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1、Thin-Walled Structures 42 (2004) of imperfections of the bucklingresponse of composite shellsMark W. Hilburger?, James H. Starnes Jr.NASA Langley Research Center, MS 190, Hampton, VA 23681-2199, USAReceived 9 June 2003; received in revised form 8 August 2003; accepted 11 September 2003AbstractThe r
2、esults of an experimental and analytical study of the effects of initial imperfectionson the buckling response and failure of unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shells are presented. The shells considered in the study have six differentshell-wall laminates two diff
3、erent shell-radius-to-thickness ratios. The shell-wall laminatesinclude four different orthotropic laminates and two different quasi-isotropic laminates. Theshell-radius-to-thickness ratios includes shell-radius-to-thickness ratios equal to 100 and 200.The numerical results include the effects of tr
4、aditional and nontraditional initial imperfec-tions and selected shell parameter uncertainties. The traditional imperfections include thegeometric shell-wall mid-surface imperfections that are commonly discussed in the literatureon thin shell buckling. The nontraditional imperfections include shell-
5、wall thickness varia-tions, local shell-wall ply-gaps associated with the fabrication process, shell-end geometricimperfections, nonuniform applied end loads, and variations in the boundary conditionsincluding the effects of elastic boundary conditions. The cylinder parameter uncertaintiesconsidered
6、 include uncertainties in geometric imperfection measurements, lamina fibervolume fraction, fiber and matrix properties, boundary conditions, and applied end loaddistribution. Results that include the effects of these traditional and nontraditional imperfec-tions and uncertainties on the nonlinear r
7、esponse characteristics, buckling loads and failureof the shells are presented. The analysis procedure includes a nonlinear static analysis thatpredicts the stable response characteristics of the shells, and a nonlinear transient analysisthat predicts the unstable response characteristics. In additi
8、on, a common failure analysis isused to predict material failures in the shells.Published by Elsevier Ltd.Keywords: Buckling; Composite shells; Imperfections; Failure?Corresponding author. Tel.: +1-757-864-3106.E-mail address: mark.w.hilburgernasa.gov (M.W. Hilburger).0263-8231/$ - see front matter
9、Published by Elsevier Ltd.doi:10.1016/j.tws.2003.09.0011. IntroductionThe increasing need to produce lighter-weight aerospace shell structures has ledto the use of advanced material systems in new structural designs, and improveddesign methods appropriate for these advanced material systems are need
10、ed. Thehigh strength-to-weight and high stiffness-to-weight ratios of advanced compositematerials offer significant weight reduction potential for aerospace structures. Inaddition, the use of advanced composite materials allows the designer to tailor thestiffness properties of composite structures t
11、o obtain structurally efficient designs.Designers often use a design-level analysis procedure with empirical data todevelop new structural designs for strength and buckling critical structures. Thetraditional approach for designing thin-walled buckling-resistant isotropic shellstructures is to predi
12、ct the buckling load of the shell with a deterministic analysis,and then to reduce this predict load with an empirical knockdown factor (e.g.,1). The empirical knockdown factor is intended to account for the differencebetween the predicted buckling load and the actual buckling load for the shelldete
13、rmined from tests. A linear bifurcation buckling analysis is often used for thedesign-level analysis, and this analysis is usually based on nominal structuraldimensions and material properties of an idealized, geometrically perfect shell. Thedesign knockdown factor used in the design of buckling-res
14、istant shells is oftenbased on the lower bound design recommendations reported in Ref. 1. Thisdesign philosophy can result in overly conservative designs for these structures, andit can potentially even result in unconservative designs if the empirical data are notrepresentative of the design of int
15、erest. While it is generally recognized that initialgeometric shell-wall imperfections are a major contributor to the discrepancybetween the predicted shell buckling loads and the experimentally measured shellbuckling loads (e.g., 26), the traditional sources of design knockdown factors donot includ
16、e data or information for shell structures made from advanced compositematerials. In addition, the traditional sources of design knockdown factors forpredicting shell buckling loads do not include information related to the sensitivityof the response of a shell to various forms of imperfections. Rec
17、ent studies (e.g.,712) have shown that traditional initial geometric shell-wall imperfections, andother nontraditional forms of imperfections or variations in geometric and materialparameters, fabrication related anomalies, loading conditions, and boundary con-ditions can significantly affect the bu
18、ckling load of a compression-loaded com-posite shell structure. The effects of these traditional and nontraditional classes ofinitial imperfections on the buckling of composite shells are generally not wellunderstood by structural engineers and designers. It has shown by Hilburger andStarnes (e.g.,
19、11,12) that highly accurate predictions of the nonlinear response andbuckling load of a compression-loaded shell can be obtained when the initial geo-metric imperfections, material properties, shell-wall thickness distribution, andother features are modeled to a high degree of accuracy. Modern high-
20、fidelity non-linear analysis procedures offer the opportunity to improve some of the engineeringapproximations that are used in the design and analysis of shell structures, and toM.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397370provide insight into the effects of traditional
21、and nontraditional imperfections onthe response of compression-loaded composite shell structures.The present paper describes the results of an experimental and analytical study ofselected unstiffened thin-walled compression-loaded graphite-epoxy cylindrical shell.The analytical results include the e
22、ffects of traditional initial geometric shell-wallimperfections, and the effects of nontraditional initial imperfections and uncertaintiesin other nontraditional geometric and material parameters, loading conditions, andboundary conditions. The results of six graphite-epoxy shells with different she
23、ll-walllaminates are presented. The shell-wall laminates include four orthotropic laminatesand two quasi-isotropic laminates. The two orthotropic shells and one of the quasi-isotropic shell have shell-radius-to-thickness ratios equal to 200, and two orthotropicshells the other quasi-isotropic shell
24、have shell-radius-to-thickness ratios equal to 100.The response of three shells with shell-radius-to-thickness ratios equal to 100 are pre-sented as examples of a thin-walled shells that exhibit material failures before or dur-ing buckling, and the material failures cause the overall failure of the
25、shell and nopostbuckling load carrying capacity. The response of three shells with shell-radius-to-thickness ratios equal to 200 are presented as contrasting examples of thin-walledshells that exhibit significant postbuckling load carrying capacity and material fail-ures occur in the postbuckling ra
26、nge of loading. A common material failure analysesis used to predict material failures in the shell. Traditional shell-wall geometric imper-fections and several nontraditional imperfections were measured, and representationsof these imperfections have been included in nonlinear analyses of the shell
