Nonlinear and Buckling Behavior of Curved Panels Subjected to Combined Loads

12 Pages·2001·0.81 MB·English

by  Hilburger, Mark W.

AIAA-2001-1398
NONLINEAR AND BUCKLING BEHAVIOR OF CURVED PANELS SUBJECTED TO
COMBINED LOADS
Mark W. Hilburger,* Michael R Nemeth, _and James H. Starnes, Jr,*
NASA Langley Research Center
Hampton, Virginia 23681-001
Abstract loads for each component of the combined loads. This
type of approximation neglects the nonlinear interac-
The results of an analytical study of the nonlinear
tion of the combined loads and results in an overly con-
and buckling response characteristics of curved panels
servative prediction of the bifurcation-buckling load.
subjected to combined loads are presented. Aluminum
and laminated composite panels are considered in the Often, empirical knockdown factors for shells
study and a flat and shallow curved panel configurations with complex geometry and subjected to combined
are considered as well. The panels are subjected tocom- loads are unavailable, and analysis-based predictions of
bined axial compression and transverse tension or com- the sensitivity of the panel response to initial geometric
pression loads or combined axial compression and in- imperfections and other complicating effects must be
plane shear loads. Results illustrating the effects of var- used. Ideally, highly accurate predictions of the re-
ious combined load states on the buckling response of sponse can be obtained when the initial geometric im-
the panels are presented. In addition, results illustrating perfections, material properties, thickness distribution,
the effects of laminate orthotropy and anisotropy and boundary conditions, and other features are known to a
panel curvature on the panel response are presented. The high level of accuracy. 2 When these quantities are not
results indicate that panel curvature can have a signifi- well known, analysis-based lower-bound estimates that
cant effect on the nonlinear and buckling behavior of the represent the effects of initial geometric imperfections
panels subjected to combined loads. Results are includ- in nonlinear analyses must be used. A common exam-
ed that show that geometrically perfect panels do not ex- ple of this a-lbirrrach is to use abifurcation-mode imper-
hibit bifurcation points for some combined loads. fection shape to represent the initial geometric
Results are also presented that show the effects of lami- imperfections. This lower-bound, nonlinear-analysis-
nate orthotropy and anisotropy on the interaction of com- based approach has been applied to the Space Shuttle
bined loads.
superlightweight external liquid-oxygen tank. 3"7 This
Introduction approach is inherently ultra-conservative and leads to
an overweight structural design, but is often the only
To achieve high performance in the next genera-
meaningful approach available to designers and ana-
tion of aerospace structures, it will be necessary to de-
lysts.
velop reliable, high-fidelity design criteria for the
buckling behavior of shell structures subjected to com- To complicate matters, recently it has been shown
bined mechanical and thermal loads. For the past thirty that geometrically perfect, shallow cylindrical panels
years, buckling loads for shell structures have been ob- that are subjected to combined axial compression along
tained for design purposes, to a large extent, by con- the generator and transverse tension or compression do
ducting linear bifurcation analyses on idealized, not exhibit bifurcation points. 8 This result was ob-
geometrically perfect shell structures and by using em- tained by using an approximate special-purpose, single-
pirical "knockdown" factors that account for the effects
mode nonlinear analysis and indicates a major short-
of initial geometric imperfections. I In many cases, bi- coming of the traditional shell stability design approach
furcation-buckling interaction curves for shells subject- that is based on linear-bifurcation buckling analyses or
ed to combined loads have been approximated by a results. The main objective of the present paper is to
linear relationship between the bifurcation-buckling examine the nonlinear behavior of curved panels sub-
jected to combined loads with a high-fidelity, nonlin-
_, , . .
+Aerospace Engrueer, Mechamcs and Durabihty Branch. Member, AIAA. ear-analysis tool. This objective is achieved in the
Senior Research Engineer, Mechanics and Durability Branch. Associate Fel-
w, AIAA. present paper by first describing the high-fidelity anal-
enior Engineer, Structures and Materials Competency. Fellow, AIAA,
ysis tool, and then by presenting results for cylindrical
Copyright © 2001 by the American Institute ofAeronautics and Astronautics, panels made of aluminum and laminated-composite
Inc. No copyright isasserted in the United States under Thle 17, U. S.Code.
