5. cables and arches

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5. cables and arches

  1. 1. Department of Civil Engineering NPIC %> kab nigeRKOgbgÁúMekagbBaÄr Cables and arches kab nig arch begáIt)anCaFatuRTbnÞúkcMbgenAkñúgRbePTeRKOgbgÁúMCaeRcIn ehIyenAkñúgCMBUk enHnwgerobrab;BIKMnitEdlTak;TgeTAnwgkarviPaKeRKOgbgÁúMTaMgenH. CMBUkenHcab;epþImCamYynwgkarerob rab;CalkçN³TUeTAénkab nigbnþedaykarviPaKkabEdlrgbnÞúkcMcMnuc nigrgbnÞúkBRgayesμI. edaysar arch PaKeRcInCaeRKOgbgÁúMminkMNt;edaysþaTic enaHmanEt three-hinged arch EdlmanlkçN³Biess nwgRtUv)anykmkerobrab;enAkñúgCMBUkenH. karviPaKeRKOgbgÁúMenHnwgpþl;nUvsBaØaNxøHBImUldæanénkar eFVIkarrbs;eRKOgbgÁúM arch TaMgGs;. %>!> kab (Cables) CaerOy² eKeRbIkabenAkñúgeRKOgbgÁúMvisVkmμsMrab;RT nigedIm,IbBa¢ÚnbnÞúkBIGgát;mYyeTAGgát;mYy. enAeBleKeRbIkabedIm,IyYrdMbUlBüÜr (suspension roof) s<anBüÜr (suspension bridge) nig trolley wheel, kabmanCaTICaFatuTTYlbnÞúkd¾sMxan;enAkñúgeRKOgbgÁúM. sMrab;karviPaKkMlaMgénRbB½n§eRKOgbgÁúM EbbenH TMgn;pÞal;xøÜnrbs;kabRtUv)anecal b:uEnþenAeBleKeRbIkabCaExSbBa©ÚnGKÁisnI b¤eRbIvaenAkñúg Gg;EtnviTüú nigm:asIunsÞÚc enaHTMgn;pÞal;rbs;kabnwgkøayCasMxan; ehIyeKRtUvEtKitbBa©ÚlvaeTAkñúg viPaKeRKOgbgÁúM. enAkñúgkfaxNÐbnÞab;nwgmanerobrab;BI³ kabrgbnÞúkcMcMnuc nigkabrgbnÞúkBRgayesμI. kñúgkrNIEdlbnÞúk nigkansßitenAkñúgbøg;EtmYy enaHkarviPaKTamTarsmIkarlMnwgRsedogKña. enAbegáItTMnak;TMngrvagkMlaMgenAkñúgkabCamYynwgCMralrbs;va eyIgRtUveFVIkarsnμt;fakabman lkçN³bt;Ebn nigminGaclUt)an (perfectly flexible and inextensible). edaysarPaBbt;Ebn (flexibility) rbs;va kabminTb;Tl;nwgkMlaMgkat; nigm:Um:g;Bt; dUcenHkMlaMgEdlmanGMeBIenAkñúgkab EtgEtb:H (tangent) eTAnwgkabRKb;cMnucTaMgGs;tambeNþayRbEvgrbs;va. edaysarvaminGac lUt)an dUcenHkabmanRbEvgefrTaMgmun nigeRkaykarGnuvtþbnÞúk. CalT§pl enAeBleKGnuvtþbnÞúk eTAelIkab eKGacKitfakab b¤kMNat;rbs;vaCaGgÁFaturwg. %>@> kabrgbnÞúkcMcMnuc (Cables subjected to concentrated loads) enAeBlEdlkabEdlecalTMgn;pÞal;xøÜnyYrbnÞúkcMcMnucCaeRcIn enaHkabmanragCakMNat;bnÞat; Rtg;CaeRcIn EdlkMNat;nImYy²kMlaMgTajefr. ]TahrN_ eKmankabdUcbgðajenAkñúgrUbTI 5-1. enATI enH θ CamMurvagkab AB CamYynwgG½kSedk ehIy L CaRbEvgrbs;kab dUcbgðajkñúgrUb. RbsibebIeK kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -148
  2. 2. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa sÁal;cMgay L1 / L2 / L3 nigbnÞúk P1 / P2 enaHeKRtUvkMNt;GBaØatcMnYn 9 EdlrYmmankMlaMgTajenAkñúg kMNat;TaMgbI bgÁúMkMlaMgRbtikmμTaMgbUnenARtg;cMnuc A nig B nigRbEvgdk; (sag) yC nig y D enARtg; cMnuc C nig D . sMrab;dMeNaHRsay eyIgGacsresrsmIkarlMnwgcMnYnBIrenARtg;cMnuc A / B / C nig D enaHeyIg)ansmIkarcMnYn 8 . edIm,IbMeBjdMeNaHRsay eKcaM)ac;RtUvkarsmIkarmYyeTotEdlTak;Tg nwgragFrNImaRtrbs;kab. ]TahrN_ RbsinebI