27、s. Inaddition, selected uncertainties in several geometric, material, and loading parameterswere characterized and were also included in the analyses. The effects of initial geo-metric shell-wall imperfections, shell-wall thickness variations, shell-end geometricimperfections, nonuniform applied end
28、 loads, and variations in the boundary con-ditions, including the effects of elastic boundary conditions, on the buckling responseof these thin-walled composite shells are discussed in the present paper. The nonlinearanalyses were conducted with the geometrically nonlinear STAGS finite elementanalys
29、is code 13. The results of the study are used to illustrate the significance ofinitial imperfections and uncertainties on composite shell response characteristics.The nonlinear shell analysis procedure used to predict the nonlinear response andbuckling loads of the shells is described, and the analy
30、sis results are compared withthe experimental results. The use of this nonlinear shell analysis procedure for deter-mining accurate, high-fidelity design knockdown factors for shell buckling, collapse,and failure, and for determining the effects of variations and uncertainties in shellgeometric and
31、material parameters on shell response is discussed.2. Test specimens, imperfection measurements, and test apparatus and tests2.1. Test specimensThe specimens tested in this investigation were fabricated from 12-in.-wide,0.005-in.-thickAS4/3502graphite-epoxypreimpregnatedunidirectionaltape371M.W. Hil
32、burger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397material made by Hercules, Inc. The nominal unidirectional lamina properties of atypical 0.005-in.-thick ply with a fiber volume fraction of 0.62 are as follows: longi-tudinal compression modulus E1 18:5 Msi, transverse modulus E2 1:64 Ms
33、i,in-plane shear modulus G12 0:87 Msi, and major Poissons ratio m12 0:30. Thematerial was laid up on a 15.75-in.-diameter mandrel and cured in an autoclave toform six shells with different shell-wall laminates and include an 8-ply axially stiff?45/02Slaminate, an 8-ply circumferentially stiff?45/902
34、Slaminate, an 8-plyquasi-isotropic ?45/0/90Slaminate, a 16-ply axially stiff?45/022Slaminate, a16-ply circumferentially stiff?45/9022Slaminate, and a 16-ply quasi-isotropic?45/0/902Slaminate. The resulting six shells are referred to herein as shells orspecimens C1C6, respectively. These specimens ha
35、d a nominal length of 16.0 in.and a nominal radius of 8.0 in. The 8- and 16-ply specimens had a nominal shell-wall thickness of 0.04 and 0.08 in., respectively, and had shell-radius-to-thicknessratios of 200 and 100, respectively. Both ends of the specimens were potted in analuminum-filled epoxy res
36、in to assure that the ends of the specimens did not failprematurely during the test. The potting material extended approximately 1.0 in.along the length of the specimens at each end resulting in a test section that wasapproximately 14.0 in. long. The ends of the specimens were machined flat andparal
37、lel to facilitate proper load introduction during the tests. A photograph of atypical specimen and the specimen coordinate system used to represent the corre-sponding geometry is shown in Fig. 1. The shell length, test-section length, radius,and thickness are designated as L, LT, R and t, respective
38、ly.2.2. Imperfection measurementsThree-dimensional surveys of the inner and outer shell-wall surfaces of the speci-mens were made prior to testing the specimens to determine their initial geometricshell-wall imperfection shapes and shell-wall thickness distributions. Measurementswere taken over a un
39、iform grid with increments of 0.125 in. in the axial directionand 0.139 in. (approximately 1vof arc) in the circumferential direction over theexposed surfaces of the specimens. The inner surface measurement was used todetermine the initial geometric shell-wall imperfection shape of a specimen, and t
40、hedifference between the outer and inner surface measurements was used to deter-mine the shell-wall thickness distribution. A contour plot of the nondimensionalinitial geometric shell-wall mid-surface imperfections ? w w0x;h for specimen C3 isshown in Fig. 2. The measured shell-wall imperfection w0i
41、s nondimensionalized bythe average measured shell-wall thickness tave 0:0381 in: These results indicatethat the initial geometric shell-wall imperfection periodic in the circumferentialdirection and exhibits slight variations in the axial direction. The amplitude of theimperfection varies from +1.34
42、1taveto ?1.535tave. A contour plot of the non-dimensionalized shell-wall thickness variation?t t0x;h for specimen C3 is shown inFig. 3, where the measured thickness value t0is nondimensionalized by the averagemeasured shell-wall thickness tave. These results indicate that the shell-wall thick-ness,
43、and hence the laminate stiffnesses, varies significantly over a short distance.The thickness varies from 0.928 to 1.321 times tave. Most of the thickness variationM.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397372Fig. 2. Typical measured inner-surface imperfection shape for sh
44、ell specimen C3.Fig. 1. Typical specimen, finite-element model geometry and loading conditions.373M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397is attributed to local variations in the resin content of the laminate associated withthe fabrication process. However, the darker a
45、ngular pattern in the thickness distri-bution is attributed to small gaps between adjacent pieces of graphite-epoxy tape insome of the laminate plies that were generated during the lay-up and curing pro-cesses. Such a region is referred to herein as a lamina ply-gap or a ply-gap. Theselocally thin s
46、hell-wall regions have a significant shell-wall mid-surface eccentricity,and have reduced stiffnesses relative to the rest of the shell wall. Lamina ply-gapswith gap widths as large as 0.15 in. have been observed in some of the shell speci-mens. The lighter angular patterns in the thickness contour
47、plot are caused bylocally thickened regions of the outermost plies of the laminate that develop duringthe curing process to form outer shell-wall surface ridges. Some of the shell-wallthickness features, such as lamina ply-gaps, are smaller than the imperfection-measurement grid spacing used in this
48、 study, and as a result, some of the smallerthickness variation features may not have been included in the measurements.Typical magnified cross-sectional views illustrating the microstructure of typicalply-gaps and outer surface ridges in a composite laminated shell wall are presentedin Ref. 12.Meas
49、urements of the top and bottom loading surfaces of the specimens weremade every degree around the circumference of the specimens to determine thevariation in the shell-end or loading-surface geometry. Typical top and bottomshell-end or loading-surface geometry variations for specimen C3 are denoted
50、bydtop(h) and dbot(h), respectively, and are shown in Fig. 4. The maximum amplitudeof this shell-end or loading-surface variation is approximately 0.0015 in., which isapproximately 4% of taveor 0.01% of the specimen length.2.3. Test apparatus and testsThe specimens were instrumented with electrical
51、resistance strain gages, anddirect-current differential transducers (DCDTs) were used to measure displace-ments. Three non-colinear DCDTs were positioned at three corners of the upperFig. 3. Typical measured wall thickness variation for a shell specimen C3.M.W. Hilburger, J.H. Starnes / Thin-Walled
52、Structures 42 (2004) 369397374loading platen of the test machine and used to measure the end-shortening dis-placement D and the rotations /yand /zof the loading platen as illustrated inFig. 1. A typical set of measured loading platen rotations is shown in Fig. 5. Theseresults indicate that significa