The U. $. Government has aroyalty-free license to exercise all rights under the materials. In particular, the results address the behavior
copyright claimed herein for Governmental Ptirposes. All other rights are
reserved by the copyright owner. of simply supported cylindrical panels that arc subject-
I
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iP; O, c, ,.
edtocombinationosfaxialcompressiotrna,nsverse
vanishes.
tensionorcompressioann,dshealroads.Resultfsor
The prebuckling responses were determined using
flatplateasndcurvedpanelasreincluded.
the geometrically nonlinear quasi-static analysis capabil-
Finite-Element Model and Analysis ity in STAGS. The ET processor was used to obtain so-
lutions beyond the bifurcation point and to branch onto
The panels considered in this study were analyzed the postbuckling equilibrium path of the panels. Once
with the STAGS finite-element code. 9 The geometry
the initial postbuckling path was established, the analysis
and loading conditions and a typical finite-element mod-
was continued into the postbuckling range by using the
el of a curved panel are shown in Fig. 1. Points on the quasi-static analysis capability in STAGS. Linear bifur-
panel mid-surface are located by an x-y-z coordinate
cation analysis were also conducted using the STAGS
frame whose origin is at the lower left corner of the pan- code. A typical finite-element model contained approx-
el. The panel length, width, and radius are defined as L, imately 10,000 degrees of freedom.
Wand R respectively. The panel length and width are L
Results and Discussion
= W= 14.0 in., for all panels. Flat plates (R = oo)and a
shallow curved panel with a radius ofR = 60 in. are con- The buckling and nonlinear response of panels sub-
sidered in the present study. Simply supported boundary jected to combined loads are presented. The results are
conditions are applied to the edges of the panels. The presented to illustrate the nonlinear behavior of curved
panels are loaded by uniform applied traction loads. panels subjected to combined loads. First, results illus-
Aluminum and laminated composite panels are consid- trating the response of flat aluminum plates subjected to
ered in the investigation. The aluminum panels have a combined loads are presented. Then results illustrating
nominal thickness of t= 0.12 in. The nominal material the effects of panel curvature on the nonlinear response
properties are as follows: Young's modulus E = 10.0 are presented. Finally, results illustrating the effects of
Msi, and Poisson's ratio v = 0.33. The composite pan- laminate orthotropy and anisotropy on the buckling loads
els are 24-ply laminates. The lamina plies have a nomi- of the panels subjected to combined loads are presented.
nal thickness equal to 0.005 in., which corresponds to a
Aluminum Flat Pl_,[e_
nominal panel thickness of t= 0.12 in. The nominal uni-
directional properties of a typical 0,005-in-thick lamina Nonlinear response curves for geometrically per-
ply are as follows: longitudinal modulus El = 19.5 Msi, fect, flat plates subjected to axial compression and trans-
transverse modulus E2= 1.45 Msi, in-plane shear modu- verse tension or compression loads are presented in Fig.
2. In particular, five curves are presented in Fig. 2a that
lus GI2 = 0.813 Msi, and major Poisson's ratio Vl2 =
show the nondimensional load as a function of the aver-
0.30. Three laminate stacking sequences are considered
age end shortening (referred to herein as load-end-short-
inthe present study and include aquasi-isotropic [__.45/0/
ening curves for convenience) that correspond to the load
9013s laminate, an axially stiff [_45/0213s laminate, and a
ratio Ny/N x= -1,-0.5, 0, 0.5, and I. Similarly, five cor-
transversely stiff [+_45/90213s laminate. The finite-ele-
responding curves are presented in Fig. 2b that show the
ment models consist of four-node STAGS 410 quadrilat- nondimensional load as a function of the out-of-plane or
eral shell elements. The STAGS 410 element is a flat
normal displacement at the center of the plate. These
facet-type element that is based on the Kirchoff-Love curves are referred to herein as load-normal-displace-
shell theory and the nonlinear Lagrangian-strain tensor. ment curves for convenience. The ordinate isnondimen-
The element nodes include three rotational degrees of sionalized bythe value of the unidirectional compression
freedom and three translational degrees of freedom.
stress resultant (i.e. Ny/N x= 0) in the plate at the bifurca-
Large rotations are accounted for by the use of aco-rota- tion point; that is, (Nx)0bif 325.5 Ibf/in. The end-short-
tional algorithm. ening and panel normal displacements are
The STAGS code uses both the modified and full nondimensionalized by the panel length L =!4 in. and
Newton methods for its nonlinear solution algorithms. thickness t = 0.12 in., respectively. Negative and posi-
The equivalence transformation (ET) bifurcation proces- tive values of the load ratio correspond to transverse ten-
sor is used to continue beyond bifurcation points into the sion and compression loads, respectively. The filled
postbuckling range of the solution. The ET processor al- circle on each curve represents the bifurcation point.
