omprtive Investigtion of ril lod nd Finite Element ethods in Anlysis of Arch Dms Vhid Nourni Abstrct: Becuse of importnt role of dms nd dm construction in humn life, in the present pper the method of nlysis of n importnt kind of dm (rch dm) hs been presented in two different scientific wys nd their results hve been compred. In the method presented herein, the dm hs been divided into horizontl elements of rcs nd verticl elements of cntilevers, nd using comptibility of displcements nd tril nd error (tril lod method) the shre of cntilevers nd rcs from pplied lods on dm hve been determined. hen nother nlysis hs been performed using Finite Element ethod (FE) by indicting stiffness mtrix using iso-prmetric hexhedrl elements with eight nodes. Using the vilble equtions, the displcements of nodes hve been clculted. Becuse of high volume of clcultions, computer hs been used nd softwre hs been prepred. he results of these two methods hve been compred to ech-other. he results show tht the tril lod method is relible method in spite of the fct tht simplifying ssumptions hve been used in its theory. As result, n rch dm cn be esily nlyzed by tril lod method. Also, to get more ccurte results, more complete methods re necessry to solve FE equtions. Index erms: Arch dm, ril lod method, Finite Element ethod, Arc nlysis, ntilever nlysis. I. INRODUION Dm nd dm construction hve been of gret importnce in humn life for optimum use of surfce wter reservoirs. Engineers hve been trying to construct this hydrulic structure by the best vilble method. Arch dm is of much more importnce mongst ll other types of dms. An rch dm is constructed on nrrow vlley with strong foundtion. his type of dms is more economicl, becuse of less depth nd volume in comprison with other types of concrete dms, but its nlysis is of specil difficulty nd hs strict need for knowledge of mthemtics nd strength of mterils. For this purpose reserchers hve presented vrious methods of nlysis. Becuse of complexity of the structure ll methods re pproximte methods so the ccurte output is not presented by these methods. Amongst the different methods, cylinder theory, rc nlysis method, method of tril lod [1] nd Finite Element ethod [], cn be mentioned. In the present pper it will be tried to explin the lst two methods nd compre the results nd through tht the method of rc nlysis will be presented. Revised Version nuscript Received on July 9, 16. Prof. Vhid Nourni, Deprtment of ivil Engineering, University of briz, briz, Irn. II. EORY OF ANALYSIS OF AR DAS BY RIAL LOAD EOD wo ssumptions cn often be considered to nlyze rch dms. At first the rch dm is divided into horizontl rcs with known lengths, ech rc supported on rocky foundtions of the nrrow vlley t two ends. hen the forces re cted upon ny rc nd the rcs re nlyzed nd the efficiency of its cross section is investigted. In this cse only the rcs trnsfer the lods towrds lterl wlls of the dm. he second method of nlysis, which is more ccurte nd is not fr from relity, is tht the dm is divided into verticl elements in the form of cntilever bems s well s horizontl rcs. hese cntilevers re fixed t the bse of the dm. In this cse frction of externl forces is trnsferred in the rocky wlls through horizontl rcs nd the remining prt is trnsferred through cntilevers in the foundtion of the dm. Both methods cn be used. he only difference is tht, in the first method one of the chrcteristics of the structure hs been ignored so tht the fctor of sfety is higher thn the second method nd s result the designed dm will not be n economicl. In second method the importnt point is how to divide the lod between the rcs nd cntilevers. As mentioned before it is ssumed tht rcs support prt of totl lod nd the other prt by cntilevers, but how much is it? o indicte the shre of rcs nd cntilevers from the lods, method of tril nd error is used. At first prt of the lod is given to cntilevers nd the remining prt is given to rcs nd then t prticulr points which re common in cntilever nd rc, the tngentil, rdil, nd rottionl deflections re clculted for both systems nd their differences re compred to ech other. he ttributed mgnitudes of lods for rcs nd cntilevers re vried until the deflections become