Thn-Walled Structures Group JOHNS HOPKINS UNIVERSITY RESEARCH REPORT Towards optmzaton of CFS beam-column ndustry sectons TWG-RR02-12 Y. Shfferaw July 2012 1
Ths report was prepared ndependently, but was motvated from, the Amercan Iron and Steel Insttute sponsored proect: Drect Strength Predcton of Cold-Formed Steel Beam Columns. The proect also receved supplementary support and fundng from the Metal Buldng Manufacturers Assocaton. Proect updates are avalable at www.ce.hu.edu/bschafer/dsmbeamcol. Any opnons, fndngs, and conclusons or recommendatons expressed n ths publcaton are those of the author and do not necessarly reflect the vews of the Amercan Iron and Steel Insttute, nor the Metal Buldng Manufacturers Assocaton. Acknowledgements: The authors would lke to acknowledge AISI and MBMA for ther support throughout the duraton of ths proect. The author would also lke to acknowledge Jzahen Leng and Dr. Zhane L for ther nput on ths work. 2
Towards optmzaton of CFS beam-column ndustry sectons 1. Introducton The obectve of ths research s to develop ndustry sectons optmzed for maxmum strength subected to shape and dmensonal constrants that can be used n low-cost sustanable md-rse cold-formed steel housng constructon. The optmzaton of common ndustry sectons that nclude channel, zee, hat and sgma sectons used n framng, truss and metal buldng cold-formed steel applcatons (Fgure 1) and nonstandard sectons subected to combned loadng of compresson and bendng s examned. The effect of the complex stablty behavor of thn-walled cross-secton when subected to combned loadng s ncorporated nto the optmzaton code va yeldng, elastc crtcal bucklng and strength envelopes. Prelmnary Drect Strength Method approach for combned loadng n cold-formed steel members subected to combned loadng s adopted for strength predcton n place of current lnear nteracton approach. The methods of gradent-based optmzaton and stochastc optmzaton va genetc algorthm are explored for optmal soluton. Based on strength envelopes of the prelmnary Drect Strength Method n the axal-bendng space, effcent members beamcolumn use can be readly selected. Fgure 1 Beam-column ndustry applcatons that motvated the research a) Framng b) Truss c) Metal buldng 2. Optmzaton problem formulaton The optmzaton of cold-formed steel ndustry sectons subected to combned loadng s examned. The maxmum combned strength s sought gven a rectangular-szed sheet of steel. A 1.57 mm thck and 215 mm wde sheet of steel s consdered based on the average dmensons of SSMA standard channel sectons. Sectons selected for 3
optmzaton study are n the sprt of the standard sectons used n the framng, truss and metal buldng ndustry. The cross-sectonal dmensons of the varous applcatons are determned based on the average dmenson for that partcular standard shape. The materal propertes are Young s modulus (E=203000 MPa) and yeld strength (fy=344.5 MPa). The member length s chosen as 3000mm. The optmzaton problem s formulated n the standard form as: Max n Mn 1 n Subect to the followng dmensonal constrants: Framng applcatons Channel secton b" d" h" h + 2b + 2d! w 2d < h b" d" h/4$ h/3$ b$ Sgma secton d$ h/12$ h/12$ 1.069h + 2b + 4d! w 2d < h h/4$ b$ d$ Truss applcatons Hat/U secton 4
4.76" 4.76" 0.625h" 0.25h" d" d" 0.5d" h/8" h/8" 2.059h + b + 5d + 3.943! w 2d < h b" b/2" Metal buldng applcatons Zee secton d" b" h" h + 2b + 2d! w! 2d < h b" d" Non-standard S secton b$ d$ h/2$ b$ h + 3b + 2d! w 2d < h h/2$ d$ b$ where w s the wdth of the sheet of steel and n s the predcted combned load capacty of the secton: where { } n = mn ne, nl, nd 5
for λ 0. 776 n = ne for λ > 0. 776 0.4 0.4 cr cr n = 1 0.15 y y y where λ = y / cr cr = Crtcal elastc local bucklng magntude under combned P-M resultant = Frst yeld under combned P-M resultant. for λ d for λ d where y 0.673c nd = y > 0.673c nd = 0.5b 0.5b crd y y γ / π 2γ / π 2γ / π, and 1 0.22a crd y 2 a = ( 1.136) b = (1.2) c = (0.834) λ d = y / crd crd = Crtcal elastc dstortonal bucklng P-M resultant γ = Angular drecton (n radans) n the P-M space measured from the postve x- axs for the frst quadrant and from the negatve x-axs for the second quadrant. For a γ! = fxed angle n P M space fxed for each run! s f(l = 3000, t = 1.57, fy = 344.5, h, b, d) 6
