1、“1i1“,L.“f-i1., “i1IINATIONALADVISORYCOMMllTEEFOR AERONAUTICS.- ._. _ .TECHNICAL NOTE 3784HANDBOOK OF STRUCTURAL STAJ311JTYPART IV - FAILURE OF PLATES AND COMPOSITE ELEMENTSBy George GerardNew York UniversityWashingtonAugust 1957.I- -Provided by IHSNot for ResaleNo reproduction or networking permitt
2、ed without license from IHS-,-,-TECHLIBRARYKM=, NMu .:,.LNACA TN 3784TAME OF CONTENTSPage1124677991032titi1719202223252627%323536;:42SUMMARY .*INTRODUCTION .moma71a71aa71a71a15a. .a71 ea71 *0.00a71a71a150000.a710.0.00a71 .*eea71 80000.0.e000.a71a15a9aea71a15a71aeee00.a71.0a71.0.0a71.00a7100a.a71a15a
3、71a15000a71.a71a71a71a15a15a15em00.aa71a150.0.9e9a71a71.a71a15a15e.e.0.e00.0.a71a71009a710.0.0.a710a71a15e9a719a71.0.e.a71.0.m.a71.a71.000a71-0.a71.a71.00.a71.a719a7100.a718a71BASIC PRINCIPLESFAILURE QF FLANGESPostbucklingBehaviorFailure of Flanges. .FAILURE (IFCOMPRESSEDFIAT PLATESPostbuckkng Behat
4、ior of Flat Plates a71 .9Effective Width . . . . . . . . . . .Failure of Flkt.Plates . . . . . . .a718a719e00.a7100.a71a719me0.ea710POE?CBUCKUNG BEHAVIOR AND FAILURE CO?COMPRESSED CURVED PIATESPostbucl+ingBehavior . . . . . . . . . . . . .Effective Width . . . . . . . . . . . . . . . .Failure-of Cur
5、ved Plates . . . . . . . . . . .CRmLING (EEXEWDED Z-, CHANNEL, AND H-SECTIONSSemiempiricalAnalysis of cripJwng o . . . 0 sAnalysis of Z-, Channel-, and H-Extrusion DataDesign ta for Z-, Channel, and H-Extrusions .a710e.0ma71e.m.“a00a71a710a71000.0a710a710.e.a71.00.a71e.9.me.a71a71a710.a0a710.e.aa71a
6、71e.0a71ma71a71.s0.e.a710.a710.aea719090.CRIH?LING CEFORMED ANGIE, Z-, AND CHANNEL SECTIONSSemiempiricalAnalysis of Angles and Sqpare !hibesCrippling of Square K13ibesand Equal Flange AnglesSemiempiricalAnalysis of Z- and Channel SectionsCriping , Z- and Channel Sections . . . . .Cladding Correction
7、 .o. o .OO. O.Increased Corner Properties of Formed Sections.00ea71.0CR13?I?IXNG(II?SECTIONS OF GENERAL SHAPEOne-CornerElements . . . . . . . . .Two-CornerElements . . . . . . . . .MuJ.ticornerElements . . . . . . . .Lips and Bulbs . . . m . . .0.a71 .meO*. .000.Ii. _- _.- _ _ - . - - - .- . .Provid
8、ed by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- . -.JAPPENDIX A - APP13CMl!IONSECTIONFlat and Curved PlatesEffective width . . .FailWe. . . . . . .Crippling . . . . . . .One-corner elements .Two-corner elements .Multicorner elementsImmENcEsTABLES 0 .FIGUR
9、ES . .a71a71a710a71a71.9.0a71 ma71e0.a710a71a71a15a15a71a71a15a15a719a71a71a15a719a7100a71ea710a71a71a71a71a71.a71m.900a71.090aa71a71a71a15a710sa71s0a71a71a710a71a71a71a71a.0a71a71a71a71a71a71a71a71a71a71a71a71a71a15a15ma71a71.a71.0.e.a71.m.0a71e.0s.a7100.a71a71a71a15a15a71a71a1509a71a71a719a71a719.
10、a7100a71a15a15a15a15a719a71a71e9a71a7109a71m.n0a71a719a71.a710a71a710.a71a71a71ma710.a71a71a71s9.b.wa710a71a71a71.a71a7199a71m9a71a71a710.* iia71a7190a71a71m.m.m .a71a71a71a71a71a71a71a71a15ma71Page434343$k546474051.61.Provided by IHSNot for ResaleNo reproduction or networking permitted without lice
11、nse from IHS-,-,-NATIONAL ADVISORY COMMITTEEFORONAUTICS.u+. a.-TEcENIcALJm!m 3784. .HAND600K OF STRUCTURAL STABILLJ?XPART IV - FAILURE OF PIATES AND COMPOSITEAvailable theoriesterms of the results ofBy George GerardSUMMARYExEMENTson failure of flat plates are reviewed. Inthese theories, available te
12、st data on the post-buckling behavior, effectivewidth, and failure of flat and curved platesare correlated.Test data on the crippling strength of various formed and extrudedshapes are reviewed, from which a generalizedmethod of crippling analysisis formikted. The effects upon the crippling strength
13、of alclad coathgsand the increasedproperties in the corners of formed sections are inves-tigated by use of this analysis.The generalized crippling analysis was applied to a variety of sec-tions and materials in common use. It was found that the cripplingstrength of all the sections is governed in a
14、sleoped length-thiclmessratio and mniber of corners ofINTRODUCTIONmanner by thethe section.devel-The present report is concernedwith the failure of flat and curvedplates an composit= elements which are subject to buckling under compres-sive loads. For composite elements such as 2- or channel section
15、s,thefailure mode considered is that commonly referred to as crippling. Thus,the effective slendernessratio of the composite elements is in the neigh-borhood of 20, a region in which variations in length result in negligiblechmges in the crippling strength.In the section entitled “Basic Principles,!