lows the user to select a solution branch in the direction The results in Fig. 2a indicate that the bifurcation
of growth of a bifurcation mode that has been computed point of the plate considered increases with increasing
for the nonlinear stress state in the neighborhood of the transverse tension load and decreases with increasing
bifurcation point. This bifurcation mode is held constant transverse compression load, as expected. The values
with a magnitude specified by the user, and the load fac- of the nondimensional buckling loads range from 0.5 to
tor in the solution is adjusted until the solution residual 2.08. In addition, the results show a decrease in preb-
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American Institute of Aeronautics and Astronautics
i ¢"
uckling axial stiffness with increasing transverse ten- substantially with the amount of shear load applied to
sion load and the opposite trend is shown for an the panel. In addition, the results indicate that the non-
increase in transverse compression load. The results in linear buckling loads for all load ratios agree well with
Fig. 2b indicate that the magnitude of the postbuckling the linear bifurcation buckling loads and are summa-
normal displacement at the center of the plate increases rized inTable 2.
with an increase in the transverse compression load and
Typical bifurcation mode shapes for square alumi-
decreases with and increase in the transverse tension
num flate plates subjected to combined axial compres-
load. The displacement response is characterized by a
sion and transverse tension or compression loads or
monotonically increasing center displacement with in- axial compression and in-plane shear loads are shown
creasing applied load after buckling. In addition, the
in Fig. 4. The solid and dashed contour lines in the fig-
shapes of the curves indicate that the stiffness of the
ure indicate positive and negative normal displace-
panel increases with decreasing Ny (tension). More- ments, respectively and the density of the lines
over, the results in Figs. 2a and 2b indicate that the indicates the severity of the displacement gradient The
buckling loads obtained from a nonlinear analysis (re- results indicate that, for panels subjected to uniaxial and
ferred to herein as nonlinear-buckling loads) are identi- biaxial compression loads, that is, values of the load ra-
cal to the linear bifurcation buckling loads for all load tio Ny/N x= -I, -0.5, and 0, the mode shape is charac-
ratios, and these results are summarized in Table 1.
terized by asingle half-wave response, as shown in Fig.
Nonlinear response curves for geometrically per- 4a. In contrast, the results indicate that the mode shape
fect flat plates subjected to axial compression and in- is characterized by a single half-wave across the width
plane shear loads are presented in Fig. 3. In particular, of the plate and two half-waves along the panel length,
three load-end-shortening response curves and three for values of the load ratio Ny/N x= 1,and 0.5, as shown
load-normal-displacements response curves are shown inFig. 4b. A typical bifurcation mode shape for a panel
for values of the load ratio Nxy/N x= __.1,_+.0.5,and 0 in subjected to combined axial compression and in-plane
Figs. 3a and 3b, respectively. The ordinate is nondimen- shear loads (Nxy/N x= 1) is shown in Fig. 4c, and is
sionalized by the value of the uniaxial compression stress characterized by askewed single half-wave mode shape
resultant (i.e. Nxy/N x= 0) in the plate at the bifurcation which exhibits a central point of inversion symmetry
.....point; that is, (Nx)0bif. The end-shortening and panel
(polar symmetry). The buckling displacements predict-
normal displacements are nondimensionalized by the ed from a nonlinear analysis of the aluminum flat plates
panel length L and thickness t,respectively. Positive and
are identical to the predicted bifurcation-buckling mode
negative values of the load ratio correspond to positive shapes, for all loading cases considered.
and negative in-plane shear loads, respectively. The
filled circle on each curve represents the bifurcation Aluminum Curved Pan¢l_;
point.
Nonlinear response curves are shown in Fig. 5 for
The results in Fig. 3a indicate that the bifurcation the curved panels with R/t = 500 (R = 60 in.) subjected
point of the plate considered decreases with increasing tocombined axial compression and transverse tension or
in-plane shear load, as expected. The normalized buck- compression. In this figure, six load-end-shortening re-
ling loads are 0.86, 0.96 and 1.0 for load ratios Nxy/N x sponse curves and six load-normal-displacement re-
= ±1, +0.5, and 0, respectively. The decrease in buck- sponse curves are shown for values of the load ratio Ny/
ling load is attributed to the diagonal compression gen- Nx= -1, -0.5, -0.1,0, 0.5 and 1. The ordinate is nondi-
erated by the shear load, however, the shear load has a mensionalized by the value of the uniaxial compression
smaller effect on the response than an applied trans- stress resultant Ny/N x= 0 (i.e. Ny/N x= 0) in the plate at
verse compression load Ny because itproduces a small- the bifurcation point; that is, (Nx)0bif= 723.4 lb/in. The
end-shortening and panel normal displacements are non-
er-magnitude compression load than the transverse
dimensionalized by the panel length L =14 in. and thick-
compression load. The results also show that the preb-
uckling axial stiffness is the same for all values of in- ness t = 0.12 in., respectively. Negative and positive
values of the load ratio correspond to transverse tension
plane shear loads. The results in Fig. 3b indicate that
and compression loads, respectively. The open circle
the magnitude of the normal displacement at the center
and square symbols on each curve indicates the bifurca-
of the plate decreases with an increase in the in-plane
tion point and limit point, respectively.
