MINIMUM COST DESIGN OF A BUNKER CONSTRUCTED FROM WELDED STIFFENED PLATES

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1 Frks,J.,Jármi,K.: Minimum cost design of bunker constructed from welded stiffened pltes, 53 rd Annul Assembl of Interntionl Institute of Welding, Florence, Jul 9-4,, IIW-Doc. XV- 38-, 6 p. MINIMUM COST DESIGN OF A BUNKER CONSTRUCTED FROM WELDED STIFFENED PLATES J. Frks nd K. Jármi, Universit of Miskolc, Hungr Abstrct: The min structurl prts of steel bunker re sstemticll optimized for minimum cost. The pressure distribution due to the stored mteril in bin wlls is nerl hdrosttic, therefore the optimum positions of horizontl stiffeners re clculted using the condition tht ll the prts of the bse plte strips should be stressed to ield strength. The optimum number of stiffeners is determined in bin nd hopper wlls to chieve the cost minimum. Trpezoidl stiffeners re designed for bending. The trnsition bems re loded b hopper rections nd should be designed for bending s horizontl welded I-bems. The totl mteril nd fbriction costs re determined for three optimized bunker structurl solutions hving different rtios of bin height/width. In the investigted numericl exmple of cement bunker the structurl version of bin height/width rtio of nd height of 6 m hs the minimum totl cost. The cost of bunkers with rtios of.5 nd.5 is 63% nd 6% higher, respectivel. Kewords: minimum cost design, steel bunkers, stiffened pltes, welded structures, structurl optimiztion IIS/IIW- (ex.doc.xv-38-) Clss A, recommended for publiction b IIW Commission XV Fundmentls of design nd fbriction for welding

2 - -.Introduction The im of the structurl optimiztion is to chieve svings in weight nd cost in design stge b chnging some significnt structurl chrcteristics. A cost function should be minimized, which contins vribles expressing the structurl chrcteristics. These chrcteristics re s follows: lods, mterils, profiles, geometr, topolog, fbriction, trnsporttion, erection, mintennce. Our im is to show this design procedure in the cse of welded stiffened bunker (Fig.). The structurl chrcteristics of bunker re s follows. Lods: self weight, pressure of the stored mteril, wind nd erthquke effects. The specilt of the pressures from stored mteril is tht the vr cross the height of the bin, this vrition is ner hdrosttic. Fig.. Welded squre bunker with horizontl stiffeners Structurl mteril is steel of ield strength 35 MP. Structurl tpes: single or combined in group, squre, rectngulr, polgonl or prismtic.

3 - 3 - Structurl prts (Fig.): stiffened plte wlls for bin nd hopper, columns, verticl edge bems, trnsition bems. Profiles: we use trpezoidl stiffeners for bin nd hopper wlls nd squre hollow sections for columns nd verticl edge bems s well s welded I-profiles for trnsition bems. Geometr is determined b min dimensions s follows: width nd height of bin, slope nd height of hopper, width of outlet, height of columns. Fbriction: ll connections re welded. Fig.. Min structurl prts of welded squre bunker nd the pressure components From these chrcteristics we select the following: we consider single squre bunker. We use onl horizontl stiffeners, since in our previous stud [] it is shown tht pltes loded b hdrosttic pressure re more economic with horizontl stiffeners thn with verticl ones. The distnces between stiffeners for bin wll should be non-equidistnt, since in our previous stud [] it is shown tht the non-equidistnt rrngement is more economic thn the equidistnt one. The hopper wlls cn be stiffened b equidistntl rrnged horizontl stiffeners, since it cn be ssumed tht the hopper wlls re loded with constnt norml pressure. The thickness of trpezoidl stiffeners should be vried, since the re loded b different bending moments. Furthermore the bin width nd height cn be vried so tht the volume of the stored mteril should be constnt. Similr vrition hs been studied in the cse of circulr silos [, 3].

