Table Of Content-.f AIAA-2001 -I3.92
ON A HIGH-FIDELITY HIERARCHICAL APPROACH
TO BUCKLING LOAD CALUCATIONS
Johann Arbocz*
Delft University of Technology, The Netherlands
and
James H. Starnes** and Michael P. Nemeth***
NASA Langley Research Center, Hampton, VA 23681-001
ABSTRACT INTRODUCTION
Itis generally agreed that, in order to make
As a step towards developing a new design
the development of the Advanced Space
philosophy, one that moves away from the
Transportation System a success and to
traditional empirical approach used today in achieve the very ambitious performance goals
design towards a science-based design (like every generation ofvehicles 10x safer and
technology approach, a recent test series of 5 10x cheaper than the previous one), one must
composite shells carried out by Waters [1] at make full and efficient use of the technical
NASA Langley Research Center is used. It is expertise accumulated in the past 50 years or
shown how the hierarchical approach to so, and combine it with the tremendous
buckling load calculations proposed by Arbocz computational power now available. It is
et al [2] can be used to perform an approach obvious that with the strict weight constraints
often called "high fidelity analysis", where the used in space applications these performance
uncertainties involved in a design are goals can only be achieved with an approach
simulated by refined and accurate numerical often called "high fidelity analysis", where the
methods. The Delft Interactive Shell DEsign uncertainties involved in a design are
COde (short, DISDECO) is employed for this simulated by refined and accurate numerical
models. In the end the use of "high fidelity"
hierarchical analysis to provide an accurate
numerical simulation will also lead to overall
prediction of the critical buckling load of the
cost reduction, since the analysis and design
given shell structure. This value is used later
= phase will be completed faster and only the
as a reference to establish the accuracy of the
reliability of the final configuration needs to be
= Level-3 buckling load predictions. As a final
verified by structural testing.
= step in the hierarchical analysis approach, the
The light-weight shell structures used in
critical buckling load and the estimated
aerospace applications are often buckling
imperfection sensitivity of the shell are verified
critical. The buckling load calculations are
by conducting an analysis using a sufficiently
usually carried out by one of the many
refined finite element model with one of the
currently available finite element based
current generation two-dimensional shell
computer codes [e.g., 3,4]. In order to reduce
analysis codes with the advanced capabilities computer execution time, buckling analyses
needed to represent both geometric and are often done using only the small
material nonlinearities.
displacement stiffness matrix Ko. This
approach is used, despite the fact that the
"initial stability problem" so formulated can only
give physically meaningful answers if the
elastic solutions based on Ko (at least
approximately) are identically equal to zero [5].
Professor, Faculty of Aerospace
When the qualitative nature of the expected
Engineering, Associate Fellow AIAA
behavior is completely unknown, the stability of
Senior Engineer, Structures and Materials
the structure must be investigated using the full
Competency, Fellow AIAA
tangent stiffness matrix KT in order to
Senior Research Engineer, Associate
Fellow AIAA guarantee accurate and reliable buckling load
Copyright © 2001 by J.Arbocz, and buckling mode predictions. In order to
J.H. Starnes and M.P. Nemeth discover the load level at which KT ceases to
Published byAIAA with permission be positive definite (that is, the load level when
American Instituteof Aeronautics andAstronautics
buckling occurs), a step-by-step analysis (Level-I) methods used by the early
procedure is needed. investigators to the more sophisticated
In addition, it is imperative (though often analytical and numerical (Level-2 and Level-3)
completely neglected), that at the beginning of methods used presently.
any stability investigation, the accuracy of the
discrete model used should be checked Level-1 Perfect Shell Buckling Analysis
against available analytical or semi-analytical
results. This step is part of a mandatory study The geometric and material properties of
needed in order to establish the dependence of the 8-ply, composite shell with symmetrical lay-
the buckling load predictions on the mesh up of Ref. [1] are listed in Table 1.
distribution used. Furthermore, as has been
pointed out in the past by Byskov [6], if one Table 1.