shear load. The displacement response is characterized
by a monotonically increasing center displacement with The results inFig. 5a indicate that only the curved
increasing applied load after buckling occurs. Howev- panel loaded by uniaxial compression, that is Ny/N x=
er, unlike the results for the transverse compression- 0, exhibits a bifurcation point. The nonlinear-buckling-
loaded plates, the postbuckling stiffness does not vary load value obtained from the nonlinear analysis (Nx=
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American Institute of Aeronautics and Astronautics
720.3Ib/in.)iswithin1%oftheloadvalueobtained
uniaxial compression stress resultant (i.e. Nx_/N x= 0) in
fromthelinear-bifurcatioannalysis.Moreoverth,e
the plate at the bifurcation point; that is, (Nx)'_Jbif= 723.4
curvedpanewl ithNy/Nx=-0.1exhibitaslimitpointat
lbf./in. The end-shortening and panel normal displace-
aloadleveol f383.5Ib/in.Thelimitpointforthispanel ments are nondimensionalized by the panel length L and
issignificantlleyssthanthecorrespondbinifgurcation- thickness t,respectively. Positive and negative values of
bucklingloadequatlo727.2lb./in.Theremaindeorf
the load ratio correspond to positive and negative in-
thepanelsexhibitabenignm, onotonicailnlycreasing plane shear loads, respectively. The open circle on each
nonlinearresponseT.henormalizendonlineanrormal curve represents the bifurcation point.
displacemernetsponsceurvefsorthepanelasreshown
The results inFig. 6a indicate that the bifurcation
inFig.5b.Theresultisndicattehatthepaneslubjected
point of the compression-loaded panel considered de-
touniaxiaclompressio(Nny/Nx= 0)doesnotexhibit
creases with increasing in-plane shear load, as expect-
anyprebucklinngormadleformationAsf.terbuckling,
ed. The normalized buckling loads are 0.79, 0.93 and
howevetrh,epaneel xhibitssignificanintwardradial
1.0 for values of the load ratio Nxy/N x= _+1,_.+0.5,and
deformationassindicatebdythenegativvealueosfthe
0, respectively. In particular, the curved panel with a
normalizecdentedrisplacemernetsponseIn.contrast,
load ratio value of Nxy/N x = _.+1exhibits a normalized
thepanelssubjectetdocombineadxialcompression
buckling load of 0.79 as compared to the corresponding
andtransversteensioonrcompressidoonnotexhibiat
fiat plate which exhibits a normalized buckling load
bifurcatiotnyperesponsrea,therth, eyexhibimt onoton-
equal to 0.86. This result suggests that panel curvature
icallyincreasinlogadversusnormadlisplacemeren-t
can have asignificant effect on the response of the pan-
sponsesfromtheonsetof loading. Thepanels
el to combined loads. The results also show that the
subjectetdobiaxiaclompressieoxnhibitpositiveout-
prebuckling axial stiffness is the same for all values of
wardnormadleformationasn,dthepaneslubjectetod
in-plane shear loads. The bifurcation point is followed
combineadxialcompressiaonndtransvertseensioenx-
by an unstable buckling response. In addition, the re-
hibitnegativeinwardnormadleformationthsathave
sults indicate that the curved panel subjected to com-
thecharacteristoicfsinitialgeometriimc perfections.
bined axial compression and in-plane shear loads
Thenonlinearm, onotonicalliyncreasinlgoad-end-
exhibit bifurcation-buckling type responses for all load
shorteninbgehavioorfthepanelss,howninFig5a,is
ratios considered. In contrast, the corresponding panels
attributetdothenormadleformatiorensponseexhibited
subjected to combined axial compression and trans-
bythepanelsM. orespecificallya,sthepaneldseform
verse tension or compression loads exhibit monotoni-
duringloadingt,hegeometrisctiffnesosfthepanel
cally increasing, stable responses. The nonlinear
changewshichcausetsheeffectivaexiasltiffnesosfthe
normal displacement response curves for the panels are
panetlochangienagraduanlonlineamrannearsindi-
shown in Fig. 6b. The results indicate that during the
catedbythenonlinearersponssehowinnFig.5a.This
prebuckling response the panels do not exhibit any
graduaclhangienaxiasl tiffnesissincontratsottherap-
normal deformations, however, the post buckling dis-
idchangienaxiasl tiffnesosfapanetlhatexhibitasbi-
placement response of the panels is characterized by
furcation-bucklitnygperesponseO.veraltlh,esehigh-
large magnitude inward normal displacements of the
fidelityresultvserifytheapproximaretesultpsresented
central region of the panels.