16、 the postbucklingbehavim -of columus,plates, and cylinders is briefly reviewed in order to deter-mine those elements for which the failure lead may considerablyexceedthe buckling load. Because of the mathematical ccuuplexitiesinherent inany theoretical treatment of the failure phenomenon, the availa
17、ble theoriesconstitute an important contributionto the understanding of the factorsoperativeat failure. Therefore, a comprehensivereview of resultsobtained for hinged ilanges is presented in the next section. This anal-ysis serves as a reference frme frum yhich semiempiricalmethods can bedevised to
18、treat failure of elements not amenable to theoretical solution.- _ . . _. _. - Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 NAcAl?N 3784 ,In the next two sections,the postbucklingbehavior, effectivewidth, .Tw membrane stressesSubscripts:av avera
19、gecr criticale edgef fIangeo. initialP flit plater reduced valuew webEASIC PRINCIPLES .InP I of this Handbook (ref. 3), the elastic and plastic bucklingof flat plates was considered, in Part II (ref.4) bucli13ngof compositeelements was summarized, and in Part III (ref.5) buckling of curvedplates and
20、 shelh was treated. In certain cases, buckling terminates theability of the element to carry additional loads and, therefore, bucklingand failure are essentially coincident. In uther cases, primarily in flatand slightly curved plates which buckle elastically,failure occurs atloads considerablyin exc
21、ess of the buckling load. It is the purpose hereto delineatethe physical principles involved in the postbucklingbehavior -of various elements in order to examine in detail those elements for whichthe failure load may considerablyexceed the buckling load. -. .Provided by IHSNot for ResaleNo reproduct
22、ion or networking permitted without license from IHS-,-,-I!llcllm 3784 5Except for those cases in which the buckle form itself is unstablea71. as discussed in reference 5 failure is generally a combination of large-/ deflection effects initiated at buckling and plasticity effects. Becauseof the non1
23、3nearitiesassociatedwith bath large deflections and plastic-ity, the problem of determining the theoretical failing load of anybuckled element is mathematically complex, if not intractable. However,since buckling initiatesthe processes leading to eventual failure, itis of utmost importancethat analy
24、ses should exist for accuratelypre-dicting buckling stresses. _ses presented in references 3t05 pro-vide a key role in constructinga theory of failure, which, because ofmathematical complexities,is ofien semiempiricalin nature.Failure in individual cases is coincidentwith, or occurs consider-ably af
25、ter, buckling. Hence, it is importantto.examine the postbucklingbehavior of various elements to determine the conditionsunder which thefailure load can exceed.thebuckling load. For this purpose, figure 1has been prepared. In figure l(a) the schematicpostbuckling behaviorof flat plates, columns, and
26、cylindersunder axial compression is shownwhen elastic buckling occurs. Figure l(b) indicates, schematically,thebehavior of flat plates and columus after plastic buclding.It canbe observed from figure”l(a) that, for flat plates and columns,failure occurs at values of w/t well remuved from the region
27、Wherebuckling initiates. Thus, small initial imperfectionsare unimportant.After buckling, the lateral deflection of the column is nti restrained inany manner. Therefore, no transverse membrane stresses exist and the post-buckling behavior is representedby a horizontal line. aeyond a certain+alue of
28、w/t, plasticity effects become important and the /ucr Mnedecreases. Thus, fdlure occurs at the onset of significantplasticityeffects.For flat plates, the boundary constraintsat the unloaded edges per-mit significanttension menibranestressesto develap after bucli13ngwhichact to restrain lateral defle
29、ction. Thus, flat plates can support loadsconsiderably in excess of the ebstic buckling load. As for columns,failure occurs at the onset of significantplasticity effects.By contrastwith flat plates and columus, axially compressed cylin-ders develop transverse compressivemenibranestresses ef%er buckl
30、ing andthus the buckle form itself is unstable. As discussed in some detail inreference 5, small initial imperfectionsare imp ()u 1/2 = 0.19 + 0.81 aeFor a square plateand free”For long plates with rtiationally restrainedand free to move laterally (Ar/at = O)unloaded edges:= ”2(%Y-0”5(*r+0”45(*rto m
31、ove(5)held straight(6)Effective WidthThe results of large-deflectionanalyses are often convenientlygivenin terms of the effectiw width of the plate acting at the edge stress.The average stress on the plate was defined as .3= P/bt (7). . .-. . . . . . - - .- - - .=_Provided by IHSNot for ResaleNo rep
32、roduction or networking permitted without license from IHS-,-,-IiACATN 3784 u. .For the same load P, it is desired to find an effectivewidth at each.edge be acting at the edge stress:,P = Ue2bet (8)By substitutingequation (8) into equation (7),:. aee (9)By rearranging eqyation (4) and by use of equa
