Effective Widths of Compression-Loaded Plates With a Cutout

Transcription

1 NAA/T Effective iths of Compression-Loae lates ith a Cutout Mark. Hilburger, Michael. Nemeth an James H. tarnes, Jr. Langley Research Center, Hampton, Virginia October 2

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3 NAA/T Effective iths of Compression-Loae lates ith a Cutout Mark. Hilburger, Michael. Nemeth an James H. tarnes, Jr. Langley Research Center, Hampton, Virginia National Aeronautics an pace Aministration Langley Research Center Hampton, Virginia October 2

4 Available from: NAA Center for Aeropace Information (CAI) National Technical Information ervice (NTI) 72 tanar Drive 5285 ort Royal Roa Hanover, MD pringfiel, VA (3) (73) 65-6

5 Introuction Abstract Results from a stuy of the effects of cutouts an laminate construction on the prebuckling an initial postbuckling stiffnesses, an the effective withs of compression-loae laminate-composite an aluminum square plates are presente. An effective-with concept is erive for plates with an without cutouts, an experimental an nonlinear finite-element analysis results are presente. Behavioral trens are compare for seven plate families an for cutout-iameter-to-plate-with ratios up to.66. A general compact esign curve that can be use to present an compare the effective withs for a wie range of laminate constructions is also presente. A iscussion of how the results can be use an extene to inclue certain types of amage, cracks, an other structural iscontinuities or etails is given. everal behavioral trens are escribe that initially appear to be nonintuitive. The results emonstrate a complex interaction between cutout size an plate orthotropy that affects the axial stiffness an effective with of a plate subjecte to compression loas. Efficient, lightweight, stiffene panels that are subjecte to compression loas are often esigne to take avantage of the postbuckling stiffness an strength of their plate elements. More specifically, the flat plate elements exhibit stable postbuckling behavior an, as a result, can often support loas significantly greater than their buckling loa. rior to buckling, the axial stresses in each plate element are typically uniform across the with of a compression-loae plate. hen a flat plate element buckles, the axial stiffness of the central region of the plate element is reuce as a result of the corresponing bening eformations, an the internal axial loa path is reistribute from the central region of the plate element towar the ajacent stiffeners of the stiffene panel. Thus, the stiffene panel behaves as if the loa is supporte by the stiffeners an by a portion of the plate elements ajacent to each stiffener. The with of a plate element that effectively supports part of the applie loa is known as the effective with of the plate. -4 Historically, the effective-with concept appears to have originate in the esign of ship plating. 2 Aroun 932, von Kármán, echler, an Donnell became the first to apply the effective-with concept to all-metal aircraft construction. The motivation for this work was to establish a simple esign-oriente analysis that approximates the stiffness of the buckle plate elements in the calculation of the stiffener strength an the ultimate loa capacity of the stiffene panel. Designing a panel to allow plate buckling above a specifie esign loa provies a lighter weight panel because the plate will be thinner than a plate esigne to resist buckling. Currently, the effective-with concept remains in use as an important preliminary esign tool, not only for metallic, stiffene aircraft panels, but also for laminate composite stiffene aircraft panels. A common structural etail that is present in many stiffene panels is a cutout. For example, cutouts are use to reuce weight, to provie access to various locations insie an aircraft, an to permit hyraulic an electrical lines to pass through the structure. Thus, the effect of a cutout on the postbuckling behavior of a plate an its effective with is important in the esign of aircraft structures. A recent survey of the postbuckling behavior of plates with a cutout is given in Ref. 5 an inicates that the effect of a cutout on the effective-with of a plate has receive relatively little attention, compare to plate buckling behavior. The present paper escribes the results of a stuy of the effects of cutouts an laminate construction on the prebuckling an initial postbuckling responses an the effective with of compression-loae, symmetrically laminatecomposite plates an aluminum plates. In particular, the behavior of square plates that are subjecte to a uniform ege isplacement is presente. First, the effective-with concept is presente in terms of prebuckling an initial postbuckling stiffnesses of a plate, an is then extene to inclue plates with a cutout. Then, results are presente for the effective with of aluminum an graphite-epoxy plates that are base on the experimentally measure prebuckling an postbuckling stiffnesses that have been presente in Refs. 6 an 7. Next, results obtaine by using the TAG finite-element analysis coe 8 are presente for corresponing geometrically perfect plates with nominal imensions an iealize bounary an loaing conitions. Behavioral trens that are exhibite by the experimental an finite-element analysis results are then presente an compare for aluminum plates an for composite plates with six laminate families an cutout-iameter-to-plate-with ratios / ranging from to.66. In aition, a general, compact esign curve that can be use to present an compare the effective withs of a wie range of laminate-composite plate constructions is presente. A iscussion of how the results can be use an extene to inclue certain types of centralize amage, cracks, an other structural iscontinuities or etails that affect the axial stiffness of a plate is also presente.

