1、f:,. . . .c. ITIECEEICAL MEMOW7DUliS. .L-.-NATIONAL ADVISORY CObWITTiiE FOR UROHAUTICS. -.- - . .TWISTING FAILURE OF CJHJTRALJiYLOADED 0PE17.SECTIORCOLUMNS HI TEE ELASTI(J RAHGEByRobert KappusLuftfahrtforschuingVolq 14 Ho. 9, Septemer 20, 1937Verlag von R. oldenhourg, liunbhen und Berliri k Washingt
2、onMardh”1938.:-.-.,/1Provided by IHS Not for ResaleNo reproduction or networking permitted without license from IHS-,-,-r ; Iqlgpgipiii1111i“.-_-._ . :MATIOI?AL ADVISORY COMMITTEE POE AERONAUTICS “TEOHITICAL MEMORANDUM HO. 851. -,- * -. - -,.,.TWISTING FAILURE Or OENTEALLY LOADED OPEN-SECTIOM00LUMHS
3、 IllTHE ELASTIC RA19GE*By Robert KappusSUMKARYThe result of the present Investigations is substan-tially as follows. The buckling of centrally loaded col-umns of open section is always accompanied by a twist Ifthe cross section discloses neither axial nor point symme-try. Then the cross sections twi
4、st about an axis of ro- .tation D, the locatlon of which depends upon the shapeof the median line of the section, the wall thickness andcolumn length, and the limiting conditions. There arethree such axes and consequently three different critical”compressive tresses (twisting failure stresses). With
5、point symmetry of the cross section (wherein the case ofdouble-column symmetry IS containod as a spoclal case) thethree critical compressive stresses are given in two Eulerstrossos for (twist-free) buckling in diroctlon of the twoprincipal axes of inertia and one twisting failure stressfor twisting
6、about an axis of rotation passing through thecontor of gravity. With simple cross-section symmetry, itfinally affords one Muler streos for buckling In directionof the axis of symmetry and two twisting failure stressesfor twisting about two axes of rotation in the plane ofsymmetry, Buckling perpendic
7、ular to the axis of symmetryis therefore connected with a twlet of the column, Thethicker the wall and the greater the length of the columns “the more the effect of the twist is neutralized, as thedistanoe between center of rotation and center of gravitycontinues to increase until finally, the obser
8、ved buck-ling Is practically free from twist.This holds for symmetrical as well as for unsymmetri-oa”l sections: the Euler formula gives, .in this case, good(slightly too high) approximate values.*“Drillknicken zentrisch gedr;ckter Stbe mit offenen pro-fil. im”elastischen Bereich.” Luftfahrtforschun
9、g, vol.14, no. 9, September 20, 1937, pP. 444-467. “Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 E.A. C.A. Technical Memorandum no. 851_INTRODUCTIONFor centrally loaded columns of solid cross eectlon”or thick-walled, hollow cross eection, ouly o
10、ne type ofinstability of the straight” equilibrium pattern is rec-ognized Y namely, twist-free buckling, which is usuallytermed flexural buckling or (In tho elastic range) Eulerbucklin. For cross sections of thin-walled open orcloned section and short column length, another type ofinstability in the
11、 original equilibrium pattern Is of i.xn-porteace, namely, the buckling of the less bending-resis-tant walls due to Inferior wall thiclmess. Lastly, thereis yet a third instability phenomenon observed on thin-walled columns of open section - that is, ti=.=lin or, as It is also called, twisti failurQ
12、.under n certain critical compressive force, the stralg;tequilibrium pattern may be accompanied by infinitely ad-jacent tmistod eqnilibriurn patterns whereby, in contrastto thos case of buckling, tho cross-sectional shape is pre-served and the strains due to reciprocal twisting of thecross sections
13、about a =,Tell-defined axis of rotation aregiven.Tho romson for tho Importance of twisting failure bo-Ing rostrictod to columns of thin-walled open section, iqduo to the fact thnt opon sections - espocinlly, whenthin-walled - possess an oxtrenoly low twisting strength.Tvisting fniluro ig most often
14、and most clearly observedIn airplane designs where the employed sections usuallyhave much thinner walls than iS customary on other struc-tures. Aviation literature on this stability problem issubstantially exhausted with a theoretical treatiso by H.Wagner (reference 1), and a further report by H. Wa
15、gnerand V- Pretschner (reference 2) which also contains testdata for angle sections. In structural engineering lit-erature, the twisting failure of centrally loaded columnsdoos not ayoar at all - as far as the writer known - ex-cept for ono recent article by H. and F. Bleich (referenco3). In this re
16、port tho writers use tho energy method forthe derivation of the differential equations for the strainquantities, while Wagner attains a clearer differentialequation for the angle of twist from a consideration ofthe moment equilibrium about the column axis. In spite ofthe fundamentally Identical assu
17、mptions, the results donot agree, and for the following reasons:Wagner defines the axis of rotation, to begin with,on the inconclusive assumption that the center of rotation. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-f l?.A.O.A. Technical Memo
18、randum No. ,851 3on twisting. failure is coincident with the so-oalled “cen-i- t-er of”shear,i.whiLe-H. and Z. .Bleich us,-formula for the1 work of the. external foroes, fails t-oinoitide the t-e”rm which accounts for the twisting of the moss sections.,In the following a omplete theory of twisting f
