Catenary Analysis and Calculation Method of Track Rope of Cargo Cableway with Multiple Loads

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1 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME 016 Cateary Aalysis ad Calculatio Method of Track Rope of Cargo Cableway with Multiple Loads Jia Qi a, Ju Che, Liag Qiao, Jiacheg Wa ad Yogju Xia Chia Electric Power Research Istitute, , Beijig, Chia Abstract. Accordig to actual workig coditio, the cateary equatios of elastic track tope for settig-up is proposed based o the cableway erectio requiremet, such as the erectio agle of ed, the iitial cable legth ad midpoit positio of cable. The mechaics equilibrium equatios, the loads spa equatios ad the cosistet equatios are preseted by aalysis of track rope stress state uder loads. The oliear equatios are costructed for elastic track tope with multiple loads ad the iitial values of ewto iteratio method are obtaied to solve the oliear equatios. The results of this method are compared with the testig results ad umerical results i other literatures ad the cotrast verifies the reliability of this method. The method is more cocise ad has smaller amout of calculatios with a uified form. It ca provide effective meas to desig the cargo cableway ad to check the egieerig safety durig the erectio stage ad ruig stage of cableway. 1 Preface The overhead cargo cableway has may characteristics, easy lie selectio, high trasportatio capability ad strog adaptability, etc. The problems of the cableway about desig, security assessmet, ad compoet selectio are more ad more importat. Calculatio methods of the track rope of the cableway are maily the aalytic method ad the fiite elemet method. The aalytic method maily cosists of the cateary method [1-3] ad the parabola method[4-5]. The cateary method ca really reflect the cable shape of the actual suspesio rope, the results obtaied through theoretical calculatio of the cateary method are cosidered as true solutio. Dead weight of the rope structure is assumed to distribute evely i the parabola theory, it is a approximate calculatio method which takes the former two items i the cateary theory. The fiite elemet method maily icludes the two ode rod elemet [6-8] ad the multiple odes curve elemet [9,10], based o the complete Lagrage descriptio or Euler descriptio, calculatio method approximately cosists of the icremet method ad the superpositio method. Because it is ecessary to carry out simplificatio or approximate durig establishig the fiite elemet format of the rope structure, the obtaied calculatio models are also differet. For the cargo freight cableway, it is a commo ackowledge that track rope without elastic elogatio is the cateary. May refereces have give out the theoretical reductio process ad the relevat calculatio equatio of the balace equatio whe the rope structure oly bears dead weight[11,1]. For the cargo cableway with the great spa, the track rope will geerate elastic a Correspodig author: The Authors, published by EDP Scieces. This is a ope access article distributed uder the terms of the Creative Commos Attributio Licese 4.0 (

2 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME 016 elogatio uder dead weight ad cetralized load, the correspodig tesio i the rope ad the drawig force of the support poit will also chage. Whe the track rope has several cocetrated loads, these cetralized loads are coverted to the evely distributed load i some approximate method, i which tesio of track rope is calculated as rope without load. But there is big error with accurate result. I this paper studies the calculatio method of track rope uder multiple cetralized loads is established to provide practicable theory istructio o calculatio ad selectio of the cargo cableway track rope based o cateary equatio of the elastic cable. Cateary equatio of elastic cable without cetralized load The followig assumptios are itroduced i basic theory for desig ad calculatio of the cableway track rope: (1) The track rope is absolutely flexible, i.e., ay sectio oly bears drawig force, ad it ca t bear bedig momet; () The material of the track rope material follows ooke s law, i.e., the stress ad strai relatioship of the track rope material is liear elasticity; (3) The dead load of the track rope is distributed evely alog the arc. l y q 0 T V B B B B h A T A V A A x Figure 1. Forcig Schematic Figure of Track Rope without Elasticity The track rope sectio show as figure 1 oly bears dead weight load of the rope without elasticity elogatio. I which T A, T B, V A, V B, θ A ad θ B are tagetial tesio at ed poit A ad ed poit B, vertical compoets of tagetial tesio, agles betwee tagetial tesio ad horizotal directio respectively, is horizotal compoet of tagetial tesio, l ad h are horizotal spa ad height differet betwee the ed poits respectively, q 0 is dead weight itesity of the track rope which is distributed alog the arc. Assume the iitial sectio area of the track rope is A 0, the sectio area is chaged to A with elasticity elogatio, itesity of the dead weight distributed alog the arc becomes q. Show as figure 1, the micro-sectio legth of the elastic track rope is ds, the iitial arc legth is ds 0. Set E as elastic module of the track rope material, the followig equatio is obtaied from ooke s law: ds=[1+t/(ea 0 )]ds 0. Accordig to the mass coservatio of the track rope micro-sectio after ad before deformatio: q 0 ds 0 =qds. Ad it is obtaied: q=q 0 /[1+T/(EA 0 )] (1) Cosiderig forcig coditio of the track rope micro-body, the force balace equatios at x directio ad y directio are obtaied: X 0 : d 0 () dy dy dy Y 0 : d( ) qd s 0 (3) dx dx dx

