1、IIANATIONALADVISORYCOMMITTEE ;FOR AERONAUTICS JTECHNICAL NOTE 2232STRESS AND DISTORTION ANALYSIS OF A SWEPT BOX BEAMHAVING BULKHEADS PERPENDICUMR TO TBE SPARSBy Richard R. Heldenfels, George W. Zender,and Charles LiboveLangley Aeronautical LaboratoryLangley Air Force Base, VaWashingtonProvided by IH
2、SNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-11.UILIuMY KAFB,NMI Illllllllllluu!lllllluuuoOb5flbq .NATIONAL ADVISORY GOMKD5XE FOR AEEIONfiICIT “ .TECmCAL NOTE2232. STRESS AND DISTORTION ANALYSIS OFA SWWT Mx = .HAKU?G EUMRMDS PERPENDICULAR TO THE SPARSBy Ri.ard
3、R. Heldenfels, George W. Zender,and Charles Libove,SUMMARY “ ,A method is presented for the approximate calculation of the -stressesand distortions in a box beam representing the main structuralcomponent of a swept considerationis also given to the relationshipsbetweenthe idealizedand actual structu
4、res and a comparisonbetween the stressesand distortions calculatedby this method and the experimental data ofreferences 1 and 2. h the discussion, the effects of shear lag, whichthe method cannot give, are considered and an extension of the basicapproach to permit their in+sion is indicated; also, t
5、he importance tothe analysis of includingthe shear and bending flexibility of the bulk-heads bordering the triangular region is demonstrated. A completenumerical exsmple is worked out in an apndix.Aa,anaij,Cn,bb:SYM20LSPrincipal Conceptsarea, sgyare incheslength of bsy, inchescoefficientsof matrixar
6、bitrary constants h solutionwidth ofwidth ofouter section, inchescsrry-through section,.of a differential equationinches,.#. = .-, . . . -. . - ,:;-JPGhIJkiLzMN,N1P-QRTtuuvwxdepth of box beam or bulkhead, fichesmodulus of elasticity,psiforce pounds er inchratio which has the value +1 for symmetrical
7、sntisymmetricalloadstorque, inch-poundssheet thickness, inches(strain energy, inch-poundsdisplacement in the x direction, inchesverticsl shearing force, pouudssectionsloads and -1 fordownuard displacement or deflection, inches.sel.f-equilibrating,statically indeterminateforce group, pounds.-. - -.-:
8、7-7-l-. . - -,. - - - -. - -. .- - . .-. -:. , . . .,? . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.4. NMA 2232.X,yz rectangular coordinates,inchesT distance fran front spar to a specified center, inchesY sheex strainb? “ ndimensional parame
9、ters used in discussion of idealizatione angle of twist, kadians A angle of sweep, degrees.L effective tidth, inchesv Poissoritsratio (assumedto be 1/3)v effectivenessfactora normal stress,psiT shear stress,psi.0 stress function .I+,$ joint rotations, radiahs (see fig. 4) .SubscriptsSubscripts also,
10、stiffnessfactorsThe single exception to the foregoing convention is:e ef-fectivewhen applied to area, thiclmess,inertiaSqerscripts are used toSuperscriptsdesignate stressesandproduced by different types of action, as follows:B, bendingF F-force groupR rigid-body displacementsT.torsionw m-x X-force g
11、roupa flexureT shearANALYSISThe type of idealizedis a four-flenge box besm,. . - . . . _ - . .“ ,.OF THE IDEALIZED STRUCTURE.qmbers to identifyor moment ofdistortionsB thus the nuniberof equilibrium equations neededis reduced and the analysis is simplified.b an analysis of this t many of the factors
12、 involved dependupon the nature of the applied load (symmetricalor autisymmetrical,bending or torsion) and it may therefore be advantageousto make aseparate analysis for each type of load and then superimposethe resultsto obtain the desired solution. For convenience in the detailed develop-ment whic
13、h follows, however, provisions for both bendhg and torsionare included stiultaneouslybut with restrictions that they are eithersynnnetricalor antisymetricsl about the csrry-through section.Joint-EquilibriumEquationsIf the three joints shown in figure 3 are considered as free bodies,a total of nine e
14、quilibrium equations.can be written, two for nmmentsand one for vertical shesr at each joint, as follows:Joint 1:.P7 - P -PI COSA=OVl+(plg-q)=o(1)(2)(3)o. . - - - . . - - .-.?- _._ _ _ - - y- - - - -. -., . . . . . . .Provided by IHSNot for ResaleNo reproduction or networking permitted without licen
15、se from IHS-,-,-8Joint 2:NACA w 2232,tJoint 3: .,P2 -Plo=o . (4) (5)(6) ,.P8+P4cos A- P3sin A=0 (7)P6 - P5 -P4sA - P3 cos A = O-. (8)(%V+c -+qlo-3 +q3). o (9)hsmuch as the number of unknown forces appearing is greater thanthe number of equations,the problem is staticaIthe modifications required for
16、different types ofsupportssxe discussed in aThe force-displacementwritten ag indicated in thesubsequent section.relationshipsfor each co,prponentcan befollowing sections.- _ . -. - . . .7- . :-Y.-:.-v- -,.-.= .-. .,.: “.”, - - : . . . “:. . . Provided by IHSNot for ResaleNo reproduction or networkin
17、g permitted without license from IHS-,-,-2 NACAsparm 2232 9Beams.- The two bulkheads 1-3 and 2-3 and that gwrt of the front1-2 bordering the triangur section can be ai.yzed as beams sub-jected to end shears andmomentk plus a runn3ng shear along the flanges.This running shear results from the shear f
18、lows in the covers adjacentto the flanges. The loading and distortion of a beam of this type isillustrated in figure 5. ti.appendixA, this type of beam is analyzedand the following general expressions are obtained for the end loads in terms of the end-parsmeters whichbeam:displacements,the running s
19、hear, and certain stiffnessinclude both the shear and bending resistance of thewhere.()+4EIZ3 Gctz2:=l; ) .12+4EI.-. 3 GetZ2()5=12=-().1C2GfD3EI.=g(!+)EI bending stiffness of besmGet shear stiffness of beam .(lo)() .(12),.-. .-. .,. -, .,-. .-, - - -.-,-=. - - -7. -. , ., .,. . ., ”,.: . . . -. .- .