LT CaRbEvgsrubrbs;kab enaHeKGaceRbIRTwsþIBItaK½r (Pythagorean theorem) edIm,IP¢ab;TMnak;TMngrvag LT eTAnwgRbEvgkMNat;TaMgbI edaysresrCa GnuKmn_eTAnwg θ / yC / y D / L1 / L2 nig L3 . EteKminGacedaHRsaycMeNaTRbePTenHeday éd)anedaygayRsYleT. buEnþeKmanlT§PaBpSgeTotKWkMNt;RbEvgdk; yC b¤ y D CMnYseGayRbEvg kab. edayeFVIEbbenH enaHeKGacTTYl)ansmIkarlMnwgRKb;RKan;sMrab;TTYlkMlaMgEdlCaGBaØat nig RbEvgdk;EdlenAsl;. enAeBleKTTYl)anRbEvgdk; (sag) enARtg;cugnImYy²rbs;bnÞúkehIy eKGac kMNt;RbEvg LT edayRtIekaNmaRt. enAeBlGnuvtþkarviPaKlMnwgsMrab;RbePTcMeNaTEbbenH eKk¾GacTTYl)ankMlaMgkñúgrbs;kab edaysresrsmIkarlMnwgsMrab;kabTaMgmUl b¤sMrab;kMNat;kabk¾)an. ]TahrN_CatMélelxxageRkam enHnwgbgðajBIKMnitTaMgenH. ]TahrN_ 5-1³ kMNt;kMlaMgTajenAkñúgkMNat;kabEdlbgðajenAkñúgrUbTI 5-2a. ehIykMNt;TMhM h ? dMeNaHRsay³ edaykarGegát eKmaneKmanGBaØatbYnCa kMlaMgRbtikmμ ¬ A / A / D nig D ¦ nig x y x y GBaØatbICakMlaMgTajkñúgkab ¬enAkñúgkMNat;kabnImYy²¦. Cables and Arches T.Chhay -149
  3. 3. Department of Civil Engineering NPIC eKGackMNt;GBaØatTaMg 7 CamYynwgRbEvg dk; h BIsmIkarlMnwg EdlGacrk)ancMnYn 8 ¬ ∑ Fx = 0 / ∑ Fy = 0 ¦EdlGnuvtþeTA elIcMnuc A rhUtdl; cMnuc D . viFIkñúgkaredaHRsayEdlmanlkçN³edaypÞal;KWkMNt; CMralrbs;kMNat;kab CD ehIydüaRkamGgÁesrIénkabTaMgmUl RtUv)anbgðajenAkñúgrUbTI 5-2b. eyIgGacTTYl)ankMlaMgTaj enAkñúgkMNat;kab CD dUcxageRkam³ + ∑M A = 0 TCD (3 / 5)(2m ) + TCD (4 / 5)(5.5m ) − 3kN (2m ) − 8kN (4m ) = 0 TCD = 6.79kN LÚveyIgGacviPaKlMnwgéncMnuc C nig B CabnþbnÞab;. sMrab;cMnuc C ¬rUbTI 5-2c¦ + → ∑ Fx = 0 6.79kN (3 / 5) − TBC cos θ BC = 0 + ↑ ∑ Fy = 0 6.79kN (4 / 5) − 8kN + TBC sin θ BC = 0 θ BC = 32.3o TBC = 4.82kN sMrab;cMnuc B ¬rUbTI 5-2d¦ + → ∑ Fx = 0 − TBA cos θ BA + 4.82kN cos 32.3o = 0 + ↑ ∑ Fy = 0 TBA sin θ BA − 4.82kN sin 32.3o − 3kN = 0 θ BC = 53.8 o TBA = 6.90kN dUcenH BIrUbTI 5-2a h = (2m ) tan 53.8 o = 2.74m %>#> kabrgbnÞúkBRgayesμI (Cables subjected to a uniform distributed loads) kabpþl;nUvmeFüa)ayd¾manRbsiT§PaBkñúgkarRTTMgn;efrrbs;r:t b¤kMrals<anEdlmanRbEvg ElVgEvg². s<anBüÜr (suspension bridge) Ca]TahrN_EdlkMrals<anRtUv)anyYredaykabedayeRbI kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -150
  4. 4. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa sMnuMénkabBüÜr (hanger) EdlmanKMlatesμI²Kña nigEk,r²Kña. edIm,IviPaKeRKOgbgÁúMEbbenH dMbUgeyIgRtUvkMNt;rUbragrbs;kabEdlrgbnÞúkBRgayesμIbBaÄr wo ¬rUbTI 5-3a¦. enATIenH eKalrbs;G½kS x nig y manTItaMgenAcMnuceRkambMput dUcenHCMralenA Rtg;cMnucenHesμIsUnü. düaRkamGgÁesrIrbs;kMNat;kabd¾tUcEdlmanRbEvg Δs RtUv)anbgðajenAkñúg rUbTI 5-3b. edaysarkMlaMgTajenAkñúgkabpøas;bþÚrCabnþbnÞab;TaMgTMhM nigTistambeNþayRbEvgkab enaHeKtagbMErbMrYlenHeday ΔT enAelIdüaRkamGgÁesrI. kMlaMgpÁÜbrbs;bnÞúkBRgayesμIRtUv)antag