inRef.8andindicategeneralltyhatlinear-bifurcation
analysemsaybemisleadinfogrcurvedpanelssubjected A comparison of bifurcation mode shapes and
tocombineadxiacl ompressioanndtransversteension buckling deformation patterns predicted from a nonlin-
orcompressiloonadsI.nadditionth,eseresultssuggest ear analysis for square aluminum curved panels sub-
thatcurvedpanelssubjectetdothistypeofcombined jected to combined axial compression and transverse
loadincgonditiomnaynothaveahighdegreoefimper- tension or transverse compression loads or axial com-
fectionsensitivityA. summaroyfthepredictebdifur- pression and in-plane shear loads are shown in Fig. 7.
cation-bucklinlogadsandthenonlinear-bucklloinagds The solid and dashed contour lines inthe figure indicate
forthecurvedaluminumpanelasrepresenintTable1. positive and negative normal displacements, respec-
tively, and the density of the lines indicates the severity
Nonlinearresponsceurvesforgeometricaplleyr-
of the displacement gradient. The results indicate that
fectcurvedpanelssubjectetodaxiaclompressiaonndin-
the predicted bifurcation mode shape for a panel sub-
planeshealroadsarepresenteindFig.6. Threeload-
jected to uniaxial compression exhibits a single half-
end-shortenicnugrvesandthreeload-normal-displace-
wave response and is characterized by a deformation
mentcsurvesareshownforvalueosftheloadratioNxy/ pattern ellipse-shaped contours, as shown in Fig. 7a.
Nx=
+1, __.0.5,and 0 in Figs. 6a and 6b, respectively. The results also indicate that the buckling deformation
The ordinate is nondimensionalized by the value of the pattern predicted from a nonlinear analysis of the panel
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American Institute of Aeronautics and Astronautics
,J
agrees well with the bifurcation mode shape. The re- plates and curved panels subjected to combined loads are
sults indicate that, for a panel subjected to biaxial com- presented in this section. Results for fiat plates and
pression with a load ratio Ny/N x = 1, the panel curved panels subjected to combined axial compression
deformation response is characterized by a single half- and transverse tension or compression loads are present-
wave response, as shown in Fig. 7b. However, these re- ed in Figs. 8a and b, respectively. Bifurcation-buckling
sults indicate that there is a slight difference in the de- loads and buckling loads predicted from a nonlinear
formation pattern determined by a linear bifurcation analysis are denoted by the square and triangle symbols,
analysis compared tothe nonlinear deformation pattern. respectively. The bifurcation-buckling values of the load
More specifically, the bifurcation buckling mode shape Nxare denoted by (Nx)bif in the figure, and (Nx)0bif cor-
is characterized by a deformation pattern with ellipse- responds to the buckling load value for a given panel
shaped contours and is in contrast to a deformation pat- with Ny/N x= 0. Similarly, the buckling load predicted
tern predicted with circular-shaped contours from a from a nonlinear analysis are denoted by (Nx)cr in the
nonlinear analysis. The results indicate that, for apanel figure and (Nx)0cr corresponds to the buckling value for
subjected to axial compression and transverse tension a given panel with Ny/N x= 0.
with a load ratio Ny/N x= -1, the bifurcation buckling
The results inFig. 8show that panel curvatureand
mode shape is characterized by two half-waves across
laminate orthotropy can have a significant effect on
the width of the plate and one half-wave along the panel
buckling response of the panels subjected to combined
length as shown in Fig. 7c. However, the predicted de-
axial compression and transverse tension loads (load ra-
formation pattern from a nonlinear analysis is charac-
tio Ny/N x= -1, and -0.5). More specifically, the results
terized by a single half-wave response. A typical
in Fig. 8a indicate that axially stiff composite fiat plates
bifurcation mode shape for a panel subjected to com-
have higher values of normalized buckling loads that
bined axial compression and in-plane shear loads with
range from 1.0 to 2.46 followed by the results for the
a load ratio Ny/N x= 1is shown in Fig. 7d, and ischar-
aluminum and quasi-isotropic composite panels, which
acterized by a skewed single-half wave mode shape
exhibit nearly identical nondimensional buckling loads,
which exhibits a central point of inversion symmetry.