6 Backgroun an Analysis The effective-with concept is base on the behavior of plates that are subcomponents of a stiffene structural configuration, such as a wing or fuselage, that are loae in compression. In particular, the concept was evelope for a plate subcomponent that is attache to two stiff frames an two less stiff stringers, an that is surroune by other similar panels, frames, an stringers. For this arrangement, the effective-with concept assumes that the plate eges remain straight uring loaing; because of the stiffness provie by the ajacent structure. However, the effective with concept can also be applie to plates that are supporte such that the unloae eges are free to eform in the plane of the plate. This type of support conition is often simulate in the laboratory (e.g., Ref. 6) an is the focus of the present stuy. The effective-with concept, at the plate subcomponent level, is base on the reistribution of the axial stress resultant, N x, at the loae eges when a compression-loae plate buckles, as illustrate in Fig.. For example, a perfectly flat compression-loae plate with simply-supporte eges remains flat an N x is uniformly istribute across the loae eges prior to buckling (Fig. (c)). hen the plate buckles, a significant portion of the plate s central region exhibits relatively large out-of-plane isplacements which reuce the effective axial stiffness (Fig. (b)). hen the stiffness of the central region of the plate is reuce, the internal loas are reistribute towar the sie supports an N x becomes nonuniform across the loae eges (Fig. ()). This change in internal loa istribution correspons to the abrupt change in slope of the loa versus en-shortening curve shown in Fig. (b). A common practice in the preliminary esign of stiffene panels is to replace the nonuniform N x istribution at the loae eges with a uniform loa istribution (N x ) max over a reuce with e (the effective with), as shown in Fig. 2. This approximation is often use to estimate the require strength of stringers for postbuckling strength an represents a simple way to incorporate nonlinear behavior into a esign. The symbol (N x ) max represents the maximum value of N x that occurs at the loae eges of a plate (Fig. 2(a)). The value of e is etermine by equating the loa that acts on the plate with the loa given by (N x ) max times the effective with (see Ref. 4); that is = N x y =(N x ) max e () in which the term (N x ) max is interprete as the loa in an unbuckle plate, corresponing to a given value of postbuckling en-shortening, if the plate was unbuckle. This interpretation of Eq. () is illustrate in Fig. 3, where =(N x ) max. Thus, the ratio of the effective with to the actual with is expresse as The experimental results that are presente in Refs. 6 an 7 inicate that in many cases the postbuckling branch of the loa versus en-shortening curves for elastic plates remains essentially linear for loas up to nearly twice the buckling loa. ith the ai of Fig. 3, the ratio of the effective with to the actual with for bilinear loa versus enshortening curves can be expresse in terms of the prebuckling stiffness,, an the postbuckling stiffness by ei- ther of the following equations e = e = e = ε cr ε + + ε cr ε cr where cr is the buckling loa, ε cr = cr L is the average strain at buckling, an ε = L is the average strain in the plate for the loa. Compression-loae plates with a centrally locate cutout exhibit a reuction in their effective axial prebuckling stiffness an a reistribution of the internal loas towar the unloae eges, as illustrate in Fig. 4. Moreover, this effect is amplifie when a plate with a cutout buckles. Thus, it is possible to introuce an effective-with concept for plates with a cutout in both the prebuckling an postbuckling ranges by following the same rationale that has been presente above. The relationship between the axial stiffness of an unbuckle plate without a cutout,, an the stiffnesses of an unbuckle an buckle plate with a cutout is illustrate in Fig. 5. In the prebuckling range, the effective-with ratio efine by Eq. (3) can be expresse by (see Fig. 5) (3) (4) (5) Following Ref. 9, Eq. () is written as e = (N x ) max (2) 2 e = where is the prebuckling stiffness of the corresponing plate with a cutout. In the postbuckling range, the ratio of (6)