53、nt upper platen rotations occur from the onset of load-ing up to a load value of approximately 6000 lbs. These rotations are attributed toan initial misalignment of the upper loading platen and the specimen. The rota-tions of the movable upper loading platen reach a steady state at a load value of60
54、00 lbs, and the loading of the specimen, for the most part, continues withoutadditional upper loading-paten rotations from 6000 lbs up to the buckling or col-lapse load. During the collapse event, the upper loading platen undergoes anadditional amount of rotation.The specimens were loaded in compres
55、sion with a 300,000-lb hydraulic univer-sal-testing machine by applying an end-shortening displacement to the shell ends orFig. 4. Typical measured shell-end or loading-surface imperfections for a shell specimen C3.Fig. 5. Typical experimentally measured loading platen rotations.375M.W. Hilburger, J
56、.H. Starnes / Thin-Walled Structures 42 (2004) 369397loading surfaces of the specimens. To help facilitate uniform load introduction intothe specimens, the upper loading platen was aligned with the loading surface of thespecimen as well as possible before the test by adjusting leveling bolts in the
57、cor-ners of the upper loading platen until strains measured by selected strain gages onthe specimens indicated a uniform axial strain distribution in the shell wall. Theshadow moire interferometry technique was used to observe the shell-wall prebuck-ling, buckling and postbuckling radial (perpendicu
58、lar to the shell outer surface)deformation patterns. All data were recorded with a data acquisition system, andthe moire patterns were recorded photographically and on videotape. The speci-mens were loaded until general instability or failure of the shells occurred.3. Finite-element models and analy
59、ses3.1. Nonlinear analysis procedureThe shells considered in this study were analyzed with the STAGS (STructuralAnalysis of General Shells) nonlinear shell analysis code 13. STAGS is a finite-element code developed for the nonlinear static and dynamic analysis of generalshells, and includes the effe
60、cts of geometric and material nonlinearities in the analy-sis. The code uses both the modified and full Newton methods for its nonlinearsolution algorithms, and accounts for large rotations in a shell by using a co-rotational algorithm at the element level. The Riks pseudo arc-length path-following
61、method 14 is used to continue a solution past the limit points of anonlinear response. With this strategy, the incrementally applied loading parameteris replaced by an arc-length along the solution path, which is then used as the inde-pendent loading parameter. The arc-length increments are automati
62、cally adjustedby the program as a function of the solution behavior. The transient analysisoption in STAGS uses proportional structural damping and an implicit numericaltime-integration method developed by Park 15. Additional information on thetransient analysis procedure can be found in Ref. 16.The
63、 prebuckling, buckling and postbuckling responses of the shells were determ-ined using the following analysis procedure. The prebuckling responses weredetermined using the geometrically nonlinear quasi-static analysis capability inSTAGS. The Riks pseudo arc-length path-following method in STAGS was
64、usedto compute the initial shell response until just before buckling occurred. Theunstable buckling response of the shell was predicted using the nonlinear transientanalysis option of the code. The transient analysis was initiated from an unstableequilibrium state close to the limit point by increme
65、nting the end displacement by asmall amount. An initial time step of 1:0 ? 10?8s was used in the analysis, and thetime step was automatically adjusted by the program as a function of the solutionbehavior. The transient analysis was continued until the kinetic energy in the shellhad dissipated to a n
66、egligible level, which indicated that the transient responsehad attenuated. Once the transient analysis had attenuated to a near-steady-statesolution, the load relaxation option of the code was used to establish a staticM.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397376equilib
67、rium state. Conventional linear bifurcation buckling analysis results werealso determined with STAGS for comparison with the nonlinear response results.3.2. Finite-element modelsA typical finite-element model of a specimen is illustrated in Fig. 1. The standard410 quadrilateral element from the STAG
68、S element library was used in the models.The elements of the finite-element mesh are approximately 0.2-in. by 0.2-in. square.Each element has four integration points, which are distributed in such a way as toprovide a modeling resolution of approximately 0.1-in. by 0.1-in. square. This inte-gration-
69、point spacing is on the order of the measurement-point spacing used whenmeasuring the initial geometric imperfections of the specimens. This highly refinedmesh is necessary to model rapidly varying geometric and material parameters suchas nonuniform shell-wall thickness and lamina stiffness properti
70、es. A typical finite-element model contained approximately 120,000 degrees of freedom.Geometrically perfect and imperfect shells were analyzed in the present investi-gation. Nominal shell geometry, laminate thickness, lamina mechanical properties,and boundary conditions were used for the finite-elem
71、ent models of the geome-trically perfect shells. The nominal boundary conditions consist of setting the cir-cumferential and normal displacements v and w equal to zero in the 1.0-in.-longpotted boundary regions of the shell illustrated in Fig. 1, setting uL=2;h 0, andapplying a uniform end-shortenin
72、g u?L=2;h D.The geometrically perfect finite-element models were modified to include theeffects of the measured shell imperfections in order to simulate more closelythe response of the specimens. These modeling modifications include the effects ofthe measured initial geometric shell-wall mid-surface
73、 imperfections, shell-wallthickness variations, local shell-wall lamina ply-gaps, thickness-adjusted laminaproperties, elastic boundary conditions, shell-end geometric imperfections, andnonuniform end loads.The initial geometric shell-wall mid-surface imperfection w0(x, h) is included inthe finite-e
74、lement models by introducing an initial normal perturbation to eachnode of the mesh by using a user-written subroutine with STAGS for that purpose.The user-written subroutine uses a linear interpolation algorithm that calculatesthe value of the imperfection for the coordinates of each finite-element
75、 node basedon the measured shell-wall data.The shell-wall thickness t, mid-surface eccentricity ecz, and lamina materialproperties E1, E2, G12and m12are adjusted at each integration point of eachelement in the finite-element models. The shell-wall mid-surface eccentricity is cal-culatedrelativetothe
76、averageshell-wallmid-surface;thatis,eczx;h ?0:5tave? t0x;h, where taveand t0denote the average shell-wall thickness andactual measured thickness, respectively. The lamina properties are adjusted byusing the rule of mixtures. In the rule-of-mixtures calculations, it is assumed thatany variation in th
77、e lamina ply thickness from the nominal thickness is due to avariation in resin volume only, and that the fiber volume remains constant for eachply. However, several assumptions and approximations related to modeling the377M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397geometr
78、y and stiffnesses of a lamina ply-gap detail were used in the analysis. First,the finite-element models are limited to modeling the shell-wall thickness variationas discrete step-changes and the resolution of the thickness variation is limited bythe finite-element integration point spacing (i.e., 0.