with normalized buckling load values that range from
The corresponding predicted buckling displacements of
1.0 to2.06 for the loading conditions considered, and fi-
the panel from a nonlinear analysis agree well with the
nally the results for the transversely stiff composite flat
predicted bifurcation buckling mode shape. The results
plates with normalized buckling load values that range
inFig. 7indicate that the predicted bifurcation buckling
from 1.0 to 1.52. In contrast, the results indicate that
mode shapes agree with buckling deformation patterns
the flat plates exhibit the same nondimensional buck-
predicted from a nonlinear analysis when the nonlinear
ling loads when subjected to uniaxial and biaxial com-
response exhibits a bifurcation response, e.g., panels
subjected to uniaxiai compression loads or combined pression loads (load ratio Ny/N x= 0, 0.5, and 1). In
axial compression and in-plane shear loads, and do not addition, the results indicate that the nonlinear-bucking
exhibit a prebuckling normal deformations. However, loads agree well with the predicted bifurcation-buck-
for curved panels subjected to combined loads that do ling loads for these flat plates and the values are gener-
ally with in 1%. The results presented inFig 8b indicate
not exhibit a bifurcation-buckling type response, the
predicted bifurcation mode shapes tend to differ from that the curved panels with R/t = 500 exhibit a similar
the deformation patterns predicted from a nonlinear trend; that is, the nondimensional buckling loads are
analysis and may be misleading. generally higher for the axially stiff composite panels
followed by the results for the aluminum and quasi-iso-
Effects of Laminate Orthotropy and Anisotropy tropic composite panels and finally the results for the
transversely stiff composite panels. However, the de-
Results for three laminates including aquasi-isotro- gree of variation in the values of the normalized bifur-
pic [___45/0/9013slaminate, an axially stiff [+_45/0213slam- cation-buckling loads is much less than that exhibited
inate, and a transversely stiff [__.45/90213slaminate were by the flat plates. These results suggests that a nonlin-
obtained to illustrate the effects of laminate orthotropy ear coupling exists between the panel curvature and
and anisotropy on the nonlinear and buckling response of laminate orthotropy for the panels considered. In addi-
the panels. Similar results to those presented for the alu- tion the results indicate that the curved panels subjected
minum panels were obtained and indicate that, in gener- to the combined axial compression and transverse ten-
al, the nonlinear and buckling response characteristics sion or compression loads do not exhibit a bifurcation
for the composite laminated panels are similar to those point, thus these bifurcation-buckling results may be
exhibited by the aluminum panels. Results illustrating misleading since bifurcation buckling may have no
the effects of orthotropy on the buckling loads of the flat meaning in the context of curved panels subjected to
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American Institute of Aeronautics and Astronautics
thesetypesofcombineldoads.
the nonlinear response of shells subjected to combined
Resultasrepresentefodrflatplatesandcurved
loads is needed. The present paper addressed this basic
panelssubjectetodcombineadxiaclompressaionndin-
need by presenting results and discussion for flat plates
planeshealroadsinFigs.9aandb,respectivelTyh. e
and curved panels subjected to a variety of combined
resultsindicatethatpanelcurvatureandlaminate
loading conditions. In particular, results for loading
orthotropayndanisotropcyanhaveaneffectonthe
conditions that involve combinations of axial compres-
nondimensionbaulcklingloadvaluesfortheloading
sion, transverse tension or compression, and shear loads
conditioncsonsidereMd.orespecificalltyh,eresultisn
have been presented. Results that illustrate the effects
Fig.9asuggesthtat,fortheflatplatesp,aneolrthotropy
of laminate orthotropy and anisotropy have also been
haslittleeffectonthenondimensiobnuacl klingload
presented. The results indicated that panel curvature
leveal sisevidencebdythatfactthatalltheresultfsor
can have a significant effect on the response of panels
thecomposiftleatplateasrenearlyidenticafolrallval-
subjected to combined loads. In particular, curved pan-
uesofthetoadratioNxy/Nx.Howeverth, eresultasIso
els subjected to combined axial compression and trans-
suggesthtattheresponsiseaffectebdylaminataenisot- verse tension or compression loads do not exhibit a
ropy.Morespecificallyth,eresultsshowthatthebuck- bifurcation-buckling type response. As aresult, a linear
lingloadvalueisdependeonnt thesignoftheload bifurcation analysis could misrepresent the physical be-
ratio,thatisthebucklingloadvalueisdependeonnt havior of the panel. The results also indicated that lam-
whethetrhein-planeshealroadispositivoernegative. inate orthotropy and anisotropy can have a substantial
ForexampleT,hecompositpelateswithaloadratio effect the buckling loads of the flat plates and curved
Nxy]Nx=-1,corresponditnoganegativine-plansehear panels subjected to combined loads. These results
load,exhibiat nondimensiobnuaclklinlgoadof0.835, should be of interest to engineers designing practical
howevert,hecomposiptelateswithaloadratioNxy/ shell structures.