7 the effective with to the actual with for bilinear loa versus en-shortening curves can be expresse in terms of the prebuckling stiffnesses an an the postbuckling stiffness of the corresponing plate with a cutout by either of the following equations e = e = ε cr ε + ε cr ε + cr Finite-Element Moel an Analyses Nonlinear finite-element analyses were conucte in the present stuy to provie insight into the experimental response trens that are subsequently presente herein. The plates were analyze with the TAG finite-element coe. 8 A typical finite-element moel of a perfectly flat plate with a centrally locate, circular cutout that was use in the stuy is illustrate in Fig. 6. oints on the plate mi-surface are locate by an x-y-z coorinate frame whose origin is at the top left corner of the plate. The plate length an with are efine as L an a, respectively, an the cutout iameter is enote by. The nominal plate lengths an withs were. in. for all plates. Cutout iameters range from to 6.6 in. Both aluminum an graphite-epoxy composite plates were consiere in this stuy. The aluminum plates were mae of 66- T6 alloy sheets an were moele with a uniform, average measure thickness of.647 in. The nominal material properties for 66-T6 aluminum alloy are: Young s moulus E =. x 6 psi an oisson s ratio ν =.33. The laminatecomposite plates were moele by using a uniform thickness equal to the average measure plate thicknesses that are given in Table. All of the laminate-composite plate test specimens were mae of Hercules, Inc. A4/352 graphite-epoxy material. The nominal lamina properties reporte in Ref. 6 an use herein inclue a longituinal moulus E = 8.5 x 6 psi, a transverse moulus E 2 =.6 x 6 psi, an inplane shear moulus G 2 =.832 x 6 psi, an a major oisson s ratio ν 2 =.35. Iealizations of the test-fixture support conitions were use in the finite-element moel. To simulate clampe loae eges, the out-of-plane isplacements, w, an the tangential isplacements, v, were set equal to zero in the bounary regions of the finite-element moel that exten /4 in. from each loae en of a plate (see Fig. 6). The compression loa was introuce into the plate by applying a uniform en-isplacement to one en of the moel while holing the other en of the panel fixe on the bounary, as illustrate in Fig. 6; that is, u(,y) = an u(l,y) =. The simply supporte, (7) (8) 3 unloae eges (knife-ege supports) were simulate by setting the out-of-plane isplacement equal to zero on a line /4 in. from each unloae ege of a panel. The corresponing unsupporte with of the plate that is between the knife eges is given by = 9.5 in. The tangential isplacements, u, were unrestraine on the unloae eges. In aition, the unloae eges are free to translate in the y-coorinate irection. The finite-element meshes use in the present stuy were efine by using user-written subroutines that are compatible with the TAG finite-element coe. 8 These userwritten subroutines facilitate the generation of moels with ifferent cutout sizes an mesh ensities, an provie a convenient means for assessing the convergence of a given finiteelement moel. Convergence stuies were one for all finiteelement moels an a typical converge finite-element mesh is shown in Fig. 6. The moels consist of four-noe TAG 4 quarilateral shell elements. The 4 element is a flat, facet-type element that is base on the Kirchoff-Love shell hypotheses an the nonlinear Lagrangian strain tensor. The element noes inclue three rilling egrees of freeom an three translational egrees of freeom. Large rotations are accounte for at the element level by the use of a corotational algorithm. The element theory is ocumente in Ref. 8. The TAG coe uses both the moifie an full Newton methos for its nonlinear solution algorithms. The TAG equivalence transformation (ET) processor is use to continue beyon bifurcation points into the postbuckling response range of a solution. The ET processor allows the user to select a solution branch in the irection of growth of a bifurcation moe that has been compute for the nonlinear stress state in the neighborhoo of the bifurcation point. This bifurcation moe is hel constant, with a magnitue specifie by the user, an the loa factor in the solution is ajuste until the solution resiual vanishes. The prebuckling responses of the plates consiere herein were etermine by using the geometrically nonlinear, quasi-static analysis capability in TAG. The ET processor was use to get beyon the bifurcation point an branch onto the postbuckling equilibrium path of the panels. Once the initial postbuckling path was establishe for a plate, the analysis was continue into the postbuckling response range by using the quasi-static analysis algorithm in TAG. A typical finite-element moel containe approximately, egrees of freeom. The initial prebuckling stiffnesses of the geometrically perfect, iealize test specimens were obtaine by computing the slope of a line passing through two solution points on the primary branch of the loa versus en-shortening curve. imilarly, the initial postbuckling stiffnesses were obtaine by computing the slope of the line that passes through the bifurcation point on the loa versus en-shortening curve for a given specimen an that is tangent to the stable seconary equilibrium path. Nonlinear finite-element analyses of geometrically

