Self - supporting Dome Roof on Tank with V = m 3 capacity. New approaches to design

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1 Proceedings of he 4 TH INTENATIONAL CONFEENCE ADVANCED CONSTUCTION 9 Ocober, 4, Kaunas, Lihuania Kaunas Universiy of Technology, Faculy of Civil Engineering and Archiecure Self - suoring Dome oof on Tank wih V = 7 m 3 caaciy. New aroaches o design Lyubomir A. Zdravkov PhD, Associae Professor, Civil Engineer Universiy of Archiecure, Civil Engineering and Geodesy, Faculy of Civil Engineering, Hriso Smirnensky sr., floor 7, Sofia 46, Bulgaria zdravkov_fce@uacg.bg A French comany, in erformance wih rogram for exansion of is roducion, assigns design of ank for molasses wih V = 7 m 3 caaciy and self-suoring dome roof. The ank will be ereced in comany s sorage area, near o Dobrovice, Czech eublic, nex o oher wo, he same anks, wih V = 7 m 3 caaciy. One of he requiremens of Invesor is o reduce weigh of seel srucure, esecially weigh of he roof. Main reason for ha are very heavy secions, according o he oin of view of Invesor. Tha requiremen forces design o be as ligh as ossible, bu on safe side. Of secial ineres is calculaion of he elemens of roof sruc ure and ossible new aroaches o demonsrae heir bearing caaciy. Keywords: design, Lalace, roof cover laes, roof's srucure, self-suoring dome roof. Basic geomerical daa of he ank Diameer of shell - D = 64, m; Heigh of shell - Н =, m; Heigh of roof - f = m; adius of bending of dome roof - r =,5.D=,5.64=96,m; Number of radial girders in he roof - n r = cs.; radial girders in cenral ring of he roof - cs. According o real rojec, radial girders are wih secion IPE. Circular elemens - none, see Fig. ; Thickness of roof cover laes - r = 5 mm; To angle on he shell - angle secion Lx; oof srucure and cover laes are made by seel S355. s μi. Ce. C. sk,8...,75,6 kn/m () μ i is he snow load shae coefficien; C e he exosure coefficien; C he hermal coefficien; s k he characerisic value of snow load on he ground for given locaion. - wind - basic wind seed v b, = 36, m/s; The eak velociy ressure q (z) a heigh z, which includes mean and shor - erm velociy flucuaions, could be calculaed using formula as follow, see sandard EN :5: q ( z ) 7. I ( z )..ρ. v ( z ) v m, N/m () ρ =,5 kg/m 3 is he air densiy; v m (z) - mean wind velociy a heigh z above errain level; I v (z) - he urbulence inensiy a heigh z. Fig.. Aboveground seel ank wih V = 7 m 3 caaciy for molasses. Loads on dome roof - overressure - р о = 5 mbar; - negaive ressure (vacuum) - v =,5 mbar; - snow load on he ground of area of ank s exloiaion (Dobrovice) - s k =,75 kn/m ; Snow load s on he roof of ank could be calculaed by formulae, see sandard EN 99--3:6: Coefficien for exernal ressure с ре should be defined o deermine wind ressure acing on he exernal surfaces. The coefficien с ре should be accouned according o Fig. 7. of sandard EN 99--4:5, in resec of geomerical values of he ank h, d и f: h,,344 c d 64 c f, c d e,b e,c,35,6,4 (3)

2 Wind ressure acing on he exernal surfaces w e, should be obained from exression: w (4) e q( ze). ce for.а w for.b w for.c w e,b e,c q q q ( z ( z ( z e,b e,c ). c ). c ). c e,b e,c 346,95.,35 368,4 N/m 47,95.,6 483,8 N/m 346,95.,4 938,8 N/m Mean characerisic value of wind ressure w e,m by whole roof could calculaed using formula: w e,m,3. w,5. w e,b,. w,3.368,4,5.483,8,.938,8 88, N/m e,c (5) As a resul of research, acceed hickness of roof cover laes is r = 5 mm of seel S355, as i is a consrucive minimum. So, he hickness by rojec is he same. b) roof srucure As he radial girders which are he ars of he dome are eccenric ressured, mos imoran should be verificaion for general sabiliy loss. Used mehod in he research is so called General Mehod which has been a lile bi modified. Mehod has been described in sandard EN A saial design model of he dome has been creaed wih he sofware SAP. Oion Buckling Analysis is acive. This oion gives chance o calculae he value of bearing caaciy of he consrucion before i losses sabiliy, arially or enirely, see Fig.3. Used radial girders in saial roof s model have secion IPE 33 from seel S Loads combinaions Load combinaions could be defined on wo main caegories, as follow: а) combinaion of loads acing from o o boom q γfg,su. gn γq. s γq.