Finding Finding The For The For ces In A T ces In A russ Truss V? Next Contents
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- Dale Newman
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1 inin Th Th ors In Truss In this 19th ntury truss ri, th sizs of th iniviul mmrs r sin in proportion to th mnitus n hrtrs of th fors thy must rry. In this lsson, you will lrn th rphil thniqu for nlyzin struturs suh s ths. To pro with this lsson, lik on th Nxt utton hr or t th top of ny p. Whn you r on with this lsson, lik on th ontnts utton hr or t th top of ny p to rturn to th list of lssons.
2 Th Prolm: This is th form irm for six-pnl truss spnnin 1trs. Th truss is trs p n is lo symmtrilly with six los. Wht is th for in h mmr of th truss 1
3 1 Stp 1: pply intrvl nottion to th form irm. To filitt rphil solution, w pply intrvl nottion to th form irm. Strtin from th lft n n workin lokwis, pitl lttrs r pl in th intrvls twn th xtrnl fors. Noti tht th lttr nots th ntir intrvl twn th lft-hn rtion n th first lo on th top of th truss.
4 Nxt, workin from lft to riht, numrs r pl in th intrior sps of th truss. 1
5 1 h xtrnl for is sint y th lttrs on ithr si of it. Th first lo on th top of th truss t th lft, for xmpl, is lo. Th riht-hn rtion is.
6 1 h mmr of th truss is lso sint y th hrtrs on ithr si of it. Th lftmost vrtil mmr, for xmpl, is 1. Noti on in tht sints th ntir intrvl from th lft-hn rtion to th first lo on th top of th truss. Th rihtmost ionl mmr is
7 Rtion 1 Rtion Stp 2: in th rtions usin qutions of quilirium. Th first stp in th solution is to fin th mnitu of th two rtions, n. This is on most sily y tkin momnts out h of th rtions. lokwis momnts r ssum to positiv. Th sum of th momnts out h support must qul 0: M = 0 M = 0
8 Rtion 1 Rtion Th sum of th momnts out is foun y multiplyin h for y th prpniulr istn from its lin of tion to th point of pplition of th rtion. Th first for,, hs mnitu of. Its lin of tion is from th xis of rottion t support. Th for uss lokwis rottion, so it is positiv trm in th qution. Thus th first trm in th qution of quilirium is: (positiv) 2m(15kN). Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN)... M = 0
9 4 m Rtion 1 Rtion Th nxt for is lso, ut it ts t prpniulr istn of 4 m from th point of pplition of th rtion. Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN)... M = 0
10 Th nxt for is. It ts t prpniulr istn of 6 m from rtion. 6 m Rtion 1 Rtion Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN)... M = 0
11 Th nxt for is. It ts t prpniulr istn of 8 m from rtion. Rtion 8 m 1 Rtion Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN) + 8m(15kN)... M = 0
12 Th nxt for is. It ts t prpniulr istn of 10 m from rtion. Rtion 10 m 1 Rtion Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN) + 8m(15kN) + 10m(15kN)... M = 0
13 Th nxt for is. It ts t prpniulr istn of 1 from rtion. Rtion 1 1 Rtion Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN) + 8m(15kN) + 10m(15kN) + 12m(15kN)... M = 0
14 Rtion 1 1 Rtion Th lst for is th unknown rtion. Th lin of tion of is 1 from th xis of rottion. us uss ountrlokwis rottion, it is ntiv trm in th qution. Noti tht rtion os not ppr in th qution. us its lin of tion psss irtly throuh th xis of rottion, its istn from tht point is zro, n it hs no fft on th momnt quilirium. Th momnt qution is now omplt. Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN) + 8m(15kN) + 10m(15kN) + 12m(15kN) - 12m() M = 0
15 W solv th momnt qution to fin th mnitu of rtion. Rtion Rtion 1 Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN) + 8m(15kN) + 10m(15kN) + 12m(15kN) - 12m() = M = 0
16 ollowin th sm prour, ut tkin momnts out th rtion, w solv for. Rtion Rtion 1 Th sum of th momnts out h support must qul 0: M = 0 = 2m(15kN) + 4m(15kN) + 6m(15kN) + 8m(15kN) + 10m(15kN) + 12m(15kN) - 12m() = M = 0 = 12m() - 10m(15kN) - 8m(15kN) - 6m(15kN) - 4m(15kN) - 2m(15kN) - 0m(15kN) =
17 s hk on th ury of our work, w sum th fors in th vrtil irtion to trmin whthr thy up to zro: V = 6(15kN) kN kN V = 0 OK Rtion 1 Rtion It is lwys worth hkin th ury of th rtions you hv lult for prformin ny furthr nlysis. If th rtions r not orrt, th rminr of th nlysis will wron n you will hv to strt ovr from th innin.