carries out imperfection sensitivity Geometric properties of NASA layered composite
investigations, which involve an extension of shell AW-CYL-I-1 [1] - [+45 /0 / 90] s
the solution into the postbuckling response
region, further mesh refinement may be
ttotal(= h) =0.039976 in (= 1.01539 mm )
needed since the wavelength of the dominant
L = 14.0 in (= 355.600 mm )
large deformation pattern may often decrease
significantly. R =7.99945 in (= 203.18603 mm )
Finally, whenever one is engaged in shell E11 =18.5111x106 psi (= 12.7629x104 N/mm 2 )
stability analysis it is especially important that E22 =1.64x106 psi (= 1.13074x104 N/mm 2 )
one is aware of the possible detrimental effects
G12 =0.8706x106 psi (= 6.00257x103 N/mm 2 )
of a whole series of factors, that have been
investigated extensively in the late 1960s and v12 =0.300235
the early 1970s. Thus for an accurate and
reliable prediction of the critical buckling load Note: Symmetrical lay-up with 8 plys of equal
of a real structure, one must account not only thicknesses (= 0.004997 in)
for the influence of initial imperfections [e.g.,
7,8] and of the boundary conditions [e.g. 9], but Assuming a perfect shell(W=0) and the
one must also consider the effects of stiffener
following membrane prebuckling state
and load eccentricity [e.g., 10] and the
prebuckling deformations caused by the edge
restraints [e.g. 11,12]. W (°) = hWv = hA12 X
c
A recent test series of 5 composite shells
(1)
carried out by Waters [1] at NASA Langley
Research Center is used to illustrate how such
F(O)= Eh 2 1Xy 2
a hierarchical approach to buckling load cR 2
calculations can be carried out. The platform where
for the multi-level computations, needed for an
accurate prediction of the critical buckling
Nx Eh
loads and a reliable estimation of their
=__ ;Oc_ =_ ;Nc_ =oc_h
imperfection sensitivity, is provided by Ocf Nc._
DISDECO [13]. With this open ended,
hierarchical, interactive computer code the and c = #3(1 -v 2)
user can access from his workstation a
succession of programs of increasing
then the nonlinear equations governing the
complexity.
prebuckling state are identically satisfied and
the linearized stability equations reduce to a
set of equations with constant coefficients. It
SOLUTION OF THE BUCKLING PROBLEM
has been shown in Ref. [14] that by assuming
7 an asymmetric bifurcation mode of the form
In the following it will be shown that with the
help of DISDECO, the Delft Interactive Shell
DEsign COde, the shell designer can study the W(1) = hsinm_ _xcos _n (y _ "CKX) (2)
buckling behavior of a specified shell, calculate
where
its critical buckling load quite accurately and
make a reliable prediction of the expected
m = k= number of axial half waves
degree of imperfection sensitivity of the critical
n = t. = number of circumferential full waves
buckling load. The proposed procedure
consists of a hierarchical approach, where the _K = Khot's skewedness parameter
analyst proceeds step-by-step from the simpler
2
American Institute of Aeronautics and Astronautics
one can reduce the solution of the linearized with m=l half-waves in the axial direction and
stability equations to an algebraic eigenvalue n=7 full waves in the circumferential direction.
problem. Notice that the eigenvalue Xmn_ In order to provide a quick overview of the
distribution of eigenvalues, the values
depends besides on the wave numbers m and
displayed inthe contour plot are re-normalized.
nalso on Khot's skewedness parameter "cK ,a
Thus in Fig. 1 the following re-normalized
real number. The critical load parameter Xc is eigenvalues are plotted
the lowest of all possible eigenvalues. Thus
finding Xc involves not only a search over the
integer valued wave numbers m and n but the
search has to be repeated over a range of
possible positive and negative real numbers for Notice that the critical buckling load can be
"_K. Using the Level-1 computational module calculated using asimple multiplication
AXBIF [14] a search over integer valued axial = =
half-wave numbers m and over a range of
possible positive and negative real numbers 0.365992 (-2238.325) = -819.209 Ib/in
"_K yielded the lowest eigenvalues listed in
Level-2 Perfect Shell Buckling Analysis
Table 2 for the specified circumferential wave
numbers n.
To investigate the effects of edge constraint
Table 2. and of different boundary conditions on the
critical buckling load of the perfect shell
Buckling loads of the NASA layered
composite shell AW-CYL-I-1 (W=0) one has to switch to the Level-2
Buckling load map for the perfect shell computational module ANILISA [15]. In this
using AXBIF [14] (Nc_ =- 2238.325 Ib/in) module the axisymmetric prebuckling state is
represented by
n=4 _,cm=0.371613 (m=7,'_K=3.320)
W(°) =hWv +hwo(x)
n=5 Xm= 0.370302 (m =7, "cK = 2.680) (3)
n=6 Xm= 0.370029 (m =7, '_K= 2.178) F(O) Eh2 . 1_ 2 a2fo(X)]
=-_L-_ _'Y +
n=7 ;Lm=0.365992 (m=l, "_K=0-011)
It has been shown in Ref. [15] that with these
n=8 _.m=0.370131 (m=6, '_K=1.681)
assumptions the prebuckling problem is
n= 9 _.mc = 0.371980 (m =6, "_K=1.488) reduced to the solution of a single fourth order
ordinary differential equation with constant
n= 10 Xmc = 0.372880 (m =5, "_K=1.345) coefficients, which always admits exponential
solutions. Closed form solutions for simply
n= 11 _,mc = 0.375261 (m =4, "cK =1.214)
supported and clamped boundary conditions
have been published inthe literature [16].
