5. cables and arches

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

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


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


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)


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



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


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 .



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



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


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.


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.



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;.


]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



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.



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.



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


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


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.



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


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.


5. cables and arches

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (9)

Similar a 5. cables and arches

Similar a 5. cables and arches (20)

Más de Chhay Teng

Más de Chhay Teng (20)

5. cables and arches