1. Calculation of shear forces and bending moments

Transcription

1 Lectre 16(18). CALCULATION OF SHEAR FORCE AND BENDING MOMENT FOR BIG ASPECT RATIO WING Plan: 1. Calclation of shear forces and bending oents. Diagra of redced oents. Featres of load calclations for sept ing 4. Check of loading calclation. 1. Calclation of shear forces and bending oents In previos lectre e considered calclation of distribted loads. In this lectre e shall consider calclation of shear force and bending oent b ing span. We consider a ing as a cantilever bea. In beginning fnctions shear force Q d ( ) and bending oent M d ( ) fro the total distribted load q t () are fond b ing span. Fro echanic of aterials sie of shear force is eqal to: q ( )d G n q t ci.5 L t ( )d M ci g Q n, (1) L here G ci - eans eight of i-th cargo, M ci is ass of i-th cargo. As o can see e have different forlas for shear force fro distribted and concentrated forces. Bending oent is eqal: M L Q()d () For this prpose integrals are calclated b tablated a ith trapeoids ethod. Yo st deterine orself signs for Q, M according to sign convention fro echanic of aterials (see fig. 1). Q> M> Fig. 1. Sign conventions for shear force Q and bending oent M. Usall shear forces and bending oents are calclated separatel fro distribted and concentrated forces and then e shold sarie the. 1

2 A calclation schee is given in the tab. 1 for shear force and bending oent fro distribted forces. Table 1 The Q d () shear forces and the М d () bending oent are affected b the q t () distribted load. i Δ i i, q t, Q di Q, di, M, di M, di q t ΔQ d Q d ΔM d M d Δ 9 q t9 ΔQ d9 Q d9 ΔM d9 M d Δ 1 q t1 ΔQ d1 Q d1 ΔM d1 M d q t11 First and second colns are reritten fro table 1 fro previos lectre. In third coln of the table 1 e shold copte distance beteen cross sections i fro i-th cross section p to i+1 cross section in eters b the forla: Δi.5(i1 i)l; Δ 11 =, (i =1, , ). We shold calclate fro tenth cross section here i=1, becase 11 =. The forth coln o shold rerite fro previos table 1 last coln. Reslt of previos table is initial data for those calclations. Integration is begn fro a ing tip of a console as for cantilever bea. This integration is ade b to steps. First step is calclation of increent for shear force. Increent of shear force fro distribted loads Q di is fond as area of trapee so: q q t,i t,i1 Q, Q di i d11 =, (i =1, , ), () Those vales e shold rite don in fifth coln. A second step is calclation of shear force fro distribted loads as area of fnction q t =f(): Q Q Q, Q di di 1 di d11 = ; (i = 1, , ), (4) here Q d11 = is shear force in cross-section nber 11 fro distribted loads in a tip ing. Q d1 - is shear force in cross-section nber 1 fro distribted loads on site beteen 1 and 11 cross-sections and so on. Those vales e shold rite don in the sixth coln. B the sae ethodic e shold calclate bending oent fro distribted loads:

3 , M d11 =, (i =1, 9 1, ), M d11 = ; (i = 1, 9 1, ) (5) here ΔM d11 = - is increent of bending oent in cross-section nber 11 fro distribted loads in tip ing, ΔM d1 - is increent of bending oent in cross-section nber 1 fro distribted loads on site beteen 1 and 11 cross-sections and so on, hich e shold rite don in seventh coln; M d11 = is bending oent in cross-section nber 11 fro distribted loads in tip ing; M d1 - is bending oent in cross-section nber 1 fro distribted loads on site beteen 1 and 11 cross-sections and so on, hich e shold rite don in eighth coln. It also is necessar to reslt shear forces and bending oents affected b P,agr concentrated ass forces (in the sae coordinate sstes that Q d and M d and in the sae scale of diagras). Hoever the sign of these diagras is opposite to sign Q d and M d. On fig. diagra Q c fro concentrated forces is shon (table ) and on fig. is shon M c. To concentrated ass forces o st inclde all aggregates of ing engines, landing gears, fel tanks and so on. Ultiate loads fro aggregates, as for concentrated loads are eqal:, =n G agr,i =n M agr,i g= Q ic (6) here G agr,i is aggregate eight in i-th cross section, M agr,i - is aggregate ass in i-th cross section, i is nber of cross section, g is gravit acceleration. Yo shold copte these vales onl for cross sections ith aggregates, in an cross sections o shold rite don. Calclation schee is given in the tab., hich incldes folloing vales: Q ic =, fro (6) here i - is nber of cross section in hich this nit is placed; in an cross sections Q ic =. Yo shold rite don this vale in forth coln. In table for exaple concentrated force is given onl in cross section i= 9. Yo can rerite colns 1, and fro previos table. Sation of the concentrated cargoes is condcted fro the tip of ing to cross section ith coordinate. In practice there are carr ot nerical integration, procedre consists in the folloing. On the ing tip in the cross section N=11 e have 11. In this forla it takes into accont a cargo is located on i-th site., (i = 1, , ), (7) here Q c11 = is shear force in cross-section nber 11 fro concentrated loads in the tip ing. Q c1 - is shear force in the cross-section nber 1 fro concentrated loads. For exaple e have concentrated load in cross section nber 9. In this case Q c9 - is shear force in cross-section nber 9 fro concentrated loads hich has jp in this crosssection and to vales one previos vale - and ne vale ΔQ c9 and so on. In cross sections ith concentrated load e have jp of shear force and to vales of the. Yo shold rite don this vale in fifth coln. Bending oent fro concentrated loads is received as reslt of integration shear forces fro concentrated loads, integration condct fro the ing tip as for cantilever bea:

4 4 M с Qсd. (8) Integration is carried ot nericall, tablated a b the ethod of trapees. On a tip of a console ing, as it is knon: M t =M t11 =ΔM t11 =. Table The Q ic () shear forces and the М ic () bending oent are affected b concentrated loads. i, Q iс, i Δ i. Q iс M iс, M iс, Q ΔM c с M с Q 7c =Q 9c 8... Q 8c =Q 9c ΔM 8c ΔM 8c =M 8c 9.9 Δ 9 Q 9 Q с 9с =Q 9 с / 1.95 Δ In a cross-section nber i an increent of bending oent is eqal b the ethod of trapees:, M 11c =, (i =1, , ), (9) here ΔM c,11 = - is increent of bending oent in cross-section nber 11 fro concentrated loads ot tip ing. ΔM c,1 - is increent of bending oent in cross-section nber 1 fro concentrated loads in cross section 11 and so on. Bending oent fro concentrated loads is eqal:, M c,11 = ; (i = 1, , ) (1) M c,11 = is bending oent in the cross-section nber 11 fro concentrated loads ot tip ing. M c,1 - is bending oent in cross-section nber 1 fro concentrated loads in cross section 11 and so on. Yo st kno that increent of bending oent fro concentrated force and bending oent fro concentrated force o can calclate for next cross-section ith nber i-1=8 in or exaple see fig. and table. Folding appropriate diagras algebraicall (table ), e shold plot total diagras Q t and M t (on fig. are shon b continos lines). The calclation schee is given in a tab., hich incldes folloing vales: Q di - is shear force fro distribted loads fro table 1; Q ci - is shear force fro concentrated loads fro table ;