eday wo Δx EdlmanGMeBIenAcMgay Δx / 2 BIcMnuc O . edayGnuvtþsmIkarlMnwg eyIg)an + → ∑ Fx = 0 − T cos θ + (T + ΔT ) cos(θ + Δθ ) = 0 + ↑ ∑ Fy = 0 − T sin θ − wo (Δx ) + (T + ΔT )sin (θ + Δθ ) = 0 + ∑MO = 0 wo (Δx )(Δx / 2) − T cos θΔy + T sin θΔx = 0 edayEcksmIkarTaMgenHnwg Δx ehIyeGaylImItén Δx → 0 rYceFVIdUcKñasMrab; Δy → 0 / Δθ → 0 nig ΔT → 0 eyIgTTYl)an d (T cos θ ) =0 (5-1) dx d (T sin θ ) = wo (5-2) dx dy = tan θ (5-3) dx edayeFVIGaMgetRkalsmIkar 5-1 Edl T = FH enARtg; x = 0 eyIgman T cos θ = FH (5-4) EdlbgðajbgÁúMkMlaMgedkenARKb;cMnucTaMgGs;tam beNþaykabenArkSatMélefr. edayeFVIGaMgetRkalsmIkar 5-2/ nigedaydwgfa T sin θ = 0 enARtg; x = 0 eyIg)an T sin θ = wo x (5-5) Cables and Arches T.Chhay -151
  5. 5. Department of Civil Engineering NPIC edayEcksmIkar 5-5 CamYynwgsmIkar 5-4 kMlaMg T RtUv)ansMrYl. bnÞab;mkedayeRbIsmIkar 5-3 eyIgGacTTYl)anCMralenARKb;cMnucTaMgGs; dy wo x tan θ = = (5-6) dx FH edayeFVIGaMgetRkalelIkTIBIrCamYynwg y = 0 enAeBl x = 0 eyIg)an wo 2 y= x (5-7) 2 FH vaCasmIkarGnuKmn_)a:ra:bUl. eKGacTTYltMélrfr FH edayeRbIlkçxNÐRBMEdn y = h enARtg; x = L . wo L2 dUcenH FH = 2h (5-8) cugbBa©b; edayCMnYstMél FH énsmIkar 5-8 eTAkñúgsmIkar 5-7 eyIg)an h y= 2 x2 (5-9) L BIsmIkar 5-4 kMlaMgTajGtibrmaenAkñúgkabekItmanenAeBl θ mantMélGtibrma eBalKWenAeBl x = L . dUcenH BIsmIkar 5-4 nig 5-5 Tmax = FH + (wo L )2 2 (5-10) b¤ edayeRbIsmIkar 5-8 eyIgGacsresr Tmax CaGnuKmn_én wo eBalKW Tmax = wo L 1 + (L / 2h )2 (5-11) eKRtUvcMNaMfa eKminKitTMgn;rbs;kabEdlBRgayesμItambeNþayRbEvgkab minEmntamRbEvgcMeNal edkrbs;vaeT. Cak;Esþg kabEdlRKan;EtrgbnÞúkpÞal;xøÜnedayminrgbnÞúkxageRkAk¾manragekag (catenary curve) Edr. b:uEnþRbsinebIpleFobRbEvgdk;elIElVg (sag-to-span ratio) mantMéltUc EdlCakrNIsMrab;eRKOgbgÁúMGnuvtþn_PaKeRcIn ExSekagenHxiteTArkrag)a:ra:bUl dUcEdl)ankMNt;enA TIenH. BI;lT§plénkarviPaKenH ExSekagenAEtrkSarag)a:ra:bUldEdl kñúgkrNIbnÞúkefrrbs;kMral deck sMrab;s<anBüÜr b¤sMrab;r:tEdlBüÜrBRgayesμIelIRbEvgcMeNaledkrbs;kab. dUcenHRbsinebIr:t kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -152
  6. 6. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa enAkñúgrUbTI 5-4a RtUv)anBüÜredayes‘rIénkabBüÜrEdlmanKMlatesμIKña nigEk,rKña enaHbnÞúkenAkñúgkab BüÜrnImYy²RtUvEtesμIKñaedIm,IFananUvrag)a:ra:bUlrbs;kabsñÜ. edayeRbIkarsnμt;xagelIenH eyIgGacGnuvtþkarviPaKeRKOgbgÁúMénr:t b¤énrcnasm<½n§déTepSg eTotEdlBüÜredayesrI. CaBiess RbsinebIr:tRTedayTMrsamBaØehIyRtUv)anBüÜredaykabeTotenaH karviPaKnwgkøayCaminkMNt;edaysþaTicdWeRkTI1 ¬rUbTI 5-4b¦. b:uEnþRbsinebIr:tmansnøak;enARtg; cMnucNamYytambeNþayr:t ¬rUbTI 5-4c¦ enHvanwgpþl;lkçxNÐm:Um:g;sUnü ehIyeKGacviPaKvaCar:t kMNt;edaysþaTic. ]TahrN_ 5-2³ kabenAkúñgrUb 5-5a yYrr:tEdlmanTMgn; 12kN / m . kMNt;kMlaMgTajenAkñúgkab enARtg;cMnuc A / B nig C . dMeNaHRsay³ eKalrbs;G½kSkUGredaenRtUv)ankMNt;ykenARtg;cMnuc B ¬cMnucTabCageKenAelIExSkab¦ Edlman CMral (slope) sUnü ¬rUbTI 5-5b¦. BIsmIkar 5-7 smIkar)a:ra:bUlsMrab;kabKW³ wo 2 12kN / m 2 6 2 y= x = x = x (1) 2 FH 2 FH FH edaysnμt;facMnuc C mancMgay x' BI B eyIgman 6 2 6= x' FH FH = 1.0 x' 2 (2) dUcKña sMrab;cMnuc A Cables and Arches T.Chhay -153