19、ail-ure hy the energy method IS developed, based on substan-tially the same assumptions aq those” employed by Wagnerand Blelch. Problqms treated. in detail are: the stressand strain condition under St. Venant twist and in twist.with axial oonatralnt: the concept of shear center andthe energy method
20、for problems of elastio stability.10 CENTER OF ROTATION AND WARPING UNDERST. VENA?IT TWIST: SHXAR CENTER M, TWISTINGWITH MKIAL CONSTRAINT3im?e 1 shows the median line of an open sectionwith several essential symbols. A rectangular system ofcoordinates x, y, E is passed through the oentroid Sof one e
21、nd cross section of the correlated column; x, yare arbitrary centroido,l axes of the cross section, and zthh column axis; axes x, y, z aro to form c right-handsyotem; i.e., the z axis in fi-re 1 points toward theobcerver. We arbitrarily fix a direction of rotation andthereby .lloce.te to each point
22、of the section center line .a circumferential coordinate u. Suppose the direction of the positive sense of rotation indicates the positive tan-1gentinl direction t; at right anglqs to it, the posi-tive normal direction n points toward the right, as ob-,served when looklng In t dlrectlon. The distanc
23、es rtmna n of the tangent and of the normal, equal the dis-tance r of the particular.point from the centroid S.1A line drawn from S In posittve n direction indicatestho positive rt direction. This a the x dlrectton!form an cngle a to be measured in”the positive sense ofrotation (rotating from +x tOw
24、ard j+y). If the t di-* rection relativq to % has, sa, a positive sense of ro-tation, rt Is positive according to the preceding notm-tlon. Correspondingly, rn Is cotited positive when nrotates positive in relation to S.Consider a column, as in figure 1, under the effectProvided by IHSNot for ResaleN
25、o reproduction or networking permitted without license from IHS-,-,-14 N.A.C.A. Technical Memorandum No. 851. .,of tvistlng moments T applled at Its two free end crosssections in “conformity vith St. Venantts theory of twist.Jl?hecross-sectional shape is preserved and all stressesand strains other t
26、han the angle of twist cp are unaf-fected by z. *The Ireforredllangle of twist V1 = isconstant. With G = Z7% as shear modulus and JT =“(), U- S3 as twisting strength 1) whereby U Is the devel-oped contour of the center line of the section and s Isthe wall thickness, it is T = GJTql. The linear dis-t
27、ribution of the shearing (twisting) stresses TT overthe mall thickness in any cross section is exactly thesame as in a small rectangular strip. In particular, theshearing stress in the center llne of the section is zero -as a result of which there in no agular change between itand tho elements of th
28、e surface (longitudinal fibers).Eve= lement of the median line of the sectton remainsperpendicular to the correlated fiber, which remainssmm, according to the linearized theory. Then sincethe fibers, depending on their distance from the contor ofrotation, are differently inclined, the cross section
29、doesnot remain flat. If the rotation is, for instance, aboutan axis passing through S, the displacement of a pointin the cross-sectional plane is given with V = r CP (fig.2): te component in tangent direction is accordingly:Vt =rtq (1)There being no anlr change between the circumferentialand the fib
30、er element, the displacement is:(2)where W is the cross-sectional warping (positive In diroction of the poitivc z axis). With the introductionof the unit warning w (of the dimension of an area, fig.1) ESU=UR1) 1For variable wall thickness JT = , r S3duma71.U=uoIProvided by IHSNot for ResaleNo reprod
31、uction or networking permitted without license from IHS-,-,-. . - . -.N.A.GA. Technical Memorandum Ho. 851 5.W=-qfw “ (3). .,-+“ . . . . . ,. -.eqtiatlon (2) (with equation 1) gives - while ob-dervihg “that according to assumption, W,cp, w are to be inde-2) that S: “pendent of z, . rtsuw =Wo+ 1rt du
32、 (4)If the rotation Instead”of about S is mround anyOther point of rotation D with the coordinates xD andyD* through whioh a coordinate sVstem , is placedparcllel to x, y, it correspondingly givesandand(5)ii. -Ps (6)ui= Go +J;t du (7)Figure 2 illustrates te”folloming relations:du cos a = ay, du sin
33、a = - dxIhenceeo =W-wo + %S(7-YO) - ?S(x-xo)I i.eI . :=W+%ay - ;s x+ KI or %= W -XD y+ YD X+ K(8)1.-. b. tric%ly smeaking, these relations are valid fox theucenter line only; w = grt du + rn n is more exact.Since n, at the most, is equal to s/2, the cross-sec-tional warping rn n, superposing themsel
34、ves on the cir-cumferential warping j“ rt du, oan be neglected.o_ -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1 I6 N.A. C.A. !T!echnlcalMemorandum No. 851The unit warping R and w for one and the samesection differ accordingly by one linear expre
35、ssion in xand y. In both caes the same stress condition prevailsand the. reciprocal strains aro complotoly the same. Thodifforonco In warping lies merely in tho relation to dif-forcnt rcforonco pianos, rhich aro porpondiculnr to thomonontary axes of rotation of tho twisted column. Accord-ing to St.
36、Vonant?s theory of twist, tho position of thoaxis of rotation 1s, on principle, undotormlnod because,since r.11 fibers remain straight, each fiber and each onopn.rcllol to it, ma7 serve as a,xis of rotr.tion. Fixingtho c.xis of rotation - say, thrbugh hinges - and prescrib-ing tho ncrping of any.poi
37、nt, establishes the warping ofall points if the load in the end sections is applied inaccord with tho theory. But the axis of rotation canalso %0 cctmblimhod, ns seen from equatLon (9),by pre-scribing the warping of threo (not lying on one straightline) points arbitrarily, bOcaUSe threo of such equa
38、tionssuffice for tho correct olution of tho unknown factors K,XTJ, .nd YD- The froquontly entertained opinion that inSt. Vonantts twist the nXiS of rotation would always haveto pass through ono definite point, the center of shear M,is thoroforo untenable.Since the concopt of shomr center Is bolng us
39、ed re-pe:todly in this report, m brief discussion of tho formu-las defining its position is glvon. Putting c thin-walled,open-section column under transverse load results, in gon-crnl, in twisting in addition to warping. Warping Is notaccompanied by twist (definition of shear center) if thetrc.nsvcr
40、sc force passes through tho center of hear MaAssuming linear distribution of tho bending stresses latnist-frco tending cnd dofi.ning the shearing etresoes dueto trailsvcrso forco in the usl manner from oquillbriuncondftioas, the sttOmeat that in this caso tho transvermoforco rolr.ttm to the shear ce
41、nter may have no momont .leads to theshear cner:gbsequent 5efinition equations for the .1 SW * ii.F I= O and yw* dF = O(lo)“F a71 F- .-.- - -3)For the derivation of these formulas as well as of thOi3eused for integration of arbitrary sections, see the arti-cle ntitled: fShear Center of Thin-Walled S
42、ectione, llbyW. Luckor, published by the Static Test 3ranch of thoD.V.L.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1-ii.A. C.A. Teehnical Memorandum I?o. 851Hereand the unit -wampi, w, rej?erredto shearIM are given with . . ,-f.u*“w = W.*+ rta71
43、 du7throughoenter. .(11) “) whereby rt* equals the die?anoe of the clroumferentlal; tangent from the shear cente (fig. 2). On the basis ofthe coordinate system Z, parallel to x, y placedthrough the arbitrary (to be chosen properly) referenoepoint O, figure 2 indicatesFt = rt* + 5M 00s a + M sln a (1
44、2)Fromfollot7s o(14)The Introduction of (14) in (10) then gives the two equa-tions (i5) for tho coordinates : M of tho shear cen-ter Mh While-no particular importance attaches,to M in theease of St. Venantfs twist, it plays a significant part1“In,tnist with axial constraint, where Q = that is, ifF i
45、s the cross-soctlonnl areaIa /3U* dF = O, J x U* dF = O, ryu*dF=OF “. (18a,b,c)Since ff(!z) is constant over F, and x and y are cen-trotdal axes, the equations (10) follow direct from (18b,c), .which proves the coincidence of center of rotation and cen-t“er of shear In this case. Equation (18a) glve
46、e:J3ff(z) II.Cp F W* dF (19)Complinnco with 1J.4* ,H.”A.C.A. Teohnioal Hemorand Ho 851 9,The shearing btresses T*; follow from the equilibriumIeond,itioa fw. foroosin ,the.,z .d*roctton at a small ele-!Iment s dlldz: . . . . -,“a they are ohtalnodfrom (151), (17), and (23). With complote cozlstrafnt
47、,for example, we have cpi = O: with no constraint, it isP “ = o,2. STRESS AND STRAIN CONDITIONS AT TTISTING FAILURE;ENERGY T3RMS Ai AWD Arw; DIFFERENTIALEQUATIONS FOR THE STRAIN QUTITIES fl, Vss 1?To grin Cn insight into tho forces involvod at twist-ing fc.ilure, considor a centrally loaded open-sec
48、tion col-umn (fig. 1) under the effect of the critical compreReiveforco P. In nontvited condition ench fi%er is under thosamo comproesivc stress; that 1s, in the critical case thetwisting-failure strens is flD= P/F. While D and, con- .sequcntly, the compression of tho column repreoent flnitoquantities, tho strossos nnd strains incurred on transitionfrom tho this is accomplished through.the slope of the longitudinal fibers toward the axis ofrotation at angle :Tg so that the components of the.Provided by IHSNot for ResaleNo re
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