5 Q t,i = Q di + Q ci = Q c,i - Q d,i ith accont signs is total shear force; M di - is bending oent fro distribted loads fro table 1; M ci - is bending oent fro concentrated loads fro table ; M t,i = M di + M ci = M di - M ci ith accont signs - is total bending oent. Table Total Q t () shear forces and total М t,i () bending oent are affected b all forces. i Q di, Q ci, Q t,i, M d,i, * M c,i, KN* M t,i, On diagra of shear forces fro concentrated loads and the total shear forces e have discontinos jp in places of application for concentrated forces and have to vales shear force before and after application of concentrated force.. Diagra of redced oents For strength analsis of a ing it is necessar to find position of shear force in cross section. So e plot the diagra of the redced oents that is the oents of shear force rather an a chosen axis. Knoledge of a diagra of redced oents allos finding a point of attack for shear force in ing cross sections. This coordinate is sed at designing and checking calclations. For realiation of calclations e st do a draing of a ing. As an axis of redction it is convenient to choose a straight line taking begin at a point of intersection of a ing leading edge ith an axes of a fselage and perpendiclarl to axes of a fselage (see fig.). On the draing e indicate position of centers of pressre, centers of gravit of a ing cross sections and fel tanks, e shon centers of gravit for concentrated cargoes. Let's consider an cross section ith coordinate. We condct calclation fro a tip of a ing console. In each cross section b the draing e find distances fro the axis of redction to points of the attack for distribted and concentrated forces х a, х W, х f and х c (see fig.5). 5

6 engine fel tank front spar leading edge rredced axis b r b t L / rear spar rear spar q center of gravit q t q a q f Z Q Z Q c Q t Q d Fig.. Diagras of distribted loads and shear forces 6

7 M c Z M tot M d M,, Z M, M, c Z M,t M, d Fig.. Diagras of bending oent M and redced oent M. 7

8 Не удается отобразить связанный рисунок. Возможно, этот файл был перемещен, переименован или удален. Убедит есь, чт о ссылка указывает на правильный файл и верное размещение. Calclation schee for redced oents fro distribted loads Table 4. i Δ i, q a / x a q / x q f / x f i ΔM di M di q a x a q x q f x f ΔM d M d Δ 1 q a1 x a1 q 1 x 1 q f1 x f1 1 ΔM d1 M d1 11 Δ 11 q a11 x a11 q 11 x 11 q f11 x f11 11 We shold plot diagra of distribted redced oent. For this prpose at beginning crrent distribted redced oent in each ing cross section is calclated b the forla: = - q a x a + q x +q f x f (11) A total redced oent M is eqal: M ( )d G n x ( )d g n x, (1) ci ci here x ci is distance fro i-th concentrated cargo p to redced axis. The redced oent is considered like positive if it acts to pitching relative to the redced axis. Integrating the diagra e receive the redced oents M d affected b the distribted loads. A schee of calclation is shon in tab. 4 in hich designations is entered: M di.5( i 1,i ) i, M,d,11 M,d, 11 ; (1), (i = 1, 9..., ). At ing tip e have M, M d1 = M d1. M d 11 d 11 Also it is necessar to copte and to plot the diagra of the redced oents affected b concentrated asses (on fig. it is shon b the light line). Affected b concentrated ass of i-th aggregate increent of oent Δ M,c,i is fond ot b the forla: ΔM,c,i =, r i =M agr,i g r i, (14) ci ci 8

9 here the r i is a distance fro i-th concentrated ass gravit center to redced axis (it is easred on the draing)., is ltiate inertia force b forla (6). This increent e have onl in point here e have aggregates. In an points this increent is eqal ero. Redced oent M.c.i is calclated b the forla: ΔM,c,11 M,c,11 ; M,c,i M,c,i1 ΔM,c,i. (i = 1, 9..., ). (15) In a point ith aggregate e have jp of redced oent (see fig. ). For this table e take M di fro table 4 and total redced oent o shold copte ith accont of signs b the forla: M,t = M,d + M,c (16) A cg of cargo redced axis cp cg cg of fel A Fig. 4. Calclation of redced oents Distance fro the axis of redction to a point of attack of a resltant force X a,i in a cross section ith coordinate eqal to: M ti Х a,i (17) Qti It is necessar coordinate position of shear force in designing cross section fro a leading edge. Distance fro a leading edge p to a point of attack for a resltant force d e take fro the draing. Calclation of the total redced oents is carr ot in the table 5 b the ethod of trapees. It also is necessar to plot the M,t total redced oent diagra (on fig it is shon b a solid line). 9

10 ra x a q a cg of fel cg of ing x f x Fig. 5. Calclation of crrent redced oents Table 5 Calclation schee of redced oent fro concentrated loads and fro all loads. I,.i r i ΔM.c.i M.c.i M,d,i M,t,i * * * * X r, M ra Fig. 6. Calclation of a point of attack for resltant force 1

11 . Featres of load calclations for sept ing For sept ing his longitdinal axis is not perpendiclar axes of a plane, and is rejected back b flight. We dra a straight line taking place throgh.5 chords and e estiate a seep angle b this line for accont of seep back fro a draing of a ing. Frther e calclate an aendent b seep back: Г Г45 45 L c Fig. 7. Calclation settleent seep. We dra a diagra: t f, here circlation of a straight flat ing Г f e calclate for straightened sept ing. With this prpose e dra so-called a straightened sept ing concerning a line hich are taking place throgh.4b() chords that area st be eqal to the area of a sept ing becase approxiatel rigidit axes is placed in this position.4b(). Then Г f e find ot b the techniqe described above for a straight flat ing in vie of aspect ratio and taper for the straightened ing. At frther diagra is dran for the straightened ing, as ell as for straight flat ing. 4. Check of load calclation For check of load calclation an approached calclation of loads and coparison ith earlier received reslts st be carried ot. Fro previos lectre o kno that approxiatel aerodnaic force can be distribted proportionall to chords: 1.5n G 1.5n Mg qa b( ) b( ) (18) S S, ith accont load on stabilier. Sies of ing chords can be calclated b linear interpolation b the forla: 11

12 b b b( ) b r t r (19) Lc here в() is a crrent chord, в r a root chord, b t - a tip chord, L c a length of a ing console, - coordinate of cross section. We cont fel as a concentrated cargo, and then a total distribted loading q t eqal to: n (`1.5G G ) n g( 1.5M M ) p p qt b( ) b( ) () S S here G p eight of a plane, G is a ing eight, M p, M are asses of plane and ing accordingl. Shear force can be calclated b integration fro a ing tip as for a cantilever bea: q d G n q d t ci t l Q gm fn ; (1) ci C S c Centre of gravit b b() b t 1 L c Fig. 6. Circit of approached calclation of loading We st sbstitte q t fro previos lectre: n (G G ) p Q Sc Gcin, () S here S c b( ) d () - is an area of a ing copartent fro considered cross section p to a ing tip. At point =L c in root cross section Q th is eqal to: n ( 1.5G G ) n з Q th Gcin, (4) k 1

13 here n is qantit of concentrated cargoes, G p, G - eight of a plane and eight of a ing, inclding to consoles. Approxiatel bending oent in a cross section is eqal to: Sc M ар n ( (G G )C GciC i ) (5) S i here C i - distance fro a crrent cross-section to a center of gravit of a concentrated cargo, С - distance fro a crrent cross-section to a center of gravit of a ing copartent, hich is eqal: С b( ) bt b( ) bt, as center of gravit for trape. Ths those cargoes hich are located p to section ith coordinate shold be taken into accont onl fro a ing tip. Miscalclation for shear force st be no ore 1% or.1 b forlas: Qta Qth δq.1 Qth here Q ta is vale of shear force fro table, Q th theoretical vale of shear force fro (4). Miscalclation for bending oent st be no ore 1% or.1 b forlas: δm M ta Map M ta.1 here M ta is vale of bending oent fro table, M ap vale of bending oent fro (5). D:\Документы\Users\kir\SA18\T6DesStrnAn\L16(18)ABendMo 1