1、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-REPORT No. 739SHEAR LAG IN BOX BEAMSMETHODS OF ANALYSIS AND EXPERIMENTALINVESTIGATIONSBy PAUL KUHN and P
2、ATRICK T. CHIARITOLangley Memorial Aeronautical LaboratoryLANGLEY FIELD, VA.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSHEADQUARTERS, 1500NEW HAMPSHIRE AVENUE NW., WASHINGTON, D. C.Createdby actof Congre
3、ssapproved March 3,1915,forthesupervisionariddirectionofthe scientificstudy ofthe problemsofflight(U.S.Code, title50,sec.151). Itsmembership was increasedto 15 by actapproved March 2,1929. The members areappointed by the President, and serve as such without compensation.JEROME C. HUNSAKER, Sc. D., C
4、hairman, O.P. ECHOLS, Major General, United States Army, Corn-Cambridge, Mass. manding General, The Mat6riel Command, Army AirGEORGE J. MEAD, Se. D., Vice Chairman, Forces, War Department.Washington, D.C. SYDNEY M. XRAUS, Captain, United States Navy, Bureau ofCHARLES G. ABBOT, Sc. D., Aeronautics, N
5、avy Department.Secretary, Smiths0nian Institution. FRANCIS W. REICHELDEBFER, So. D.,HENRY I_I. ARNOLD, Lieut. General, United States Army, Chief, United States Weather Bureau.Commanding General, Army Air Forces, War Depart- JOIIN H. TOWEHS, Rear Admiral, United States Navy,merit. Chief, Bureau of Ae
6、ronautics, Navy Department.LYMAN J. BBIGGS, Ph. D., EDWARD WARNER, Sc. D.,Director, National Bureau of Standards. Civil Aeronautics Board,W. A. M. BURDEN, Washington, D. C.Special Assistant to the Secretary of Commerce. ORVILLE WRIGHT, Sc. D.,VANNEVAR BUSH, Sc. D., Director, Dayton, Ohio.Office Scie
7、ntific Research and Development, THEODORE P. WRIGHT, Sc. D.,Washington, D.C. Assistant Chief, Aircraft Branch,WILLIAM F. DURAND, Ph.D., War Production Board.Stanford University, Calif.GEORGE W. LEWIS, Director of Aeronautical Research JOHN F. VIeTOaY, SecretaryHENRY J. E. REID, Engineer-in-Charge, L
8、angley Memorial Aeronautical Laboratory, Langley Field, Va.x SMITH J. DEFRANCE, Engineer-in-Charge, Ames Aeronautical Laboratory, Moffett Field, Calif.EDWARD R. SHARP, Administrative Oficer, Aircraft Engine Research Laboratory, Cleveland Airport, Cleveland, OhioTECHNICAL COMMITTEESAERODYNAMICS AIRCR
9、AFT MATERIALS INVENTIONS analytical solu-but the treatment of these methods presented in this paper is tions based on the assumption of constant cross sectionconsolidated and improved in several respects. The are therefore of little practical value, and methods ofmethods are sui_ciently general to c
10、over any arbitrary span- analysis have had to be developed to cope with the con-wi_;e variation of cross section and loading as well as ditions found in actual structures. The development ofchordwise variations of stringer area, stringer spacing, and such methods has been continued over a period of
11、severalsheet thic/cness. Methods o/analyzing the effects of cut- years (references 1 to 3) and it is now possible to giveouts are also given, a reasonably well-rounded presentation of practicalmethods of analysis.The second part of the paper describes strain-gage testsmade by the NACA to verify the
12、theory. Three tests were The paper is divided into three parts. The first partmade on axially loaded panels oJ variable cross section, six discusses the methods of analysis. The second partwere made on beasts (_ variable cross section, and three describes tests made by the NACA and shows eompari-wer
13、e made on beams of constant cross section for extreme sons between experimental and calculated results foror limiting cases. Three tests published by other investi- the NACA tests as well as for tests made elsewhere.gators are also analyzed by the proposed method. Numerical examples to illustrate th
14、e methods of analysisIn order to make the test of the theory as severe as possible, are presented in the third part.the NACA specimens were designed to show larger shear- The method of presentation chosen is intended tolag e_ects than may be expected in typical present-day meet the needs of the prac
15、ticing stress analyst. Theconstruction. The agreement was quite satisfactory even paper eontmns the information actually needed in stressin extreme cases such as very short wide beams. Satis- analysis. Detailed derivations and discussions havefactory agreement was also found in tests on the limiting
16、 been omitted, but they may be found in several of thecase of a cover without stiffeners; this agreement shows cited references.that the theory is applicable to the case oj heavy cover I. METHODS OF ANALYSISplates used without stitching or to cases in which contin-uous stiffening in the form (_ corr
17、ugated sheet is used. DEFINITION OF THE PROBLEM AND BASICThe third part of the paper gives numerical examples ASSUMPTIONSitlustrating the methods of analysis. An appendix gives Reduced to its simplest form the problem may becomparisons with other methods, particularly with the stated as follows: A s