3 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME 016 From equatio (), is costat which meas horizotal compoet of tesio of the track rope is equal at ay locatio. Accordig to equatio (3), there is d y q ds (4) d x d x Substitutig equatio (1) ito equatio (4), the basic cateary equatio of the elastic track rope without cetralized load is obtaied by ds dx 1 (dy d x) ad T 1 (d y/d x) accordig to deductio process i referece [11]: x t a b a b ( ) [sih( ) sih( )] (5) y t[cosh( a) cosh( b)] [cosh ( a) cosh ( b)] (6) s0 t[sih( a) sih( b)] (7) st[sih( a) sih( b)] [sih( a) sih( a)] ( a b) (8) I which, K q0 /( EA0) (costat), t / q0, a sih ( VA / ), b sih ( V / ). V is vertical compoet of tagetial tesio at ay poit. s 0 is iitial rope legth, s is legth of the rope after elastic elogatio. 3 Balace equatio of elastic track rope uder multiple cetralized loads After the settig-up of cableway, the height differece h, the horizotal spa l ad the iitial legth s 0 of the whole track rope are determied. The track rope bearig multiple cetralized loads is show as figure. l V B B q p i hi h y 1 A h 1 p i-1 l i V A l 1 p 1 Figure. The Track Rope uder Multiple Cetralized Loads The rope sectio splited by cetralized load oly bears the dead weight of the rope, so the whole rope ca be cosidered as combiatio of the catearies. Assumig the track rope has -1 cetralized loads, which are p 1, p,, p -1 respectively, the track rope is divided ito sectios, the ed poits of every sectio are A i ad B i, iitial arc legth is s i, horizotal spa is l i ad height differece is h i (i=1,,, ). Established the local coordiatio system for every sectio of the track rope, the cateary equatio for the every rope sectio without cetralized load is obtaied, show as equatios (5)-(8). The oliear equatios of the whole track rope uder multiple cetralized loads are established i the followig. 3

