Finding a Funicular Curve Through Two Points

Transcription

1 This is th glss pyrmi t th Louvr Musum in Pris, sign y rhitt I.M. Pi. It is support from nth y stl ls. In signing strutur suh s this, it is oftn most usful to slt l of rtin siz n tnsil strngth, n thn to fin shp for it tht will utiliz fully th givn tnsil strngth. In this lsson w will lrn to fin th form for l or rh tht psss through ny two points n xprins signt mximum tnsil or omprssiv for. To pro with this lsson, lik on th Nxt utton hr or t th top of ny pg. Whn you r on with this lsson, lik on th Contnts utton hr or t th top of ny pg to rturn to th list of lssons.

2 50 ft Form Digrm 20ft 20ft 20ft 90 ft 190 ft spn Cl 60 ft Th Prolm: Suppos tht w r signing l to support th rig of n xhiition tnt. Th tnt xrts four los on th l of 1200 l h. Although th los r sp t 20-foot intrvls, th ns of th l r 190 ft prt t lotions n, n thir vrtil lvtions iffr y 10 ft. Th mximum sf tnsil for in th l is 6600 l. W must fin th form for th l tht mts ths ritri.

3 Stp 1: St up th solution. W gin our work y onstruting th Loing Digrm. Th igrm is pl nr th top of th pg to lv room for th grphil onstrutions low.

4 A W pply intrvl nottion to th. Bginning t th lft, w pl pitl lttrs in th intrvls twn th los.

5 Lo DE Working from lft to right, w ontinu lling th intrvls twn h pir of fors. Los r nm y th lttrs on ithr si. For xmpl, th rightmost lo is nm DE.

6 W xtn vrtil lins of tion from th four lo vtors ownwr on th pg. rtil lins r xtn from th n points of th s wll.

7 Th lotions of th npoints of th l, n, r pl towr th ottom of th igrm, lving sp ov for onstrution of Tril Funiulr Polygon. Th rltiv iffrn in hight twn n is mintin. 10 ft

8 Stp 2: Construt th Lo Lin. Nxt w onstrut Lo Lin to ny sl tht fits omfortly on th pg. Th Lo Lin is tip-to-til ition of th los ting on th strutur. Lowr s lttrs on th Lo Lin orrspon to th pitl lttrs on th. W gin y plotting, vtor of lngth 1200 l, on th Lo Lin, orrsponing to for AB on th Loing Digrm l Lo Lin 2000 l

9 A B C D E Working from lft to right, w ontinu to plot th los from th onto th Lo Lin. Lo Lin 2000 l

10 A B C D E Lo Lin 2000 l

11 A B C D E On ll th los hv n plott, th ovrll lngth of th Lo Lin sls to 4800 l, th totl lo on th strutur. 0 l Lo Lin 2000 l

12 Stp 3: Construt th Tril Funiulr Polygon. Th solution pros in two stps, Tril Stp n Finl Stp. For th Tril, w slt tril pol lotion,, ritrrily. Th Lo Lin now oms prt of Tril For Polygon. Tril pol lotion Tril For Polygon 2000 l

13 W onstrut st of rys, through, on th Tril For Polygon. Tril For Polygon 2000 l

14 Prlll to h ry on th Tril For Polygon, w rw th orrsponing sgmnt of th Tril Funiulr Polygon. This is onstrut in th mpty sp twn th n th Finl Funiulr Digrm. Sgmnt lis low intrvl A of th Loing Digrm. Tril Funiulr Polygon Tril For Polygon 2000 l

15 A B C D E Sgmnt lis low intrvl B of th Loing Digrm. Tril Funiulr Polygon Tril For Polygon 2000 l

16 By onstruting sgmnts,, n, w omplt th Tril Funiulr Polygon. Tril Funiulr Polygon Tril For Polygon 2000 l

17 ' Th ns of th Tril Funiulr Polygon r ll ' n '. ' n ' lign vrtilly with n points n of th Finl Funiulr Polygon. ' Tril Funiulr Polygon Tril For Polygon 2000 l

18 ' ' Stp 4: Fin th Finl Pol Now w r in position to fin th pol lotion tht will gnrt th Finl Funiulr Polygon tht psss through n with mximum intrnl for of 6600 l. First w rw th losing string, of th Tril Funiulr Polygon. Tril Funiulr Polygon Tril For Polygon 2000 l

19 ' Prlll to losing string, w rw ry through tril pol on th Tril For Polygon. This ry intrsts th Lo Lin t point q. ' Tril Funiulr Polygon Point q q Tril For Polygon 2000 l

20 ' Tril Funiulr Polygon ' For ny polygon tht is funiulr for this loing pttrn, ry prlll to its losing string must pss through point q. Thus, th ry prlll to losing string of th Finl Funiulr Polygon must pss through q. W n us this ft to lot th pol of th Finl For Polygon, n from thr, to onstrut th Finl Funiulr Polygon. q Tril For Polygon 2000 l

21 ' ' W onstrut lin twn n, th ns of th Finl Funiulr Polygon tht w sk. Th slop of is trmin y th rltiv hights of points n stlish t th ginning of th prolm. will th losing string of this Funiulr Polygon. Tril Funiulr Polygon q Tril For Polygon 2000 l