20、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-_10 NACA TN 2232,Specific force-displacementrelationships for each of the beams canhe ol these forces, too, can be expressed in terms of the nine joint dis-placements when the other structural component
21、s are considered.Trimgular cover sheet.- The triangular cover sheet is assumed tocarry a uniform shear flow q5 along its mutually perpendicular edges(l-2 and 2-3). In order that this element be in equilibrium, shear andnormal.stresses are required along the hypotenuse and the correspondingforces are
22、 shown (fig. 2) as a uniform shear flow q act- Qowthat edge and a pair of concentratedforces acting at the joints.The equilibrium equations ae:(23) Force-displacementlationships are obtained by assuming that themaximum shear strain in the sheet is equal to the amount by which theright angle 1-2-3 is
23、 changed. In terms of the joint rotations, thisshea strain is:Then,and the relations for qU and P5 followtiO (22) SDI (23).Acot A- $*+ !33)immediately from equa-(24)(25)Outer section.- That mrt of the structure outboard of bulkhead 2-3acts as au unswept cantile=r box beam supported on a flexible roo
24、t and,as such, csn be analyzed by existing methods of analysis. The stresses-and distortions at any point csa be essed in terms of the appliedloads, the distortions of the root, and certain elastic stiffnessfactors. Then, the force-displacementrelationships required to definethe internal forces at t
25、he root rxre:. .- - 7 . -., .-.- - - -,. ., -. . :. : .-. . . . . . . . . .,Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . - .4E 9NACA 2232#-.P3 = lq_M- k2T + k3(2 - 3)%0 =k4M+T- - flk3(2 , = v + 6T + (2 - 3)q3 = -klOv + + k7(ti2- 3)q4=k8V+ k9T-
26、(2 -$3)(27)(28)(29)(30)h these equations V, M, and T represent, respectively,the appliedvertical shear,bending moment, and torqde (about some reference axis)at the rmt of the outer section and the kts represent elastic stiff-nesses of the outer section. The stiffnessfactors kl, , andthe like are fun
27、ctions of the distribution of the applied loads aud thedimensions and material of the outer section,whereas k3 and k7depend only on the latter. The gpautity $2- 3 is a measure of theq= of the root cross section and is the only root distortionappearing in the equations, since the others are rigid-bod
28、y movementswhich do not affect the stress distribution. Thus, effectively,theroot bulkhead is assumed rigid in its own plane as far as the outer-section analysis is concerned.Any method of analysis can be used to determine the stiffnessfactors provided that cross-sectionalwarping and its restraint a
29、re “taken into,account. This provision requires a more refined approachthan is made in elementarybending theory. The stiffness factors arethe same for symmetricaland antisymmetricslloadings but, since bendingand torsion produce different typesof effects, they have been separatedin the equations M or
30、der to evaluate the torque T, the loads must bereferred to a reference axis. The most desirable axis is one which makesthe stresses at the root due to the bending moment M equivalent tothose given by elementary theory, al.$houghit is not generallypossibleto achieve this relationship at all stations.
31、 The so-called “shearcenter” does not locate such an axis. The choice of a reference axiswill be treated at greater length in the section on idealization.Csxry-through section.- The carry-through section, like the outersection, is a box beam that can be analyzed by existing methods. .Inthis case, ho
32、wever, the stress distribution is expressed in terms ofonly the end distortions since internal end forces are the only loadsapplied. The force-displacementrelationships are then:P6 = U(*1 + $3 sfiA + #3 COSA )+kM(*l -*3sfiA:3cosA)+.k13pl + q) + w(kl - G) (31). . . -+-;.-. : . - .:-,. :.,. .-. . . .