n=12 Xm=0.369089 (m=l, "cK=0.579)
For anisotropic shells the linearized stability
n= 13 X_q= 0.370076 (m = 1, '_K =0.653) equations admit separable solutions of the
form
n=14 z,m=0.376748 (m=l,'c K=0.734)
W(1)=h[wl(X) cosne+w2(x)sinne]
Notice that besides the absolute minimum
(4)
of;Lm= 0.365992 at n = 7 there is a local
minimum of ;Lmc=0.369089 at n= 12. F(1)= ERh2 [fl(x)cos n8+f2(x)sinnO]
To facilitate the interpretation of the C
numerical results obtained, DISDECO provides
where 0 Y
the user with various graphical interfaces. R
Thus the results of the search for the critical
(lowest) buckling load ;Lc can be displayed in Using a generalization of Stodola's method
a contour map as shown in Fig. 1. Using [17] first published by Cohen [18] the resulting
membrane prebuckling the critical eigenvalue nonlinear eigenvalue problem is reduced to a
is (see also Table 2) sequence of linearized eigenvalue problems.
The resulting ordinary differential equations are
solved numerically by a technique known as
X_ = 0.365992
"parallel shooting over N-intervals" [19]. Notice
3
American Institute of Aeronautics and Astronautics
that by this approachthe effectof edge In fig. 2 the critical buckling modes using
restraintandthespecificboundaryconditions membrane and rigorous nonlinear prebuckling
aresatisfiedrigorouslyT.ofindthecriticatload are depicted. Notice that the solutions with
parameterXc an n-searchmustbecarried nonlinear prebuckling differ significantly from
out,wherebyonemustbecarefultofindnota the ones obtained using membrane
localminimumbuttheabsoluteminimumA. s prebuckling, especially at n = 11 where one
can be seenfromthe resultspresentedin observes a typical edge buckling type
Table 3 the n-search using membrane behavior.
prebucklingandarigoroussatisfactioonfSS-3
Level-3 Perfect Shell Buckling Analysis
(Nx=v=w=Mx=0) boundaryconditionfsor
the stabilityproblemnow yields a local
To verify the earlier predictions the finite
minimumof Xc=0.364715 at n = 7 and an difference version [20] of the well known shell
absolute minimum of Xc =0.364370 at n= 11. analysis code STAGS [21] will be used. Due to
the slightly skewed buckling pattern predicted
Table 3 by the Level-1 and Level-2 computations one
is forced to model the whole shell.
Buckling loads of the NASA layered composite
Initially a convergence study must be
shell AW-CYL-I-1 (Nc_ =-2238.325 Ib/in)
carried out in order to establish the mesh size
Buckling load map for the perfect shell using needed for accurate modeling of the buckling
ANILISA [15](B.C. Nx = v = w = Mx = 0) behavior of the shell in question. For this
purpose the asymmetric bifurcation from a
nonlinear prebuckling path option was used,
Prebuckling: Membrane Nonlinear whereby the earlier results obtained with the
Level-2 module ANILISA listed in Table 3
n=6 Xmc = 0.371920 _nl= 0.347378 serve as a reference.
Inthe convergence study, at first, for afixed
n=7 Xm= 0.364715 _nl= 0.337088
number of mesh points in the axial direction
n=8 Xm= 0.372592 _nl= 0.339701 (NR = 161) the number of mesh points in the
circumferential direction (NC) was increased
n=9 Xmc =0.371460 ;_cnl=0.330957 until the bifurcation load approached a
horizontal tangent. As can be seen from Fig. 3
n= 10 Xcrn=0.367479 _nl= 0.329163
the results converge to a limiting value from
below at about NC = 201. Next, for a fixed
n= 11 Xm =0.364370 _,_1=0.328594
number of mesh points in the circumferential
n= 12 Xmc =0.364551 _nl= 0.330271 direction (NC = 201) the number of rows (NR)
was varied. This time convergence is from
n= 13 _,mc =0.368528 _,cnl=0.334044 above and as can be seen from Fig. 3 the
horizontal tangent is reached at about NR =
n= 14 Xm=0.376314 xnl= 0.339388
201.