79、1 in.). Results from a study illus-trating the potential effects of these modeling approximations and mesh refinementon the response of a shell with a ply-gap indicate that this modeling approach canaffect the predicted response. In particular, it was found that these modelingapproximations can lead
80、 to an artificial increase or decrease in the size of the mea-sured thickness details, e.g., the width of a lamina ply-gap, resulting in misrep-resentation of the local bending stiffnesses and the mid-surface eccentricity of theshell wall at that particular point. In addition, the mesh refinement an
81、d integrationpoint spacing used in the present models tend to provide models that are overlystiffin bending by a small amount when they are used for modeling a ply-gapdetail. These modeling approximations can have a direct influence on the localbending response of the shell and can result in as much
82、 as a 2% variation in thepredicted buckling loads.The second assumption is related to the modeling of the stiffnesses of the laminaply-gap. Two modeling approaches were considered. One approach is to modeleach ply-gap with a local reduction in thickness, as measured, including the localeccentricity,
83、 ecz, and reducing the appropriate number of lamina plies in the shell-wall laminate model, hence, reducing the local stiffnesses associated with the localply-gap detail. The second approach is to model the ply-gap with the local as-measured reduced shell-wall thickness and the corresponding local m
84、id-surfaceeccentricity, but assuming that the local thickness reduction is due to a reductionin the resin volume only and, consequently, keeping the fiber volume constant.This approach neglects the local stiffness reduction associated with the ply-gapdetails. The former modeling approach is very tim
85、e consuming to implement sinceit requires the manual definition of each shell-wall laminate associated with eachspecific ply-gap in the finite-element model. The latter modeling approach is mucheasier to implement in the model by using a user-written subroutine compatiblewith the STAGS finite-elemen
86、t analysis code. As a result, the latter modelingapproach was used in the present study. Results from a study of this modelingassumption indicate that neglecting the local stiffness reduction associated with aply-gap would cause only slight differences in the magnitude of the local bendingresponse a
87、nd no more than a 2% variation in the predicted buckling load for acompression-loaded shell. Moreover, the results of this modeling study indicatedthat the local shell-wall mid-surface eccentricity is the most important feature ofthe ply-gap detail for these stability critical problems, and this ecc
88、entricity wasincluded in the models for the results presented herein. Results from a numericalparametric study of the effects of ply gaps on the nonlinear and buckling responseof compression-loaded cylinders are presented in Ref. 11.To provide a better simulation of the elastic boundary constraints
89、provided by thepotting material at the ends of the specimens, effective axial and radial potting-support stiffnesses were determined for each shell specimen using a two-dimensionalgeneralized plane-strain finite-element analysis of the potting-material-shell-wallM.W. Hilburger, J.H. Starnes / Thin-W
90、alled Structures 42 (2004) 369397378detail. The predicted results indicate that the effective axial potted-shell stiffnessrange from 1.1 to 2.4 times the nominal shell-wall stiffness and the nominaleffective radial potting-support stiffness was predicted to be approximately equalto 1.0E5 lbf/in. In
91、the present study, the nominal effective axial potted-shellstiffnesses are equal to 1.2, 2.0, 1.3, 1.1, 1.4, and 1.2 times the nominal shell-wallstiffness of shells C1C6, respectively. The predicted results also indicate that theincrease in the effective axial potted-shell stiffness is inversely pro
92、portional to thenominal shell wall stiffness. Details on the boundary stiffness analyses and effectsof the boundary stiffness on the response of the shells are given in Ref. 12.Nonuniform end loading of a specimen is attributed to initial specimen-end orlocating-surface imperfections and to upper lo
93、ading-platen rotations that are mea-sured during the experiment. First, the measured upper and lower specimen-end orloading-surface imperfections dtop(h) and dbot(h), respectively, were included in thefinite-element model by introducing an initial in-plane axial perturbation to thenodes at the loade
94、d ends of the shell. Then, the compression load was applied tothe shell in two parts. The nonuniform specimen-end imperfections, ?dtop(h) and?dbot(h), were applied as displacements to the upper and lower ends of the shell,respectively, at the beginning of the analysis to simulate a full contact cond
95、itionbetween the shell ends and the loading platens. Then, the experimentally measuredend-shortening displacement D and upper loading-platen rotations /yand /zwereapplied to the upper shell end or loading surface while holding the lower loadingsurface fixed as illustrated in Fig. 1; that is, u?L=2;h
96、 D Rcos/ycosh Rcos/zsinh ? dtoph and u?L=2;h ?dboth.3.3. Failure analysesA common Tsai-Wu tensor failure criterion was used to predict material failurein the shells. Two failure criteria were defined and include a delamination failurecriteria given bys213 s223S2 11and an in-plane failure criteria gi
97、ven byr21X2?r1r2X2r22Y2s212S2 12The transverse shear stresses s13and s23were approximated using a strength ofmaterials approach which assumes that the transverse shear stress resultant distri-bution is parabolic through the thickness of the shell wall. The material allowablesare as follows: longitud
98、inal strength X 124:0 ksi, transverse strength Y 8:4 ksi,and shear strength S 11:6 ksi. When either of the failure indices equals orexceeds a value of one, the material is assumed to have failed. Each stress compo-nent of the failure indices is examined to determine the mode of failure. A pro-gressi
99、ve failure analysis approach was not used in the present study since it wasoutside the scope of the present study, however, the failure indices were used to379M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397indicate the possibility of material failure and to establish failure t
100、rends associatedwith the compression-loaded cylinders considered herein.4. Parameter uncertainty characterizationSeveral cylinder parameter values exhibit a significant amount of uncertainty andan attempt to characterizes these uncertainties and include them in the presentanalyses was make. The cyli
101、nder parameter uncertainties considered include uncer-tainties in geometric imperfection measurements, lamina fiber volume fraction,fiber and matrix properties, applied end load distribution, and boundary con-ditions.Imperfection measurement uncertainty is attributed to the accuracy tolerances ofthe
102、 coordinate measurement device used to measure the shell-wall geometry andend-surface imperfection and this tolerance is equal to ?0.0006 in. This tolerancecorresponds to less than 0.01% uncertainty in the shell-wall imperfection measure-ment (e.g., Fig. 2). In contrast, the measurement tolerance co
103、rresponds to a ?3.0%uncertainty in the thickness measurement (e.g., Fig. 3) and approximately ?6%uncertainty in the shell-end imperfection measurement (e.g., Fig. 4).The extent of uncertainty in fiber and matrix properties and fiber volume frac-tion were characterized using published data contained
104、in Volume 2 of the MIL 17Handbook and from the material manufacturer. It was determined that the nom-inal fiber ad matrix properties can vary ?5% and the nominal fiber volume fractioncan vary ?3%. The nominal fiber properties used in the present study are as fol-lows: longitudinal modulus of 31.19 M
105、si, transverse modulus of 3.49 Msi, shearmodulus of 1.81 Msi, and Poissons ratio of 0.27. The nominal matrix propertiesare as follows: Youngs modulus of 0.53 Msi, shear modulus of 0.22 Msi, andPoissons ratio of 0.35. The nominal fiber volume fraction is equal to 0.62.Applied load distribution uncert
106、ainty is attributed to shell end-surface imperfec-tion uncertainty and uncertainty in the orientation of the loading platen withrespect to shell-ends or loading surfaces of the cylinder specimen during the test.Applied load distribution uncertainties are characterized indirectly by comparingthe meas
107、ured and predicted axial strains at selected points near the top and bottomloading surfaces of the shell. A correction to the applied displacement distributionwas determined from the differences in the measures and predicted strains as fol-lows. A user-written program external to the STAGS code was
108、used to analyze thedifferences in the measured and predicted strains for a specified applied load value.This program used an iterative predictorcorrector method to determine a correc-tion to the applied shell-end displacements. A new finite-element model was con-ducted with this displacement correct