22 ' Prlll to losing string, w rw ry through point q on th Lo Lin. Th finl pol must li somwhr on this lin. ' Tril Funiulr Polygon Th finl pol must li somwhr long this lin q Tril For Polygon 2000 l

23 ' Th mximum for in th l ws spifi s 6600 l. This mns tht th longst ry on th Finl For Polygon must 6600 l long. Th longst ry will ithr o or o, it is not yt lr whih. ' Tril Funiulr Polygon Rys o n o will intrst t finl pol o, whih lis sompl on ry. Nithr n x 6600 l in lngth. q Tril For Polygon 2000 l

24 ' W strik n r of rius 6600 l from to intrst ry, n nothr r of 6600 l rius from. W opt th losr of ths two intrstions to th Lo Lin s th finl pol, o. ' Tril Funiulr Polygon q rius = 6600 l Finl Pol o rius = 6600 l Tril For Polygon 2000 l

25 ' In this s, o is lot t th intrstion with ry of n r out point whos rius is 6600 l. This mns tht ry o will 6600 l long, n ry o slightly lss. ' Tril Funiulr Polygon o rius = 6600 l q o o Tril For Polygon 2000 l

26 ' Stp 5: Construt th Finl Funiulr Polygon. Now w r l to omplt th Finl For Polygon. From Finl Pol o w onstrut st of rys. ' Tril Funiulr Polygon q o o o o o o Finl For Polygon 2000 l

27 A B C D E ' Prlll to h ry, w rw th orrsponing sgmnt of th Finl Funiulr Polygon. Sgmnt o flls in th intrvl low lttr A on th. ' Tril Funiulr Polygon o Finl Funiulr Polygon q o o o o o o Finl For Polygon 2000 l

28 A B C D E ' Sgmnt o flls in th intrvl low lttr B on th. ' Tril Funiulr Polygon o o Finl Funiulr Polygon q o o o o o o Finl For Polygon 2000 l

29 ' Continuing from lft to right n top to ottom, w omplt th Finl Funiulr Polygon. ' Tril Funiulr Polygon o o Finl Funiulr Polygon o q o o o o o o Finl For Polygon 2000 l

30 ' ' Tril Funiulr Polygon o o Finl Funiulr Polygon o o q o o o o o o Finl For Polygon 2000 l

31 ' Sgmnt o loss th Finl Funiulr Polygon t point, vrifying th ury of our onstrution. ' Tril Funiulr Polygon o o Finl Funiulr Polygon o o o q o o o o o o Finl For Polygon 2000 l

32 ' ' W my sl h ry on th Finl For Polygon to trmin th for in th orrsponing sgmnts of th l. Noti tht th highst for is in in sgmnt o, n tht th mximum for in th l is limit to 6600 l s sir. Tril Funiulr Polygon o 6600 l o 6200 l Finl Funiulr Polygon o 6020 l o 6080 l o 6370 l q o o 0 l o o o o Finl For Polygon 2000 l

33 ' Th Finl Funiulr Polygon is now omplt. It is th shp tht th l will tk twn support points n whil xprining mximum tnsion of 6600 l. ' Tril Funiulr Polygon o 6600 l o 6200 l Finl Funiulr Polygon o 6020 l o 6080 l o 6370 l q o o o o o Mximum for = 6600 l o Finl For Polygon 2000 l

34 This is th omplt strutur with th form of th l tht w hv trmin. Now tht w hv trmin th form n fors in th min l, w will rifly invstigt th fors in th l towrs n kstys. Form Digrm o Finl For Polygon 2000 l

35 A B C D E Stp 6: Invstigt th towr n ksty fors. W gin y onstruting fr oy igrm of th l. Th fr oy igrm shows ll fors ting on th l, inluing th fors xrt on th l y th towrs n kstys. Form Digrm o Finl For Polygon 2000 l

36 g G og A B C O D E O is th whol sp twn th towrs of f F W us intrvl nottion to kp trk of th fors. Noti tht pol o on th For Polygon orrspons to th intrvl low th min l on th Form Digrm. W will now th towr n ksty fors to th For Polygon. Form Digrm o Finl For Polygon 2000 l

37 g g G og of O f F For in ksty g g og For in towr og Form Digrm o o o o o o f For in ksty f Complting th For Polygon, w isovr tht th fors in th towrs n kstys r quit lrg in omprison to thos in th min l. Evn whn th sl of th for polygon is ru y hlf, w n no longr fit th ntir igrm on th pg! Finl For Polygon 4000 l of For in towr of

38 g g G og of O f F For in ksty g g og For in towr og Form Digrm Chnging th slop of g n f movs thir intrstion with og n of so tht ll of ths vtors om shortr. o o o o o o Th r lins on th For Polygon tll us tht w oul ru th fors in th olumns n kstys y ruing th slop of th kstys. f For in ksty f Finl For Polygon 4000 l of For in towr of

39 g A Form Digrm B C D G og of O E F f This For Polygon shows th rsults of this hng. Dos it suggst to you nothr wy of ruing th fors in th kstys n towrs For in ksty g g o o o o o f For in ksty f g o For in towr og og For in towr of of f Finl For Polygon 4000 l

40 Cl in Tnsion How woul you go out signing n rh onrt roof shll twn two givn support points so tht it woul xprin givn mximum xil intrnl for Arh In Comprssion Clik on th Contnts utton to gin nw lsson. Clik on th img of th Louvr Pyrmi to rturn to th ginning of this lsson.