Analysis and Design of Warren Trusses

Transcription

1 Alyss d Desg of Wrre Trusses Preprt Amer Hque Abstrct Formuls for member forces re derved for deck d troug Wrre trusses. Stregt d deflecto costrt desg procedures re developed for te trusses. Te desg procedures re ppled to ypotetcl rlrod brdge. Preprt submtted to Elsever July, 07

2 . Formulto.. Itroducto Ts rtcle provdes formuls for te member forces d deflectos for Wrre trusses. Te procedures descrbed ts rtcle follow tt of Hque [,. Hece, muc of te text of ts rtcle s smply coped from tose ppers. Te metod of sectos d te metod of jots re used to compute member forces. Deflectos re computed usg te ut dummy lod teorem. Tese cocepts re expled y stdrd text o structurl lyss suc s Hbbeler [3, Kssml [4, Tmoseko d Youg [7. Hstorcl formto d termology for truss brdges c be foud Ketcum [5, Mllery [6, Trotsky [8. Computto of structurl wegt of truss s detled ts pper. We ssume fxed sp legt L d kow uform lod w log te sp. Specfyg te umber of pels d truss egt determes te geometry of te truss. For kow geometry, te members re szed to stsfy eter stregt or deflecto costrts. Members c be szed dvdully or ccordg to ter member type. Te szg completely determes te desg of te truss. Te wegt of te desg s gve by te fucto W, were s te umber of pels for lf te sp d s te truss egt. Te optml desg of truss s te desg wc mmzes W,. For stregt desg we derve formuls for optml vlues of gve kow. For deflecto costred desg, we fx oe of te vrbles d determe te optml vlue of te oter vrble. It sould be oted tt stregt desg usully results structure wt excessve deflectos. Members of suc desg c be uformly reszed to stsfy deflecto costrts... Truss Geometry I order to properly descrbe truss, ts geometry d lodg must be specfed. Te deck d troug Wrre trusses re sow fgures,, d 3. Let L be te sp legt d be te truss egt. Symmetry mples tt we eed oly cosder lf te sp of te truss. s te umber of complete pels for lf te sp. We sll ssume tt. Te pel umberg depeds o te type of truss. Idex =,..., for deck trusses sow fgure. Idex = 0,..., for deck trusses of fgure troug trusses of fgure 3. Te pel legt s defed s: L Deck Truss = L + Alterte Deck/Troug Truss. Fgure : Deck truss dmesos d lodg Fgure : Alterte deck truss dmesos d lodg Fgure 3: Troug truss dmesos d lodg Defe θ to be te gle betwee te dgols d te cords. Some bsc trgoometrc fuctos re te esly defed. cosθ = sθ = + + tθ = cotθ =