18 1 Stp 3: onstrut th Lo Lin. Th Lo Lin is rphil summtion of th los tin on th strutur. W opt onvnint sl of lnth-tofor, on tht will llow th Lo Lin to fit omfortly on th p. us ll th xtrnl fors tin on th truss r vrtil, th Lo Lin will vrtil s wll. Lo Lin
19 1 Lo Lin 0kN Th xtrnl fors on th r plott from lft to riht onto th Lo Lin. h smnt on th Lo Lin is ll t its ns with lowr s lttrs. Noti how intrvls on th orrspon to points on th Lo Lin. Th first for,, ts ownwr with mnitu of. Thus w plot vrtil smnt on th Lo Lin, sl to in lnth. Sin this for ts ownwr s w mov lokwis from to on th, is pl low on th Lo Lin.
20 Nxt, for is plott s smnt on th Lo Lin. Lik, lso ts ownwr with mnitu of. 1 Lo Lin
21 W ontinu lokwis roun th, plottin los onto th Lo Lin. 1 Lo Lin
22 1 Lo Lin
23 1 Lo Lin f
24 1 Lo Lin f
25 s w mov lokwis from to on th, th rtion for ts upwr, so h is lot ov on th Lo Lin. h is sl to. 1 Lo Lin 0kN h f
26 1 inlly, movin from to on th, w plot smnt h on th Lo Lin. h loss th Lo Lin urtly, rturnin to th oriinl point,, from whih it ws strt. This furnishs nothr hk tht th rtion fors wr lult orrtly. Lo Lin 0kN h f
27 1 Mmr 1 Mmr 1 Stp 4: in th fors in th truss mmrs. To fin th mmr fors, w onstrut pirs of intrstin lins off th Lo Lin to form or Polyon. W work from lft to riht. h pir of mmr fors shrs th sm numr, innin t th lft, with 1. Thus w look first on th orm irm for two mmrs whos nms inlu th numr 1. Ths r 1 n 1. h or Polyon f
28 1 Mmr 1 Mmr 1 Mmr 1 is vrtil. Th for in mmr 1 is rprsnt on th or Polyon y vrtil lin throuh point, whih lis lon th Lo Lin. Sin w o not yt know th lotion of point 1, w know th irtion of this lin, ut not yt its lnth. Lin 1 of known irtion ut unknown lnth (shown in r) or Polyon h f
29 1 Mmr 1 Mmr 1 Mmr 1 is horizontl. Th for in mmr 1 is rprsnt on th or Polyon y horizontl lin throuh point h. Th two lins tht w hv onstrut prlll to mmrs 1 n 1 intrst t point h. Th intrstion of ths two lins is lso point 1. Thrfor h n 1 r th sm point. Point 1 h or Polyon f
30 or Polyon 0kN h,1 f Now tht w hv lot point 1, w n fin th mnitus of th fors in ths two mmrs. us point 1 oinis with point h, th sl istn twn h n 1 is zro, n mmr 1 hs no for in it. Th mnitu of th for in mmr 1 sls to, qul to th rtion for. Th mnitus of th fors in 1 n 1 r to th.
31 Point 2 h,1 Nxt w sk th lotion of point 2. Th truss mmrs whos nms inlu th numr 2 r 2 n is horizontl, n 1-2 slops own to th riht t 45 o. or 2 is rprsnt on th or Polyon y horizontl lin throuh point. or 1-2 is rprsnt y lin throuh point 1 prlll to truss mmr 1-2. Th intrstion of ths two lins is point 2. or Polyon f
32 kN kN 0kN kN h,1 W sl ths two lins to fin th mnitus of th mmr fors, n th rsults r trnsfrr to th. or Polyon f
33 In similr fshion, w lot point 3 on th or Polyon y rwin lin prlll to mmr 2-3 throuh point 2, n lin prlll to mmr 3 throuh h. Th intrstion of ths two lins is point 3. W sl ths lins to fin th mnitus of th fors in th two mmrs tht thy rprsnt. 3 h,1 or Polyon f
34 kn Point 4 on th or Polyon lis t th intrstion of lins prlll to mmrs 4 n 3-4, th truss mmrs whos nms oth inlu th numr 4. Noti how w r finin our wy throuh th truss y followin th numrs in squn. 3 h,1 or Polyon f
35 kn kn h,1 or Polyon f
36 kn kn h,1 or Polyon f
37 kn kn h,1 or Polyon f
38 kn kn kn h,1 or Polyon f
39 kn kn kn kn ,9 3 h,1 Points 5 n 9 shr th sm lotion. or Polyon f
40 kn kn kn kn , or Polyon h,1 f
41 kn kn kn kn , ,11 or Polyon h,1 f Points 3 n 11 shr th sm lotion.
42 kn kn kn kn , ,11 or Polyon h,1 f,12 Lin intrsts lin f12 xtly t point f on th Lo Lin, so f n 12 shr th sm lotion. This tlls us tht mmr 12 rris no for.
43 kn kn kn kn , ,11 or Polyon h,1 f,12 Sin point lry xists on th or Polyon, plottin th lst for, 12, llows us to vrify th ury of th rphi onstrution. If th or Polyon wr inurtly onstrut, lin strtin from point 12 n plott prlll to 1iht not intrst with th lry stlish point. If this wr th s, th istn y whih th finl smnt miss th known point oul sl to trmin whthr th inury ws suffiintly lr to rquir omplt rrwin of th or Polyon.