ψ. v (6),6.,35,5.,6,5.,6.,5,935 kn/m γ fg,su =,35 is self-weigh overloading coefficien, according o EN 99; γ Q =,5 - overloading coefficien for emorary loads; ψ - coefficien for simulaneously working wo or more emorary loads, according o EN 99. b) combinaion of loads acing from boom o o q γq. we,i γq.ψ. o γfg, inf. gn (7) where coefficiens of overloading are: γ fg,inf =, self-weigh overloading, when acs favourably, see sandard EN 99. for.а q,5.3,68,5.,6.,5.,6 4,6kN/m mean q,m,a γ Q Fw γ. w. w e,m γ γ γ γ Fg,inf,5.,88,5.,6.,5.,6,67kN/m Q Q.ψ..ψ. o o Fg,inf. g. g 4. Tradiional aroach for design of roof's elemens a) roof cover laes Acceed saic scheme of roof laes is a muli-san girder on many suors, as is shown on Fig.. Disance beween suors is measured where riangle field, ransmiing loads o shell, ends. Fig.. Scheme of roof's cover laes n n Fig. 3. Deformed shae of dome roof, ha looses sabiliy, due o loading of self-weigh g n, snow s and vacuum v According o done research, acceed radial girders on he roof wih secion IPE 33 will no loss sabiliy. Obviously, he calculaed secion IPE 33 is bigger several imes comaring o he real used secion IPE. Somehing more he sherical roof of he ank V = 7 m 3 in Dobrovice is execued and u ino exloiaion several years ago and is sill in a good condiion. The ossible reasons for his difference in he resuls are as follow: - he emorary wind, snow and vacuum loads did no reach for now heir criical values, which would desroy he sherical roof; - new, more accurae aroach in deermining he bearing caaciy of he roof s elemens. Second oion seems more reasonable. Anyhow, design of such big anks wih V = 7 m 3 caaciy is no given o he inexerienced comanies. 5. New aroach for design of roof's elemens The resuls of done analysis gives o me reason o conclude ha classical aroaches for design of saial domes are oo conservaive. Because of i I begun o look for anoher alernaive aroaches for analyses and design of such ye of consrucions. а) design of roof's cover laes as a saial shell

3 For he roof's cover laes which should be seen as a hin shell wih consan hickness, he equaion of Lalace could be used: σ m σr (8) m r σ m is a normal sress in meridional direcion, see Fig. 4; σ r normal ension in he radial (annular) direcion ; m radius of bending in meridional direcion; r - radius of bending in radial (annular) direcion; he value of he ressure on he shell which can be a funcion only on he coordinae z; - hickness of he shell. Limiaion of he equaion (8) is ha i can be used in hin shells only, which can be researched according o membrane heory. q,a 4,6 (). r.96,65m, 65mm f y γ,5 M I should be noed ha when considering work of he roof's cover laes as a 3-D shell, loaded by combinaion q,i, radial girders are no necessary, i.e. heir sress check is saisfied auomaically. Obviously such an aroach for considering he work of he roof's cover laes as a 3-D shell is ossible when he deails for is joins are aroriae for his urose. For insance, he roof shees should be joined beween hemselves hrough bu welds wih full eneraion. Deermined by he formula () minimum hickness of he roof's cover laes r does no corresond wih he cases wih oin loads, caused by eole or equimen on he roof. In his case he roof shees should be considered and designed as laes working on he bending. b) design of roof