Numerical Analysis of Gravity and Parabolic Catenaries

Transcription

1 Numericl Anlysis of Grvity nd Prbolic Ctenries J. Vsek, O. Suchrd Abstrct This pper nlyses grvity nd prbolic ctenries. It discusses development of n lgorithm for tsks nd optimising of the clcultion. Different itertion methods hve been used in order to optimise the clcultion. The itertion methods re: direct itertion, regul flsi, bisection nd Newton methods. Development of the lgorithm is used in different tsks. For ech method, ttention is pid to suitbility of ppliction, needed time nd number of clcultion steps needed in order to chieve the correct result. Mtlb ws used for development of lgorithms for tsks. Keywords grvity ctenry, prbolic ctenry, itertion methods, development of n lgorithm T I. INTRODUCTION HIS pper nlyses the grvity nd prbolic ctenries. Ctenries re commonly used nlysis methods for cble structures discussed in [1, ]. Severl pproches bsed on numericl methods with specil ctenry element re vilble [3, 4] when nlysing the ctenry or cble structures. Nonliner nlysis of cble structures is shown in [5, 6, 7]. The solution cn be mde on the bsis of discrete nlysis [8]. The Finite Element Method is used when deling with cble structures for bridges [9]. This pper uses four interction methods for nlysis of grvity nd prbolic ctenry. Computtionl complexity is compred for those methods. The methods re direct itertion, regul flsi, bisection nd tngentil methods. The itertion method hs been used for clcultion of some model tsks. Then, the optimum method which is suitble for generl solution hs been chosen. The Mtlb softwre [10] ws used for computtions. Algorithms for civil engineering tsks were developed in Acknowledgements: This outcome hs been chieved with the finncil support of the Ministry of Eduction, Youth nd Sports of the Czech Republic with use of the Institutionl support of conceptul development of reserch in 014. This work ws lso supported from the results SP013/99. Jkub Vsek, VSB-Technicl University of Ostrv, Fculty of Civil Engineering, Deprtment of Structurl Mechnics, L. Podeště 1875, Ostrv, Czech Republic (e-mil: jkub.vsek.st@vsb.cz). Oldrich Suchrd, VSB-Technicl University of Ostrv, Fculty of Civil Engineering, Deprtment of Structurl Mechnics, L. Podeště 1875, Ostrv, Czech Republic (phone: ; e- mil: oldrich.suchrd@vsb.cz). Mtlb, for instnce, in [11] nd [1]. The development of n lgorithm ws bsed on methods described in [13]. When nlysing the steel structures, it is recommended in some cses to use the probbilistic pproch [14]. II. THEORY OF THE PARABOLIC CATENARY The studied problem ws prbolic ctenry in the Fig. 1 with n dditionl condition the totl length. H (1) L = [ λ 1 ln( 1 ) 1 ln( 1 )] + λ + λ + + λ + λb + λb + λb + + λb q = qx d H q = ( l x ) d λ, λ () nd b H l Hh x d = + (3) ql The cble which is suspended in two joints nd loded with continuous lod pplied onto the horizontl projection is referred to s prbolic ctenry [15] nd [16]. Becuse of very low bending stiffness, the lodcrrying cble is considered in clcultions to be n element which does not ber bending moments. The only internl force which rises in the structure is tensile force. All geometric nd sttic quntities re expressed by mens of horizontl rection. Of importnce for description of the structure is determintion of the horizontl force. The horizontl force cnnot be described using conditions of blnce only. It is necessry to choose n dditionl condition. In this cse, the dditionl condition is the specified length of the cble. When clculting the horizontl force, the eqution for the cble length (1) is tken s bsis. For more detils bout clcultion see () nd (3). The horizontl force H ws determined using itertion methods. Fig. 1 Prbolic ctenry ISSN:

2 III. THEORY OF THE GRAVITY CATENARY The grvity ctenry cn be regrded s perfectly flexible cble which is not ble to trnsfer other internl forces thn tensile norml forces. In this cse, cble without prolongtion hs been considered. The difference between the prbolic ctenry nd grvity ctenry is the wy of loding. The fibre in Fig. is loded with continuous lod pplied onto the fibre xis. l L =..sinh. (7) h Fig. Loding of the loded ctenry This type of lod cn be represented by the ded lod. Effects of the ded lod depend on cble deflection. In generl, the higher the deflection is, the bigger the effects of the ded weight re. For this reson, it is not lwys possible to simplify the sitution nd use prbolic ctenry. In this cse, it is lso ssumed tht the both ends of the fibre re fixed to non-displceble supports. x z. cosh,, 1, z 1+ z = (4) = (5) The ded weight of ech cble mke the cble to shpe s ctenry. From the mthemticl point of view, the ctenry shpe cn be described using (4). Becuse it is mthemticlly esy to derive differentil eqution (5) of genuine grvity ctenry, such coordinte system is chosen where the grvity ctenry crosses the verticl xis in the lowest point. The coordinte of the point of intersection of the ctenry nd the xis is () - this prmeter is used then to derive ll other geometric quntities which chrcterise the ctenry. Fig. 3 shows this loction in the coordinte system. As this is not stticlly determined tsk, it is essentil to include n dditionl condition into the clcultion. In the cse, the dditionl condition is the know length of the cble. Mthemticl description of the cble length is obtined by (6). If coordintes nd nd cble length re known, the equtions (4) nd (6) cn be used to derived (7). Becuse the prmeter () cnnot be expressed explicitly, itertion methods should be used. x A x B x x B A (6) dz x x l = 1 + dx = dx dx 1 + sinh =. sinh x B x A Fig. 3 Loction of the ctenry in the coordinte system IV. CALCULATING THE PARABOLIC CATENARY A. A model cse of the prbolic ctenry The distnce between the suspended points for this structure is l = 30 m nd the difference in height is h = 1 m. The continuous lod pplied onto the cble projection is q = 0.8 kn/m. The dditionl condition the length of the cble is L = 33 m. B. Itertion methods All itertion methods re bsed on the equtions (1) nd (). Another condition for those methods is selection of specific criteri, for instnce, the vlue of the first pproximtion or the termintion condition. In order to compre the solutions, sme vlues for identicl criteri were mintined. The use of the numericl methods ws bsed on [17] nd [18]. C. Direct itertion For grphicl representtion of this method see Fig. 4. The initil eqution (1) ws modified nd one side shows the horizontl force only tht side of the eqution represents the liner function (dotted), while the other side of the eqution comprises other input prmeters. This is the intersection of the solid line nd dotted curve. Then, the zero pproximtion nd itertion cycles result in the finl vlue. This is the intersection of the solid line nd dotted curve. ISSN:

3 Fig. 4 Prbolic ctenry results Direct itertion Fig. 5 Prbolic ctenry results for regul flsi The zero pproximtion needed for itertion is 1 kn. Two termintion conditions were specified. The first termintion condition is the exct number of itertion steps being 100. The second termintion condition is the devition between two subsequent clculted vlues being kn. With this method, divergence ws n issue. Becuse the curve which represented the modified eqution (1) ws convex, it ws moving towrds infinity in ech subsequent itertion step. Therefore, the lgorithm ws modified in order to use n inverse function (dshed curve). It ws not necessry to determine the entire inversion function. It is, however, more efficient for the clcultion to dd double of the difference between the originl function nd I nd III qudrnt xis to the originl function. This resulted in convergence. For the required devition the horizontl force ws kn. For the required number of steps, 100, the horizontl force ws kn. ISSN:

4 D. Regul flsi This method is sometimes referred to s the flse position method or the chord method. For generl bckground see Fig. 5. Once the eqution (1) is djusted to be homogeneous, this method gives intersection of curve with horizontl xis. The initil condition is the intervl in which the required vlue is locted. In every subsequent itertion step, chord line is creted between the outer points. Then, the vlue of the outer point in the intervl is replced with the vlue obtined by intersection of the chord nd horizontl xis. The first three itertion steps re represented by the chords of the curve. Becuse of the shpe of the curve under investigtion, this method itertes very slowly. In order to ccelerte convergence of this method, it would help mking the input intervl nrrower so tht the outer point could be s close s possible to the required vlue. The input intervl comprises the required result nd is limited by the lower boundry 1 kn nd by the upper limit 40 kn. The condition which will stop the itertion is the devition between the two subsequent clculted vlues. In order to keep the input conditions, this devition is gin kn. The horizontl force clculted using this method is kn. E. Bisection method The bisection method or the intervl dividing method is similr to the regul flsi becuse of its input criteri. The intervl under investigtion is limited gin by 1 kn nd 40 kn. Fig. 6 Prbolic ctenry results Bisection Fig. 7 Prbolic ctenry results Newton method ISSN:

5 The termintion condition is gin the difference between two subsequent itertion vlues: kn. The eqution for the cble length (1) ws modified nd is homogeneous now. In the next itertion step, the previous intervl with the required vlue is reduced. The vlue of the outer point of the intervl chnges fter the functionl vlue of the function under investigtion is compred in the hlf which precedes the intervl. If the difference ginst the next vlue is within the specified termintion devition, the clcultion will be interrupted. The bisection method is described in Fig. 6. The first five itertion steps re described there s horizontl lines. It is evident tht the intervl with the required vlue ( kn) becomes smller more quickly. Unlike the regul flsi method, the bisection method does not depend much on the shpe of the curve under investigtion nd itertes considerbly fster. F. Newton method This itertion method is shown in Fig. 7. The initil nd termintion conditions re identicl with those used in the direct itertion method. In the zero pproximtion point, the tngent to the curve under investigtion is found. Then, the intersection with the horizontl xis is found. The next tngent is constructed in the functionl vlue of tht point. The tngent represents the next itertion step. In the chrt, the first four itertion steps re visible. The pproch the finl vlue, kn, reltively quickly. In order to develop n lgorithm for this method it is necessry tht derivtions should be clculted in ech itertion point [5]. The model ws clculted using the three-point forwrd formul (8) with the 0.01 differentition. If other methods were used, the time needed for the modelling by mens of the Newton method did not extend. The number of itertion steps were not be influenced too. -3.f (x) + 4.f (x + dif) - f (x +.dif) f, (x) =.dif G. Comprison of the itertion methods When using the methods described bove, the resulting H ws kn for the given input vlues. Using this vlue, other geometric nd force prmeters cn be determined. Fig. 8 shows deflection of the cble for the specified vlues of the structure. Fig. 9 shows how the clculted horizontl forces depend on the number of steps of ech method. The zero step represents the initil pproximtion vlues. In terms of necessry steps, the regul flsi method is the most demnding it requires 76 steps. The reson for such high vlue is the shpe of the function under investigtion nd the initil vlues. The result is lso proved by the chrt which shows the itertion steps used in the regul flsi method, see Fig. 5. The lest number (8) of itertions (7 steps) ws needed by the Newton method. Tb. 1 shows the number of itertion methods in ech method s well s the time needed for the clcultion. Except for the Newton method, the time correltes with the number of steps. The reson for more time needed in the Newton method is rther long opertion in one step, the reson being clcultion of derivtions in ech point. The shortest time needed for clcultion of the horizontl rection ws for the bisection method. Unlike the Newton method, the bisection method does not hve enough input conditions. It is necessry to specify the intervl where the required vlue is locted. For this reson, it is recommended to use the tngent method. In this method, it is sufficient to determine the zero pproximtion nd the difference. The direct itertion needed lso shorter time thn the Newton method. But the direct itertion fces similr problem s the bisection method. H. Comprison of the itertion methods The Mtlb softwre [10] ws used to model by mens of itertion methods the horizontl rection of plnr prbolic ctenry. Method H [kn] Clc. Time (s) steps Itertion (step) Itertion (devition) Regul flsi Bisection Newton method Tble 1. Comprison of the methods results It is generlly ssumed tht the cble is perfectly bendble nd non-flexible fibre. The structure is supported in two suspended points by mens of solid joints the difference in the height of which is 1 m. The pln distnce between the supports is 30 m. The cble structure is loded with continuous lod where 0.8 kn/m is pplied onto ground projection. Algorithms were developed for the following methods: the direct itertion, bisection, regulr flsi nd Newton method. Attention ws pid to the time needed for clcultions nd for the number of itertion steps. The lest time needed for obtining the result ws mesured for the bisection method. This method, however, requires tht the intervl be known where the vlue is locted. Therefore, the Newton method is better choice for generl development of n lgorithm. The number of itertion steps is the lowest for the tngent method, even if the time needed for the clcultion is rther long. The dvntge of the tngent method over the bisection method is tht it needed only to enter the first pproximtion method nd difference. The reson for more time needed by the Newton method is the more extensive itertion step which clcultes derivtions in ech point. ISSN:

6 Fig. 8 Prbolic ctenry cble deflection Fig. 9 Prbolic ctenry comprison of the methods ISSN:

7 V. A MODEL CASE OF THE GRAVITY CATENARY A. A model cse of the grvity ctenry A model cse hs been used to support the ppliction of the itertion method for ctenry. Only three input prmeters were used in the clcultion. Difference in heights between the supports ws m. The ground distnce between the supports ws 40 m nd the length of the cble ws 45 m. It is necessry to know the lod in order to clculte the dimensioning force. This quntity, however, does not need to be known for determintion of () tht is why, this quntity ws not tken into ccount. B. Conditions for itertion methods The initil nd end conditions re essentil for the itertion methods. Becuse the gol is to compre the itertion methods, identicl conditions hve been used for ll methods. The initil condition of 10 is the zero pproximtion vlue. This condition is necessry for following methods: direct itertion nd Newton method. The regul flsi nd bisection methods require n intervl which comprises the solution. Limit vlues in the intervl were 10 nd 100. The Newton method required for derivtion three-point forwrd formul with 0.1 difference. The end condition ws identicl in ll cses. The difference between the two subsequent steps should be below C. Comprison of the itertion methods It is essentil to modify (7) in order to clculte () nd pply the itertion methods. The dependence of () ws determined for the direct itertion using (9). Fig. 10 shows the grphic chrt. The solution is point which is locted in the intersection of the curve nd the xis in the first nd third qudrnts. In cse of other methods, it ws necessry to djust (7) into homogeneous form (10). Fig. 11 shows this eqution the point of intersection with the horizontl xis is the solution. l h L.sinh. = (9) L..sinh l h = 0 (10). Tble shows the vlues of () nd complexity of clcultion in terms of needed clcultion steps. Fig. 1, which corresponds with results in Tble, shows development of the method, depending on the number of steps. Fig. 10 Function direct itertion ISSN:

8 Fig. 11 Function homogeneous Fig. 1 Course of the itertions ISSN:

9 Method Prmeter () Clculti on steps Time (s) Direct Itertion Regul flsi Bisection Newton method Tble. Comprison of the methods - results It follows from the results bove tht the rounded off vlue of () is Regrding the clcultion complexity, the most itertion steps were needed in order to rech the result for the direct itertion method. The lest clcultion steps were needed for the Newton method. However, big issue is there the input condition for the zero pproximtion. The curve in Fig. 11 shows tht the initil vlue should be bigger thn the minimum vlue of (10). For this reson, tht method cnnot be pplied in generl cses. Therefore, the most suitble method seems to be the regul flsi itertion method. A disdvntge is knowledge of n intervl with the known vlue. The clcultions included simultions with the incresing limit vlue of the input intervl up to Even with such n extremely high vlue, the time needed for clcultion or the number of clcultion steps hve not incresed considerbly. VI. CONCLUSION Algorithms bsed on the following methods were developed in Mtlb [10]: direct itertion, regul flsi, nd bisection nd Newton methods. The methods were used for clcultion of the prbolic nd grvity ctenries. The best method suitble for the grvity ctenry in generl pplictions hs ppered to be the regulr flsi method. As fr s the prbolic ctenry is concerned, the choice of n optimum method depends on criteri. The lest number of itertions ws needed for the Newton method. The shortest time needed for clcultion of the horizontl rection ws reched for the bisection method. The uthors will focus on further reserch on the use of specil cble elements [19] nd ppliction of probbilistic methods [0]. [6] M. S. A. Abd, A. Shooshtri, V. Esmeili, A.N. Ribi, Nonliner nlysis of cble structures under generl lodings. Finite Elements in Anlysis nd Design, 013, Vol. 73, pp DOI: /j.finel [7] H.T. Thi, S.E. Kim, Nonliner sttic nd dynmic nlysis of cble structures. Finite Elements In Anlysis nd Design, Vol. 47, 011. Iss. 3 pp DOI: /j.finel [8] M. Kiis, J. Idnurm, S. Idnurm, Discrete Anlysis Of Elstic Cbles. Bltic Journl of Rod nd Bridge Engineering. Vol. 7, 01, Iss., pp DOI: /bjrbe [9] I. Both, M. Ivn, A. Ivn, Anlysis of Suspension Crossing for Pipeline Using Finite Element Method. Influence of Elsticity Modulus, Proceedings Of The 3rd Wses Int. Conference On Finite Differences - Finite Elements - Finite Volumes - Boundry Elements, Buchrest, WSEAS Press. 010, ISBN [10] Progrm Mtlb, Informtion the progrm, Avilble: [11] O. Suchrd, o., J. Kubošek, Anlysing the Slbs by Mens of the Finite Difference Method. In Proceeding of the interntionl conferences : AMATH'13, Budpest, Hungry, December 10-1, 013. WSEAS Press, 013, p [1] P. Lehner, P. Konecny, P. Ghosh. Finite Element Anlysis of - D Chloride Diffusion Problem Considering Time-dependent Diffusion Coefficient Model, AMATH'13: Budpest, Hungry, December 10-1, 013., WSEAS Press, 013, p [13] M. Krejs, Algoritmizce inženýrských výpočtů (In English: Algorithmics engineering computtionl). [online]. 013 [cit ]. Avilble: [14] V. Křivý, P. Mrek, Probbilistic design of steel frme structures, [Zur Probbilistischen Bemessung von Sthlrhmen]. Sthlbu, Vol. 76, 007, Iss. 1, pp [15] J. Bend, Stvební sttik II. (In English: Building sttics II). VŠB-TUO, Ostrv, 005. [16] J. Kdlčák, J. Kytýr, Sttik stvebních konstrukcí I. Stticky určité prutové konstrukce. (In English: Sttics of structures I. Stticlly determinte bem structures.). Brno: VUTIUM, 001. ISBN X. [17] K. Rektorys kol. Přehled užité mtemtiky I. (In English: Overview of Applied Mthemtics I). 7. edition, Prh, Prométheus, p. ISBN [18] K. Rektorys kol. Přehled užité mtemtiky II. (In English: Overview of Applied Mthemtics II). 7. edition, Prh, Prométheus, p. ISBN [19] Y. B. Yng, J.Y. Tsy, Geometric nonliner nlysis of cble structures with two-node cble element by generlized displcement control method. Interntionl Journl of Structurl Stbility nd Dynmics, Vol.7, 007. Iss. 4, pp DOI: /S [0] M. Krejs, P. Jns, V. Krejs, Direct Optimized Probbilistic Clcultion, Recent Advnces in Systems Science nd Mthemticl Modelling, Proceedings of the 3rd Interntionl Conference on Mthemticl Models for Engineering Science (MMES '1), North Atlntic University Union, WSEAS Press, Pris, 01. ISBN REFERENCES [1] H. Ozdemir, A finite element pproch for cble probléme. Int. J. Solid Struct., Vol. 15, 1979, pp [] A.H. Pevrot, A.M. Goulois, Anlysis of cble structures. Comput. Struct., Vol. 10, 1979, pp [3] A. Andreu, L. Gil, P. Roc, A new deformble ctenry element for the nlysis of cble net structures, Comput. Struct., Vol. 84, 006, pp [4] H.B. Jyrmn, W.C. Knudson, A curved element for the nlysis of cble structure. Comput. Struct., Vol. 14, 1981, pp [5] T.V. Vu, H.E Lee, Q.T. Bui, Nonliner nlysis of cblesupported structures with sptil ctenry cble element. Structurl Engineering And Mechnics. Vol. 43, 01. Iss. 5, pp ISSN: