1、. “ rf tl “ “p_-/ c;Ii2_CHxCZCA:LIfl_NORANDUMNO. 806 FLAT SHEET .t_TAL GIRDERS _TtTH VERy THIN LIETAL 1,7_*%By Herbert WagnerJi PART ZZZi Sheel _eta! Girders Witli Spars _esistant to Bending _; The “Stress in Uprlghts - Diagofial Tension FieldsLThe actually occurring form change is of Cour.se not id
2、en- ttical w“ I,_.mtn our arbitrarily assumed one TL,.e principal differ- i;il, ence l:lay!oethe change in specific numlber of wrinkles from di- Fii!-o rectlon x to z, so that b and, according to equation 29b) _,(See Part Z - Technical ie:norandumNo. 606, page 32), f, be- _:Come variable in directio
3、n z. l_lereover,it seems likely thatb and f increase from the edge toward the cen_er if the sheet +is infinitely thin.The work of deformation A, actually produced is, however,certainly less (equal in the li2:itingcase) than in that arbi-trarily asstuned, and r,_usttherefore (Cempare eq. 31, Technica
4、l. _emor_ndum No. 605, page 36), equal Amin. But from this itfollows (eq. 31a) that this actual deformation is Under constant_ tension stress a Az = _-E := E and that there _.reno additionalstresses of finite magnitude i_ “_.-,._ ,% t_llS zone“ “Ebene _,.,_,-,_iC_- _ _ , that ms,-.,-_,.,_anc_t,;, _,
5、 , .- -_ -C_qk andsch “. , . _zager mit sent . , “ _-rzft zu_ Flu_,+_.-., :. dun,e,_SteO thezc cq_a, tions _!_e c.,o,_plie,d wit: whenr“_ I) every ratio o.f. _, whicl_ oa, tisfJ, cr, (_“ k_.lconformable re,“ every finite _ for lr_:_t;,uyu:eProvided by IHSNot for ResaleNo reproduction or networking p
6、ermitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-%N.A.C.A. Technical Memorandum I_o.80( 5 ,edges than in the middle. In this way the depth of the wrinkles decreases toward the edge,/without_effecting_a .like incre
7、aseIn !“_, cross stress _ ._: oqk. It is only at the cflge itself that the _i stresses parallel to the spar (aside from o sin_)_ _ equal thamstress in the spar. j:,$;. Determination of _“ “ ii_xper imcnta.lError Now we figure back from the deformations determined by . test, in order, to see.,the ext
8、ent of applicability of our as- _!_;d_ _ sumption aq = 0 to practical cases; then wc comnute %. _,_ From (Fig. 7 of my report given at Danzig, Jahrbuch 1928, i“!_,i, tier Wissenschaftlichen Gesellschaft f_zLuftf_d_rt, p 115,) ?. the w- _a_n b of a wr“ _Inic.LeWith respect to its length I andgirder -
9、“ ?_nclght h can be measured quite accurately. In the _left panel we have _i= 18 (=. .en_,th,- of wrinkle from ul)zi,.,._t ._. to upri!_;ht), i_._,. b=2.8 t s/b = 0.0089s :=0.0215 “JIn addition:E = 700,000 kg/cm _= 2,4-00 “_. consequent.y = 0.0034ii):. Oy = - I,600 “ “ Cy 4) 00,23!:i x. = - ,500 “ ,
10、c x -.,.:-0.0()044.2%Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I _,) the contract/on -q im the midd.e ran_):je o the nhee, tmu_t he)v,“difSer from the original -qo“ This meanm lo_;er e,!;v_ be,._u_o _,-,ero the fiber_ are /a,rth(;r a p), that i
11、nequation (2) is s.a.ti_iod at “ “_in th_ . “ _ “_ ,.,; 5ted lt./llf_ _ ZlZOII!q(I _ iLL w_ich ,.;,:_e ()fir as,_umotio_swodld i:_u,._/e to be di:_r_.ga.rd:.d, for otn_ mor com.:)li_a.ted one_,But in ou.m iurth_r di,_,oL_;J, on we oonsider (L) to be e,:mp.l.i,_d, v_.i,thF:i.(_._.re 36 show;:_ o _;on
12、stant _md -_ va.:riable a._._.n-._-“_, th,qwrink.e,s_ c,)ns_-_,quently the depth with r . p(,(,$ to the wid.bh othe wrinkle, must vary ator.!._; its length accordinf: to (29b) (see,( Part I, ,T.A.C.A. _ ;_tl_Ii,“,_fl Iqe,_lor_md_u, No. t;,05), th_:,_ wrink,l.cdsh(_. , _:l.,.:r:Tt_,“,. , _I_, (I_ V(_
13、LO)_T_“.(_; l_l.l_:t_O ,l_i“_ .t d1:IS“ c 1_, ., “C:that t!.(; LJ!(;Ot W_/_l.d. re;._i, ni, tni, type, o% _v: li _:_: l._ .l . be proved that in an infinite_ly thin _,he,t with v_:ry (L:_:iin point ,_and_m_ = Cm + 8nm dnm (34)in point i_.First we define the width dn of the tension diagonal atz (poin