equl t common points. Usully for the simplicity of clcultions, only the comptibility of rdil deflections is tken into ccount. he nlysis of cntilevers is similr to the nlysis of fixed end bem with vrible cross section under linerly distributed lod. For nlysis of rcs the comptibility equtions, which re usully pplied in stticlly indeterminte structures, re used. A. Anlysis of the rcs Any rc from dm is n indeterminte structure with three degrees of indetermincy. For nlysis, it is divided into two hlves from the crown nd the internl forces re considered in both sides of the structure nd principles of comptibility equtions re used. It mens tht: L R L R δ = δ, δ = δ, θ = θ (1) L R V V in which L nd R indicte the left nd right sides of the rc 1
omprtive Investigtion of ril lod nd Finite Element ethods in Anlysis of Arch Dms nd,v nd θ denote to horizontl,verticl nd rottion displcements respectively. Solving the system of equtions, the internl forces t the crown of the rc re determined. he rc is divided into smll elements nd the resultnt of externl forces cting on ech element will be exerted s concentric lod. he rottion resulting from the moment () cn be obtined from eqution () in element No (s n exmple), of Fig. 1 s follows: S α = () EI in which S = length of the element E = modulus of elsticity I = moment of inerti of the section If the depth of the rc is ssumed s unit, then I = 1 he totl cting on ech element will be s follows: = y V x () E Where,,, nd V re the internl forces t the crown of the rc (Fig.1), nd E is the moment produced by externl forces t the element. By summing up the eqution () on elements t left hnd side of the crown of the rc, we cn write: S S y S V X S α (4) EI EI EI EI E = Similrly this cn be done on ll elements t left hnd side of the rc nd the mount of rottion produced by moment cn be computed t crown of the rc. It should be noted tht the concentric lods cting on ech element re becuse of the distributed lod of wter nd hydrodynmic lod of erthquke tht hve been multiplied in length of externl rc of the element nd cts on the middle of externl surfce of the element. If the rottion produced by therml grdient ( f) nd lso the rection of butments re dded to eqution (4), then the totl rottion produced t the crown of the rc cn be computed s follows: = α α α (5) f α In similr wy the displcements in x nd y directions cn be computed t the crown of the rc considering the effect of left side elements s follows []: x = x x x x x S f f y α S osα n Sinα (6) y = y y y y y S f f x α S Sinα n osα (7) in which,, S, f, nd f represent effects of moment, tngentil force, sher force, constnt temperture nd therml grdient to produce deflections respectively. t Subscript represent the effect of rections of elstic butments of dm which cn be computed using Fredrick Wogutt method []. his effect depends on thickness of dm in butments, ngle of dm with the bse, modulus of elsticity of butments, the rtio of dm cross section to bse cross section t the intersection point, nd Poison s rtio of the bse. Similr procedure cn be used for right hnd side of the rc. Using the comptibility equtions of (1) nd simplifying them, set of equtions with the internl forces of the rc t the crown s unknowns cn be obtined s follows. A BαV α Dα = B V D = α (8) A (9) x x x x ByV y Dy A = (1) y In which A, B, nd re the coefficients of forces, V, nd α considers the effect of rottion, the indices x nd y show the effects in displcements in x nd y directions, nd D prmeters re the coefficients for externl forces of the rc. After finding these unknowns from equtions (5) to (1), the displcements produced in ll elements cn be obtined. For this purpose, it is enough to sum up from the crown of the rc to the element under considertion. ence three displcement components will be obtined for ll elements of the rc. If the thickness of the rc is constnt Integrtion cn be used in plce of summing up, nd the problem will be simpler; for this purpose U.S.B.R [1] hs produced tbles nd simple reltionships, which cn be used insted of integrtion. Using U.S.B.R. tbles nd hving the bove-mentioned prmeters for unit lod, the displcements cn be computed for totl lod under considertion. In reference [4] some more