srucure for loss of sabiliy - mehodology of Evoluion Grou for EN Proosed on by Evoluion Grou for EN mehodology should be alied o smooh sherical shells. For his urose siffened shell of he sherical roof shall be ransformed o equivalen smooh shell which has he same bending siffness. On he base of equilibrium beween he momens of ineria of he roof's cover laes and radial girder we calculae: Fig. 4. Shell wih axis of symmery (roaion) z and consan hickness In he sherical shells in which radius of bending in all direcions is one, i.e. m = r =, normal ensions in all direcion would be equal. They can be deermined according o he formula: σr σm. (9). Using i we can calculae minimum of he necessary hickness of he roof's cover laes: q,m,67 (). r.96,4m, 4mm fy γ,5 M f y = 355 MPa - yield srengh of seel S355; γ M =,5 - coefficien of safey by maerial, according o EN I could be seen ha difference in necessary hickness of roof's cover laes, when i is considered as a siff lae working on ure bending, and as a smooh shell working on ure ension, is much bigger. Several imes bigger. Even if we are measuring he roof's cover laes wih load combinaion q, A for oin A, i should be more economical: 3. an. I () And afer clear ransformaion :. I cm (3) a n a n is a disance beween he radial girders in oin of join o he ank s shell ; equivalen hickness of he roof's cover laes; I common momen of ineria of he girder and roof shee wih hickness a n. Here, in his research, conribuion of roof's cover laes is no considered. Acceed in formula (3) value for I = I y = 77 cm 4, as much as is he momen of ineria of he radial girder IPE. To be valid he alied in Evoluion Grou for EN mehodology, he following condiion should be mached: r 3 (4)

4 r ,6 3 he shown saemen could be used. Criical value of he alied ressure, in elasic range, will be calculaed according o: r,l,365 λ,54 λ,58 r,cr λ =, is squash limi relaive slenderness., () r,cr 3.(. C. E. c ) r (5).,7.. 3.(,3 ) 96,58 kn/cm С с =,7 is a coefficien, accouning condiions of shell's suors in is erihery; Е = kn/cm module of he seel elasiciy; = = cm equivalen hickness of he smooh sherical shell; ν =,3 coefficien of Poisson. Criical value of he alied exernal ressure, in lasic range, will be calculaed according o: Plasic limi relaive slenderness of he shell calculaed according o: λ λ will be α,47,6897 λ,54 () β,7 β =,7 is a lasic range facor in buckling ineracion; η =, - ineracion exonen. The loss of he sabiliy coefficien χ will be calculaed according o: α,47,8 () λ,54 χ r, l fy,k. Cl. 35,5.,9.,365 kn/cm (6) r 96 f y,k = 35,5 kn/cm is a characerisic value of yield srengh of seel S355; C l =,9 coefficien, accouning suoring condiions of he shell. The amliude of characerisic imerfecion Δw k will be calculaed as follow: w k. r ,348 cm (7) Q 6 Q is a arameer, accouning roducion qualiy. Q = 6 in case of normal qualiy. The imerfecion reducion facor α I will be calculaed according o: I w,9. k,75 4,348,9.,75 (8),38 Coefficien accouning he geomery imerfecion α G =,7. The elasic imerfecion facor α deends on direcly by α I and α G, and can be calculaed by formula:,38.,7,47 (9) I. G In lasic range: r, l r, l,365. l 88,63 (3) q,935 Ed In elasic range: r,cr r,cr,58. el 3,8 (4) q,935 Ed The coefficien k, indicaes how many imes we can increase he loads before ha shell will lose sabiliy, characerisically, will be calculaed according o he formula: χ.,8.88,63 4,3 (5) k l Design value of he reserve of bearing caaciy d, is deermined below: k 4,3 d 4,95 (6),5 γ M From he equaion (6) i follows ha he roof dome consruced by ieces of radial girders IPE, considered as a equivalen smooh shell will no lose sabiliy when he loads are symmerically and evenly disribued. - mehodology of Вольмир (956) Wih sufficien correcness for racice, for sherical shells wih raio r / = 4 and cenral verical angle of he dome θ =.α n = 4, criical values q cr of exernal ressure, when shell will lose sabiliy, can be deermined. The formula is as follows: elaive slenderness λ will be calculaed according o:

5 ra cr,3... k E r q (7) k is coefficien, calculaed by formula: θ 4 r k,75.., , ,7., (8) q cr,3. k. E. 4,3kN/m qcr q (9) r q,935kn/m According o used mehodology of Вольмир (956), sherical shell should no loses sabiliy loaded by load combinaion q. - mehodology, wrien in "Справочник проектировщика" (973) According o ha mehodology, he criical values q cr of exernal ressure, wherein sherical shells will lose sabiliy could, be deermined by exression: Fig. 5. Model of dome roof, where cover laes are included Values of inernal forces are accouned on some oins along he radial girders. Srengh check is made o he sufficiency of secions: М max = -8, 779 kn.m N c = + 63,39 kn 63,39 8,779. σmax 33,4 85 (3) f y 35,5 8,49kN/cm 33,8kN/cm γ,5 M 4 E. qcr. K 4. f f K.,3kN/cm kn/m f K3. K ,5.,576.,47., N max = + 98,64 kn M c = - 3,438 kn.m 98,64 3,438. σmax 33,4 85 f y 35,5 7,67kN/cm 33,8kN/cm γ,5 M (3) is hickness of he shell; - radius of circular base of sherical shell; f - heigh of dome roof; K, K, K 3, K 4 - coefficiens deending on suoring condiions of dome roof. By analysis of he sresses in roof shees of load combinaion q, see Fig. 6, i is seen ha resence of radial girders IPE on racice does no change membrane sress of roof laes. In oher words, has a reason o use Lalace s equaion, shown on formula (8). Comaring values of load combinaion q and criical values q cr of exernal ressure, accouned by formula (3) is clear ha shell should no lose sabiliy. I should be noed ha according o his mehodology, calculaed values for q cr are unrealisically heigh. 6. Deermining inernal forces in radial girders wih secion IPE, when roof laes are loaded by combinaion q A saial comuing model is creaed in order o deermine inernal forces in radial girders IPE and heir influence on hin seel laes. Sofware roduc SAP v.4. is used. The model includes all elemens of he roof - radial girders, cover laes, cenral ring and o angle, see Fig. 5. Fig. 6. Normal sresses by von Mises in roof cover laes, kn/cm

6 7. Conclusions Self-suoring seel dome roofs are buil by girders and cover laes. When hese elemens are reliable conneced, dome roofs could be considered as a saial smooh shells. And aly o hem mehods and analyzes of number of researches working on shells. As a resul, aking ino accoun saial work of laes and srucural elemens, can significanly reduce amoun of seel. Basic disadvanage of used here aroach is ha all researches, resecively normaive documens, consider only cases of a uniformly disribued loads. A snow on he roof, blowing by wind, can be moved, o accumulae, wherein he load becomes asymmerical, i.e., less favourable. On oher hand, Euroean sandard for snow loads EN 99--3:6 do no considers case of an uneven load on sherical domes. eferences EN 99:, Eurocode - Basis of srucural design. EN 99--3:6, General acions - Snow loads. EN 99--4:5, General acions - Wind acions. EN 993--:5, Design of seel srucures - Par -: General rules and rules for buildings. Evoluion Grou for EN Summary of Amendmens o EN :7 for 3. SAP v.4.. Srucural analysis rogram. Comuers and Srucures, Inc. Вольмир А. С., Гибкие пластинки и оболочки. Техтеоретиздат, 956. Справочник проектировщика промишленных, жилых и общественных зданий и сооружений. Под редакцией проф. д-р А. А. Уманского, Издание второе. Москва, 973. Lyubomir ZDAVKOV PhD, Associae Professor, Civil Engineer, Dearmen Seel and Timber srucures Main research area: seel srucures Address: Universiy of Archiecure, Civil Engineering and Geodesy, Faculy of Civil Engineering, Hriso Smirnensky sr., floor 7, Sofia 46, Bulgaria zdravkov_fce@uacg.bg