44 60.0 kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 In this s, th or Polyon loss urtly, n w n s tht 12 is qul in mnitu to th lo irtly ov it.
45 60.0 kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 Stp 5: in th hrtr of th mmr fors. lthouh w now know th mnitus of ll th mmr fors, w o not yt know thir hrtrs: tnsion or omprssion. Intrvl nottion hlps us to trmin th hrtr of h for y usin simpl lokwis onvntion.
46 60.0 kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 s n xmpl of this, onsir th joint unr th lftmost lo. This joint is nm y rin th lttrs n numrs roun it in ny lokwis orr. W my ll it , or , or ny othr lokwis squn.
47 kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 To fin th hrtr of th for in mmr 4, w not tht in lokwis orr, it must ll 4, n not 4. s w r from to 4 on th or Polyon, w r movin from riht to lft. This mns tht on th, th for in mmr 4 ts from riht to lft, pushin inst joint Thus th mmr is in omprssion. W init this with on th.
48 2 t kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 ontinuin to r lokwis roun th sm joint , th ionl mmr is ll 4-3, not 3-4. s w r from 4 to 3 on th or Polyon, w r movin from uppr lft to lowr riht. On th, this mns tht th for in mmr 4-3 pulls own to th riht on joint This mmr is in tnsion, n t is pl on th orm irm.
49 2 t kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 ontinuin lokwis, th vrtil mmr is ll 3-2, not 2-3. s w r from 3 to 2 on th or Polyon, w r movin upwr. On th orm irm, this mns tht th for in mmr 3-2 pushs up on joint This mmr is in omprssion, n is pl on th.
50 2 t kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 Th lst mmr t joint is ll 2-, not -2. s w r from 2 to on th or Polyon, w r movin from lft to riht. On th orm irm, this mns tht th for in mmr 2- pushs inst joint This mmr is in omprssion, n is pl on th.
51 kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 Noti how th nmin of for my hn pnin on th prtiulr joint invstit. onsir joint -2-1, shown hr in lu. Rltiv to this joint, th horizontl r is nm -2, not 2- s it ws whn onsir rltiv to joint s w r from to 2 on th or Polyon, w r now movin from riht to lft, th opposit irtion from th prvious joint. owvr, this is not ontritory rsult. Rltiv to joint -2-1, th for tin from riht to lft pushs on this joint. On in, w fin tht this mmr is in omprssion.
52 t 2 t 4 t 6 8 t 10 t kn kn 0 t t t t t kn kn , ,11 or Polyon 15.0 kn h,1 f,12 In similr fshion, w n xmin othr joints to trmin th hrtrs of ll th mmr fors in th truss. W t or to h mmr in th form irm to init tnsion or omprssion.
53 60.0 kn kn 0 t 2 t 4 t 6 8 t 10 t 12 t t t t t kn kn 15.0 kn Th solution is now omplt. Nxt w will rviw wht w n lrn from this nlysis ,9 10 3,11 or Polyon h,1 f,12
54 t 2 t 4 t 6 8 t 10 t kn kn 0 t t t t t kn kn , ,11 or Polyon 15.0 kn h,1 f,12 Wht n w lrn from th omplt nlysis Th for polyon shows us tht fors in th top n ottom hors r hihst t th ntr n iminish to minimum vlus t th ns of th spn. ors in th intrior mmrs (th ionls n vrtils) follow th opposit pttrn: Thy r hihst t th ns n miniml t th ntr of th spn.
55 t 2 t 4 t 6 8 t 10 t kn kn 0 t t t t t kn kn , ,11 or Polyon 15.0 kn h,1 f,12 Th hrtrs of th fors in th intrior mmrs rvrs t mispn. This is us th truss hs th sm orinttion of ionls throuhout its lnth, n is thus omtrilly unsymmtril. If th ionls in on hlf of th truss wr mirror ims of thos in th othr hlf, whih is th mor usul rrnmnt, th hrtrs of th fors in th intrior mmrs woul follow th sm pttrn throuhout. Tht is, ll ionls woul in ithr tnsion or omprssion, n ll vrtils woul hv th opposit for hrtr.
56 h h/2 2 or Polyon hnin th pth of th truss. Wht hppns if th pth of th truss is ru y hlf, whil th spn n los sty th sm Th or Polyons tll th story: Mmr fors in th top n ottom hors of th truss xtly oul. ors in th ionls inrs, ut y lssr mount. ors in th vrtil mmrs rmin unhn. In nrl, mmr fors in th top n ottom hors r invrsly proportionl to th pth of th truss. or Polyon
57 60.0 kn kn kn kn , ,11 or Polyon 15.0 kn h,1 f,12 n you think of wy to us th rphil mtho to fin pth for this truss suh tht th mximum for in its top hor is 75 kn, rthr thn 6 lik on th ontnts utton to in nw lsson. lik on th im of th truss ri to rturn to th innin of this lsson.
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