109、ion included in the model. This process wasrepeated iteratively until the difference in the predicted and measured strainsreached a predetermined tolerance. A typical predicted displacement correction ispresented in Ref. 12 and the amplitude of the displacement correction is on theorder of ?0.0005 i
110、n.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397380Boundary condition uncertainty is attributed to uncertainties in the pottingmaterial stiffness and uncertainty in the integrity of the bond between the pottingmaterial and the shell wall. Visual inspection of the specimens be
111、fore and aftertesting indicated that the potting material has a tendency to separate from the shellwall. This boundary condition uncertainty was not rigorously characterized, how-ever, results from several numerical experiments indicate that variations in theboundary stiffness can have a significant
112、 effect on the displacement and strainresponse near the shell ends and effect the character of the collapse response of theshells, e.g., Ref. 12. Therefore, it was arbitrarily assumed that the effective axialand radial boundary stiffnesses might vary ?10%.5. Results and discussionNumerically predict
113、ed and experimentally measured results for the six com-pression-loaded graphite-epoxy cylindrical shells considered in this study are pre-sented in this section. The shell-wall laminates of the six shells include fourdifferent orthotropic laminates and two different quasi-isotropic laminates. The8-p
114、ly shells, C1C3, have shell-radius-to-thickness ratios equal to 200 and the 16-ply shells, C4C6, have shell-radius-to-thickness ratios equal to 100. The predictedresults were obtained from finite-element models of geometrically perfect shells andshells that include initial geometric shell-wall mid-s
115、urface imperfections, shell-wallthickness variations and thickness-adjusted lamina properties, local shell-walllamina ply-gaps, elastic boundary support conditions, and nonuniform loadingeffects. In addition, uncertainties in geometric and material properties, loadingdistribution, and boundary condi
116、tions were included in the analyses. These resultsare presented to illustrate the overall behavior of compression-loaded graphite-epoxy shells and the effects of imperfections and parameter uncertainties on theirresponse. First, results illustrating a typical nonlinear response of a compression-load
117、ed quasi-isotropic 8-ply shell are presented. Then, comparison between selectednumerically predicted results and experimentally measured results for the 8- and16-ply shells are presented. The results include predicted and measured loadend-shortening response curves, predicted prebuckling, buckling a
118、nd postbucklingdeformation response patterns, predicted axial and circumferential stress resultantpatterns, and predicted material failures. The six composite shells considered in thisstudy are interchangeably referred to herein as shells or specimens C1C6.5.1. Typical nonlinear response of an imper
119、fect compression-loaded cylindrical shellResults from a nonlinear analysis of an imperfect compression-loaded 8-ply?45/0/90Squasi-isotropic cylindrical shell C3 are presented in this section. Thenonlinear analysis results are from a shell model that includes that effects of themeasured initial shell
120、-wall geometric and thickness imperfections, thickness-adjustedmaterial properties, measured loading variations, and elastic radial supportconditions. The predicted loadshortening response of shell C3 is shown inFig. 6a. The axial load P and end-shortening D are normalized with respect to381M.W. Hil
121、burger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397the linear bifurcation buckling load of the geometrically perfect nominal shell,Pbif 42:59 Kips, and the nominal shell-wall thickness, t = 0.04 in., respect-ively. The loadend-shortening curve indicates a linear prebuckling response. Theg
122、eneral instability occurs at a normalized axial load level of P=Pbif 0:977,marked by the letter A. The general instability response is followed by a suddenreduction in the axial load supported by the shell and is associated with thetransient collapse response of the shell. The corresponding loadtime
123、 history ofthe transient collapse response is shown in Fig. 6b. The loadtime history curveexhibits a sudden reduction in axial load until the collapse response attenuatesand the axial load achieves a steady-state value. The kinetic energy in the shellobtains a maximum value during the transient coll
124、apse response and dissipatesover time and the shell reaches a stable postbuckling equilibrium state afterapproximately0.0070.008s.Theshellexhibitspostbucklingloadcarryingcapacity, however, the effective axial stiffness of the specimen is significantlyreduced in the postbuckling range of loading as i
125、ndicated by the reduction inthe slope of the loadshortening response curves. This reduction in effective axialstiffness is caused by large magnitude radial deformations that develop in thespecimen during buckling which result in significant load redistribution in thespecimen and reduces the effectiv
126、e, load-carrying cross-section of the cylinder.The transient deformation responses for selected time steps during the transientcollapse response of shell C3, indicated by the letters AF in Figs. 6a and b, arepresented in Figs. 7af, respectively. Just before buckling occurs, the shell-walldeformation
127、s are characterized by several localized ellipse-like buckles as indicatedin Fig. 7a. The localization in the deformation pattern is caused by the combi-nation of a local geometric shell-wall imperfection that is in the form of a signifi-cant variation in the shell-wall mid-surface geometry, and the
128、 intersection of ahelical ply-gap and a circumferentially aligned ply-gap in the shell at x=LT 0:25Fig. 6. Numerically predicted nonlinear response of an imperfect, compression-loaded, quasi-isotropicshell C3.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397382and h 210v. The lo
129、calized deformations occur in regions with destabilizing com-pressive axial and circumferential stresses. After approximately 0.0012 s haveelapsed in the transient response, a single ellipse-like buckle has grown in ampli-tude and couples with the destabilizing stresses in the shell wall to cause th
130、e gen-eral instability and collapse of the shell. The magnitude of the shell-wall radialdisplacement varies between ?0.5 times the shell-wall thickness. After additionaltime has elapsed in the transient collapse response, additional local buckles haveformed around the circumference and along the len
131、gth of the shell as indicated inFig. 7c, and the normalized axial load has decreased from 0.974 to 0.759. The mag-nitude of the shell-wall radial displacement varies between +2 and ?4 times theshell-wall thickness. As the buckling process continues, the normalized axial loadhas decreased further to
132、0.554 and the deformation pattern in the shell wall con-tinues to evolve and additional ellipse-like buckles have formed around the circum-ference of the shell as indicated in Fig. 7d. In addition, some of the buckles in theshell begin to coalesce into larger diamond-shaped buckles. The magnitude of
133、 theFig. 7. Numerically predicted prebuckling, buckling, and postbuckling response of an imperfect, com-pression-loaded, quasi-isotropic shell C3.383M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397shell-wall radial displacement varies between +3 and ?7 times the shell-wall thic
134、k-ness. After approximately 0.01 s have elapsed in the transient response, the kineticenergy in the shell has dissipated to a negligible level indicating that the transientresponse has attenuated, and the shell has deformed into a stable postbucklingmode-shape as indicated in Fig. 7e. As loading con
135、tinues in postbuckling region,the diamond-shaped buckles increase in size and the magnitude of the radial defor-mations of the buckles and the outer-surface ridges increase to between +4 and ?9times the shell-wall thickness as shown in Fig. 7f.Numerically predicted prebuckling, transient buckling, a
136、nd postbuckling axialand circumferential membrane stress contours of the imperfect quasi-isotropic shellC3 are presented in Figs. 8ac, respectively. Just before buckling occurs, the stressdistributions are nonuniform and exhibit several localized regions of biaxial com-pression stress in the bending
137、 boundary near the ends of the shell as shown inFig. 8a. In particular, compressive axial, an circumferential membrane stressesoccur in the shell at x=LT 0:25 and H 210vand these stresses couple with thelocal bending deformations in the shell wall to cause the general instability and col-lapse of th
138、e shell. During the transient collapse of the shell, the axial load decreasesrapidly and a significant redistribution of the stresses in the shell occurs. Forexample, at time 0:0024 s and P=Pbif 0:554, redistribution of the axial andcircumferential membrane stresses occurs and high-magnitude, locali