8 imperfect plates were also conucte in the present stuy primarily to etermine the effect of initial geometric imperfections on the prebuckling stiffness of a plate an because initial geometric imperfections are known to affect the effective with of a compression-loae plate. Because no full-fiel or other type of initial imperfection measurement ata are available, an eigen-moe imperfection characterize by a single half-wave shape along the coorinate axes an a variable amplitue was use. For these geometrically imperfect plates, the prebuckling stiffnesses were obtaine in the same manner as escribe previously for the geometrically perfect plates. Results an Discussion The experimental results presente in Refs. 6 an 7 for square aluminum plates an for square [, [9, [(/ 9) 5, [(±3) 6, [(±45) 6, an [(±6) 6 laminate-composite plates with or without a central circular cutout an subjecte to a uniform axial ege isplacement are escribe in this section. For these results, the compression-loae eges of the plates were clampe in a test fixture an the unloae eges were supporte by knife eges, which are intene to simulate simply supporte eges. The istance between the knife eges is the unsupporte with of a plate an is esignate herein by the symbol, an the relative size of the cutout is inicate by the ratio /. The nominal value of is 9.5 in. for each specimen an the actual, nominal plate with a is. in. pecific etails about the test specimens an apparatus an etails about the test proceure are escribe in Refs. 6 an 7 an are not repeate, for the most part, herein. All of the laminate-composite plates were mae of Hercules, Inc. A4/352 graphite-epoxy material an the aluminum plates were mae of 66-T6 aluminum alloy material. The nominal material properties have been escribe in the previous section of the present paper. First, the experimental results of Refs. 6 an 7 are presente in a manner that illustrates the effect of cutout size on the prebuckling an postbuckling axial stiffnesses of the aluminum plates an the laminate-composite plates. Because these experimental results exhibit some unusual trens, corresponing results that were obtaine from nonlinear finite-element analyses are also presente to provie insight into the behavior. These finite-element analysis results are base on iealize geometry, bounary conitions, an loaing conitions; uniform thickness istributions; an the nominal material properties that were previously escribe. Because of these iealizations an approximations, the finite-element analysis results are not intene to be use for test-analysis correlation. The finite-element results are use to provie a means for contrasting the behavioral trens from iealize analyses that a esigner might use with the behavior that may exist in real structures. Next, effective withs are presente for selecte experimental results as a function of the average axial-strain level to 4 emonstrate the effects of laminate construction an cutout size. Finally, a compact esign chart is presente that provies a means for representing effective withs of plates with or without a cutout in a general manner. Aitional charts, base on finite-element analyses, that illustrate the effects of plate geometry an material properties on the parameters that appear in the compact esign chart are also presente. Overall tiffness Trens The test results of Refs. 6 an 7 are presente together in Figs. 7-9, an the corresponing analysis results are presente in Figs. -2. Graphs of the prebuckling stiffness ratio /, the postbuckling stiffness ratio /, an the postbuckling-to-prebuckling stiffness ratio / that were measure experimentally are presente in Figs. 7-9, respectively. In Fig. 7, is the experimentally measure preb- uckling stiffness of the specimen without a cutout for each family of test specimens. In Fig. 8, is the experimentally measure postbuckling stiffness of the specimen without a cutout for each family of test specimens (values of an are given in Refs. 6 an 7). Corresponing graphs of the prebuckling stiffness ratio /, the postbuckling stiffness ratio /, an the postbuckling-to-prebuckling stiffness ratio / that were obtaine from finite-element analyses are presente in Figs. -2, respectively. In Fig., = E x h for each family of plates, an in Fig., is the postbuckling stiffness of the plate without a cutout that was obtaine from the finite-element analyses for each family of plates. Values of E x for each of the laminates without a cutout an their average thicknesses h are given in Table. In all of these figures, results for the aluminum plates are given by the black line with the fille black circles. imilarly, results for the [, [9, an [(/9) 5 plates are given by the soli gray lines with the unfille triangles, unfille iamons, an unfille circles, respectively. Results for the [(±3) 6, [(±45) 6, an [(±6) 6 plates are given by the ashe black lines with the unfille triangles, unfille iamons, an unfille squares, respectively. The results in Fig. 7 generally show that the largest reuctions in prebuckling stiffness with increasing cutout size are exhibite by the [ specimens, followe by the [(/ 9) 5 specimens. This tren is also exhibite by the finiteelement analysis results for the corresponing iealize plates (Fig. ). The smallest reuctions in prebuckling stiffness with cutout size are generally exhibite by the [9 specimens, followe by the [(±3) 6 specimens. The finite-element analysis results of the corresponing iealize plates show that the smallest reuctions in prebuckling stiffness with cutout size are exhibite by the [(±6) 6 specimens, followe by the [(±45) 6 specimens. However, only three ata points are given for the specimens for values