33、, - ., “,:,:- ,. ,.,. . .I.,“Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.mm m 2232, 13.p7 = 15(*1 + 3 fi A + 3 cos )+ kl - *3 sti A - 3 COS A) +(32)h7(a + w3) + k18(wI - w3) ,%=%9($1 +$3sh A+3cOs A)+ k20(vl-$3SA-f$3COSA)+%lYs .*I)and they can th
34、en be solvedThe equations obtainedcontainingmany terms whichnumerically for these unlmowns.by direct substitutionhave coefficientsare tediuus to evaluate; however, a nuriber- .- .-. ,=,.-.s -”, .: -. . :7.-,-”- - - -. . . . . . ,-,. - .,. .Provided by IHSNot for ResaleNo reproduction or networking p
35、ermitted without license from IHS-,-,-14 NACA TN 22327of combinationscan be made which substantiallysimpli the finalequations. Equations (1) to (9) are combined as follows to obtain ninesimpler equations: mEl) - (8I =. A - (4) + (5) t- A (36)p)+ (7rJ csc A + (4) + (5) cot A:3) + (6) + (gj(4) ,1(5)%(
36、37)(38)(39).(40)b 6 t-fS=( ) (41)(7) = A + (5) (42)(8) sec A - (5) tan A 43). %(9) sec A (44) ,The resulting system of egyations is written in matrix form asf01.1.ows:% %22 %3 %4 %5 %26 %27 %28 %29a31 a32 a33 a34 a35 a36 a37 a38 a39a4 ah a43 a44 a45 au a4 a48 a49 92 53 “a54 a55 Y6 97 98 suort stiffn
37、essfactorcomponent of loads appliedCalculating.Stresses andto triangular sectionDistortionsthe Idealized StructureThroughoutthe completebox beamThe stress end distortion distributionsforhave been defined in terms of the applied loads and the nine joint dis-placements. Once these joint displacementsh
38、ave been determined bysolving equation (45), the procedures outlinedpreviously can be reversedand all of the forces at the joints can be calculated. 7. = - -. :-; - - - - -.- . .-. . . - . . . . . . . . . . . .-.Provided by IHSNot for ResaleNo reproduction or networking permitted without license fro
39、m IHS-,-,-.-16 NACA TN 2232 .Determination of the detailed distributions is slightlymore com-plicated. For the front spar and the two bulkheads bordering the tris section,the equations of appendti A canbe used. For the outerand carry-through sections,the complete stress distribution can bedetermined
40、 from the analysis that was used to obtain the stiffnessfactors kl to k30. The effects of rigid body motions, which do notaffect the stresses,must be included in the calculation of distortions.The relationshipsbetween the computed stresses in the idealizedstructureandidealization,the actual structur
41、eare discussed in the section onwhich follows.IDEALIZATIONOF AN ACTUAL STRUCTUREOuter and Csrry-Throu SectionsrThe outer and c-through sections are unswept box beems whi that is,equivalent cover is therefore:the.17adding, to thethe moment-carryingarea of theA. + nn(47)(3) camte the moment of inertia
42、 of the equivalent cover abouta vertical axis through its centroid: .(4) The effective area of each front flange is then:(a) For bending stresses:where I is the moment of inertia of the entire cross section aboutthe horizontal.axis of symmetry.(b) For warping stresses:Ib/b2$=-F(48)(49)(50)-, - . - -
43、r . . . ;- = . -. -. . . . . ., . . . . . . . . -,. . .”,. . . . .,. .“Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- - _ . . . .(5) The effective area-of the resx flange in each case is: 1.(51)“/H many equally spaced stringers are used, satisfacto
44、ryresultscan be obtained by treating them as an etguivalentsheet and,thuselhinating theevaluation of lengthy SunImationsa71It is hnportantto note that a different effective area isassociatedwith each type of stress distribution,as should be expected,since each is associated with a different typ of p
45、hysical action;therefore, if accurate results are to be obtained, the two types ofstress distributionsmust be completely separable in the analysis, thatis, they do not appear simtaneously h the evaluation of any onestiffnessfactor in equations (26) to (35). s separation is notgenerallypossible; howe
46、ver, one way to accomplish complete separationin the outer sectionwill be described. Similar considerationsapply i,to the csrry-through section.The outer section is an unswept cantileverbox besm on a flexible.root and the forces on any cross-sectionas given h equations (26)to (30) canbe expressed as
47、 the sum of: (1) forces that exist in theloaded cantilever-ona rigid root and (2) forces that exist in arunloaded cantileverhaving the root warped an amount (2 - 93). Sinceroot warping produces only warping stresses,the effective areas forwarping stresses (equations (50) and (51) are used for the determina-tion of the stiffnessfactors k3 and . The choice of effectiwareas for the anelysis of the loaded cantilever is more difficult becausethe application of vertical loads till, in general, produce both bendingand warping stresses;however, since torque loa