Using a mesh of 161 rows and 201 columns
(a model with 99268 D.O.F.'s and a maximum
The most accurate Level-2 solutions are semi-bandwidth of 795) and SS-3 boundary
obtained when one employs a rigorous conditions (Nx=-N O,v=W=M x=0) the
nonlinear prebuckling analysis. As can be seen following 3 lowest eigenvalues were obtained:
from the results listed in Table 3, for this
particular shell the critical buckling loads with
X(c1)= 0.327759 (n= 11)
nonlinear prebuckling are always lower than
the corresponding results obtained using a N(c1)= X(cl)Nc_= - 733.630 Ib/in
membrane prebuckling analysis. Specifically,
nl
the local minimum of _,c = 0.337088 at n = 7
is about 8% lower, whereas the absolute ;L(2c) =0.328652 (n =10) -->
minimum of ;_ncl =0.328594 at n = 11 is about
N(c2) = X(c2)Nc_=- 735.631 Ib/in
11% lower. Notice that the critical load Nc can
be calculated easily by multiplying the lowest
eigenvalue )_c by the normalizing factor k(c3) =0.329421 (n= 12) -->
Nc_ = -2238.325 Ib/in yielding
N(3) = X(c3)Nc_=, -737.352 Ib/in
Nc = XcNc_ = -735.500 Ib/in (n= 11)
4
American Institute ofAeronautics and Astronautics
Detailsof the critical bucklingmodeare where
displayedinFig.4.Noticethatthesequencoef
the 3 lowest buckling loads and the
Wo(X)= _' hE1cosi= x
correspondingbucklingmodesagreeclosely
Xci -X L
withthepredictionsobtainedwiththeLevel-2
moduleANILISA(seealsoTable3).
_. (1+B21 c_2) Eh3 - x
fo(x)= _1cos i_--
;Lci-_ 20c2_22 c L
IMPERFECTION SENSITIVITY STUDY
(7)
That initial imperfections may decrease the
load carrying capacity of thin-walled shell
structures is by now widely known and _'ci =1{°c2Dl1+ 2oc2A22 }
accepted. However, in order to calculate the
effect of initial imperfections one must know
Notice that the linearized stability equations
their shape and amplitude, an information that
become now a set of equations with variable
is rarely available.
coefficients. The reduced wave number cq
In the absence of initial imperfection and the normalized stiffness coefficients A22,
measurements, as a first step one must
establish whether a given shell-loading B21 and 1are all listed in Ref. [14].
combination is imperfection sensitive, and if It has been shown in Ref. [14] that by
the answer is positive to estimate how assuming an asymmetric bifurcation mode of
damaging certain characteristic imperfection the form
shapes are.
X n
(8)
Single Axisymmetric Imperfection W(1)=hsinm__-cos_ (y - _Kx)
where
Based on Koiter's pioneering work on the
effect of initial imperfections [7,22] the simplest m =k = number of axial half waves
imperfection model consists of a single n= _ = number of circumferential full waves
axisymmetric imperfection _K = Khot's skewedness parameter
: h_lcosi_x (5) a Galerkin type approximate solution yields for
L the eigenvalues (read, buckling loads) ;Lof the
problem a characteristic equation inthe form of
where i is an integer denoting the number of a cubic polynomial
half-waves in the axial direction and _1 is the
amplitude of the axisymmetric imperfection
;L3 -(Xmn.: +2_.ci - C1_16i=2m)X2 (9)
normalized bythe shell wall-thickness h.
If one assumes that both the axial load and the +{2Xmn'_+ ;Lci +((_2- (_1)_1Si=2m}Xci_
boundary conditions are independent of the
circumferential coordinate, then the prebuck-
-{Xmnl: +(_2_15i=2m +((_3- (_45im)_2} ;L2 = 0
ling solution will also be axisymmetric, a fact i
that simplifies the solution considerably.
where
Level-1 Analysis of Axisymmetric
Imperfection 6i=2m =I if i= 2m ;
Neglecting the effect of the prebuckling
=0 otherwise
boundary conditions the nonlinear equations
governing the prebuckling state admit the
following axisymmetric solutions 6im=1 ifi=m
=0 otherwise
W(°) =hW v +hwo(x)
(6)
^ ^
and the constants C1,C 2.... are listed in
F(O)= Eh2 1Xy2+fo(x ) Appendix C of Ref. [14].
cR 2
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American Institute of Aeronautics and Astronautics
Hereit mustberememberetdhatonewill the solution of the nonlinear partial differential
onlygetanynoticeabledegradinginfluenceof equations governing the prebuckling state can
theassumedaxisymmetriicmperfectionif _1 be reduced to the solution of a single fourth
order ordinary differential equation with
is negativeandifthecouplingconditioni=2m
constant coefficients, which can be solved
is satisfied.Thephysicalexplanationforthis
routinely.
can be foundin Koiter's1963paper[22].
For anisotropic shells the resulting
Furthermorei,n orderto obtainthesmallest
linearized stability equations admit separable
realrootof Eq.(9),fora givenaxisymmetric
solutions of the form
imperfection_1ann-searchmustbecarried
out.It shouldalsobenoticedthattheterms
W(1)=h[wl(X) cosne+w2(x)sinne]
involvingthe Kroneckerdelta5i=2mareall
(12)
linearin _1, and thus they dominatethe
F(1)= ERh2 [fl(x)cosn 8+f2(x) sinne]
buckling behavior of the shell with
C
axisymmetriicmperfection.
Y
Assumingthatthemostlikelyaxisymmetric where e=--.
R
imperfectionofthesteelmandreul sedtolay-
uptheNASAcompositeshellAW-CYL-I-1is
Solution proceeds as outlinedon Ref. [23].
givenby
Using an updated version of the Level-2
computational module ANILISA [24] and SS-3
=h_lcos2_L (10) (Nx=v=w=M x=0) boundary conditions
one obtains the results presented in Table 4.
Notice that a rigorous nonlinear prebuckling
the Level-1DISDECOcomputationaml odule
analysis was used and an n-search was
AXBIFgeneratedthesolidcurveshowninFig.
carried out for each specified axisymmetric
5. Noticethatthe curveis re-normalizebdy
imperfection amplitude El-
Xn_=0.366892,the criticalLevel-1buckling
loadoftheperfecsthellcomputedusingAXBIF
The values of Table 4 are plotted as the
[14]withmembraneprebucklinfgor I:K=0.0
dashed curve in Fig. 5. A comparison of the
andn=7.Noticealsothataninitialimperfection
results obtained via the Level-1 module AXBIF
amplitudeequalto thewallthicknessofthe
(solid curve) and the Level-2 module ANILISA
shell (_1=-1.0) generatesa "knockdown (dashed curve) shows that also in the case of
factor"ofPc=Xc/Xmc =0.486,resultinginthe axisymmetric imperfections a rigorous pre-
followingratherlowbucklingload buckling analysis should be used. Especially
for very small initial imperfection amplitudes
Xc=0.486Xmc=0.486(0.36689=2)0.178310 (El <0"1) the Level-1 predictions are
inaccurate and overestimate the critical
Nc=XcNc#=. buckling load. Notice further that both curves
0.178310(-2238.32=5-)399.115Ib/in have been normalized by ;Lm =0.366892, the
critical Level-1 buckling load of the perfect
shell computed using membrane prebuckling
with "_K = 0.0 and n = 7 full waves in the
by AXBIF [14] for 'cK =0.0 and n = 7. This way
circumferential direction.
the effect of using a rigorous prebuckling
analysis becomes easily discernible.
Level-2 Analysis of Axisymmetric
it is interesting to see that for small enough
Imperfection
Since the external loading, the boundary initial imperfection amplitudes (_1<0.07,
conditionsand the assumed initialimperfection say) the critical buckling load of {he shell is
are axisymmetric, therefore the prebuckling insensitive to the initial imperfection shape
solutionwill also be axisymmetric. It has been specified by Eq. (10). Notice that the critical
shown inRef.[23] thatbyassuming buckling modes have n = 11 full waves in the
circumferential direction, and as can be seen in
W(°) =hWv +hwo(x) Fig. 6, a somewhat skewed buckling mode
(11) shape which is dominated by edge buckling.
However, larger initial imperfections
F(o) Eh2. 1_ 2
( -_11_0>.>07, say) force the shell to respond in
=---_--(.--_ JLy +R2fo(x)}
another mode shape with n = 7 full waves in
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American Institute ofAeronautics and Astronautics
the circumferential direction and with (remember STAGS defines W positive
practically straight nodal lines. Interestingly outward) the following critical bifurcation load
enough now the critical buckling load of the was found
shell is sensitive to the axisymmetric initial
imperfection shape specified by Eq. (10) and Nc = -333.449 Ib/in
an initial imperfection amplitude equal to the
wall thickness of the shell (_1=-1"0) As can be seen from Fig. 7 the critical buckling
generates a "knockdown factor" of mode has n =7 full waves in the circum-
Pc =Xc/Xm =0.418. It predicts thus the ferential direction and no visible skewedness.
following rather low buckling load The nondimensional bifurcation load of the
shell with axisymmetric imperfection is for
Xc = 0.418X m = 0.418(0.366892) = 0.153361
_1 = 1.0
Nc = XcNc_ = Xc = N_£_c -333.449 = 0.148973
0.153361(-2238.325) = -343.271 Ib/in Nce -2238.325
with a very slight skewedness of the buckling The re-normalized bifurcation load is
pattern and n = 7 full waves in the
circumferential direction.
pnl= _----Ncwhere Nc4' = _.nl Ncf.
Nce
Table 4
Buckling loads of the NASA layered composite thus
shell AW-CYL-I-1 (Nc_ = -2238.325 Ib/in)
Axisymmetric imperfection using ANILISA [24] p_l= X_.E_=c0.148973 _ 0.4545
(B.C.:N x =v =w =M x =0) X_I 0.327759
Notice that the Level-3 re-normalization is
_1 Xcnl _1 Xnl done using xn1=0.327759, the critical
0. 0.328594 (n=l 1) -0.2 0.299830 (n=7) Level-3 buckling load of the perfect shell
- 0.01 0.328671 (n=11) -0.3 0.274773 (n=7) computed using STAGS-A with nonlinear
prebuckling and a 161x201 mesh.
- 0.02 0.328745 (n=l 1) -0.4 0.251275 (n=7)
- 0.05 0.328939 (n=l 1) -0.5 0.229976 (n=7)
Notice also that the Level-2 ANILISA prediction
- 0.06 0.328994 (n=11) -0.6 0.210876 (n=7)
(pnl= 0.4664, n = 7) agrees closely with the
-0.07 0.328332 (n=7) -0.7 0.193827 (n=7)
Level-3 STAGS-A prediction (p_l_=0.4545,
-0.08 0.326652 (n=7) -0.8 0.178655 (n=7)
n = 7). The slight difference is partly due to
-0.09 0.324860 (n=7) -0.9 0.165187 (n=7)
the fact that ANILISA uses the Donnell type
-O.10 0.322960 (n=7) - 1.0 0.153263 (n=7) nonlinear shell equations, whereas STAGS-A
employs the higher order Marlowe-FISgge
equations.
Level-3 Analysis of Axisymmetric
Imperfection
Single Asymmetric Imperfection
Recalling that since both the axial load and
the boundary conditions are independent of the
The effect of a single asymmetric initial
circumferential coordinate, therefore the
imperfection can be investigated either by
prebuckling solution will also be axisymmetric,
solving the full nonlinear response problem or
one can use once again the asymmetric
by employing the well known Lyapunov-
bifurcation from a nonlinear prebuckling path
Schmidt-Koiter [7] reduction technique. When
option. By modeling the full shell the code can
investigating the degrading effect of a single
choose itself the critical number of full waves in
mode asymmetric imperfection
the circumferential direction. No n-search must
be carried out. Using a uniformly spaced mesh
of 161 rows and 201 columns and the user -W- = t_2 sin. m_--xcos ny (13)
L R
written subroutine option WlMP to introduce
the following axisymmetric imperfection
where m and n are integers denoting the
number of axial half-waves and the number of
= 1.0hcos2_ x
circumferential full waves, respectively,
L
7
American Institute of Aeronautics and Astronautics
instability occurs at the limit point of the certain critical eigenvalues chosen one at the
prebuckling state in the generalized load- time, can at best give an indication as to the
deformation space. Assuming that the severity of the expected imperfection
eigenvalue problem for the critical (lowest)
sensitivity.
buckling load Ac will yield a unique asym- Assuming initially an asymmetric
imperfection affine to the critical buckling mode
metric buckling mode W (1), then for an
of the perfect NASA composite shell AW-CYL-
imperfect shell (_2 _ 0) the shape of the
1-1 as computed by the Level-1 computational
generalized load-deflection curve inthe vicinity module AXBIF (see also Table 2)
of the bifurcation point A = Ac is given by the
following asymptotic expansion
=hE2sin_x cos7 (y_0.01lx) (16)
L I-1
(A- Ac)_ = Aca_ 2+Acb_ 3+...
(i4) and using the Level-1 computational module
- Aca_2 - (A- Ac)1_2+O(_2) BFACT to carry out the initial postbuckling
analysis yields the following results
Expressions for the postbuckling coefficients
"a" and "b" and the imperfection forms factors Xc =Xm.c= 0.365992 (m=1,n=7, "_K=0.011)
"(z" and "[5" are derived in References [25,26].
Ifthe limit point is close to the bifurcation point, b=- 0.048844 _ = #=1.0
then the maximum load As that the structure
can carry prior to buckling can be evaluated
Substituting these values into Eq. (15), one
from Eq. (14) by maximizing A with respect to
can plot the degrading effect of an asymmetric
_. For cases where the first postbuckling
imperfection of the shape given by Eq. (16) as
coefficient "a" is zero, this analysis yields the
afunction of its amplitude _2 • As can be seen
modified Koiter formula [26]
from Fig. 8 an initial imperfection amplitude
equal to the wall thickness of the shell
(_2 =1.0) generates a "knockdown factor" of
Ps = X,s/;tcm = 0.541, resulting inthe following
where Ps= As/Ac" rather low buckling load
Notice that, if the second postbuckling
;ts =0.541Xm, = 0.541(0.365992) =0.198002
coefficient "b" is positive, Eq. (15) has no real
solutions. Thus the buckling load of the
specified shell-loading combination is not Ns =XsNc_ =
sensitive to small asymmetric initial
imperfections of the shape given by Eq. (13). 0.198002(-2238.325) =-443.192 Ib/in
tf, however, the second postbuckling
coefficient "b" is negative, the equilibrium load Notice that the imperfection form factors
A decreases following buckling and the "c¢' and "I_" are identical equal to 1.0 because
buckling load of the real structure As is BFACT uses membrane prebuckling to
sensitive to the asymmetric initial imperfection calculate the necessary first and second order
specified by Eq. (13). fields and the assumed asymmetric
imperfection shape of Eq. (16) is affine to the
Level-1 Analysis of Asymmetric buckling mode. Please notice that in Fig. 8 the
Imperfection collapse load is re-normalized by
For the composite shell under investigation,
Xm, = 0.365992, the critical Level-1 buckling
as can be seen from the partial results listed in
load of the perfect shell computed using
Table 2, there are many eigenvalues only
membrane prebuckling by AXBIF [14] for
slightly higher than the critical one of
"cK =0.011 and n=7.
X,c =0.365992 for m = 1, n = 7 and
XK = 0.011. Hence, strictly speaking, the
Level-2 Analysis of Asymmetric
proposed form of the perturbation expansion
Imperfection
given by Eqs. (14) is not applicable, since the
To investigate the effects of edge-constraint
nonlinear interaction between the many nearly
and/or different boundary conditions on the
simultaneous eigenmodes is not accounted for.
imperfection sensitivity of the critical buckling
Thus the following results, where one
load ofthe NASA composite shell AW-CYL-I-1
considers the eigenfunctions corresponding to
American Institute of Aeronautics andAstronautics
one has to switchto the Level-2module
Notice that up to about _2 =0.3 the
ANILISA[15]andrunitspostbucklinagnalysis
option.Inthismodulea,sdescribedearliert,he asymptotic predictions from ANILISA and the
nonlinear results of COLLAPSE agree very
axisymmetripcrebucklingstateisrepresented
closely. Thus one can say that in this case the
by Eqs.(3),thebucklingmodesbyEqs.(4)
range of validity of the asymptotic solution is
andthepostbucklinsgtateby
0_>_2 >_0.3.
W(2)=h[woc()x+wl3(x)cosne+wy(x)sinn0]
DISCUSSION OF THE RESULTS
(18)
When comparing and analyzing the results
F(2) = ERh2 [fa (x) +f13(x)cos 2nO+ fy(x)sin 2ne] obtained sofar it is important to keep in mind
C
that all Level-1 and Level-2 solutions are
based on approximate representations of the
where e=y/R. Details of the computational
unknown functions. As pointed out in the
procedures used are reported inRefs. [15,23]. previous sections Level-1 solutions use a
Next, let us assume that the specified single term double Fourier series
asymmetric imperfection is affine to the critical approximation to reduce the solution of the
buckling mode obtained by the rigorous Level- stability problem, formulated in terms of partial
2 perfect shell analysis discussed earlier differential equations, to algebraic eigenvalue
problems. The effect of edge restraint is
= h_2 [Wl(X)COS11_)+w2 (x)sin11 e] neglected (one uses a membrane prebuckling
solution) and the assumed field functions
satisfy approximately SS-3 (Nx =-N o,
where 0= y/R and the component functions
Wl(X) and w2(x ) are shown in Fig. 2b. v =w=Mx =0) boundary conditions.
Running ANILISA with rigorous prebuckling Level-2 solutions eliminate the y-
and SS-3 boundary conditions (Nx =-No, dependence by a truncated Fourier
v = w =Mx = 0)yields the following results decomposition in the circumferential direction.
The resulting system of nonlinear ordinary
differential equations are solved numerically,
Xc =Xn_ =0.328594 (n=11)
whereby both the specified boundary
conditions and the effect of edge restraint are
b= -0.37605 ; oc= 0.46663 ; 13= -0.22174 rigorously satisfied. Thus by this approach the
only approximation is that one represents the
variation of the solution in the circumferential
Using Eq. (15) to plot the degrading effect
of the asymmetric imperfection specified by direction by a single harmonic with n full
waves, whereby an n-search is used to
Eq. (18) as a function of its amplitude _2 one
establish which wave number is the critical
obtains the results displayed in Fig. 9 as a one. The Level-2 module ANILISA can also be
solid line. Obviously the fact that for an used to investigate the effect of using different
imperfection shape affine to (similar to) the boundary conditions. In Table 5, the results for
buckling mode with an amplitude of _2 =1.0 four different boundary conditions are
presented. As expected the fully clamped C4
one obtains a negative load carrying capacity
is unrealistic. boundary conditions has the highest critical
Here one must remember that Koiter's buckling load. The increase in load carrying
capacity with respect to the weaker SS3
Sensitivity Theory is asymptotically exact, that
is, it yields accurate predictions for sufficiently boundary conditions is about the same as for
an isotropic shell of similar characteristic
small imperfections, whereby what is
sufficiently small may vary from case to case. dimensions (same L/R and R/t ratios) of Ref.
[28].