  7. 7. Department of Civil Engineering NPIC 12 = 6 [− (30 − x')]2 FH 12 = 6 2 [− (30 − x')]2 1.0 x' x' 2 +60 x'−900 = 0 x' = 12.43m dUcenH BIsmIkar 2 nig 1 ¬b¤smIkar 5-6¦ eyIg)an FH = 1.0(12.43)2 = 154.4kN dy 12 = x = 0.07772 x (3) dx 154.4 enARtg;cMnuc A x = −(30 − 12.43) = −17.57 m = 0.07772(− 17.57 ) = −1.366 dy tan θ A = dx x =−53.79o θ A = −53.79 o edayeRbIsmIkar 5-4 FH 154.4 TA = = = 261.4kN ( cos θ A cos − 53.79 o ) enARtg;cMnuc B / x = 0 dy tan θ B = =0 θ B = 0o dx x =0 FH 154.4 TB = = = 154.4kN cos θ B cos 0 o enARtg;cMnuc C / x = 12.43m = 0.0772(12.43) = 0.966 dy tan θ C = dx x =12.43 θ C = 44.0 o FH 154.4 TC = = = 214.6kN cos θ C cos 44.0 o ]TahrN_ 5-3³ s<anBüÜrenAkñúgrUbTI 5-6a RtUv)ansg;edayeRbI stiffening truss BIrEdlP¢ab;Kña edaysnøak;enARtg;cMnuc C enAmanTMrsnøak;enARtg;cMnuc A nigTMrkl;enARtg;cMnuc B . kMNt;kMlaMgTaj GtibrmaenAkñúgkab IH . kabmanrag)a:ra:bUl ehIys<anrgbnÞúkeTal 50kN . kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -154
  8. 8. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ düaRkamGgÁesrIénRbB½n§kab nig truss RtUv)anbgðajenAkñúgrUbTI 5-6b. eyagtam smIkar 5-4 ¬ T cosθ = FH ¦ bgÁúMkMlaMgedkrbs;kMlaMgTajenAkñúgkabenARtg;cMnuc I nig H RtUvEt mantMélefrKWesμInwg FH . Kitm:Um:g;Rtg;cMnuc B eyIgman + ∑MB = 0 − I y (24m ) − A y (24m ) + 50kN (9m ) = 0 I y + Ay = 18.75 RbsinebIeKKiteRKOgbgÁúMBüÜrEtBak;kNþal ¬rUbTI 5-6c¦ enaHkareFVIplbUkm:Um:g;Rtg;snøak; C eyIg)an + ∑MC = 0 FH (14m ) − FH (6m ) − I y (12m ) − A y (12m ) = 0 I y + A y = 0.667 FH BIsmIkarTaMgBIr 18.75 = 0.667 FH FH = 28.125kN Cables and Arches T.Chhay -155
  9. 9. Department of Civil Engineering NPIC edIm,ITTYlkMlaMgTajGtibrmaenAkñúgkab eyIgnwgeRbIsmIkar 5-11 b:uEnþdMbUgeyIgcaM)ac;kMNt;tMélbnÞúk BRgaysnμt; wo BIsmIkar 5-8³ 2 FH h 2(28.125kN )(8m ) wo = = = 3.125kN / m L (12m )2 dUcenH edayeRbIsmIkar 5-11 eyIgman Tmax = wo L 1 + (L / 2h )2 = 3.125(12m ) 1 + (12m / 2(8m ))2 = 46.9kN %>$> eRKOgbgÁúMekagbBaÄr (Arches) dUckrNIkabEdr eKeRbI arch edIm,Ikat;bnßym:Um:g;Bt;enAkñúgeRKOgbgÁúMEdlmanElVgEvg². Ca Biess arch eFVIkardUcCakabEdlmanlkçN³bRBa©as dUcenHvargkugRtaMgsgát;CacMbgenAeBlEdl bnÞúkxageRkAmanGMeBIelIva. edaysarPaBrwgmaMrbs;va vak¾RtUvTb;Tl;nwgm:Umg;Bt; nigkMlaMgkat;xøH : edayGaRs½yeTAnwgrebobénkarrgbnÞúk nigrUbragrbs;va. RbsinebI arch manrag)a:ra:bUl ehIyvarg