18、heet, stiffened or unstiffened, ismethod of Ebner and Kdller. fastened to a foundation along one edge and loadedINTRODUCTION along the two edges perpendicular to the foundation bydistributed or concentrated forces as indicated inThe bending stresses in box beams do not always figure 1. The sheet may
19、 be a structure in itselfconform very closely to the predictions of the engineer- (fig. 2 (a) or it may be the cover of a box beam (fig. 2 (b).ing theory of bending. The deviations from the theory The problem is to find the stresses in the sheet.1Provided by IHSNot for ResaleNo reproduction or netwo
20、rking permitted without license from IHS-,-,-2 REPORT NO. 739-NATIONAL ADVISOP_Y COMMITTEE FOR AERONAUTCS/, P and this extension is therefore givem An approximatemethod for dealing with moderate amounts of camberis given in reference 2.ANALYSIS OF SINGLE-STRINGER STRUCTURESStructures like those show
21、n in figure 2, having but asin.gle stringer, are rarely encountered in praeffcc.Nevertheless, the analysis of single-stringer structureswill be fully discussed for several reasons. The immedi-x ,_,x_,x,_, ate reason is that the fundamental relations as well asfm_,_ _. all the methods of analysis can
22、 be easily demonstratedAs shown in figure 1, stiffeners are theoretically on this type of structure. A more important reason isthe fact that the most rapid method of analyzing multi-necessary along the loaded edges if concentrated forcesP arc introduced because the stresses would otherwise stringer
23、structures is based on the temporary reductionof the multistringer structure to a single-stringerbecome infinite. These edge stiffeners will be referred struetme.to throughout this paper as “corner flanges“ or simply StuN CONVENTIONS“flanges.“ Other stiffeners parallel to the loaded edgeswill be ref
24、erred to as “longitudinals“ or “stringers“; Tile sign conventions adopted are as follows: Normalthese stiffeners may or may not exist in any given case stresses and strains in the stringers and the flanges areand may or may not be attached to the foundation.It will be assumed that the structure is a
25、lwayssymmetrical about a longitudinal plane (y=0). Thisassumption materially simplifies the problem withoutdecreasing the practical usefulness of the theory verymuch because most practical structures are at leastapproximately symmetrical. On account of the sym-metry, it will be sufficient to conside
26、r one-half thestructure in all derivations and computations.It will be assumed that infinitely many ribs of infin.itcextensimtal (ehordwise) stiffness are distributed “fiche Fi(tmE3.()onvelltionforeoordinateaxes.the spall. An equivalent assumption is frequentlymade in theoretical solutions of stress
27、 problems. The positive when they are t,cnsilc. Shear stresses andassumption is plausible in this ease because it is fairly strains ill the cover sheet arc positive when they areobvious that the extensional stiffness of tile ribs togedw, caused by positive strains in the flange. Shear stresseswith t
28、he lateral bending stiffness of the flanges between in the web arc positive when they are cmsing positivethe ribs is sufficient to take care of such transverse strains in tile flange.stresses as might arise from longitudinal forces and The compression side of the beam is analyzed inde-pendendy of th
29、e tension side. It is therefore permis-sible and convenient to retain the sign convention just: f-_fN given for the analysis ofthe compression side, changingonly the definition of stringer stresses to positive whencompressive.,_the relative longitudinal displacement (u_-uL)of two . -,corresponding p
30、oints on the flange and on the longitu“ _3 _- 5hFIGURE 4.-Convention for syml3ols on crosssections. _nal divided by the width b defines the shear strain _, mid _-_ similar sohltions have been given, by variable cross section and loading.other authors. These analytical solutions are of some Recurrenc
31、e formula for shear lag. As stated in thevalue in making comparative studies and in studying preceding section, the beam is divided into a numbervarious aspects of the shear-lag problem. For praeti- of bays; the cross section and the web shear Sw/h arecal stress analysis, however, numerical methods
32、capable assumed to be constant within each bay. The lengthsof dealing with arbitrary variations of cross section and of the bays need not be equal nor need they be small,loading are required. Two such methods will be as is often required in similar methods. In the limit,described: The solution, by m
33、eans of a recurrence a single bay may span the entire length of the beam.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 REPORT NO. 739-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSThe system of numbering the stations and the bays given in reference