Also, Eq. (15) was obtained by using the
The Level-3 solutions are based either on a
perturbation expansion given by Eq. (14),
2-dimensional finite difference orfinite element
where terms of order (_) are neglected. As
formulation. In both cases, if one uses the
can be seen from the dotted curve plotted in appropriate meshes, one can obtain rigorous
Fig. 9, by using more advanced computational solutions where all nonlinear effects are
modules such as COLLAPSE [27], where a full properly accounted for. The only real problem
nonlinear solution is used and terms upto and with Level-3 type solutions is that for each
including order (_2) are kept, one obtains problem one must establish the appropriate
mesh size. Coarse meshes yield inaccurate
more reasonable predictions.
solutions. What is coarse depends on the
particular problem under investigation. Thus,
9
American Institute of Aeronautics and Astronautics
forageneranl onlineasrolutionaconvergence Xc_0.328594-0.895615~0.896
studymustalwaysbecarriedout. Pc- •
Xc 0.366892
Usinga hierarchicaslimulationplatform
such as DISDECO(Delft InteractiveShell
This value representsthe effect of edge
DEsignCode),wheretheanalysthasathis
restraintS. ee alsotheresultsofTable4 and
disposalcomputationaml odulesof different
Fig. 5. Notice that using membrane
levelof sophistications,ucha convergence
prebucklingth,usneglectintgheeffectofedge
studycanbecarriedoutrelativelyquicklyand restraintt,he normalizedbucklingloadofthe
accuratelyIn. Table5asummaryoftheresults perfecsthellis1.000.
obtainedin this studyis presentedusing Turningnowto the effectof asymmetric
normalizedvariables.In Table6 the same imperfectionfsro, mtheresultslistedinTable5
resultsarerepeatedbutthistimetheimperfect itisevidentthatforanimperfectioanmplitude
bucklingloadsare printedas re-normalized equalto one wall-thicknessthe rangeof
variablesp.Lookingatthefirstcolumnw, here validity of the asymptoticsolutions is
exceeded,and the predictionsof Koiter's
the criticalbucklingloads and the critical
imperfectionsensitivitytheorycomputedby
bucklingmodeshapesoftheperfectshellare
ANILISA[15]arenolongervalid(seealsoFig.
listed,oneseesthatusingthe Level-3code
9).Ontheotherhand,thereseemstobegood
STAGS-Aonemustindeedusearelativelfyine
agreemenbtetweenthe resultsobtainedby
mesh(161rowsand201columns)inorderto COLLAPSE[27], a Level-2computational
obtainan accuratepredictionof the critical modulewhichcomputesa nonlinearsolution
buckling load. Remember,all preceding basedonatwomodesapproximatioann,dthe
computationamlodulesarebasedonDonnell STAGS-A[20]solutionobtainedsofarforthe
type anisotropicshell equations,however, two modal imperfectionsand the affine
STAGS-Ausesthe moreaccurateMarlowe- imperfectionlistedinTable5. Ingeneralthe
moreaccurateSTAGS-AL,evel-3solutionsare
FILiggetypeequationsU. suallythe useofa
slightlylowerthanthe Level-2COLLAPSE
morerefinedtheoryimpliesa lowerbuckling
solutions,withexceptionof the asymmetric
load.The valueof the normalizationfactor
imperfectioanffinetotheperfectshellbuckling
used, Nc_=. Eh2/cR, is printed in the
mode.Thereasonforthisanomalyliesinthe
headingof the table. The lowestcritical
factthatthisimperfectiontriggersmorethan
bucklingloadwasfoundusingSTAGS-Aand onecircumferentiahlarmonicclosetothelimit
modelingthe whole shell with a mesh point(ascanbeseeninFig.66ofRef.29).
consistingof161rowsand201columnsT.he
asymmetricbifurcationfrom a nonlinear
prebucklingpathoptionwithSS-3boundary CONCLUSIONS
conditionysielded
By relying on a series of theoretical results of
various degree of sophistication published in
NIcn=0.327759(-2238.325)
the literature, the hierarchical approach used in
=-733.630Ib/in(n=11)
this paper has resulted in a series of buckling
load predictions of increasing accuracy. It was
Consideringnowthe effect of different shown that in order to be able to arrive at a
types of imperfectionshapes a second reliable prediction of the critical buckling load
normalizatioins introducedw, herebythenew and to make an estimate of its imperfection
normalizatiofnactorsarechosensuchthatfor sensitivity which can be used with confidence,
one must proceed step by step from simple to
vanishinglsymallimperfectiontshenormalized
more complex models and solution proce-
variablep approachesunity(i.e.1.000)T. he
dures.
onlyexceptionto thisruleisthecaseofthe
In particular one can state, that in order to
axisymmetriimc perfection
predict the critical buckling load accurately and
to make a reliable estimate of its imperfection
sensitivity, the nonlinear effects caused by the
=-h_lcos2_L
edge restraint conditions must be included in
the analysis. Any solution procedure which
whereby usingas the normalizationfactor fails to account for these effects, should be
X_=0.366892 (the asymmetricbifurcation suspect of having provided incorrect results.
perfectshell bucklingload with membrane The most approximate of the here
p_rebucklin"g_,K=0.0andn=7)inthelimitas described analyses, the Level-1 solutions
which neglect the effects caused by the edge
_1_ 0, Pcapproachetshevalue
restraints, can still be used to great advantage
10
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