4 - 4 - From the relevnt literture the following should be mentioned. Books [ ] s well s rticles [9, ]. The present uthors hve not found n studies on minimum cost design of bunkers. Summrizing the bove mentioned design spects, we optimize the positions of horizontl stiffeners in bin wlls, clculte the required stiffener thicknesses, determine the optimum number of bin nd hopper stiffeners b minimizing the cost, nd clculte the totl cost of bunker. This procedure is repeted for three bunkers with different widths nd heights (for rtios H/ =.5, l. nd l.5) nd the optimum vlue of H/ is determined, which gives the minimum totl cost. The cost function contins the mteril nd fbriction cost s it hs been shown in our recent studies [3, ].. Optimum positions nd number of horizontl stiffeners of bin wlls Bin wlls re loded b nerl hdrosttic pressure due to stored mteril (Fig.3). The mximum pressure intensit cn be clculted using the Jnssen formul [6 or ] p kp () h mx v mx p m v mx exp H / 4k x ; x 4k () where m is the densit of stored mteril, is the coefficient of friction of the mteril on the wll, k is the pressure coefficient, is the bin width (Fig.). The optimum positions of horizontl stiffeners cn be obtined using the condition tht ech prt of the bse plte of constnt thickness should be loded till ield stress. Since it cn be seen tht the bse plte prts hve side rtios lrger thn 3, the cn be clculted s strips with fixed edges (Fig.3). Thus, the stress constrint for bse plte strip is p( x x )( x x ) i i i i mx f ; i = n (3) 4Htb where p p h mx, is the sfet fctor, tb is the thickness of bin, f is the ield stress. Fig.3. Bse plte strip subject to bending, xi+ nd xi re the stiffener positions

5 - 5 - The optimum positions of stiffeners cn be clculted solving the following set of nonliner equtions expressing tht ll the bse plte prts should hve the sme thickness / 3 4 px t i (4) Hf ( x x x )( x x )... 3 ( x x 3 i xi )( xi xi ) i = n (5)... 3 ( H xn )( H xn ) x Hving obtined the optimum positions of n stiffeners, ech stiffener should be designed s simpl supported bem for bending moment of p M Si ( xi xi )( xi xi xi ) (6) 64H Fig.4. Cross-section of trpezoidl stiffener We consider trpezoidl stiffeners ccording to [3] with given dimensions of = 9 nd 3 = 3 mm nd ppl the locl buckling constrint for the inclined webs ccording to Eurocode 3 [4] 35 / ; / 38t S f (7) Considering stiffener cross-section s shown in Fig.4 the distnce of the grvit centre G is where z ht ( ) S G (8) 4t F ( ) t S 38 5 / h (9) t S The moment of inerti is I t 4 F z G 3 ts sin h ts ( h zg ) ts zg 6 ()

6 - 6 - where sin 5 38 t S The required stiffener thickness ts cn be clculted from the stress constrint M I Si f M f Si ( h zg ) or zg ; M. I M M () () The optimum number of stiffeners cn be determined from the condition tht the cost of the whole bin wll should be minimum. The cost function contins the mteril nd fbriction costs s we hve used it in our recent studies [3, ] K k m k f.5 V C d V.3T (3) k m where is the mteril densit, kf nd km re the fbriction nd mteril cost fctors, respectivel, is the number of structurl elements to be ssembled, V is the volume of the structure, d is the difficult fctor expressing the complexit of structure (plnr or sptil, using simple pltes or profiles), the coefficient for the preprtion time is C =. min/kg.5. To give interntionll usble results, the rtio of kf/km is vried in wide rnge. For steel km =.5-.4 $/kg, for fbriction including overheds kf = - 6$/mnhour = -$/min, thus, the rtio m vr in the rnge of - kg/min, the vlue of corresponds to the minimum weight design. The welding time is given b T i n C iwi LWi (4) the fctor of.3 expresses tht the dditionl time for chipping, deslgging nd chnging the electrode is pproximted b T3 =.3T.. The formule for weld tpes. W is the weld size, LW is the weld length []. n C W re given for vrious welding technologies nd It is ssumed tht the bse plte is butt welded from plte strips of width 5 mm or less. In the following numericl clcultions for bunker of H/= nd H = 6 mm re onl given in detils, for other vlues of H/ onl the results re summrized to show the optimum rtio corresponding to the minimum cost of the whole bunker. Numericl results for bin of H/ = nd H = 6 mm. Cement is selected s the stored mteril with densit of m 6 kg/m 3 =.6x -5 N/mm 3. With the vlues of k =.6 nd. 4, using Eqs. () nd () one obtins x = 65 mm nd phmx =.36 N/mm, p.54 N/mm. p h mx A stiffener is welded to the bse plte with fillet welds of size W =.7tS. The welding times re clculted with the following dt: use GMAW-M welding technolog (Gs Metl Arc Welding with mixgs). For (4) the following formule re used: W= 4-5 mm V butt welds CW n =.86W