139、zed stress con-centrations develop near the buckles in the shell wall as indicated in Fig. 8b. Morespecifically, the axial stress distribution exhibits a significant unloading of the shellnear the center of the buckles and the majority of the axial compression load in thebuckled region of the shell
140、is supported by the ridges between the adjacent buckles.In addition, the results indicate a significant increase in the magnitude of the cir-cumferential compression stresses in the buckles with the maximum value of thecompressive stress increasing from ?240 to ?430 lbf/in., as compared to the initi
141、albuckling stresses shown in Fig. 8a. The results indicate that, in the postbucklingconfiguration, the shell exhibits significant membrane stress redistribution through-out the entire shell. For example, at D=t 1:3 and P=Pbif 0:554, the results indi-cate that the majority of the axial compression lo
142、ad is supported by the ridgesbetween adjacent buckles in the shell wall as shown in Fig. 8c. In addition, alter-nating bands of circumferential tension and compression membrane stresses arepresent in the shell. However, the magnitude of the maximum axial and circumfer-ential in the postbuckling conf
143、iguration shown in Fig. 8c are 18.5% and 23.2%lower than the corresponding stresses during the transient collapse response shownin Fig. 8b. This result indicates that the maximum stresses during the buckling pro-cess are not necessarily obtained in the stable postbuckling configuration, rather,the l
144、arge magnitude displacement gradients and internal stress redistribution cancause higher stresses to occur in the shell during the transient collapse process.Similar results were obtained for the other shells considered and indicated similarresponse characteristics.Corresponding numerical results pr
145、esented in Ref. 12 indicate that the defor-mation and internal stress response of a compression-loaded geometrically perfectidealized shell is characterized by a uniform axisymmetric prebuckling responseM.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397384Fig. 8. Numerically pred
146、icted prebuckling, buckling, and postbuckling axial and circumferential stresscontours of an imperfect, compression-loaded, 8-ply quasi-isotropic shell C3.385M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397and uniform asymmetric transient buckling and postbuckling response. The
147、seresults are in contrast to the present results in which the prebuckling and initialtransient buckling responses of the imperfect shells are characterized by nonuni-form localized behavior caused by the nonlinear coupling of localized imperfectionsin the shell. Results from a numerical parametric s
148、tudy illustrating the nonlinearcoupling between selected imperfections and its effects on the buckling load of acompression-loaded quasi-isotropic cylindrical shell is discussed in Ref. 12.Numerically predicted prebuckling, transient buckling, postbuckling material fail-ure contours of the imperfect
149、 quasi-isotropic shell C3 are presented in Figs. 9ac,respectively. The contour plots correspond to the values of the failure indices fordelamination failures, FI-1, and in-plane fiber or matrix, FI-2, defined in Eqs. (1)and (2), respectively. Material failure is assumed when the failure index equals
150、 orexceeds a value of one. Incipient to buckling, both failure indices are less than oneindicating no material failure occurs during the prebuckling response of the shellas shown in Fig. 9a. However, during the transient collapse response the in-planefiber/matrix failure index FI-2 exceeds a value o
151、f one as shown in Fig. 9b, andindicates a potential for failures to occur in regions of large magnitude displace-ment gradients and large magnitude compressive membrane stresses in the shellwall, e.g., see Fig. 8b. The material failures are characterized by matrix com-pression failures. The results
152、indicate that, in the postbuckling configuration,delamination failures and in-plane matrix and fiber compression failures are pre-dicted to occur along the ridges between the adjacent buckles in the shell wall(sometime referred to as nodal lines) as shown in Fig. 9c. These results are con-sistent wi
153、th the observed failures in the test specimens. Similar results wereobtained for shells C1 and C2 and indicated similar failure characteristics.5.2. Predicted and measured response comparisonsSelected results from nonlinear analyses of the six compression-loaded cylindri-cal shells are compared to t
154、he experimentally measured results in this section. Thenonlinear analysis results are for shell models that include the effects of the mea-sured initial geometric and thickness imperfections, thickness-adjusted materialproperty variations, measured loading variations, elastic radial support conditio
155、ns,and selected specimen parameter uncertainties. The specimen parameter uncertain-ties considered include uncertainty in the imperfection measurement accuracy, fiberand matrix properties, fiber volume fraction, applied load, and boundary stiffness,as characterized in an earlier section. Upper and l
156、ower response bounds weredetermined based upon the results of a traditional combinatorial analysis of theeffects the selected parameter uncertainties. Predicted and measured loadend-shortening response curves and postbuckling displacement contours are presentedin this section. In addition, typical n
157、umerically predicted prebuckling, buckling,and postbuckling displacement contours, axial and circumferential stress contours,and failure index contours are presented for imperfect 8- and 16-ply quasi-isotropicshells.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397386Fig. 9. Num
158、erically predicted prebuckling, buckling, and postbuckling failure contours of an imperfect,compression-loaded, 8-ply quasi-isotropic shell C3.387M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 3693975.3. Eight-ply shellsThree sets of numerically predicted and experimentally measured
159、 loadend-shortening response curves for the 8-ply shells; C1 ?45/02s, C2 ?45/902s, andC3 ?45/0/90s, are shown in Fig. 10. The axial load P is normalized by the quan-tity EA, i.e. the effective axial stiffness of the shell, denoted by E, multiplied by thenominal shell length L 16:0 in: The solid and
160、dashed lines in the figure representexperimentally measured and numerically predicted results, respectively. Each shellhas two predicted response curves representing predicted upper and lower boundsto the response based on specimen parameter uncertainties and the regions betweenthe response bounds a
161、re shaded for clarity. The measured buckling point of eachshell is marked by a filled circle and the ultimate failure of each shell is markedwith an X. In addition, each shell has one or more numerically predictedfailure boundaries represented by the dark grey solid lines in the figure. Each fail-ur
162、e boundary is labeled with the number 1, 2, or 3, and denotes matrix failureinitiation, fiber failure initiation, and delamination failure initiation, respectively.The measured results indicate that the prebuckling responses are linear up to thegeneral instability point indicated in the figure for e
163、ach specimen. General insta-bility occurs at normalized load levels of P=EA 0:00122, 0.0044, and 0.0022 forspecimens C1, C2, and C3, respectively, and are 7.8%, 13.7%, and 17.6% lowerthan the predicted linear bifurcation buckling loads for the corresponding geome-trically perfect, nominal shells, re
164、spectively. The general instability points are fol-lowed by a sudden and significant reduction in the axial load supported by thespecimens and is associated with the unstable transient collapse response of the spe-cimens. During collapse, the specimens buckled into the classical diamond-shapedgenera
165、l instability mode-shape and the collapse response was accompanied by anaudible snapping sound. In addition, no significant visible failures were observed inFig. 10. Numerically predicted and experimentally measured loadend-shortening response curves for8-ply compression-loaded shells; predicted res
166、ults represent response bounds.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397388the specimen as a result of the collapse response. The specimens obtained stablepostbuckling equilibrium state and exhibited additional load carrying capacity inthe postbuckling range. Additional
167、audible popping sounds were heard during theloading of the specimens in the postbuckling range suggesting a progressiveaccumulation of material failures in the specimens and the accumulation ofmaterial failures continued until the ultimate failure of the specimen occurred. Inaddition, the progressiv
168、e accumulation of material damage in the specimens mayaccount for the discontinuous jumps in the loadshortening response curves shownin Fig. 10. The results in Fig. 10 indicate that, for the most part, the measuredresponses fall within the numerically predicted response bounds. In particular, theres