nUvbnÞúkBRgayesμIbBaÄr enaHBIkarviPaKkabbgðajfa arch RtUvTb;Tl;EtnwgkMlaMgsgát;b:ueNÑaH. rUbrag arch kñúglkçxNÐenHRtUv)aneKeGayeQμaHfa funicular arch edaysarminmanm:Um:g; nigkMlaMgkat; ekItmanenAkñúg arch. eRKOgbgÁúM arch KMrURtUv)anbgðajenAkñúgrUbTI 5-7 EdlbgðajBIeQμaHsMKal;epSg². GaRs½ynwg karGnuvtþ eKGaceRCIserIsRbePT arch CaeRcInedIm,IRTbnÞúk. Fixed arch ¬rUbTI 5-8a¦ CaerOy² RtUv)anplitBIebtugGarem:. eTaHbICavaTamTarsMPar³ticsMrab;karsagsg; arch EbbenHCagkarsag sg; arch RbePTepSgeTot EtvaRtUvmanRKwHtan; (solid foundation) edaysarvaCaeRKOgbgÁúMminkMNt; edaysþaTicdWeRkTIbI. Cavi)ak kugRtaMgbEnßmGacekItmanenAkñúg arch edaysarsMruténTMrrbs;va. Two-hinged arch ¬rUbTI 5-8b¦ CaFmμtaRtUv)anplitBIEdk b¤eQI. vaCaeRKOgbgÁúMminkMNt;eday sþaTicdWeRkTImYy. eTaHbIvaminrwgdUc fixed arch EtvaGacrgplb:HBal;edaysarsMruttictYc. eyIg kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -156
  10. 10. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa GaceFVIeGayvakøayCaeRKOgbgÁúMkMNt;edaysþaTicedayCMnYsTMrsnøak;mYyedayTMrkl;. b:uEnþ kareFVIEbb enH eFVIeGayvadklT§PaBrbs;vaedIm,ITb;Tl;nwgkarBt;tambeNþayElVgrbs;va ehIyCalT§plvanwg eFVIkarCaFñwmekag (curved beam) minEmnCa arch eT. Three-hinged arch ¬rUbTI 5-8c¦ EdleKGacsg;vaBIEdk b¤eQI ehIyvaCaeRKOgbgÁúMkMNt;edaysþaTic. mindUc arch minkMNt;edaysþaTic vaminrgT§iBlBIsMrut b¤bMErbMrYlsItuNðPaBeT. cugeRkay RbsibebIeKsg; two-hinged arch b¤ three- hinged arch edayminRtUvkarRKwHFM nigRbsinebIminmanbBaðaerOglMh enaHeKGacP¢ab;TMrrbs;vaeday tie rod ¬rUbTI 5-8d¦. Tie rod eFVIeGayeRKOgbgÁúMeFVIkarCaeRKOgbgÁúMrwg edaysar tie rod Tb;Tl;nwg bgÁúMkMlaMgtamTisedkenARtg;TMr. vaminrgT§iBledaysMrutrbs;TMreT. %>%> eRKOgbgÁúMekagbBaÄrsnøak;bI (Three-hinged arches) edIm,IeXIjBIrebobEdl arch bBa©ÚnkMlaMg eyIgnwgBicarNaBIkarviPaK three-hinged arch dUc bgðajenAkñúgrUbTI 5-9a. enAkúñgkrNienH snøak;TIbImanTItaMgsßitenAcMnucx<s;bMputrbs; arch (crown) ehIyTMrmannIv:UxusKña. edIm,IkMNt;kMlaMgRbtikmμenARtg;TMr eKRtUvbMEbk arch enH ehIydüaRkam GgÁesrIrbs;GgÁt;nImYy²RtUv)anbgðajenAkñúgrUbTI 5-9b. enATIenH eKmanGBaØatcMnYn 6 ehIyeKGac mansmIkarlMnwgcMnYn 6 Edr. viFImYykñúgkaredaHRsaycMeNaTenHKWGnuvtþsmIkarlMnwgm:Um:g;Rtg;cMnuc A nigRtg;cMnuc B . eKGacTTYl)ankMlaMgRbtikmμ C x nig C y BIRbB½n§smIkarenH. eKGacTTYl)ankMlaMg Ekg kMlaMgkat; m:Um:g;Bt;enARKb;cMnucTaMgGs;tambeNþay arch edayeRbIviFImuxkat;. eKRtUvkat;Ggát; eGayEkgnwgG½kSbeNþayrbs;vaRtg;cMnucEdlRtUvBicarNa. ]TahrN_ düaRkamGgÁesrIsMrab;kMNat; AD RtUv)anbgðajenAkñúgrUbTI 5-9c. Cables and Arches T.Chhay -157