3. Genetc algorthm (GA) approach to beam-column dmensonal optmzaton 3.1 Genetc Algorthm [3, 4, 5] The constraned optmzaton problem s frst transformed nto an unconstraned problem by makng use of penalty functons. Among the varous penalty methods, the addtve form s chosen [3]. Usng the general formulaton of an exteror penalty functon, the new obectve functon to be optmzed s gven as Mnmze f = 1/ Pcr + q = 1 r G + m c = q+ 1 L where G and are the functons of g ( X ) and h ( X ),nequalty and equalty L constrants respectvely, and L are, G max[ 0, g L = = h (X ) γ ( X )] r and c are penalty parameters. General formulas of G and where and γ are commonly 1 or 2. If the nequalty holds, g ( X ) 0and max[0, g ( X )] wll be zero. Therefore the constrant does not affect the g new obectve functon. If the constrant s volated ( ( X ) or h ( X ) 0 ), a bg term wll be added to the new obectve functon such that the soluton s pushed back towards the feasble regon. The severty of the penalty depends on the penalty parameters r and c. If ether the penalty s too large or too small, the problem could be very hard for genetc algorthm. A bg penalty prevents t from searchng an nfeasble regon. In ths case, the genetc algorthm (GA) wll converge to a feasble soluton very quckly even f t s far from the optmal. A small penalty wll cause t to spend too much tme n searchng n an nfeasble regon; thus GA would converge to an nfeasble soluton. The basc operatons of natural genetcs-reproducton, crossover, and mutaton-are mplemented by use of genetc algorthm. It starts wth a set of desgns, randomly generated usng the allowable values for each desgn varable. Each desgn s also assgned a ftness value, usng the penalty functon for constraned problems. From the 7
current set of desgns, a subset s selected to generate new desgns usng the selected subset of desgns. The sze of the set of desgns s kept fxed. Snce more ft members of the set are used to create new desgns, the successve sets of desgns have a hgher probablty of havng desgns wth better ftness values. The process s contnued untl a stoppng crteron s met. Reproducton s a process n whch the ndvduals are selected based on ther ftness values relatve to that of the populaton. In ths process, each ndvdual strng (desgn vector) s assgned a probablty of beng selected as F P = where Np s the sze of populaton and s the ftness value for each desgn that defnes ts relatve mportance n the set, and s gven by F = ( 1+ ε ) f max f where s the cost functon( penalty functon value for constraned problems) for the f th desgn, f s the largest recorded cost(penalty) functon value, and ε s a small value max to prevent numercal dffcultes when becomes 0. After reproducton, the crossover operaton s mplemented n two steps. Frst two ndvdual strngs (desgns) are selected from the matng pool generated by the reproducton operator. Next, a crossover ste s selected at random along the strng length, and the bnary dgts (alleles) are swapped between the two strngs followng the crossover ste. Mutaton s the next operaton whch n terms of a genetc strng, corresponds to selectng a few members of the populaton, determnng a locaton on each strng randomly, and swtchng 0 to 1 or vce versa. The number of members selected for mutaton (I m ), s based on heurstcs, and the selecton of locaton on the strng for mutaton s based on a random process. I m =nteger(p m *N p ) where Pm-fracton of populaton selected for mutaton In each generaton, whle the number of reproducton operatons s always equal to the sze of the populaton, the amount of crossover (I max ) and mutaton (I m ) can be adusted to fne-tune the performance of the algorthm. If the mprovement for the best cost (penalty) ' functon value s less than ε for the last I g consecutve teratons, or f the number of teratons exceeds a specfed value, then the algorthm termnates. Genetc algorthm operatons Selecton-Roulette wheel (ftness proportonate) selecton [ Spall p. 243] F F N p = 1 F 8