simplifying ssumptions re presented to nlyze the rcs. B. Anlysis of the cntilever At first it is ssumed tht similr to nlysis of the rc, the lod is given to cntilevers nd fter nlysis, the displcements of points re obtined nd the results re compred for common nodes in two structures. As mentioned before, nlysis of cntilever is similr to fixed end bem with combined section. When shring the lods between rcs nd cntilevers, verticl forces of wter nd the weight of concrete (ded lod) re crried by cntilever nd the horizontl force of erthquke by rcs. he horizontl force of wter is shred between the cntilever nd the rc on the bse of comptibility of displcements. he cntilever, which is stticlly determinte structure, is nlyzed esily using theories of strength of mterils. he detils cn be seen in reference [4].. he process of tril lod method As he nlysis cn be performed in three methods: Anlysis of centrl cntilever: In this type of nlysis, only the nlysis of centrl cntilever nd the rcs re considered nd the displcements computed in the crown of rcs re compred with the displcements of centrl cntilever. In this method the comptibility of rdil displcements is considered. his method is usully used in symmetric rch dms. Anlysis of rdil displcement: It is similr to (). he
difference is tht some other cntilevers in neighborhood of the centrl cntilever re nlyzed. he complete tril lod method: his is similr to (b). he difference is tht, rther thn rdil displcements, tngentil nd rottionl displcements re computed s well. It is obvious tht the method (c) will result in more relible nd confident output becuse ll displcements nd ll nodes of dm re considered. owever the method is time consuming nd more expensive thn other methods, but it will give more economicl cross sections for the dm. III. ANALYSIS OF AR DAS USING FINIE ELEEN EOD In this method, three dimensionl, isoprmetric elements re used to nlyze rch dms (similr to element solid in SAP softwre) (Fig. ). he reltionship between internl forces nd nodl displcements cn be written in mtrix form s follows: kq Pb P = (11) P Where, k is stiffness mtrix, q is the nodl displcement mtrix, P is externl point lod mtrix pplied in nodes, P b is the volume forces mtrix, nd P is the mtrix for nodl equivlent forces becuse of initil strins (such s effect of vrition of temperture) which re written s follow: k = B EBdv P P = F bdv (1) b (1) = B E dv ε (14) B = df (15) In which, B nd F re shpe functions mtrix; d is differentil opertor to trnsform displcement function mtrix to nodl strin mtrix. E is mtrix-relting stresses to strins (σ = E. ε ) nd b is the volume force pplied on the element (such s weight of the element) nd dv is volumetric differentil. In eqution (14) if the initil strin is due to uniform vrition of temperture ( f ) in the element, we will hve: ε = f f ( f is the coefficient of temperture drop) (16) In the next step, to determine the stuffiness mtrix of ech element, it is necessry to trnsform the elements into stndrd elements with unit dimension, (Fig. ), which cn be done using Jcobin mtrix (J). In this cse the stiffness mtrix is computed from the following reltionship: k = 1 1 1 B ( ξ, η, ζ ) EB( ξ, η, ζ ) J ( ξ, η, ζ ) dξdηdζ (17) In which ( ξ, η, ζ ) re the coordintes in the stndrd coordinte system for elements hving eight nodes, mtrix B is 4 6, mtrix E is 6 6 nd mtrix B is 6 4 then the mtrix k will be 4 4. o solve the eqution (17), the so-clled numericl method of Guss-Newton [5] my be used. Similrly, the nodl volume forces nd initil strin for ech element is produced nd the stiffness mtrix of the structure is obtined nd then the eqution (11) is solved nd mtrix of nodl displcements is obtined t different points. IV. PREPARAION OF E SOFWARE FOR ANALYSIS OF E AR DA he most used softwre for nlysis of rch dms is (ADAP) tht hs been prepred in U.S.B.R. It nlyzes the rch dms using two-dimensionl (plne strin) nd three-dimensionl elements. o compre the bove-mentioned methods, three softwres were prepred by the uthors using the theories outlined bove for nlysis of rch dms. For tril lod method, becuse of high volume of repetition of opertions, computer progrm ws prepred in Visul Bsic, in which the tril lod opertions could