169、ults indicate that the measured response curves tend to correlate with the mid-point between the upper and lower predicted response bounds. The predictedresults indicate that, in most cases, material failure in the specimens is likely tooccur at load levels near the general instability point and in
170、the postbuckling rangeof loading. More specifically, matrix compression failure is predicted to occur inshell specimens C2 and C3 near the general instability point, followed by fiber com-pression failures and delamination type failures in the postbuckling range of load-ing. In contrast, the numeric
171、al results predict matrix and fiber compression failuresto occur in specimen C1 in the postbuckling range of loading. These failure predic-tions correlate well with the failure trends observed in the tests.Predicted initial post-collapse radial displacement contours and the correspond-ing observed m
172、oire fringe patterns for specimen C1 are shown in Fig. 11. Thedashed contour lines in the predicted displacement contour plots represent inwarddisplacements and the solid lines represent outward displacements. The density ofthe contour lines indicates the severity of the displacement gradients in th
173、e speci-men. These results indicate that the specimen collapses into a general-instabilitydiamond-shaped buckling pattern with 16 half-waves around the circumferenceand one half-wave along the length, as predicted by the finite-element analysis.Fig. 11. Observed and predicted initial postbuckling no
174、rmal displacements for specimen C1.389M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397However, the numerical results predict that the mode-shape pattern is 1520vout-of-phase with the observed mode-shape pattern. Predicted initial post-collapse nor-mal displacement contours and
175、the corresponding observed moire fringe patternsfor specimen C2 are shown in Fig. 12. These results indicate that the specimen col-lapses into a general instability diamond-shaped buckling pattern with 14 half-waves around the circumference and 2 half-waves along the length, as predicted bythe finit
176、e-element analysis. In addition, the predicted mode-shape is in-phase withthe observed mode-shape. Similar results for shell C3 indicate that the shell col-lapses into a general-instability diamond-shaped pattern with 16 circumferentialhalf-waves and 2 axial half-waves. However, the analytical resul
177、ts predicted thatthe mode-shape pattern is approximately 15vout-of-phase with the observed moire fringe pattern.5.4. Sixteen-ply shellsNumericallypredictedandexperimentallymeasuredloadend-shorteningresponse curves for the 16-ply shells C4C6 are shown in Fig. 13. The axial load Pis normalized by the
178、quantity EA, i.e. the effective axial stiffness of the shell, deno-ted by E, multiplied by the nominal shell cross-sectional area, denoted by A, andthe end-shortening D is normalized by the nominal shell length L 16:0 in: Thesolid and dashed lines in the figure represent experimentally measured and
179、numeri-cally predicted results, respectively. Each shell has two predicted response curvesrepresenting predicted upper and lower bounds to the response based on specimenparameter uncertainties and the regions between the response bounds are shadedfor clarity. The measured buckling point of each shel
180、l is marked by a filled circleand the ultimate failure of each shell is marked with an X. In addition, each shellhas one or more numerically predicted failure boundaries represented by the darkgrey solid lines in the figure. Each failure boundary is labeled with the number 1,2, or 3, and represents
181、matrix failure initiation, fiber failure initiation, andFig. 12. Observed and predicted initial postbuckling normal displacements for specimen C2.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397390delamination failure initiation, respectively. The measured results indicate that
182、 theinitial loadshortening responses are, for the most part, linear up to the limit loadfor each specimen as indicated in the figure. However, the loadshortening respon-ses for specimens C5 and C6 exhibit slight nonlinear behavior at end-shorteninglevels greater than D=L 0:004. General instability o
183、ccurs at normalized loadlevels of P=EA 0:0027, and 0.0049 for specimens C4, and C6, respectively, andare 16.8%, and 18.4% lower than the predicted linear bifurcation buckling loadsfor the corresponding geometrically perfect, nominal shells, respectively. Theresults show that the general instability
184、points of specimens C4 and C6 coincidewith the ultimate failure of the specimens and the specimens do not exhibit post-buckling load carrying capacity. More specifically, experimental results indicatedthat, upon collapse, specimens C4 and C6 exhibited a significant amount ofmaterial failure includin
185、g fiber and matrix compression failures and delaminationfailures, and these material failures caused the ultimate or complete failure of thespecimens. The predicted results indicate that, in most cases, the initiation ofmaterial failure is likely to occur during the initial portion of the transient
186、collapseresponse as shown in Fig. 13 and these results explain the observed failure trendsin specimens C4 and C6. In contrast, specimen C5 does not exhibit a general insta-bility point, rather, the specimen exhibits complete failure at a load level ofP=EA 0:0062, and is 43.9% lower than the predicte
187、d linear bifurcation bucklingload for the corresponding geometrically perfect, nominal shell. The complete oroverall failure of the specimen is characterized by a significant amount of delami-nation failures and fiber and matrix compression failures around the entire circum-ference of the shell. Pos
188、t-test inspection of the specimen indicated that the over-allfailure of the shell may have been initiated by a material failure near an axiallyaligned ply-gap in the cylinder wall.Fig. 13. Numerically predicted and experimentally measured loadend-shortening response curves for16-ply compression-load
189、ed shells; predicted results represent response bounds.391M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397Numerically predicted results of the prebuckling, transient buckling, and post-buckling deformation and internal stress responses were obtained for the 16-plyshells and ind
190、icated similar response trends to those presented for the 8-ply quasi-isotropic shell in Figs. 7 and 8. In particular, the results indicate that the prebuck-ling responses in the shells are characterized by localized displacement and internalstress distributions. The localized shell-wall deformation
191、s couple with destabilizingstresses in the shell to cause the general instability and collapse of the shell. In gen-eral, the localized deformations and stress distributions are caused by the combi-nation of local geometric shell-wall geometric and thickness imperfections, materialproperty variation
192、s, and shell-end-loading nonuniformities.Predicted initial post-collapse radial displacement contours and the correspond-ing observed displacement response for specimen C4 are shown in Figs. 14a and b.The dashed contour lines in the predicted displacement contour plots representinward displacements
193、and the solid lines represent outward displacements. Thedensity of the contour lines indicates the severity of the displacement gradients inthe specimen. These results indicate that the specimen collapses into a general-instability diamond-shaped buckling pattern with 16 half-waves around the circum
194、-ference and 1 half-wave along the length, as predicted by the finite-element analy-sis. The numerical results predict that the mode-shape pattern is in phase with theobserved mode-shape pattern. In addition, the specimen exhibits a significantamount of damage near the ridges between the adjacent bu
195、ckles in the shell wall.The material failures include fiber and matrix compression failures and delami-nation type failures. These failures are consistent with the predicted material fail-ures in the shell. In particular, the numerical results indicate that, incipient tobuckling, the in-plane fiber/
196、matrix failure index FI-2 exceeds a value of one andindicates that matrix compression failures are likely to occur in the bending bound-ary regions of the shell. During the transient collapse response, the delaminationfailure index FI-1 and the in-plane fiber/matrix failure index FI-2 exceed a value
197、 ofFig. 14. Observed and predicted initial post-collapse normal displacements for specimen C4.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397392one and indicate a potential for significant failures to occur. More specifically, theresults indicate that fiber and matrix compress
198、ion failures and delamination typefailures may occur in regions of large magnitude displacement gradients and largemagnitude membrane compression stresses and transverse shear stresses associatedwith the ridges that form between adjacent buckles in the shell wall. Similar resultswere obtained for sp