34、3. Equating the deformations atbetween stations is shown in figure 6. the adjoining ends of successive bays yields theEach individual bay can now be treated as a free recurrence formulabody subjected to certain forces (fig. 7). These forces X, _q,-X,_(l_,+p,_+Oq iV, t lq,-t_ v,+v,H (9)_can be split
35、into two groups (fig. 8): One group consistsof the forces calculated by the ordinary bending theory, wheretwhich assumes no shear deformation; the other group K (3a)represents the differences between the actual forces and P_ Gtt,niil KLKq_= Gt sinh KL (3b)S.:A_, S.,Q_ (:_c)- / /_ . / / A v=htA_,G=-D
36、-G n_+/ / where K is a shear-lag parameter appearing in allT . / 77/“ - 9 / _/_/ (_+/ . .3 . 2 / oFIGURE 6.-Convention for nulllberillg bays and stations of a beam. _*z _changes in forces caused by the shear deformation of /_the cover sheet, s_ll II _-_ _: iThe first group of forces will t)e designa
37、ted P-forces _- “_-JL I -_/ Jto indicate that they are calculated by tile theory that J -_k_J!_“ _ assumes plane sections to remain plane. Individual (a)forces and stresses belonging to this group will be de-noted by a superscript P. The calculation of these ,_forces and stresses is familiar to ever
38、y engineer andconsequently need not be discussed in detail.The second group of forces will be designated X-forces. _*_Because the P-forces on any one bay are in static equilib- _ _: _rium, the X-forces at any one station must be a self- -4“_._(b) X-forces.G, G“ _L._/ FIGURE 8. Separation of forces a
39、cting on bays.I_1_ _ _ _ _ _ _ t1_/_, s . / analytical solutions for single-stringer structures_ IR _“- 11 1“-. r. : I (references 1 and 2) and is defined by The magnitudes just en.umcrated should be separatelytabulated because they will rcmain constant; whereas,tile m_fin part of the calculation is
40、 repeated a n.umber l Iof times. The details of the proc(_dure are learned mosteasily by following column, for columu the mlmericalexample given in part III, table 10. (a) (b)Cohmm 1 in table l0 gives assumed values for _ /z _ 7 /_ ,/_H,/_/,lit assuming these stress values, the analyst must be F_,_E
41、 10. Shearfaultandshcar-faultcorrecIion.guided by pre:vious experien.cc. It is possible to us_entirely arbitrary values but, if the assumcd values Column 8 gives the increments of shear forccdiffer too much from the true ones, a large number of ASeE-rtAx (SS-2)cycles of the computation will be requi
42、red. The Cohnnn 9 gives tile increments AFL, obtained by sub-simplest procedure for general use is to multiply the tracting the value of FL at the outboard end of the baystresses obtained from the ordinary bending theory by from the value of FL at the inboard end of the bay.a factor slightly larger
43、than unity. With some ex- According to the basic relation (le), AF_ shouldperience, this factor can be estimated reasonably wall equal ASc_ in each bay. The differences in each bayfrom a knowledge of the average of the shear-lag p_.- constitute the shear faultsrametcr KL and the loading condition. S
44、F=ASc, E-AF,. (SS-3)Column 2 gives the foiees/;_,=z_.A_,._V/ _ and tile shear faults SF are given in cotmnn 10.Cohmm 3 gives the forces ZL=-h-/%, it1the case of Consider now figure 10 (a), which shows one bay witha beam or I+)=P-F_, in the ease of an axially loaded a positive shear fault SF and the
45、corresponding shear-panel, fault correction SFC; SFC is in the form. of externalColumn 4 gives the stiesses _=F_/Ar,. forces distributed uniformly along the bay.Colunm 5 gives the differences between cohmms 1 and The length of a bay is small compared with the length4 (_-L). of the structure; it may
46、therefore be assumed that the47_as._ 4a.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 REPOI_T NO. 739-NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSproperties Of the structure just outboard and just ill- than tile sum of tile faults in tlie precedin
47、g cycle. Thisboard of the bay considered arc the same. Under this criterion is not sufficiently sensitive to prove the absenceassumption, one-half of tile shear-fault correction SFC of any numerical error, but it is sometimes a welcomewill bc absorbed by the structure outboard of the bay; help when
48、starting ellculations.the other half will be absorbed by the structure inboard A complication arises when the longitudinal is notof the bay. As previously stated, the total shear-fault conlooted at the root. In this case, the stress zLis equalcorrective force will be taken as one-half tile shear fau
49、lt, to zero at the root but tile shear stress r is not equal t,oTotal SFC=-IsF_ (SS-4) zero. It is therefore impossible to proceed directlywith the summation of the increments zXr. In order toTherefore the corrective force at the outboard end of overcome this difficulty, a trial value 70 for 7 at x 0the bay will be