4 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME Cetralized load i1 d y d x Ai 1 i B i Ai 1 i 1 d y i d x Bi p i Figure 3. Forcig Aalysis at Actio Poit of Cetralized Load Show as figure 3, from the force o the actio poit betwee every rope sectio, the force balace coditios at every load poit is obtaied: i i 1 0 (9) dy dy i i 1 pi (i=1,,, -1) (10) dx B dx i Ai 1 It is kow from the balace coditios that the horizotal force of every rope sectio i is same, which is marked as. Ad equatio (10) ca be writte as pi t[sih( bi) sih( ai 1)] (i=1,,, -1) (11) q ere t=/q 0, a i =sih -1 (V Ai /), b i =sih -1 (V Bi /) (i=1,,, ) Legth of suspesio rope Durig operatio of the cableway, the rope legth of the track rope betwee the cetralized load (heavy objects) is fixed ad uchaged, iitial legth s i of every rope sectio appears i the equatio as the kow quatity, the horizotal spa l i ad height differece h i of the rope sectio will chage followig differet loads ad differet positios. Therefore, whe iitial rope legth of every rope sectio s i (i=1,,, ) is kow, the rope sectio legth equatio is obtaied accordig to equatio (7): t[sih( a ) sih( b)] s (i=1,,, ) (1) i i i 3.3 Compatibility coditio 1) Every sectio of the track rope shall meet with the compatibility equatio of the whole horizotal spa l i i1 l, accordig to equatio (5), it ca be obtaied: ( i i) [sih( i) sih( i)] i1 i1 (13) t a b a b l ) Accordig to deformed compatibility equatio i height hi h ad equatio (6) i1 4

5 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME 016 ere K=q 0 /(EA 0 ). t a b a b h [cosh( i) cosh( i)] [cosh ( i) cosh ( i)] i1 i1 (14) 3.4 Noliear equatio set Summarizig the above equatios, the oliear equatio set of the elastic track rope uder multiple cetralized loads is obtaied: pi t[sih( bi) sih( ai 1)] 0 ( i 1,,, 1) q0 t[sih( ai) sih( bi)] si 0 ( i 1,,, ) t( ai bi) [sih( ai) sih( bi)] l 0 i1 i1 t a b a b h [cosh( i) cosh( i)] [cosh ( i) cosh ( i)] 0 i1 i1 (15) The equatio set icludes +1 oliear equatios, +1 ukow umber ai, bi(i=1,,, ) ad t, therefore the oliear equatio set is closed, which ca be expressed as: FX ( ) 0 (16) I which X=[a 1, a,, a, b 1, b,, b, t] T. After X is calculated, the vertical compoet V x ad tagetial tesio T x at ay rope legth s x poit of the track rope are obtaied respectively: m Vx tq0sih( am) q0sx si (17) i1 I which, m is umber of cetralized load i the [0,s x ]. x T tq V (18) 0 x 4 Examples I order to validate calculatio accuracy of the cargo cableway track rope, testig data i the referece documet [1] ad calculatio data i referece documet [8] are applied for compariso. Load weight i the referece[1] is P=6130N. The cross area of the track rope is A 0 =89.95mm, weight of uit legth is q 0 =7.0N/m, elastic module E=90GPa. The cableway set up for testig is two spa with sigle track rope. Testig data are complete i the first spa of the cableway. The horizotal spa of first spa is l=70.9m, height differece h=15.m. This paper selects the testig data with deflectio coefficiet S f =f/l=0.05 for compariso ad validatio. Table 1. Testig Compariso ad Validatio Compariso items Test value Calculatio value Error % 5

6 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME 016 Noload O load Drawig force of lower support poit /N Torsio at 0.3l locatio /m Torsio at 0.5l locatio /m Torsio at 0.8l locatio /m Load at 0.3l locatio Load at 0.5l locatio Load at 0.8l locatio Drawig force of lower support poit /N Torsio of loadig poit /m Drawig force of lower support poit /N Torsio of loadig poit /m Drawig force of lower support poit /N Torsio of loadig poit /m See from compariso ad validatio data i Table 1, calculated drawig force of the lower support poit of the track rope ad deflectio value at differet positio are almost same as testig value. Whe there is cetralized load, drawig force of the lower support poit of the track rope obtaied through calculatio is greater tha testig value, deflectio value at differet positio is smaller tha testig value. Frictio force geerated betwee the saddle ad the track rope at o-load belogs to static frictio force, o displacemet occurs i the track rope. Whe the cableway is loaded, frictio force geerated betwee the saddle ad the track rope is isufficiet to overcome tesio differece betwee two sides of the saddle. The track rope moves, the sag icreases, ad tesio reduces, correspodig deflectio at differet positio icreases. The referece [8] calculates the large sag sigle spa cableway with the oliear fiite elemet method i the two ode ropes uit with Euler descriptio. The horizotal spa of the cableway l=305m, height differece is 0, ad the middle deflectio coefficiet S f =0.1. The elastic module of the rope is E=130GPa, the sectio area is A 0 =550mm, cetralized load actig o C poit 1m away from the ed poit is P=35.6kN. No load O load Table. Value Calculatio Compariso ad Validatio Compariso item orizotal tesio /kn Value i referece documet Value i text Error % Sag at C poit /m Displacemet of C poit after load acts /m See from Table, the calculatio value i this paper after ad before actig of cetralized load are basically same as the results i the referece documets, it meas the calculatio method i this paper has extesive adaptability. 6