3 .3. Lods Te mxmum uform lve lod s ssumed costt, but te ded lod depeds o te pel legt. Ts s due to te fct tt more structurl mterl s requred for loger deck d rod legts. We sll ssume tt te uform ded+lve lod s te followg smple form: w = w 0 +ω 0. werew 0, ω, d 0 re costts. 0 s te mmum fesble pel legt. Structurl wegt of te truss s ot cluded s prt of te ded lod. Ts s resoble procedure for steel structures were te structurl wegt s muc less t te ded d lve lods. Te uform lod w s trsfered to te jots s seres of pot lods: p = w.3.4. Wegt Suppose tere re N structurl members.e. cords, dgols, d posts. Let member k ve cross-sectol re A k d legt L k. Te totl structurl wegt of te truss s computed to be: N W = γ A k L k.4 k= were γ s te specfc wegt of te structurl mterl. Clerly W depeds o te umber of members d ter geometry..5. Stregt Desg.5.. Stregt Costrts If te force F k member k s kow, te member c be szed ccordg to stregt crter. All members re ssumed to ve yeld stregt bot teso d compresso. Te requred cross-sectol re s tus: A k = F k.5 Te wegt fucto c be rewrtte terms of member forces: W = γ N F k L k.6 k=.5.. Stregt Desg Procedure Te followg procedure s employed for gve truss geometry d lodg:. Clculte te member forces for te truss. Explct expressos for member forces re possble becuse te trusses re sttclly determt.. Use te stregt costrt to sze ec member type ccordg to equto Compute te wegt W usg equto.6 for te desg determed by step Stregt Desg Optmzto Te stregt desg procedure determes te member szes d wegt W,. We ws to compute te truss egt, for gve umber of pels, wc mmzes wegt W,. Ts s wrtte mtemtclly s: optml for gve = rg m R + W,.7 were R + = 0,. Te optml s computed lytclly by solvg te followg equto for :.6. Deflecto Desg W, = Deflecto Costrts Szg members ccordg to stregt costrts wll usully produce structure wc deflects excessvely. It s ecessry to lmt deflectos by cresg member sze. Te frst step s to compute te md-sp deflecto usg te ut dummy lod teorem. = N k= Lk F k ˆFk EA k.9 ˆF k s te force member k due to ut lod t te mdsp. Te mxmum llowble deflecto lmt s typclly stted terms of sp legt. Te deflecto costrt s: lmt = L b.0 were b s costt. Assumg tt lmt, te members c be uformly reszed usg: A k := A k. lmt Ts cuses te deflecto to be reduced to exctly lmt..6.. Deflecto Desg Procedure Te followg procedure s employed for gve truss geometry d lodg:. Clculte te member forces for te truss. Explct expressos for member forces re possble becuse te trusses re sttclly determt.. Use te stregt costrt to sze ec member type ccordg to equto Compute te wegt W usg equto.6 for te prelmry desg determed by step. 4. Compute te deflecto usg te ut dummy lod teorem formul Apply te deflecto costrt.0 to te prelmry desg. Equto. s used to resze te members. 6. Compute te wegt W for te reszed truss from step 5 usg equto.4. 3

4 .6.3. Deflecto Desg Optmzto Fxg eter or llows W be fucto of oly oe vrble. We perform te followg optmztos: optml for gve N = rg m R +W,. subject to = lmt optml for gve H = rg mw,.3 N subject to = lmt were H 0, d N N. Tese procedures re too cumbersome to perform lytclly. Isted, we geerte plots for te desg exmple descrbed ext Desg Optmzto Exmple Sp Lodg Yeld stregt Youg s modulus L = 00 ft w 0 = 9.5 kp/ft ω = kp/ft 3 0 = 0 ft = 50 ks E = 9, 000 ks Specfc wegt γ = 0.8 lb/ 3 Deflecto lmt lmt = L/800 Tble : Brdge prmeters A deflecto desg optmzto study s ppled to double trck steel rlrod brdge wt truss o ec sde of te brdge. Wrre trusses bot te deck d troug cofgurtos re vestgted. Te fxed prmeters for ec truss re gve tble. Ts exmple s for teoretcl purposes d s ot teded to stsfy te requremets of ctul desg codes. Cooper s E80 lodg s used to compute uform lve lod of w lve = 8 kp/ft. Cooper s lodg s descrbed Mllery [6, Tmoseko d Youg [7, Wlg [9. Ded lods vry wt pel legt d re pproxmted by: w ded =.5 kp/ft kp/ft 3 0 ft Te combed lod w = w lve +w ded prmeters for equto. re gve tble. Note tt te lods re ufctored d stregts re ot reduced. Te vrble truss prmeters re te umbers of pels for lf te sp d te truss dept. Te optmzto procedure. uses N =,3,4,5,6,7} for te deck truss d N =,,3,4,5,6} for te lterte deck d troug truss. Procedure.3 uses H = 4 ft, 6 ft, 8 ft, 30 ft} for ll truss types..7. Szg by Member Type Te procedures descrbed te prevous sectos wll sze ec member dvdully. For smplcty of fbrcto, trusses re sometmes szed by member type. I oter words, members of te sme type ll ve detcl cross-sectol res. We modfy our desg procedures to sze members by type. We frst ctegorze member forces by member type. Let F be te set of cotg N member forces: F = F,...,F N } = D+T +C +P D re te dgols, T re te bottom cords, C re te top cords, d P re te posts: Clerly we must ve: D = D,...,D ND } T = T,...,T NT } C = C,...,C NC } P = P,...,P NP } N = N D +N T +N C +N P We sze member ccordg to te lrgest force of tt member type. Let Y D,T,C,P} be symbolc dex d defe: Y mx = Y µ, µ = rg mx Y k.4 k=,...,n Y µ s te dex to te lrgest bsolute force te member sety werey D,T,C,P}. Note tt Y mx my be egtve, but s te lrgest mgtude of ll te forces Y. All members set Y re te ssged cross-sectol re of:.8. Szg Zero Force Members A Y = Y mx.5 Zero force members my occur truss. Suc members re stll ecessry to esure stblty of te truss. Tey wll lso cqure o-zero forces uder oter lodg codtos. A smple wy to sze tese members s to ssg tem te smllest sze of te o-zero force members of te sme type. Let K Y be te set of dces to o-zero force members set Y d defe: Y m = Y ν, ν = rg m k K Y Y k.6 Te we set te zero force member to ve sze: A Y0 = Y m.7 Ts procedure s ppled oly to dvdul member szg. Szg by member type lwys ssgs zero force members to o-zero cross-sectol re. 4

5 . Deck Wrre Truss.. Member Forces Fgure 5: Free body dgrm; odd pel dex Fgure 4: Deck Wrre truss Member Force Idex V = + p =,..., V + D = V + =,..., T = 0 = p [ T = =,..., T + Fgure 6: Free body dgrm; eve pel dex f s odd f s eve p T = C + p C = p C + p C = [ T p C = f s odd f s eve p = =,..., P = p = 0 P = =,...,+ p Tble : Member forces Mxmum Member Force Idex V mx = p = D mx = V mx + = T mx = T C mx = C P mx = p = Tble 3: Mxmum member forces 5

6 Te ser tt exsts te -t pel s computed by cosderg equlbrum te vertcl drecto: Fy = 0 p p p V = 0 p p+ p V = 0 + p V = 0 Solvg for V d corportg te vlue of p usg equto.3 gves: V + p. Te locto of te mxmum member forces s determed usg prtl dervtves. Ts procedure s descrbed Hque [ d we sll oly stte te results. Te mxmum ser s locted te ed pels: V mx = V = p. Te force te dgol depeds o weter te pel dex s odd or eve. Fgure 5 sows tt te force s tesle for odd pel umbers: D = V sθ = V +.3 For eve pel umbers, fgure 6 te gves te compressve forces: D = V sθ = V +.4 Te lrgest force dgol member occurs t te ed pels: D mx = D = V mx +.5 Te forces te posts re esly computed by specto for =,...,+. 0 P =.6 p For te ed post, we ve: P = p Hece te mxmum post force s te ed post: P mx = P.7 For odd umbered pels, te tesle force te bottom cord s computed by summg te momets bout te jot coectg te post, dgols, d top cords fgure 5. For odd dces, we ve: M = 0 T + p p = 0 [ T + p = 0 [ T + + p = 0 [ T + p = 0 Solvg for T d computg eve umbered pels by specto: p [ T =.8 T + For te frst pel: T = 0. Te mxmum tesle force for bottom cords s locted t te -t pel: p T mx = T = f s odd C + p.9 f s eve For eve umbered pels, te compressve force te top cord s computed by summg te momets bout te jot coectg te post, dgols, d bottom cords fgure 6. For eve dces, we ve: M = 0 C + p p = 0 [ C + p = 0 [ C + + p = 0 [ C + p = 0 Solvg for C d computg odd umbered pels by specto: C + C = p [.0 Te frst pel s specl cse d te force s determed by equlbrum te frst top jot: C = p. Te mxmum s locted t pel : T p C mx = f s odd f s eve p. 6

7 .. Ut Dummy Lod Fgure 8: Free body dgrm; odd pel dex Fgure 7: Ut dummy lod Member Force ˆV = Idex =,..., ˆD = sθ sθ =,..., ˆT = 0 = ˆT = ˆT + =,..., ˆT = f s odd Ĉ + f s eve Ĉ = = Ĉ = Ĉ = Ĉ+ ˆT f s odd f s eve =,..., ˆP = = ˆP = 0 ˆP + = f s odd 0 f s eve Tble 4: Member forces =,..., + Fgure 9: Free body dgrm; eve pel dex Te ut dummy lod teorem s used to compute te mdsp deflecto. Te lodg d free body dgrms for te ut dummy lod re gve fgures 7 to 9. Te member forces for te ut dummy lod re computed usg te metod of sectos. Te ser ec pel s gve by: Fy = 0 ˆV = 0 ˆV = Te force te dgol s te computed to be: ˆD = sθ sθ.3.4 To compute te teso te bottom cords, we sum te momets bout te jot coectg te post, dgol, d top cords. Ts s doe for odd pel dces: M = 0 ˆT = 0 Solvg for ˆT d cludg te eve dces: ˆT = ˆT +.5 Te sme procedure s ppled to te bottom cords for eve dces. Te results re: Ĉ = Ĉ+.6 Te pels = d re specl cses. Te forces te posts re esly computed by specto. Te results re lsted tble 4. 7

8 .3. Stregt Desg Optmzto.3.. Idvdul Member Szg I order to fcltte te computto of totl wegt, we orgze te clcultos ccordg to member type. Equto.6 s ppled to ec member order to stsfy te stregt costrts. We lso ws to clerly wrte te wegts s fuctos of. Ts wll llow us to compute te optml vlue of. Te totl wegt of te dgols s: W D = γ D +.7 = Ts formul s rewrtte usg.3: W D W D W D W D Te desred form s: = γ = γ = γ = γ W D = γ D + = D + = = [ V + [ + = V + V.8 = We computg te combed wegt of te bottom cords, we must sze te zero force member. Te smllest o-zero sze s gve by T. Rememberg tt T = 0, we c wrte: W T = γ [ T + T.9 = We rewrte ts te more coveet form: W T = γ [ T + T =.0 For top cords, te wegt fucto s: We c lso wrte: usg te defto: W C = γ W C = γ C = C C. = C. = Te wegt of te posts must be dled more crefully sce te umber of posts s ot equl to te umber of truss pels. W P = γ [ P + + P + P.3 We must sze te zero force members to o-zero crosssectol res. Te totl wegt of te posts c be smplfed s follows: W P = γ [ P + + P + P W P = γ [p+p+ W P Tus we ve: = p = = = γ [p+p+ p γ W P = 4 p.4 Te totl wegt of te truss s te sum of te member wegts: W = W D +W T +W C +W P.5 were T s defed usg equto.8: T = T 8

9 Assume tt we fx te umber of pels. Te, wc mmzes totl wegt W, s computed by settg: Preprt W = 0.6 d solvg for. We otce tt ec term of te resultg expresso wll cot fctor of γ/ d ts term s dvded o te ec sde of te equto. We tus ve: [ V T + = T = = Ts equto s esly solved for / : [ T + V + T + C p+ = = = C + p = 0 = V p+ = V T + = V + T + C T + = V + T + C p+ = V Tkg te squre root of bot sdes gves te fl result: = T + = V + T + C p+ = V.7 We ote tt every term sde te squre root s fctor of p. Te dvso esures tt.7 does ot deped o p. We prefer to rewrte ts equto te formul terms of /L: T + = V + T + C L = p+ = V.8 9

10 .3.. Szg by Member Type Te totl wegts for ec type of member s gve by: W D = γ [+ V mx W T W C W p = = γ = γ γ T mx C mx + P mx Preprt Recll tt V mx = p, d Pmx = p. Te vlues of T mx d C mx from tble 3 deped o weter s odd or eve. Te totl truss wegt s g: W = W D +W T +W C +W P Settg te dervtve of W, wt respect to, to zero: V mx T mx C mx + Solvg for / produces: V mx + T mx + C mx + P mx = 0 = V mx + P mx + = V mx + + Pmx V mx + T mx + C mx V mx + T mx + C mx = V mx + + Pmx Tkg te squre roots of bot sdes: = V mx + T mx + C mx V mx p Rewrtg equto.9 terms of /L results : L = V mx + T mx + C mx V mx p 0