14、t P). We obto.in il8_ recurs many times in the subsequent .cal-Since the term z 8-_mculation, we abbreviated it to r, so that8_r = z 8n-m (36). where, of course, we ;,lustalws,Fs.bear in ;_indin tho _mb_e_,_,_.:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from
15、 IHS-,-,-18 I,.A.C.,A.Technical llemorandum No. 606 differentiations and integrations, that r is a function of z. iConsequently, equation (35)becomes nowdn = dnm C1- r) (35a)iThen we calculate thestress attitude of the tension diag- _o.i onal. In conformity with the f_reedom from sources in the fiel
16、dof the principal stresses (See Part I - Technical l,emorandum No.604, theorem 2, page l!)- Cm dnm = a dn (37)is valid, that is, wltll (35a):1 (37a)“ _ =_ml_r8_Now we confute .-, so that (38) yields: i:_r_= _m Oz _ 1 )_8z (i - r)z = _m nm (1 - r In particular, we obtain with z = 0 and r = 0 for poin
17、t _ iiliz=o= %1_nmThen the stress becomes ,.“iii) O:L= e + _C._d n8n ii;in point P_. Taking into account the infinitely small term.;of the first order, we obtaini oldn = Om, dnm8aThen we make c_ - _ = _-_dn (Fig. 38), ,thatis, with the pre-Provided by IHSNot for ResaleNo reproduction or networking p
18、ermitted without license from IHS-,-,-11 , % %-. I;.AC A Tcc,mloal _bmorandum _o.606 17ceding equation8(_ dn = _z G - o_n - = Cm_-!i(_ and obtain with equations (34), (37) and (35a):8n 8n_._(1 - r)_ (os) ,.i_Jq_ t_! Now o and its derivations oonfo2ma, to z and n are known at _Ii;_;every ooint of the
19、 tension diagonal. _“ With -:(Tc = _ , (39)E_ i!.ithat is, partioulctrly wi_en, for example, i_m _ca = -E- etc., (39a) iit becomes apparent that all these equations are applicable to ;the elongations, providing we write e instea,d of o.Now we calculate the form into which the ori_:inallystraight _In
20、 Fib_urc 43 the center axis of the gi%der is denoted byi !:xm. Axes xo and xu a,rcparallel to xm and h/2 di._tantthan where the spars arc to be. Stres._es and deformations are ,indicg_tcd by subscripts o, u, and m.:Zow,first, we assume the intensity and the direction ofthe principal tension stress q
21、 and its ensuing elongation as known; that is, om and _m“ This gives us the derivations of .: da dc da _ .these quantities accord:r.g to Xm (n_une:ly, d_xm :-z,:d Xml “No:“it follows from FiF_re 44 that d nm = d Xm sin a. :_With 8_a = O, we h_,ve i_:!_z.88.:.= I d._ (47)8 nm sin _ d xmRThe elongatio
22、n in direction z at point _ is l(iid xm Cm+ ifm+ d xm+ 8_: d sin a.+ _c_ d cos awith equations (37b) and (47),Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-J _ IT.A.O.A Teohn1_al Mor_lorandumNo. 506 25:! 8c I ck cos_ da - . Cm (48)8nm sinc_ d xm si
23、rf_ d xmNow we know ._.g_ and ._ with resoeot to c m and a, thereby8 nm 8ninthe whole staess a.nddeformation attitude of the diagonal ten-!. sion field (aside from the tro.usver_;econtraction - Cq). In par-i tioular, we are given the _t:LeSS_nd. the deformation in 0 and U, _ - ,but it must be emphas
24、ized that the values of Co, Cu, etc., oal- !:i!,eulated for cry. and a at xm are at _“xo = Xr_+ h oot_I“ h (49)xU xm - _-cot aWe designate by X_,_,Xo, and Xu the displacements of thepoirts - origina.lyon axes Xm, Xo, and xu - perpendicul_ tothe direction of these a.nes, By virtue of the different el
25、onga-tions Cxo and c_:u the girder is bent (Fig. 43) at a contin- L!gent angle _!_“_;co - cx_, d_ d? _ d;X i_i :. _m_.-_0 _V i,_h dx2 dxo _“ axe U i:_ii ,!With _o a.-_d Zu as the 4eflections of the spars between two up-rights, _:_easured f-_om the oileular are with above curvature (Fig.43), we have:
26、d.0 dX0 c._ -_._ = ct,._O+ Xo o CXu-J- h. (50)Then we rlefine (Sue Figs. 42 u.nd 4.3)angle _, at _,_hic_ the “_e;,_-. nearing line OU of the tennion diagonal through _r is dis-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for Res