7 MATEC Web of Cofereces 8, (016) DOI: / mateccof/ DME Coclusios The track rope is a importat weight bearig member i the cargo cableway, its shape ad tesio calculatio relatioship is directly related to speciatio ad model selectio of the track rope, etc. This paper establishes the balace equatios of the elastic track rope uder the multiple cetralized loads based o the cateary equatio of the elastic track rope. The formatio of equatios is brief ad the structure is clear. The Newto iteratio method is applied to calculate the equatios, i which calculatio quatity is small, ad accuracy of result is high. Through compariso with result data i other referece documets, reliability of the calculatio method i this paper is validated, which ca meet with requiremet of desig ad egieerig safety i the cargo cableway. Refereces 1. Zheg Lifeg. Theory study of cableway o the cateary method [J]. Fujia: Fujia Agriculture ad Forestry Uiversity CEN Chagsog, CEN Zhegqig, YAN Doghuag. Accurate iteratio method to calculate the iitial [J]. Egieerig Mechaics, 006, 3(8): 6~ QIN Jia, XIA Yogju. The matrix iteratio method for aalysis of suspesio cable based o segmetal cateary theory [J]. Chiese Joural of Egieerig Desig, 013, 0(5): 404~ Qia Chagzhao, Che Chagpig, Che Zili. Direct solutio procedure obtaiig the lie shape of cable with segmetatio parabolic fuctio [J]. Joural of Xiame Uiversity of Techology, 01, 0(3): 71~ Zhou Xiia, Zha Zhegyi. The system of multi-spa cableway desig based o parabola method (JIA SI) [J]. Joural of Zhejiag Forestry College, 000, 17(1):50~ Ahmadi-Kashai, K. Represetatio of cables i space to uiformly distributed loads [J]. Iteratioal Joural of Space Structures, 1988, 3(4): 1~ Zhag Qili, Zhag Li, Zhou Dai, U.Peil. Fem with Splie Fuctio for Noliear Aalysis of Cotiuous Log Cables [J]. Egieerig Mechaics, 1999, 16(1): 115~1. 8. Tag Jiami, Zhao Yi. A Euler Geometrically No-liear Fiite Elemet Method with Twoode Cable Elemet for the Aalysis of Cable Structure [J]. Shaghai Joural of Mechaics, 1999, 0(1): 89~ Yua Xigfei, Dog Shili. A two-ode curved cable elemet for oliear aalysis [J]. Egieerig Mechaics, 1999, 16(4): 59~ Yag Meggag, Che Zhegqig. Noliear aalysis of cable structure usig a two-ode curved cable elemet of high precisio [J]. Egieerig Mechaics, 003, 0(1):4~ Zhag Zhiguo, Ji Migju, Zou Zhezhu. Static solutio of suspesio cables uder tare load [J]. Chia Railway Sciece, 004, 5(3): 67~ Bai Xuesog, Miao Qia. Research o calculatio model based o suspesio cable theories i costructio cargo cableway [J]. Power System Techology, 008, 3(0): 90~94. 7

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