be performed only for rdil displcements nd ll cntilevers of dm. In the progrm, the detils, which were explined in section, hve been considered; the other two progrms were prepred by FE. he difference between these two progrms ws the wy of reservtion of stiffness mtrix nd the method of solution of eqution (11). he min difficulties in FE progrmming re the utomtic selection of elements, the construction of generl stiffness mtrix nd lck of memory to sve it nd solve the eqution (11). o encounter the first difficulty, it is possible to use n utomtic mesh genertion when the dm is symmetricl. Otherwise, ll elements should be defined. For second difficulty, the numericl methods re used. So, in one of the progrms, the bnd mtrix nd in the other one the frontl method [6] were used. For detils of generl structure of the progrm nd subroutines the reference [4] cn be seen. V. RESULS AND DISUSSION OF A NUERIAL EXAPLE o compre the methods, symmetricl dm is considered nd only one hlf of the dm is to be nlyzed. o nlyze the dm, four rcs nd four cntilevers hve been considered (Fig. ) nd the comptibility of rdil displcements hs been stisfied by tril lod method using the prepred softwre. he min purpose is to determine the shre of hydrulic force portion tht is divided between cntilevers nd rcs nd lso mount of the rdil displcements in common points. he specifictions of the dm re s follows: - Unit weight of concrete 4 Kg / m Kg / m - Unit weight of wter 1 - he ccelertion of erthquke prllel to xis.1 g - emperture drop 4 = t 8 (t is section thickness) - Rdil drop of temperture is neglected f = - he coefficient of temperture drop f =.6 1/ - Allowble tensile stress - Allowble compressive stress 5 - he modulus of elsticity of dm 1 - he modulus of elsticity of bed rock 8 It is necessry the geometricl chrcteristics of ech
omprtive Investigtion of ril lod nd Finite Element ethods in Anlysis of Arch Dms cntilever nd lso rcs should be given s dt into the softwre. he geometricl properties of the dm re given in ble 1. he profiles of the dm s well s two cntilevers A nd B re shown in Fig.. For geometry of rcs the centrl ngle, externl rdius, the depth of the rc nd lso the number of elements should be given. Using these dt, the results of tril lod method for cntilevers A nd B will be s shown in Figs. 4 nd 5. In these digrms the horizontl contributions of wter lod tht re crried by these cntilevers re drwn. In Fig. 6 the contribution of the rc is drwn in height h=4 m. he dm is nlyzed by FE using the softwre produced by the uthors s well. o compre these two methods, the rdil displcements (in y direction) of centrl cntilever (A) t points of intersection re given in ble. he mximum displcements of the dm will be produced in this direction nd t this plce (cntilever A). (he results re on the bse of the ssumption tht the dm is solid.) As there is no ccess to true strins, one cnnot judge bout the ccurcy of these two methods, but by comprison of the results shown in ble, the comprbility of the results shows the relibility of the methods. Also, it cn be concluded from ble, tht the results of FE ner the bse of the dm gives lesser displcements in comprison to tril lod method nd this is inverse in upper prts. he reson cn be found in the structure of the nlysis. As described in section 1- in tril lod method to nlyze the rcs, it is ssumed s curved bem with two fixed ends in which cse the rc of height h=6 m will hve only two fixed ends nd the effect of fixity of the bse will not ffect on this rc. In finite element method, becuse of constructing the generl stiffness of the structure, nd interring the fixity of the bse by reducing the degrees of freedom of elements, the rc of h = 6 m is fixed from sides s well s down side nd so the displcements resulted in this method re lesser in rc in comprison to tril lod method. VI. ONLUSIONS AND REOENDAIONS. In spite of the fct tht some simplifying ssumptions hve been considered in tril lod method, still rch dms with different geometricl shpes cn be nlyzed by this method nd in spite the progress of other methods such s finite element method, this method cn compete with them. he rch dms tht hve been nlyzed by tril lod method in the pst cn be ccepted nd