7 - 7 - W= -5 mm fillet welds CW n =.358W nd LW is clculted in mm. The difficult fctor is chosen s d 3. The results of computtions for n = 7 nd 8 re summrized in Tble. Tble. Optimum positions of horizontl stiffeners nd costs of the stiffened bin wlls in the cse of n = 7 nd 8 considering n dditionl top stiffener for different vlues of kf/km (kg/min) n xi (m) ts (mm) tf (mm) K/km(kg) for kf/km= K/km(kg) for kf/km= K/km(kg) for kf/km= It should be noted tht n dditionl stiffener on the bin wll top is lso considered, which hs the sme thickness s the uppermost one. It cn be seen tht the minimum cost is chieved b using n = 8. A further increse of stiffeners number is limited b the minimum distnce of stiffeners, which should be lrger thn 3 = 3 mm.. Optimum number of horizontl stiffeners of hopper wlls The hopper wlls pressures re clculted with formule given b [] (Fig.5). Pressures from the mteril in the hopper (Fig.5) re given s p n.6 mk sin ; q p / n / Pressures from the mteril bove the hopper cn be clculted s (Fig.5b) p p sin p c p sin ; q p / (5) cos h ; cb =.5 (6) 4 n v b n pvcb cos ; q p / (7) n n Summtion of two pressures results in pressure distribution shown in Fig.5c, where p n mx pn pn pn pn (8) 4

8 - 8 - Fig.5. Pressure distribution on hopper wll. () pressure from the stored mteril in hopper; (b) pressures from the mteril bove the hopper; (c) summrized norml pressure distribution Insted of this distribution we clculte pproximtel with constnt norml pressure pnmx, this pproximtion is the side of sfet. For constnt norml pressure n equidistnt rrngement of horizontl stiffeners cn be used (Fig.6). Thus, we should determine the optimum number of stiffeners, which gives the minimum cost of hopper wll. Fig.6. Equidistnt rrngement of horizontl stiffeners of hopper wll Numericl results for hopper wll of the bunker of H/ =, 6 With Eq.(5) pn =.498 N/mm.Using () nd () one obtins pv =.67 nd ph =.36 N/mm. With (6), (7) nd (8) pn =.784, pn =.3 nd pnmx =.559 N/mm. Multipling b the sfet fctor.584 N/mm. p n mx It is possible to clculte the required constnt bse plte thickness th ssuming tht plte strip hs fixed edges. From pn mx f S t / 6 M h 3 MP

9 we obtin p / n mx th S. 98 f / M S is the distnce between stiffeners (Fig.6). S The required trpezoidl stiffener thicknesses re clculted similrl to the cse of bin wll stiffeners using Eqs (7-), but, insted of Msi we use M p n mx 8 i S. The results re summrized in Tble. Note tht we use minimum stiffener thickness of 6 mm. Tble. Hopper bse plte thicknesses nd costs for different number of stiffeners. Dimensions in mm, costs in kg nd kf/km in kg/min K/km n S th kf/km= kf/km= kf/km= It cn be seen tht the costs decrese when the number of stiffeners increses, except in the cse of kf/km =. We decide tht the optimum number of stiffeners is Optimum design of trnsition bems We consider the trnsition bems (Fig.) s simpl supported ones with spn length of. The re subject to bending from the horizontl pressure cting on the lowest prt of the bin nd from the horizontl component of pressures cting on the hopper. Bending moment from the first ction is M p( H xn ) (9) 6 We pproximte the pressure distribution s shown in Fig.7. The horizontl rection from the norml pressure is H pn mx pn pn sin 3 6 () h The mximum bending moment from H, ssuming lod distribution shown in Fig.7, is M F F H F H () F F ; F ; For tngentil pressures q we ssume distribution similr s tht for norml pressure.

10 - - Fig.7. Pressure rections on trnsition bem Fig.8. Rections from the tngentil pressure on hopper wll The horizontl rection from tngentil pressure cting on the hopper is (Fig.8) cos 4 mx 3 h n n n p p p H () The mximum bending moment from H3, ssuming the sme distribution s for H, is ; 4 ; 8 6 H F H F F F F F M (3) The trnsition bem is designed s horizontl smmetric welded I-bem loded b bending in horizontl plne b the mximum bending moment of

11 - - M = M + M M3 (4) Numericl results for the bunker of H/ = Considering the bin wll with 8 stiffeners M = 55x 6 Nmm. H = 38.4 (N). M = 46x 6 Nmm, H3 = 68.9 (N), M3 = 555x 6, M = 546 knm. The dimensions of welded I-bem optimized for minimum cross-sectionl re re s follows [3] / 3 / /.5 W / ;/ 69; 35 / f ; b h / ;/ 8 h nd the thickness of web nd flnges re t h; t b. w f The required section modulus is W M. f / M Using these formule one obtins h = 645, tw =, b = 9, tf = mm. The cost of the bem is clculted with Eqs () nd (3) considering 4 fillet welds of size 5 mm, difficult fctor 3 nd the number of ssembled structurl prts 3. For kf/km = one obtins K/km = 98 kg. 4. Verticl edge bems of the bin The required width of these squre hollow section bems is determined b geometric condition tht the lrgest horizontl stiffener should be welded to them. For the bin with 8 stiffeners the mximum stiffener thickness is 8 mm, thus, using Eq.(9) we obtin hs = 86 mm. A squre hollow section of 3x3x8 is required. 5. Design of columns Clcultion of the self-weight for column Optimized bin wll with 8 stiffeners 379 kg Verticl edge bem 6x7,8 437 kg Optimized hopper wll with 7 stiffeners 348 kg Trnsition bem 64 kg Totl self-weight cting on column 854 kg Volume of bunker is given b 3 3 tn V H (5) 6 Volume of the bunker of H/ =, H = = 6 m, =.6 m, 6 is V = 78.9 m 3. Weight of the stored mteril is Q = l6x78.9 = 445x 3 kg. Q/4 =.5 kn. The effect of wind cn be neglected. The effect of erthquke is clculted ccording to [6].

12 - - The horizontl force is Ve =.ZQ For moderte dmge Z = 3/8, thus Ve = kn. This force is cting t height of H/ + H + H =.8 m. The compression force cting on column due to erthquke is Ne =.8x333.75/4 = 4 kn. According to [4] two lod combintions should be considered s follows. () Permnent ction (self weight) + the most unfvorble vrible ction (Q) multiplied b sfet fctors.35x x.5 = 778 kn () Permnent ction nd ll the unfvorble ctions including erthquke multiplied b.9.35x x.5(.5 + 4) = l8 kn. It cn be seen tht the second combintion is the leding one. We select for column the squre hollow section of 3x3x mm. Assuming tht the column is constructed with pinned ends, the length is 7.8 m. Check of the column ccording to [4] for overll flexurl buckling: x.6389; 56.87x3 74 MP, OK. 8 x Clcultion of the totl cost of the bunker of H/ = We ssume tht kf/km = kg/min nd km = $/kg. One stiffened bin wll with 8 + dditionl top stiffener 5767 $ One stiffened hopper wll with 7 stiffeners 466 $ One verticl edge bem 3x3x8, length 6 m 58 $ One trnsition welded I-bem 98 $ One column 3x3x, length 7.8 m 773 $ Totl cost of the structurl prts of the bunker 4x7 = 5844 $ The costs of the connecting welds re s follows. Welds connecting the verticl edge bems with bin wlls (Fig.9). 3 fillet welds of size 4 mm, welding technolog SMAW (shielded metl rc welding). The length of welds connecting the bse bin plte nd the stiffeners is clculted pproximtel s H, insted of this length we clculte with 5 welds hving length of H. Insted of.3 we multipl b considering the time of ssembl s well. K = x.7889x -3 x4 x5x6 = 757 $ Welds connecting the stiffened hopper wlls to n edge plte strip (Fig.9b) K = x.7889x -3 x4 x4x64 = 6 $ Welds connecting the hopper bse plte to the trnsition I-bem (Fig.9c) K = x.7889x -3 x6 xx6 = 68 $ Totl cost of connecting welds 4( ) = 896 $

13 - 3 - Totl cost of the bunker 594 $. Fig.9. Welded connections in bunker. () Connection of stiffened bin wlls to the verticl edge bem; (b) connection of stiffened hopper wlls to n edge plte strip; (c) connection of hopper bse plte, verticl edge bem nd column to the trnsition welded I-bem 7. The optimum H/ rtio In order to find the most economic structurl solution we optimize bunkers hving different H/ rtios, but the sme storge cpcit. Eq.(5) cn be written in the form tn V 78.9 m 3 (6) 6 where H / ; 6 ; =.6 m. Eq.(6) cn be solved for s follows / (7).8868 The height of the hopper is clculted with the following formul H tn (8) Tble 3 shows the min dimensions of bunkers with different H/ rtios Tble 3. Bunker dimensions in m H/ H H