3.2 Parametrc study for GA Np! Gen!! 1! 2!! 3!! 4!! 5!! 6!! 7!! 8!! 9!! 10!! 1! 117130.7869! 144864.0392! 170732.404! 164921.3477! 143457.2461! 158048.2203! 127398.3869! 147440.9101! 160942.1512! 70924.17656! 2! 143457.2461! 156356.6621! 143457.2461! 152963.8255! 147692.2162! 158048.2203! 118854.5021! 164921.3477! 160942.1512! 161905.7814! 3! 158048.2203! 154843.6104! 153401.455! 160942.1512! 171285.2437! 155994.98! 141793.6399! 125963.4056! 156356.6621! 143457.2461! 4! 138745.764! 156356.6621! 155053.6317! 160942.1512! 122893.2368! 156079.7307! 149331.2623! 154843.6104! 153769.9157! 145206.6601! 5! 160942.1512! 156356.6621! 158367.5445! 141832.5823! 158052.7092! 152853.0307! 93922.24229! 133188.8764! 166879.2983! 155053.6317! 6! 160942.1512! 105523.2717! 92925.23717! 158367.5445! 134152.8996! 155053.6317! 158367.5445! 88669.00512! 132829.7427! 163015.8436! 7! 157210.4075! 158367.5445! 160942.1512! 152853.0307! 168541.8889! 82757.7953! 151739.1627! 146645.453! 160942.1512! 135268.0959! 8! 152853.0307! 158367.5445! 160942.1512! 160942.1512! 147583.1599! 148859.3648! 143935.1209! 160942.1512! 158367.5445! 160942.1512! 9! 160942.1512! 152853.0307! 160942.1512! 160942.1512! 158367.5445! 152361.3381! 158367.5445! 160942.1512! 158367.5445! 158367.5445! Np Gn Pm Ic C_neq Bn 20 20 0.001 0.8 C1 20 20 0.005 0.8 C1 20 20 0.075 0.8 C1 20 20 0.010 0.8 C1 20 20 0.005 0.6 C1 20 20 0.005 0.7 C1 20 20 0.005 0.8 C1 20 20 0.005 0.9 C1 20 20 0.005 0.8 C1 40 20 0.005 0.8 C1 60 20 0.005 0.8 C1 80 20 0.005 0.8 C1 40 20 0.005 0.8 C1 40 20 0.005 0.8 C1 40 20 0.005 0.8 C1 40 20 0.005 0.8 C1 40 20 0.005 0.8 C2 40 60 0.005 0.8 C3 40 100 0.005 0.8 C4 40 140 0.005 0.8 C5 9
4. Gradent-based approach to beam-column dmensonal optmzaton 4.1 6""-('75* 89:;<!;=*9>4#1.%$/)3%()'4*7()30-'%)/.*)()4%)/'-*$%)%$%&'#()* 9%)*8?@A*3>2B/70*0(* 6C@DEF* GFDE@DEHF*!"#$%&'#()*+'-%'24/3* I(4/-')7/EJC/1K* 9'@%0/-ELMM* N@'$"4/*O(-*'*75'))/4*3/7#()P*,'$$'%E"%QR* @(ESJMM*LM*TCLU* @OESZMCK*RMCJ*[TU*!)(EJRMCVJWJMXY*!)OEJTRCTVWJMXY* 4.2 Parameters n gradent-based Gamma=p/4 teraton constant Beta,n 10
[h'b'd]=[70.8,47.2,24.9]' [168.6,14.9,1.2]' [80.6,40.1,27.0]' [70.8,47.2,24.9]' n=173.02*10^3' n=216.4*10^3' n=174.79*10^3' n=175.8*10^3' 11
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5. Conclusons The optmzaton of a channel cross-secton subect to combned loadng has been studed usng genetc algorthm and gradent-based methods. The effect of complex bucklng behavor of thn-walled cross-secton when subected to combned actons has been ncorporated nto the optmzaton code by usng the fnte-strp method and the drect strength method. The study ndcated that effectve use of genetc algorthm to such optmzaton problem requres examnaton of parameters such as the penalty functon, mutaton rate and suffcently large number of populaton and generatons to capture the optmal soluton. The penalty, cross-over and mutaton parameters need to be adusted for optmum soluton. In comparson, the gradent-based methods.. Future research that combnes genetc algorthm wth gradent-based methods to ensure globally optmal results s an approach that should be explored. In addton, optmzaton of beam-column sectons under combned axal and mnor-axs and baxal bendng has the potental to explot addtonal strength as predcted under newly developed Drect Strength Method for combned loadng. References 1. Schafer BW. Progress on the drect strength method. 16th Internatonal Specalty Conference on Cold-formed Steel Structures,Orlando,FL. 2002. 2. Amercan Iron and Steel Insttute (2006). Drect Strength Method Desgn Gude. Amercan Iron and Steel Insttute, Washngton,D.C. 3. Ozgur Yenay. Penalty Functon Methods For Constraned Optmzaton wth Genetc Algorthms:, Mathematcal and Computatonal Applcatons, Vol. 10,No.1,pp.45-56,2005 4. Introducton to Optmum Desgn: Jasbr S.Arora, Second Edton 2004 5. Engneerng Optmzaton: Theory and Practce, Sngresu S. Rao,Thrd ed. 1996 13