  11. 11. Department of Civil Engineering NPIC Three-hinged arch k¾GacmanTMrg;Ca truss BIrP¢ab;Kñaedaysnøak; (two pin-connected trusses) Edl truss nImYy²CMnYseGaykMNat; arch AC nig CB enAkñúgrUbTI 5-9a. karviPaKTMrg;Ebb enHGnuvtþdUckarerobrab;xagelI. ]TahrN_CatMélelxxageRkamnwgbgðajBIviFITaMgenH. ]TahrN_ 5-4³ three-hinged open-spandrel arch bridge EdleXIjenAkñúgrUbTI 5-10a man rag)a:ra:bUl nigRTbnÞúkBRgayesμI. bgðajfa arch )a:ra:bUlrgEtkMlaMgsgát;tamG½kSenARtg;cMnucxag kñúgdUcCacMnuc D CaedIm. snμt;fabnÞúkBRgayesμIRtUv)anbBa¢ÚneTAkMNat; arch. kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -158
  12. 12. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa dMeNaHRsay³ enATIenH TMrrbs;vasßitelInIv:UdUcKña. düaRkamGgÁesrIrbs; arch TaMgmUl nigrbs;kMNat; BC RtUv)an bgðajenAkñúgrUbTI 5-10b nig 5-10c. edayGnuvtþsmIkarlMnwg eyIg)an sMrab; arch TaMgmYl³ + ∑MA = 0 C y (40m ) − 320kN (20m ) = 0 C y = 160kN sMrabkMNat;; arch BC ³ + ∑MB = 0 − 160kN (10m ) + 160kN (20m ) − C x (10m ) = 0 C x = 160kN + → ∑ Fx = 0 B x = 160kN + ↑ ∑ Fy = 0 B y − 160kN + 160kN = 0 By = 0 muxkat;rbs; arch Edlkat;tamcMnuc D / x = 10m / y = −10(10 )2 / (20 )2 = −2.5m dUcbgðajenAkñúg rUbTI 5-10d. CMralrbs;kMNat; arch enARtg; D KW dy − 20 tan θ = = x = −0.5 dx (20 )2 x =10 m θ = −26.6 o edayGnuvtþsmIkarlMnwg ¬rUbTI 5-10d¦ eyIg)an + → ∑ Fx = 0 160kN − N D cos 26.6 o − VD sin 26.6 o = 0 + ↑ ∑ Fy = 0 − 80kN + N D sin 26.6 o − VD cos 26.6 o = 0 + ∑MD = 0 M D + 80kN (5m ) − 160kN (2.5m ) = 0 N D = 178.9kN VD = 0 MD = 0 cMNaM³ RbsinebI arch manragepSgBIenH b¤RbsinebIbnÞúkminBRgayesμI enaHkMlaMgkat; nigm:Um:g;Bt;nwg minesμIsUnüeT. ehIy RbsinebIeKeRbIFñwmTMrsamBaØedIm,IRTbnÞúkBRgayesμI vaRtUvTb;Tl;nwgm:Um:g;Bt; Gtibrma M = 1600kN .m . eyIgeXIjfavaBitCamanRbsiT§PaBkñgkarTb;Tl;nwgkMlaMgsgát;edaypÞal; ú ¬eTaHbICaeKRtUvBicarNaBI;lT§PaBén buckling k¾eday¦ CagkarTb;Tl;bnÞúkedaym:Um:g;Bt;. Cables and Arches T.Chhay -159
  13. 13. Department of Civil Engineering NPIC ]TahrN_ 5-5³ Three-hinged tied arch rgnUvbnÞúkdUcbgðajenAkñúgrUbTI 5-11a. kMNt;kMlaMgenAkñúgGgát; CH nig CB . Ggát;EdlmanbnÞat;dac;² GF rbs; truss CaGgát;kMlaMgsUnü. dMeNaHRsay³ eKGacTTYl)ankMlaMgRbtikmμTMrBIdüaRkamGgÁesrIén arch TaMgmUl ¬rUbTI 5-11b¦³ + ∑MA = 0 E y (12m ) − 15kN (3m ) − 20kN (6m ) − 15kN (9m ) = 0 E y = 25kN + → ∑ Fx = 0 Ax = 0 + ↑ ∑ Fy = 0 Ay − 15kN − 20kN − 15kN + 25kN = 0 Ay = 25kN eKGackMNt;bgÁúMkMlaMgEdlmanGMeBIenARtg;cMnuc C edayBicarNadüaRkamGgÁesrIrbs;kMNat; arch xag eqVg ¬rUbTI 5-11c¦. eKtagbgÁúMkMlaMgTaMgenHedaykMlaMgpÁÜbrbs;Ggát; CF nig CD EdleFVIGMeBIenA Rtg;cMnuc C . dMbUg eyIgRtUvkMNt;kMlaMgenAkñúg rod³ + ∑ MC = 0 FAE (5m ) − 25kN (6m ) + 15kN (3m ) = 0 kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -160