9 of / up to.32, an these ata points are very close to the ata points for the [9 an [(±3) 6 specimens, which show the next smallest reuctions in prebuckling stiffness with cutout size. The results in Fig. 8 shows that all of the test specimens, except the [9 specimens, exhibit a reuction in postbuckling stiffness with increasing cutout size. A somewhat similar tren is exhibite in Fig. by the analysis results. The [(/9) 5 specimens generally exhibit the largest, smooth reuction in the postbuckling stiffness with cutout size, followe closely by the aluminum specimens. In contrast, the [9 test specimens with / =.2,.32, an.42 exhibit postbuckling stiffnesses between 25% an 56% greater than the postbuckling stiffness of the corresponing plate without a cutout. imilarly, the corresponing analysis results in Fig. for the [9 plates show a marke increase in / as / increases to. followe by a very substantial reuction. The smallest reuctions in postbuckling stiffness with cutout size shown in Fig. 8 are generally exhibite by the [(±3) 6 specimens, followe by the [ specimens. In Fig., the analysis results show that the smallest reuctions in postbuckling stiffness with cutout size are generally exhibite by the [(±3) 6 plates. Although many of the test specimens exhibit a reuction in postbuckling stiffness with increasing cutout size (see Fig.8), none of the specimens show a significant reuction in the ratio of the initial postbuckling stiffness to the prebuckling stiffness, / (see Fig.9). This tren is also exhibite by the finite-element analysis results shown in Fig. 2 for the corresponing iealize plates, which initially appears to be nonintuitive. The tren is explaine by observing that the prebuckling an postbuckling stiffnesses of these plates are changing approximately the same rate as the cutout size increase; that is, the slopes of the corresponing curves in the figures are approximately the same. The [ an [(±3) 6 specimens exhibit the smallest values of / an show the largest increases with increasing cutout size, which means that the prebuckling stiffness of these plates is generally ecreasing faster with increasing cutout size than the postbuckling stiffness. The analysis results for the corresponing iealize plates show that the [ an [(/9) 5 specimens generally exhibit the smallest values of / an show the largest increases with increasing cutout size, followe by the [(±3) 6 specimens. Many of the test specimens show little change in the value of / with increasing cutout size, particularly for / >.. This tren is also exhibite by the finite-element analysis results for the corresponing iealize plates. The overall largest values of / are generally exhibite by the aluminum, [(±6) 6 an [(/9) 5 specimens. The corresponing results for the iealize plates preict that the overall largest values of / are exhibite by the [(±6) 6 an [(9) plates. In nearly every case, the values of / 5 for test specimens with a cutout are greater than the corresponing value for a plate without a cutout. This effect is most pronounce for the [9 specimens an is also preicte by the analysis results. This tren, an the nonintuitive behavior of the [9 specimens, is also exhibite to a large extent by the analysis results shown in Fig. 2. The test results in Figs. 7-9 inclue real effects like initial geometric imperfections, resiual stresses, loaing an thickness nonuniformities, an non-ieal support conitions that are likely to be encountere in real structures. This fact accounts, to a large extent, for the ifferences between the corresponing test results an the iealize finite-element analysis results that are shown in the figures. In aition, the experimentally measure values of the initial postbuckling stiffnesses are, to some egree, subjective an contribute to the ifferences between test an analysis results. In particular, the prebuckling an initial postbuckling results obtaine from the finite-element analyses are obtaine in a well-efine manner that has been escribe previously herein. imilarly, the prebuckling stiffnesses of the test specimens in Refs. 6 an 7 were obtaine by etermining the slope of the initial linear branch of the loa versus enshortening curve, which is also a relatively well-efine process. In contrast, the postbuckling stiffnesses of the test specimens in Refs. 6 an 7 were obtaine by fitting a line to the linear-most portion of the seconary branch of the loa versus en-shortening curve an fining the slope, which oes not necessarily yiel a unique result when initial imperfections are present. This process, which was use in Refs. 6 an 7, was also guie by other ata that were recore uring the tests. One significant cause of the ifferences between the test an analysis results appears to be initial geometric imperfections (eviation from flatness). The effects of an initial geometric imperfection on the normalize prebuckling stiffness /E x h of aluminum plates an [ an [9 graphite-epoxy plates without a cutout, that were obtaine from finite-element analyses (E x = E for aluminum plates), are shown in Fig. 3. For these results, an initial imperfection in the form of a half-sine wave along each coorinate irection an amplitue-to-average-plate-thickness ratio A/ h was use because no measure imperfection ata was available. The results in Fig. 3 inicate a similar reuction in axial prebuckling stiffness for all three plates as A/h increases, with a maximum reuction of approximately 33% for an imperfection amplitue equal to the average plate thickness, h (A/h = ). Thus, it is very likely that the iscrepancies between the analysis an test results curves for /Eh in Fig. 7 are cause, to some extent, by initial geometric imperfections. This observation is consistent with the results presente in Ref. that show a significant effect of initial geometric imperfections on the effective with of metallic plates. Moreover, the corresponing loa versus outof-plane eflection curves presente in Refs. 6 an 7 inicate that relatively large initial geometric imperfections may

10 have been present in many of the test specimens. Other sources of iscrepancies between the finite-element analysis an experimental results are variations in thickness an material properties, resiual stresses, an nonuniform loa introuction effects that were not measure for the test specimens. In aition, the iealization of the actual bounary conitions provie by the test fixture is likely to contribute some ifference between the test an analysis results. Regarless of the ifferences between corresponing test an analysis results presente in Figs. 7 through 2, both the test an analysis results preict several of the same trens. Most importantly, the test an analysis results inicate a complex interaction between plate orthotropy an cutout size which results in significant variations in the prebuckling an postbuckling stiffness ratios of compression-loae plates with or without a cutout. The result also show that real, imperfection-like effects can greatly affect the behavior of plates an shoul be consiere in their esign. Effective-ith Ratios lots of effective-with ratios provie a simple means for incorporating the postbuckling strength an stiffness of plate subcomponents into the esign of stiffene structures. imilarly, as shown previously herein, effective-with ratios can be efine for structural etails, like a cutout, that cause a similar loa reistribution as the structural etail increases in influence. In this section, effective-with ratios e ε are presente as a function of normalize strain ε cr for the aluminum test specimens an the [, [9, [(/ 9) 5, an [(±3) 6 laminate-composite test specimens consiere herein in Figs. 4-8, respectively. In particular, prebuckling an postbuckling effective-with ratios are presente for / values that range from /=. to / =.66 an implicitly inclue real imperfections like initial geometric imperfections an thickness nonuniformities. The prebuckling range of the figures correspons to ε ε < cr. imilar plots can be generate for the finite-element analysis results that have been presente herein, but have been omitte in favor of the compact esign curve that is presente in Fig. 9. The effective withs in Figs. 4-8 are etermine by using Eq. (7) an the inputs use in Eq. (7) are from the experimentally measure results that were presente in the previous section. As a result of using Eq. (7), the effective-with curves that are presente subsequently are base on a bilinear representation of the loa versus enshortening curves, an represent the initial postbuckling behavior of the test specimens. Material nonlinearities an large geometric nonlinearities that are associate with eviations from a bilinear loa-en-shortening curve are not consiere in the present stuy. The test results shown in Figs. 4-8 exhibit many of 6 the same general trens. For example, the results inicate that the effective withs of the plates are constant in the prebuckling response range, as expecte. In aition, with the exception of the results for the [9 specimens, the results inicate that, as / increases, the prebuckling effective with ecreases monotonically. Moreover, in the postbuckling response range, the effective withs of the plates generally ecrease monotonically with increasing strain level for all cutout sizes. The test results for the [9 specimens that are presente in Fig. 6 inicate that the prebuckling effective with of the plate with /=.42 is approximately 6% greater than the prebuckling effective with of the corresponing plate without a cutout. In the postbuckling response range, the effective withs of the [9 specimens ε ecrease monotonically with ε cr. However, the results inicate that the [9 specimen with a larger cutout oes not correspon to lower effective-with ratios. More specifically, the results show that, for values of ε slightly greater ε cr than, the [9 specimens with small values of / have higher effective-with ratios than the specimens with larger ε values of /. However, for larger values of ε this tren cr is reverse, an the [9 specimens with larger values of / exhibit larger effective-with ratios than the specimens with smaller values of /. Like the results presente in the previous section, the results in Figs. 4-8 inicate a complex interaction between plate orthotropy an cutout size which results in significant variations in the prebuckling an postbuckling effective-with ratios of compression-loae plates with or without cutouts. Equation (7) forms the basis for a very compact or concise way of representing the effective withs of the general class of laminate plates in aition to metallic plates. This property of Eq. (7) results from the fact that it applies to compression-loae plates, with or without a cutout, an can be use with measure test ata for / an / or ata from analytical or numerical methos. imilarities an ifferences in the behavior of various plates can be highlighte an compare irectly by casting Eq. (7) into the form of a generic esign chart like the one shown in Fig. 9. In this figure, stiffness-weighte, effective-with ratios are presente as a function of normalize strain for values of / o =.,.2,.3,.4,.5,.6,.7, an.8. Insight into response trens can be obtaine by generating results such as those presente in Figs. 7 an 9 for the test specimens consiere herein or such as the finite-element analysis results presente in Figs. an 2. Extrapolation of the Results The concept of effective with can be applie to any structural etail or iscontinuity that affects the apparent axial stiffness of a plate an causes the stresses on the loae

11 eges of a compression-loae plate to be reistribute towar the unloae eges. For example, loa introuction an thickness nonuniformities, initial geometric imperfections, spatial variation in material properties, bounary conitions, resiual stresses, an certain kins of amage affect the stiffness ratios / an /, an graphs that show these effects can be use with Eq. (7) an Fig. 9 to obtain values for the effective with an the corresponing behavioral trens cause by variations in the stiffness parameters. A specific example of how the results presente herein can be use is given by a amage state in a plate that oes not propagate or grow significantly with the loa level. For this case, the loa versus en-shortening response curve can be approximate by a bilinear curve an the effect of the amage on the stiffness ratios / an / can be replace by an estimate equivalent (or conservative) cutout. More precise values for the effective with coul be obtaine by establishing experimentally the effect of amage on the stiffness ratios / an /. The approach presente herein shoul not be use for extensive amage propagation with loa, material nonlinearities, geometric nonlinearities, or any other quantity that results in loa versus en-shortening response curves that cannot be approximate as bilinear curves. However, Eq. (2) forms the basis for efining an effective with for laminate-composite plates, just as it i for yieling in metallic plates. For this type of behavior, (N x ) max in Eq. (2) must be relate to the applie en-shortening by a failure criterion, such as the maximum strain criterion. Concluing Remarks A stuy of the effects of cutouts an laminate construction on the prebuckling an initial postbuckling stiffnesses an the corresponing effective withs of compressionloae, symmetrically laminate-composite an aluminum square plates has been presente. The effective-with concept has been extene to plates with cutouts, an results have been presente that inclue previously publishe experimental results an nonlinear finite-element analysis results for the corresponing geometrically perfect plates with iealize support an loaing conitions an material properties. Behavioral trens that were obtaine from the experimental results an from finite-element analysis results have been presente an compare for seven plate families an cutout iameter-to-plate with ratios ranging from to.66. In aition, a general, compact esign curve that can be use to etermine an compare the effective withs of a wie range of laminate-composite plate constructions has been presente, an a iscussion of how the results can be use an extene to inclue certain types of centralize amage, cracks, an other structural iscontinuities or etails that affect the axial stiffness of a plate has been given. The results show that some plates with cutouts can 7 have significantly higher prebuckling an initial postbuckling stiffnesses an effective withs than the corresponing plate without a cutout. These results illustrate a complex interaction between cutout size an plate membrane an bening orthotropies that affect the internal loa istribution in a plate. References. Von Kármán, T., echler, E. E., an Donnell, L. H., The trength of Thin lates in Compression, AME Transactions, AM-54-5, Vol. 54, No. 2, January 3, 932, pp Bengston, H.., hip lating Uner Compression an Hyrostatic ressure, presente at the annual meeting of the ociety of Naval Architects an Marine Engineers, New York, NY, November 6-7, 939, aper No.. 3. teinbacher, F. R. an Gerar, G., Aircraft tructural Mechanics, itman ublishing Corp., 952, pp Hu,. C., Lunquist, E. E., an Batorf,. B., Effect of mall Deviations From Flatness on Effective ith an Buckling of lates in Compression, NACA Technical Note No. 24, eptember Nemeth, M.., Buckling an ostbuckling Behavior of Laminate Composite lates ith a Cutout, NAA Technical aper 3587, July Nemeth, M.., Buckling an ostbuckling Behavior of quare Compression-Loae Graphite-Epoxy lates ith Circular Cutouts, NAA Technical aper 37, August Nemeth, M.., Buckling an ostbuckling Behavior of Compression-Loae Isotropic lates ith Cutouts, NAA Technical aper 324, eptember Rankin, C. C., Brogan, F. A., Loen,. A. an Cabiness, H. D., TAG Users Manual, Lockhee alo Alto Research Laboratory, Report LMM 32594, Dym, C. L., tability Theory an Its Applications to tructural Mechanics, Noorhoff-International, Leyen, The Netherlans, Vilnay, O. an Rockey, K. C., A Generalize Effective ith Metho for lates Loae in Compression, J. Constructional teel Research, Vol., No. 3, 98, pp Mayers, J. an Buiansky, B., Analysis an Behavior of imply upporte Flat lates Compresse Beyon the Buckling Loa into the lastic Range, NACA Technical Note No. 3368, February 955.