33、tion (9) the followingeffective width rehtion can be obtained for values of ae/Ucr less t3:(10)For vdlES Of ue/Ccr greater than 3 equations () and (6) may be useddirectly in conjunctionwith equation (9).The effective width discussed in the foregoing ccmments is concernedwith the load-carryingability
34、 of the plate after buckling. A secondtvne of effectivewidth is associated with the stiffness of the plate“.against further compression. The reduced effective width isFrom equationFor values ofbe found fromsquare plate:()2be = ddTr (4), therefore()br=Pe/Ucr greater than 3, theeqpations (5) and (6) i
35、n thereduced effectivefo12mwing forms:and for the long plate:()2be cm 1/2T. ()=0.19+ o.405q-def-=d as(U).(12)width C=For the(13)(+)r:oe7+#5-o.13 (fa acy (28)Cyvalidity of equation (28)probab3y applies for values ofthan 3/4, a value obtained in the preceding sections. Nocurved plates are availableto
36、establish this limit.Equation (28) apparently gives a reasonable fit to all the test datafor values of Zb between 10 and 125. The tits for the Zb range betweenO and 10 consiste?rtlyfall on the lower lo-percent Mmit. For this range,it is suggestedthat a value ofCRITPLING OF EXTRUDEDThe failure follow
37、ing localvaxiously referred to as fl= 1;79 be used in eqyation (28).Z-, CHANNEL, AND H-SECTIONSbucltLingof stiffening elements isaverage strength,ultimate strength,crushing-strength, or criping stre&h. The last term is in commonuse in the aircraft industry and is associatedwith the short-columu .reg
38、ion (L/p less than approximately 20) where the strength does notvary significantlywith the length of the stiffener. The term crilingstrength is used in this sense herein. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACATN 3784 21. Fran the stand
39、point of local buckling (ref. 4), the stiffener crosssection is envisioned as an asseniblageof plate elements.5e boundaryconditionsalong the unloaded edges of each of the individualplate ele-ments at a common junction are associatedwith rotational restrahts pro-vided by adjoiningplate elements. The
40、junctionmay be the filletedcorner of an extrusion or the bend line of a formed section.Early attempts to determine the crippling strength of stiffeningelements were based upon the buckling behavior of the elements. In suchanalyses, the crippling load was taken as the sum of the buckling loadsof each
41、 of the plate elements ccmpring the stiffener cross section.Such analyses are typified by methods presented in the book of .Sechlerand Dunn (ref. 18).Based on the knowledge that the failure load of a flat plate canappreciably exceed the elastic bucldlingload, later analyses attemptedto refine the me
42、thod of calculation of stiffener cripLng strengths.Such methods are in wide use in the aircrti indmtry for sections ofgeneral shape and differ possibly only in minor details from that pre-sented by Crockett (ref. 19).For equal flange stiffeners such as Z- or H-extrusions, extensivetest data are avai
43、lable for sevez%materials. The s-harpcorners andretively uniform stress-straincharacteristicsin the cross sectionsof extrusionshas permitted a synthesis d these data in a simple form.A relationship invohing the buckling stress of the section and thecompressiveyield strength of the material is relate
44、d directly to thecrippling strength.For formed Z- and channel sections,the relationship smong bucklingstress, compressiveeld strength,and crippling strength appears to beinfluencedby additional factors. These include the finite radius ofthe rounded corner and the increase in compressiveyield propert
45、ies inthe corner due to the forming process!These effects have recently led Needham to propose a revised methodof analysis for formed sections of general shape (ref. 7). In thismethod, the .crippl.ingload of the stiffener is obtainedby sunuuingthefailure loads of each of the corner elements comprisi
46、ngthe stiffenercross section.In this section of the report, a semiempiricalanalysis is presentedbased upon the knowledge gained from the preceding sections on failure offlanges and flet plates. This analysis serves to unify test results onvarious types of stiffeningelements and also provides some ph
47、ysicalinsight as to the factors operative at failure. Available criping testdata on Z-, H-, and channel extrusions are reviewed in this section.Formed angle, channel, and Z-sections are reviewed in the following. - .-. _. _. ._ .= -. . . .Provided by IHSNot for ResaleNo reproduction or networking pe
48、rmitted without license from IHS-,-,-22 NACATN 3784 -.section and methods of analysis for sections of general shape are pre- .sented in section after next.SemiempiricalAnalysis ofFrom the flat-plate analysis presentedCompressedFlat Plates,” it can be ectedwill influencethe phstbucklingbehavior of.cripplingin the section “Failure ofthat the following factorsstiffened elements:(a) The deee of rotational restraints at the unloaded edges(b) The degree of warpi