  14. 14. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa FAE = 21.0kN bnÞab;mk + → ∑ Fx = 0 − C x + 21.0kN = 0 C x = 21.0kN + ↑ ∑ Fy = 0 25kN − 15kN − 20kN + C y = 0 C y = 10kN edIm,ITTYl)ankMlaMgenAkñúg CH nig CB eyIgGaceRbIviFItMNdUcxageRkam³ tMN G ¬rUbTI 5-11d¦ + ↑ ∑ Fy = 0 FGC − 20kN = 0 FGC = 20kN (C ) tMN C ¬rUbTI 5-11e¦ + ⎛ 2 ⎞ ⎛ 3 ⎞ → ∑ Fx = 0 FCB ⎜ ⎜ ⎟ − 21.0kN − FCH ⎜ ⎟ ⎜ ⎟=0 ⎟ ⎝ 10 ⎠ ⎝ 10 ⎠ ⎛ 1 ⎞ ⎛ 1 ⎞ + ↑ ∑ Fy = 0 FCB ⎜ ⎜ ⎟ + FCH ⎜ ⎟ ⎜ ⎟ − 20kN + 10kN = 0 ⎟ ⎝ 10 ⎠ ⎝ 10 ⎠ dUcenH FCB = 26.9kN (C ) FCH = 4.74kN (T ) ]TahrN_ 5-6³ Three-hinged trussed arch EdlbgðajenAkñúgrUbTI 5-12a RTbnÞúksIuemRTI. kMNt; kMNt; h1 éntMN B nigtMN D edIm,IeGay arch manragCa funicular shape. Ggát; HG CaGgát;kMlaMg sUnü. Cables and Arches T.Chhay -161
  15. 15. Department of Civil Engineering NPIC dMeNaHRsay³ sMrab;bnÞúksIuemRTI/ funicular shape sMrab; arch RtUvEtmanragCa)a:ra:bUldUcbgðajedaybnÞat;dac; ¬rUb TI 5-12b¦. enATIenH eyIgRtUvrksmIkarEdlRtUvnwgrUbragenH. CamYynwgG½kS x / y EdlmanKl;enA Rtg;cMnuc C smIkarRtUvmanTMrg; y = −cx 2 . edIm,ITTYl)ancMnYnefr c eyIgtMrUveGay − (4.5m ) = −c(6m )2 c = 0.125 / m dUcenH y D = −(0.125 / m )(3m )2 = −1.125m BIrUbTI 5-12a h1 = 4.5m − 1.125m = 3.375m edayeRbItMélenH ¬RbsinebIeKGnuvtþviFItMNeTAelI truss¦ lT§plnwgbgðajfaGgát;xagelI nigGgát; Ggát;RTUgTaMgGs;suT§EtCaGgát;kMlaMgsUnü ehIybnÞúksIuemRTIRtUv)anRTedayGgát;xageRkam AB / BC / CD nig DE rbs; truss. kab nigeRKOgbgÁúMekagbBaÄr T.Chhay -162
  16. 16. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa cMeNaT 5>1> kMNat;kabyYrbnÞúkdUcbgðajkñúgrUbxag 5>4> RbsinebIkMNat;kabnImYy²kñúgrUbxagelI eRkam. kMNt;cMgay yc BIkMlaMgenARtg; C eTA RTkMlaMgTaj Gtibrma 375N cUrkMNt;bnÞúk P cMnuc D . yk P = 20kN . EdlGacGnuvtþ)an. 5>5> kMNat;kabyYrbnÞúkbIdUcbgðajkñúgrUbxag eRkam. kMNt;RbEvgdk; y B nig y D éncMnuc B nig D . yk P1 = 4kN / P2 = 2.5kN . 5>2> kMNat;kabyYrbnÞúkdUcbgðajkñúgrUbxagelI. 5>6> kMNat;kabyYrbnÞúkbIdUcbgðajkñúgrUbxagelI. kMNt;TMhM rbs;kMlaMgbBaÄr P edIm,IeGay yc = kMNt;TMhM P1 RbsinebI P2 = 3kN nig y B = 0.8m . 3m . dUcKñakMNt;RbEvgdk; y D . 5>3> kMNat;kMlaMgTajenAkñúgkMNat;kabnImYy² 5>7> kMNat;kMlaMgTajenAkñúgkMNat;kabnImYy² nigRbEvgsrubrbs;kab. yk P = 400 N . nigRbEvgsrubrbs;kab. 5>8> kab ABCD yYrbnÞúkdUcbgðajkñúgrUbxag eRkam. kMNt;kMlaMgTajGtibrmaenAkñúgkab nig Problems T.Chhay -163
  17. 17. Department of Civil Engineering NPIC kMNt;RbEvgdk;rbs;cMnuc B . 5>11> kabrgbnÞúkBRgayesμI w = 3.75kN / m . kMNt;kMlaMgTajGtibrma nigGb,brmaenAkñúg kab. 5>9> kMNt;kMlaMg P EdlRtUvkaredIm,IeFVIeGay kabmanTItaMgdUcbgðaj eBalKWeFVIeGaykMNat; 5>12> kMNt;bnÞúkBRgayGtibrmaEdlkabGac BC enAEtrkSaPaBedkdEdl. ehIyKNna yYr)an RbsinebIersIusþg;kMlaMgTajGtibrma RbEvgdk; y B rbs;kabKW15kN . nigkMlaMgTajGtibrmaenAkñúgkab. 5>13> eKeGaykabrgbnÞúkBRgayesμI. kMNt; 5>10> kabyYrbnÞúkdUcbgðajkñúgrUbxageRkam. smIkar y = f (x ) EdlCaGnuKmn_ragrbs;kab kMNt;cMgay xB BIcMnuc B eRkamGMeBIénkMlaMgeTA AB nigkMlaMgTajGtibrmaenAkñúgkab. cMnuc A . yk P = 200 N . 5>14> kabyYrr:tEdlmanTMgn;12.75kN / m . kMNt;kMlaMgTajenAkñúgkabRtg;cMnuc A . cMeNaT T.Chhay -164