12 Table : Apparent axial moulus an average thickness of geometrically perfect plates consiere in the present stuy. late type Apparent axial moulus, E x (Msi) Average thickness, h (in.) Aluminum..647 [() 8.5. [(9).6. [(/9) 5.3. [(±3) [(±45) [(±6)

13 L y cr x (a) late configuration an coorinate system N x (,y) cr (b) Bilinear loa versus en-shortening iagram N x (,y) N x (L,y) (c) tress resultant istribution at loae eges for < cr N x (L,y) () tress resultant istribution at loae eges for > cr Fig. Coorinate system, loaing conition, response iagram, an axial stress resultant istribution in square plates without a cutout. 9

14 N x (L,y) y (N x ) (N x ) max max y 2 e (N x ) max x N x (L,y) y (N x ) max (a) Axial stress resultant istribution at loae eges for > cr (b) tress resultant acting over effective with, e Fig. 2 tress resultant istribution, average stress result istribution, an effective with for square plates without a cutout. cr cr Fig. 3 Bilinear loa versus en-shortening iagrams use to compute the effective withs of plates without a cutout.

15 y x (a) Unloae plate with a cutout an the coorinate system N x (,y) cr cr / = / > (b) Bilinear loa versus en-shortening iagram N x (,y) N x (L,y) (c) tress resultant istribution at loae eges for < cr N x (L,y) () tress resultant istribution at loae eges for > cr Fig. 4 Coorinate system, loaing conition, response iagram, an axial stress resultant istribution in square plates with a cutout.

16 / = / > / > cr cr Fig. 5 Bilinear loa versus en-shortening iagram use to compute the effective with of a plate with a cutout. a Clampe loae ege u(, y) = y, v L imply supporte ege imply supporte ege x, u u(l, y) = Clampe ege Fig. 6 Typical finite-element moel geometry an bounary conitions (ashe lines mark the rows on which the out-of-plane bounary conitions are applie). 2

17 .2 rebuckling stiffness ratio,.8 [(±45) 6 [9 [(±6) 6 [(±3) 6 cr / = cr / >.6 Aluminum.4 [(/9) 5.2 h [ Cutout iameter-to-plate-with ratio, / Fig. 7 rebuckling stiffness ratios for square aluminum an laminate-composite plates, with an without a central circular cutout, obtaine from experiments ostbuckling stiffness ratio, [9 [(±45) 6 [(±6) 6 h Aluminum cr [ / = / > cr [(±3) 6 [(/9) Cutout iameter-to-plate-with ratio, / Fig. 8 ostbuckling stiffness ratios for square aluminum an laminate-composite plates, with an without a central circular cutout, obtaine from experiments. 3

18 .8 [(±6) 6 [9 [(/9) 5 Aluminum tiffness ratio,.6.4 [(±45) 6 [(±3) 6.2 [ h Cutout iameter-to-plate-with ratio, / Fig. 9 ostbuckling-to-prebuckling stiffness ratios for square aluminum an laminate-composite plates, with an without a central circular cutout, obtaine from experiments. [(±45) 6 [(±6) 6 [(±3) 6.8 [9 h rebuckling stiffness ratio,.6.4 cr / = / > Aluminum [(/9) 5 [.2 cr = E x h Cutout iameter-to-plate-with ratio, / Fig. rebuckling stiffness ratios for perfectly flat, square aluminum an laminate-composite plates, with an without a central circular cutout, obtaine from finite-element analyses. 4

19 ostbuckling stiffness ratio, [(±45) 6 [(±6) 6 h [9 [ / = / > cr cr [(±3) 6 [(/9) 5 Aluminum Cutout iameter-to-plate-with ratio, / Fig. ostbuckling stiffness ratios for perfectly flat, square aluminum an laminate-composite plates, with an without a central circular cutout, obtaine from finite-element analyses. tiffness ratio,.8.6 [(±6) 6 [9 [(±45) 6 h.4.2 [ [(/9) 5 Aluminum [(±3) Cutout iameter-to-plate-with ratio, / Fig. 2 ostbuckling-to-prebuckling stiffness ratios for perfectly flat, square aluminum an laminatecomposite plates, with an without a central circular cutout, obtaine from finite-element analyses. 5