relied on their stbility nd economy. b. he results of the computer progrm, tht hs been produced using finite element method nd using bnd mtrix, re of high errors. his is due to high number of equtions which in solving simultneous equtions, the errors re incresed. By investigtion in the results of the solved exmple, it cn be concluded tht the importnce of usul methods of nlysis such s Guss bnd mtrix, Guss-Seidel [7], nd other methods of nlysis by finite element, which re discussed in litertures, re clssic methods only for eduction purpose nd lerning nd not for ppliction problem. For ccurte nlysis, some other specil methods re necessry tht will be noted in 6-4. c. he reson for ccurcy of tril lod method in spite the simplifiction used, is the independence of equtions. In this method the mximum number of simultneous equtions to be solved is, which definitely will not increse the error. d. o increse the ccurcy of the nlysis by finite element method, the wy of sving the dt nd method of solving the equtions should be optimized. For this purpose we used frontl method in progrmming to reduce the memory needed so the number of elements cn be incresed nd more ccurte results cn be chieved nd by choosing right method of solving equtions, increse of error cn be prevented. It is recommended to use other numericl methods such s Skyline [7] or use combintion of different methods to increse the efficiency. For exmple to reserve the equtions the method of Skyline nd to solve them the frontl method re recommended [4]. e. As both methods re pproximte methods nd there ws not ccess to true results, it cnnot be judged on the ccurcy of the methods, therefore, it is recommended tht lbortory model is mde nd strin guges used in the body of dm model to get more ccurte nd more nturl results, then the ccurcy of the methods cn be studied. ble 1- Geometric Dimensions of Dm in eters eight Rdius hord of rc hickness (t) 18. 97.5 6 16 9 4.5 1 1 88.4 6. 18 97.7 8.8 8.8 4 9.7 94. 11.4 87.7 74.7 1.6 6 8.7 7.1 15.7 4 78 65.5 17.6 48 7.5 6. 19. 54 69 54.8.7 6 64.4 47. 1.8 66 59.5 8.1.5 7 54 1..8 ble -ngentil displcement of cntilever A eight of rc (m) ril Lod ethod (cm) Finite Element ethod (cm) 5.5 6.7 4. 4.1 4.1 1.8 6 1.7 Fig. 1- One hlf of the selected rc from dm with pplied forces 4
Lod(t/m^) 5 ξ Arc lod enter of element ζ ntilever section b Fig.- he stndrd cubic element nd iso-prmetric element 15 d 84 c 6 4 Distnce from crown of rch(m) 1 Fig. 6- Lod shre of rc (h=4 m.) Fig.- Dm profile nd the sections of cntilevers A nd B Wter pressure Arc section REFERENES 1. Design of rch dms, USBR (1977).. Zienkiewicz, O.., the finite element method, cgrw-ill (1977).. Greger W.P. & Justin J.D. & inds J., Engineering for Dms, Wiley (1964). 4. Nourni, V. A comprtive investigtion of tril lod nd finite element method in the nlysis of rch dms..sc. thesis, briz University, (in Persin) (). 5. Bthe, K. J., Finite element procedures, Prentice-ll (1996). 6. Jennings A. & ckeown J.J., trix computtion, John Wiley nd Sons (199). 7. Belegundu.D. & hndruptl D.A., Introduction to finite elements in engineering, Wiley (199). 8 ntilever lod 4 Lod(t/m^) Fig.4- Lod shre of cntilever A 4 6 7 h(m) Prof. Vhid Nourni, Received his B.Sc. nd.s. degrees in ivil Engineering from University of briz, Irn in 1998 nd, respectively. e then continued his grdute study in ivil nd Environmentl Engineering in the field of ydrology t Shirz University, Irn nd ohoku University, Jpn nd ws grduted in 5. Prof. Nourni ws with the Fculty of ivil Engineering, University of briz s n Assistnt Professor from 5-9; s Associte Professor from 9-14; s Professor from 14 nd with Dept. of ivil Eng., University of innesot, USA t 11 s visiting ssocite professor. In this period, 6 Ph.D. nd.s. students were grduted under his technicl supervision. is reserch interests include rinfll-runoff modeling, Artificil Intelligence pplictions to wter resources engineering, ydroinformtics nd computtionl hydrulics. is reserches outcomes hve been published s 1 Journl rticles, books, 7 book chpters nd more thn 1 ppers presented in interntionl nd ntionl conferences. Wter pressure Arc section ntilever lod h(m) 4 6 8 4 lod(t/m^) Fig.5- Lod shre of cntilever B 7.5 5