11 .4. Deflecto Desg Optmzto.4.. Optml for gve.4.. Optml for gve 0.35 Deck Wrre Stregt Desg - Idvdul Stregt Desg - By type Deflecto Desg - Idvdul Deflecto Desg - By type Deck Wrre - Idvdul =4 ft =6 ft =8 ft =30 ft /L W [kp Fgure 0: Optml /L vs. Fgure : Idvdul member szg: mmum W vs..8.6 Stregt Desg - Idvdul Stregt Desg - By type Deflecto Desg - Idvdul Deflecto Desg - By type Deck Wrre =4 ft =6 ft =8 ft =30 ft Deck Wrre - By type /.8 W [kp Fgure : Optml / vs. Fgure 3: Szg members by type: mmum W vs. Fgures 0 d sow ow truss egtsould be cose for gve umber of pels. Fgure demostrtes tt 4 pels for lf te sp re optml for dvdul member szg. Fgure 3 sows tt 3 or 4 pels re optml we szg members by type. Te wegt creses cosderbly we usg oly pels. Ts s due to te cresed deck wegt cused by log pel legt.

12 3. Alterte Deck Wrre Truss 3.. Member Forces Fgure 5: Free body dgrm; eve pel dex Fgure 4: Alterte deck Wrre truss Member Force Idex V = + p = 0,..., V + D = V + = 0,..., p [ T = + =,..., T + + f s eve f s odd p T = C + p C 0 = p + C + p C = [ + T p C = f s eve + f s odd P = p 0 p = 0 =,..., =,...,+ Fgure 6: Free body dgrm; odd pel dex Te deck truss of te prevous secto d ed pels wt lower cords s zero force member. Also, very lrge compressve lod s tke by te ed post. Tese members c be elmted to produce lgter truss. Fgure 4 sows te ltertve deck truss. Te formuls for member forces tbles d 3 rem vld for ts truss. However, te dexg must be djusted sce we ow dex from = 0 d ve + totl pels. Te followg vrble replcemets re mde tose formuls: +, + 3. Te ew formuls re gve tbles 5 d 6. Tble 5: Member forces Mxmum Member Force Idex V mx = + p = 0 D mx = V mx + = 0 T mx = T C mx = C P mx = p = Tble 6: Mxmum member forces

13 3.. Ut Dummy Lod Fgure 8: Free body dgrm; eve pel dex Fgure 7: Ut dummy lod Member Force ˆV = ˆD = sθ sθ ˆT = ˆT + ˆT = f s eve Ĉ + f s odd Ĉ 0 = Idex = 0,..., = 0,..., =,..., = 0 Fgure 9: Free body dgrm; odd pel dex Ĉ+ Ĉ = =,..., ˆT Ĉ = f s eve f s odd ˆP = 0 ˆP + = f s eve 0 f s odd =,..., + Tble 7: Member forces 3

14 3.3. Stregt Desg Idvdul Member Szg We frst compute wegt ccordg member type. Te stregt costrts.6 re used to compute te wegt of ec member. We wrte wegts s fuctos of order to perform te optmzto. Te totl wegt of te dgols s: W D = γ D + 3. Usg te vlue for D tble 5, we compute wegt s: Te wegt of te posts s computed s: W P = γ [ P + + P 3.8 Te term te brckets c be smplfed furter sce ll posts must be te sme sze. W P = γ [p+ p W P = = = γ [p+p W D W D W D W D = γ = γ = γ = γ D + D + [ V + [ + V Tus we ve: γ W P = +p 3.9 Te totl wegt of te truss s te sum of te member wegts: W = W D +W T +W C +W P 3.0 Te expresso for W D s: W D = γ Te totl wegt of te bottom cords s: Ts s rewrtte s: W T = γ W T = γ + V 3.3 T 3.4 = T 3.5 = were T s defed usg: T = T For top cords, te wegt fucto s: W C = γ C 3.6 Or equvletly: W C = γ [ C0 + C = 3.7 usg te defto: C = C 4