20 .8 tiffness ratio,.6 E x h.4 h.2 Half sine wave initial geometric imperfection shape Imperfection amplitue-to-plate-thickness ratio, A/h Fig. 3 Effect of imperfection amplitue on prebuckling stiffness ratio for square aluminum plates an [ an [9 graphite-epoxy plates without a central circular cutout..2 =..2.3 Effective with ratio, e Ratio of average axial strains, ε/ε cr Fig. 4 Experimentally etermine effective-with ratios for square aluminum plates with an without a central circular cutout. 6

21 .2 = Effective with ratio, e Ratio of average axial strains, ε/ε cr Fig. 5 Experimentally etermine effective-with ratios for square [ graphite-epoxy plates with an without a central circular cutout..2 =, Effective with ratio, e Ratio of average axial strains, ε/ε cr Fig. 6 Experimentally etermine effective-with ratios for square [9 graphite-epoxy plates with an without a central circular cutout. 7

22 .2 = Effective with ratio, e Ratio of average axial strains, ε/ε cr Fig. 7 Experimentally etermine effective-with ratios for square [(/9) 5 graphite-epoxy plates with an without a central circular cutout. Effective with ratio, e = Ratio of average axial strains, ε/ε cr Fig. 8 Experimentally etermine effective-with ratios for square [(±3) 6 graphite-epoxy plates with an without a central circular cutout. 8

23 .2 =.8 tiffness weighte effective with ratio,.8 e cr cr = / = / > Ratio of average axial strains, ε/ε cr Fig.9 tiffness weighte effective-with ratios for plates with an without a central circular cutout. 9

24 REORT DOCUMENTATION AGE Form Approve OMB No ublic reporting buren for this collection of information is estimate to average hour per response, incluing the time for reviewing instructions, searching existing ata sources, gathering an maintaining the ata neee, an completing an reviewing the collection of information. en comments regaring this buren estimate or any other aspect of this collection of information, incluing suggestions for reucing this buren, to ashington Heaquarters ervices, Directorate for Information Operations an Reports, 25 Jefferson Davis Highway, uite 24, Arlington, VA , an to the Office of Management an Buget, aperwork Reuction roject (74-88), ashington, DC AGENCY UE ONLY (Leave blank) 2. REORT DATE October 2 4. TITLE AND UBTITLE Effective iths of Compression-Loae lates ith a Cutout 3. REORT TYE AND DATE COVERED Technical ublication 5. FUNDING NUMBER AUTHOR() Mark. Hilburger, Michael. Nemeth an James H. tarnes, Jr. 7. ERFORMING ORGANIZATION NAME() AND ADDRE(E) NAA Langley Research Center Hampton, VA ERFORMING ORGANIZATION REORT NUMBER L ONORING/MONITORING AGENCY NAME() AND ADDRE(E) National Aeronautics an pace Aministration ashington, DC ONORING/MONITORING AGENCY REORT NUMBER NAA/T ULEMENTARY NOTE resente at the 4st AIAA/AME/ACE/AC/AH tructures, tructural Dynamics, an Materials Conference, April 3-6, 2, Atlanta, Georgia. Hilburger: NRC-Resient Research Associate, Langley Research Center, Hampton, VA; Nemeth an tarnes: Langley Research Center, Hampton, VA 2a. DITRIBUTION/AVAILABILITY TATEMENT 2b. DITRIBUTION CODE Unclassifie-Unlimite ubject Category 39 Distribution: tanar Availability: NAA CAI (3) ABTRACT (Maximum 2 wors) A stuy of the effects of cutouts an laminate construction on the prebuckling an initial postbuckling stiffnesses, an the effective withs of compression-loae, laminate-composite an aluminum square plates is presente. The effective-with concept is extene to plates with cutouts, an experimental an nonlinear finiteelement analysis results are presente. Behavioral trens are compare for seven plate families an for cutoutiameter-to-plate-with ratios up to.66. A general compact esign curve that can be use to present an compare the effective withs for a wie range of laminate constructions is also presente. A iscussion of how the results can be use an extene to inclue certain types of amage, cracks, an other structural iscontinuities or etails is given. everal behavioral trens are escribe that initially appear to be nonintuitive. The results emonstrate a complex interaction between cutout size an plate orthotropy that affects the axial stiffness an effective with of a plate subjecte to compression loas. 4. UBJECT TERM effective with, buckling, post buckling, composite plates, cutouts, esign 7. EC U RITY CL AIF IC AT ION O F REO R T Unclassifie 8. EC U RITY CL AIF IC AT ION O F TH I A GE Unclassifie 9. ECURITY CLAIFICATION OF ABTRACT Unclassifie 5. NUMBER OF AGE RICE CODE A3 2. LIMITATION OF ABTRACT UL NN tanar Form 298 (Rev. 2-89) rescribe by ANI t. Z