N o, v = W =M x = 0; Nct = -49.51517 Ib/in; STAGS-A results.

2416

J

i°I/l/l/J/l/tllll/ttl/l/t/l/l/l/l/l/l/l/l/ttl/l/ti,t/.oI/I/Jill/I/I/I/[1I/I/I/I/I1I/i/!/I/I/i/I/!tI/_ VV _V VV _VV

o.oo 6o.o0 i zo.oo I oo.oo =14.o.oo 3oo.oo 34o,_

cr_:_umfof'ont|ol :totlon (clegroeu)

c. Circumferential trace of the critical (lowest) buckling mode

; J ii i i_

I "iii ii

°"Ti i_ i.=i

• l! ::&: :

1

t=ti: : ! _ : =-o : :i

Fig. 7

Perfect IROII A--8 -- 2(51 _OWO x 2(51 col_lr'_mll -- $$--.3 B-C. -- u--O ol; rnic_--plone

® y _ ® 9

i_i..'+i i i ;_fi i i !i._i _iii i!_

' ; ' " : : : ' ; i A ' " '. : ' " ' " ' ,'. ! ' ! - ' : " • i 6

,i, i ._i _ ii iii$ ii ._i ii i_ i! i_, iiCJ ii Jiii i

" i_ '" ,_' ,',i ,: !: ' w : ., " i.: " _ _" _ *

o.oo I_oo t =o.oo t io.oo J_o. oo _oo,oo _eo.oo

(=t'mkJmferemt|OI Itotion (de.Tee=)

d. Circumferential trace of the critical (lowest) buckling mode

Buckling mode of the axially compressed isotropic shell A-8

SS-3 B.C. N x = -N o, v = W = M x = 0; Nc£ = -49.51517 Ib/in; STAGS-A results.

2417

Fig 8 Imperfection sensitivity for long wave axisymmetric imperfection under axial compression

(SS-3 B,C.: N x = -N o, v = W =M x =0; Nct =-49.515171b/in).

Fig 9

oo

I

\

I

EO

"o_oo_o"_¢

_o[

oO _, , = , I , i , , a ,o0o _o=o o.... o.0 -o%

Imperfection sensitivity for short wave axisymmetric imperfection under axial compression

(SS-3 B.C.: N x = -N o, v = W = M x = 0; Nc£ = -49.51517 Ib/in).

2418

ColtQ(::h i=otrop[c shol.I A--8 -- 1 6 1 row8 x 1 8 1

Criticel buckling modo ot NC I -- _2. 582 Ib/_n

o .oo io_oo 1 Io.oo 2_o.oo 31o.oo

circ. coc_rdfnote y ro. _ y _i 360.00]

columns -- SS--3 B.C.

-- Wbor/h= 1.0-coi(2--pi*.x/L)

contour

levels

1 --0.80

2 --0.62222

3 --0.44444

4 --0.26667

5 --0.0888£

6 0.08889

7 0.26667

8 0.44444

9.0.62222

10 0.80

Fig. 10 Contour map of the critical buckling mode for axisymmetric imperfection.

2419

Fig. 11

\

no

_0ocOO

E

I , = , I , = , I , i , I

O.10 o,.4,O O.41o O.IIO

nori.y_oillld |mD(lrflQtle_ a_=Dartudl -- =Sb,=r2

Imperfection sensitivity for long wave asymmetric imperfection under axial compression

using Level-1 routine BFACT (Nct = -49,51517 Ib/in).

Fig. 12 Imperfection sensitivity for short wave asymmetric imperfection under axial compression

using Level-1 routine BFACT (Nc£ = -49.51517 Ib/in).

2420

Fig. 13

r . .,=_lyre_et_a I_l_me"t_¢t;c_n Se_llltlvlt_y ¢=_ blot o_J= =hell A--8 _ SS_*._ El C

r-

;o

I

\

I 0

i:

e_¢

_d

8Ol

Wb<=r/h -- =lbc=r2-1"=l_(¢n.,q/L).=.=,m(e-.y'/R)3

= ANILI_,_=9 =(_P*_* COLL.A, PS¢.n=g

""O- ..... "O--...

• , , i , a , I , , I * , = , , ,

,.0o OJO O,4.o o-ao. 0.80 1 O0

.z d ; I= ,'f tx _lrt wo -- im 2_rmo e _ o eo o_ _m u

Imperfection sensitivity for long wave asymmetric imperfection under axial compression

(SS-3 B.C.: N x = -N o, v = W = M x = 0; Nct = -49.515171b/in).

Fig. 14

"1;8

I

\

,o

¢

:if0,00

""0. ........ _. ....... 0 ....... -E) ......

f , i , J , i , I , * , I , *

R¢•moll|Icl [mlll_fl=t[on omplltu4e -- =|i_Ot2

Imperfection sensitivity for short wave asymmetric imperfection under axial compression

(SS-3 B.C.: N x = -N o, v = W =M x = 0; Nct = -49.51517 Ib/in).

2421

?

Fig. 15

2_

ORCl.I4FERENllAL /U_3LE (ind.]

Measured initial imperfections of Caltech isotropic shell A-8 [1].

g=y/R

Fig. 16

x

T Fully restrained:U =V=W=W,,

11"8751 Brass: v=o.3E=15=106psi

8.5 I__R

Aluminum:

l E • 10=1015psi

V=0.3

3.0,

Steel:E =30=106psi

v=0.3

0

V=W=_

Reference surface

Cerroiow

Oevcon B

Spacer

Devcon B

1IN Steel end plate

Caltech test set-up and corresponding computer model.

2422

Fig. 17

-0.4 ! l

0 00 1110

dmmc=,=m_ p,_cm_, On_=U,,_)

Measured flatness of the Caitech load-cell end-ring.

!

27O 3ffi0

Fig. 18

Cor-f'll:_it'_ld effo¢_, of n"lOOlur41cI r'rlodol ond boundor"y imogrfect_O_l

Coltech _sotrop_c lllhell A--8

.._..(_..._. gw;mpb.4,.gwlrnO c -- _xpgrimentoI

- • - , l ,. J i

o _ "a-- •

U "'...

_. ....

•v ",'f" O

o q 1

I

d i . , . i , r , I , * , I , i , ,

0.00 0.02 0.04 0.08 0.08 O. 10

Amplitude of the bounclory imperfection -- xibound

Combined effect of measured modal and boundary imperfections.

2423