15 Settg te dervtve of W wt respect to equl to zero: Preprt W = 0 3. Solvg ts equto for / gves te optml vlue of. Dvg bot sdes of te equto by γ/ produces: V = [ T C 0 + = C + + p = 0 Ts equto s esly solved for / : [ V 0 + C 0 + V + T + C = = = = + p+ V + p+ V V 0 + C 0 + = V + T + C V + T + C V 0 + C 0 + = + p+ V We use te fct tt V 0 = + p to get te fl result: = + p+ C0 + = V + T + C +p+ = V 3. We observe tt every term sde te root s fctor of p. Te dvso esures tt 3. does ot deped o p. Rewrtg te equto te formul terms of /L: L = + p+ C0 + = V + T + C +p+ = V 3.3 5

16 3.3.. Szg by Member Type Te totl wegts of ec member type re: W D = γ [+ +V mx W T W C W p = = γ = γ γ T mx + C mx + P mx Preprt were V mx = + p d Pmx = p. Te totl truss wegt s g: W = W D +W T +W C +W P Settg te dervtve of W wt respect to to zero d dvdg by γ/ : +V mx + C mx + Solvg for / produces: + [ + V mx + C mx + T mx = = T mx P mx = 0 = +V mx + + P mx +V mx + + Pmx + V mx + C mx + T mx + V mx + C mx + T mx +V mx + + Pmx Tkg te squre roots of bot sdes substtutg te vlues of V mx d P mx : = [ + + p+ Cmx + T mx p Every term sde te root g cots fctor of p d tus t c be ccelled. I terms of /L: L = + [ + + p+ Cmx + T mx p 6

17 3.4. Deflecto Desg Optmzto Optml for gve Optml for gve 0.4 Alt Deck Wrre Stregt Desg - Idvdul Stregt Desg - By type Deflecto Desg - Idvdul Deflecto Desg - By type 60 Alt Deck Wrre - Idvdul =4 ft =6 ft =8 ft =30 ft /L W [kp Fgure 0: Optml /L vs. Fgure : Idvdul member szg: mmum W vs. 3.8 Stregt Desg - Idvdul Stregt Desg - By type Deflecto Desg - Idvdul Deflecto Desg - By type Alt Deck Wrre =4 ft =6 ft =8 ft =30 ft Alt Deck Wrre - By type /. W [kp Fgure : Optml / vs. Fgure 3: Szg members by type: mmum W vs. Fgures 0 d sow te optml truss egt for gve umber of pels. Fgures d 3 mply tt 3 pels produce te truss wt te lowest wegt. Ts s true for bot dvdul member szg d szg by member type. 7

18 4. Troug Wrre Truss 4.. Member Forces Fgure 5: Free body dgrm; odd pel dex Fgure 4: Troug Wrre truss Member Force Idex V = + p = 0,..., V + D = V + = 0,..., T 0 = p + = 0 p [ T = + =,..., T + p T = + f s odd C + p f s eve C + C = p [ + T p C = f s odd + f s eve P = p 0 p Tble 8: Member forces =,..., =,...,+ Fgure 6: Free body dgrm; eve pel dex We computg member forces, we eed to cosder te frst pel = 0 s specl cse. Cosderg force equlbrum te vertcl drecto gves expresso for ser pel : Fy = 0 +p p p V = 0 p p+ p V = 0 + p V = 0 Te formul for ser s detcl to tt of te deck truss. V + p 4. Mxmum Member Force Idex V mx = + p = 0 D mx = V mx + = 0 T mx = T C mx = C P mx = p = Tble 9: Mxmum member forces Ts formul s lso vld for = 0. Te mxmum ser occurs t te ed pel = 0: V mx = V 0 = Te force te dgols re: + p 4. V + D = V Te mxmum force te dgols occurs t te ed pels: D mx = D 0 = V mx