199、ecimen C6 and indicate similar failure trends (Fig. 16). Pre-dicted initial post-collapse normal displacement contours and the correspondingobserved displacement response for specimen C5 are shown in Figs. 15a and b. Theobserved material failures in specimen C5 are clearly visible in Fig. 15a. Inadd
200、ition, a 0.15-in.-wide ply-gap is shown in the figure and the overall failure ofthe shell may have been initiated by a material failure near this ply-gap. Numericalresults illustrating the effects of a ply-gap on the internal stress distribution in theshell wall are presented in Ref. 12. The results
201、 indicate that the ply-gap can causesignificant stress concentrations to develop within the laminate and that the stresslevels can equal or exceed the stress allowables for the material at relatively lowapplied load levels. In particular, predicated membrane compression stress andtransverse shear st
202、ress values at buckling can be on the order of two to three timesthe allowable values for the material. These results suggest that delamination fail-ures and fiber and matrix compression failures may occur near ply gaps in the shellwall for load values less than the predicated buckling load of the s
203、hell.6. Concluding remarksThe results of an experimental and analytical study of the effects of imperfec-tions on the nonlinear response and buckling loads of unstiffened thin-walled com-pression-loaded graphite-epoxy cylindrical shells with six shell-wall laminates areFig. 15. Observed and predicte
204、d initial post-buckling normal displacements for specimen C5.393M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (2004) 369397presented. The shell-wall laminates considered in this study include two quasi-isotropic laminate and four different orthotropic laminates. Shell-radius-to-thicknessr
205、atios equal to 100 and 200 were also considered. Numerical results for the nonlinearFig. 16. Numerically predicted prebuckling, buckling, and postbuckling failure contours of an imperfect,compression-loaded, 16-ply quasi-isotropic shell C6.M.W. Hilburger, J.H. Starnes / Thin-Walled Structures 42 (20
206、04) 369397394prebuckling, transient buckling, and postbuckling response of shells with mea-sured imperfections are presented. The numerical results include the effects oftraditional initial geometric shell-wall mid-surface imperfections and the effectsof other nontraditional imperfections. These non
207、traditional imperfections includeshell-wall thickness variations, material property variations, shell-end geometricimperfections, local shell-wall ply-gaps associated with the fabrication process,variations in loads applied to the end of the shell, and elastic boundary supportconditions. In addition
208、, upper and lower bounds to the nonlinear response ofthe shells are presented which were determined from a combinatorial analysis ofthe effects of uncertainties in several shell parameters. The uncertainties con-sidered in the present study include uncertainties in the geometric imperfectionmeasurem
209、ents, lamina fiber volume fraction, lamina fiber and matrix properties,boundary conditions, and applied load distribution. A high-fidelity nonlinearshell analysis procedure has been used to predict the nonlinear response andfailure of the shells, and the analysis procedure accurately accounts for th
210、eeffects of these traditional and nontraditional imperfections and parameteruncertainties on the nonlinear response and failure of the shells. The analysisresults generally correlate well with the experimental results indicating that it ispossible to predict accurately the complex nonlinear response
211、, buckling loads,and failure for compression-loaded composite shell structures.The numerical results indicate that the effects of the traditional and nontradi-tional imperfections, and selected parameter uncertainties considered in this studycan be important for predicting the buckling loads of comp
212、osite shells since theycan significantly affect the nonlinear response and buckling loads of the shells. Theresults indicate that the measured imperfections can couple with the in-planecompressive stress resultants in a nonlinear manor to affect the shell response. Inparticular, typical results that
213、 illustrate the response of a compression-loadedquasi-isotropic shell were presented and indicated that a complex nonlinear interac-tion between localized shell-wall prebuckling deformations in the bending bound-ary region of the shell and compressive axial and circumferential stresses causedthe ove
214、rall buckling of the shell to occur. The localized deformations were causedby the combination of a local geometric shell-wall imperfection that was of theform of a significant variation in the shell-wall mid-surface geometry, and theintersection of helical ply-gap and a circumferentially aligned ply
215、-gap in the shellwall.The numerically predicted and experimentally measured results indicate that the8-ply shells considered in this study exhibit linear prebuckling responses followedby a general instability response. The general instability response corresponds withthe overall collapse of the shel
216、l in which the shell exhibits a significant reduction inaxial load carried by the shell and the shell wall deforms into a classical diamond-shaped general instability mode-shape and have significant postbuckling load car-rying capacity. The shells exhibited significant material failures in the postb
217、ucklingregion of loading and a progressive accumulation of these material failures causedthe ultimate failure of the shells. The material failures in the shell included matrixand fiber compression failures and delamination type failures and the material395M.W. Hilburger, J.H. Starnes / Thin-Walled S
218、tructures 42 (2004) 369397failures typically occurred in regions of large magnitude displacement gradientsand large magnitude compressive membrane stresses and transverse shear stressesin the shell wall. In contrast, the 16-ply shells exhibited significant material failuresupon buckling and had no p
219、ostbuckling load carrying capacity. More specifically,as the 16-ply shells buckled, the internal stresses exceeded the allowable stresseslevels of the material and caused the overall failure of the shell. In one case, how-ever, a 16-ply specimen exhibited premature failure during the test. The overa
220、ll fail-ure of the specimen was characterized by significant fiber and matrix compressionfailures and delamination type failures around the circumference of the specimen.Post-test inspection of the shell indicated that the overall failure of the specimenmay have been initiated by a material failure
221、near an axially aligned ply-gap in thespecimen wall.The results indicate that, for the most part, the measured response of the shellsfalls mid-way between the predicted upper and lower bounds to the response thatare associated with the uncertainties or variations in the shell parameters con-sidered
222、in the study. In addition, numerically predicted material failure initiationagreed well with experimentally observed failure trends. These results indicate thatthe nonlinear analysis procedure used in this study can be used to determine accu-rate, high-fidelity design knockdown factors and response
223、bounds that can be usedfor predicting composite shell buckling and failure loads in the design process. Thetraditional and nontraditional imperfections considered in this study could be usedto formulate the basis for a generalized imperfection signature of a compositeshell that includes the effects
224、variations or uncertainties in the shell-geometry,fabrication-process, load-distribution and boundary stiffness parameters. The high-fidelity nonlinear analysis procedure used in this study can be used to form thebasis for a shell analysis and design approach that includes this generalized imper-fec
225、tion signature and addresses some of the critical shell-buckling design criteriaand design considerations for composite shell structures without resorting to thetraditional empirical shell design approach that can lead to overly conservativedesigns. Since the nonlinear analysis procedure can be used
226、 to predict local shell-wall stresses and strains at any point in the shell load-response history, the analysisprocedure can also be used to form a robust failure analysis for composite shellstructures with nonlinear response characteristics.References1 Anonymous. Buckling of thin-walled circular cy
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232、es Jr. JH. Buckling of an axially compressed imperfect cylin-drical shell of variable thickness. Proceedings of the 35th AIAA/ASME/ASCE/AHS/ASC Struc-tures, Structural Dynamics and Materials Conference, Hilton Head, SC, 1994. 1994 AIAA PaperNo. 94-1339.10 Starnes Jr. JH, Hilburger MW, Nemeth MP. The
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