Nx=+I,whichcorrespontdosapositivien-plansehear
References
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plateinwhichthebucklingloadsareindependeonftthe Monographs on Shell Stability Design Recommenda-
signoftheloadratio.Theresultfsorthecurvepdanels tions - A Review and'Suggested Improvements," NASA
subjectetodcombineadxialcompressiaonndin-plane TP- 1998-206290, January 1998.
shealoradsarepresenteindFig.9b.Theresulitndicate
2. Hilburger, M. W. and Starnes, J. H., Jr., "Effects
asimilartrendtothatexhibitedbythecorresponding
of Imperfections on the Buckling Response of Compres-
flatplatesH. owevetrh,eresultsindicattehati,ngener-
sion-Loaded Composite Shells," AIAA Paper No. 2000-
al,thecurvedpanelesxhibiltowernormalizebduckling
1387, April 2000.
loadswhensubjectetdocombineldoadsthandothe
correspondifnlagtplatesI.nadditiont,heresultisndi- 3. Nemeth, M. E, Britt, V. O., Young, R. D., Collins,
T. J., and Starnes, J. H., Jr., "Nonlinear Behavior of
catethatlaminataenisotropayffecttshebucklingloads
ofthecurvedpanelisnasimilarmannetorthatexhibit- Space Shuttle Superlightweight Liquid-Oxygen Tank
edbytheflatplatesasevidencebdythedependence Under Prelaunch Loads," Journal of Spacecraft and
of
Rockets, Vol. 36, No. 6, November-December, 1999, pp.
the buckling load on the sign of the in-plane shear load.
788-803.
Moreover, the results indicate that laminate orthotropy
of the panels considered also has a slight effect on the 4. Young, R. D. and Rankin, C. C., "Modeling and
buckling response. More specifically, the results indi- Nonlinear Structural Analysis of a Large-Scale Launch
cate that the transversely stiffpanels have higher values Vehicle," Journal of Spacecraft and Rockets, Vol. 36,
of normalized buckling loads which range invalue from No. 6, November-December, 1999, pp. 804-811.
0.8 to 1.0 followed by the axially stiff, quasi-isotropic, 5. Nemeth, M. E, Young, R. D., Collins, T. J., and
and aluminum panels with values of normalized buck- Starnes, J. H., Jr.," Effects of Welding-Induced Imper-
ling loads that range from 0.77 to !.0. fections on Behavior of Space Shuttle Superlightweight
Concluding Remarks Tank," Journal of Spacecraft and Rockets, Vol. 36, No.
6, November-December, 1999, pp. 812-819.
High-fidelity analysis tools and advanced compu-
tational capability now make it possible todevelop reli- 6. Young, R. D., Nemeth, M. P., Collins, T. J., and
able, high-fidelity design criteria for the buckling Starnes, J. H., Jr., "Nonlinear Behavior of Space Shuttle
behavior of shell structures subjected to combined me- Superlightweight Tank Under Booster Ascent Loads,"
chanical and thermal loads. As part of updating and de- Journal of Spacecraft and Rockets, Vol. 36, No. 6,
veloping such design criteria, in-depth examination of November-December, 1999, pp. 820-827.
f... ¢
7. Nemeth, M. E, Young, R. D., Collins, T. J., and 9. Rankin, C. C., Brogan, F. A., Loden, W. A. and
Starnes, J. H., Jr., "Nonlinear Behavior of Space Shuttle Cabiness, H. D., STAGS Users Manual, Lockheed Palo
Superlightweight Tank Under End-of-Flight Loads,"
Alto Research Laboratory, Report LMMS P032594,
Journal of Spacecraft and Rockets, Vol. 36, No. 6,
1999.
November-December, 1999, pp. 828-835.
8. Librescu, L., Nemeth, M. P., Starnes, .l.H., Jr.,
and Lin, W., "Nonlinear Response of Flat and Curved
Panels Subjected to Thermomechanical Loads," Journal
of Thermal Stresses, Vol. 23, No. 6, August, 2000, pp.
549-582.