19 Te bottom cords ve tesle forces wc re computed by summg te momets bout te jot coectg te post, dgol, d top cords fgure 5. For odd pel dces: M = 0 Preprt T s gve by: T + p + p = 0 [ T + + p = 0 [ T + p = 0 [ T + p = 0 T = p [ + T Te frst bottom cord force s: T 0 = p + Te mxmum occurs t pel : p T mx = T = p + = + f s odd C + p f s eve 4.6 Te dervto for te force te top cords s ccomplsed by summg te momets bout te jot coectg te post, dgol, d bottom cords fgure 6. For eve pel dces: M = 0 C + p + p = 0 [ C + + p = 0 [ C + p = 0 [ C + p = 0 Te formul for C s: C + C = p [ + Te mxmum vlue of force te top cord s: T p C mx = C = f s odd + f s eve p Te forces te posts re esly determed by specto: 0 P = 4.9 p 9

20 4.. Ut Dummy Lod Fgure 8: Free body dgrm; odd pel dex Fgure 7: Ut dummy lod Member Force ˆV = ˆD = sθ sθ ˆT 0 = ˆT = ˆT + Idex = 0,..., = 0,..., = 0 =,..., Fgure 9: Free body dgrm; eve pel dex Te ut dummy lod teorem s g used to compute te md-sp deflecto. Te dervto proceeds te exct sme mer s for te deck truss. Te sers re detcl to tt of te deck truss: ˆT = f s odd Ĉ + f s eve Ĉ = Ĉ+ =,..., ˆT Ĉ = f s odd f s eve ˆP = 0 ˆP + = f s eve 0 f s odd =,..., + ˆV = Te forces te dgols re: ˆD = sθ sθ To compute te teso te bottom cords, we sum te momets bout te jot coectg te post, dgol, d top cords fgure 8. For odd pel dces: M = 0 ˆT = 0 Tble 0: Member forces Te fl result s: ˆT = ˆT + 4. Te compressve force te top cords re computed usg fgure 9 d te results re: Ĉ = Ĉ+ 4.3 Te mddle post c crry tesle lod. All oter posts re zero force members. Te results re summrzed tble 0. 0

21 4.3. Stregt Desg Optmzto Idvdul Member Szg Te totl wegt of te dgols s g: W D = γ D Te lgebr s detcl to te prevous trusses d te expresso for W D s: W D = γ Te totl wegt of te bottom cords s: Ts s rewrtte s: W T = γ W T = γ + V 4.5 T 4.6 [ T0 + T 4.7 Te wegt of te posts s te sme s for te lterte deck truss: W P = γ [ P + + P 4.0 Te term te brckets c be smplfed furter sce ll posts must be te sme sze. W P = γ [p+ p Tus we ve: W P = = = γ [p+p γ W P = +p 4. Te totl wegt of te truss s te sum of te member wegts: W = W D +W T +W C +W P 4. were T s defed usg equto 4.5: T = T For top cords, te wegt fucto s: Or equvletly: W C = γ W C = γ C 4.8 = C 4.9 = usg te defto: C = C

22 Settg te dervtve of W wt respect to equl to zero: Preprt W = Solvg ts equto for / gves te optml vlue of. Dvg bot sdes of te equto by γ/ produces: [ V T 0 + T = = C + + p = 0 Solvg for / : [ V 0 + T 0 + V + T + C = = = = + p+ V + p+ V V 0 + T 0 + = V + T + C V + T + C V 0 + T 0 + = + p+ V We g use te fct tt V 0 = + p to get te result: = + p+ T0 + = V + T + C +p+ = V 4.4 Every term sde te root s fctor of p. Te dvso esures tt 4.4 does ot deped o p. Rewrtg te equto te formul terms of /L: L = + p+ T0 + = V + T + C +p+ = V 4.5

23 4.3.. Szg by Member Type Te totl wegts of ec member type re: W D = γ [+ +V mx W T W C W p = = γ = γ γ + T mx C mx + P mx Preprt were V mx = + p d Pmx = p. Te totl truss wegt s g: W = W D +W T +W C +W P Settg te dervtve of W wt respect to to zero d dvdg by γ/ : +V mx C mx + Solvg for / produces: + T mx + [ + V mx + T mx + C mx = = P mx = 0 = +V mx + + P mx +V mx + + Pmx + V mx + T mx + C mx + V mx + T mx + C mx +V mx + + Pmx Tkg te squre roots of bot sdes substtutg te vlues of V mx d P mx : = [ + + p+ Tmx + C mx p Every term sde te root g cots fctor of p d tus t c be ccelled. I terms of /L: L = + [ + + p+ Tmx + C mx p 3