  18. 18. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 5>17> eKeGaykabrgbnÞúkBRgayesμI. Rbsin ebICMralrbs;kabenARtg;cMnuc O esμInwgsUnü cUrkM Nt;smIkarragrbs;ExSekag nigkMlaMgenAkñúg kabenARtg;cMnuc O nig B . 5>15> kabyYrbnÞúkBRgayesμI wo = 12kN / m . kMNt;kMlaMgTajenAkñúgkabRtg;TMr A nig B . 5>18> kMNt;kMlaMgTajGtibrma nigGb,brma enAkñúgkabrag)a:ra:bUl nigkMlaMgenAkñúgkabBüÜr nImYy². r:trgbnÞúkBRgayesμI nigRtUv)anP¢ab; 5>16> kab AB yYrbnÞúkBRgayesμI 200 N / m . edaytMNsnøak;enARtg; B . RbsinebIeKecalTMgn;rbs;kab nigmMuCMralenA Rtg;cMnuc A nig B KW 30o nig 60o erogKña. cUr kMNt;GnuKmn_ragrbs;kab nigkMlaMgTaj GtibrmaEdlekItmanenAkñúgkab. 5>19> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; sMrab;r:tEdlP¢ab;edaysnøak; AB nig BC . kab manrag)a:ra:bUl. rUbxagelI. 5>20> Fñwm AB nig BC RtUv)anyYredaykabEdl manrag)a:ra:bUl. kMNt;kMlaMgTajenAkñúgkab Rtg;cMnuc D / F nig E ehIykMNt;kMlaMgenAkab BüÜrEdlmanKMlatesμI²Kña. Problems T.Chhay -165
  19. 19. Department of Civil Engineering NPIC snμt; A / B nig C CatMNsnøak;. 5>21> sg;düaRkamkMlaMgkat; nigdüaRkamm:Um:g; sMrab;Fñwm AB nig BC . kabmanrag)a:ra:bUl. rUbxagelI. 5>25> Laminated-wood three-hinged arch rg 5>22> Truss RtUv)antP¢ab;edaysnøak; nigRtUv bnÞúkdUcbgðaj. kMNt;bgÁúMkMlaMgRbtikmμedk nig )anyYredaykabrag)a:ra:bUl. kMNt;kMlaMgTaj bBaÄrenARtg;snøak; A / B / C nigKUrdüaRkam GtibrmaenAkñúgkab enAeBleRKOgbgÁúMrgbnÞúkdUc m:Um:g;sMrab;Ggát; AB . bgðaj. 5>26> eRKOgbgÁúM arch rgnUvbnÞúkdUcbgðaj. kMNt;bgÁúMkMlaMgRbtikmμedk nigbBaÄrrenARtg; 5>23> kMNt;kMlaMgpÁÜbenARtg;snøak; A / B nig snøak; A nig D rYckMNt;kMlaMgTajenAkñúg AD . C rbs; truss dMbUl arch snøak;bI. 5>27> Three-hinged trussed arch rgbnÞúkdUc bgðaj. kMNt;bgÁúMkMlaMgRbtikmμedk nigbBaÄr 5>24> kMNt;bgÁúMkMlaMgRbtikmμedk nigbBaÄrenA enARtg;snøak; A / B nig C . Rtg;cMnuc A / B nig C rbs; arch snøak;bI. cMeNaT T.Chhay -166
  20. 20. mhaviTüal½ysMNg;sIuvil viTüasßanCatiBhubec©keTskm<úCa 5>31> kMNt;bgÁúMkMlaMgRbtikmμedk nigbBaÄr enARtg;TMr A nig C . 5>28> s<anRtUv)ansg;edayeRbI three-hinged trussed arch. cUrkMNt;bgÁúMkMlaMgRbtikmμedk nigbBaÄrenARtg;snøak;; A / B nig C . Ggát; 5>32> kMNt;bgÁúMkMlaMgRbtikmμedk nigbBaÄr bnÞat;dac; DE mankMlaMgkñúgsUnü. enARtg;TMr A nig C . 5>29> kMNt;kMBs;sikSaKNna h1 / h2 nig h3 rbs;Ggát;xageRkamén truss s<anxagelIEdlCa funicular arch. 5>30> Tied three-hinged arch rgbnÞúkdUc bgðaj. cUrkMNt;bgÁúMkMlaMgRbtikmμedk nig bBaÄrenARtg;snøak;; A nig C nigkMNt;kMlaMg TajenAkab. Problems T.Chhay -167

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