Table 1: Buckling loads for aluminum panels subjected to combined axial compression and transverse
tension or compression loads.
R/t = oo(Fiat plate) R/t = 500
Loading Bifurcation- Nonlinear- Bifurcation- Nonlinear-
Ratio, buckling buckling buckling buckling
Ny/Nx load (Sx)bif, load (Nx)cr, load (Sx)bif_ load (Nx)c_,
lbf/in, lbf/in, lbf/in, lbf/in.
-1.0 677.7 677.7 1215 a
-0.5 581.1 581.1 1051 a
-0.1 - 727.2 383.5 b
0.0 325.6 325.6 723.4 720.3
0.5 217.0 217.0 496.5 a
1.0 162.8 162.8 376.3 a
a Monotonically increasing response which does not exhibit abucking point
b Limit point
Table 2: Buckling loads for aluminum panels subjected to combined axial compression and in-plane shear
loads.
R/t = o0(Flat plate) R/t = 500
Loading Bifurcation- Nonlinear- Bifurcation- Nonlinear-
Ratio, buckling buckling buckling buckling
Nxy/Nx load (Nx)bif, load (Nx)cr, load (Nx)bif, load (Nx)cr,
lbf/in, lbf/in, lbf/in, lbf/in.
-1.0 281.I 281.1 570.9 568.5
-0.5 312.0 312.0 674.8 666.1
0.0 325.6 325.6 723.4 720.3
0.5 312.0 312.0 674.8 666.1
1.0 281.1 281.1 570.9 568.5
x W
N_
_R
Fig. 1 Typical panel geometry and loading conditions and finite-element model.
"1
3 Ny__1 J -0.5 rx 3 Ny=_
2
N, °
1
0 , , , , ! , 1 l , I i i i J I , , , , I .... t .... t .... | A A i I I i i i i |
0 0.0005 0.001 0.0015 0.002 1 2 3 4 5
A/L _/t
a)Load-end-shortening response b) Load-normal-displacement response
Fig. 2 Nonlinear response of square aluminum flate plates subjected to combined axial compression and transverse ten-
sion or compression loads (L= 14.0 in., t=0.12 in., (Sx)0bif = 325.5 lbf/in.).
8
American Institute of Aeronautics and Astronautics
N x
-¢,-y
A
/',5 N_y
Ttt B
3
rl
Nxy
-if:-=0
2
Nx
(NJ°.
1 Ii
/ o Bifurcationpoints
0 I i i L I I i i 1 I , , , , I , , , , I i i | i i i i ! _ I
0 0.0005 0.001 1 2 3 4
A/L _t
a) Load-end-shortening response b) Load-normal-displacement response
Fig. 3 Nonlinear response of square aluminum flate plates subjected to combined axial compression and in-plane shear
loads (L = 14.0 in., t = 0.12 in., (Nx)0bif = 325.5 lbf/in.).
f...:=:::::iE===.--._.:..=:::=:::.._.
_._.-=i:_ :::'--:_--_:':_--.:..-_-
a) Ny/N x=0, 0.5, ! b) Ny/N x= -1,-0.5 c) Nxy/N x= 1
Fig. 4 Typical bifurcation mode shapes for square aluminum flat plates subjected to combined axial compression and
transverse tension or in-plane shear loads.
9
American Institute of Aeronautics andAstronautics
ts
N= R
1.25
1.25 NN-y_=0 / =o\
1 0.5
0.75
N= 0.75 Nx
(N=)O; (N,)=0,
0.5 0.5
0.25 0.25
[If o Bifurcatio n points
0 o Umit points 0
0 0.0005 0.001 0.0015 0.002 -5 -2.5 0 2.5 5
_/L 8/t
a) Load-end-shortening response b) Load-normal-displacement response
Fig. 5 Nonlinear response or square aluminum curved panel subjected to combined axial compression and transverse ten-
sion or compression loads (L= 14.0 in., R = 60 in., t = 0.12 in., (Sx)0bif -- 723.4 lbf/in.).
Nx !R
1.25 1.25
±0.5
0.75
Nx° 0.75 N,
(N,)_,
0.5 0.5
0.25 0.25
o Bifurcation points
0 0.0005 0.001 0.0015 0.002 -5 -2.5 0 2.5 5
A/L 6/t
a) Load-end-shortening response b) Load-normal-displacement response
Fig. 6 Nonlinear response of square aluminum curved panel subjected to combined axial compression and in-plane shear
loads (L = 14.0 in., R = 60 in., t = 0.12 in., (Nx)°bir = 723.4 Ibf/in.).
10
American Institute of Aeronautics and Astronautics
C ,, (:_.