27、aleNo reproduction or networking permitted without license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-28 :.A.C.A. Technical l_lemoraudtumI1o.806 :l“r, speot to xm and the spar deflections. If, in addition, theielastic lines of the s
28、pars are Given and it is desired to findtl_eproduced diagonal tensionfield, the two equations above -(53) and (Z3a) - together with (49), suffice for conlputing c:_I and _ with respect to Xm.Lastly, if the elastic lines of the spars are not givenJI but are to be defined conformal to stresses PYo and
29、 PYu(Conrpa_,ePart I - Technical Memorandum No. 604 - page 23, andFigure 9, for example, PYu = s sin_ _ (_u= s sin_ a E ().n1 +r_/we aoply in addition to (53), (53a) and (49) the two differen-tial equations for the elastic lines of the sparsd_r_ _ s sin2 (_ Cm(l - ) Jo (54a)id4_U s sin 2 c_ cmdxu-C
30、= + (-1+ rl) JU (54.)(Jo, JU = inertia moments of the spar cross sections; s =sheet ti:.ickness.)Altogether, ae have now 7 equations for xO, Xu, r_,cm, _:._a, _0 and rjU.For later purposes, we subtract the two equations (53), and ihave : Jd _0 dDU h d: 1 Ld-x0 dxU - sin2 c_ dxm 1 - r_ (53b)I have no
31、t yct carz-ied through the ntuucrical cv_lu;,L_ion:_iProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-v “,.A.C.A.Technical Memorandttm No. 606 29 of equations (53) etc.* We have yet to consider the case where a and _m are constant. Then c = cm becomes
32、 constant in thewhole range of the sheet; further, d-_m = 0 and r_ = O, which,writ.ten in (53) and added becomes:+1,_O+( =-? - 2 cots ,. + i2 dxuJ 2 !i!+ 2 cot a = constsmt (55a)When subtracted,- I = o (55b)“ For two linear differential equations of the first orderwe usually apply two limit equation
33、s. But equations (53) for_0 and _U already satisfy one limit equation ina.smuch as wewrote the value for % into (52) according to (40a) instead oferpressing it generally, by. Cxo, CXu,k, av and _o, Du“ Theother limiting condition is expressed by observing the connec-tion (460) or (46d) and complying
34、 with (.6a)a“, (46b) at one i:_place. In o,:.=case of (55a), (55b) it yields: _id_0 d_U _“*!f, by given elastic lines of the spars, thesedeflections _o iii!:;iand Dr. are small enough so that the stress fluctuations- /do h _m/ and the variations in direction of wrinkling (r_) become _i_(infinitely)
35、small qu_ntities, then _o and _u as well as cTmand a with respect to xm ;.z_.ybe CXOiessed Ly Fouricr ;,:_“i_:-_which inserted in the two equa.tions( 53) yield the :i.nd,t, ._“_,t-,“ L. 1_.,_._ , .t.coefficlr.;nt;, of the series for o_ a.nd a _ T:,.(.,result i, _;_rstress distzibution, i!h_tthe m_:_
36、t_cm dxu O; = 2 cot a_ - Cxo + Cxg_, _, 2 j (550) ,This equation (55c) (Compare eau:_tion (6), - Part I (Technica.li:emorandum i_o. 604, page 22) proves the accuracy of our str.Lte-ment made at the end.of an earlier section (Part I, page 29), _:nmnely, that _ depends on the mean elongation of both s
37、parsonly.In particular, we want to point out that, through (53) and _.:_ (55), spars which by the deformation reraain straight and paral-lel, are given constant wrinkling direction, which in previoussections we had assumed as obvious (See Part I, pages 16 and 23).Sheet Metal Girder with Spars not Ri
38、gid in Bending;Simplifying Assumptions :_Again weassume the angle 0f displacement _, the elonga- :tion Cx0 and CXu of the .two spars and the elongation Cv of !:the uprights as given.ca_se. :What are the results if the direction of the _rinkles isconstant (oc = constant)? I_oting (37a) (0 = _U - m-“ a)equation (53b) becomes:dx0 dxU E sin2 a dx_n,we integrate with (49) asProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
