TRIANGULAR AND SQUARE BRACED TUBULAR COLUMNS Cost comparison of optimized column structures

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1 EUROSTEEL 4, Septemer -, 4, aples, Ital TRIAGULAR AD SQUARE BRACED TUBULAR COLUS Cost comparison o optimize column structures Józse Farkas, Károl Jármai Universit o iskolc, H-55 iskolc, Egetemváros, Hungar altar@uni-miskolc.hu, altjar@uni-miskolc.hu ITRODUCTIO Columns or masts are important parts o inustrial uilings an sport staiums. Various crosssections can e use or them. Tues are oten use or their large stailit. In the present stu the column is constructe with three or our main circular hollow section (CHS) tues an CHS racings. A m high cantilever column is loae a compression orce o F = x 6 [] an a horizontal orce H =.F (Figures -4). The aim o the present stu is to compare the costs o the two ierent optimize structural versions. The avantage o triangular column is that it oes not nee transverse iaphragms to avoi torsional eormation o the cross-section. The truss columns work as a uilt-up memers, the eect o shear orce uring the uckling is consiere accoring to Eurocoe []. The racings are constructe rom CHS as trusses with wele overlap K joints. Unknowns to e optimize are as ollows: CHS proile imensions o the main tues (chors) (D, t ), tuular racings (, t ) an iaphragms (, t ), istance etween chors (h ), numer o spacings (q) (istances etween K joints). The ollowing constraints shoul e ulille: overall an local uckling o CHS chors an racings, strength o overlap K joints accoring to ISO-IIW esign rules or staticall loae hollow section joints (local ieling o overlapping race, local chor ieling an race shear). Special arication constraints are ormulate to make it possile the weling o joints: the angle etween chors an racing struts shoul e minimum as well as the istance etween strut ens in joints o chors, racings an iaphragms shoul e minimum t (t is the thickness o racings or iaphragms) The cost unction to e minimize contents the cost o material, cutting an grining o CHS strut ens, weling o chors an joints as well as painting. The minima are oun a sstematic search using a athcad algorithm. The cost comparison o the two optimize structural versions in a numerical prolem shows that the triangular column is more economic than the square one. Although the imensions o CHS struts are smaller or the square column, the racing lengths are larger, thus the cost is also larger. IIU COST DESIG OF A TRIAGULAR TUBULAR TRUSS COLU Detaile optimization is treate or triangular column onl.. Given ata F, H, L,, E, =F/, H =.F, F = x 6 [], L = m, = 55 Pa, E =.x 5 Pa. Variales h (istance etween chors), D,t (A,r,I rom tale)(or CHS chors),,t (A,r rom tale)(or CHS races), q=l/a numer o segments (istances etween wele joints in a chor).. Buckling constraint o a chor Calculating the column as a uilt-up compression memer accoring to Eurocoe []

2 A Fa HL () F F W FE Sv cos Fig.. Cross section o the triangular tuular column Fig.. A race tuular column Fig.. A part o the triangular tuular column Fig.4. Geometric ata o the racing a L / 5 () EI e For a cantilever column the Euler uckling orce FE 4L () Eective moment o inertia I e I A h (4) The section moulus I e W h (5) The coeicient o overall uckling accoring to Eurocoe [] E a L,, E,. 9, a E r q (6)., (7)

3 Since α =.4 is or a = L/, thus x (8) The actor consiering the eect o shear uring the overall uckling EA h a h S v, h, cos a h (9) 4.4 Buckling constraint o a compression race V, V, V A, () h cos E or a cantilever column the shear orce V () L Fa HL E () F F FE S v cos The coeicient o overall uckling o a race,. 4.,,. 7 () E r.5 Constraints on strength o overlap K joints o races It shoul e mentione that the use o gap joints woul e less economic than the overlap ones. Constraints accoring to Static esign []. Local ieling o overlapping race (Ov= %) t s (4) max e se e. ov 4t, e. ov t (5) 4 / t Local chor memer ieling. 7.pl.pl (6).pl A (7) D a D H (8). pl W pl (9) Brace shear a s cos s max () e t h s max. 58 u, u = 5 Pa, sin () 4 sin t t e () D / t t Dt.6 Farication constraint to allow the weling o races to chor t D () 6

4 .7 Cost unction The cost is calculate accoring to the arication sequence [,4,5]. Fig. 5. Geometric ata o the overlap K joint Accoring to Figure 5 D Dh u sin (4) Length o the overlappe race is h L u D (5) Length o the overlapping race is calculate as c h L u D (6) 4 cos ah Cost o material K ( k V k V ), ρ = 7.85x -6 kg/mm (7) V, V qa L LA L The k material cost actors or CHS accoring to Price list o the British Steel [6] are given in Tale. Tale. aterial cost actors (mm) k ($/kg) 88.9,.6, , 68., 77.8, , 44.5, 7., , , step: weling o the main tues. It shoul e note that the joints occasionall neee or transportation an asseml o parts o length m or smaller are not treate. (a) Cost o cutting an grining o CHS chor ens (together ens) (8)

5 K CG. 5D 5 t. (9) () Weling o chor elements o x5 m (together 9 elements) with utt wels 958. KW kw 4 V. x. 5x t D, () V A, L 5 mm., k.$ / min, Θ = () L W (c) Weling o chors o total length with utt wels (together chors). 958 K k 8V. x. 5x t D () W W. step: weling o all the overlappe iagonals to the main tues. (a) Cutting an grining o overlappe CHS race ens (total numer o ens 6q) Accoring to Figure 5 c sin h. 5 K CG 6q 5 t. h () (4) () Weling o overlappe races to the chors (total numer o iagonals q) K W kw q V. x. 7889x t 6q, Θ = (5) h V LA qa u LA qa L (6).step: weling o all the overlapping iagonals to the previous structure. (a) Cutting an grining o overlapping CHS grace ens (total numer o ens 6q). 5 K CG 6q ( 5 t ). h (7) () Weling o overlapping races (total numer o ens 6q) K W kw q V. x. 7889x t 6q, Θ = h (8) where V V qa L, (9) Cost o painting K P k P S P = 4.4x -6 $/mm (4) The surace to e painte S LD q L L (4) The total cost K K K 9K K K K K K K (4) CG W W CG CG W W P OPTIIZATIO RESULTS The optimization is carrie out a sstematic search using a athcad program. The CHS proile thicknesses are selecte taken into account the local uckling constraint / t 5. The cost is

6 calculate or selecte chor an race CHS proiles an or q = -6 values, ater the etermination o h consiering the chor uckling constraint. The race proile is checke or uckling. The wele overlap K-joints are checke or constraints given Eqs.(4,6,). The optimization results are given in Tale. Data o the optimum structure: q = 5, h = 476 mm, D xt =.9x8, xt = 9.7x4. Check o the constraints: (): 5.8<5. Pa, (): 8<7 Pa, (4):.9x 5 <4.x 5, (6):.678<, ():.x 5 <5.x 5, (): 48<7 mm, OK. Tale. Cost in $ or ierent chor proiles an q-values. Optimum is marke ol letters, optima or another chor proiles are given in italics. Chor proile q x K x8 K x8 K unreal The race proile is 9.7x4 except or 46.4x as well as.9x8 an q =. ote that the structure or chor proile o.9x8 an q =, as well as chor proile o 7.x6 cannot e realize It can e seen that the cost increases when the chor proile increases, thus, larger chor proiles o not give cost minimum. All the optima all in the range o q = 4-5, thus, it is enough to investigate values in the range o q = -6. The optimization o the square tuular column is perorme similarl. The result is K min = 984$, which is (988-65)/98.8 = 7% larger than that or triangular column. It can e conclue that the triangular tuular column is more economic, since the ierence o cost minima is 7%. ACKOWLEDGET The research was supporte the TÁOP 4..4.A/---- priorit project entitle ational Excellence Program - Development an operation o omestic personnel support sstem or stuents an researchers, implemente within the ramework o a convergence program, supporte the European Union, co-inance the European Social Fun. The research was supporte also the Hungarian Scientiic Research Fun OTKA T 986 projects an was partiall carrie out in the ramework o the Center o Excellence o Innovative Engineering Design an Technologies at the Universit o iskolc. REFERECES [] Eurocoe. E 99--: 9. Design o steel structures. Part -: General structural rules. Brussels, 9. [] Static esign proceure or wele hollow section joints Recommenations. ISO 446, IIWoc. XV-4-.. [] Jármai K., Farkas J., 999. Cost calculation an optimization o wele steel structures. J Constr Steel Res Vol.5, pp [4] Farkas J., Jármai K., 8. Design an optimization o metal structures. Chichester, UK, Horwoo. [5] Farkas J., Jármai K.,. Optimum esign o steel structures. Heielerg, etc. Springer. [6] Price List. Steel tues, pipes an hollow sections. Part. Structural hollow sections. British Steel Tues an Pipes, 995.