14 - 4 - Fig.. Min dimensions of bunkers in m with different H/-rtios: ().5, (b)., (c).5 Results for H/ =.5 Bin wlls with 3 stiffeners, bse plte thickness 9 mm cost 436 $ Hopper wlls with 9 stiffeners, bse plte thickness 6 mm 47 $ Trnsition I-bems web 875x3, flnges 395x4 79 $ Verticl edge pltes 64x mm 85 $ Columns squre hollow section (SHS) 35x8 mm 73 $ Connecting welds 88 $ Totl cost 9655 $ Results for H/ =.5 Bin wlls with 7 stiffeners, bse plte thickness 8 mm cost 836 $ Hopper wlls with 6 stiffeners, bse plte thickness 4 mm 46 $ Trnsition I-bems, web 57x9, flnges 6x 944 $ Verticl edge bems, SHS 35x8 756 $ Columns SHS 35x8 36 $ Connecting welds 736 $ Totl cost 65 $ The min cost dt for the three structurl versions re summrized in Tble 4. It cn be seen tht the bunker of rtio H/ = cn be built with minimum cost. The bunker of H/ =.5 results in high cost becuse of lrge hopper dimensions.

15 - 5 - Tble 4. Comprison of the min cost dt in $ H/.5..5 Bin Hopper Other prts nd welds Totl Conclusions The bin nd the hopper of welded squre bunker cn be optimized for minimum cost. The pressure distribution on the bin wlls is nerl hdrosttic, therefore the optimum position of horizontl stiffeners should be clculted. The hopper wlls re subjected to nerl constnt norml pressure nd, on the bsis of the detiled cost clcultion, the optimum number of horizontl stiffeners cn be determined. Importnt structurl prts re the trnsition bems loded in bending b hopper rections nd designed s horizontl welded I-bems. The most economic solution is chieved b comprison of costs for bunkers of different rtios of H/. The best version is the bunker with the rtio of H/ =. The difference between the costs of bunkers with rtios H/ = nd.5 is 63%. Acknowledgements This work hs been supported b grnt OTKA 846 of the Hungrin Fund for Scientific Reserch nd the Fund for Higher Eduction grnt 8/. References [] Frks,J. & Jármi,K. Optimum design of welded stiffened pltes loded b hdrosttic pressure. 3 rd WCSMO World Congress of Structurl nd Multidisciplinr Optimiztion, Bufflo, New York, 999. Short pper proceedings Vol [] Frks,J. & Jármi,K. Fbriction cost clcultion nd optimum design of welded steel silos. Welding in the World. 37 (996) No (IIW doc. XV ). [3] Frks,J. & Jármi,K. Anlsis nd optimum design of metl structures. Rotterdm, Blkem, 997. [4] Lmbert,F.W. The theor nd prcticl design of bunkers. London, The British Constructionl Steelwork Assoc. Ltd [5] Metllicheskie Konstruktsii (Metl structures). Ed. Belen,E.I. 4. Ed. Moskv, Stroizdt (in Russin) [6] Glord,E.H.jr. & Glord,Ch.N. Design of steel bins for storge of bulk solids. Prentice Hll, Englewood Cliffs, New Jerse. 984.

16 - 6 - [7] Sfrin,S.S. & Hrris,E.C. Design nd construction of silos nd bunkers. New York, Vn Nostrnd Reinhold [8] Silo-Hndbuch I-II. Ed. Mrtens,P. Berlin, Ernst&Sohn.988. [9] Lightfoot,E. & Michel,D. Prismtic col bunkers in structurl steelwork. The Structurl Engineer 44 (966) No [] Shmel,P. Konstruktive Ausbildung der Wndussteifungen bei geschweissten Bunkern und Silos. Schweissen und Schneiden 3 (97) [] Jármi,K. & Frks,J. Cost clcultion nd optimistion of welded steel structures. J.Constructionl Steel Reserch 5 (999) [] DIN 55/6.(987). Lstnnhmen für Buten. Lsten in Silozellen. [3] Sthlbu Hndbuch, Bnd. Sthlbu Verlg, Köln [4] Eurocode 3 Design of steel structures. Prt.: Generl rules nd rules for buildings. Europen Committee for Stndrdiztion, Brussels, 99.