Optimum Design of Steel Frames Considering Uncertainty of Parameters

Transcription

1 9 th World Congress on Structural and Multdscplnary Optmzaton June 13-17, 211, Shzuoka, Japan Optmum Desgn of Steel Frames Consderng ncertanty of Parameters Masahko Katsura 1, Makoto Ohsak 2 1 Hroshma nversty, Hgash-Hroshma, Japan, ohsak@hroshma-u.ac.jp 2 Hroshma nversty, Hgash-Hroshma, Japan, b74962@hroshma-u.ac.jp 1. Abstract A hybrd approach of optmzaton and ant-optmzaton s presented for worst-case sesmc desgn of steel buldng frames. ncertantes are consdered n yeld stresses and cross-sectonal geometres of members. The concept of unknown-but-bounded s ntroduced, and the parameters are assumed to exst n a bounded regon of uncertanty. An ant-optmzaton problem s formulated for fndng the worst value of the global response defned by the maxmum value of the nterstory drft among all the stores. Optmal cross-sectons of beams and columns are then selected from the lst of avalable sectons under constrants on worst responses. The regons of uncertanty of parameters are dscretzed nto several values, and a heurstc approach called tabu search s used for optmzaton and ant-optmzaton. It s shown n the numercal examples that a good approxmate soluton of the global optmal soluton, whch can be found by enumeraton, s successfully found by tabu search wthn practcally acceptable number of analyses. A random selecton approach s also carred out to effcently fnd approxmate solutons. 2. Keywords: ncertanty, Steel frame, Sesmc response, Ant-optmzaton, Tabu search 3. Introducton Most of the optmzaton methods of buldng frames are developed for applcaton to mnmum weght desgn under constrants on elastc responses aganst statc loads [1]. The lmt desgn of regular frames subjected to statc proportonal loads s also a tradtonal problem n 196 s, because analytcal solutons can be found, or the solutons can be found by solvng a lnear programmng problem. However, recent buldng codes n those countres that are prone to sesmc hazards demand nonlnear dynamc analyss for relatvely tall buldngs. Another mportant aspect n structural optmzaton s that uncertanty n structural and problem parameters should be taken nto account approprately for applcaton to practcal desgn. There are varous approaches to such purpose; namely, relablty-based approach, probablstc approach, and worst-case desgn. It should also be noted that evaluaton of elastoplastc dynamc responses usng tme-hstory analyss requres much computatonal tme; therefore, we cannot carry out structural analyss many tmes for optmzaton. In ths study, we present a new approach to worst-case desgn of buldng frames. ncertantes are consdered n yeld stresses and cross-sectonal geometres of members. The concept of unknown-but-bounded s ntroduced [2], and the parameters are assumed to exst n a bounded regon wth prescrbed upper and lower bounds. The analyss program called OpenSees [3] s used for evaluaton of maxmum responses under sesmc motons that are compatble to the specfed desgn response spectrum. An ant-optmzaton problem s formulated for fndng the worst value of the global response defned by the maxmum value of nterstory drfts of the frame. Optmal cross-sectons of beams and columns are then selected from the lst of avalable sectons under constrants on worst responses. The regons of uncertanty of parameters are dscretzed nto several values. Therefore, the ant-optmzaton problem as well as the optmzaton problem turns out to be an nteger programmng problem, whch s also called a combnatoral problem. Hence, the heurstc approach called tabu search (TS) [4] s used for optmzaton and ant-optmzaton. It s shown n the numercal examples that a good approxmate optmal soluton can be successfully found by TS wthn practcally acceptable number of analyses. A random search approach [5] s also carred out to show that the approxmate solutons can be successfully found by a smple algorthm wthout any problem-dependent parameter. 4. Optmzaton Problem Consder a problem for optmzng the cross-sectons of steel buldng frames. The sectons are selected from the pre-assgned lst of standard sectons. The members are classfed nto m groups, where members n each group have the same secton. The desgn varable vector s denoted by J = ( J1,, J m ), whch has nteger values. For example, f J = k, the the secton of the th group has the k th secton of the lst. Let F( J ) denote the objectve functon, whch represents, e.g., the total structural volume. The constrant 1

2 functons representng, e.g., the maxmum nterstory drfts, are denoted by G ( J ) ( = 1,, n), where n s the number of constrants. Then optmzaton problem s formulated as Mnmze F( J) subject to G ( J ) G, ( = 1,, n) J {1,, s }, ( = 1,, m) (1) where G s the upper bound for G ( J ), and s s the number of admssble values for J. Snce the structural and materal parameters have uncertanty, we assgn constrants on the worst values G worst ( J ) of responses, and formulate the optmzaton problem as. Mnmze F( J ) subject to worst G ( J ) G, ( = 1,, n) J {1,, s }, ( = 1,, m) (2) Let X denote the vector of parameters that have uncertanty. Then, the constrant functon s wrtten wth parameter vector as G ˆ ( JX, ), and G worst ( J ) s obtaned by solvng the followng antoptmzaton problem: Fnd G subject to ( J) = max Gˆ ( J, X) worst X L X X X (3) L where X and X are the upper and lower bounds for X, whch can be obtaned from measurements and experments. Hence, the optmal soluton consderng the worst values of responses can be found by solvng a two-stage hybrd optmzaton-antoptmzaton problem. 5. Tabu search The smplest heurstc approach s a local random search that consecutvely selects the best soluton n the neghborhood of the current soluton. The convergence property to the global optmal soluton may be enhanced f many solutons are searched before movng to a neghborhood soluton, or preferably, all neghborhood solutons are searched to select the best neghborhood soluton. A neghborhood soluton that does not reduce (for a mnmzaton problem) the objectve value can also be selected to mprove the possblty of reachng the global optmal soluton. However, n ths case, a so-called cyclng or loop can occur, where a set of neghborng solutons s chosen teratvely. TS has been developed to prevent cyclng utlzng the tabu lst contanng the prohbted solutons that have already been searched. The algorthm of TS s summarzed follows, where the superscrpt ( k ) denotes a value at the k th teraton: () () Step 1: Assgn an ntal soluton J, and ntalze the tabu lst T by J. Set the teraton counter k =. N N ( k ) * Step 2: Generate neghborhood solutons J ( j = 1,, n ) of J and move to the best soluton J, satsfyng the constrants, among them that s not ncluded n the tabu lst T. * Step 3: Add J to T. Remove the oldest soluton n T f the length of the lst exceeds the specfed value. ( 1) * Step 4: Let J k + = J and k k+ 1. Go to Step 2 f the termnaton condton s not satsfed; otherwse, output the best soluton satsfyng the constrants, and termnate the process. 2

3 RFL 36 4FL Sa(m/s 2 ) Desgn response spectrum Artfcal moton 3FL acceleraton perod A B C 2FL 1FL 36 Fgure 1: Desgn acceleraton response spectrum Fgure 2: A 4-story plane frame 6. Frame model and sesmc motons Consder a 4-story sngle-span plane frame model as shown n Fg. 2. The steel sectons of beams and columns of the standard desgn s lsted n Table 1. The steel materal has a blnear stress-stran relaton, where Young s modulus s 25 kn/mm 2, the nomnal value of hardenng rato s.1, and the nomnal values of yeld stresses of beams and columns are 235 N/mm 2. The artfcal sesmc motons are generated by the standard superposton method of snusodal waves, where the phase dfference spectrum of El-Centro 194 (EW) s used. The target spectrum s the desgn acceleraton response spectrum, as shown n Fg. 1, for 5 % dampng specfed by Notfcaton 1461 of the Mnstry of Land, Infrastructure and Transport (MLIT), Japan. The amplfcaton factor for the ground of 2nd rank s used as defned n Notfcaton 1457 of MLIT. The sesmc moton s appled at the base of the frame n horzontal drecton. The acceleraton response spectrum of an artfcal moton s also shown n Fg. 1.The wave s scaled by 7.5 to desgn the frame wth sesmc performance of rank 3 for the lfe-safety level. The mean value of maxmum response aganst fve artfcal waves s taken as the representatve response. A general purpose frame analyss software called OpenSees [3] s used for sesmc response analyss of the frame. The members are dvded nto fber sectons. Snce only plane frame analyss s carred out, the flange and web of the beam are dscretzed to 4 and 16 fbers n the drectons of thckness and depth, respectvely. The ntegraton s carred out usng the Gauss-Lobatto rules that has ntegraton ponts at the ends of the element, where the number of ntegraton ponts s 8; thus, the plastfcaton at the member ends can be accurately detected. The standard Newmark-β method ( γ =.5, β =.25) s used for ntegraton n tme doman wth the ncrement.1 sec. The stffness proportonal dampng s used wth the dampng rato.2 for the frst mode. The duraton of sesmc moton s 2 sec. The fundamental perod of the standard model s.71 sec, whch means from Fg. 1 that the response acceleraton reduces as the perod becomes larger as the result of plastfcaton. Table 1: Sectons of standard frame model beam column floor secton story secton 4th, roof H st 4th nd, 3rd H Optmzaton results In the followng optmzaton problem, a constrant s gven for the maxmum value among the maxmum nterstory drfts, whch s smply denoted by maxmum nterstory drft, of stores so that the frame does not collapse under severe earthquakes. 3

4 7.1 Antoptmzaton We frst carry out parametrc study to nvestgate the effect of varous parameters on the maxmum nterstory drft. It has been found through the prelmnary parametrc study that the maxmum nterstory drft s a monotoncally decreasng functon of the yeld stress of columns and the hardenng ratos of beams and columns. Therefore, the smallest possble values should be chosen for these parameters to obtan the worst response; hence, the yeld stress of column s the lower bound value 325 N/mm 2, and the hardenng coeffcent s the small value nterstory drft yeld stress Fgure 3: Relaton between yeld stress (N/mm 2 ) of beams and maxmum nterstory drft (m) b In contrast, the maxmum nterstory drft s not a monotonc functon of the yeld stress σ Y of the beam as shown n Fg. 3. Ths s because a stronger beam leads to a column collapse mechansm that has small energy dsspaton and larger local nterstory drft. In addton to materal parameter, the cross-sectonal geometry also has b uncertanty. Therefore, we consder uncertanty n σ Y and flange thcknesses n two groups of beams; hence, there are four uncertan parameters. The ranges of uncertanty are 2 % of the nomnal value for all parameters, whch are dscretzed nto fve equally spaced values. Snce the nomnal value 235 N/mm 2 b of yeld stress ndcates the lower bound, σ Y can take 235, 248, 261, 274, and 287 N/mm 2. The nomnal value of flange thckness s assumed to be a mean value; hence, the possble values 4 are detrmned by multplyng.95,.975, 1., 1.25, and 1.5 to the nomnal value. Therefore, we have 5 = 625 possble combnatons of parameters. Although the dstrbuton of each parameter can be modeled parametrcally usng, e.g., normal dstrbuton, we use the unform dstrbuton, because no nformaton s avalable for the dstrbuton. Note that the followng methods and results can be extended to any dstrbuton by modfyng the samplng spaces..15 probablty nterstory drft Fgure 4: Probablty densty functon by enumeraton Fg. 4 shows the dscretzed probablty densty functon of the roof dsplacement for the 625 samples. The maxmum and mnmum values are.518 m and.475 m, respectvely. The purpose here s to fnd the 4

5 approxmate maxmum value wthn small number of analyses. For ths purpose, we assume the parameter set correspondng to the maxmum nterstory drft up to the 5th maxmum value are assumed to be approxmate worst parameter sets. Suppose we select parameter sets 5 tmes randomly from 625 samples. Then, the probablty that no approxmate worst parameter set s found through ths random selecton s ( 575 / 625) 5 =.154, whch s very small. Therefore, we carry out 5 analyses n the followng;.e., for TS, the number of neghborhood solutons s 5, and the number of steps s 1. We fnd approxmate optmal solutons four tmes startng wth dfferent ntal values of random number for verfcaton purpose. Comparson s also done wth random samplng of 5 parameter sets probablty nterstory drft Fgure 5: Probablty densty functon by TS Table 2 Results of TS Intal value of random number Maxmum Mnmum Average Standard devaton ( 1 4 ) Order 1st 9th 3rd 5th The maxmum value, mnmum value, mean value, and the standard devaton of the maxmum nterstory drft obtaned through the process of TS s lsted n Table 2, where the last row s the order of the worst (maxmum) value n the orgnal lst of 625 parameter sets. As s seen, a parameter set wthn the 8th has been found for all cases. Furthermore, the standard devaton s very small, whch s less than 1/1 of the dfference between the maxmum and mnmum values n the orgnal lst, whch means that TS searches the solutons wth lmted range of functon value. Ths way, a good approxmate worst value can be found through analyses of less than 1/1 of the sze of the orgnal lst. Fg. 5 shows the probablty densty functon of 5 solutons for a sngle run of TS wth ransom seed 5. It s verfed from Fgs. 4 and 5 that TS searches the solutons wth larger responses. Table 3 Results of random selecton Intal value of random number Maxmum Mnmum Average Standard devaton ( 1 4 ) Order 28th 3rd 18th 3rd 5

6 Table 3 shows the results of random selecton, n whch the worst order of the soluton s 28th. Ths result s worse than TS; however, a soluton wthn 5th has been successfully found for all cases wthn the analyses of 1/1 of the total number of solutons. Furthermore, there s no problem-dependent parameter for ransom selecton, whch s superor to TS..2 probablty nterstory drft Fgure 6: Probablty densty functon by randomzaton Table 4: Lst of standard sectons Beam 1 (2F, 3F) H-4x2x9x12 H-4x2x9x16 H-4x2x9x19 H-4x2x9x22 H-4x2x12x22 Column -3x3x9-3x3x12-3x3x16-3x3x19-3x3x22 Beam 2 (4F, RF) H-25x125x6x9 H-3x15x6.5x9 H-35x175x7x11 H-4x2x8x13 H-45x2x9x14 Beam1 (2F, 3F) Beam2 (4F, RF) Column Table 5 Optmzaton result H-4x2x9x12 H-3x15x6.5x9-3x3x12 Yeld stress: Beam N/mm 2 Beam N/mm 2 Thckness of flange: Beam mm Beam mm 7.2 Optmzaton of frame secton In the upper-level optmzaton problem, we select member sectons from the pre-assgned lst of standard sectons n Table 4. A constrant s gven such that the worst value of maxmum nterstory drft s not more than.72 m, whch s equvalent to 2% of the nterstory drft angle. The objectve functon s the total structural volume. In the lower-level antoptmzaton problem, uncertanty s consdered n the yeld stress and flange thcknesses of beams. TS s used for both optmzaton and antoptmzaton, where the numbers of neghborhood solutons and steps are 5 and 1, respectvely. shear force nterstory drft Fgure 7: Relaton between nterstory drft angle and shear force of 1st story 6

7 The optmal sectons and the worst parameter set at optmum are shown n Table 5. The relaton between the nterstory drft and shear force of the 1st story s plotted n Fg. 7 for the optmal desgn wth worst parameter set subjected to one of the fve sesmc motons. The dotted lnes are the result for the standard secton wth nomnal parameter set. The envelopes of maxmum nterstory drft angles are plotted n sold lne n Fg. 8. The dotted lne s the result of the standard frame model wth nomnal parameter values. Note that the maxmum nterstory angles of the optmal frame s almost equal to 2%. 4 story number /2 1/1 1/5 nterstory drft angle Fgure 8: Maxmum nterstory drft angles 8. Conclusons A method has been presented for cross-sectonal optmzaton of buldng frames consderng uncertanty of parameters for materal and cross-sectonal geometry. The objectve functon s the total structural volume, and a constrant s gven for the worst value of the representatve response of the frame subjected to sesmc motons. The cross-sectons are selected from the lst of avalable standard sectons. The parameters are also descretzed nto nteger values. Therefore, the optmum desgn problem and the antoptmzaton problem for fndng the worst response are formulated as combnatoral problems. A new concept s ntroduced for the antoptmzaton problem, where the accuracy of the soluton s defned by the order of the response rather than the value of the response among the possble combnatons of the parameters. Hence, the approxmate values for the optmzaton and antoptmzaton problems are found usng a heurstc approach called TS. The performance of TS s compared wth that of random selecton. It has been shown that TS s better than random selecton n vew of accuracy; however, the latter has superorty because t does not have any parameter that should be assgned by ntuton. 9. References [1] M. Ohsak, Optmzaton of Fnte Dmensonal Structures, CRC Press, 21. [2] I. Elshakoff and M. Ohsak, Optmzaton and Ant-Optmzaton of Structures under ncertanty, Imperal College Press, 21. [3] Open System for Earthquake Engneerng Smulaton (OpenSees), PEERC, CB, 26. (avalable at [4] F. Glover: Tabu search: Part I, ORSA J. on Computng, Vol. 1(3), pp.19-26, [5] M. Mtzenmacher and E. pfal, Probablty and Computng: Randomzed Algorthm and Probablstc Analyss, Cambrdge nversty Press, 25. 7