24 4.4. Deflecto Desg Optmzto Optml for gve Optml for gve Tru Wrre Stregt Desg - Idvdul Stregt Desg - By type Deflecto Desg - Idvdul Deflecto Desg - By type Ler Ft by Wlg Tru Wrre - Idvdul =4 ft =6 ft =8 ft =30 ft /L 0.5 W [kp Fgure 30: Optml /L vs. Fgure 3: Idvdul member szg: mmum W vs. 3.8 Stregt Desg - Idvdul Stregt Desg - By type Deflecto Desg - Idvdul Deflecto Desg - By type Tru Wrre =4 ft =6 ft =8 ft =30 ft Tru Wrre - By type /. W [kp Fgure 3: Optml / vs. Fgure 33: Szg members by type: mmum W vs. Fgures 30 d 3 sow te optml truss egt for gve umber of pels. Fgure 30 lso sows ler ft 4.8 by Wlg [9 to ctul truss dt. Fgures 3 d 33 mply tt 3 pels produce te truss wt te lowest wegt. Ts s true for bot dvdul member szg d szg by member type. = N, N = L 4

25 5. Coclusos Explct formuls were developed for stregt desg of te Wrre truss. Te coclusos of ts pper re detcl to tt of Hque [ d ow wll ow be quoted verbtm for coveece. Stregt desg usully produces trusses wc do ot stsfy deflecto requremets. Members c be uformly reszed to stsfy deflecto costrts. Te deflecto desg procedure s performed by umerclly evlutg te resultg lgebrc expressos. It s see tt cresg truss egt geerlly reduces wegt. However, tere re lmts to suc pproc. Te prmry cocer s tt deep trusses ve log members compresso. Te bucklg of suc members requres ever member or ddtol brcg. For our desg exmple, we lmted members to legts wt resoble desg prctce. Te trusses were desg by ssumg uform lod. Movg lods must be cosdered we desgg brdges. Suc lods my lter te desg te followg wys: Forces cert members my be lrger for prtl lodg t for uform lodg. Ifluece les must be used to determe tese codtos. Stress reversls occur for movg lods. Members desged for teso my be compresso d must be desged for bucklg. Refereces [ Amer Hque. Alyss d desg of k trusses. 07. [ Amer Hque. Alyss d desg of prtt trusses. 07. [3 R.C. Hbbeler. Structurl Alyss. Pretce Hll, Upper Sddle Rver, New Jersey, egt edto, 0. [4 Aslm Kssml. Structurl Alyss. PWS-Ket, Bosto, 993. [5 Mlo Smt Ketcum. Te Desg of Hgwy Brdges d te Clculto of Stresses Brdge Trusses. Te Egeerg News Publsg Compy, New York, 909. [6 Pul Mllery. Brdge d Trestle Hdbook. Crstes, Newto, New Jersey, fourt edto, 99. [7 S.P. Tmoseko d D.H. Youg. Teory of Structures. McGrw-Hll, New York, secod edto, 965. [8 M.S. Trotsky. Plg d Desg of Brdges. Wley, New York, 994. [9 J.L. Wlg. Lest-wegt proportos of brdge trusses. Teccl Report 47, Uversty of Illos, 953. Bullet Seres. Horzotl d oter lods wll be duced we tr eters te brdge. Self-wegt of te truss ws gored ts rtcle, but t eeds to be cosdered te lyss of te fl desg. Furtermore, ufctored lods d ureduced stregts were used for te desg exmple. Tus te desg procedures descrbed ts pper sould be vewed s mes to pproxmtg lower bouds o te truss wegt. Te totl wegt of te brdge cludes two trusses, lterl brcg, deck wegt, gusset pltes, d oter coecto detls. 5